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Curved and Layer. Struct. 2014; 1:1–10

Research Article

Open Access

S. Natarajan*, A.J.M. Ferreira, and Hung Nguyen-Xuan

Analysis of cross-ply laminated plates using isogeometric analysis and unified formulation Abstract: In this paper, we study the static bending and free vibration of cross-ply laminated composite plates using sinusoidal deformation theory. The plate kinematics is based on the recently proposed Carrera Unified Formulation (CUF), and the field variables are discretized with the non-uniform rational B-splines within the framework of isogeometric analysis (IGA). The proposed approach allows the construction of higher-order smooth functions with less computational effort. Moreover, within the framework of IGA, the geometry is represented exactly by the Non-Uniform Rational B-Splines (NURBS) and the isoparametric concept is used to define the field variables. On the other hand, the CUF allows for a systematic study of two dimensional plate formulations. The combination of the IGA with the CUF allows for a very accurate prediction of the field variables. The static bending and free vibration of thin and moderately thick laminated plates are studied. The present approach also suffers from shear locking when lower order functions are employed and shear locking is suppressed by introducing a modification factor. The effectiveness of the formulation is demonstrated through numerical examples. Keywords: unified formulation; isogeometric analysis; non-uniform rational B-splines; shear locking; sinusoidal shear deformation theory DOI 10.2478/cls-2014-0001 Received July 30, 2014 ; accepted September 3, 2014

*Corresponding Author: S. Natarajan: Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai 600036, India, Email: [email protected]; [email protected] A.J.M. Ferreira: Faculdade de Engenharia da Universidade do Porto, Porto, Portugal and Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Hung Nguyen-Xuan: Department of Mechanics, Faculty of Mathematics and Computer Science, University of Science, HCMC, Vietnam

1 Introduction The need for high strength-to-weight and high stiffnessto-weight ratio materials has led to the development of laminated composite materials. Typical laminated composites consist of layers of fibrous composite materials, which are combined to provide required engineering properties, such as in-plane stiffness, bending stiffness and coefficient of thermal expansion. This class of material has seen increasing utilization in structural elements, because of the possibility to tailor the properties to optimize the structural response. Some of the advantages exhibited by composite over alloys are light weight, reduced corrosion, reduced noise generation, lack of magnetic signature and shape adaptability. Moreover, by changing the orientation of the fibre between the laminae various coupling effects can be achieved, such as extension-shear, bendtwist and bend-extension. In the literature, different approaches have been employed to study the response of such laminated composite plates, ranging from complete 3D analysis to two-dimensional theories. Recently, a formulation based on three-dimensional consistency is being investigated [1, 2]. The main advantage of such a formulation is that it does not suffer from the shear locking syndrome and the through-thickness behaviour is represented analytically, whilst the in-plane behaviour can be described by any displacement-based formulations. However, in this paper, we restrict ourselves to conventional two dimensional plate theories. A brief overview of the development of different plate theories is given in [3, 4]. The various two dimensional plate theories can be further classified into three different approaches: (a) equivalent single layer theories [5], (b) discrete layer theories [6] and (c) mixed plate theory [7]. Among these, the equivalent single layer (ESL) theories, viz., the first order shear deformation theory [8], the second and the higher order accurate theory [5, 9] are the most popular theories employed to describe the plate kinematics. Although the plate kinematics can be described by various aforementioned theories, a systematic study on the influence of various theories on the structural response of laminated plates could be computationally intensive. Thanks to the recent deriva-

© 2014 S. Natarajan et al., licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

2 | S. Natarajan, A.J.M. Ferreira, and Hung Nguyen-Xuan

tion of a series of axiomatic approaches by Carrera [10], called the Carrera Unified Formulation [11], for the general description of two-dimensional formulations for multilayered plates and shells. With this unified formulation, it is possible to implement in a single software a series of hierarchical formulations, thus affording a systematic assessment of different theories, ranging from simple equivalent single layer models up to higher order layerwise descriptions. Ferreira et al., [12, 13] studied the static and dynamic response of isotropic and cross-ply laminated composites by employing the unified formulation and the differential quadrature method. Existing approaches in the literature to study plate and shell structures made up of laminated composites include the finite difference method [14], the singular convolution method [15], the finite element method based on Lagrange basis functions [16], non-uniform rational B splines (NURBS) [17], meshfree methods [18, 19] or the differential quadrature method [20, 21]. Not only do these approaches suffer from shear locking when applied to thin plates, these techniques do not provide a single platform to test the performance of various theories. Recent interest in the unified formulation has led to the development of discrete models such as those based on the finite element method [22, 23], and more recently, meshless methods [19]. Nevertheless, even with the unified framework, there is an important shortcoming when applied to thin plates. With lower order basis functions within the finite element framework, the formulation suffers from shear locking. Intensive research over the past decades has led to the development of robust methods to suppress the shear locking syndrome. These includes: (a) reduced integration [24]; (b) use of the assumed strain method [25]; (c) using field redistributed shape functions [26]; (d) the mixed interpolation tensorial components (MITC) technique with strain smoothing [27] and (e) very recently, the twist Kirchhoff plate element [28].

1.1 Objective

locking is to employ higher order basis functions [30]. In this study, to alleviate shear locking, a simple modification is made to the shear term when lower order NURBS basis functions are used. However, the draw back of this approach is that the shear correction factor becomes problem dependent. The influence of various parameters, viz., the ply thickness, the ply orientation, the plate geometry, the material properties and the boundary conditions on the global response is studied numerically.

1.2 Outline The paper commences with a brief discussion on the unified formulation for plates and the description of spatial discretization. Section 3 describes the isogeometric approach employed in this study, followed by a technique to address shear locking when lower order NURBS functions are used to discretize the field variables. The efficiency of the present formulation, numerical results and parametric studies are presented in Section 4, followed by concluding remarks in the last section.

2 Carrera Unified Formulation 2.1 Basis of the CUF Let us consider a laminated plate composed of perfectly bonded layers with coordinates x, y along the inplane directions and z along the thickness direction of the whole plate, while z k is the thickness of the kth layer. The CUF is a useful tool to implement a large number of two-dimensional models with the description at the layer level as the starting point. By following the axiomatic modelling approach, the displacements u(x, y, z) = (u(x, y, z), v(x, y, z), w(x, y, z)) are written according to the general expansion as: u(x, y, z) =

N X

F τ (z)uτ (x, y)

(1)

τ=0

The main objective of this manuscript is to investigate the potential application of the NURBS-based isogeometric finite element method within the Carrera Unified Formulation (CUF) to study the global response of cross-ply laminated composites. The present formulation also suffers from shear locking when lower order basis functions are applied to thin plates. To address the shear locking problem with lower-order NURBS elements for plates, the introduction of a stabilization technique for shear locking has been studied [29]. The other approach to suppress shear

where F(z) are known functions to model the thickness distribution of the unknowns and N is the order of the expansion assumed for the through-thickness behaviour. By varying the free parameter N, a hierarchical series of twodimensional models is obtained. The strains are related to the displacement field via the geometrical relations: iT h ε pG = ε xx ε yy γxy = Dp u h iT ε nG = γxz γyz ε zz = (Dnp + Dnz ) u (2)

Analysis of cross-ply laminated plates using isogeometric analysis and unified formulation

where the subscript G indicates the geometrical equations, and Dp , Dnp and Dnz are differential operators given by:     ∂x 0 0 0 0 ∂x     Dp =  0 ∂ y 0  , Dnp =  0 0 ∂ y  , ∂y ∂x 0 0 0 0   ∂z 0 0   Dnz =  0 ∂ z 0  . (3) 0 0 ∂z The 3D constitutive equations are given as: σ pC = Cpp ε pG + Cpn ε nG σ nC = Cnp ε pG + Cnn ε nG

(4)

with 

Cpp

C11  =  C12 C16



Cnp

0  = 0 C13

0 0 C23

C12 C22 C26

 C16  C26  C66 

0  0  C36



Cpn

0  = 0 0



C55  Cnn =  C45 0

0 0 0 C45 C44 0

 C13  C23  C36  0  0  C33

| 3

and in the case of free vibrations, we have: k kτs k ¨τ Kkτs u uu uτ = M

(9)

where the fundamental nucleus Kkτs uu is: h k T k k k k Kkτs uu = (−Dp ) (Cpp Dp + Cpn (DnΩ + Dnz )+ i (−DknΩ + Dknz )T (Cknp Dkp + Cknn (DknΩ + Dknz )) F τ F s (10) and Mkτs is the fundamental nucleus for the inertial term given by: ( ρ k F τ F s if i=j kτs (11) M ij = 0 if i ≠ j where Pkuτ are variationally consistent loads with applied pressure. For a more detailed derivation and for the explicit form of the fundamental nuclei, interested readers are referred to [11, 31].

3 Non-uniform rational B-splines

(5) where the subscript C indicates the constitutive equations. The Principle of Virtual Displacements (PVD) for a multilayered plate subjected to mechanical loads is written as: Nk Z Z n o X (δε kpG )T σ kpC + (δε knG )T σ knC dΩ k dz = k=1 Ω

k

Ak

Nk Z X

Z

k=1 Ω

k

Ak

T

¨ k dΩ k dz + ρ k δuks u

Nk X

δLke

(6)

k=1

where ρ k is the mass density of the kth layer, Ω k , A k are the integration domains in the (x, y) and the z direction, respectively. Upon substituting the geometric relations (Equation (2)), the constitutive relations (Equation (4)) and the unified formulation into the PVD statement, we have: Z Z T n o Dkp F s δuks Ckpp Dkp F τ ukτ + Ckpn (DknΩ + Dknz )F τ ukτ +

In this study, the finite element approximation uses the NURBS basis functions. Here we give only a brief introduction to NURBS. More details on their use in FEM are given in [32]. The key ingredients in the construction of NURBS basis functions are: the knot vector (a non decreasing sequence of parameter values, ξ i ≤ ξ i+1 , i = 0, 1, · · · , m − 1), the control points P i , the degree of the curve p and the weight associated with a control point, w. The ith B-spline basis function of degree p, denoted by N i,p , is defined as: ( N i,0 (ξ ) = N i,p (ξ ) =

1 0

if ξ i ≤ ξ ≤ ξ i+1 else

ξ i+p+1 − ξ ξ − ξi N (ξ ) + N (ξ ) (12) ξ i+p − ξ i i,p−1 ξ i+p+1 − ξ i+1 i+1,p−1

The B-spline basis functions have the following properP ties: (i) non-negativity, (ii) partition of unity, N i,p = 1, i

(iii) interpolatory at the end points. As the same set of Ωk Ak h io functions is also used to represent the geometry, the exact (DknΩ + Dknz )f x δuks )T (Cknp Dkp F τ ukτ + Cknn (DknΩ + Dknz )F τ ukτ ) × representation of the geometry is preserved. It should be Nk Z Z Nk noted that the continuity of the spline functions can be taiX X T ¨ k dΩ k dz + dΩ k dz = ρ k δuks u δLke (7) lored to the needs of the problem. Also, the spline function k=1 Ω A k=1 has limited support. When employed to approximate the k k After integration by parts, the governing equations for the FE solution space, the resulting stiffness matrix has similar properties to the stiffness matrix computed by employplate are obtained: ing Lagrange shape functions. Given n + 1 control points kτs k k Kuu uτ = Puτ (8) (Po , P1 , · · · , Pn ) and a knot vector Ξ = {η o , η1 , · · · , η m },

4 | S. Natarajan, A.J.M. Ferreira, and Hung Nguyen-Xuan the piecewise polynomial B-spline curve of degree p is defined as: n X C(η) = Pi N i,p (η) (13)

1 0.9 0.8

i=0

C(ξ , η) =

n X m X

N i,p (ξ )M j,q (η)Pi,j

where Pi,j is the bidirectional control net and N i,p and M j,q are the B-spline basis functions defined on the knot vectors over an m × n net of control points Pi,j . Despite the flexibility offered by the B-splines, they cannot exactly represent some shapes such as circles and ellipsoids. To improve this, non-uniform rational B-splines (NURBS) are constructed through rational functions of B-splines. The NURBS thus form the superset of B-splines. The key ingredients in the construction of NURBS basis functions are: the knot vector (a non decreasing sequence of parameter values, η i ≤ η i+1 , i = 0, 1, · · · , m − 1), the degree of the curve p and the weight associated to a control point, w. A p th degree NURBS basis function is defined as follows: N i,p (η)w i N i,p (η)w i = n P W(η) N i,p (η)w i

(15)

i=0

where w i are the weights for the i th basis function N i,p (η). Figure (1) shows the third order NURBS for an open knot vector Ξ = {0, 0, 0, 0, 1/3, 1/3, 1/3, 1/2, 2/3, 1, 1, 1, 1}. The NURBS surface is then defined by: Pn Pm i=1 j=1 N i,p (ξ )M j,q (η)Pi,j w i w j R(ξ , η) = (16) w(ξ , η) where w(ξ , η) is the weighting function. The displacement field, uτ (x, y) (see Equation (1)) within the control mesh is approximated by: uτ (x, y) = R(ξ , η)qτ (x, y),

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Ξ

Figure 1: Non-uniform rational B-splines with an open knot vector, order of the curve = 3

(14)

i=1 j=1

R(η) =

Basis functions

0.7

where Pi are the control points. A B-spline curve contains the following information: n +1 control points, m +1 knots and a degree p. It is noted that n, m and p must satisfy m = n + p + 1. The B-spline functions also provide a variety of refinement algorithms, which are essential when employing B-spline functions to discretize the unknown fields. The analogous h and p refinement can be done by the process of ‘knot insertion’ and ‘order elevation’. The B-spline surfaces are defined by the tensor product of the basis functions in two parametric dimensions ξ and η with two knot vectors, one in each dimension:

locking appears when lower order NURBS basis functions are employed [30, 33], for example with quadratic, cubic and quartic elements¹. One approach to alleviate the shear locking is to employ interpolation functions of order 5 or higher [30], but this inevitably increases the computational cost. A stabilization technique for several lowerorder NURBS elements for plates was reported in [29]. In this paper, we adopt a stabilization technique proposed in [34] and later used in [33] to study the response of Reissner-Mindlin plates. In this approach, the material matrix related to shear terms are multiplied by the following factor: h2 (18) shearFactor = 2 h + α 2 `2 where ` is the largest length of the edges of the NURBS element, and α is a positive constant limited to values 0.05 ≤ α ≤ 0.15. From numerical experiments of NURBS-based isogeometric plate elements it is found that setting α = 0.1, yields reasonably accurate solutions.

4 Numerical Results In this section, we present the static response and the natural frequencies of laminated composite plates using the combined IGA and CUF framework. In this study, we use a hybrid displacement assumption, where the in-plane displacements u and v are expressed as sinusoidal expan-

(17)

where qτ (x, y) are the nodal variables and R(ξ , η) are the basis functions given by Equation (16). Similar to the finite element method based on Lagrange basis functions,

1 Linear NURBS basis functions are same as the linear Lagrange basis functions and are not discussed here. Approaches employed for Lagrange basis functions can readily be applied to NURBS basis functions with order 1


sions in the thickness direction, and the transverse displacement w is quadratic function in the thickness direction. We refer to this theory as SINUS-W2. The displacements are expressed as: πz u(x, y, z, t) = u o (x, y, t) + zu1 (x, y, t) + sin u2 (x, y, t) h πz v(x, y, z, t) = v o (x, y, t) + zv1 (x, y, t) + sin v2 (x, y, t) h w(x, y, z, t) = w o (x, y, t) + zw1 (x, y, t) + z2 w2 (x, y, t) (19) where u o , v o and w o are translations of a point at the midsurface of the plate, w2 is the higher order translation, and u1 , v1 , u3 and v3 denote rotations [35]. The effect of the plate aspect ratio, the ply angle and the ratio of Young’s modulus E1 /E2 on the static bending and free vibration is numerically studied.

| 5

in [22, 23]. In this study, the results from the present formulation are denoted by Quadratic, Cubic and Quartic, which corresponds to the order of shape functions employed, referred to as p−refinement. Three different mesh discretizations, viz., 5×5, 7×7 and 9×9 are considered, called h−refinement. Table 1 shows the convergence of the central deflection and stresses of a simply supported crossply laminated square plate. It can be seen that with both h− and p−refinement, the results from the present formulation converge, and that highly accurate results are obtained from the present formulation even with a coarse mesh. A comparison with other approaches and an elasticity solution is given in Table 2. Table 1: Convergence of the central deflection w

=

3 2h w(a/2, a/2, 0) 100E of a simply supported cross-ply laminated Pa4 square plate [0◦ /90◦ /90◦ /0◦ ] with E1 = 25E2 , G12 = G13 = 0.5E2 ,

G23 = 0.2E2 , ν12 =0.25.

4.1 Static bending Method The static analysis is conducted for cross-ply laminated plates with three and four layers under the following sinusoidal load: πx πy p z (x, y) = P o sin sin (20) a a where P o is the amplitude of the mechanical load. The origin of the coordinate system is located at the lower left corner of the midplane. The physical quantities are nondimensionalized by the following relations, unless otherwise mentioned: 100h3 E2 ; P o a4 h2 σ xx = σ xx (a/2, a/2, h/2) ; P o a2 h2 σ yy = σ yy (a/2, a/2, h/4) ; P o a2 h τ xz = τ xz (0, a/2, 0) . Po a

w

σ xx

w = w(a/2, a/2, 0)

σ yy

(21) τ xz

Validation Before proceeding with a detailed numerical study of the effect of various parameters on the global response of cross-ply laminated composites, the results from the proposed formulation are compared with available results pertaining to static bending of laminated plates. In this study, we consider three orders of NURBS basis functions, viz., quadratic, cubic and quartic. It is noted that, in this study, we do not consider first order NURBS basis functions. This is because, the first order NURBS basis functions are similar to the conventional bilinear shape functions. The performance of which is discussed in detail

Quadratic Cubic Quartic HSDT [5] Elasticity [36] Quadratic Cubic Quartic HSDT [5] Elasticity [36] Quadratic Cubic Quartic HSDT [5] Elasticity [36] Quadratic Cubic Quartic HSDT [5] Elasticity [36]

5×5 1.9207 1.9076 1.9045 1.8937 1.9540 0.6966 0.7074 0.7062 0.6651 0.7200 0.6179 0.6277 0.6268 0.6322 0.6660 0.2293 0.2210 0.2205 0.2064 0.2700

Meshes 7× 7 1.9100 1.9038 1.9020

9×9 1.9058 1.9021 1.9010

0.7009 0.7063 0.7060

0.7029 0.7061 0.7058

0.6221 0.6270 0.6267

0.6239 0.6268 0.6266

0.2246 0.2205 0.2202

0.2227 0.2202 0.2201

4.1.1 Four layer (0◦ /90◦ )s square cross-ply laminated plate under sinusoidal load A square simply supported laminate of side length a and thickness h, composed of four equally thick layers oriented at (0◦ /90◦ )s is considered. The plate is subjected to a sinu-

6 | S. Natarajan, A.J.M. Ferreira, and Hung Nguyen-Xuan

soidal vertical pressure given by Equation (20). The material properties are as follows: E1 = 25E2 ; G12 = G13 = 0.5E2 ; G23 = 0.2E2 ; ν12 = 0.25. For this example, a threedimensional exact solution by Pagano [36] is available. The central deflection and the corresponding stresses for the SINUS-W2 theory with an isogeometric approach are shown in Table 2. We compare the results with higher order plate theories [5, 37], a first order theory [38], an exact solution [36] and also with the strain smoothing approach with SINUS-W2. The effect of the plate thickness is also shown in Table 2. It is shown clearly that the first order shear deformation theories (FSDT) cannot be used for thick laminates. Further, it can be seen that the results from the present formulation are in very good agreement with those in the literature and very precise transverse displacement and stress values are obtained.

4.1.2 Three layer (0◦ /90◦ /0◦ ) square cross ply laminated plate under sinusoidal load A square laminate of side a and thickness h, composed of three equally thick layers oriented at (0◦ /90◦ /0◦ ) is considered. It is simply supported at all edges and subjected to a sinusoidal vertical pressure of the form given by Equation (20). The material properties for this example are: E1 =132.38 GPa, E2 = E3 =10.756 GPa, G12 =3.606 GPa, G13 = G23 = 5.6537 GPa, ν12 = ν13 = 0.24, ν23 = 0.49. In Table 3, we present results for the SINUS-W2 theory with an isogeometric approach with quadratic, cubic and quartic NURBS basis functions having a 9×9 NURBS patch. The results from the present formulation are compared with the analytical solution [10, 39] and the MITC4 formulation with and without strain smoothing [22, 23]. It can be seen that the numerical results from the present formulation are in good agreement with the existing solutions. Moreover, it is noted that with the isogeometric approach, the geometry of the domain can be represented exactly. Although only a simple geometry is considered, the proposed formulation can easily be extended to complex geometies. The main features of the present formulation are: (1) theories from ESL to higher order layer descriptions can be implemented within a single program (since it is based on the CUF); (2) the isogeometric approach provides flexibility to construct higher order smooth functions and provides accurate solutions even for a coarse NURBS mesh and (3) the present formulation is insensitive to shear locking.

4.2 Free vibration - cross-ply laminated rectangular plates Next, we study the fundamental frequencies of cross-ply laminated plates based on the proposed formulation. In this example, all layers of the laminate are assumed to be of the same thickness, density and made up of the same linear elastic material. The following material parameters are considered for each layer E1 = 10,20,30, or 40; G12 = G13 = 0.6E2 ; E2 G3 = 0.5E2 ; ν12 = 0.25. The subscripts 1 and 2 denote the directions normal and the transverse to the fibre direction in a lamina, which may be oriented at an angle with respect to the plate axes. The ply angle of each layer is measured from the global x−axis to the fibre direction. The example considered is a simply supported square cross-ply laminated plate[0◦ /90◦ ]s . The thickness and length of the plate are denoted by h and a, respectively. A thickness-to-span ratio of h/a = 0.2 is employed in the computations. In this study, we present the non-dimensionalized free flexural frequencies as: r ρ a2 Ω=ω h E2 unless specified otherwise. Table 4 shows the convergence of the normalized fundamental frequency of a simply supported cross-ply laminated square plate based on the current isogeometric approach. The performance of various basis functions with NURBS mesh refinement is studied. It can be seen that, as expected with h− refinement, the solutions converge and with p− refinement, the accuracy increases for the same mesh size. Table 5 lists the fundamental frequency for a simply supported cross-ply laminated square plate with h/a = 0.2 and for different Young’s modulus ratios, E1 /E2 . It can be seen that the results from the present formulation are in very close agreement with the values reported in [41] based on higher order theory, the meshfree results of Liew et al., [40] and Ferreira et al., which are based on FSDT and higher order theories with radial basis functions [42]. The effect of plate thickness on the fundamental frequency is shown in Table 6. It can be seen that the results agree with the results available in the literature. The present formulation is insensitive to shear locking.


3

| 7

h h 2h Table 2: The normalized central deflection w = w(a/2, a/2, 0) 100E , stresses, σ xx = σ xx (a/2, a/2, h/2) Pa 2 , σ yy = σ yy (a/2, a/2, h/4) Pa 2 Pa4 h /a2, 0) Pa

and τ xz = τ xz (0, G23 = 0.2E2 , ν12 =0.25.

2

2

of a simply supported cross-ply laminated square plate [0◦ /90◦ /90◦ /0◦ ], with E1 = 25E2 , G12 = G13 = 0.5E2 ,

a/h

10

100

Method HSDT [5] FSDT [38] Elasticity [36] RBF [37] CS-FEM Q4 (4 subcells) [23] Present (Quadratic 9× 9) Present (Cubic 9×9) Present (Quartic 9×9) HSDT [5] FSDT [38] Elasticity [36] RBF [37] CS-FEM Q4 (4 subcells) [23] Present (Quadratic 9× 9) Present (Cubic 9×9) Present (Quartic 9×9)

w 0.7147 0.6628 0.7430 0.7325 0.7195 0.7250 0.7203 0.7187 0.4343 04337 0.4347 0.4307 0.4304 0.4383 0.4336 0.4317

σ xx 0.5456 0.4989 0.5590 0.5627 0.5597 0.5571 0.5596 0.5594 0.5387 0.5382 0.5390 0.5431 0.5368 0.5334 0.5368 0.5366

σ yy 0.3888 0.3615 0.4030 0.3908 0.3905 0.3908 0.3913 0.3907 0.2708 0.2705 0.2710 0.2730 -

τ xz 0.2640 0.1667 0.3010 0.3321 0.2952 0.2985 0.2983 0.2967 0.2897 0.1780 0.3390 0.3768 0.3285 0.4069 0.3271 0.3275

Table 3: Transverse displacement w = w(a/2, a/2, h/2) at the center of a multilayered plate [0◦ /90◦ /0◦ ] with E1 = 132.38 GPa, E2 = E3 = 10.756 GPa, G12 = 3.606 GPa, G13 = G23 = 5.6537 GPa, ν12 = ν13 = 0.24, ν23 = 0.49.

w Analytical (ESL-2) [10, 39] MITC4 [22] CS-FEM Q4 (4 subcells) [23] Present (Quadratic 9×9) Present (Cubic 9×9) Present (Quartic 9×9)

10 0.9249 0.9195 0.9235 0.9252 0.9226 0.9217

Table 4: Convergence of the normalized fundamental frequency p Ω = ωa2 /h ρ/E2 of a simply supported cross-ply laminated square plate (0◦ /90◦ )s with h/a = 0.2, EE1 = 40, G12 = G13 = 0.6E2 , 2 G23 = 0.5E2 , ν12 = 0.25..

Method Quadratic Cubic Quartic

5×5 10.6926 10.7340 10.7498

Meshes 7× 7 10.7295 10.7517 10.7598

9×9 10.7454 10.7590 10.7640

50 0.7767 0.7713 0.7703 0.7713 0.7704 0.7695

a/h 100 0.7720 0.7666 0.7655 0.7650 0.7656 0.7646

500 0.7705 0.7650 0.7639 0.7624 0.7640 0.7631

1000 0.7704 0.7650 0.7639 0.7624 0.7639 0.7630

4.3 Free vibration - cross-ply laminated circular plates In this example, consider a circular four layer [θ/−θ/−θ/θ] laminated plate with fully clamped boundary conditions. The influence of the fiber orientations on the free vibration of a clamped circular laminated plate is studied. The following material properties are used: E1 = 40; G12 = G13 = 0.6E2 ; E2 G3 = 0.5E2 ; ν12 = 0.25. The subscripts 1 and 2 denote the directions normal and the transverse to the fibre direction in a lamina. The circular plate has a radius-to-thickness ratio of 5 (R/h =5). For this problem, a NURBS quadratic basis function is sufficiently accurate to model the circular geometry. Any

8 | S. Natarajan, A.J.M. Ferreira, and Hung Nguyen-Xuan p Table 5: The normalized fundamental frequency Ω = ωa2 /h ρ/E2 of a simply supported cross-ply laminated square plate (0◦ /90◦ )s with h/a = 0.2, EE1 = 10, 20, 30 or 40, G12 = G13 = 0.6E2 , G23 = 0.5E2 , ν12 = 0.25. 2

Method Liew [40] Reddy, Khdeir [41] HSDT [42] (ν23 = 0.18) CS-FEM Q4 (4 subcells) [23] Present (Quadratic 9×9) Present (Cubic 9×9) Present (Quartic 9×9)

E1 /E2 20 30 9.5613 10.3200 9.5671 10.3260 9.5411 10.2687 9.5793 10.2973 9.5437 10.2572 9.5532 10.2691 9.5566 10.2734

10 8.2924 8.2982 8.2999 8.3642 8.3358 8.3417 8.3439

40 10.8490 10.8540 10.7652 10.7887 10.7454 10.7590 10.7640

p Table 6: Variation of fundamental frequencies, Ω = ωa2 /h ρ/E2 with a/h for a simply supported square laminated plate p [0◦ /90◦ /90◦ /0◦ ], Ω = ωa2 /h ρ/E2 , with E1 /E2 = 40, G12 = G13 =0.6E2 , G23 =0.5E2 , ν12 = ν13 = ν23 = 0.25.

Method FSDT [43] Model-1 (12dofs) [9] Model-2 (9dofs) [9] HSDT [5] HSDT [44] CS-FEM Q4 (4 subcells) [23] Present (Quadratic 9×9) Present (Cubic 9×9) Present (Quartic 9×9)

2 5.4998 5.4033 5.3929 5.5065 6.0017 5.4026 5.3931 5.3945 5.3951

a/h 10 20 15.1426 17.6596 15.1048 17.6470 15.0949 17.6434 15.1073 17.6457 15.9405 17.9938 15.1766 17.7540 15.0660 17.5781 15.1086 17.649 15.1239 17.6749

4 9.3949 9.2870 9.2710 9.3235 10.2032 9.2998 9.2701 9.2785 9.2815

further refinement, would will only improve the accuracy of the solution. The following knot vectors for the coarsest mesh with one element are defined as follows: Ξ =[0,0,0,1,1,1]; and H =[0,0,0,1,1,1]. The parameters for the circular plate is given in Table 7. In this study, 13×13 NURBS cubic elements are used. The first three fundamental frequencies for a clamped circular laminated plate are given in Table 8. The fibre orientation of each layer is considered to be the same, and the influence of the fibre orientation on the first three fundamental frequencies is given in Table 8. The numerical results from the present approach are compared with the moving least square differential quadrature method (MLSDQ), which is based on FSDT [40] and IGA with inverse trigonometric shear deformation theory [45]. It can be seen that the results from the present formulation agree well with the results in the literature.

50 18.6742 18.6720 18.6713 18.6718 18.7381 18.7947 18.5913 18.6711 18.7024

100 18.8362 18.8357 18.8355 18.8356 18.8526 18.9611 18.7579 18.8343 18.8665

Table 7: Control points and weights for a circular plate with radius R = 0.5.

i xi yi wi

1

√ - √42 2 4

1

2

3

4

5

6

0

2 2

1

0 0 1

0

0

2 √4 - 42

√

-

2 2

√

√

√ 2 √2 2 2

√

- √22 2 2

7

√ 2 √4 2 4

1

8

√

2 2

9

√

0

2 √4 - 42

2 2

1

√

5 Conclusions In this article, the isogeometric approach was combined with the unified formulation to study the static bending and the free vibration of laminated composites. The present approach allows us to achieve a smooth approximation of the unknown fields with arbitrary continuity. When employing lower order elements, the method suffers from the shear locking syndrome, which is alleviated by multiplying the shear term with a correction factor. The results from the present formulation are in very good agreement with the solutions available in the literature. We believe that the present formulation is an effective compu-


Table 8: Influence of fiber orientations on the fundamental frequencies, Ω = ωa2 /h

θ

0

π/12

π/6

π/4

Method MLSDQ-FSDT [40] IGA [45] Present MLSDQ-FSDT [40] IGA [45] Present MLSDQ-FSDT [40] IGA [45] Present MLSDQ-FSDT [40] IGA [45] Present

tational formulation for practical problems. On one hand, the unified formulation allows the user to test different theories within a single framework, and the isogeometric approach provides flexibility in constructing higher-order smooth basis functions, and the geometry is accurately described.

[9]

[10]

References [12]

[2]

[3]

[4]

[5] [6]

[7]

[8]

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ρ/E2 for clamped circular laminated plates.

Ω 1 22.2110 23.5781 22.6663 22.7740 23.6090 23.0024 24.0710 24.2081 23.9749 24.7520 24.6607 24.5253

[11]

[1]

p

| 9

[13]

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[19]

2 29.651 30.7459 30.3485 31.4550 31.7743 31.5752 36.1530 35.6047 35.2577 39.1810 37.8980 37.4311

3 41.1010 42.0042 41.7294 43.350 43.9569 43.7671 43.9680 46.5406 44.2964 43.6070 46.2560 44.0796

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Research Article

Open Access

D. A. Hadjiloizi, A. L. Kalamkarov, Ch. Metti, and A. V. Georgiades*

Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates: Part I – Model Development Abstract: A comprehensive micromechanical model for the analysis of a smart composite piezo-magneto-thermoelastic thin plate with rapidly-varying thickness is developed in the present paper. A rigorous three-dimensional formulation is used as the basis of multiscale asymptotic homogenization. The asymptotic homogenization model is developed using static equilibrium equations and the quasi-static approximation of Maxwell’s equations. The work culminates in the derivation of a set of differential equations and associated boundary conditions. These systems of equations are called unit cell problems and their solution yields such coefficients as the effective elastic, piezoelectric, piezomagnetic, dielectric permittivity and others. Among these coefficients, the so-called product coefficients are also determined which are present in the behavior of the macroscopic composite as a result of the interactions and strain transfer between the various phases but can be absent from the constitutive behavior of some individual phases of the composite material. The model is comprehensive enough to allow calculation of such local fields as mechanical stress, electric displacement and magnetic induction. In part II of this work, the theory is illustrated by means of examples pertaining to thin laminated magnetoelectric plates of uniform thickness and wafer-type smart composite plates with piezoelectric and piezomagnetic constituents. The practical importance of the model lies in the fact that it can be successfully employed to tailor the effective properties of a smart composite plate to the requirements of a particular engineering application by changing certain geometric or material parameters. The results of the model constitute an important refinement over previously established work. Finally, it is shown that in the limiting case of a thin elastic plate of uniform thickness the derived model converges to the familiar classical plate model. Keywords: smart composite piezo-magneto-thermoelastic thin plate; asymptotic homogenization; effective properties; product properties

DOI 10.2478/cls-2014-0002 Received August 7, 2014 ; accepted September 11, 2014

1 Introduction Significant advancements in the production and application of composites coupled with emerging technologies in the fields of sensors and actuators have permitted the integration of smart composites in an increasingly larger number of engineering applications. Of particular interest among smart composites is the class of structures which include both piezoelectric and piezomagnetic phases. The strain transfer and general interactions between the various phases of these composites give rise to the so-called product properties, see Newnham et al. [1]. These properties are found in the behavior of the macroscopic composite but are usually absent from the constituent behavior of the individual phases. Examples of product properties are

*Corresponding Author: A. V. Georgiades: Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus and Research Unit for Nanostructured Materials Systems, Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus; E-mail: [email protected]; Tel.: 357-25002560 D. A. Hadjiloizi: Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus and Research Unit for Nanostructured Materials Systems, Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus A. L. Kalamkarov: Department of Mechanical Engineering, Dalhousie University, PO Box 15000, Halifax, Nova Scotia, B3H 4R2, Canada Ch. Metti: Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus and Research Unit for Nanostructured Materials Systems, Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus

© 2014 D. A. Hadjiloizi et al., licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

12 | D. A. Hadjiloizi et al. the magnetoelectric, pyromagnetic and pyroelectric properties, see Nan et al. [2], Bichurin et al. [3]. The magnetoelectric property is the behavior which governs the generation of an electric displacement when a magnetic field is applied and vice-versa. In particular, applying a magnetic field induces a mechanical strain in the piezomagnetic phase. In turn, provided that there is satisfactory degree of bonding between the two constituents, this magnetically induced strain is transferred to the piezoelectric phase which then produces an electric field. Likewise, the pyroelectric and pyromagnetic product properties refer to the generation of an electric or magnetic field when a thermal load is applied. The unique properties of magnetoelectric composites render them suitable candidates for a broad range of novel practical applications in the form of components, devices and systems. For example their sensitivity to external stimuli (electric and magnetic fields, temperature etc.) can be exploited for frequency tunable devices such as resonators and filters, magnetic field sensors, energy harvesting transducers, miniature antennas, etc. [3–9]. Other attractive potential applications of some classes of magnetoelectric composites include data storage devices and spintronics [10], biomedical sensors for EEG/MEG and other relevant equipment [11, 12] etc. In view of the aforementioned (and many more) practical applications, the main objective of this work is to develop accurate micromechanical models that can be used to design magnetoelectric and general smart composite and reinforced plates. The model must be comprehensive enough to afford the designer significant flexibility with regards to both the structural make-up and the overall geometry of the given structures. The use of composites and smart composites in new engineering applications is often limited due to the lack of reliable data concerning their long-term behavior. This disadvantage could be successfully mitigated if the behavior of such structures could be determined at the design stage. This can be effectively achieved via the development of accurate micromechanical models. To be useful, these models must be comprehensive enough to capture all the important behavioral characteristics of the composite structure. At the same time, they must not be too complicated to be used effectively, efficiently and expediently. Ideally, and in order to be readily amenable to design, such models should lead to closed-form expressions for the determination of effective properties in terms of the material and geometric parameters of the constituents and the macroscopic structure. Despite the increased interest in the magnetoelectric effect and other product properties, little research work

pertaining to the micromechanical modeling of this behavior exists. As expected, both analytical and numerical (principally finite element-based) approaches have been examined and implemented. Noteworthy among the analytical models are the works of Harshe et al. [13, 14] and Avellaneda and Harshe [15], Huang et al. [16–19], Bichurin et al. [20, 21], Soh and Liu [22], Bravo-Castillero et al. [23], Ni et al. [24] Akhabarzadeh et al. [25], and others. Harshe et al. [13, 14] and Avellaneda and Harshe [15] obtain the magnetoelectric coefficients of 2-2 piezoelectric/magnetostrictive multilayer composites for mechanically free and clamped structures. Huang and Kuo [16] developed a comprehensive model pertaining to piezoelectric/piezomagnetic composites on the basis of the classical works of Eshelby [26] and Mori-Tanaka [27]. In particular, their model incorporated reinforcements in the form of ellipsoidal inclusions which allowed the reinforcement geometry to vary from thin flakes to long continuous fibers. The determination of Eshelby-like tensors [26] allowed the authors to compute not only the effective properties, but also the local fields around the inclusions and the pertinent stress concentration factors. Using the Mori-Tanaka approach [27] the interactions between the constituent phases were also examined. In an extension of this work, Huang [17] obtained closed-form solutions for a transversely isotropic matrix and reinforcements in the shape of elliptic cylinders, circular cylinders, disks and ribbons. The resulting expressions are functions of the inclusion properties and geometry, as well as the pertinent volume fractions. In another work, Huang et al. [18] examined the magnetoelectric effect in piezoelectric/piezomagnetic bilayers under coupled bending and stretching loading conditions. They discovered that the magnetoelectric coupling coefficients in this case were significantly higher than in the case of pure stretching. In an interesting new study based on their earlier works, Huang et al. [19] obtained the magnetoelectric coefficients in composites of continuous piezoelectric fibers embedded in a piezomagnetic matrix. The authors also obtained an analytical expression for the optimized fiber volume fraction for maximizing the magnetoelectric coupling coefficients. Surprisingly, their results indicated that the optimum volume fraction is a function of the elastic properties of the constituents and is independent of the magnetic and electric properties. Bichurin et al. [20] investigated the magnetoelectric effect in ferromagnetic/piezoelectric multilayer composites using a two-step approach. In the first step they wrote down the constitutive relationships of the individual phases and in the second they used the corresponding expressions of the macroscopic composite. The same au-

Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates | 13

thors, Bichurin et al [21], extended their work to magnetoelectric nanocomposites. Soh and Liu [22] adopted a new approach in deriving eight sets of constitutive equations characterizing magnetoelectric composites directly from eight thermodynamic potentials. The theoretical framework of their study is important in that it established the necessary relationships that must exist between the various material constants that appear in the constitutive laws. Bravo-Castillero et al. [23] used generalized test functions to avoid singularities that occur due to the discontinuity at the interphase between the constituents. To illustrate their model they obtained closed-form expressions for the effective properties of piezoelectric/piezomagnetic laminates. Ni et al. [24] investigated the magnetoelectric properties of 3-ply polycrystalline multiferroic laminates consisting of a piezoelectric lamina sandwiched between two ferromagnetic ones. In their modeling approach, the authors applied a magnetic field and determined the induced electric displacement. They computed the magnetoelectric coupling coefficients as ratios of applied magnetic field to induced polarization. Their work showed that the magnetoelectric constants depend strongly on the orientation of the magnetic fields. Akbarzadeh et al. [25] considered, among others, the pyroelectric coefficients when analyzing the thermo-electro-magneto-elastic behavior of rotating functionally graded piezoelectric cylinders. Other works can be found in Kirchner et al. [28], Pan and Heyliger [29], Benveniste [30], Nan et al. [31], Spyropoulos et al. [32] and others. Naturally, the finite element technique has proven a popular method for analyzing magnetoelectric composites. In this respect, special consideration must be given to the works of Tang and Yu [33, 34], who employed the variational asymptotic method to investigate periodic twophase and three-phase structures. Starting from the total electromagnetic enthalpy or thermodynamic potential, the authors then applied constraint minimization. The pertinent equations were solved using the finite element technique. The authors illustrated their model by considering two types of fiber-reinforced composites; one consisted of piezoelectric fibers embedded in a piezomagnetic matrix and the other of piezoelectric and piezomagnetic fibers embedded in an elastic matrix. Their calculated effective coefficients agreed well with other reported values. Sunar et al. [35] used the finite element method to examine piezoelectric/piezomagnetic composites. The authors began by defining two energy functionals and then applied Hamilton’s principle to derive the constitutive equations for the smart structure. The authors then employed a finite element approach to a barium titanate/cobalt ferrite twolayer composite. In particular they examined the genera-

tion of a magnetic field when an applied electrostatic field induces a piezoelectric mechanical strain. The authors’ results conformed fairly well to those obtained via a simple analytical technique. Other work can be found in Lee et al. [36], Liu et al. [37], Mininger et al. [38], Sun et al. [39] and others. The micromechanical modeling of periodic composites and smart composites is characterized by rapidly varying material coefficients with period “ε”, the characteristic dimension of the periodicity or unit cell. At the same time, the dependent local fields such as mechanical stress, magnetic induction and electric displacement are functions of both periodic (microscopic) and nonperiodic (macroscopic) variables. The coupling of the microscopic and macroscopic scales renders even the numerical analysis of the aforementioned structures rather cumbersome. Further, an analytic solution is unattainable in all but the simplest geometries. These problems could be overcome if the two scales were decoupled and each handled separately. An effective technique which can be used to achieve precisely this is that of asymptotic homogenization. The mathematical framework of asymptotic homogenization can be found in Bensoussan et al. [40], Sanchez-Palencia [41], Bakvalov and Panasenko [42] and Cioranescu and Donato [43]. Many problems in elasticity, thermoelasticity and piezoelasticity have been solved via asymptotic homogenization. Examples can be found in Kalamkarov [44], Kalamkarov and Kolpakov [45], Kalamkarov and Georgiades [46], Georgiades et al. [47], Hassan et al. [48], Saha et al. [49], Guedes and Kikuchi [50], Sevostianov and Kachanov [51] and many others. Currently, the preponderance of uses of composite materials is in the form of plate and shell structural members, the strength and reliability of which, combined with reduced weight and concomitant material savings, offer the designer very impressive possibilities in many applications. It often happens that the reinforcing elements such as fibers form a regular array with a period much smaller than the characteristic dimension of the composite structure; consequently asymptotic homogenization analysis is applicable. Homogenized models of plates with periodic nonhomogeneities in tangential coordinates have been developed in this way by Duvaut [52], Duvaut and Metellus [53], Adrianov and Manevitch [54], Adrianov et al. [55] and others. Particularly noteworthy is the modified technique employed by Caillerie [56, 57] in his conduction studies. In particular, two sets of microscopic variables were introduced, one of which pertained to the tangential directions (characterized by periodicity) while the other variable re-

14 | D. A. Hadjiloizi et al. lated to the transverse direction in which no periodicity exists. Kohn and Vogelius, [58, 59] adopted this approach in their study of the pure bending of a thin, linearly elastic homogeneous plate. Kalamkarov [44], developed general homogenized composite shell models by applying the modified twoscale asymptotic method directly to three-dimensional elastic and thermoelastic problems for a thin curvilinear composite layer with rapidly varying thickness. Challagulla et al. [60] employed this methodology to develop rigorous asymptotic homogenization models for thin smart composite shells and illustrated their results by means of interesting and practically important examples including single-walled carbon nanotubes. Kalamkarov and Kolpakov [61] developed a new model for the analysis of clamped piezoelastic plates. Hadjiloizi et al. [62] implemented a general model (based on the time-varying form of Maxwell’s equations and the dynamic force balance) for the micromechanical dynamic analysis of magnetoelectric thin plates with rapidly varying thickness. In [62] however, only an in-plane temperature variation is taken into consideration and therefore any out-of-plane thermal effects are ignored. Thus, unlike in the present work, the out-ofplane thermal expansion, pyroelectric and pyromagnetic coefficients were not captured in [62]. As a further consequence, in the general field expressions for the force and moment resultants, electric and magnetic fields etc. the influence of the out-of-plane temperature variation is neglected. More importantly however, the micromechanical model in [62] was only applied to simple laminated plates. One can certainly not argue against the practical significance of such structures; on the other hand it is evident that laminates offer little design flexibility with respect to geometry which limits their application potential. In contrast, the model developed in the present paper allows for explicitly different periodicity in the tangential directions of the structure. This feature makes the model much more amenable to the analysis and design of not only laminated plates, but also reinforced plates such as wafer- and ribreinforced structures shown in Section 7. Such structures were not at all considered in [62]. Also relevant to the present papers are the works of Hadjiloizi et al. [63, 64] and Kalamkarov and Georgiades [65], Georgiades and Kalamkarov [66]. In [63, 64], Hadjiloizi et al. developed two general three-dimensional models for magnetoelectric composites. One model used dynamic force and thermal balance and the time-varying form of Maxwell’s equations to determine closed-form expressions for the effective properties. The second model used the quasi-static approximation of the aforementioned constitutive equations. However, these models are

three-dimensional in nature and as such cannot capture the mechanical, thermal, etc. behavior that is related to bending, twisting and general out-of-plane deformation as well as electric and magnetic field generation. The model developed in the current work and its companion paper [67] however, accomplishes precisely this; it employs a modified asymptotic homogenization technique, which makes use of two sets of microscopic variables (and is therefore quite different than the “classical” schemes of [63], [64]) that permit the decoupling of the in-plane and out-of plane behavior of the structure under consideration. For example, the elastic coefficients are separated into the familiar extensional, bending and coupling coefficients. This is not possible to achieve with the 3D models in [63] and [64]. Essentially, the two models are applicable to entirely different geometries. The 3D models in [63, 64] can be used to analyze structures of comparable dimensions (such as thick laminates) but cannot be used for thin structures such as wafer- and rib-reinforced plates. The micromechanical models developed in the current works however are applicable to structures with a much smaller dimension in the transverse direction than in the other two directions. Thus, it can be used in the design and analysis of composite and reinforced plates such as the aforementioned wafer- and rib-reinforced structures (see Section 7), three-layered honeycomb-cored magnetoelectric plates, thin laminates etc. In [65] and [66], Kalamkarov and Georgiades performed only a semi-coupled analysis of a composite or reinforced plate and therefore the resulting expressions of the effective coefficients do not reflect the influence of such parameters as the electric permittivity, magnetic permeability, primary magnetoelectricity etc. The present work and its companion paper [67] however, perform a fully coupled analysis and as a consequence the expressions for the effective coefficients involve all pertinent material parameters. As an example, the effective elastic coefficients depend not only on the elastic properties of the constituent materials, but also on the piezoelectric, piezomagnetic, magnetic permeability, dielectric permittivity and other parameters. The same holds true for the remaining effective coefficients. In a sense, the thermoelasticity, piezoelectricity and piezomagnetism problems are entirely coupled and the solution of one affects the solutions of the others. This feature is captured in the present papers, but not in previously published works, such as [65] and [66]. Thus, the results presented here represent an important refinement of previously established results. To the authors’ best knowledge, this is the first time that completely coupled piezo-magneto-thermo-elastic effective coefficients for reinforced plates have been presented and analyzed.


In view of the practical applications mentioned earlier on in this Section, the primary importance of this work lies in the fact that it develops a novel micromechanical model that leads to closed-form design-oriented equations easily integrated in MATLABTM or other similar software packages. These equations can be used to analyze and design magnetoelectric and other smart composite and reinforced plates with a broad range of geometries. The results of the developed models show improved accuracy as compared to previously published results. In particular, the present paper deals with the development of appropriate micromechanical models to examine the quasi-static plane stress solution of the aforementioned magnetoelectric composite and reinforced plates. The work is implemented in two parts. In part II [67] the derived models are applied to the practically important cases of thin magnetoelectric laminates and wafer-reinforced magnetoelectric plates. Our overall objectives are to: (a) obtain expressions for the dependent field variables, (b) derive closed-form expressions for all effective coefficients including product properties, (c) compare the results of our model with those of other models and illustrate, where possible, the improvements over previously established results. Following this introduction, the basic relations of the three-dimensional problem are formulated in Section 2 and the two-scale asymptotic expansions for the displacement and stress fields are introduced in Section 3. The equilibrium equations and pertinent boundary conditions are derived in Section 4 followed by the determination of the unit cell problems in Section 5. The governing equations and the effective coefficients of the homogenized plate are obtained in Section 6 in which a comparison of the homogenized plate with a smart composite lamina is also performed. Section 7 is a brief overview of the structures to be considered in Part II [67] of this work, and finally section 8 concludes this paper.

2 Problem Formulation Consider a thin smart layer representing an inhomogeneous solid with wavy surfaces, containing a large number of periodically arranged actuators as shown in Fig. 1. This periodic structure is obtained by repeating a certain small unit cell Ω δ in the x1 − x2 plane (Fig. 1). In the parlance of asymptotic homogenization the unit cell is thus sometimes referred to as a “periodicity cell”, [46–48]. All three pertinent coordinates are assumed to have been made dimensionless by division by a certain characteristic dimension of the body, L. Note that the shape of the top and bot-

tom surfaces of the layer is determined by the type of the surface reinforcement (for example by the shape of stiffeners or reinforcing ribs) or actuator (when the actuator is surface attached and not embedded within the structure). The surfaces can be plane if surface reinforcements or actuators are not used. In the context of the present work, the meaning of actuator (piezoelectric, piezomagnetic, magnetostrictive etc.) is a device that can be used to induce stresses and strains in a coordinated fashion [64–66]. Furthermore, it is assumed that stress concentrations and/or property variations at the interphase region between the matrix and the reinforcements and/or actuators are negligible. Essentially, it is assumed that the interphase regions are highly localized and do not contribute significantly to the integral over the entire unit cell domain. In practical terms, the error incurred will be negligible if the dimensions of the actuators/reinforcements are much smaller than the spacing between them. As an indication, we note that for the purely elastic case, Kalamkarov [44] showed that if the spacings between the unit cells are at least ten times bigger than the thickness of the reinforcements then the error in the values of the effective elastic coefficients incurred by ignoring the regions of overlap between the reinforcements is less than 1%. The unit cell of the problem is defined by the following inequalities (see Fig. 1), }︁ {︁ − δh2 1 < x1 < δh2 1 , − δh2 2 < x2 < δh2 2 , S− < x3 < S+ , (︁ )︁ x1 x2 where S± = ± 2δ ± δF ± δh , δh 1 2 (2.1) and the microscopic behavior of this smart structure is characterized by means of the following boundary value problem: (︁ )︁ x1 x2 )︂ (︂ ∂σ ij x1 , x2 , x3 , δh , δh x1 x2 1 2 = P i x1 , x2 , x3 , , ∂x j δh1 δh2 (2.2a) (︁ ∂D i x1 , x2 , x3 ,

x2 x1 δh1 , δh2

)︁

x1 x2 δh1 , δh2

)︁

∂x i (︁ ∂B i x1 , x2 , x3 , ∂x i

=0

(2.2b)

=0

(2.2c)

As well, the irrotational electric and magnetic (in the absence of free conduction currents) fields may be written down as the gradients of scalar functions, φ and ψ. (︁ )︁ x1 x2 (︂ )︂ ∂φ x , x , x , , 1 2 3 δh1 δh2 x x E i x1 , x2 , x3 , 1 , 2 =− ∂x i δh1 δh2 (2.2d)

16 | D. A. Hadjiloizi et al. represents the surface tractions (external forces per unit area acting on the top and bottom surfaces of the plate, see Kalamkarov [44]). On the lateral surfaces we will assume the following boundary conditions (where ui is the mechanical displacement): (︂ )︂ x1 x2 2 , u i = 0, φ = δ e x1 , x2 , x3 , , δh1 δh2 (︂ )︂ x x ψ = δ2 h x1 , x2 , x3 , 1 , 2 (2.3c) δh1 δh2

Figure 1: Thin smart composite plate with rapidly varying thickness and its periodicity cell.

(︂ Hi

x1 , x2 , x3 ,

x1 x2 , δh1 δh2

(︁ ∂ψ x1 , x2 , x3 ,

)︂ =−

x2 x1 δh1 , δh2

)︁

∂x i

(2.2e) In Eqs. 2.2a- 2.2e, σ ij is the mechanical stress, Di and Bi are, respectively, the electric displacement and magnetic induction, Ei and Hi are the electric and magnetic fields and Pi represents a generic body force. Eq. 2.2a represents the static equilibrium equations and Eqs. 2.2b and 2.2c represent the quasi-static approximation of Maxwell’s Equations. It should be noted that all field variables defined thus far are characterized by both periodic (dependence on xi /δhi ) and non-periodic components (dependence on xi ) as is expected for the periodic structure of Fig. 1, see for example Kalamkarov and Georgiades [65]. We will further assume that the top and bottom surfaces of the plate, S± , have the following boundary conditions: σ ij n j = p i ,

D i n i = 0,

B i n i = 0,

on S±

(2.3a)

where for the surfaces x3 = S± (x1 , x2 ) we have the following unit normal vector, see Kalamkarov [44]: ]︃−1/2 )︂ [︃(︂ ± )︂2 (︂ ± )︂2 (︂ ∂S± ∂S± ∂S ∂S ± n = ∓ ,∓ ,1 + +1 ∂x1 ∂x2 ∂x1 ∂x2 (2.3b) The first expression in 2.3a is the familiar Cauchy’s Law, the second implies that we have no free surface electrical charge and the third indicates that the normal component of the magnetic induction field is continuous at the top surface. In Eq. 2.3b, pi (not to be confused with body forces, Pi )

The boundary value problem of Eqs. 2.2a – 2.2c must be complemented by the appropriate constitutive equations in the form of: (︁ )︁ xα )︁ (︁ )︁ (︁ ∂u x , i k δh xα xα α =C ijkl x3 , σ ij x i , ∂x l δh α δh α (︁ )︁ xα (︁ )︁ ∂φ x , i δh α xα + e kij x3 , ∂x k δh α (︁ )︁ xα (︁ )︁ ∂ψ x , i δh α xα + Q kij x3 , ∂x k δh α (︁ x α )︁ (︁ x α )︁ − δΘ ij x3 , T xi , (2.4a) δh α δh α {︂ (︁ (︁ x α )︁ ∂u k (︁ x α )︁ x α )︁ Di xi , =δ e ijk x3 , xi , δh α δh α ∂x l δh α (︁ )︁ xα (︁ x α )︁ ∂φ x i , δh α −ε ij x3 , ∂x δh α (︁ k )︁ xα (︁ x α )︁ ∂ψ x i , δh α −λ ij x3 , ∂x k δh α (︁ x α )︁}︁ x α )︁ (︁ T xi , (2.4b) +δξ i x3 , δh α δh α {︂ (︁ (︁ x α )︁ ∂u k (︁ x α )︁ x α )︁ Bi xi , =δ Q ijk x3 , xi , δh α δh α ∂x l δh α (︁ )︁ xα (︁ x α )︁ ∂φ x i , δh α −λ ij x3 , ∂x δh α (︁ k )︁ xα (︁ )︁ ∂ψ x , i δh xα α −µ ij x3 , ∂x k δh α (︁ x α )︁}︁ x α )︁ (︁ +δη i x3 , T xi , (2.4c) δh α δh α Here, e kl = ∂u k /∂x l is the second order strain field, and Cijkl , eijk , Qijk , and Θ ij are the tensors of the elastic, piezoelectric, piezomagnetic and thermal expansion coefficients respectively. Finally, ε ij , λ ij , µ ij , ξ i and η i represent, respectively, the dielectric permittivity, the magnetoelectric, the magnetic permeability, the pyroelectric and the pyromagnetic tensors. We reiterate that as a consequence of the fact that the composite layer is periodic only in the


tangential directions the material parameters are dependent on x α /δ hα and x3 while the dependent field variables are also dependent on x α = (x1 , x2 ). Eqs. 2.4b and 2.4c show that the constituents of the structure under investigation may, if desired, exhibit magnetoelectric, pyroelectric and pyromagnetic characteristics; Newnham et al. [1] refer to this as the “primary” effect.However, it is more likely that these product properties only appear in the behavior of the macroscopic composite as a consequence of the interactions between the various phases as explained in the previous section. In Eq. 2.4a and in the sequel Roman letters, i, j, k, . . . will vary from 1 to 3, while their Greek counterparts, α, β, 𝛾 , . . . will assume values of 1 or 2 only.

3 Asymptotic analysis and basic assumptions The overall objective of this paper is to obtain general expressions for the effective coefficients (including product properties) and the dependent field variables for magnetoelectric thin plates with rapidly varying thickness. In order to be able to obtain the aforementioned expressions in a form that is immediately comparable to other works, and, more importantly, readily applicable to composite and reinforced structures of different geometrical and compositional make-up, the following procedure will be adhered to in Sections 3-6 of this work. We begin with the asymptotic expansions of all field variables of interest in terms of the dimensionless thickness of the plate δ. Realizing that we can obtain all desirable information via the determination of the mechanical displacement and electric and magnetic potentials, we recast each of the three governing equations 2.2a – 2.2c and associated boundary conditions 2.3a as functions of the leading terms of the asymptotic expansions of the mechanical stress, electric displacement and magnetic induction. The resulting three equations/boundary conditions are then expressed in terms of the desired mechanical displacement, ui , and electric and magnetic potentials, φ and ψ respectively. The mathematical form of these latter expressions is such that it permits us to write down the solutions for ui , φ and ψ as linear combinations involving a set of so-called local functions. The independence of the local functions allows us to group them together into different sets of problems called unit cell problems which are solved entirely on the domain of the unit cell and are independent of the global formulation of the original problem. The solution of the unit cell problems eventually yields the effective or homogenized coefficients, after application of an associated aver-

aging or homogenization procedure. At this stage, we have managed to essentially smooth out the sub-structural variations that exist in the original inhomogeneous composite plate and generate an equivalent homogeneous structure characterized by a single set of material parameters called effective coefficients. Once these effective coefficients are determined, a wide variety of boundary value problems involving a given composite geometry can be studied with relative ease. It would not be remiss to mention at this point that this methodology or some variant thereof has been followed in many of the authors’ previous works, see for example [44–49, 60–65]. It is apparent from the preceding analysis that the smart composite structure under consideration is characterized by two scales; the microscopic scale which is a manifestation of periodicity in the tangential directions, and the macroscopic scale which arises from the global formulation of the problem, see Kalamkarov [44], Challagulla et al [60]. To this end, we begin our analysis with the introduction of the microscopic or “fast” variables, x x x (3.1) y1 = 1 , y2 = 2 , z = 3 δh1 δh2 δ remembering that δ is the thickness of the smart layer. Hence, in terms of these variables, the unit cell Ω δ is defined by {︂ }︂ 1 1 1 1 − + − < y1 < , − < y2 < , Z < z < Z , 2 2 2 2 1 where Z ± = ± ± F ± (y) 2 and y = (y1 , y2 ), x = (x1 , x2 ) (3.2) and the unit normal vector from Eq. 2.3b becomes, (︂ )︂ 1 ∂F ± 1 ∂F ± ,∓ ,1 × n± = ∓ h1 ∂y1 h2 ∂y2 [︃ (︂ ± )︂2 (︂ )︂2 ]︃−1/2 1 ∂F 1 ∂F ± 1+ 2 + 2 h1 ∂y1 h2 ∂y2

(3.3)

Let us now make the following asymptotic assumptions: p α ± = δ2 r α (x,y) , P α = δf α (x,y,z) ,

p±3 = δ3 q±3 (x,y) P3 = δ2 g3 (x,y,z)

(3.4a)

Further, let us assume the following through-the-thickness linear relationships for T, following the commonly adopted assumption in the treatment of heat conduction of plate and shell structures, see for example Podstrigach and Shvets [68], Podstrigach et al. [69]. T (x,y,z) = T1 (x,y,z) + zT2 (x,y,z)

(3.4b)

The reason for the asymptotic forms of Eqs. 2.4b, 2.4c and 3.4a is to ensure convergence of the developed model to its classical plate counterpart as δ → 0.

18 | D. A. Hadjiloizi et al. The introduction of the fast variables necessitates the transformation of the derivatives according to: ∂ ∂ 1 ∂ → + ∂x α ∂x α δh α ∂y α

and

∂ 1 ∂ = ∂x3 δ ∂z

(3.4c)

It is noted that the transformations involving x1 and x2 have the form shown in Eq. 3.4c in accordance with the two-scale expansion formalism, see e.g. [65], whereas for those involving x3 we have an ordinary coordinate transformation. One can therefore express the dependent field variables in powers of δ in the form of: (i) Basic expansions (1) u i (x,y,z) =u(0) i (x,y,z ) + δu i (x,y,z ) (︁ )︁ 3 + δ2 u(2) i (x,y,z ) + O δ

Here, and in the sequel, we adopt the following short-hand convention (except in a few instances where the original format is maintained for clarity): (3.5a)

{︁ φ (x,y,z) =δ φ(0) (x,y,z) + δφ(1) (x,y,z) (︁ )︁}︁ +δ2 φ(2) (x,y,z) + O δ3

(3.5b)

{︁ ψ (x,y,z) =δ ψ(0) (x,y,z) + δψ(1) (x,y,z) (︁ )︁}︁ δ2 ψ(2) (x,y,z) + O δ3

(3.5c)

T1 (x,y,z) = T1(0) (x,y,z) + δT (1) 1 (x,y,z ) (︁ )︁ + δ2 T1(2) (x,y,z) + O δ3 T2 (x,y,z) =T2(0) (x,y,z) + δT (1) 2 (x,y,z ) (︁ )︁ + δ2 T2(2) (x,y,z) + O δ3

(3.5d)

(ii) Derived expansions (1) σ ij (x,y,z) =σ(0) ij (x,y,z ) + δσ ij (x,y,z ) (︁ )︁ 3 + δ2 σ(2) ij (x,y,z ) + O δ

∂φ α = φ α,βy , ∂y β

∂φ α = φ α,βx , ∂x β

∂φ α = φ α,z ∂z

(3.6b)

To obtain equivalent expressions for the general terms pertaining to the asymptotic stress field expansion we substitute expressions 3.5a- 3.5e into the constitutive equation 2.4a and compare terms with the same power of δ to obtain: [︁ ]︁ σ ij (0) =C ijkα u k,αx (0) + h α −1 u k,αy (0) [︁ ]︁ + C ijk3 u k,z (0) + e αij φ ,αx (0) + h α −1 φ ,αy (1) [︁ ]︁ + e3ij φ ,z (1) + Q αij ψ ,αx (0) + h α −1 ψ ,αy (1) + Q3ij ψ ,z (1)

[︁ ]︁ σ ij (n) =C ijkα u k,αx (n) + h α −1 u k,αy (n+1) [︁ ]︁ + C ijk3 u k,z (n+1) + e αij φ ,αx (n) + h α −1 φ ,αy (n+1) [︁ ]︁ + e3ij φ ,z (n+1) + Q αij ψ ,αx (n) + h α −1 ψ ,αy (n+1) [︁ ]︁ + Q3ij ψ ,z (n+1) − Θ ij T1 (n−1) + zT2 (n−1) , n ≥ 1 (3.6c)

(3.5e)

{︁ (1) B i (x,y,z) =δ B(0) i (x,y,z ) + δB i (x,y,z ) (︁ )︁}︁ +δ2 B(2) x,y,z + O δ3 ( ) i

(3.5f)

{︁ (1) D i (x,y,z) =δ D(0) i (x,y,z ) + δD i (x,y,z ) (︁ )︁}︁ 3 +δ2 D(2) i (x,y,z ) + O δ

(3.5g)

(1) e ij (x,y,z) =e(0) ij (x,y,z ) + δe ij (x,y,z ) (︁ )︁ + δ2 e(2) x,y,z + O δ3 ( ) ij

Eq. 3.5h can be used in conjunction with Eq. 3.4c and the familiar strain-displacement relationships to obtain the following expressions for the terms of the mechanical strain expansion: 1 (︁ u α,βx (m) + u β,αx (m) + h β −1 u α,βy (m+1) e αβ (m) = 2 )︁ +h α −1 u β,αy (m+1) , )︁ 1 (︁ e3β (m) = u3,βx (m) + h β −1 u3,βy (m+1) + u β,z (m+1) 2 (m) e33 =u3,z (m+1) m = 0, 1, 2 . . . (3.6a)

(3.5h)

Similarly, substituting Eqs. 3.5a – 3.5d and 3.5f – 3.5g into the constitutive relations 2.4b and 2.4c, gives the corresponding terms for the electric displacement and magnetic induction: [︁ ]︁ D i (0) =e ikα u k,αx (0) + h α −1 u k,αy (1) [︁ ]︁ (0) (1) + e ik3 u k,z (1) − ε iα φ ,αx + h α −1 φ ,αy [︁ ]︁ (1) − ε i3 φ ,z − λ iα ψ ,αx (0) + h α −1 ψ ,αy (1) − λ i3 ψ ,αz (1)

[︁ ]︁ D i (n) =e ikα u k,αx (n) + h α −1 u k,αy (n+1) [︁ ]︁ (0) (n) (n+1) + e ik3 u k,z (n+1) − ε iα φ ,αx φ ,αx + h−1 α φ ,αy


[︁ ]︁ − λ iα ψ ,αx (n) + h α −1 ψ ,αy (n+1) ]︁ [︁ − λ i3 ψ ,αz (n+1) + ξ i T1 (n−1) + zT2 (n−1) , n ≥ 1 − ε i3 φ ,z

(n+1)

(3.6d) [︁ ]︁ B i (0) =Q ikα u k,αx (0) + h α −1 u k,αy (1) + Q ik3 u k,z (1) [︁ ]︁ − λ iα φ ,αx (0) + h α −1 φ ,αy (1) − λ i3 φ ,z (1) [︁ ]︁ − µ iα ψ ,αx (0) + h α −1 ψ ,αy (1) − µ i3 ψ ,z (1)

]︁ [︁ B i (n) =Q ikα u k,αx (n) + h α −1 u k,αy (n+1) [︁ ]︁ + Q ik3 u k,z (n+1) − λ iα φ ,αx (n) + h α −1 φ ,αy (n+1) [︁ ]︁ − λ i3 φ ,z (n+1) − µ iα ψ ,αx (n) + h α −1 ψ ,αy (n+1) [︁ ]︁ − µ i3 ψ ,z (n+1) + η i T1 (n−1) + zT2(n−1) , n ≥ 1 (3.6e) We note that for reasons of compactness, in Eqs. 3.6a and in the sequel, we will forego the arguments of the functions except when it is deemed necessary to include them for the sake of clarity. It is also important to mention that in the process of deriving expressions 3.6a to 3.6e it is readily discovered that the leading terms in the asymptotic expansions for mechanical displacement and electric and magnetic potentials are independent of the macroscopic variables, yα and z. Likewise, consideration of a thermal conductivity boundary value problem examined in Kalamkarov [44] and Hadjiloizi et al. [62], leads to the same conclusion for the leading terms of the asymptotic expansion of the temperature field, Eq. 3.5d. Collectively, these observations are summarized as: u j (0) = u j (0) (x) ; φ(0) = φ(0) (x) ; ψ(0) = ψ(0) (x) ; T1 (0) = T1 (0) (x) ; T2 (0) = T2 (0) (x)

4 Balance laws, boundary conditions and homogenization Our ultimate objective is to derive the so-called unit cell problems from which the effective coefficients may be extracted. To this end we substitute Eq. 3.5e into Eq. 2.2a and then compare terms with the same power of δ to obtain the

following system of differential equations: h β −1 σ iβ,βy (0) + σ i3,z (0) = 0 σ iβ,βx (0) + h β −1 σ iβ,βy (1) + σ i3,z (1) = 0 σ iβ,βx (1) + h β −1 σ iβ,βy (2) + σ i3,z (2) = f i σ iβ,βx (2) + h β −1 σ iβ,βy (3) + σ i3,z (3) = g i and σ iβ,βx (n) + h β −1 σ iβ,βy (n+1) + σ i3,z (n+1) = 0, n ≥ 3 (4.1) where we define f3 = g1 = g2 = 0. Each of these differential equations must be accompanied by the appropriate boundary condition. To this end, we write Cauchy’s expression in 2.3a as, σ ij n j ± = ± p i

(4.2a)

where the negative sign on the right-hand side corresponds to an inward unit normal vector. We then substitute expansion 3.5e into Eq. 4.2a to obtain, in view of Eq. 3.4a, the following boundary conditions to be satisfied at the top and bottom surfaces of the smart composite plate: σ αj (0) N j ± + δσ αj (1) N j ± + δ2 σ αj (1) N j ± + δ3 σ αj (2) N j ± + · · · = ±ω± δ2 r α ± (0) ± σ3j N j + δσ3j (1) N j ± + δ2 σ3j (1) N j ± + δ3 σ3j (2) N j ± + · · · = ±ω± δ3 q3 ± (4.2b) Here, for the sake of convenience, the following definitions are made: (︁ )︁ ± 1 ∂F ± N± = ∓ h11 ∂F and ∂y1 , ∓ h2 ∂y2 , 1 √︂ (︁ ± )︁2 (︁ ± )︁2 (4.2c) 1 ∂F + ω± = 1 + h12 ∂F 2 ∂y1 ∂y2 h 1

2

We recall that functions F± define the geometric profiles of the top and bottom surfaces of the plate as shown in Fig. 1. Finally, equating like powers of δ gives the final form of the appropriate stress boundary conditions, namely: ⎫ ⎪ σ ij (m) N j ± = 0, m = 0, 1 ⎪ ⎪ σ ij (2) N j ± = ±ω± r i ± , r3 ± = 0 ⎬ on Z ± (4.2d) σ ij (3) N j ± = ±ω± q i ± , q α ± = 0 ⎪ ⎪ ⎪ ⎭ σ ij (n) N j ± = 0, n ≥ 4 We proceed in much the same way for the electric displacement problem. Accordingly, keeping Eq. 3.4c in mind, we substitute Eq. 3.5g into the governing Eq. 2.2b and compare terms with the same powers of δ to get the following set of differential equations: h β −1 D β,βy (0) + D3,z (0) = 0 D β,βx (n) + h β −1 D(n+1) + D3,z (n+1) = 0 β,βy

n≥0

(4.3a)

The pertinent boundary conditions are obtained by substituting Eq. 3.5g into the second expression in Eq. 2.3a and comparing terms to obtain: D i (n) N i ± = 0 for n ≥ 0

on Z ±

(4.3b)

20 | D. A. Hadjiloizi et al. Similar expressions are readily obtained for magnetic induction from Eqs. 2.2c, 3.4c, 3.5f and the third expression in Eq. 2.3a: h β −1 B β,βy (0) + B3,z (0) = 0 B β,βx (n) + h β −1 B β,βy (n+1) + B3,z (n+1) = 0

n≥0

(4.4a)

and B i (n) N i ± = 0

on Z ±

(4.4b)

We next introduce the averaging procedure, ∫︁ ⟨. . .⟩ = (. . .) dy1 dy2 dz

(4.5a)

for n ≥ 0

Ω

Gibson [70] etc., are given by: ⟨︀ ⟩︀ ⟨︀ ⟩︀ N αβ = δ σ αβ , ⟨Q α ⟩ = δ ⟨σ α3 ⟩ ,

⟩︀ ⟨︀ ⟩︀ M αβ = δ2 zσ αβ (4.6) To obtain the force resultants we average the expressions in Eq. 4.1 in the sense of Eq. 4.5a and apply at the same time the boundary conditions 4.2d and the general result 4.5b. We get ⟨︀

N αβ,x β (0) = 0, N αβ,x β (1) + δr α * (x𝛾 ) = δ ⟨f α ⟩ , N αβ,x β (n) = 0 where n ≥ 2 Q β,xβ (1) = 0, Q β,xβ (2) + δq3 * (x𝛾 ) = δ ⟨g3 ⟩ , Q β,xβ (n) = 0 where n ≥ 3

(4.7a)

where we define:

defined over the volume |Ω| of the unit cell Ω with boundary surface ∂Ω, and proceed to show the following relationship,

∫︁1/2 ∫︁1/2

*

r α (x) =

(︀

)︀ ω+ r α + + ω− r α − dy1 dy2

−1/2 −1/2

⟨

⟩

h α −1 Q α,αy + Q3,z =

∫︁1/2 ∫︁1/2

(︀

)︀ Q i + N i + − Q i − N i − dy1 dy2

*

q 3 ( x) =

−1/2 −1/2

(︀

)︀ ω+ q3 + + ω− q3 − dy1 dy2

(4.7b)

1/2 −1/2

(4.5b) where N is defined in Eq. 4.2c and Q i are the values Qi takes on the surfaces Z± . Starting from the divergence theorem we have ⟨ ⟩ )︁ ∫︀ (︁ −1 h α −1 Q α,αy + Q3,z = h α Q α,αy + Q3,z dv Ω δ (︁ )︁ ∫︀ = h α −1 Q α n yα + Q3 n y3 dA, ±

±

∫︁1/2 ∫︁1/2

∂Ω δ

(4.5c) where n y + (n y − ) is the outward (inward) unit normal vector defined with respect to the (y1 , y2 , z) coordinate system of the unit cell and is given by: (︂ )︂ √︃(︂ )︂2 (︂ )︂2 ∂F ± ∂F ± ∂F ± ∂F ± ± ny = ∓ ,∓ ,1 / + +1 ∂y1 ∂y1 ∂y1 ∂y2 (4.5d) Now, periodicity considerations stipulate that the first integral in Eq. 4.5c reduces to )︁ ∫︀ (︁ −1 + + h α Q α n yα + Q3 + n y3 + ds Ω + S+ (︁ )︁ (4.5e) ∫︀ − h α −1 Q α − n yα − + Q3 − n y3 − ds Ω − S−

where ds Ω + (ds Ω − ) is given by: √︃(︂ )︂2 (︂ )︂2 ∂F ± ∂F ± + + 1dy1 dy2 ds Ω ± = ∂y1 ∂y2

To obtain the moment resultants we multiply the expressions in Eq. 4.1 by z and then integrate over the volume of the unit cell to obtain, on account of the boundary conditions 4.2d and Eq. 4.5b: ⟨ ⟩ M αβ,xβ (0) + δ Q α (1) = 0, ⟨ ⟩ M αβ,xβ (1) + δ2 ρ α * (x) −δ Q α (2) = δ2 ⟨zf α ⟩ ⟨ ⟩ (4.7c) M αβ,xβ (2) −δ Q α (3) =0, ⟨ ⟩ zσ3β,xβ (2) + σ3 * (x) = ⟨zg3 ⟩ , M αβ,xβ (n) = 0 where Here we define: *

∫︁1/2 ∫︁1/2

ρ α (x) =

Finally, substituting Eq. 4.5f into Eq. 4.5e proves the result in Eq. 4.5b on account of 4.2c. Within the framework of the terminology adopted in this paper, the force resultants, Niα , Qα , and moment resultants, Miα , of the homogenized plate, see Kalamkarov [44],

(︀

)︀ z+ ω+ r α + + z− ω− r α − dy1 dy2

(︀

)︀ z+ ω+ q3 + + z− ω− q3 − dy1 dy2

−1/2 −1/2

*

∫︁1/2 ∫︁1/2

σ 3 ( x) =

(4.7d)

−1/2 −1/2

It will later on be seen that as a consequence of the plane stress assumption and the fact that σ αβ (0) will turn out to be 0, two important consequences of Eqs. 4.7a and 4.7c are: ⟨

(4.5f)

n≥3

⟩ ⟨ ⟩ σ i3 (1) = zσ i3 (1) = 0

(4.7e)

We finally repeat this homogenization procedure on Eqs. 4.3a and 4.4a to give the governing equations for the averaged electric displacement and magnetic induction in the form of: ⟨ ⟩ D β,xβ (n) = 0, n ≥ 0 ⟨ ⟩ (4.7f) B β,xβ (n) = 0, n ≥ 0


5 Unit cell problems for homogenized plate

A j * u j (1) −L* φ(1) −M * ψ(1) = −G kα * (y, z) u(0) x k,αx ( ) x +I α * (y, z) φ k,αx (0) (x) + K α * (y, z) ψ(0) ( ) k,αx

Our first objective in this section is to determine the leading terms in the asymptotic expansions of mechanical displacement and electrical and magnetic potential. To achieve this, we substitute the first term of the asymptotic expansion (3.5e) given in Eq. (3.6c) for n = 0 into the first expression in (4.1) and the stress boundary condition in (4.2d) for m = 0. After some straightforward albeit tedious algebraic manipulations we obtain the following expression and accompanying boundary condition: D ik u k (1) +C i φ(1) +F i ψ(1) = −C ikα (y, z) u k,αx (0) (x) −P αi (y, z) φ k,αx (0) − R αi (y, z) ψ(0) k,αx (5.1)

{︁

L ijk u k (1) +M ij φ(1) +N ij ψ(1) +C ijkα u k,αx (0) (x) }︁ +e αij φ k,αx (0) + Q αij ψ k,αx (0) N j ± = 0 on Z ±

L ij * u k (1) −M i * φ(1) −N i * ψ(1) +e ijα u k,αx (0) (x) }︁ −ε iα φ k,αx (0) (x) − λ iα ψ k,αx (0) (x) N j ± = 0 on

(5.2a)

Z±

(5.3) Here, for the sake of convenience, we define the following differential operators ∂ 1 ∂ ∂ 1 ∂ + e ij3 , M i * = ε iα + ε i3 h α ∂y α ∂z h α ∂y α ∂z 1 ∂ ∂ ∂ 1 ∂ + λ i3 , A i * = L *+ L * N i * = λ iα h α ∂y α ∂z h α ∂y α αi ∂z 3i 1 ∂ 1 ∂ ∂ ∂ L* = Mα * + M3 * , M * = Nα * + N3 * h α ∂y α ∂z h α ∂y α ∂z (5.4a) and the following parameters L ij * = e ijα

1 ∂e βkα ∂e3kα + , h β ∂y β ∂z 1 ∂λ αβ ∂λ3β + = h β ∂y β ∂z

G kα * =

Here, for economy of notation, we defined the following differential operators 1 ∂ ∂ L ijk = C ijkα + C ijk3 , h α ∂y α ∂z 1 ∂ ∂ M ij = e αij + e3ij h α ∂y α ∂z 1 ∂ ∂ N ij = Q αij + Q3ij , h α ∂y α ∂z ∂ 1 ∂ L + L D ij = h α ∂y α iαj ∂z i3j 1 ∂ ∂ Ci = M + M , h α ∂y α iα ∂z i3 1 ∂ ∂ Fi = N + N h α ∂y α iα ∂z i3

{︁

Kα *

Iα * =

1 ∂ε αβ ∂ε3β + , h β ∂y β ∂z

(5.4b) Finally, substitution of Eq. 3.6e for n = 1 into the first expression in Eq. 4.4a and the associated boundary condition in Eq. 4.4b for n = 0 results in the following system: A j u j (1) −Lφ(1) −Mψ(1) = −G kα (y, z) u(0) x k,αx ( ) (0) +I α (y, z) φ(0) x + K y, z ψ x ( ) ( ) ( ) α k,αx k,αx {︁

L ij u k (1) −M i φ(1) −N i ψ(1) +q ijα u k,αx (0) (x) }︁ −λ iα φ k,αx (0) (x) − µ iα ψ k,αx (0) (x) N j± = 0

on

Z± (5.5)

and the following parameters Here we define the following differential operators 1 ∂C iβkα ∂C i3kα + , h β ∂y β ∂z 1 ∂Q iαβ ∂Q iα3 R αi = + h β ∂y β ∂z C ikα =

P αi =

1 ∂e iαβ ∂e iα3 + , h β ∂y β ∂z

(5.2b) It should be noted that each term on the right-hand side of the first expression of Eq. 5.1 is a product of a function of x and a function of y, z. This will play a significant role in the general form of the solution of u k (0) , φ(0) and ψ(0) , as we will see shortly. Similarly, substitution of Eq. 3.6d for n = 0 into the first expression in Eq. 4.3a and the associated boundary condition in Eq. 4.3b for n = 0 results in the following system:

∂ 1 ∂ ∂ 1 ∂ + q ij3 , M i = λ iα + λ i3 h α ∂y α ∂z h α ∂y α ∂z 1 ∂ ∂ 1 ∂ ∂ N i = µ iα + µ i3 , A i = L + L h α ∂y α ∂z h α ∂y α αi ∂z 3i 1 ∂ ∂ 1 ∂ ∂ L= Mα + M3 , M = Nα + N3 h α ∂y α ∂z h α ∂y α ∂z (5.6a) and the following parameters L ij = q ijα

1 ∂q βkα ∂q3kα + , h β ∂y β ∂z 1 ∂µ αβ ∂µ3β Kα = + h β ∂y β ∂z G kα =

Iα =

1 ∂λ αβ ∂λ3β + = Kα * , h β ∂y β ∂z

(5.6b) The separation of variables on the right-hand sides of the differential equations in (5.1), (5.3) and (5.5) allows us to write down the solution of u k (0) , φ(0) and ψ(0) in the form of:

22 | D. A. Hadjiloizi et al.

u i (1) (x, y,z) = N i kα (y,z) u k,αx (0) (x) +M α i (y,z) φ k,αx (0) (x) + N α i (y,z) ψ k,αx (0) (x) + ω i (x) (5.7a) φ(1) (x, y,z) = A kα (y,z) u k,αx (0) (x) +Ξ α (y,z) φ k,αx (0) (x) + O α (y,z) ψ k,αx (0) (x) + 𝛾 (x) (5.7b) ψ(1) (x, y,z) = Λ kα (y,z) u k,αx (0) (x) +Z α (y,z) φ k,αx (0) (x) + Γ α (y,z) ψ k,αx (0) (x) + ω (x) (5.7c) Excluding functions ω i (x), 𝛾 (x), ω (x) which are the homogeneous solutions Eqs. 5.7a, 5.7b and 5.7c, contain 9 unknown functions, N i kα , M i α , N i α , Akα , Ξ α , Oα , Λ kα , Zα , Γ α which are solved by back-substitution into Eqs. 5.1, 5.3 and 5.5 to generate the unit cell problems. These are: kα h β −1 b kα (y, z) = 0 iβ,βy (y, z ) + b i3,z kα with b ij (y, z) N j ± = 0 on Z ± −1

iβ

iβ

b ij kα = L ijm N m kα + M ij A kα + N ij Λ kα + C ijkα

(5.11a)

b α ij = L ijm M α m + M ij Ξ α + N ij Z α + e αij

(5.11b)

a α ij = L ijm N α m + M ij O α + N ij Γ α + Q αij

(5.11c)

η j kα = L ji N i kα −M j A kα −N j (y) Λ kα +Q jkα

(5.12a)

a jα = L ji M α i −M j Ξ α −N j Z α −λ jα

(5.12b)

𝛾jα = L ji N α i −M j Oα −N j Γ α −µ jα

(5.12c)

δ j kα = L ji * N i kα −M j * A kα −N j * Λ kα +e jkα

(5.13a)

(5.8a)

i3

h β b α,y (y, z) + b α,z (y, z) = 0 with b α ij (y, z) N j ± = 0 on Z ± −1

in “classical” homogenization schemes, see for example Bakhvalov and Panasenko [42], the unit cell problems in Eqs. 5.8a- 5.10c also involve boundary conditions on the upper and lower surfaces of the unit cell. The following definitions are used in the aforementioned equations.

(5.8b) δ jα = L ji * M α i −M j * Ξ α −N j * Z α −ε jα

(5.13b)

ξ jα = L ji * N α i −M j * O α −N j * Γ α −λ jα

(5.13c)

i3

h β a α,βy (y, z) + a α,z (y, z) = 0 with a α ij (y, z) N j ± = 0 on Z ±

(5.8c)

iα h β −1 η iα β,βy (y, z ) + η 3,z (y, z ) = 0 iα with η j (y, z) N j ± = 0 on Z ±

(5.9a)

h β −1 a βα,βy (y, z) + a3α,z (y, z) = 0 with a jα (y, z) N j ± = 0 on Z ±

(5.9b)

h β −1 𝛾βα,βα (y, z) + 𝛾3α,z (y, z) = 0 with 𝛾jα (y, z)N j ± = 0 on Z pm

(5.9c)

h β −1 δ β,βy iα (y, z) + δ3,z iα (y, z) = 0 with δ j iα (y, z) N j ± = 0 on Z ±

(5.10a)

h β −1 δ βα,βy (y, z) + δ3α,z (y, z) = 0 with δ jα (y, z) N j ± = 0 on Z ±

(5.10b)

h β −1 ξ βα,βy (y, z) + ξ3α,z (y, z) = 0 with ξ jα (y, z) N j ± = 0 on Z ±

(5.10c)

The unit cell problems can be viewed as being grouped in three separate sets. The first set of three unit cell problems, Eqs. 5.8a – 5.8c, pertains to the mechanical stress problem (force balance equation), the equations of the second set, Eqs. 5.9a – 5.9c, stem from Maxwell’s Law for the magnetic field, and the third set of three equations, Eqs. 5.10a – 5.10c, are related to Maxwell’s Law involving electric displacement. We also note that unlike unit cell problems

As their name suggests, the unit cell problems are solved entirely on the domain of the unit cell and are entirely independent of the macroscopic variable. As we will see later on, nine more unit cell problems will be generated six of which will relate to the out-of-plane deformation of the homogenized plate. In the analysis of the homogenized plate model the possibility of finding an exact solution often plays a significant role. In our case an exact solution for k, α = 3, 1 and 3, 2 can be readily found from Eqs. 5.8a, 5.9a and 5.10a and is of the form: N1 31 = −z; N2 31 = 0; N2 32 = −z; N3 31 = 0; Λ31 = Λ32 = 0 A31 = A32 = 0;

N1 32 = 0; N3 32 = 0;

(5.14a)

With these results in mind it is readily shown that: b ij 31 = b ij 32 = η j 31 = η j 32 = δ j 31 = δ j 32 = 0

(5.14b)

Since the “3” superscript in the local functions b ij kl , η j kl , and δ j kl becomes obsolete, only b ij αl , η j αl , and δ j αl need to be considered. Consequently,


the 1st , 4th and 7th unit cell problems, Eqs. 5.8a, 5.9a and 5.10a may be simplified as follows: µα µα h−1 β b iβ,βy (y, z ) + b i3,z (y, z ) = 0 µα with b iβ (y, z) N j ± = 0 on Z ± µα h−1 β η β,βy

with

µα η3,z

(y, z) + (y, z) = 0 b ijα (y, z) N j ± = 0 on Z ±

(5.15a)

in Eqs. 3.6d and 3.6e to obtain, in view of definitions 5.12a – 5.12c, 5.13a – 5.13c and the results in Eq. 5.14b, the leading terms in the asymptotic expansions for the electric displacement and magnetic induction. They are: D i (0) = δ i αβ u α,βx (0) + δ iβ φ ,βx (0) + ξ iβ ψ ,βx (0)

(5.18b)

B i (0) = η i αβ u α,βx (0) + a iβ φ ,βx (0) + 𝛾iβ ψ ,βx (0)

(5.18c)

(5.15b)

Finally, the mechanical strain is computed from Eq. 3.6a:

µα µα h−1 β δ β,βy (y, z) + δ 3,z (y, z) = 0 µα with δ j (y, z)N j ± = 0 on Z ±

(5.15c)

Furthermore, expressions 5.7a – 5.7c are simplified as follows: u1 (1) = −zu3,1x (0) +N 1 βα u β,αx (0) +M α 1 φ ,αx (0) +N α 1 ψ ,αx (0) +ω1 u2 (1) = −zu3,2x (0) +N 2 βα u β,αx (0) +M α 2 φ ,αx (0) +N α 2 ψ ,αx (0) +ω2 u3 (1) = N 3 βα u β,αx (0) +M α 3 φ ,αx (0) +N α 3 ψ ,αx (0) +ω3

φ(1) = A βα u β,αx (0) + Ξ α φ ,αx (0) + O α ψ ,αx (0) + 𝛾

(5.16a)

2e αβ (0) = Q µν αβ u α,βx (0) + I αβ µ φ ,βµ (0) + S αβ µ ψ ,βµ (0) ˜ µαβ u α,βx (0) + ˜I βµ φ ,βµ (0) + S˜ βµ ψ ,βµ (0) 2e3β (0) = Q e33 (0) = N3,z αβ u α,βx (0) + M µ,z 3 φ ,βµ (0) + N µ,z 3 ψ ,βµ (0) (5.18d) wherein the following definitions are used: [︁ ]︁ Q µν αβ = h ν −1 N µ,µy αβ +h µ −1 N ν,µy αβ + δ αµ δ βν +δ αν δ βµ ]︁ [︁ I αβ µ = h β −1 M µ,βy α + h α −1 M µ,αy β , ]︁ [︁ S αβ µ = h β −1 N µ,βy α + h α −1 N µ,αy β (5.19a)

(5.16b) [︁

ψ

(1)

= Λ βα u β,αx

(0)

+ Z α φ ,αx

(0)

+ Γ α ψ ,αx

(0)

+ω

T1 (1) = β α T1,αx (0) + λ1 (1) ,

zT 2 (1) = β α (1) T2,αx (0) + λ2 (1) (5.17) where β α , β α (1) are local functions similar to, for example, A βα in Eq. 5.16b, and λ1 (1) , λ2 (1) are the solutions of the homogeneous thermal conductivity boundary value problems, see Kalamkarov [44], similar to the 𝛾 or ω functions in Eqs. 5.16b and 5.16c. We are now in a position to compute the leading terms in the expansions of mechanical stress, electric displacement, magnetic induction and mechanical strain. We recall that these variables were defined in Eqs. 3.5e – 3.5h as the “derived” variables. To this end we substitute the results of Eqs. 5.7a – 5.7c into expression 3.6c for n = 0, and keeping Eq. 5.14b in mind, we obtain for σ ij (0) : σ ij (0) = b ij αβ u α,βx (0) + b α ij φ ,βx (0) + a α ij ψ ,βx (0)

˜ µαβ = h β −1 N3,βy µα + N β,z µα , Q ]︁ [︁ ˜I βµ = h β −1 M α,βy 3 + M α,z β , ]︁ [︁ S˜ βµ = h β −1 N α,βy 3 + N α,z β

(5.16c)

Further, an examination of the thermal conductivity boundary value problem and associated Fourier’s Law of conduction, see Kalamkarov [44], reveals that second terms of the asymptotic expansion of the temperature field are given by

(5.18a)

Here, definitions 5.11a – 5.11c were used. In a similar manner, we substitute Eqs. 5.7a – 5.7c into the first expressions

]︁

(5.19b)

Averaging Eqs. 5.18a- 5.18c in the sense of Eq. 4.5a gives the pertinent homogenized expressions: ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ σ ij (0) = b ij αβ u α,βx (0) + b α ij φ ,αx (0) + a α ij ψ ,αx (0) (5.20a) ⟨

⟩ ⟨ ⟩ D i (0) = δ i αβ u α,βx (0) + ⟨δ iα ⟩ φ ,αx (0) + ⟨ξ iα ⟩ ψ ,αx (0) (5.20b)

⟨

⟩ ⟨ ⟩ ^ αβ B i (0) = a u α,βx (0) + ⟨a iα ⟩ φ ,αx (0) + ⟨𝛾iα ⟩ ψ ,αx (0) i

(5.20c) If we subsequently take the expansions in 3.5a- 3.5c and substitute them into the boundary conditions on the lateral surfaces of the thin plate in Eq. 2.3c, we arrive at the following expressions after comparing terms with like powers of δ: ⎫ ⎪ on the u i (n) = 0, n ≥ 0 ⎬ (0) (1) (n) φ = 0, φ = e, φ = 0, n ≥ 2 lateral ⎪ ψ(0) = 0, ψ(1) = h, ψ(n) = 0, n ≥ 2 ⎭ surfaces (5.21) If we substitute the homogenized fields given in Eqs. 5.20a – 5.20c into both the first expression of Eq. 4.7a and the governing equations in 4.7f for n = 0 we readily see

24 | D. A. Hadjiloizi et al. that the solution of the resulting differential equations, in conjunction with the boundary conditions in Eq. 5.21, is: u α (0) = 0;

φ(0) = 0 ;

ψ(0) = 0

(5.22)

It then follows from Eqs. (5.18a)-(5.18d) that: σ ij (0) = 0 ;

D i (0) = 0 ;

B i (0) = 0 ;

e ij (0) = 0 (5.23) With these results in mind, the expressions in Eqs. 5.16a5.16c simplify to: u1 (1) = −zu3,1x (0) +ω1 (x) u2 (1) = −zu3,2x (0) +ω2 (x) u3 (1) = ω3 (x)

(5.24a)

φ(1) = 𝛾 (x)

(5.24b)

ψ(1) = ω (x)

(5.24c)

In view of the results in Eq. 5.23, the leading terms in the asymptotic expansions for mechanical stress, electric displacement, magnetic induction and mechanical strain are, respectively, σ ij (1) , D i (1) , B i (1) and e ij (1) . We now proceed to calculate them. To this end, we substitute Eqs. 5.24a – 5.24c into Eq. 3.6c for n = 1 to obtain: {︁ }︁ σ ij (1) = C ijαβ −zu3,αβx (0) + ω α,βx + C ij3β ω3,βx + (2) +C ijkβ{︁h β −1 u k,βy (2) + C ijk3 }︁ u k,z −1

(2)

Finally, the mechanical strain is obtained from Eqs. 3.6a and 5.24a: e αα (1)

= −zu3,ααx (0) + ω α,αx + (2) +h−1 no summation on α α u α,αy , (1) (0) 2e12 = −2zu3,x1 x2 + ω1,x2 + ω2,x1 + h1 −1 u2,y1 (2) + +h2 −1 u1,y2 (2) (1) 2e3β = ω3,βx + u β,z (2) + h β −1 u3,βy (2) , (1) e33 = u3,z (2) (5.25d) The next step is to solve for the next terms in the asymptotic expansions for the mechanical displacement and the two potential functions. We begin by substituting Eq. 5.25a into the second expression in Eq. 4.1 and into the boundary condition in Eq. 4.2d for m = 1. In view of Eq. 5.23 and the definitions in Eqs. 5.2a and 5.2b, we arrive at the following expressions: D ij u j (2) +C i φ(2) +F i ψ(2) = −C ijα ω j,αx − P αi 𝛾,αx − R αi ω ,αx {︁ )︀ (︀ (2) + U i T1 (0) + (Θ i3 +zU i ) T2 (0) + C i3αβ +zC iαβ u(0) 3,x α x β L ijk u k +M ij φ(2) +N ij ψ(2) +C ijkα ω k,αx + e αij 𝛾,αx + Q αij ω ,αx [︁ }︁ ]︁ ± −Θ ij T1 (0) + zT 2 (0) − zC ijαβ u(0) 3,x α x β N j = 0 on

(5.26a) Here, we also make the following definition: U i = h β −1 Θ iβ,y β + Θ i3,z

(5.26b)

(2)

+e αij 𝛾,αx +h α φ αy +e3ij φ z + {︁ }︁ −1 (2) +Q αij ω ,αx +h α ψ αy +Q3ij ψ z (2) + {︁ }︁ −Θ ij T1 (0) + zT 2 (0) (5.25a) Similarly, Eqs. 5.24a- 5.24c are substituted into Eqs. 3.6d and 3.6e for n = 0 to obtain the corresponding expressions for the electric displacement and magnetic induction. They are of the form {︁ }︁ D i (1) = e iαβ −zu3,αβx (0) + ω α,βx + e i3β ω3,βx + +e ijβ{︁ h β −1 ∂u j,βy (2) + e ij3}︁u j,z (2) +

−ε iα 𝛾,αx +h α −1 φ ,αy (2) −ε i3 φ z (2) + {︁ }︁ −λ iα ω ,αx +h α −1 ψ ,αy (2) −λ i3 ψ ,z (2) + {︁ }︁ +ξ i T1 (0) + zT 2 (0)

We repeat this procedure by substituting Eq. 5.25c into the second expression in Eq. 4.4a and the boundary condition in Eq. 4.4b for n = 1. Keeping the results in Eq. 5.23 and the definitions in Eqs. 5.6a and 5.6b in mind, we arrive at the following expressions: A j u j (2) − Lφ(2) − Mψ(2) = −G jα ω j,x α + I α 𝛾,x α + K α ω ,x α {︁ (︀ )︀ (2) − VT 1 (0) + − (η3 +zV ) T2 (0) + Q3αβ +zG αβ u(0) 3,x α x β L ij u j −M i φ(2) − N i ψ(2) +Q ijα ω j,x α − λ iα 𝛾,x α + −µ iα ω ,x α [︁ ]︁ }︁ +η i T1 (0) + zT 2 (0) − zQ iαβ u3,x α x β (0) N i± = 0 on

(5.25b)

Z± (5.26c)

Here, we use the following definition: V=

B i (1)

Z±

{︁ }︁ = Q iαβ −zu3,αβx (0) + ω α,βx + Q i3β ω3,βx + +Q ijβ{︁h β −1 u j,βy (2) + Q ij3}︁ u j,z (2) +

−λ iα 𝛾,αx +h α −1 φ ,αy (2) −λ i3 φ ,z (2) + {︁ }︁ −µ iα ω ,αx +h α −1 ψ ,αy (2) −µ i3 ψ ,z (2) + {︁ }︁ +η i T1 (0) + zT 2 (0) (5.25c)

1 ∂η β ∂η3 + h β ∂y β ∂z

(5.26d)

The last differential equation and its associated boundary condition are determined in the same way, by substituting Eq. 5.25b into the second expression in Eq. 4.3a and into the boundary condition in Eq. 4.3b for n = 1. Recalling the results in Eqs. 5.4a, 5.4b and 5.23 we obtain A j * u j (2) − L* φ(2) − M * ψ(2) = −G jα * ω j,x α + I α * 𝛾, xα + K α * ω , xα


(︁ )︁ (︁ )︁ − V * T1 (0) + − ξ3 +zV * T2 (0) + e3αβ +zG αβ * u(0) 3,x α ∂x β {︁ × L ij * u j (2) − M i * φ(2) − N i * ψ(2) + e ijα ω j,x α − ϵ iα 𝛾,x α }︁ −λ iα ω ,x α + ζ i [T1 (0) + zT2 (0) ] − ze iαβ u3,x α ∂x β (0) N j ± = 0 on Z ± (5.26e) where we use the following definition: 1 ∂ξ β ∂ξ3 V = + h β ∂y β ∂z *

(5.26f)

Again, we observe that all terms on the right hand side of the differential equations in 5.26a, 5.26c and 5.26e are products of a function of x and a function of y. This separation of variables suggests that we can write down the solution of u k (2) (x,y, z), φ(2) (x,y, z) and ψ(2) (x,y, z) in the form of the following linear combinations: N i jβ ω j,x β + M α i 𝛾,x β +N α i ω ,x α +G i T1 (0) + (*) ^ (0) +G i (1) T2 (0) − N i (1)αβ u (x) 3,x α x β + ω i

u i (2) =

(1) h β −1 𝛾β,βy (y, z) + 𝛾3,z (1) (y, z) = 0 with 𝛾j (1) (y, z) N j ± = 0 on Z ±

(5.29b)

h β −1 a(1)µα (y, z) + a3,z (1)µα (y, z) = 0 β,βy with a i µα (y, z) N j ± = 0 on Z ±

(5.29c)

h β −1 τ β,βy (y, z) + τ3,z (y, z) = 0 with τ j (y, z) N j ± = 0 on Z ±

(5.30a)

y, z) + τ3,z (1) (y, z) = 0 h β −1 τ(1) β,βy ( with τ j (1) (y, z) N j ± = 0 on Z ±

(5.30b)

h β −1 δ(1)µα (y, z) + δ3,z (1)µα (y, z) = 0 β,βy with δ i µα (y, z) N j ± = 0 on Z ±

(5.30c)

Here, we use the following definitions: b ij = L ijm G m + M ij Π + N ij ∆ − Θ ij

(5.27a) b ij (1) = L ijm G m (1) + M ij Π (1) + N ij ∆(1) − zΘ ij

φ

(2)

=

A kα ω k,x α + Ξ α 𝛾,x α ∂x∂ α + O α ω ,x α + ΠT1 (0) + * ^ (0) +Π (1) T2 (0) − A αβ (1) u 3,x α x β + 𝛾 ( x ) (5.27b)

ψ(2) =

Λ kα ω k,x α + Z α 𝛾,x α + Γ α ω ,x α + ∆T1(0) + ^ (0) + ω* (x) u +∆(1) T2 (0) − Λ(1) αβ 3,x α x β

(5.31a)

(5.27c)

As was the case in Eqs. 5.7a – 5.7c, functions ω i (*) (x), 𝛾 * (x) and ω* (x) are the homogeneous solutions and will not affect our subsequent results, particularly the effective coefficients. Excluding these functions, Eqs. 5.19a – 5.19b contain 18 unknown functions, N i kα , M i α , N i α , G i , G i (1) , Akα , , and Λ αβ (1) . Ξ α , Oα , Π, Π (1) , Λ kα , Zα , Γ α , ∆, ∆(1) , N i (1)αβ , A(1) αβ Their solution is obtained by back substitution of these functions into Eqs. 5.27a – 5.27c and comparing similar terms. We arrive at a set of eighteen unit cell problems the first nine of which have already been determined and are defined in Eqs. 5.8a – 5.10c. The remaining nine local functions, N i (1)αβ , Λ αβ (1) and Aαβ (1) , satisfy the following unit cell problems: h β −1 b iβ,βy (y, z) + b i3,z (y, z) = 0 with b ij (y, z) N j ± = 0 on Z ±

(5.28a)

h β −1 b iβ,βy (1) (y, z) + b i3,z (1) (y, z) = 0 with b ij (1) (y, z) N j ± = 0 on Z ±

(5.28b)

h β −1 b(1)µα y, z) + b i3,z (1)µα (y, z) = 0 iβ,βy ( with b ij µα (y, z) N j ± = 0 on Z ±

(5.28c)

h β −1 𝛾β,βy (y, z) + 𝛾3,z (y, z) = 0 with 𝛾j (y, z) N j ± = 0 on Z ±

(5.29a)

(5.31b)

b ij (1)kα = L ijm N m (1)kα + M ij A kα (1) − N ij Λ kα (1) + zC ijkα (5.31c)

𝛾j = L ji G i −M j Π − N j ∆ − η j

(5.32a)

𝛾j (1) = L ji G i (1) −M j Π (1) −N j ∆(1) +zη j

(5.32b)

a i (1)kα = L im N m (1)kα − M i A kα (1) − N i Λ kα (1) + zQ ikα (5.32c) τ j = L ji * G i −M j * Π − N j * ∆ + ξ j

(5.33a)

τ j (1) = L ji * G i (1) −M j * Π (1) −N j * ∆(1) +zξ j

(5.33b)

δ i (1)kα = L im * N m (1)kα − M i * A kα (1) − N i * Λ kα (1) +ze ikα (5.33c) The presence of the z coordinate in the unit cell problems 5.28b, 5.28c, 5.29b, 5.29c as well as 5.30b, 5.30c implies that these problems are related to out-of-plane deformation and electric and magnetic field generation in the homogenized plate.


6 Effective properties of homogenized plate and relationships with classical plate Substitution of Eqs. 5.27a – 5.27c into Eqs. 5.25a – 5.25c gives the leading terms of the asymptotic expansions of mechanical stress, electric displacement and magnetic induction in terms of the local functions obtained via the 18 unit cell problems in Eqs. 5.8a – 5.10c and 5.28a – 5.30c. σ ij (1) =

D i (1) =

B i (1) =

b ij αβ ω α,x β − b ij (0)αβ u3,x α x β (0) + b β ij 𝛾,x β + +a β ij ω ,x β + b ij T1 (0) + b ij (1) T2 (0) δ i αβ ω α,x β − δ ij (1)αβ u(0) 3,x α x β + δ iβ 𝛾,x β + +ξ iβ ω ,x β + τ i T1 (0) + τ i (1) T2 (0)

(6.1b)

η i αβ ω α,x β − a i (1)αβ u(0) 3,x α x β + a iβ 𝛾,x β + +𝛾iβ ω α,x β + 𝛾i T1 (0) + 𝛾i (1) T2 (0)

(6.1c)

(6.2a) ⟨ ⟩ ⟩︀ ⟨︀ δ2 zb αβ µν ε µν − δ3 zb αβ (1)µν u(0) 3,x µ x ν + ⟨ ⟩ ⟨ ⟩ αβ αβ 2 2 * * +δ zb µ φ x µ + δ za µ ψ x µ + ⟨ ⟩ ⟨︀ ⟩︀ (0) 3 +δ zb αβ T1 + δ3 zb αβ (1) T2 (0)

(6.2b) To arrive at Eqs. 6.2a and 6.2b we use the following definitions: ψ* = δω,

φ* = δ𝛾 ,

v α = δω µ ,

ε µν =

∂v µ ∂x ν

(6.2c)

Likewise, we can write down the averaged electric displacement and magnetic induction by applying the homogenization procedure directly to Eqs. 6.1b and 6.1c. Thus we arrive at: ⟨ ⟩ ⟨︀ µν ⟩︀ ⟨D α ⟩ = δ δ α ε µν − δ2 δ α (1)µν u(0) 3,x µ x ν + 2 (0) * * +δ ⟨δ⟨αµ ⟩ φ⟩ x µ + δ ⟨ ξ αµ ⟩ ψ x µ + δ ⟨ τ α ⟩ T 1 +

+δ2 τ α (1) T2 (0)

(6.2d)

⟩

+δ ⟨a⟨αµ ⟩ 𝛾x*⟩µ + δ ⟨𝛾αµ ⟩ ω*x µ + δ2 ⟨𝛾α ⟩ T1 (0) +

+δ

2

𝛾α

(1)

λ2

(6.2e)

*(1)

Next, in view of definitions 6.2c, the expressions for the mechanical displacement, Eqs. 5.24a and 5.27a, can be written down as: {︁ }︁ uβ = v β − x3 u(0) +δN µν ε − δ2 N β (1)µν u(0) 3,x µ x ν + 3,βx β µν +δM βµ φ*x µ + δN βµ ψ*x µ + δ2 G β T1 (0) + +δ2 G(1) T2 (0) + δ2 ω β * β

(6.1a)

Recalling that σ ij (0) = 0 from Eq. 5.23, then we may use Eq. 4.6 to write down the in-plane force and moment resultants pertaining to the homogenized plate in a form which is reminiscent of the classical composite laminate theory, see for example Gibson [56]. Thus we have: ⟨ ⟩ ⟩︀ ⟨︀ N αβ = δ b αβ µν ε µν − δ2 b αβ (1)µν u(0) 3,x µ x ν + ⟨ ⟩ ⟨ ⟩ ⟨︀ ⟩︀ ψ*,x µ + δ2 b αβ T1 (0) + +δ b µ αβ φ*,x µ + δ a αβ µ ⟨ ⟩ +δ2 b αβ (1) T2 (0)

M αβ =

⟨

⟨B α ⟩ = δ ⟨η α µν ⟩ ε µν − δ2 a α (1)µν u(0) 3,x µ x ν +

(6.2f)

u3 =

{︁

}︁ u3 (0) +v3 +δN 3 µν ε µν − δ2 N3 (1)µν u(0) 3,x µ x ν +

+δM 3µ φ*x µ + δN µ 3 ψ*x µ + δ2 G3 T1 (0) + +δ2 G3 (1) T2 (0) + δ2 ω3 * (6.2g) Finally, the expressions for the electric and magnetic potentials, Eqs. 5.24b, 5.24c and 5.27b, 5.27c, may be conveniently written down as: φ=

^ (0) + u δφ* + δA µα ε µα − δ2 A(1) αβ 3,x α x β +δΞ α φ*x α + δO α ψ*x α + δ2 ΠT1(0) + δ2 Π (1) T2(0) + δ2 𝛾 * (6.2h)

ψ=

^ (0) + δZ α φ*x α + u δψ* + δΛ µα ε µα − δ2 Λ(1) αβ 3,x α x β +δΓ α ψ*x α + δ2 ∆T1(0) + δ2 ∆(1) T2(0) + δ2 ω*

(6.2i)

Careful ⟨ ⟩examination of Eqs. 6.2a – 6.2i readily reveals that δ b µν are the extensional effective elastic coefficients, ⟩ ⟨ ⟨ αβ ⟩ µν 2 are the coupling effective elas= δ zb δ2 b(1)µν αβ αβ ⟩ ⟨ 3 tic coefficients and δ zb(1)µν are the bending effective αβ elastic coefficients. In fact, the following correspondence is evident, see Gibson [68]. ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ A11 = δ b11 11 , A12 = δ b11 22 , A16 = δ b11 12 , ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ A22 = δ b22 22 , A26 = δ b22 12 , A66 = δ b12 12 , ⟨ ⟩ ⟨ ⟩ B11 = δ2 zb11 11 = δ2 b11 (1)11 , ⟨ ⟩ ⟨ ⟩ B12 = δ2 zb11 22 = δ2 b11 (1)22 , ⟨ ⟩ ⟨ ⟩ B16 = δ2 zb11 12 = δ2 b11 (1)12 , ⟨ ⟩ ⟨ ⟩ B22 = δ2 zb22 22 = δ2 b22 (1)22 , ⟨ ⟩ ⟨ ⟩ B26 = δ2 zb22 12 = δ2 b12 (1)22 , ⟨ ⟩ ⟨ ⟩ B66 = δ2 zb12 12 = δ2 b(1)12 , 12


⟨

⟩

⟨

⟩

, D12 = δ3 zb11 (1)22 , D11 = δ3 zb(1)11 ⟩ ⟨ 11 D16 = δ3 zb11 (1)12 , ⟩ ⟩ ⟨ ⟨ D22 = δ3 zb22 (1)22 , D26 = δ3 zb22 (1)12 , ⟩ ⟨ D66 = δ3 zb12 (1)12

(6.3)

⟨ ⟩ ⟨︀ ⟩︀ Furthermore, δ b αβ ,δ δ µν are the effective in-plane µ α ⟨ ⟩ ⟨︀ ⟩︀ αβ piezoelectric coefficients, δ a µ ,δ η µν the effective inα ⟨ ⟩ ⟨︀ ⟩︀ plane piezomagnetic coefficients, −δ b αβ and − b(1) αβ the effective in-plane thermal expansion coefficients related to the mid-plane temperature variation and the through-the-thickness linear temperature variation, re⟨ ⟩ ⟨ ⟩ spectively, see Eq. 3.4b, δ2 zb αβ = δ2 δ(1)µν µ α are⟨ the effective out-of-plane ⟩ ⟨ ⟩ piezoelectric coefficients, αβ (1)µν 2 2 δ za µ = δ aα the effective out-of-plane ⟩ ⟨ ⟩︀ ⟨︀ piezomagnetic coefficients, −δ2 zb αβ and −δ2 zb(1) αβ the effective out-of-plane thermal expansion coefficients, −δ ⟨δ αµ ⟩ and −δ ⟨𝛾αµ ⟩ are, respectively, the effective dielectric permittivity and magnetic permeability, −δ ⟨ξ αµ ⟩,−δ ⟨a αµ ⟩ are ⟨the effective magnetoelectric coef⟩

ficients, δ ⟨τ α ⟩ and δ τ(1) are the effective pyroelectric α ⟨ ⟩ coefficients, and δ ⟨𝛾α ⟩ and δ 𝛾α(1) the effective pyromagnetic coefficients. Further, v1 , v2 and u(0) 3 represent the displacements of the middle plane of the plate and consequently ε11 , ε22 and ε12 are the mid-surface strains, ∂2 u(0)

∂2 u(0)

see Gibson [68]. Similarly, ∂x23 = κ x1 and ∂x23 = κ x2 are 1 2 the bending curvatures associated with bending of the middle surface in the x1 x3 and x2 x3 planes, respectively, ∂2 u(0)

3 and 2 ∂x1 ∂x = k xy is the twisting curvature associated with 2 torsion of the middle surface. It should be pointed out that in the case of the purely elastic case, the results of this model converge exactly to those of Kalamkarov [44], Kalamkarov and Kolpakov [45] and Kalamkarov and Georgiades [65]. In the present work, however, the authors adhere to a completely coupled approach, which results in significantly refined expressions as compared to previously published results, such as those in [65, 66, 71, 72]. All these previously published papers employed a semi-coupled approach, resulting in expressions for the effective coefficients, which do not reflect the influence of all material parameters. For example, the effective elastic coefficients of smart laminates as well as wafer-reinforced plates (as obtained via the semi-coupled approach) given in [65] and [66] depend only on the elastic parameters of the constituents. However, an examination of, say, unit cell problem 5.8a and associated definition 5.11a will reveal that in the completely coupled

approach followed in the present work the effective inplane elastic coefficients are dependent on not only the elastic properties of the constituent materials, but also on the piezoelectric, piezomagnetic, magnetic permeability, dielectric permittivity and other parameters. The same is true for all remaining effective coefficients, as expressions 5.11a – 5.13c and 5.31a – 5.33c reveal. In this sense, the thermoelasticity, piezoelectricity and piezomagnetism problems are entirely coupled, and the solution of one affects the solutions of the others. This feature is captured in the present works, but not in [65, 66, 71, 72]. For the same reasons, if applied to the case of simple laminated structures, the work presented here represents an extension of the classical composite laminate theory (see e.g. [70, 73]) to magneto-piezo-thermo-elastic structures. More important, however, is the fact that the model developed in the present work explicitly allows for different periodicity in the lateral directions. As such, it is readily amenable to the design and analysis of magnetoelectric reinforced plates, such as the wafer-reinforced and rib-reinforced structures shown in the next section. To the authors’ best knowledge, this is the first time that completely coupled piezo-magneto-thermo-elastic effective coefficients for reinforced plates are presented and analyzed.

7 Examples of magnetoelectric composite and reinforced plates The mathematical model developed in Sections 1-6 can be used in analysis and design to tailor the effective elastic, piezoelectric, magnetoelectric and other coefficients of composite and reinforced plates (Figs. 2 and 3) to meet the criteria of specific engineering applications. The main objective of Part II of this work [67] is precisely that.

Figure 2: Laminated magnetoelectric composite plate


(a)

(b) Figure 3: (a) Wafer-reinforced and (b) rib-reinforced magnetoelectric composite plates

At this point, it is worthwhile to reiterate that the nature of the reinforced structures is such that it would be more efficient if we first considered a simpler type of unit cell consisting of only a single reinforcement/actuator. Having dealt with this situation, the effective coefficients of more general structures with multiple families of reinforcements/actuators can readily be determined by superposition. In following this procedure, one must naturally accept the error incurred at the interphase and/or regions of overlap between the actuators/reinforcements or between the matrix and the actuators/reinforcements. However, our approximation is quite accurate, since these regions are highly localized and do not contribute significantly to the integral over the entire unit cell domain. Essentially, the error incurred will be negligible if the dimensions of the actuators/reinforcement are much smaller than the spacing between them. We note for example the asymptotic homogenization model developed by Kalamkarov [44] for the purely elastic case of thin composite plates reinforced with mutually perpendicular ribs. In that work, he determined that if the spacing between the unit cells is at least ten times bigger than the thickness of the reinforcements the error in the values of the

effective elastic coefficients incurred by ignoring the regions of overlap (between the reinforcements) is less than 1%. A complete mathematical justification for this argument in the form of the so-called principle of the split homogenized operator has been provided by Bakhvalov and Panasenko [42]. Furthermore, we note that the general purely mathematical aspects of the asymptotic homogenisation procedure can be found in [41, 42]. The pertinent mathematical details for the asymptotic homogenization of thinwalled inhomogeneous structures can be found in [56][59]. The convergence of the two-scale asymptotic method is proven in these papers on the basis of G-convergence. These purely mathematical aspects of asymptotic homogenization are beyond the scope of our papers, which are aimed at developing micromechanical models and deriving results for effective properties of magnetoelectric and other smart structures of practical importance.

8 Summary and concluding remarks The method of asymptotic homogenization is used to analyze a periodic smart composite plate of rapidly varying thickness with elastic, piezoelectric and piezomagnetic constituents. A set of eighteen fully-coupled threedimensional local unit cell problems is derived. However, unlike classical homogenization schemes, the derived unit cell problems are shown to depend on boundary conditions rather than periodicity in the transverse direction. The solution of the unit cell problems yields a set of functions which, when averaged over the volume of the periodicity cell, can be used to determine the effective elastic, piezoelectric, dielectric permittivity and other coefficients of the homogenized anisotropic smart plate. Of interest among these coefficients are the so-called product coefficients, which are present in the behavior of the macroscopic composite as a result of the interactions between the various phases but can be absent from the constitutive behavior of the individual phases of the composite material. The effective coefficients are substituted into the governing equations of the structure, which in turn yield a set of local functions. These functions allow us to make very accurate predictions concerning the three-dimensional local structure of the mechanical stress and displacement fields, electric and magnetic potentials etc. The local problems are expressed in a form that shows that they are completely determined by the geometrical and material characteristics of the unit cell of the smart


plate and are totally independent of the global formulation of the original problem. It follows that derived effective coefficients are universal in nature and may be used to analyze different types of boundary value problems associated with a given smart structure. Finally, it is shown that in the limiting case of a thin elastic plate of uniform thickness the derived model converges to the familiar classical plate model. Appropriately, in part II of this work, Hadjiloizi et al. [67], illustrate the theory developed here using the practically important examples of magnetoelectric thin laminates and magnetoelectric wafer-reinforced composite plates. In both cases it is shown that the developed model can be used to tailor the properties of a given structure to conform to the requirements of a particular engineering application by changing appropriate geometrical or material parameters.

[8]

Acknowledgement: The authors would like to acknowledge the financial support of the Cyprus University of Technology (1st , 3rd and 4th authors), the Research Unit for Nanostructured Materials Systems (1st , 3rd and 4th authors) and the Natural Sciences and Engineering Research Council of Canada (2nd author).

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Research Article

Open Access

D. A. Hadjiloizi, A. L. Kalamkarov, Ch. Metti, and A. V. Georgiades*

Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates: Part II – Applications Abstract: A comprehensive micromechanical model for the analysis of a smart composite piezo-magneto-thermoelastic thin plate with rapidly varying thickness is developed in Part I of this work. The asymptotic homogenization model is developed using static equilibrium equations and the quasi-static approximation of Maxwell’s equations. The work culminates in the derivation of general expressions for effective elastic, piezoelectric, piezomagnetic, dielectric permittivity and other coefficients. Among these coefficients, the so-called product coefficients are determined which are present in the behavior of the macroscopic composite as a result of the interactions between the various phases but can be absent from the constitutive behavior of some individual phases of the composite structure. The model is comprehensive enough to also allow for calculation of the local fields of mechanical stresses, electric displacement and magnetic induction. The present paper determines the effective properties of constant thickness laminates comprised of monoclinic materials or orthotropic materials which are rotated with respect to their principal material coordinate system. A further example illustrates the determination of the effective properties of wafer-type magnetoelectric composite plates reinforced with smart ribs or stiffeners oriented along the tangential directions of the plate. For generality, it is assumed that the ribs and the base plate are made of different orthotropic materials. It is shown in this work that for the purely elastic case the results of the derived model converge exactly to previously established models. However, in the more general case where some or all of the phases exhibit piezoelectric and/or piezomagnetic behavior, the expressions for the derived effective coefficients are shown to be dependent on not only the elastic properties but also on the piezoelectric and piezomagnetic parameters of the constituent materials. Thus, the results presented here represent a significant refinement of previously obtained results. Keywords: smart composite piezo-magneto-thermoelastic thin plate; asymptotic homogenization; effective properties; product properties

DOI 10.2478/cls-2014-0003 Received August 7, 2014 ; accepted September 11, 2014

1 Introduction The incorporation of composites and, more recently, nanocomposites, into new engineering applications has been restricted to some extend by the lack of a reliable data-base of their long-term performance characteristics under different loading conditions. This problem has been offset to a significant degree by the recent development and optimization of elaborate sensor and actuator systems. On one hand, sensors can provide valuable data on the current state and serviceability of the structure and on the other hand the actuators can respond effectively to changes in external conditions/stimuli such as mechanical loading, electric and magnetic field intensity, temperature etc. To enhance a structure’s ability to fulfill the re-

*Corresponding Author: A. V. Georgiades: Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus and Research Unit for Nanostructured Materials Systems, Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus; E-mail: [email protected]; Tel.: 357-25002560 D. A. Hadjiloizi: Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus and Research Unit for Nanostructured Materials Systems, Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus A. L. Kalamkarov: Department of Mechanical Engineering, Dalhousie University, PO Box 15000, Halifax, Nova Scotia, B3H 4R2, Canada Ch. Metti: Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus and Research Unit for Nanostructured Materials Systems, Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus

© 2014 D. A. Hadjiloizi et al., licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.


quirements of a particular engineering application and guarantee long-term reliability and sustenance, it is necessary to integrate composites/nanocomposites with the aforementioned sensors/actuators; this gives rise to smart composites and smart nanocomposites, see for example Kalamkarov et al. [1]. Jain and Sirkis [2] aptly compared smart structures to biological systems. In their work they defined the goal of smart structures as being able to reproduce biological functions in load-bearing mechanical systems; thus smart structures should be endowed with a “skeletal system” to provide load-bearing capabilities, a “nervous system”, which is a system of integrated sensors or actuators to assess the structural health state, a “motor system” to provide adaptive response, an “immune system” for self-healing capabilities and a “neural system” for promoting learning and decision making [2]. Suitable candidates for use as sensors and actuators in smart composite materials systems include electro/magneto-rheological fluids, shape-memory alloys, piezomagnetics and magnetostrictives, piezoelectrics and others. Of particular interest among smart materials are those consisting of piezoelectric and piezomagnetic constituents. Such magnetoelectric composites have attracted attention both because of their significant potential for engineering applications as well as the unique properties they exhibit. In particular, magnetoelectric composites are primarily characterized by the so-called product properties which are present in the behavior of the overall macroscopic structure but are usually absent from the constitutive behavior of the individual constituents, see Newnham et al. [3], Nan et al. [4]. Examples of product properties are the magnetoelectric effect, pyromagnetism and pyroelectricity. According to Nan et al. [4] we can succinctly write the product properties as follows: Magnetoelectric Effect = Pyroelectric Effect =

Magnetic Mechanical

Thermal Mechanical

Pyromagnetic Effect =

×

Thermal Mechanical

×

Mechanical Electric

Mechanical Electric

×

Mechanical Magnetic

Thus, applying an electric field to a magnetoelectric composite generates mechanical displacement in the piezoelectric phase. Provided there is a satisfactory degree of bonding between the different constituents, this mechanical deformation is transferred to the piezomagnetic phase and in turn induces a magnetic field. Thus, overall, an applied electric field generates a magnetic field and, conversely, an applied magnetic field produces electric displacement. This is the magnetoelectric phenomenon. Similarly, changing the temperature produces mechanical strain through thermal expansion. In turn this strain generates magnetic induction in the piezomagnetic phase (py-

romagnetism) and electric displacement (pyroelectricity) in the piezoelectric phase. As a result of their unique properties magnetoelectric composites are continuously coming into the forefront of an increasing number of engineering applications. Typical examples include resonators, phase shifters, delay lines and filters, magnetic field sensors, energy harvesting transducers, miniature antennas, data storage devices and spintronics, biomedical sensors for EEG/MEG devices and other relevant equipment, see [5–14]. The incorporation of smart materials and, particularly, magnetoelectric composites into new engineering applications is significantly facilitated if their properties and coefficients can be predicted at the design stage; thus micromechanical models are needed. These models must be comprehensive enough to capture both the individual behavior of the different constituents as well as the influence of the macroscopic composite. At the same time, if the developed models are too complicated to be used in an effective and efficient manner then they are of little use beyond the obvious academic interest. Ideally, these models should lead to closed-form design-oriented equations that can be programmed into a simple spread-sheet. That way they can facilitate engineering analysis to obtain a preliminary design and, if any fine-tuning is needed, the designer can turn to a numerical model based on, for example, the finite element technique. To this end, the main objective of this work is the development of multiphysics micromechanical models for the analysis and design of thin piezo-magneto-thermo-elastic composite and reinforced plate structures. The nature of the developed models is such that they permit the designer to readily customize the effective properties of a given smart structure by varying one or more geometrical or material parameters such as the stacking sequence in a magnetoelectric laminate, thickness and composition of the perpendicular ribs of wafer-reinforced plates etc. Among the earlier works on magnetoelectric composites are those of Harshe et al. [15, 16] and Avelaneda and Harshe [17], who developed theoretical models to determine the magnetoelectric coefficients of 0-3 and 2-2 piezoelectric-magnetostrictive multilayer composites. However, deviations are observed between their results and corresponding experimental values. Osaretin and Rojas [18] purport that this discrepancy could be attributed to poor interface coupling between the layers, as far as the experimental results are concerned, and failure to properly incorporate the appropriate electromagnetic boundary conditions as far as the theoretical models are concerned. The authors then go on to develop a different modeling methodology by solving the constitutive equations in

34 | D. A. Hadjiloizi et al. each phase and then applying a field-averaging method, see e.g. Getman [19], with pertinent boundary conditions, to obtain the composite effective properties. Their results agree with their counterparts from other theoretical models such as the one developed by Nan [20], who employed a Green’s Function approach. However, a discrepancy with experimental results still remained which was corrected by applying an interface coupling parameter of 0.4 (logical from a practical point of view) to the theoretical model. Particularly noteworthy are the works of Huang and collaborators, [21, 23, 24]. In particular, Huang and Kuo [21] developed a comprehensive micromechanical model for piezoelectric-piezomagnetic composites containing ellipsoidal inclusions. As in most other theoretical models, the authors assumed perfect bonding between the inclusion and the matrix and ignored any other interfacial defects. Their modeling methodology allowed them to obtain the coupled magneto-electro-elastic analogue of the Eshelby tensors [22] which, as expected, are a function of the geometric characteristics of the ellipsoidal inclusions as well as the properties of the host matrix. In an extension of this work, Huang [23] obtained closed-form solutions for reinforcements in the shape of elliptic cylinders, circular cylinders, disks and ribbons embedded in a transversely isotropic matrix. Other related work on magnetoelectric composites consisting of long piezoelectric fibers embedded in a piezomagnetic matrix can be found in [24]. Hadjiloizi et al. [25, 26] employed the asymptotic homogenization technique to develop two general threedimensional models for magnetoelectric composites. One model used dynamic force and thermal balance and the time-varying form of Maxwell’s equations to determine closed-form expressions for the complete set of the effective properties of the structure including electrical conductivity and the product properties. The second model used the quasi-static approximation of the aforementioned constitutive equations. The models were applied to the case of thick laminates. It was shown that the results of the quasi-static model agreed with those of other models, see for example Bravo-Castillero et al [27]. Ni et al [28] examined the effect of the orientation of the applied electric and magnetic fields on the magnetoelectric coupling of polycrystalline multiferroic laminates. The authors based their theoretical model on a three-layer sandwich laminate consisting of a polycrystalline piezoelectric layer between two piezomagnetic laminae. Their model assumed the application of dc poling electric and magnetic fields because the constituent materials do not exhibit an appreciable degree of electromechanical and magnetomechanical behavior in their unpoled states. In the model an ac magnetic field, δHj, is applied and the induced polariza-

tion, δPi , is calculated. The effective magnetoelectric coefficients were then calculated as aij = δPi /δHj . The model showed that these coefficients depended on the sum of the in-plane piezomagnetic strains, ε11 +ε22 , which in turn depended strongly on the orientation of the applied magnetic fields relative to the laminate’s interface. Tsang et al. [29] developed an effective medium-based micromechanical model for 3-phase magnetoelectric composites consisting of spherical magnetostrictive and piezoelectric particles embedded in a conductive polymer matrix. Based on their model, the authors calculated the magnetoelectric charge and voltage coefficients, and the results agreed reasonably well with corresponding experimental data. In their modeling approach, Pan et al. [30] transformed a cylindrical layered composite, consisting of piezoelectric and piezomagnetic laminae, into an equivalent planar one. This modeling approach proved useful in the analysis and design of magnetoelectric devices and structures. Bichurin et al. [31] developed a micromechanical model pertaining for ferromagnetic/piezoelectric bilayered and multilayered composites with the electric and magnetic fields applied in varying orientations with respect to the laminates’ interface. Longitudinal and transverse magnetoelectric coefficients were calculated. Regarding the former coefficient, the authors illustrated that earlier results [15] were a limiting case of their theory. The same authors, Bichurin et al. [32], extended their work to magnetoelectric nanocomposites. Other relevant results can be found in the works of Akhabarzadeh et al. [33], Soh and Liu [34], Kirchner et al. [35], Pan and Heyliger [36], Benveniste [37], Spyropoulos et al. [38], Tang and Yu [39, 40] and others. The constitutive and governing equations describing the micromechanical behavior of periodic composites and smart composites (including the magnetoelectric composites that are of interest to us in this work) are characterized by rapidly varying material coefficients. In other words, the material coefficients are periodic with a small period, of the order of a few micrometers or nanometers. This spatial scale is typically portrayed in literature as the microscopic or “fast” scale. Superimposed on this scale however, one encounters a macroscopic or “slow” scale, which is a function of the global formulation of the problem (external loads, boundary conditions etc.) and is “oblivious” to the substructural phenomena that take place at the level of the reinforcement or general inclusion. Consequently, any attempt to solve a problem related to a smart composite must successfully decouple the two scales and treat the two problems (macroscopic and microscopic) independently. One such technique that has enjoyed significant success for many years is that of asymptotic homogenization. The pertinent mathematical details


of the technique can be found in Bensoussan et al. [41], Sanchez-Palencia [42], Bakvalov and Panasenko [43] and Cioranescu and Donato [44]. Many problems in elasticity, thermoelasticity, and piezo-magneto-elasticity have been solved via asymptotic homogenization. We want to mention particularly the works of Kalamkarov [45], who analyzed a wide variety of problems, such as composite and reinforced plates and shells, network-reinforced shells, plates with corrugated surfaces and other structures, Kalamkarov and Kolpakov [46] who used these models to design and optimize various composite structures on account of strength and stiffness requirements, the pioneering work of Guedes and Kikuchi [47] on computational aspects of homogenization, the modification of asymptotic homogenization for problems related to elasticity and thermal conductivity of thin plates appearing in the works of Duvaut [48, 49], Andrianov et al. [50, 51], Caillerie [52, 53], Kohn and Vogelius [54–56] and many others. Recent years have witnessed the emergence of smart composite plates and shells as the preeminent structural members for many practical applications. Enhanced strength, reduced weight, materials savings and ease of fabrication are among the reasons that make these structures attractive. More recently, advancements in the field of nanotechnology and the increasing popularity of nanocomposite thin films, plates and shells [57] have further enhanced the application potential of such structures. The periodic or nearly periodic nature of smart composite and nanocomposite plates and shells renders asymptotic homogenization a valuable tool in their analysis, design and optimization. The “classical” asymptotic homogenization approach however cannot be applied directly to a thin plate or shell if the scale of the spatial inhomogeneity is comparable to the thickness of the structure. In that case, a refined approach developed by Caillerie [52, 53] in his heat conduction studies is needed. In particular, a two-scale formalism is applied, whereby a set of microscopic variables is used for the tangential directions in which periodicity exists and another microscopic variable is used for the transverse direction in which periodicity considerations do not apply. Kohn and Vogelius [54–56] adopted this approach in their study of the pure bending of a thin, linearly elastic homogeneous plate. Kalamkarov [45] and Kalamkarov and Kolpakov [46] applied this modified twoscale methodology to determine the effective elastic, thermal expansion and thermal conductivity coefficients of thin curvilinear composite layers. Challagulla et al. [58], Georgiades et al. [59] employed this methodology to develop comprehensive asymptotic homogenization models for network-reinforced thin smart composite shells. These

authors illustrated their results by means of practically important examples including single-walled carbon nanotubes, which can be treated as network-reinforced composite shells in which the covalent bonds between the carbon atoms play the role of isotropic reinforcements embedded in a matrix of zero rigidity. Kalamkarov and Georgiades [60] and, Georgiades and Kalamkarov [61] developed comprehensive micromechanical models for smart composite wafer- and rib-reinforced plates. Saha et al. [62] determined the effective elastic constants of orthotropic honeycomb-like sandwich composite shells. Hadjiloizi et al. [63] implemented a general model (based on the timevarying form of Maxwell’s equations and dynamic force balance) for the micromechanical dynamic analysis of magnetoelectric thin plates with rapidly varying thickness. In that work only an in-plane temperature variation was considered and therefore any out-of-plane thermal effects were ignored. Thus, unlike in the present work, the out-of-plane thermal expansion, pyroelectric and pyromagnetic coefficients were not captured in [63]. More important, however, is the fact that the micromechanical model in [63] is only applied to the case of simple laminated plates. In contrast, the model developed in the present work explicitly allows for different periodicity in the lateral directions. As such, it is readily amenable to the design and analysis of magnetoelectric reinforced plates such as the wafer-reinforced structures shown in Section 4. To the authors’ best knowledge, this is the first time completely coupled piezo-magneto-thermo-elastic effective coefficients for reinforced plates are presented and analyzed. Also relevant to the present papers are the works of Kalamkarov and Georgiades [60], Georgiades and Kalamkarov [61] and Hadjiloizi et al. [25], [26]. In [60] and [61], Kalamkarov and Georgiades developed and illustrated the use of an asymptotic homogenization model for the analysis of reinforced piezoelectric plates. In their work, [60], [61] the authors adopted only a semi-coupled analysis, which results in expressions for the effective coefficients that do not reflect the influence of such parameters as the electric permittivity, magnetic permeability, primary magnetoelectricity etc. In the present work and its companion paper [64], however, a fully coupled analysis is performed, and as a consequence the expressions for the effective coefficients involve all pertinent material parameters. As an example, the effective extensional elastic coefficients are dependent on not only the elastic properties of the constituent materials, but also on the piezoelectric, piezomagnetic, magnetic permeability, dielectric permittivity and magnetoelectric coefficients. The same holds true for the remaining effective coefficients. In

36 | D. A. Hadjiloizi et al. a sense, the thermoelasticity, piezoelectricity and piezomagnetism problems are entirely coupled, and the solution of one affects the solutions of the others. This feature is captured in the present papers, but not in previously published works, such as [60] and [61]. Thus, the results presented here represent an important refinement of previously established results. To the authors’ best knowledge completely coupled piezo-magneto-thermo-elastic effective coefficients for reinforced plates have not been presented and analyzed before. In [25], [26], Hadjiloizi et al. developed general quasistatic and dynamic three-dimensional models for magnetoelectric composites. However, these models employed the “classical” homogenization approach, see [43] for example, and consequently could not capture the mechanical, thermal, piezoelectric and piezomagnetic behavior that is related to bending, twisting and general out-ofplane deformation and electric and magnetic field generation. The model developed in the current work and its companion paper [64], however, accomplishes precisely this; it employs the modified asymptotic homogenization technique (discussed earlier) which makes use of two sets of microscopic variables that permit the decoupling of inplane and out-of-plane behavior of the structure under consideration. For example, the elastic coefficients can be distinguished into the familiar extensional, bending and coupling coefficients, which is not possible to achieve with the 3D models in [25] and [26]. What this amounts to is the fact that the two modeling approaches are essentially applicable to entirely different structures and geometries. The 3D models in [25], [26] can be used to analyze structures of comparable dimensions in the x, y, z directions (such as thick laminates) but cannot be used for thin structures such as wafer- and rib-reinforced plates. The micromechanical models developed in the present work, however, are applicable to structures with a much smaller dimension in the transverse direction than in the other two directions. Thus, they can be used in the design and analysis of an impressive range of composite and reinforced plates such as the aforementioned waferand rib-reinforced structures (see Section 4), three-layered honeycomb-cored magnetoelectric plates, thin laminates (Section 3) etc. To summarize, the present paper deals with the development and applications of appropriate plane stress micromechanical models for thin magnetoelectric composite and reinforced plates. The work is implemented in two parts. In part I [64] the pertinent micromechanical models are derived, and the unit cell problems, from which the effective coefficients (including the product properties) can be extracted, are obtained. The applications of the de-

veloped models to the practically important cases of thin composite laminates and wafer-reinforced magnetoelectric plates are presented herein. Following this introduction the basic mathematical model and the pertinent unit cell problems are reviewed in Section 2. Sections 3 and 4 present, respectively, the solution of the unit cell problems for magnetoelectric laminates of constant thickness and for wafer-reinforced magnetoelectric plates. Section 4 also compares the results of the developed model with previously reported results, and, finally, Section 5 concludes the work. In view of the applications mentioned earlier in this section, the most important aspect of this publication is the development of closed-form design-oriented equations that can be used in the analysis and design of magnetoelectric composite and reinforced plates. It is shown that thermoelasticity, piezoelectricity and piezomagnetism are entirely coupled and the solution of one affects the solutions of the others.

2 Problem Formulation The boundary value problem characterizing the thin smart composite plate of rapidly varying thickness, Fig. 1 in Hadjiloizi et al. [64], is given by: σ ij,jx = P i , D i,ix = 0, B i,ix = 0 E i = −φ ,ix , H i = −ψ ix

(2.1a)

σ ij n j = p i , D i n i = 0, B i n i = 0, on S± u i = 0, φ = δ2 e, ψ = δ2 h, on lateral surfaces (2.1b) σ ij = C ijkl u k,xl + e kij φ ,xk + Q kij ψ ,xk −δΘ ij T {︀ }︀ D i = δ e ijk u k,xl − ε ij φ ,xk − λ ij ψ ,xk + δξ i T {︀ }︀ B i = δ Q ijk u k,xl − λ ij φ ,xk − µ ij ψ ,xk + δη i T

(2.2a) In these equations we use the following short-hand notation for the derivatives: ∂φ α = φ α,βy , ∂y β

∂φ α = φ α,βx , ∂x β

∂φ α = φ α,z ∂z

(2.2b)

Here σ ij is the mechanical stress, Di and Bi are, respectively, the electric displacement and magnetic induction, Ei and Hi are the electric and magnetic fields, Pi represents a generic body force, pi represents the surface tractions, and ui is the mechanical displacement. Finally, T is the change in temperature with respect to a suitable reference. Eq. 2.1a represents the static equilibrium equations


and the quasi-static approximation of Maxwell’s Equations. The irrotational electric and magnetic fields can be expressed as gradients of scalar potential functions, φ and ψ, respectively. Furthermore, e kl = ∂u k /∂x l is the second order strain field, Cijkl , eijk , Qijk , and Θ ij are the tensors of the elastic, piezoelectric, piezomagnetic and thermal expansion coefficients respectively. Finally, ε ij , λ ij , µ ij , ξ i and η i represent, respectively, the dielectric permittivity, the magnetoelectric coefficients, the magnetic permeability, and the pyroelectric and pyromagnetic tensors. We note that because the composite layer is periodic only in the tangential directions, see Fig. 1 of Hadjiloizi et al [64], the material parameters are dependent on xα /δ hα and x3 , while the dependent field variables are also dependent on xα = (x1 , x2 ). In all equations we adopt the convention that Greek letters, α, β, 𝛾 etc. assume values of only 1,2, while Latin letters, a, b, c etc. take on values 1,2,3. We finally note that the overall thickness of the structure must be small compared to the other two dimensions. For the analysis of a thick piezo-magneto-thermo-elastic laminate, one should consider an appropriate 3D model, e.g. [25, 27]. The in-plane force and moment resultants pertaining to the homogenized plate are given in Hadjiloizi et al. [64] and are: ⟩ ⟨ ⟨ ⟩ (1)µν 2 u(0) + ε − δ b N αβ = δ b µν µν ⟨ αβ ⟩ ⟨ αβ ⟩ 3,x µ x ν ⟨︀ ⟩︀ +δ b αβ φ*,x + δ a αβ ψ*,x µ δ2 b αβ T1(0) + µ µ ⟩ µ ⟨ T2(0) +δ2 b(1) αβ (2.3a) M αβ =

⟨

⟩

⟩

⟨

u(0) + ε µν − δ3 zb(1)µν δ2 zb µν ⟨ αβ ⟩ 3,x µ x ν ⟨︀ ⟨ αβ ⟩ ⟩︀ ψ*x µ + δ3 zb αβ T1(0) + +δ2 zb αβ φ*x + δ2 za αβ µ µ ⟩ µ ⟨ +δ3 zb(1) T2(0) αβ

(2.3b) Likewise, the averaged electric displacement and magnetic induction, see Hadjiloizi et al. [64], are given by: ⟨ ⟩ ⟨︀ µν ⟩︀ ⟨D α ⟩ = δ δ α ε µν − δ2 δ(1)µν u(0) α 3,x µ x ν + (0) 2 * * +δ ⟨δ⟨αµ ⟩ φ ⟩x µ + δ ⟨ξ αµ ⟩ ψ x µ + δ ⟨τ α ⟩ T1 +

+δ2 τ(1) T2(0) α

(2.3c) ⟨B α ⟩ =

⟨ ⟩ ⟨︀ ⟩︀ δ η µν ε µν − δ2 a(1)µν u(0) α α 3,x *

*

µ

+ ∂x ν

+δ ⟨⟨a αµ ⟩⟩𝛾x µ + δ ⟨𝛾αµ ⟩ ω x µ + δ ⟨𝛾α ⟩ T1(0) +

(2.3d)

+δ 𝛾α(1) T1(0)

Finally, the expressions for the mechanical displacement and the electric and magnetic potentials can be written

down as: uβ =

{︁

}︁ v β − x3 u(0) +δN µν ε − δ2 N β(1)µν u(0) 3,x µ x ν + 3,βx β µν

+δM βµ φ*µx + δN βµ ψ*µx + δ2 G β T1(0) + δ2 G(1) T2(0) + β +δ2 ω*β (2.3e) u3 =

{︁

}︁ µν 2 (1)µν (0) u(0) u3,x µ x ν + 3 +v 3 +δN 3 ε µν − δ N 3

(0) +δM 3µ φ*x µ + δN 3µ ψ*x µ + δ2 G3 T1(0) + δ2 G(1) 3 T2 + 2 * +δ ω3 (2.3f)

φ=

^ (0) + δΞ α φ*x α + δφ* + δA µα ε µα − δ2 A(1) u αβ 3,x α x β

(2.3g)

^ (0) + δZ α φ*x α + u δψ* + δΛ µα ε µα − δ2 Λ(1) αβ 3,x α x β

(2.3h)

+δO α ψ*x α + δ2 ΠT1(0) + δ2 Π (1) T2(0) + δ2 𝛾 * ψ=

+δΓ α ψ*x α + δ2 ∆T1(0) + δ2 ∆(1) T2(0) + δ2 ω*

⟩ ⟨ ⟨ ⟩ , δ ⟨𝛾αµ ⟩, etc. are the effective co, δ2 b(1)µν Here,δ b µν αβ αβ efficients to be determined from the following set of eighteen unit cell problems, see Hadjiloizi et al. [64]: µα µα h−1 β b iβ,βy (y, z ) + b i3,z (y, z ) = 0 ± ± with b µα ij (y, z ) N j = 0 on Z

(2.4a)

iβ i3 h−1 β b α,βy (y, z ) + b α,z (y, z ) = 0 ij ± with b α (y, z) N j = 0 on Z ±

(2.4b)

iβ i3 h−1 β a α,βy (y, z ) + a α,z (y, z ) = 0 ij ± with a α (y, z) N j = 0 on Z ±

(2.4c)

h−1 β b iβ,βy (y, z ) + b i3,z (y, z ) = 0 with b ij (y, z) N j± = 0 on Z ± (1) (1) h−1 β b iβ,βy (y, z ) + b i3,z (y, z ) = 0

with

± b(1) ij (y, z ) N j = 0

on Z ±

(2.4d)

(2.4e)

(1)µα (1)µα h−1 β b iβ,βy (y, z ) + b i3,z (y, z ) = 0 ± ± with b µα ij (y, z ) N j = 0 on Z

(2.4f)

iα iα h−1 β η β,βy (y, z ) + η 3,z (y, z ) = 0 iα with η j (y, z) N j± = 0 on Z ±

(2.5a)

h−1 β a βα,βy (y, z ) + a 3α,z (y, z ) = 0 with a jα (y, z) N j± = 0 on Z ±

(2.5b)

h−1 β 𝛾βα,βy (y, z ) + 𝛾3α,z (y, z ) = 0 with 𝛾jα (y, z) N j± = 0 on Z ±

(2.5c)

h−1 β 𝛾β,βy (y, z ) + 𝛾3,z (y, z ) = 0 with 𝛾j (y, z) N j± = 0 on Z ±

(2.5d)


(1) (1) h−1 β 𝛾β,βy (y, z ) + 𝛾3,z (y, z ) = 0

with 𝛾j(1) (y, z) N j± = 0

on Z ±

(2.5e)

(1)µα (1)µα h−1 β a β,βy (y, z ) + a 3,z (y, z ) = 0 µα ± with a i (y, z) N j = 0 on Z ±

(2.5f)

iα iα h−1 β δ β,βy (y, z ) + δ 3,z (y, z ) = 0 iα with δ j (y, z) N j± = 0 on Z ±

(2.6a)

h−1 β δ βα,βy (y, z ) + δ 3α,z (y, z ) = 0 with δ jα (y, z) N j± = 0 on Z ±

(2.6b)

h−1 β ξ βα,βy (y, z ) + ξ 3α,z (y, z ) = 0 with ξ jα (y, z) N j± = 0 on Z ±

(2.6c)

h−1 β τ β,βy (y, z ) + τ 3,z (y, z ) = 0 with τ j (y, z) N j± = 0 on Z ±

(2.6d)

(1) (1) h−1 β τ β,βy (y, z ) + τ 3,z (y, z ) = 0

with

± τ(1) j (y, z ) N j = 0

on Z ±

(1)µα (1)µα h−1 β δ β,βy (y, z ) + δ 3,z (y, z ) = 0 µα ± with δ i (y, z) N j = 0 on Z ±

(2.6e) (2.6f)

where, kα b kα ij = L ijm N m + M ij A kα + N ij Λ kα + C ijkα , ij m b α = L ijm M α + M ij Ξ α + N ij Z α + e αij a ijα = L ijm N αm + M ij O α + N ij Γ α + Q αij , b ij = L ijm G m + M ij Π + N ij ∆ − Θ ij (1) (1) b(1) + N ij ∆(1) − zΘ ij , ij = L ijm G m + M ij Π (1)kα (1)kα + zC ijkα + N ij Λ(1) = L ijm N m + M ij A(1) b ij kα kα

(2.7a)

kα η kα j = L ji N i −M j A kα −N j ( y ) Λ kα +Q jkα , i a jα = L ji M α −M j Ξ α −N j Z α −λ jα 𝛾jα = L ji N αi −M j Oα −N j Γ α −µ jα , 𝛾j = L ji G i −M j Π − N j ∆ − η j (1) (1) 𝛾j(1) = L ji G(1) i −M j Π −N j ∆ +zη j , (1)kα (1)kα (1) ai = L im N m − M i A kα − N i Λ(1) + zQ ikα kα

(2.7b)

* kα * * δ kα j = L ji N i −M j A kα −N j Λ kα +e jkα , i * * * δ jα = L ji M α −M j Ξ α −N j Z α −ε jα ξ jα = L*ji N αi −M *j O α −N *j Γ α −λ jα , τ j = L*ji G i −M *j Π − N *j ∆ + ξ j * (1) * (1) * (1) τ(1) j = L ji G i −M j Π −N j ∆ +zξ j , (1)kα (1)kα * * (1) δi = L im N m − M i A kα − N i* Λ(1) +ze ikα kα

(I-6.6a). Note that in the previous sentence and from this point onwards, for the sake of convenience, all equations that are referenced from Hadjiloizi et al. [64] will be denoted by the uppercase letter I preceding the corresponding equation number. For example, Eq. (I-5.27a) will denote Eq. 5.27a in Hadjiloizi et al. [64].

3 Applications of the General Model – Constant Thickness Laminates We will illustrate our work by means of several examples. The first examples are for laminates of constant thickness, as shown in Fig. 1. As shown in the unit cell of Fig. 1, each layer is completely determined by the parameters δ1 , δ2 ,. . . δ M where M is the total number of layers. The thickness of the mth layer is therefore δ m -δ m−1 with δ0 = 0 and δ M = 1. The real thickness of the mth layer as measured in the original (x1 ,x2 ,x3 ) coordinate system is δ(δ m -δ m−1 ), where δ is the thickness of the laminate (again with respect to the original coordinate system).

Figure 1: Unit cell of a laminated magnetoelectric composite plate.

(2.7c)

kα and N m , A kα , Λ kα , etc. are local functions, which define the asymptotic expansions of the mechanical displacement and electric and magnetic potentials, respectively, see Eqs. 2.3e – 2.3h. Furthermore, L ijm , M ij , N ij etc. are differential operators defined in Eqs. (I-6.2a), (I-6.4a) and

It is apparent that all material parameters are independent of y1 and y2, and consequently, all partial derivatives in Eqs. 2.4a- 2.7c become ordinary derivatives with respect to z. We will consider laminates made up of perfectly bonded laminae of piezoelectric and piezomagnetic materials with the poling and magnetization directions along the z-axis. The “perfect bonding” assumption is akin to neglecting the interphase layers between adjacent plies. (Because the interphase regions might be important in the case of nano-laminates, application of the derived models to such structures might need to take the interphase lay-


ers into consideration, see for example Sevostianov and Kachanov for particulate-reinforced nanocomposites [65]). Furthermore, the overall thickness of the laminate is considered to be small compared to the in-plane dimensions. For the sake of generality, we will also assume that the constituent materials are made of orthotropic materials, with the principal material coordinate axes not necessarily coinciding with the y1 , y2 , z system but with a system that has been rotated by an arbitrary angle with respect to the z axis. As such, the pertinent coefficient matrices (tensors) are the same as those of a monoclinic material, as far as the number and location of the non-zero coefficients is concerned, see Reddy [68]. Thus: ⎡ ⎤ C11 C12 C13 0 0 C16 ⎢ C12 C22 C23 0 0 C26 ⎥ ⎢ ⎥ ⎢ C 0 0 C36 ⎥ ⎢ 13 C23 C33 ⎥ ⎢ ⎥ ⎢ 0 0 0 C44 C45 0 ⎥ ⎢ ⎥ ⎣ 0 0 0 C45 C55 0 ⎦ C16 C26 C36 0 0 C66 ⎤T ⎡ 0 0 e31 ⎢ 0 ⎡ ⎤ ⎡ ⎤ 0 e32 ⎥ ⎥ ⎢ Θ11 Θ12 0 ξ1 ⎥ ⎢ 0 0 e33 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎦ ⎣ ξ2 ⎦ ⎥ ⎢ ⎣ Θ12 Θ22 ⎢ e14 e24 0 ⎥ ⎥ ⎢ 0 0 Θ33 ξ3 ⎣ e15 e25 0 ⎦ 0 0 0 (3.1) In Eq. 3.1 the first matrix is the elasticity tensor, the second matrix represents the tensor of piezoelectric or piezomagnetic coefficients, the third matrix pertains to the thermal expansion (or dielectric permittivity/magnetic permeability/magnetoelectric) tensor, and the last vector is the pyroelectric or pyromagnetic tensor. We will now proceed with the solution of the unit cell problems and the determination of general expressions for the effective coefficients. (a) Unit Cell Problems 2.4a, 2.5a and 2.6a To obtain the effective properties, the following procedure will be adhered to: The unit cell problems 2.4a2.6f will be reduced to ordinary differential equations in z due to the aforementioned independency on y1 and y2 . The pertinent boundary conditions will also be simplified, because the normal vector N becomes (0,0,1). Subsequently, the reduced unit cell problems will be solved in a straight-forward manner, giving the coefficient funcµν kα tions b kα ij , η j , δ j etc. as defined in Eqs. 2.7a – 2.7c. Because each of these functions is in turn a function of three local functions (for example Eq. 2.7a shows that b kα ij is a kα function of N m , A kα , and Λ kα ), we will need a total of three unit cell problems to solve for the three unknowns. Hence, we will look for the three unit cell problems that use

these same local functions. For example, local functions kα Nm , A kα , and Λ kα appear in unit cell problems 2.4a, 2.5a µν kα and 2.6a via coefficient functions b kα ij , η j and δ j . Finally, after obtaining these local functions, we will backsubstitute them into the appropriate expressions for the coefficient functions which in turn yield the effective coefficients after applying the homogenization procedure of Eq. (I-4.5a). Based on the above procedure, Eq. 2.4a reduces to: db kα i3 ( z ) =0 dz

with

b kα i3 ( z ) = 0

on Z ±

(3.2a)

Solving 3.2a leads to b kα i3 = 0 everywhere in the unit cell. From Eq. (I-5.2a) and the first expression in Eq. 2.7a we have: b µα i3 = C i3n3

dA µα dΛ µα dN µα n +e3i3 +Q3i3 +C i3µα = 0 (3.2b) dz dz dz

Letting i = 1, 2 in Eq. 3.2b and bearing in mind the orthotropy of the constituents, see Eq. 3.1, we readily see that: ∂N1µα ∂N2µα = =0 (3.2c) ∂z ∂z For i = 3, Eq. 3.2b yields: C33

dN 3µα dΛ µα dA µα +e33 +Q33 = −C33µα dz dz dz

(3.2d) dN

µα

dA

Eq. 3.2d contains three unknown functions, dz3 , dzµα , dΛ µα dz . Hence, we will need two more equations which will come from unit-cell problems 2.5a and 2.6a. Following the same procedure as above, and keeping Eq. 3.2c in mind, these unit-cell problems yield the following equations: Q33

dN 3µα dA µα dΛ µα −λ33 −µ33 = −Q3µα dz dz dz

dN 3µα dAµα dΛ µα e33 −ε33 −λ33 = −e3µα dz dz dz

(3.2e)

Solving Eqs. 3.2d and 3.2e as a system yields the following solution: (︀ )︀ dN 3µα e3µα λ33 − Q3µα ε33 Q33 = + dz ^1 Π λ2 C33µα +λ33 e33 Q3µα − µ33 e33 e3µα − µ33 ε33 C33µα + 33 ^1 Π (3.2f) (︀ )︀ 2 dA µα Q3µα e33 +C33µα λ33 Q33 +Q33 e3µα = + dz ^1 Π −λ33 C33 Q3µα − µ33 e33 C33µα +µ33 C33 e3µα + ^1 Π

(3.2g)


(︀ )︀ dΛ µα −e3µα e33 − C33µα ε33 Q33 +e233 Q3µα = + dz ^1 Π ε33 C33 Q3µα +λ33 e33 C33µα − λ33 C33 e3µα + ^1 Π

(3.2h)

^ 1 = µ33 C33 ε33 − C33 λ233 +Q233 ε33 + where Π −2λ33 Q33 e33 +µ33 e233

(3.2i)

model: it can be tailored to meet the specific requirements of an engineering application by changing one or more geometric, physical or material parameters. (b) Unit Cell Problems 2.4b, 2.5b and 2.6b Eq. 2.4b reduces to:

Using these solutions, the in-plane elastic, piezoelectric and piezomagnetic functions (from which the effective coefficients will be computed in the sequel) may be calculated as follows: dN 3µν dΛ µν dA µν +e3αβ +Q3αβ +C αβµν dz dz dz [In − plane elastic functions] (3.3a) b µν = C αβ33 αβ

η µν = 0 [In − plane piezoelectric functions] β

(3.3b)

= 0 [In − plane piezomagnetic functions] δ µν β

(3.3c)

Two features are worth mentioning here. First of all, it can be seen that the elastic functions (and as a consequence the effective extensional elastic coefficients) depend not only on the elastic parameters of the constituent phases, but also on the piezoelectric, piezomagnetic, and magnetoelectric coefficients, as well as the dielectric permittivities and magnetic permeabilities. This is in marked contrast with previous simpler models, see Kalamkarov and Georgiades [60], which predicted that the extensional elastic coefficients depend only on the elastic parameters of the constituents. Therefore, the present work constitutes an important refinement over previously established results. Also, for the case of simple laminates such as the ones considered in this section, the work presented here represents an extension of the classical composite laminate theory (see e.g. [66], [67]) to piezo-magneto-thermoelastic structures. Of course, if piezoelectric and piezomagnetic effects are completely ignored, then Eq. 3.3a reduces to: C33µν b µν = −C αβ33 +C αβµν (3.3d) αβ C33 which conforms exactly to the results of Kalamkarov and Georgiades [60], Georgiades and Kalamkarov [61] as well as the classical composite laminate theory. The second feature that is evident in Eqs. 3.3b and 3.3c is that the in-plane piezoelectric and piezomagnetic functions are zero. Clearly, this is because of the fact that the polarization/magnetization directions are the same as the stacking orientation. Had we chosen polarization and magnetization directions along y1 or y2 the results would be drastically different. Herein lies a significant advantage of our

db i3 α (z) =0 dz with b i3 α (z) = 0

(3.4a)

on Z ±

Solving 3.4a leads to b i3 α = 0 everywhere in the unit cell. From the second expression in Eq. 2.7a and Eq. (I-5.2a) we have: b i3 α = C i3n3

dΞ α dZ α dM nα +e3i3 +Q3i3 +e αi3 = 0 dz dz dz

(3.5a)

Letting n =1, 2 in Eq. 3.5a while keeping Eq. 3.1 in mind, dM 1 dM 2 leads to two simultaneous equations in dzα and dzα . Their solution is readily found to be: dM 2α C45 e α13 −C55 e α23 = dz C44 C55 −C245 (3.5b) Letting n = 3 in Eq. 3.5a leads to a homogeneous equation in three unknown functions: dM 1α C45 e α23 −C44 e α13 = , dz C44 C55 −C245

C33

dΞ α dM 3α dZ α +e33 +Q33 =0 dz dz dz

(3.5c)

As we need two more equations, we resort to unit cell problems 2.5b and 2.6b. Following the same procedure as above we end up with: dM 3α dZ α dΞ α −µ33 −λ33 =0 dz3 dz dz dZ α dΞ α dM α −λ33 −ε33 =0 e33 dz dz dz Q33

(3.5d)

The solution of Eqs. 3.5c and 3.5d gives: dM 3α dZ α dΞ α = = =0 dz dz dz

(3.5e)

Using Eqs. 3.5b and 3.5e we obtain the in-plane piezoelectric, magnetoelectric and dielectric permittivity functions from which their effective coefficient counterparts may easily be determined (as will be shown shortly): 45 e α23 45 e α13 −a µα = Q µ13 C44Ce α13C −C−C +Q µ23 C55Ce α23C −C−C + λ µα 2 2 44 55

45

[Magnetoelectric functions]

44 55

45

(3.6a)

45 e α23 45 e α13 −δ µα = e µ13 C44Ce α13C −C−C +e µ23 C55Ce α23C −C−C + ε µα 2 2 44 55

45

[Dielectric permittivity]

44 55

45

(3.6b)

Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates |

b µν α = 0 [In − plane piezoelectric functions]

(3.6c)

The reason why the in-plane piezoelectric functions vanish was explained above. Also, we note the appearance of the magnetoelectric functions (first product properties). (c) Unit Cell Problems 2.4c, 2.5c and 2.6c Similarly to the previous three unit cell problems, Eqs. 2.4c, 2.5c and 2.6c are easily solved, yielding: dN 1α

C Q −C Q = 45 α23 442 α13 , dz C44 C55 −C45 dN 3α dO α dΓ α = = =0 dz dz dz

dN 2α dz

=

C45 Q α13 −C55 Q α23 , C44 C55 −C245

(3.7a) Using these results, the in-plane piezomagnetic, magnetic permeability and magnetoelectric functions are derived as follows: C44 Q α13 −C45 Q α23 C Q −C Q +Q µ23 55 α23 452 α13 +µ µα C44 C55 −C245 C44 C55 −C45 [Magnetic permeability] (3.7b)

−𝛾 µα= Q µ13

41

With these results we can calculate the thermal expansion, pyroelectric and pyromagnetic functions related to the mid-plane temperature variation. We recall from Eq. (I-3.4b), that (as is customary in heat conduction studies of thin plates and shells) we assume a temperature variation that is the superposition of a mid-plane term and a linear through-the-thickness term. The set of coefficients stemming from Eqs. 3.8a- 3.8d is related to the mid-plane term. Thus: dΠ d∆ dG3 − e3αβ − Q3αβ + Θ αβ dz dz dz [In − plane thermal expansion functions] (3.9a) −b αβ = −C αβ33

𝛾α = η α

τα = ξα

[In − plane pyromagnetic functions]

(3.9b)

[In − plane pyroelectric functions]

(3.9c)

(e) Unit Cell Problems 2.4e, 2.5e and 2.6e Following the above procedure it can be seen that: dG(1) dG 3 = z 3, dz dz

dΠ (1) dΠ =z , dz dz

d∆(1) d∆ =z (3.10a) dz dz

C Q −C Q C44 Q α13 −C45 Q α23 +e µ23 55 α23 452 α13 +λ µα Hence, the solution of the secondary pyroelectric and pyC44 C55 −C245 C44 C55 −C45 romagnetic as well as in-plane thermal expansion func[Magnetoelectric functions] tions (all related to the through-the-thickness temperature (3.7c) variation) is given simply by:

−ξ µα= e µ13

a µν α = 0 [Piezomagnetic functions]

(3.7d)

(d) Unit Cell Problems 2.4d, 2.5d and 2.6d Proceeding as above, we arrive at the following results: dG3 = dz

d∆ = dz

dΠ = dz

λ33 Q33 ξ3 − Q33 ε33 η3 −λ233 Θ33 + ^1 Π λ33 e33 η3 − µ33 e33 ξ3 +‘µ33 Θ33 ε33 + ^1 Π −ξ 3 e33 Q33 + Θ33 ε33 Q33 +e233 η3 + ^1 Π ε C η − λ33 e33 Θ33 − λ33 C33 ξ3 + 33 33 3 ^1 Π

(3.8a)

= zb αβ , b(1) αβ

𝛾α(1) = z𝛾 α ,

τ(1) α = zτ α

(3.10b)

(f) Unit Cell Problems 2.4f, 2.5f and 2.6f Similarly, solving these three unit-cell problems yields: [Elastic coupling functions] b(1)µν = zb µν αβ αβ

(3.11a)

a(1)µν = zb µν α α =0 [Out − of − plane piezoelectric functions]

(3.11b)

δ(1)µν = zδ µν α α =0 [Out − of − plane piezomagnetic functions]

(3.11c)

(3.8b)

−Q33 e33 η3 − Θ33 λ33 Q33 +Q233 ξ3 + ^1 Π −λ33 C33 η3 +µ33 Θ33 e33 +µ33 C33 ξ3 + ^1 Π

(3.8c)

∂G1 ∂G2 = =0 ∂z ∂z

(3.8d)

The last set of functions that we need to consider are the elastic bending functions shown in Eq. (I-6.2b). It is evident that: zb(1)µν = z2 b µν [Elastic bending functions] αβ αβ

(3.11d)

(g) Effective Coefficients and Numerical Examples

42 | D. A. Hadjiloizi et al. The effective coefficients are obtained in a straightforward fashion by directly applying the homogenization procedure in Eq. (I-4.5a). For example, referring to Fig. 1, one can see that the extensional effective elastic coefficients are given by: ⟨

⟩ b λµ = αβ

0.5 ∫︀

b λµ dz = αβ

−0.5 M ∑︀

=

∫︀δ m

m=1 δ m−1

∫︀1 0

b λµ dδ m = αβ

b λµ dδ m =b λµ(m) (δ m − δ m−1 ) αβ αβ

(3.12a) Likewise, the coupling and bending effective elastic coefficients are given by: ⟨ ⟩ ⟨ ⟩ (1)λµ zb λµ = b = αβ αβ M )︀ (︀ 2 ∑︀ = 21 b λµ(m) δ m − δ2m−1 − (δ m − δ m−1 ) , αβ ⟨ ⟩ ⟨ m=1 ⟩ zb(1)λµ = z2 b λµ = αβ αβ M ∑︀ λµ(m) (︀ 3 δ m − δ3m−1 + b αβ = 31 m=1 (︀ )︀ )︀ − 32 δ2m − δ2m−1 + 43 (δ m − δ m−1 ) . (3.12b) In the same manner, the remaining effective elastic coefficients may be determined. We will illustrate our work by considering a simple 4-ply laminate consisting of alternating barium titanate (top layer) and cobalt ferrite laminae. The overall thickness of the laminate is 1 mm. The pertinent material parameters are given in Table 1. For the sake of discussion we will further assume that the Barium Titanate is doped with Fe so that it exhibits primary magnetoelectricity. For illustration purposes only, we will presume that the value of the magnetoelectric coefficient of bulk Fedoped BaTiO3 is similar to that pertaining to a nanostructured counterpart, and is ∼16 mV/Oe cm, see Verma et al. [72]. Thus, in the parlance of our present work (we define the magnetoelectric coefficients slightly differently, see Section 2) and for a dielectric permittivity value of around 11.2 × 10−9 C2 /N m2 (Table 1) we assume that λ11 = λ22 = λ33 is ≈2.2 × 10−10 C/A m. We will also assume that the Fe doping does not affect the remaining properties of BaTiO3 as shown in Table 1. Likewise, we will assume that the cobalt ferrite is doped with the rare earth element Dy, see Dascalu et al. [73], so that it too exhibits primary magnetoelectricity. The pertinent magnetoelectric value ≈ −2.5 µV/Oe cm. Thus, for a dielectric permittivity value of around 0.08 × 10−9 C2 /N m2 (Table 1) we can assume that λ11 = λ22 = λ33 is ≈0.25 x 10−15 C/A m. ⟨ ⟩ Fig. 2 shows the variation of the zb(1)22 bending co22 efficient vs. the thickness of the BaTiO3 laminae. It can clearly be seen that the effective elastic coefficients change

Table 1: Material properties of BaTiO3 , and CoFe2 O4 (Li and Dunn [69], Yoshihiro and Tanigawa, [70], Cook et al., [71]).

C11 = C22 (GPa) C12 (GPa) C13 = C23 (GPa) C33 (GPa) C44 = C55 (GPa) e31 = e32 (C/m2 ) e33 (C/m2 ) e24 = e15 (C/m2 ) ε11 = ε22 (10−9 C2 /N m2 ) ε33 (10−9 C2 /N m2 ) Q31 = Q32 (N/A m) Q33 (N/A m) Q24 = Q15 (N/A m) µ11 = µ22 (10−6 N s2 /C2 ) µ33 (10−6 N s2 /C2 ) α11 = α22 (10−6 1/K) α33 (10−6 1/K) ξ1 = ξ2 = ξ3 (10−4 C/m2 K)

BaTiO3 166 77 78 162 43 −4.4 18.6 11.6 11.2 12.6 0 0 0 5 10 15.7 6.4 1.5

CoFe2 O4 286 173 170 269.5 45.3 0 0 0 0.08 0.093 580.3 699.7 550 −590 157 10 10 0

⟨ ⟩ Figure 2: Plot of effective zb(1)22 bending coeflcient vs. thick22 ness of the BaTiO3 laminae.

significantly as the relative volume fractions of the constituents change. Since cobalt ferrite is generally stiffer than its barium titanate counterpart, increasing the overall thickness of the latter causes a corresponding reduction in the value of the effective bending coefficients. ⟨︀ ⟩︀ Fig. 3 shows the variation of the effective zb11 11 coupling coefficient vs. the thickness of the BaTiO3 laminae. As expected, the absence of barium titanate renders the laminate symmetric and nullifies the effective coupling coefficients. As the thickness of this constituent increases,


43

materials are altered, if the nature of the doping materials is changed, if the stacking configuration of the laminae is rearranged etc. In essence, our derived model is comprehensive enough, in that it affords complete flexibility to the designer to customize the effective properties of the smart composite structure to conform to the requirements of a particular engineering application. This is also evident in the next example considered in this paper.

⟨︀ ⟩︀ Figure 3: Plot of effective zb11 11 coupling coeflcient vs. thickness of the BaTiO3 laminae.

Figure 5: Plot of effective ⟨ξ11 ⟩ magnetoelectric coeflcient vs. thickness of the BaTiO3 laminae.

Figure 4: Plot of effective ⟨𝛾11 ⟩ magnetic permeability coeflcient vs. thickness of the BaTiO3 laminae.

the laminate becomes more asymmetric, and the effective coupling coefficients increase. Finally, as the thickness of barium titanate approaches 0.5 (which is tantamount to having no cobalt ferrite - the entire structure is made of BaTiO3 ) the effective coupling coefficients approach zero, as the laminate approaches geometric symmetry with respect to the mid-plane. Figs. 4 and 5 show the variation of the effective magnetic permeability,⟨𝛾11 ⟩, and the magnetoelectric product coefficient ⟨ξ11 ⟩. As expected, reducing the volume fraction of Dy-doped CoFe2 O4 results in a corresponding reduction of the effective magnetic permeability coefficients and an increase in the effective magnetoelectric coefficients. What is important to emphasize here though, is that these trends may be easily changed, if the polarization and /or magnetization directions for the constituent

4 Applications of the General Model – Wafer-reinforced Smart Composite Plates The following examples will be concerned with a different type of structure, namely a wafer-reinforced magnetoelectric plate, shown in Fig. 6. For generality we will assume that the material of the base-plate is different than that of the ribs. For example, the base-plate may be elastic or piezomagnetic and the ribs may be piezoelectric. Each constituent material may be assumed to be orthotropic. We are interested in calculating the effective elastic, piezoelectric, thermal expansion, dielectric permittivity, magnetoelectric, pyroelectric etc coefficients for this structure. A solution of the local problems relevant to this kind of geometry may be found assuming that the thickness of each of the three elements of the unit cell is small in comparison with the other two dimensions, i.e. t1 ≪ h2 ,

t2 ≪ h1 ,

H ∼ h1 , h2 .

(4.1)

44 | D. A. Hadjiloizi et al. The local problems can then be approximately solved for each of the unit cell elements assuming that the discontinuities at the joints are highly localized. Consequently, the local problems can be solved independently for regions Ω1 , Ω2 and Ω3 as shown in Fig. 7. Fig. 7 also depicts the transformed unit cell, showing the microscopic coordinates y1 , y2 , and z. The analytical procedure followed in this example is similar to its counterpart in the previous example. First of all, the unit cell problems are simplified in each of the three regions of the unit cell. In particular, periodicity conditions in y1 and y2 reduce the pertinent partial differential equations in Region 3 to ordinary differential equations in z. Likewise, since Region 1 is thin and entirely oriented in the y2 direction it is characterized by independence in y2 . Hence, the corresponding unit cell problem is reduced from a partial differential equation in variables y1 , y2 , and z into one involving y1 and z only. Similarly, the appropriate differential equation for Region 2 is reduced to one involving variables y2 and z only. The solution of the unit cell problem in each region involves coefficient functions, e.g. b kα ij which in turn are functions of three unkα known local functions, e.g. N m , A kα , and Λ kα . Since we need three equations to solve for the three unknown local functions, we need to simultaneously consider all three unit cell problems which involve the given local functions. For example, unit cell problems 2.4a, 2.5a and 2.6a must be solved together. Once the local functions are determined, they are back substituted into the expressions for the coefficient functions, Eqs. 2.7a- 2.7c, from which the effective coefficients can be readily obtained after application of the homogenization procedure, Eq. (I-4.5a). The results from each region are then superimposed. As mentioned above, in following this procedure, one must naturally accept the error incurred at the regions of intersection between the actuators/reinforcements. However, our approximation will be quite accurate, since these regions of intersection are highly localized and do not contribute significantly to the integral over the entire unit cell domain. Essentially, the error incurred will be negligible, if the dimensions of the actuators/reinforcement are much smaller than the spacing between them. As an indication, we note that Kalamkarov [45] developed an asymptotic homogenization model for thin composite plates reinforced with mutually perpendicular wafers and concluded that if the spacing between the unit cells is at least ten times bigger than the thickness of the reinforcements, the error in the values of the effective elastic coefficients incurred by ignoring the regions of overlap between the reinforcements is less than 1%. A complete mathematical justification for this argument in the form of the so-called princi-

ple of the split homogenized operator has been provided by Bakhvalov and Panasenko [43]. (a) Unit Cell Problems 2.4a, 2.5a and 2.6a and Effective Extensional, Piezoelectric and Piezomagnetic Coefficients We will first tackle unit cell problem 2.4a. If we assume that the structure is piece-wise homogeneous, then the elastic coefficients in each region of Fig. 7 are uniform. As such, in each of Ω1 , Ω2 , Ω3 the unit cell problem becomes: µα

∂τ µα 1 ∂τ iβ + i3 = 0 h β ∂y β ∂z ± ± with τ µα ij N j = 0 on Z kα where τ kα ij = L ijm N m + M ij A kα + N ij Λ kα

(4.2a)

Now, recalling Eqs. (I-4.2c) and (I-4.5d), we can write down the boundary condition in Eq. 4.2a in the form of: }︀ ± 1 {︀ µα }︀ 1 {︀ µα τ +C n y1 + τ i2 +C i2µα n±y2 h1{︀ i1 i1µα h 2 }︀ ± ± + τ µα i3 +C i3µα n y3 = 0 on Z

(4.2b)

Dropping the “y” subscript and the “±” superscript for simplicity, Eq. 4.2b becomes: n

β t µα i +C iβµα h β +C i3µα n 3 = 0

where

t µα i =

on Z ±

nβ τ µα +τ µα i3 n 3 iβ h β

(4.2c)

The elastic b kα ij functions (from which the effective elastic coefficients may be readily determined) are then given by: µα b µα ij = τ ij +C ijµα

(4.2d)

In an entirely analogous manner, unit cell problem 2.5a becomes: µα ∂τ µα 1 ∂τ β + 3 =0 h β ∂y β ∂z nβ with t µα +Q βµα +Q3µα n3 = 0 on Z ± hβ kα where τ kα = L N m − M j A kα − N j Λ kα jm j n β and t µα = τ µα +τ3µα n3 β h β

(4.3a)

This problem will be solved in each of the regions Ω1 , Ω2 , and Ω3 separately and will yield the in-plane η kα j piezomagnetic functions (which will later give the effective piezomagnetic coefficients) according to: µα η µα j = τ j +Q jµα

(4.3b)


45

Figure 7: Unit cell of smart wafer and individual elements.

Figure 6: Thin magnetoelectric wafer-reinforced plate and its periodicity cell.

Finally, the third unit cell problem of the group, Eq. 2.6a becomes: µα ∂π3µα 1 ∂π β + =0 h β ∂y β ∂z nβ with r µα +e βµα +e3µα n3 = 0 on Z ± hβ kα kα * where π j = L jm N m − M *j A kα − N j* Λ kα n β µα µα and r µα = π β +π n3 hβ 3

(4.4a)

Again, this problem will be solved in each of the regions Ω1 , Ω2 , and Ω3 separately and will yield the in-plane δ kα j piezoelectric functions (which will later give the effective piezoelectric coefficients) according to: µα δ µα j = π j +e jµα

(4.4b)

As was the case of the laminate of the previous example, we expect coupled solutions of the current unit cell problems. Hence, Eqs. 4.2a- 4.2c will be solved in each region simultaneously with the corresponding problems in Eqs. 4.3a and 4.4a. We begin by setting up the boundary conditions in each region. (i) Region Ω3 . This is defined by −1/2 < y1 < 1/2, −1/2 < y2 < 1/2, −1/2 < z < 1/2, and boundary conditions must be supplied on z = ±1/2 where n1 = n2 = 0, n3 = 1. Thus, from Eqs. 4.2a, 4.2c, 4.3a and 4.4a, and keeping in mind that we are dealing with orthotropic materials, see Eq. 3.1, the

boundary conditions become: 12 t λµ α = t3 = 0 ⇒ λµ λµ = τ12 τ13 = τ23 33 = 0 11 t3 = −C13 , t22 3 = −C 23 τ11 = −C and τ22 13 33 33 = −C 23

⎫ ⎪ ⎪ ⎪ ⎬

on z = ±1/2 (4.5a)

⎪ ⎪ ⎪ ⎭

t11 = −Q31 , τ22 3 = −Q 32 ,

t22 = −Q32 , t12 = 0 τ11 3 = −Q 31 , 22 τ3 = 0 on z = ±1/2 (4.5b)

r11 = −e31 , π322 = −e32 ,

r22 = −e32 , r12 = 0 π311 = −e31 , π322 = 0 on z = ±1/2 (4.5c)

(ii) Region Ω1 . This region is defined by −δ1 /2 < y1 < δ1 /2, −1/2 < y1 < δ1 /2 < y2 < 1/2, and 1/2 < z < 1/2 + H. Therefore, boundary conditions must be supplied on z = 1/2, z = 1/2 +H, where n1 = n2 = 0, n3 = 1 and on y1 = ± δ1 /2 where n2 = n3 = 0, n1 = 1. Thus, using 4.2a, 4.2c, 4.3a and 4.4a, the boundary conditions become: ⎫ −C11 −C12 ⎪ 22 ⎪ t11 = , t = , 1 1 ⎪ ⎪ h1 h1 ⎪ ⎪ ⎪ −C66 12 12 11 ⎪ ⎪ t2 = , t1 = t2 = ⎪ ⎬ h1 λµ 22 on y1 = ±δ1 /2 = t2 = t3 = 0 ⎪ ⎪ 11 22 ⎪ τ11 = −C11 , τ11 = −C12 , ⎪ ⎪ ⎪ 11 ⎪ ⎪ τ12 τ12 12 = −C 66 , 11 = τ 12 = ⎪ ⎪ ⎭ λµ 22 = τ12 = τ13 = 0 t11 t22 3 = −C 13, 3 = −C 23 , λµ 12 t α = t3 = 0 τ11 τ22 33 = −C 13 , 33 = −C 23 , λµ 12 τ α3 = τ33 = 0

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

on z = 1/2, 1/2 + H (4.5d)


τ1λµ 12

Similarly, from the unit cell problem 4.3a and boundary conditions 4.5b we arrive at

t λµ = 0 ⇒ = 0 on y1 = ±δ1 /2 t λλ = −Q3λ , t = 0 τ11 3 = −Q 31 , 22 12 τ3 = −Q32 , τ3 = 0 on z = 1/2, 1/2 + H

(4.5e)

r λµ = 0 ⇒ π1λµ = 0 on y1 = ±δ1 /2 r λλ = −e3λ , r12 = 0 π311 = −e31 , 22 12 π3 = −e32 , π3 = 0 on z = 1/2, 1/2 + H

(4.5f)

t λµ = 0 ⇒ τ2λµ = 0 on y2 = ±δ2 /2 t λλ = −Q3λ , t12 = 0 τ11 , 3 = −Q 31 , 22 12 τ3 = −Q32 , τ3 = 0 on z = 1/2, 1/2 + H

(4.5h)

r λµ = 0 ⇒ π2λµ = 0 on y2 = ±δ2 /2 r λλ = −e3λ , r12 = 0 π311 = −e31 , , π322 = −e32 , π312 = 0 on z = 1/2, 1/2 + H

(4.5i)

We are now ready to solve the unit cell problems in Eqs. 4.2a, 4.3a and 4.4a. We will begin with the 22 22 τ22 ij , τ i , and π i problems. Region Ω3 . Because of periodicity in y1 and y2 , and considering differential equations 4.2a and boundary conditions 4.5a we have: τ22 33 = −C 23

in Ω3

(4.6a)

The latter expression in Eq. 4.6a gives, on account of the corresponding definition in Eq. 4.2a and the differential operators (I-5.2a), the following equation: C33

∂N322 ∂A ∂Λ +e33 22 +Q33 22 = −C23 ∂z ∂z ∂z

in Ω3

in Ω3

(4.6c)

which, on account of the pertinent definition in Eq. 4.3a and the differential operators (I-5.2a), gives:

(iii) Region Ω2 . This region is defined by −δ1 /2 < y2 < δ1 /2, −1/2 < y1 < 1/2 and1/2 < z < 1/2+ H. Therefore, boundary conditions must be supplied on z = ½, z = 1/2 +H, where n1 = n2 = 0, n3 = 1 and on y2 = ± δ2 /2 where n1 = n3 = 0, n2 = 1. Thus, using 4.2a, 4.2c, 4.3a and 4.4a, the boundary conditions become: ⎫ −C22 ⎪ −C12 ⎪ , t22 , ⎪ t11 2 = 2 = ⎪ h2 h2 ⎪ ⎪ ⎪ −C 66 12 ⎪ ⎪ t1 = , ⎪ ⎬ h2 on y1 = λµ 12 11 22 t2 = t1 = t1 = t3 = 0 ⎪ = ±δ1 /2 ⎪ ⎪ τ11 τ22 ⎪ 22 = −C 12 , 22 = −C 22 , ⎪ ⎪ ⎪ ⎪ τ12 12 = −C 66 , ⎪ ⎪ ⎭ λµ 22 11 12 τ22 = τ12 = τ12 = τ23 = 0 ⎫ 22 11 t3 = −C13, t3 = −C23 , ⎪ ⎪ ⎪ ⎬ 12 t λµ on z = 1/2, α = t3 = 0 ⎪ τ11 τ22 1/2 + H 33 = −C 13 , 33 = −C 23 , ⎪ ⎪ ⎭ 12 = 0 = τ τ λµ 33 α3 (4.5g)

22 τ22 13 = τ 23 = 0,

τ22 3 = −Q 32

(4.6b)

Q33

∂N322 ∂A ∂Λ −λ33 22 −µ33 22 = −Q32 ∂z ∂z ∂z

in Ω3

(4.6d)

In an analogous manner, unit cell problem 4.4a and boundary conditions 4.5c give, π322 = −e32

in Ω3

(4.6e)

and e33

∂N322 ∂A ∂Λ −ε33 22 −λ33 22 = −e32 ∂z ∂z ∂z

in Ω3

(4.6f)

The solution of the linear system defined by Eqs. 4.6b, 4.6d and 4.6f is: 2 ∂N322 (Q32 ε33 −λ33 e32 ) Q33 −λ33 C23 −λ33 e33 Q32 + = − 2 2 ∂z µ33 C33 ε33 −λ33 C23 +Q33 ε33 −2λ33 e33 Q33 +e233 µ33 µ33 e33 e32 +µ33 C33 ε33 + µ33 C33 ε33 −λ233 C23 +Q233 ε33 −2λ33 e33 Q33 +e233 µ33 ^2 Π = ^ Π1 2 ∂A22 (Q32 e33 +λ33 C32 ) Q33 +Q33 e32 −λ33 C33 Q32 = + ∂z ^1 Π ^ −µ e33 C32 +µ33 C33 e33 Π + 33 = 3 ^1 ^1 Π Π ∂Λ22 (C32 ε33 +e33 e32 ) Q33 +e233 Q32 −λ33 C33 e32 + = ∂z ^1 Π ^ Π Q ε C +λ C e + 32 33 33 33 23 33 = 4 ^1 ^1 Π Π (4.6g) We observe that these expressions are the same as those in Eqs. 3.2f – 3.2h after letting µ = α = 2. Using these soλµ λµ lutions, functions τ λµ ij , τ i , π i are readily determined as follows: ⎫ ^2 ^3 ^4 ⎪ Π Π Π ⎪ τ22 = C +e +Q , 13 31 ⎪ 11 ^1 ^ 1 31 Π ^1 ⎪ ⎪ Π Π ⎪ ⎪ ^2 ^3 ^4 ⎪ ⎪ Π Π Π 22 ⎪ τ22 = C23 +e32 +Q32 , ⎪ ^1 ^1 ^1 ⎬ Π Π Π in Ω3 (4.7) τ22 ⎪ 33 = −C 23 , ⎪ ⎪ 22 22 22 ⎪ ⎪ τ α3 = τ12 = 0, τ3 = −Q32 , ⎪ ⎪ ⎪ 22 22 ⎪ τ22 = τ = 0, π = −e , ⎪ 32 1 2 3 ⎪ ⎭ 22 22 π1 = π2 = 0

Region Ω1 . In this region we have independence of the y2 coordinate, since the element is oriented entirely in the y2 direction. Thus, from differential equations 4.2a and boundary conditions 4.5d we get: 22 22 τ22 13 = τ 23 = τ 12 = 0, 22 τ33 = −C23 in Ω1

τ22 11 = −C 12 ,

(4.8a)


The latter two expressions in Eq. 4.8a give, on account of the corresponding definitions in Eq. 4.2a and the differential operators (I-5.2a), the following equations: ⎫ ∂N 22 ∂N 22 1 ⎪ ⎪ C11 1 +C13 3 + ⎪ ⎪ h1 ∂y1 ∂z ⎪ ⎪ ⎪ ∂A22 ∂Λ22 ⎪ +e31 +Q31 = −C12 ⎬ ∂z ∂z 22 in Ω1 (4.8b) ∂N ∂N 22 1 ⎪ ⎪ ⎪ C13 1 +C33 3 ⎪ ⎪ h1 ∂y1 ∂z ⎪ ⎪ ⎪ ∂A22 ∂Λ22 +e33 +Q33 = −C23 ⎭ ∂z ∂z

⎤ (C23 e33 λ33 −C23 ε33 Q33 +C33 e32 λ33)︀+ ⎥ +Q32 ε33 C33 + e233 Q32 −e33 Q33 e32 Q31 + ⎥ ⎥ +Q33 C13 λ33 e33 +C23 Q33 λ33 e31 + ⎥ ⎥ ⎥ −C23 µ33 e33 e31 − 2e33 λ33 Q33 C12 + ⎥ C33 µ33 e32 e31 − λ233 C33 C12 +C33 C12 µ33 ε33 + ⎥ ⎥ ⎥ +e233 µ33 C12 +e33 C13 λ33 Q32 − ⎥ ⎥ e33 C13 µ33 e32 + Q233 e31 e32 + Q233 C12 ε33 + ⎥ ⎥ ⎦ −Q33 C13 Q32 ε33 + λ233 C23 C13 − C33 e31 λ33 Q32 −e33 Q33 e31 Q32 (4.9c)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ^6 = ⎢ Π ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Similarly, from the unit cell problem 4.3a and boundary conditions 4.5e we arrive at τ22 3 =

−Q32 ,

τ22 1 =

0

in Ω1

(4.8c)

which, on account of the pertinent definition in Eq. 4.3a and the differential operators (I-5.2a), gives: ∂N 22 ∂N 22 ∂A 1 Q31 1 +Q33 3 − λ33 22 + h1 ∂y1 ∂z ∂z ∂Λ −µ33 22 = −Q32 in Ω1 ∂z

(4.8d)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ^7 = ⎢ Π ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Realizing that we need one more equation, we turn our attention to unit cell problem 4.4a and boundary conditions 4.5f to get, ⎡ π322 =

−e32 ,

π122 =

0 in Ω1

(4.8e)

and ∂N 22 ∂N 22 ∂A ∂Λ 1 e31 1 +e33 3 − ε33 22 −λ33 22 = h1 ∂y1 ∂z ∂z ∂z = −e32 in Ω1 (4.8f) Eqs. 4.8b, 4.8d and 4.8f represent a system of four linear equations in four unknowns. The solution may be given as: ^ ^ ∂N322 Π ∂N122 Π = h1 6 , = 7, ∂y1 ∂z ^ ^5 Π5 Π ^8 ^9 ∂A22 Π ∂Λ22 Π = , = where ∂z ∂z ^5 ^5 Π Π ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ^5 = ⎢ Π ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(︀ 2 −e33 − ε33 C33 ) Q231 + (2C33 e31 λ33 + +2ε33 Q33 C13 − 2C13 e33 λ33 + +2e31 Q33 e33 ) Q31 + −e231 C33 µ33 − Q233 C11 ε33 + λ233 C11 C33 + +C213 µ33 ε33 − Q233 e231 − C213 λ233 + −2C13 Q33 e31 λ33 − C11 µ33 e233 + +2µ33 C13 e31 e33 + +2C11 λ33 Q33 e33 − C33 C11 ε33 µ33

(4.9a)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4.9b)

47

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ^8 = ⎢ Π ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ^ Π9 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤ (e33 λ33 C12 −2C23 e31 λ33 −e31 e33 Q32 + ⎥ +C13 e32 λ33 −C12 ε33 Q33 −e31 Q33 e32 − ⎥ ⎥ 2 Q32 ε33 C13 ) Q31 + (C23 ε33 + e33 e32 ) Q31 + ⎥ ⎥ ⎥ +C11 e33 µ33 e32 +C11 Q33 Q32 ε33 − ⎥ ⎥ C11 Q33 λ33 e32 +C11 C23 µ33 ε33 + ⎥ 2 2 −C11 e33 λ33 Q32 − λ33 C11 C23 + e31 C23 µ33 ⎥ ⎥ ⎥ ⎥ +C13 e31 λ33 Q32 − e33 µ33 e31 C12 + ⎥ ⎦ −C12 µ33 C13 ε33 −C13 µ33 e31 e32 + +λ233 C13 C12 + Q33 e31 λ33 C12 +Q32 Q33 e231 (4.9d) ⎤ (−Q33 e33 C12 − C13 e33 Q32 − C23 e31 Q33 + −C33 λ33 C12 + 2e32 Q33 C13 + e31 Q32 C33 + ⎥ ⎥ λ33 C23 C13 ) Q31 + (C23 e33 − C33 e32 ) Q231 + ⎥ ⎥ ⎥ ⎥ +C33 e31 µ33 C12 + Q233 e31 C12 + ⎥ C33 C11 λ33 Q32 − C213 λ33 Q32 +C213 µ33 e32 + ⎥ ⎥ ⎥ +C11 C23 µ33 e33 − C11 C33 µ33 e32 + ⎥ ⎥ 2 −C11 Q33 e32 + C11 Q33 e33 Q32 + ⎥ ⎥ ⎥ −C13 Q33 e31 Q32 + λ33 Q33 C13 C12 + ⎥ ⎦ −C13 µ33 e33 C12 − C11 λ33 Q33 C23 + −µ33 C13 e31 C23 (4.9e) (−e33 e31 C23 − C13 e33 e32 + +C33 C12 ε33 + C12 e233 +C33 e31 e32 − C23 C13 ε33 ) Q31 + C33 C11 λ33 e32 − C33 C11 Q32 ε33 + −e231 Q32 C33 − C13 Q33 C12 ε33 + −C33 λ33 C12 e31 + 2C13 Q32 e31 e33 + +C11 Q33 e32 e33 − C11 λ33 e33 C23 + +C11 Q33 C23 ε33 − C11 e233 Q32 + −Q33 e33 e31 C12 + λ33 e31 C13 C23 + −C213 λ33 e32 −C13 Q33 e31 e32 + +Q33 e231 C23 + Q32 C213 ε33 + λ33 C13 e33 C12

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.9f)

48 | D. A. Hadjiloizi et al. λµ λµ Using these solutions, functions τ λµ ij , τ i , π i are readily determined as follows: ⎫ ^6 Π ⎪ τ22 + τ22 11 = −C 12 , 22 = C 12 Π ⎪ ^5 ⎪ ⎪ ^9 ^7 ^8 ⎪ Π Π Π ⎪ +C23 Π^ +e32 Π^ +Q32 Π^ , ⎬ 5 5 5 22 22 22 (4.10) τ33 = −C23 , τ α3 = τ12 = 0 ⎪ in Ω1 ⎪ ⎪ 22 22 22 ⎪ τ3 = −Q32 , τ1 = τ2 = 0, ⎪ ⎪ ⎭ π322 = −e32 , π122 = π 22 2 =0

Region Ω2 . In this region we have independence of the y1 coordinate since the element is oriented entirely in the y1 direction. Thus, the solution of differential equations 4.2a and boundary conditions 4.5g gives: 22 22 τ22 13 = τ 23 = τ 12 = 0, in Ω2

τ22 11 = −C 22 ,

τ22 33 = −C 23

(4.11a)

The latter two expressions in Eq. 4.11a give, on account of the corresponding definitions in Eq. 4.2a and the differential operators (I-5.2a), the following equations: ⎫ ∂N 22 ∂N 22 1 ⎪ ⎪ C22 2 +C23 3 + ⎪ ⎪ h2 ∂y2 ∂z ⎪ ⎪ ⎪ ∂A22 ∂Λ22 ⎪ +e32 +Q32 = −C22 ⎬ ∂z ∂z 22 in Ω2 (4.11b) ∂N ∂N 22 1 ⎪ ⎪ ⎪ C23 1 +C33 3 + ⎪ ⎪ h1 ∂y2 ∂z ⎪ ⎪ ⎪ ∂A22 ∂Λ22 +e33 +Q33 = −C23 ⎭ ∂z ∂z We need two more equations in order to be able to solve for the unknown functions. Therefore, we resort to unit cell problem 4.3a and boundary conditions 4.5h to arrive at: τ22 3 = −Q 32 ,

τ22 2 =0

in Ω2

(4.11c)

On account of the pertinent definition in Eq. 4.3a and the differential operators (I-5.2a), the first expression in Eq. 4.11c gives: ∂N 22 ∂N 22 ∂A ∂Λ 1 Q32 2 +Q33 3 − λ33 22 − µ33 22 = h2 ∂y2 ∂z ∂z ∂z = −Q32 in Ω2 (4.11d) Finally, from unit cell problem 4.4a and boundary conditions 4.5i we get: π322 = −e32 ,

π222 = 0

in Ω2

(4.11e)

From the appropriate definition in Eq. 4.4a and the differential operators (I-5.2a) we arrive at: ∂N 22 ∂N 22 1 ∂A ∂Λ e32 2 +e33 3 − ε33 22 −λ33 22 = h2 ∂y2 ∂z ∂z ∂z = −e32 in Ω2 (4.11f)

The solution of system 4.11b, 4.11d, 4.11f is trivial requiring only few algebraic manipulations, and is: ∂N322 ∂A22 ∂Λ22 = = =0 ∂z ∂z ∂z

∂N222 = −h2 , ∂y2

(4.12)

λµ λµ Using these solutions, functions τ λµ ij , τ i , π i are readily determined as follows: ⎫ ⎪ τ22 11 = −C 12 , ⎪ ⎪ ⎬ 22 22 22 τ22 = −C , τ = −C , τ = τ = 0 22 23 22 33 α3 12 in Ω2 22 22 22 ⎪ τ3 = −Q32 , τ1 = τ2 = 0, ⎪ ⎪ ⎭ π322 = −e32 , π122 = π 22 2 =0 (4.13) 11 11 The solution of theτ11 ij , τ i , and π i problems proceeds in much the same way as outlined above. The expressions of the appropriate local functions are given as: ⎫ ^* ^* ^* Π Π Π 3 4 2 ⎪ +e +Q , τ11 = C ⎪ 31 13 31 11 ^* ^* ^* ⎪ Π Π Π ⎪ 1 1 3 ⎪ * * * ^ ^ ⎪ ^ Π Π Π 11 ⎪ 3 4 2 ⎬ τ22 = C23 Π^ * +e32 Π^ * +Q32 Π^ * , 1 1 1 in Ω3 11 τ33 = −C13 ⎪ ⎪ ⎪ 11 ⎪ ⎪ τ11 τ11 ⎪ α3 = τ 12 = 0, 3 = −Q 31 , ⎪ ⎭ 11 11 11 11 11 τ1 = τ2 = 0, π3 = −e31 , π1 = π 2 = 0 (4.14a)

⎫ ^* ^* Π Π 7 6 ⎪ +C + τ11 ⎪ 13 11 = C 12 Π * * ^ ^ ⎪ Π ⎪ 5 5 ⎪ * * ^ ^ ⎪ Π Π 11 ⎪ ⎬ +e31 ^ 8* +Q31 ^ 9* , τ33 = −C13 , τ11 22 = −C 12 , Π5

Π5

11 τ11 α3 = τ 12 = 0 11 τ11 τ11 3 = −Q 31 , 1 = τ 2 = 0, 11 11 π3 = −e31 , π1 = π 11 2 =0

τ11 τ11 11 = −C 11 , 22 = −C 12 , 11 11 τ33 = −C13 , τ11 α3 = τ 12 = 0 11 11 τ3 = −Q31 , τ1 = τ11 2 = 0, π311 = −e31 , π111 = π 11 2 =0

in Ω2

(4.14b)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎫ ⎪ ⎪ ⎪ ⎬

in Ω1

(4.14c)

⎪ ⎪ ⎪ ⎭

^ 1* − Π ^ 9* are obtained from the corHere, material constants Π ^1−Π ^ 9 constants given in Eqs. 4.6g, 4.9b- 4.9f, responding Π by simply switching index “1” with index “2” and index “2” with index “1” wherever they occur. 12 12 The solution of the τ12 ij , τ i , and π i problems proceeds in the same manner. In this case the algebraic systems involved are trivial and the expressions of the appropriate local functions are obtained in a straight-forward fashion as: ⎫ 12 12 12 τ12 11 = τ 22 = τ 33 = τ 13 = ⎪ ⎪ ⎪ ⎬ 12 = τ12 23 = τ 12 = 0, in Ω3 (4.15a) 12 12 12 τ1 = τ2 = τ3 = 0, ⎪ ⎪ ⎪ ⎭ π112 = π212 = π312 = 0 ⎫ 12 12 12 12 τ12 11 = τ 22 = τ 33 = τ 13 = τ 23 = 0, ⎪ ⎪ ⎪ ⎬ τ12 12 = −C 66 in Ω1 (4.15b) 12 12 ⎪ τ12 1 = τ 2 = τ 3 = 0, ⎪ ⎪ ⎭ π112 = π212 = π312 = 0


12 12 12 12 τ12 11 = τ 22 = τ 33 = τ 13 = τ 23 = 0, 12 τ12 = −C66 12 12 τ12 1 = τ 2 = τ 3 = 0, 12 12 π1 = π2 = π312 = 0

⎫ ⎪ ⎪ ⎪ ⎬

Eq. 2.6f, becomes: in Ω2

(4.15c)

⎪ ⎪ ⎪ ⎭

Before explaining how the effective coefficients may be obtained from the aforementioned local coefficient functions, we will first solve the corresponding unit cell problems associated with the out-of-plane deformation and electric and magnetic field generation of the reinforced magnetoelectric plate. (b) Unit-Cell Problems 2.4f, 2.5f and 2.6f and Effective Coupling, Out-of-plane Piezoelectric and Out-ofplane Piezomagnetic Coefficients We now turn our attention to unit cell problems 2.4f, 2.5f and 2.6f. For a piecewise homogeneous unit cell, problem 2.4f may be expressed as: (1)µα ∂τ(1)µα 1 ∂τ iβ + i3 = −C33µα δ i3 h β ∂y β ∂z n β +zC iβµα +zC i3µα n3 = 0 on Z ± with t(1)µα i hβ (1)µα (1) + M ij A(1) = L ijm N m where τ(1)µα µα + N ij Λ µα ij n β and t(1)µα = τ(1)µα +τ(1)µα n3 i iβ h β i3 (4.16a) (1)µα The coupling elastic b ij functions (from which the effective elastic coupling coefficients may be determined) are then given by:

b(1)µα = τ(1)µα +zC ijµα ij ij

(4.16b)

In an analogous manner, unit cell problem 2.5f becomes: (1)µα ∂τ(1)µα 1 ∂τ β + 3 = −Q3µα h β ∂y β ∂z nβ with t(1)µα +zQ βµα +zQ3µα n3 = 0 on Z ± hβ (1)µα (1) where τ(1)µα = L Nm − M j A(1) jm µα − N j Λ µα j n β and t µα = τ µα +τ3µα n3 β h β

(4.17a)

This problem will be solved in each of the regions Ω1 , Ω2 , and Ω3 separately and will yield the out-of-plane a(1)µα j piezomagnetic functions (which in turn give the effective out-of-plane piezomagnetic coefficients) according to: a(1)µα = τ(1)µα +zQ jµα j j

49

(4.17b)

Finally, the third unit cell problem that will be solved in conjunction with the aforementioned two problems,

(1)µα 1 ∂π β h β ∂y β

with

r

∂π3(1)µα ∂z (1)µα

+

= −e3µα n

+ze βµα h ββ +ze3µα n3 = 0

on Z ±

(1)µα * (1) where π (1)µα = L*jm N m − M *j A(1) µα − N j Λ µα j n (1)µα (1)µα β and r(1)µα = π β n3 h β +π 3

(4.18a)

Again, this problem will be solved in each of the regions Ω1 , Ω2 , and Ω3 separately and will yield the out-of-plane δ(1)µα piezoelectric functions (which in turn give the effecj tive out-of-plane piezoelectric coefficients) according to: δ(1)µα = π(1)µα +ze jµα j j

(4.18b)

Let us begin by setting up the boundary conditions: (i) Region Ω3 . As before, boundary conditions must be supplied on z = ±1/2 where n1 = n2 = 0, n3 = 1. Thus, from Eqs. 4.16a, 4.17a, and 4.18a, and remembering that we are dealing with orthotropic materials, the boundary conditions become: τ(1)λµ = τ(1)λµ = τ(1)12 = −zC13 , = 0, τ(1)11 33 13 23 33 (1)22 on z = ±1/2 τ33 = −zC23

(4.19a)

= −zQ32 , = −zQ31 , τ(1)22 τ(1)11 3 3 (1)12 =0 on z = ±1/2 τ3

(4.19b)

π3(1)11 = −zQ31 , π3(1)22 = −zQ32 , π3(1)12 = 0 on z = ±1/2

(4.19c)

(ii) Region Ω1 . Boundary conditions must be supplied on z = 1/2, z = 1/2 +H, where n1 = n2 = 0, n3 = 1 and on y1 = ±δ1 /2 where n2 = n3 = 0, n1 = 1. Thus, using Eqs. 4.16a, 4.17a, and 4.18a, the boundary conditions become: τ(1)11 τ(1)22 11 = −zC 11 , 11 = −zC 12 , (1)λµ (1)22 (1)11 (1)12 τ12 = −zC16 , τ(1)12 11 = τ 12 = τ 12 = τ 13 = 0 on y1 = ±δ1 /2 τ(1)11 τ(1)22 33 = −zC 13 , 33 = −zC 23 , (1)λµ (1)12 τ α3 = τ33 = 0 on z =1/2, 1/2+H (4.20a) τ(1)λµ =0 on y1 = ±δ1 /2 1 τ(1)11 = −Q , τ(1)22 = −Q32 , 31 3 3 (1)12 τ3 =0 on z = 1/2, 1/2 + H

(4.20b)

π1(1)λµ = 0 on y1 = ±δ1 /2 π3(1)11 = −ze31 , π3(1)22 = −ze32 , π3(1)12 = 0 on z =1/2+H

(4.20c)

(iii) Region Ω2 . Here, boundary conditions must be supplied on z = 1/2, z = 1/2 +H, where n1 = n2 = 0, n3 = 1

50 | D. A. Hadjiloizi et al. and on y2 = ±δ2 /2 where n1 = n3 = 0, n2 = 1. Thus, using Eqs. 4.16a, 4.17a, and 4.18a the boundary conditions become: τ(1)22 τ(1)11 22 = −zC 22 , 22 = −zC 12 , (1)λµ (1)22 (1)11 (1)12 τ12 = −zC66 , τ(1)12 22 = τ 12 = τ 12 = τ 23 = 0 on y2 = ±δ2 /2 τ(1)22 τ(1)11 33 = −zC 23 , 33 = −zC 13 , (1)λµ (1)12 τ α3 = τ33 = 0 on z =1/2+H (4.21a) τ(1)λµ =0 on y2 = ±δ2 /2 2 τ3(1)11 = −zQ31 , τ(1)22 = −zQ32 , 3 (1)12 on z =1/2+H τ3 = 0 π2(1)λµ = 0 on y2 = ±δ2 /2 π3(1)11 = −ze31 , π3(1)22 = −ze32 , π3(1)12 = 0 on z =1/2+H

(4.21b)

(4.21c)

We are now ready to solve the unit cell problems in Eqs. 4.17a, 4.18a and 4.19a. We will begin with the functions. Following the same , and π(1)22 , τ(1)22 τ(1)22 i i ij 22 22 methodology as per the corresponding τ22 ij , τ i , and π i functions we arrive at the following system of equations in region Ω3 . ⎡ ⎤ ∂Λ22 (1) ∂A22 (1) ∂N3 (1)22 C33 +e33 +Q33 ⎥ ⎢ ∂z ∂z ∂z ⎢ (1)22 (1) (1) ⎥ ⎥ ⎢ ∂N ∂A ∂Λ 3 22 22 ⎥= ⎢ Q33 −λ33 −µ33 ⎥ ⎢ ∂z ∂z ∂z ⎣ (1) (1) ⎦ (1)22 ∂A22 ∂Λ22 ∂N3 −ε33 −λ33 e33 ∂z ⎤ ∂z ∂z ⎡ C23 ⎢ ⎥ = −z ⎣ Q32 ⎦ in Ω3 Q32 (4.22a) Comparing this system with Eqs. 4.6b, 4.6d and 4.6f, we observe that the only difference is the presence of the “z” coordinate on the right-hand side. Clearly, this is to be expected, because the three unit cell problems under discussion pertain to out-of-plane deformation of the magnetoelectric composite. The solution of Eq. 4.22a is thus readily obtained from its counterpart in Eq. 4.6g by simply multi22 22 plying by z. In turn, the local functions τ22 ij , τ i , π i are determined as follows: ⎧ )︃ ⎫ (︃ ⎪ ⎪ ^3 ^4 ^2 Π Π Π ⎪ ⎪ (1)22 ⎪ ⎪ τ11 = z C13 +e31 +Q31 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ^ ^ ^ Π Π Π ⎪ 1 1 1 )︃ ⎪ ⎪ ⎪ (︃ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ^ ^ ^ Π3 Π4 Π2 ⎨ ⎬ (1)22 τ22 = z C23 +e32 +Q32 in Ω3 ^ ^ ^ Π Π Π 1 1 1 ⎪ ⎪ ⎪ ⎪ (1)22 (1)22 (1)22 ⎪ ⎪ ⎪ ⎪ τ33 = −zC23 , τ α3 = τ12 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (1)22 (1)22 (1)22 ⎪ ⎪ τ = −zQ , τ = τ = 0 ⎪ ⎪ 32 3 1 2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ (1)22 (1)22 (1)22 π3 = −ze32 , π1 = π 2 = 0 (4.23a)

The same conclusion, namely that τ(1)λµ = zτ(1)λµ , τ(1)λµ = ij ij i (1)λµ zτ λµ = zπ iλµ is also true in regions Ω1 and Ω2 . Thus: i , πi (1)22 = τ22 τ(1)22 11 (︃= −zC 12 , )︃ ^6 ^9 ^7 ^8 Π Π Π Π +e32 +Q32 = z C12 +C23 ^ ^ ^ ^ Π Π Π Π 5 5 5 5 (1)22 (1)22 (1)22 τ33 = −zC23 , τ α3 = τ12 = 0 =0 = τ(1)22 = −zQ32 , τ(1)22 τ(1)22 2 1 3 (1)22 (1)22 = 0 π3 = −e32 , π1 = π (1)22 2

τ(1)22 τ(1)22 22 = −zC 22 , 11 = −zC 12 , (1)22 (1)22 τ33 = −zC23 , τ(1)22 α3 = τ 12 = 0 (1)22 (1)22 = 0, τ3 = −zQ32 , τ1 = τ(1)22 2 (1)22 (1)22 (1)22 π3 = −e32 , π1 = π 2 = 0

⎫ ⎪ ⎪ ⎪ ⎬

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

in Ω1

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

in Ω2

(4.23b)

(4.23c)

⎪ ⎪ ⎪ ⎭

^ 1 -Π ^ 9 are given in Eqs. 4.6g and 4.9b – Here, functions Π 11 11 4.9f. Likewise, functions τ11 ij , τ i , π i are obtained in a similar way: (︃ )︃ ⎫ * * * ⎪ ^ ^ ^ Π Π Π ⎪ 3 4 2 τ(1)11 , ⎪ ⎪ 11 = z C 13 * +e 31 * +Q 31 * ⎪ ^ ^ ^ Π1 Π1 Π1 )︃ ⎪ ⎪ ⎪ (︃ ⎪ ⎪ * * * ⎪ ^ ^ ^ Π Π Π ⎬ (1)11 3 4 2 τ22 = z C23 +e32 +Q32 * * * in Ω3 ^ ^ ^ Π1 Π1 Π1 ⎪ ⎪ (1)11 (1)11 (1)11 ⎪ ⎪ τ33 = −zC13 , τ α3 = τ12 = 0 ⎪ ⎪ ⎪ (1)11 (1)11 ⎪ = 0, = τ = −zQ , τ τ(1)11 ⎪ 31 2 1 3 ⎪ ⎪ ⎭ (1)11 (1)11 (1)11 π = −ze , π =π =0 31

3

1

2

(4.24a)

τ(1)11 τ(1)11 = 22 (︁ = −zC 12 , 11 )︁ * ^* ^* ^* ^ Π Π Π Π = z C12 Π^ 6* +C13 Π^ 7* +e31 Π^ 8* +Q31 Π^ 9* 5

5

5

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

5

(1)11 τ(1)11 τ(1)11 α3 = τ 12 = 0 33 = −zC 13 , τ(1)11 = −zQ31 , τ(1)11 = τ(1)11 =0 3 1 2 (1)11 (1)11 = 0 π3 = −e31 , π1 = π (1)11 2

τ(1)11 τ(1)11 11 = −zC 11 , 22 = −zC 12 , (1)11 (1)11 τ33 = −zC13 , τ(1)11 α3 = τ 12 = 0 (1)11 (1)11 τ3 = −zQ31 , τ1 = τ(1)11 = 0, 2 (1)11 (1)11 (1)11 π3 = −ze31 , π1 = π 2 = 0

in Ω2 (4.24b)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎫ ⎪ ⎪ ⎪ ⎬

in Ω1

(4.24c)

⎪ ⎪ ⎪ ⎭

In dealing with the τ(1)12 , τ(1)12 , π (1)12 functions a slight ij i i complication arises. In particular from unit cell problem 4.17a and boundary condition 4.20a we readily arrive at (region Ω1 ): (1)12 ∂τ(1)12 1 ∂τ12 + 23 = 0 h1 ∂y1 ∂z

(4.25a)

Expressing τ(1)12 and τ(1)12 in terms of the N2(1)12 functions 12 23 by using the second expression in Eq. 4.16a, we arrive at


Laplace’s equation, namely: (1)12

∂2 N2 1 C66 2 h1 ∂y21

Here:

∂2 N2(1)12 + C44 =0 ∂z2

(4.25b)

^ 10 Π

The solution of Eq. 4.25b is obtained in a straightforward manner as: N2(1)12 = −

√︂ (︂ )︂ nπh1 C44 n ]︀ cosh 1− −1 y ( ) 1 ∞ ∑︀ H C66 = (︂√︂ )︂ C nπδ h n=1 44 1 1 2 n cosh [︂ (︂ )︂]︂ C66 2H nπ 1 cos z− H 2 √︂ (︂ )︂ [︀ nπh1 C44 n ]︀ cosh y 1− −1 ( ) 1 ∞ ∑︀ H C66 = (︂√︂ )︂ C44 nπδ1 h1 n=1 n2 cosh [︂ (︂ )︂]︂ C66 2H nπ 1 sin z− H 2 [︀

h1 [ H + 1] y 1 + 2

√︂ (︂ )︂ * ^ 10 ]︀ Π nπh1 C66 1− (−1)n sinh y1 H C44 2H C66 (︂ )︂ + 3 C44 π C nπδ h 1 1 66 n=1 n3 cosh C44 2H [︂ (︂ )︂]︂ nπ 1 cos z− (4.25c) ^ 11 is obtained from Π ^ 10 , with C55 replacing C44 , h2 and Π H 2 replacing h1 , δ2 replacing δ1 , y2 replacing y1 , sinh(. . . ) A similar solution (with the indices “1” and “2” inter- replacing cosh(. . . ) and sin(. . . ) replacing cos(. . . ). Likechanged) is obtained in region Ω2 . Thus, the following so- wise, Π * * ^ 11 ^ 10 is obtained from Π by making the same sublutions are obtained in the three regions of the unit cell: stitutions. It would not be remiss to mention here that, }︃ because the materials of choice in this example are or=0 τ(1)12 ij thotropic with the poling/magnetization direction choin Ω (4.26a) 3 =0 = 0, π(1)12 τ(1)12 i i problem decouples from its sen as the z-axis, the τ(1)12 ij √︂ 3

∞ ∑︁

[︀

(1)12 (1)12 (1)12 τ11 = τ(1)12 22 = τ 33 = τ 13 = 0, (1)12 (1)12 (1)12 = π3(1)12 = = 0, π1 = τ3 τ1 C 2H ^ 10 , = − 66 [H + 1] + 2 C66 Π τ(1)12 12 2 π

√︀ 2H ^ * (1)12 = − C66 C44 2 Π τ23 10 √︂ π 2H C66 ^ * τ2(1)12 = −Q24 2 Π π √︂ C44 10 2H C66 ^ * π2(1)12 = −e24 2 Π C44 10 π

⎫ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

in Ω1

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (4.26b)

⎫ (1)12 (1)12 (1)12 τ(1)12 ⎪ 11 = τ 22 = τ 33 = τ 23 = 0, ⎪ ⎪ ⎪ = 0, π2(1)12 = π(1)12 =0 ⎪ = τ(1)12 τ(1)12 ⎪ 3 3 2 ⎪ ⎪ 2H C ⎪ 66 (1)12 ⎪ ^ ⎪ τ12 = − [H + 1] + 2 C66 Π11 , ⎪ ⎪ 2 π ⎪ √︀ ⎪ 2H ^ * ⎬ (1)12 τ13 = − C66 C55 2 Π11 in Ω2 √︂ π ⎪ 2H C66 ^ * ⎪ (1)12 ⎪ τ1 = −Q15 2 Π ⎪ ⎪ ⎪ C55 11 π ⎪ ⎪ (1)12 (1)12 ⎪ ⎪ π1 = π3 = 0, ⎪ √︂ ⎪ ⎪ ⎪ 2H C ⎪ 66 ^ * (1)12 ⎭ π1 = −e15 2 Π11 C44 π (4.26c)

counterparts. Hence, the elastic coefficients , π (1)12 τ(1)12 i i in Eqs. 4.26c and 4.26d depend only on elastic parameters, but the piezoelectric and piezomagnetic coefficients in the latter two expressions in Eqs. 4.26c and 4.26d depend on both elastic as well as piezoelectric/piezomagnetic material parameters. We are now ready to calculate the effective elastic, piezoelectric and piezomagnetic coefficients. We first recall the averaging procedure defined in Eq. (I-4.5a). We can thus easily show the following formulae: ⟨1⟩Ω α =

∫︀ Ωα

⟨1⟩Ω3 = 1, ⟨ z ⟩Ω α =

∫︀ Ωα

⟨z⟩Ω3 = 0,

1dy1 dy2 dz =

Ht α = F (w) α , hα

)︀ H2 + H tα = S(w) zdy1 dy2 dz = α , 2h α (︀

(︀ 3 )︀ ⟨︀ 2 ⟩︀ ∫︀ 2 4H + 6H 2 + 3H t α z Ω = z dy1 dy2 dz = = J (w) α , α 12h α Ωα ⟨︀ 2 ⟩︀ z Ω = 1/12, 3 (4.26d) (w) (w) where F1 , F2 are cross-sectional areas (perpendicu(w) lar to the orientation of the reinforcement), S(w) 1 , S 2 are (w) (w) the first moments of the cross-sections, and J1 , J2 are the moments of inertia of the cross-sections of the reinforcing elements Ω1 and Ω2 relative to the middle surface of the plate Ω3 . Using these results, as well as Eqs. 4.7, 4.10, 4.13, 4.14a- 4.14c, 4.15a- 4.15c, 4.23a4.23c, 4.24a- 4.24c, 4.26a- 4.26d and (I-6.3), the effective extensional elastic coefficients of the magnetoelectric

52 | D. A. Hadjiloizi et al. wafer are obtained as follows:

and

}︁(3¯ ) ⟩︀ {︁ ^* ^* ^* Π Π Π 2 b11 +e31 Π^ 3* +Q31 Π^ 4* + C11 + 11 = C 13 Π ^* 1 1 1 {︁ }︁ ¯ (2) (w) ^* ^* ^* ^* Π Π Π Π F2 + C12 Π^ 6* +C13 Π^ 7* +e31 Π^ 8* +Q31 Π^ 9* + C11 5 5 5 5 }︁ {︁ ¯ 3 * * ( ) ⟨︀ 11 ⟩︀ ^ ^ ^* Π Π Π , b22 = C23 Π^ 2* +e32 Π^ 3* +Q32 Π^ 4* + C12 1 1 1 {︁ }︁ ¯ ⟨︀ 22 ⟩︀ (3) ^ ^ ^ b11 = C13 ΠΠ^ 2 +e31 ΠΠ^ 3 +Q31 ΠΠ^ 4 + C12 1 1 1 }︁(3¯ ) ⟨︀ 22 ⟩︀ {︁ ^ ^ ^ b22 = C23 ΠΠ^ 2 +e32 ΠΠ^ 3 +Q32 ΠΠ^ 4 + C22 + 1 1 1 {︁ }︁(1¯ ) ^ ^ ^ ^ + C12 ΠΠ^ 6 +C23 ΠΠ^ 7 +e32 ΠΠ^ 8 +Q32 ΠΠ^ 9 + C22 F1(w) 5 5 5 5 ⟨ ⟩ ⟨︀ 11 ⟩︀ ⟨︀ 12 ⟩︀ ⟨︀ 22 ⟩︀ ⟨︀ 12 ⟩︀ λµ b12 = b11 = b12 = b22 = b3j = 0, ⟨︀ 12 ⟩︀ ¯ b = {C66 }(3) ⟨︀

12

(4.27a) (︀ )︀ (︀ )︀ (︀ )︀ ¯ , ¯ , ¯ following a set of Here, superscripts 1 2 3 ¯ braces, e.g. {. . .}(1) denote the corresponding region of the unit cell, and the material parameters in the preceding braces pertain to the constituent material of that region. For example, referring to the first expression in Eq. 4.27a, all parameters within the first set of braces refer to the material of the base plate (region Ω3 ), and all parameters within the second set of braces refer to region Ω2 . Likewise, the effective coupling elastic coefficients are given by: ⟩ ⟨ ⟩ ⟨ (1)11 = zb11 11 = b 11 {︁ }︁(2¯ ) ^* ^* ^* ^* Π Π Π Π = C12 Π^ 6* +C13 Π^ 7* +e31 Π^ 8* +Q31 Π^ 9* + C11 S(w) 2 5⟩ 5 5 ⟩ 5⟨ ⟨ 22 (1)22 = zb22 = b22 {︁ }︁(1¯ ) ^ ^ ^ ^ = C12 ΠΠ^ 6 +C23 ΠΠ^ 7 +e32 ΠΠ^ 8 +Q32 ΠΠ^ 9 + C22 S(w) ⟩ 1 ⟩5 ⟨ ⟩5 ⟨ 5 ⟩ ⟨5 ⟨ = = b(1)12 = b(1)11 = b(1)11 b(1)11 11 ⟩ ⟨12 ⟩ ⟩ ⟨22 ⟨22 (1)λµ (1)12 (1)22 = b3j =0 = b22 = b12 (4.27b) Finally, the effective bending elastic coefficients are given as: ⟨

(1)11 zb11

⟩

1 = 12

{︁

^* ^* ^* Π Π Π C13 Π^ 2* +e31 Π^ 3* +Q31 Π^ 4*

}︁(3¯ )

+ C11 + }︁(2¯ ) + C12 +C13 +e31 +Q31 + C11 J2(w) ⟨ ⟩ }︁(3¯ ) {︁ ^* ^* ^* Π Π Π (1)11 1 zb22 = 12 C23 Π^ 2* +e32 Π^ 3* +Q32 Π^ 4* + C12 , 1 1 1 ⟨ ⟩ {︁ }︁(3¯ ) ^ ^ ^ (1)22 1 zb11 = 12 C13 ΠΠ^ 2 +e31 ΠΠ^ 3 +Q31 ΠΠ^ 4 + C12 1 1 1 ⟨ ⟩ {︁ }︁(3¯ ) ^ ^ ^ (1)22 1 zb22 = 12 C23 ΠΠ^ 2 +e32 ΠΠ^ 3 +Q32 ΠΠ^ 4 + C22 + 1 1 1 }︁(1¯ ) {︁ ^ ^ ^ ^ + C12 ΠΠ^ 6 +C23 ΠΠ^ 7 +e32 ΠΠ^ 8 +Q32 ΠΠ^ 9 + C22 J1(w) 5 5 ⟩ ⟨ ⟩ ⟨ 5 ⟩ 5⟨ zb(1)11 = zb(1)12 = zb(1)22 = 12 ⟨ 12 ⟩ ⟨ 11 ⟩ λµ (1)12 zb22 = zb3j = 0 {︁

^* Π 6 ^* Π 5

1

^* Π 7 ^* Π 5

1

^* Π 8 ^* Π 5

1

^* Π 9 ^* Π 5

(4.27c)

{︂ 3 ⟩ 1 (3¯ ) + 1 {C66 }(1¯ ) H t1 + = { C } zb(1)12 66 12 12 12 h1 {︂ 3 }︂(2¯ ) 1 H t ¯ ¯ 2 {C }(2) − K2 − K1 }(1) + 12 66⎯ h2 ⎸ ¯ ∞ 96H 4 ⎸ {C }(1) ∑︀ [1 where K1 = 5 ⎷ 66 ¯ − 5 π h1 {C44 }(1) n=1 n ⎞ ⎛⎯ ⎸ n ]︀ ⎸ {C44 }(1¯ ) nπt1 (−1) ⎠, + tanh ⎝⎷ ¯ n5 {C66 }(1) 2H ⎯ ⎸ ¯ ∞ {C }(2) ∑︀ 96H 4 ⎸ [1 and K2 = 5 ⎷ 66 ¯ − 2) n=1 n 5 π h2 ( {C55 } ⎛⎯ ⎞ ⎸ n ]︀ ¯) 2 ( ⎸ −1 ( ) {C } nπt2 ⎠ + tanh ⎝⎷ 55 ¯ . n5 {C66 }(2) 2H (4.27d) As mentioned above, because the materials of choice in this example are orthotropic with the poling/magnetization direction chosen as the z-axis, the coun, π (1)12 problem decouples from its τ(1)12 τ(1)12 i i ij terparts. Therefore, the elastic coefficients in Eqs. 4.26c and 4.26d ⟩ only on elastic parameters. Thus, ⟨ depend (1)12 coefficient alone, the expression in for the zb12 Eq. 4.27d matches exactly the corresponding expression in Kalamkarov [45], Kalamkarov and Georgiades [60], Georgiades and Kalamkarov [61]. Using Eqs. 4.26d as well as Eqs. 4.7, 4.10, 4.13, 4.14a – 4.14c, 4.15a – 4.15c, 4.23a – 4.23c, 4.24a – 4.24c, 4.26a – ⟨︀ ⟩︀ 4.26d and (I-6.3), the effective in-plane, δ µν α , and out-of⟨ ⟩ (1)µν plane, δ α , piezoelectric coefficients, and the effective ⟨ ⟩ ⟨︀ µν ⟩︀ ^ α , and out-of-plane, a(1)µν , piezomagnetic in-plane, a α coefficients of the magnetoelectric wafer are obtained as follows: ⟨︀ µν ⟩︀ ⟨︀ µν ⟩︀ ⟨ (1)µν ⟩ ⟨ (1)µν ⟩ =0 (4.28) = δα ηα = δα = aα ⟨

We observe in this example that the piezoelectric and piezomagnetic functions are zero. Clearly, this is because the polarization/magnetization directions were chosen to coincide with the z-axis for all three elements of the unit cell. Hence, the piezoelectric e11 , e12 , e13 , e21 , e22 , e23 values as well as the corresponding piezomagnetic values are zero for each constituent in each element. Had we chosen the poling/magnetization direction in, say, element Ω1 to be parallel to y1 and in element Ω2 to be parallel to y2 , then the e11 , e12 , e13 piezoelectric and corresponding piezomagnetic coefficients would be non-zero in Ω1 , and likewise the e21 , e22 , e23 piezoelectric and corresponding piezomagnetic coefficients would be non-zero in Ω2 . Thus, the effective in-plane and out-of-plane piezoelectric


and piezomagnetic coefficients would be non-zero. As with the previous example, this underlines one of the principal advantages of our model; the designer has complete flexibility to enhance, reduce or even suppress selected coefficients to conform to the design criteria of a particular application, by changing one or more geometric, physical or material parameters. (c) Unit-Cell Problems 2.4b, 2.5b and 2.6b and Effective Dielectric Permittivity, Magnetoelectric and Piezoelectric coefficients We follow the same methodology as outlined above. For the sake of brevity we give only the final expressions. {︁ }︁(3¯ ) {︁ }︁(2¯ ) e2 e2 − ⟨δ11 ⟩ = ε11 + C15 + ε11 + C15 F2(w) 55 55 }︁(3¯ ) {︁ }︁(1¯ ) {︁ e2 e2 + ε22 + C24 F1(w) , − ⟨δ22 ⟩ = ε22 + C24 44 44

(4.29a)

⟨δ12 ⟩ = ⟨δ21 ⟩ = 0

}︁ 3¯ {︁ }︁ 2¯ {︁ Q15 ( ) Q15 ( ) (w) − ⟨a11 ⟩ = λ11 + e15C55 + λ11 + e15C55 F2 {︁ }︁(3¯ ) {︁ }︁(1¯ ) Q24 Q24 − ⟨a22 ⟩ = λ22 + e24C44 + λ22 + e24C44 F1(w) , ⟨a12 ⟩ = ⟨a21 ⟩ = 0

(4.29b) ⟨︀

⟩︀ =0 b µν α

(4.29c)

Eqs. 4.29a give the effective dielectric permittivity coefficients, Eqs. 4.29b give the first product properties we encounter, the effective magnetoelectric coefficients, and Eqs. 4.29c give the effective piezoelectric coefficients, which vanish for reasons explained above. (d) Unit-Cell Problems 2.4c, 2.5c and 2.6c and Effective Magnetic Permeability, Magnetoelectric and Piezomagnetic Coefficients As above, the effective magnetic permeability, magnetoelectric and piezomagnetic coefficients are given by Eqs. 4.30a, 4.30b and 4.30c respectively. {︁ }︁ 3¯ {︁ }︁ 2¯ Q215 ( ) Q215 ( ) (w) − ⟨𝛾11 ⟩ = µ11 + C55 + µ11 + C55 F2 {︁ {︁ ¯) ¯) 2 }︁(3 2 }︁(1 Q24 Q24 − ⟨𝛾22 ⟩ = µ22 + C44 + µ22 + C44 F1(w) , ⟨𝛾12 ⟩ = ⟨𝛾21 ⟩ = 0

(4.30a)

{︁ }︁ 3¯ {︁ }︁ 2¯ Q15 ( ) Q15 ( ) (w) − ⟨ξ11 ⟩ = λ11 + e15C55 + λ11 + e15C55 F2 {︁ }︁(3¯ ) {︁ }︁(1¯ ) Q24 Q24 − ⟨ξ22 ⟩ = λ22 + e24C44 + λ22 + e24C44 F1(w) , ⟨ξ12 ⟩ = ⟨ξ21 ⟩ = 0

(4.30b)

⟨︀

⟩︀ a µν =0 α

(4.30c)

Note that the effective magnetoelectric coefficients can be determined via two unit cell problems, and as expected ⟨ξ αµ ⟩ = ⟨a αµ ⟩ in this case. (e) Unit Cell Problems 2.4d, 2.5d and 2.6d and Effective Thermal Expansion, Pyroelectric and Pyromagnetic coefficients The effective thermal expansion, pyromagnetic and pyroelectric coefficients are given by Eqs. 4.31a, 4.31b and 4.31c, respectively. We reiterate that these effective coefficients are related to the mid-plane component of the temperature variation, see Eq. (I-3.4b). {︂ ^ *(Θ) ^ *(Θ) ^ *(Θ) Π Π Π − ⟨b11 ⟩ = −C13 ^ 2*(Θ) −e31 ^ 3*(Θ) −Q31 ^ 4*(Θ) Π1 Π1 Π1 {︂ ^ *(Θ) ^ *(Θ) ¯ Π Π +Θ11 }(3) + −C12 ^ 6*(Θ) −C13 ^ 7*(Θ) Π5 Π5 }︂(2¯ ) *(Θ) *(Θ) ^ ^ Π Π F2(w) −e31 ^ 8*(Θ) −Q31 ^ 9*(Θ) + Θ11 Π5 Π5 {︂ ^ (Θ) ^ (Θ) ^ (Θ) Π Π (4.31a) Π 3 4 2 − ⟨b22 ⟩ = −C23 ^ *(Θ) −e32 ^ *(Θ) −Q32 ^ *(Θ) Π1 Π1 Π1 {︂ ^ (Θ) ^ (Θ) ¯ Π Π +Θ22 }(3) + −C12 ^ 6(Θ) −C23 ^ 7(Θ) Π5 Π5 }︂(1¯ ) (Θ) (Θ) ^ ^ Π Π −e32 ^ 8(Θ) −Q32 ^ 9(Θ) + Θ11 F1(w) Π5

Π5

⟨b12 ⟩ = ⟨b21 ⟩ = 0 ⟨𝛾1 ⟩ = {η1 }(3) + {η1 }(2) F2(w , ¯

¯

(3¯ )

⟨𝛾2 ⟩ = {η2 }

¯ + {η2 }(1) F1(w)

¯ ¯ 2 3 ⟨τ1 ⟩ = {ξ1 }( ) + {ξ1 }( ) F2(w) , ¯ ¯ 1 3 ⟨τ2 ⟩ = {ξ2 }( ) + {ξ2 }( ) F1(w)

(4.31b)

(4.31c)

^ *(Θ) are obtained ^ *(Θ) -Π In Eq. 4.31a, material parameters Π 1 9 * * ^ 1 -Π ^ 9 parameters via the followfrom the corresponding Π ing substitutions: C13 → −Θ33 , C12 → −Θ22

Q31 → η3 ,

e31 → ξ3 ,

(4.31d)

^ (Θ) -Π ^ (Θ) are obtained from the corLikewise, parameters Π 1 9 ^ 1 -Π ^ 9 parameters via the following substituresponding Π tions: C23 → −Θ33 , C12 → −Θ11

Q32 → η3 ,

e32 → ξ3 ,

(4.31e)

(f) Unit Cell Problems 2.4e, 2.5e and 2.6e and Secondary Effective Thermal Expansion, Pyroelectric and Pyromagnetic Coefficients

54 | D. A. Hadjiloizi et al. The effective thermal expansion, pyromagnetic and pyroelectric coefficients related to the linear through-thethickness variation of the temperature field are given by Eqs. 4.32a, 4.32b and 4.32c, respectively. {︂ ⟩ ⟨ ^ *(Θ) ^ *(Θ) Π Π (1) − b11 = − ⟨zb11 ⟩ = −C12 ^ 6*(Θ) −C13 ^ 7*(Θ) Π5 Π5 }︂(2¯ ) ^ *(Θ) ^ *(Θ) Π Π S(w) −e31 ^ 8*(Θ) −Q31 ^ 9*(Θ) + Θ11 2 Π5 Π5 {︂ ⟨ ⟩ (Θ) ^ ^ (Θ) Π6 Π (4.32a) − b(1) −C23 ^ 7(Θ) 22 = − ⟨ zb 22 ⟩ = −C 12 Π ^ (Θ) Π 5 5 }︂(1¯ ) ^ (Θ) ^ (Θ) Π Π 9 8 S1(w) −e32 ^ (Θ) −Q32 ^ (Θ) + Θ11 Π5 ⟩ ⟨ ⟩Π5 ⟨ (1) b(1) 12 = b 21 = 0 ¯ = ⟨z𝛾 1 ⟩ = {η1 }(2) S2(w) , ¯ = ⟨z𝛾 ⟩ = {η2 }(1) S(w)

(4.32b)

⟩ ¯ τ1(1) = ⟨zτ1 ⟩ = {ξ1 }(2) S(w 2 , ⟨ ⟩ ¯ 1 (w) (1) τ2 = ⟨zτ2 ⟩ = {ξ2 }( ) S1

(4.32c)

⟨

𝛾1(1)

⟩

𝛾2(1)

⟩

⟨

2

1

analyzed. Of course, the semi-coupled approach is fairly accurate in predicting many of the effective coefficients but may err significantly in some instances such as in the prediction of some of the product properties. This is also evident in corresponding micromechanical models of threedimensional structures. For example, the fully-coupled approach given in Hadjiloizi et al. [25, 26], predicts the effective coefficients accurately whereas the semi-coupled approach in Challagulla and Georgiades [76] does not predict correctly some of the effective product properties. It is however accurate in predicting many of the remaining effective coefficients. Finally, we reiterate that if applied to the case of simple laminates, such as the ones considered in Section 3, the work presented here represents an extension of the classical composite laminate theory (see e.g. [66, 67]) to piezo-magneto-thermo-elastic structures.

⟨

(g) Comparison With Other Works and Discussion As mentioned previously, for the case of the purely elastic case the results of this model converge exactly to those of Kalamkarov [45], Kalamkarov and Kolpakov [46]. However, examination of Eqs. 4.27a – 4.27c, reveals that in the general case of a smart composite structure, the elastic coefficients are dependent on not only the elastic properties of the constituent materials, but also on the piezoelectric, piezomagnetic, magnetic permeability, dielectric permittivity and other parameters. The same holds true for the remaining effective coefficients. In a sense, the thermoelasticity, piezoelectricity and piezomagnetism problems are entirely coupled and the solution of one affects the solutions of the others. This is captured in the present papers, but not in previously published works. Thus, the results presented here represent an important refinement of previously established results such as those in Kalamkarov and Georgiades [60], Georgiades and Kalamkarov [61], Kalamkarov [74], Kalamkarov and Challagulla [75]. In essence, in these previous studies semi-coupled analyses of composite and reinforced plates are carried out and therefore the resulting expressions of the effective coefficients do not reflect the influence of many parameters such as the electric permittivity, magnetic permeability, primary magnetoelectricity etc. In the present work however, a fully coupled analysis is performed and as a consequence the expressions for the effective coefficients involve all pertinent material parameters. To the authors’ best knowledge, this is the first time that completely coupled piezo-magneto-thermo-elastic effective coefficients of reinforced plates are presented and

⟨︀ ⟩︀ Figure 8: Plot of the effective b11 11 extensional coeflcient vs. height of the piezoelectric wafer.

It is noted that the unit-cell problems are completely characterized by the structure of the unit cell of the magnetoelectric composite. It follows that the solutions of these problems and the effective coefficients in particular, are representative of the entire macroscopic composite and, once determined, they can be used to study a wide range of boundary value problems associated with that particular geometry. Examples include, but are certainly not limited to, static problems (the composite can be used as a structural member in manufacturing and infrastructure applications), dynamic problems (aerospace, automotive, vibration-absorption applications), magnetic/electric problems (the composites can be used as resonators, phase shifters, energy-harvesting devices, biomedical sensors and actuators), thermochem-


ical problems (chemical sensors) etc. Because the effective coefficients are representative of the macroscopic composite, the resulting expressions can easily be integrated in MATLABTM or other similar software packages to accurately and expediently analyze the aforementioned magnetoelectric structures. Ordinarily, the analysis and design of such complex geometries as shown in Fig. 6, would require a time-consuming numerical technique (such as the finite element method for example). The analytical model and its accompanying closed-form expressions presented here, however, allow us to quickly perform a preliminary design in a time-efficient manner. This preliminary design can then be used in conjunction with a finite element model to refine the results if enhanced accuracy is needed (by considering stress concentration effects for example). Obtaining a preliminary design before employing a numerical technique speeds up the design cycle time significantly. We finally note that the solutions of the local and the homogenized problems, Eqs. 2.3a- 2.3h, enable us to make very accurate predictions about the threedimensional local structure of the mechanical displacements, electric and magnetic potentials, force and moment resultants, electric and magnetic displacements etc.

Figure 9: Plot of the effective ⟨τ1 ⟩ pyroelectric coeflcient vs. height of the piezoelectric wafer.

Before closing this section, let us examine a magnetoelectric wafer made up of a 1-mm-thick cobalt ferrite base plate and a barium titanate wafer with a thickness of 1 mm and height which varies from 5 to 10 mm. The material properties are taken from Table 1. Figs. 8 and 9 show, re⟨︀ ⟩︀ spectively, the effective extensional b11 11 coefficient and the effective pyroelectric ⟨τ1 ⟩ coefficient. As expected, in-

creasing the height of the piezoelectric wafer increases the value of both coefficients.

5 Conclusions The method of asymptotic homogenization was used to analyze a periodic smart composite magnetoelectric plate of rapidly varying thickness. From a set of eighteen unit cell problems the effective elastic, piezoelectric, magnetoelectric, pyromagnetic, thermal expansion and other coefficients for the homogenized anisotropic composite and/or reinforced plate were derived. These effective coefficients are universal in nature and may be utilized in studying very different types of boundary value problems associated with a given smart composite structure. To illustrate the use of the unit cells and the applicability of the effective coefficients, two broad classes of examples were considered. The first example was concerned with a magnetoelectric laminate consisting of alternating piezoelectric and piezomagnetic laminae. The other example dealt with wafer-reinforced magnetoelectric plates. These are plates reinforced with mutually perpendicular ribs or stiffeners. The most general case was examined whereby the ribs had different orthotropic properties than the base plate. The unit cell problems were solved for this unique structure by considering each of the three regions of the unit cell separately. In the solution, we ignored complications at the regions of overlap between the actuators/reinforcements because these regions are highly localized and contribute very little to the integrals over the unit cell. The solution of the unit cell problems led to the determination of the effective coefficients including the product properties. It is shown in this work that in the case of the purely elastic case, the results of the derived model converge exactly to those of Kalamkarov [45, 75], Kalamkarov and Kolpakov [46], Kalamkarov and Challagulla [76]. However, in the more general case wherein some or all of the phases exhibit piezoelectric and/or piezomagnetic behavior, the expressions for the derived effective coefficients were shown to be dependent on not only the elastic properties of the constituent materials, but also on the piezoelectric and piezomagnetic parameters. One of the most important features of the derived model is that it affords complete flexibility to the designer to tailor the effective properties of the smart composite structure to conform to the requirements of a particular engineering application by changing one or more material or geometric parameters.

56 | D. A. Hadjiloizi et al. Acknowledgement: The authors would like to acknowledge the financial support of the Cyprus University of Technology (1st , 3rd and 4th authors), the Research Unit for Nanostructured Materials Systems (1st , 3rd and 4th authors) and the Natural Sciences and Engineering Research Council of Canada (2nd author).

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Research Article

Open Access

Salvatore Brischetto* and Roberto Torre

Exact 3D solutions and finite element 2D models for free vibration analysis of plates and cylinders Abstract: The paper proposes a comparison between classical two-dimensional (2D) finite elements (FEs) and an exact three-dimensional (3D) solution for the free vibration analysis of one-layered and multilayered isotropic, composite and sandwich plates and cylinders. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the finite element method and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 2D finite element results are obtained by means of a well-known commercial FE code. The differences between 2D FE solutions and 3D exact solutions depend on the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence. Keywords: plates, shells, finite element method, exact three-dimensional solution, free vibrations, vibration modes DOI 10.2478/cls-2014-0004 Received September 23, 2014 ; accepted October 20, 2014

1 Introduction The present paper investigates low and high frequencies in the case of free vibration response of simply-supported

*Corresponding Author: Salvatore Brischetto: Salvatore Brischetto, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, corso Duca degli Abruzzi, 24, 10129 Torino, ITALY. tel: +39.011.090.6813, fax: +39.011.090.6899, E-mail: [email protected]. Roberto Torre: Postgraduate at Politecnico di Torino

one-layered and multilayered isotropic, composite and sandwich plates and cylinders. The behavior and design of vibrating shells and plates have been extensively discussed in the reports by Leissa [1, 2] and more recently in the book by Werner [3] and in the work by Brischetto and Carrera [4], among others. The main aim of this work is the comparison between results obtained by means of an exact three-dimensional (3D) solution and those obtained by means of the classical two-dimensional (2D) finite element method (FEM). The proposed exact 3D solution has been developed by Brischetto in [5]-[7] where the differential equations of equilibrium written in general orthogonal curvilinear coordinates have exactly been solved by means of the exponential matrix method. The 2D FE results have been obtained by means of the commercial finite element code MSC Nastran [8]. In the most general case of exact threedimensional analyses, the number of frequencies for a free vibration problem is infinite: three displacement components (3 degrees of freedom DOF) in each point (points are ∞ in the 3 directions x, y, z) leads to 3×∞3 vibration modes. Assumptions are made in the thickness direction z in the case of a two-dimensional plate/shell model, the three displacements in each point are expressed in terms of a given number of degrees of freedom (NDOF) through the thickness direction z. NDOF varies from theory to theory. As a result, the number of vibration modes is NDOF × ∞2 in the case of exact 2D models. For exact beam models, the number of vibration modes is NDOF×∞1 . In the case of computational models, such as the Finite Element (FE) method, the number of modes is a finite number. This number coincides with the number of employed degrees of freedom: ∑︀Node NDOF i , where Node denotes the number of nodes 1 used in the FE mathematical model, and NDOF i is the NDOF through the thickness direction z in the i-node. It is clear that some modes are tragically lost in simplified models (such as computational two-dimensional models) [4]. In order to make a comparison between the 2D FE free vibration results and the 3D exact free vibration results, the investigation of the vibration modes is mandatory in order to understand what are the frequencies that must be compared.

© 2014 Salvatore Brischetto et al., licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

60 | Salvatore Brischetto and Roberto Torre The most relevant works about three-dimensional free vibration analysis of plates are shown below. Analytical three-dimensional solutions for free vibrations of a simply supported rectangular plate made of an incompressible homogeneous linear elastic isotropic material were proposed in Aimmanee and Batra [9] and in Batra and Aimmanee [10]. A three-dimensional linear elastic, small deformation theory obtained by the direct method was developed in Srinivas et al. [11] for the free vibration of simply supported, homogeneous, isotropic, thick rectangular plates. The same method was previously proposed in Srinivas et al. [12] for the flexure of simply supported homogeneous, isotropic, thick rectangular plates under arbitrary loads. Batra et al. [13] showed useful comparisons between two-dimensional models and an exact three-dimensional solution for free vibrations of a simply supported rectangular orthotropic thick plate. Ye [14] presented a threedimensional elastic free vibration analysis of cross-ply laminated rectangular plates with clamped boundaries; the analysis was based on a recursive solution. Comparisons between 2D-displacement-based-models and exact results of the linear three-dimensional elasticity were proposed in Messina [15] for natural frequencies, displacement and stress quantities in multilayered plates. A global three-dimensional Ritz formulation was employed in Cheung and Zhou [16] for the exact three-dimensional elastic investigation of isosceles triangular plates, and in Liew and Yang [17] for the three-dimensional elastic free vibration analysis of a circular plate. A set of orthogonal polynomial series was used to approximate the spatial displacements. Theoretical high frequency vibration analysis is fundamental in a variety of engineering designs. The importance of high frequency analysis of multilayered composite plates was also confirmed in the literature [4]. Zhao et al. [18] introduced the discrete singular convolution (DSC) algorithm for high frequency vibration analysis of plate structures, the Levy method was also employed to provide exact solutions to validate the DSC algorithm. The same investigation (comparison between DSC algorithm and the Levy method) was also proposed in Wei et al. [19]. Taher et al. [20] computed the first nine frequency parameters of circular and annular plates with variable thickness and combined boundary conditions, the eigenvalue equation was derived by means of three-dimensional elasticity theory and Ritz method. Xing and Liu [21] proposed the separation of variables to solve the Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates. Vel and Batra [22] extended three-dimensional exact models to free vibration of functionally graded material plates.

The most relevant works about three-dimensional free vibration analysis of shells are shown below. The coupled free vibrations of a transversely isotropic cylindrical shell embedded in an elastic medium were studied in [23] where the three-dimensional elastic solution used three displacement functions to represent the three displacement components. Free vibrations of simply-supported cylindrical shells were studied in [24] on the basis of three dimensional exact theory. Extensive frequency parameters were obtained by solving frequency equations. The free vibrations of simply-supported cross-ply cylindrical and doubly-curved laminates were investigated in [25]. The three-dimensional equations of motion were reduced to a system of coupled ordinary differential equations and then solved using the power series method. The threedimensional free vibrations of a homogenous isotropic, viscothermoelastic hollow sphere were studied in [26]. The surfaces were subjected to stress-free, thermally insulated or isothermal boundary conditions. The exact three-dimensional vibration analysis of a trans-radially isotropic, thermoelastic solid sphere was analyzed in [27]. The governing partial differential equations in [26] and [27] were transformed into a coupled system of ordinary differential equations. Fröbenious matrix method was employed to obtain the solution. Soldatos and Ye [28] proposed exact, three-dimensional, free vibration analysis of angle-ply laminated thick cylinders having a regular symmetric or a regular antisymmetric angle-ply lay-up. Armenakas et al. [29] proposed a self-contained treatment of the problem of plane harmonic waves propagation along a hollow circular cylinder in the framework of the threedimensional theory of elasticity. A comparison between a refined two-dimensional analysis, a shear deformation theory, the Flügge theory and an exact elasticity analysis was proposed in [30] for frequency investigation. Further details about the Flügge classical thin shell theory concerning the free vibrations of cylindrical shells with elastic boundary conditions can be found in [31]. Other comparisons between two-dimensional closed form solutions and exact 3D elastic analytical solutions for the free vibration analysis of simply supported and clamped homogenous isotropic circular cylindrical shells were also proposed in [32]. Vel [33] extended exact elasticity solutions to functionally graded cylindrical shells. The threedimensional linear elastodynamics equations were solved using suitable displacement functions that identically satisfy the boundary conditions. Loy and Lam [34] obtained the governing equations using an energy minimization principle. A layer-wise approach was proposed to study the vibration of thick circular cylindrical shells on the basis of three-dimensional theory of elasticity. Wang et al. [35]

Exact 3D solutions and finite element 2D models for free vibration analysis of plates and cylinders |

proposed the three-dimensional free vibration analysis of magneto-electro-elastic cylindrical panels. Further results about three-dimensional analysis of shells, where the solutions are not given in closed form, can be found in [36] for the dynamic stiffness matrix method and in [37] and [38] for the three-dimensional Ritz method for vibration of spherical shells. The papers of the literature discussed in this introduction show the three-dimensional analysis for free vibrations of plates or shells. They separately analyze shell or plate geometries and they do not give a general overview for both structures. The proposed exact 3D model uses a general formulation for several geometries (square and rectangular plates, cylindrical and spherical shell panels, and cylindrical closed shells). The equations of motion for the dynamic case are written in general orthogonal curvilinear coordinates using an exact geometry for multilayered shells. The system of second order differential equations is reduced to a system of first order differential equations, and afterwards it is exactly solved using the exponential matrix method and the Navier-type solution. The approach is developed in layer-wise form imposing the continuity of displacements and transverse shear/normal stresses at each interface. The exponential matrix method has already been used in [15] for the threedimensional analysis of plates in rectilinear orthogonal coordinates and in [28] for an exact, three-dimensional, free vibration analysis of angle-ply laminated cylinders in cylindrical coordinates. The equations of motion written in orthogonal curvilinear coordinates are a general form of the equations of motion written in rectilinear orthogonal coordinates in [15] and in cylindrical coordinates in [28]. The present equations allow general exact solutions for multilayered plate and shell geometries as already seen in the past author’s works [5]-[7]. In the literature review proposed in this introduction, only few works analyzes higher order frequencies. Moreover, papers that discuss the comparison between 2D models and exact 3D models are even less. The present work aims to fill this gap, it proposes a comparison between the free frequencies for plates and cylinders obtained by means of the commercial FE code MSC NASTRAN and those obtained by means of the exact 3D solution. The proposed 3D exact solution gives results for plates, cylindrical and spherical shell panels, and cylindrical closed shells. However, the comparison with the commercial FE code is proposed only for plates and cylinders. This choice is made for the sake of brevity, and further investigations for cylindrical and spherical shell panels could be proposed in the future. The aim of the present paper is to understand how to compare these two

61

different methods (exact 3D and numerical 2D solutions) and also to show the limits of a classical 2D FE solution.

Figure 1: Geometry, notation and reference system for shells.

2 Exact elasticity solution for shells The three differential equations of equilibrium written for the case of free vibration analysis of multilayered spherical shells made of N L layers with constant radii of curvature R α and R β are (the general form for variable radii of curvature can be found in [39] and [40]): (︂ )︂ 2H β H α ∂σ αβk ∂σ ∂σ + H α H β αzk + + σ αzk = H β ααk + H α ∂α ∂z Rα Rβ ∂β ¨k , = ρk Hα Hβ u Hβ

(1)

∂σ αβk ∂σ βzk ∂σ ββk + Hα + Hα Hβ + ∂α ∂z ∂β

(︂

2H α H β + Rβ Rα

)︂ σ βzk =

= ρ k H α H β ¨v k , Hβ

(2)

∂σ βzk Hβ ∂σ αzk ∂σ Hα + Hα + H α H β zzk − σ − σ ∂α ∂z R α ααk R β ββk ∂β (3)

(︂ +

Hβ Hα + Rα Rβ

)︂ ¨k , σ zzk = ρ k H α H β w

where ρ k is the mass density, (σ ααk , σ ββk , σ zzk , σ βzk , σ αzk , ¨ k , ¨v k and w ¨ k indiσ αβk ) are the six stress components and u cate the second temporal derivative of the three displacement components. Each quantity depends on the k layer. R α and R β are referred to the mid-surface Ω0 of the whole multilayered shell (see Figure 1 for further details about shell geometry). H α and H β continuously vary through the thickness of the multilayered shell and they depend on the thickness coordinate. The parametric coefficients for

62 | Salvatore Brischetto and Roberto Torre shells with constant radii of curvature are: ˜z − h/2 z ) = (1 + ), Rα Rα ˜z − h/2 z H β = (1 + ) = (1 + ) , Hz = 1 , Rβ Rβ H α = (1 +

(4)

H α and H β depend on z or ˜z coordinate (see Figure 2 for further details).

Figure 2: Thickness coordinates and reference systems for plates and shells.

Figure 3: Geometries for assessments (structures a-d) and for benchmarks (structures a and b).

The geometrical relations written for shells with constant radii of curvature are obtained from the general straindisplacement relations of the three-dimensional theory of elasticity in orthogonal curvilinear coordinates proposed in [39] and [41]: 1 ∂u k wk ϵ ααk = + , (5) H α ∂α Hα Rα


ϵ ββk =

1 ∂v k wk + , H β ∂β Hβ Rβ

∂w k , ∂z 1 ∂v k 1 ∂u k = + , H α ∂α H β ∂β

63

(6)

ϵ zzk =

(7)

𝛾αβk

(8)

1 Hα 1 = Hβ

𝛾αzk = 𝛾βzk

uk ∂w k ∂u k + − , ∂α ∂z Hα Rα vk ∂w k ∂v k − . + ∂z Hβ Rβ ∂β

(9) (10)

Geometrical relations for spherical shells degenerate into geometrical relations for cylindrical shells when R α or R β is infinite (with H α or H β equals one), and they degenerate into geometrical relations for plates when both R α and R β are infinite (with H α =H β =1). Three-dimensional linear elastic constitutive equations in orthogonal curvilinear coordinates (α, β, z) for orthotropic material in the structural reference system are given for a generic k layer of the multilayered structure: σ ααk = C11k ϵ ααk + C12k ϵ ββk + C13k ϵ zzk + C16k 𝛾αβk ,

(11)

σ ββk = C12k ϵ ααk + C22k ϵ ββk + C23k ϵ zzk + C26k 𝛾αβk ,

(12)

σ zzk = C13k ϵ ααk + C23k ϵ ββk + C33k ϵ zzk + C36k 𝛾αβk ,

(13)

σ βzk = C44k 𝛾βzk + C45k 𝛾αzk ,

(14)

σ αzk = C45k 𝛾βzk + C55k 𝛾αzk ,

(15)

σ αβk = C16k ϵ ααk + C26k ϵ ββk + C36k ϵ zzk + C66k 𝛾αβk .

(16)

The closed form of Eqs. (1)-(3) is obtained for simply supported shells and plates made of isotropic material or orthotropic material with 0∘ or 90∘ orthotropic angle (in both cases C16k = C26k = C36k = C45k = 0). The three displacement components have the following harmonic form: ¯ , u k (α, β, z, t) = U k (z)e iωt cos(¯α α) sin(ββ) iωt ¯ , v k (α, β, z, t) = V k (z)e sin(¯α α) cos(ββ)

(18)

¯ , sin(¯α α) sin(ββ)

(19)

w k (α, β, z, t) = W k (z)e

iωt

(17)

where U k , V k and W k are the displacement amplitudes in α, β and z directions, respectively. i is the coefficient of the imaginary unit, ω = 2πf is the circular frequency where f is the frequency value, t is the time. In coefficients α¯ = mπ a and β¯ = nπ b , m and n are the half-wave numbers and a and b are the shell dimensions in α and β directions, respectively (calculated in the mid-surface Ω0 ). The system of equations in closed form is obtained substituting Eqs. (5)-(10), (11)-(16) and (17)-(19) in the equilibrium equations proposed in Eqs. (1)-(3): )︁ (︁ )︁ C55k H β C11k H β ¯ 2 C66k H α C ¯ 12k − α¯ βC ¯ 66k V k + − 55k − α¯ 2 −β + ρ k H α H β ω2 U k + − α¯ βC 2 Rα Rβ Hα Hβ Hα Rα (︁ C H )︁ (︁ C H (︁ C H C C C H α )︁ 11k β 55k β 55k β α¯ + α¯ 12k + α¯ + α¯ 55k W k + + 55k U k,z + α¯ C13k H β + Hα Rα Rβ Hα Rα Rβ Rα Rβ )︁ (︁ )︀ α¯ C55k H β W k,z + C55k H α H β U k,zz = 0 , (︁ )︁ (︁ )︁ ¯ 66k − α¯ βC ¯ 12k U k + − C44k H α − C44k − α¯ 2 C66k H β − β¯ 2 C22k H α + ρ k H α H β ω2 V k + − α¯ βC 2 Rα Rβ Hα Hβ Hβ Rβ (︁ C H )︁ (︁ )︁ (︁ C C H α ¯ C12k C44k H α C44k H β α ¯ 44k H α + + β¯ 44k + β¯ 22k +β Wk + + V k,z + βC β¯ 44k Hβ Rβ Rα Hβ Rβ Rα Rβ Rα )︁ (︁ )︀ ¯ 23k H α W k,z + C44k H α H β V k,zz = 0 , βC (︁ C H (︁ C H C11k H β C C )︁ C C H α ¯ C12k )︁ 55k β α α¯ − α¯ 13k + α¯ + α¯ 12k U k + β¯ 44k − β¯ 23k + β¯ 22k +β Vk + Hα Rα Rβ Hα Rα Rβ Hβ Rβ Rα Hβ Rβ Rα (︁

−

(20)

(21)

64 | Salvatore Brischetto and Roberto Torre (︁ C )︁ C11k H β 2C12k C22k H α C55k H β ¯ 2 C44k H α C 13k −β + 23k − − − − α¯ 2 + ρ k H α H β ω2 W k + 2 2 Rα Rβ Rα Rβ Rα Rβ Hα Hβ Hα Rα Hβ Rβ (︁ )︁ (︁ )︁ (︁ C H C H α )︁ 33k β ¯ 44k H α − βC ¯ 23k H α V k,z + − α¯ C55k H β − α¯ C13k H β U k,z + − βC + 33k W k,z + Rα Rβ (︁ )︁ C33k H α H β W k,zz = 0 .

(22)

The system of Eqs. (20)-(22) can be written in a compact form introducing coefficients A sk for each block 1 to 19:

(︁)︁

with s from

A1k U k + A2k V k + A3k W k + A4k U k,z + A5k W k,z + A6k U k,zz = 0 ,

(23)

A7k U k + A8k V k + A9k W k + A10k V k,z + A11k W k,z + A12k V k,zz = 0 ,

(24)

A13k U k + A14k V k + A15k W k + A16k U k,z + A17k V k,z + A18k W k,z + A19k W k,zz = 0 .

(25)

The Eqs. (23)-(25) are a system of three second order differential equations. They are written for spherical shell panels with constant radii of curvature but they automatically degenerate into equations for cylindrical shells and plates. The system of second order differential equations proposed in Eqs. (23)-(25) can be reduced to a system of first order differential equations using the method described in [42] and [43] (further details can also be found in past author’s works [5–7]): ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

A6k 0 0 0 0 0

0 A12k 0 0 0 0

0 0 A19k 0 0 0

0 0 0 A6k 0 0

0 0 0 0 A12k 0

0 0 0 0 0 A19k

⎤⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Uk Vk Wk U k′ V k′ W k′

⎤′

⎡ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ =⎢ ⎥ ⎢ −A1k ⎥ ⎢ ⎦ ⎣ −A7k −A13k

0 0 0 −A2k −A8k −A14k

0 0 0 −A3k −A9k −A15k

A6k 0 0 −A4k 0 −A16k

0 A12k 0 0 −A10k −A17k

0 0 A19k −A5k −A11k −A18k

⎤⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Uk Vk Wk U k′ V k′ W k′

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎥ ⎦

(26)

Eq. (26) can be written in a compact form for a generic k layer: Dk where

∂U k ∂˜z

∂U k = Ak U k , ∂˜z

(27)

= U ′k and U k = [U k V k W k U k′ V k′ W k′ ]. The Eq. (27) can be written as: D k U ′k = A k U k ,

(28)

U ′k U ′k

Uk ,

(29)

,

(30)

= =

D−1 k Ak * Ak U k

with A*k = D−1 k Ak . In the case of plate geometry coefficients A3k , A4k , A9k , A10k , A13k , A14k and A18k are zero because the radii of curvature R α and R β are infinite. The other coefficients A1k , A2k , A5k , A6k , A7k , A8k , A11k , A12k , A15k , A16k , A17k and A19k are constant in each k layer because parametric coefficients H α = H β = 1 and they do not depend on the thickness coordinate ˜z. Therefore, matrices D k , A k and A*k are constant in each k layer of the plate. The solution of Eq. (30) for the plate case can be written as [43], [44]: U k (˜z k ) = exp(A*k ˜z k )U k (0) with ˜z k ϵ [0, h k ] ,

(31)

where ˜z k is the thickness coordinate of each layer from 0 at the bottom to h k at the top (see Figure 2). The exponential matrix for the plate case (constant coefficients A sk ) is calculated with ˜z k = h k for each k layer as: * * A** k = exp(A k h k ) = I + A k h k +

A*3 A*N A*2 k h2k + k h3k + . . . + k h Nk , 2! 3! N!

(32)

where I is the 6×6 identity matrix. This expansion has a fast convergence as indicated in [45] and it is not time consuming from the computational point of view. In the case of N L layers for shell geometry A*k is not constant in each k layer because H α (˜z) and H β (˜z) are not constant.


65

In the case of multilayered plates, N L − 1 transfer matrices T k−1,k must be calculated using for each interface the following conditions for interlaminar continuity of displacements and transverse shear/normal stresses: u bk = u tk−1 , v bk = v tk−1 , w bk = w tk−1 , σ bzzk

=

σ tzzk−1

,

σ bαzk

=

σ tαzk−1

,

σ bβzk

(33) =

σ tβzk−1

,

(34)

each displacement and transverse stress component at the top (t) of the k-1 layer is equal to displacement and transverse stress components at the bottom (b) of the k layer. The Eqs. (33)-(34) can be grouped in a system (details can be found in [5], [6] and [7]): ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

U V W U′ V′ W′

⎤b

⎡

⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ =⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ k

1 0 0 T1 0 T7

0 1 0 0 T4 T8

0 0 1 T2 T5 T9

0 0 0 T3 0 0

0 0 0 0 T6 0

0 0 0 0 0 T10

⎤

⎡

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ k−1,k

U V W U′ V′ W′

⎤t ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

(35)

k−1

Eq. (35) in compact form is: U bk = T k−1,k U tk−1 .

(36)

The calculated T k−1,k matrices allow to link U at the bottom (b) of the k layer with U at the top (t) of the k − 1 layer. Eq. (36) can also be written as: U k (0) = T k−1,k U k−1 (h k−1 ) , (37) where U k is calculated for ˜z k = 0 and U k−1 is calculated for ˜z k−1 = h k−1 . U at the top of the k layer is linked with U at the bottom of the same k layer by means of the exponential matrix A** k : U k (h k ) = A** k U k (0) ,

(38)

Eq. (37) can recursively be introduced in Eq. (38) for the N L − 1 interfaces to obtain: ** ** ** U N L (h N L ) = A** N L T N L −1,N L A N L −1 T N L −2,N L −1 . . . . . . A 2 T 1,2 A 1 U 1 (0) ,

(39)

the definition of the matrix H m for the multilayered plate allows Eq. (39) to be written as: U N L (h N L ) = H m U 1 (0) ,

(40)

that links U calculated at the top of the last N L layer with U calculated at the bottom of the first layer. In the case of multilayered plates, matrices D k , A k and A*k are constant in each k layer because R α and R β are infinite and H α and H β equal 1. In the case of shell geometry matrices D k , A k and A*k are not constant in each layer because of parametric coefficients H α and H β that depend on ˜z coordinate (see Figure 2). A first method could be the use of hypothesis Rzα = z R β = 0 (it is valid only for very thin shells) that means H α = H β = 1. In this case the solution is the same already seen for the multilayered plate because matrices D k , A k and A*k are constant in each k layer. This method is not used in this paper because it is an approximation that is valid only for very thin shells, and it does not consider the exact geometry of the structure. The second method (used in this paper) is the introduction of several j fictitious layers in each k physical layer where H α and H β can exactly be calculated. Matrices A** j are constant in the j layer because they are evaluated with R α , R β , α¯ and β¯ calculated in the mid-surface Ω0 of the whole shell, and with H α and H β calculated in the middle of each j fictitious layer. Matrices T j−1,j are also constant because they are calculated with R α , R β , α¯ and β¯ calculated in the mid-surface Ω0 of the shell, and with H α and H β calculated at each fictitious interface. In the present paper each physical k layer of the multilayered shell is divided in j fictitious layers where we can recursively apply the Eqs. (36)-(40) with index q=k× j in place of index k. The thickness of each fictitious layer is h q . The index q considers all the fictitious and physical layers and it goes from 1 to P. N=3 for the exponential matrix in Eq. (32) for each q layer guarantees the exact convergence for each shell investigated. The total number of mathematical layers that will be used for multilayered shell investigations will be P=102 or P=100 (it will depend on the analyzed case).

66 | Salvatore Brischetto and Roberto Torre The structures are simply supported and free stresses at the top and at the bottom of the whole multilayered shell, this feature means: σ zz = σ αz = σ βz = 0

for

z = −h/2, +h/2 or ˜z = 0, h , (41)

w = v = 0, σ αα = 0

for

α = 0, a ,

(42)

w = u = 0, σ ββ = 0

for

β = 0, b .

(43)

Eqs. (41)-(43) in compact form to express the free stress state at the top and bottom of the whole shell are (further details can be found in [5], [6] and [7]): B P (h P ) U P (h P ) = 0 ,

(44)

B1 (0) U 1 (0) = 0 ,

(45)

Eq. (40) can be substituted in Eq. (44) considering a total number of layers equals P (both physical and fictitious layers, and not only the physical layers N L ): B P (h P ) H m U 1 (0) = 0 ,

(46)

B1 (0) U 1 (0) = 0 ,

(47)

Eqs. (46) and (47) can be grouped in the following system: [︃

B P (h P ) H m B1 (0)

]︃ U 1 (0) = 0 ,

(48)

in the 6×6 matrix E, this last matrix has six eigenvalues. We are interested to the null space of matrix E that means to find the 6 × 1 eigenvector related to the minimum of the six eigenvalues proposed. This null space is, for the chosen frequency ω l , the vector U calculated at the bottom of the whole structure: U 1ω l (0) = [︁ = U1 (0)

V1 (0)

W1 (0)

U1′ (0)

V1′ (0)

W1′ (0)

]︁T ωl

,

(51) T means the transpose of the vector and the subscript ω l means that the null space is calculated for the circular frequency ω l . It is possible to find U qω l (˜z q ) (with the three displacement components U qω l (˜z q ), V qω l (˜z q ) and W qω l (˜z q ) through the thickness) for each q layer of the multilayered structure using Eqs. (37)-(40) with the index q (in place of k) from 1 to P. The thickness coordinate ˜z can assume all the values from the bottom to the top of the structure. For the plate case the procedure is simpler because there are not the j fictitious layers and the index q coincides with the index k of the physical layers (in this case, the total number of layers is N L and it is not P).

and introducing the 6 × 6 E matrix, the Eq. (48) is: E

U 1 (0) = 0 .

(49)

The Eq. (49) is also valid for plate case where the fictitious layers are not introduced and B N L (h N L ) = B P (h P ). Matrix E has always 6 × 6 dimension, independently from the number of layers P, even if the method uses a layer-wise approach. The solution is implemented in a Matlab code where only the spherical shell method is considered, it automatically degenerates into cylindrical open/closed shell and plate geometries. The free vibration analysis means to find the nontrivial solution of U 1 (0) in Eq. (49), this means to impose the determinant of matrix E equals zero: det[E] = 0 ,

(50)

Eq. (50) means to find the roots of an higher order polynomial in λ = ω2 . For each pair of half-wave numbers (m,n) a certain number of circular frequencies are obtained depending on the order N chosen for each exponential matrix A** q . A certain number of circular frequencies ω l are found when half-wave numbers m and n are imposed in the structures. For each frequency ω l , it is possible to find the vibration mode through the thickness in terms of the three displacement components. If the frequency ω l is substituted

2.1 Validation of the 3D exact model Before the comparison study between the 3D exact solution and the 2D FE solution, the proposed 3D exact model has been validated by means of several comparisons with other 3D results already given in the literature. The validation considers plates, cylinders, cylindrical panels and spherical panels (see Figure 3 for further details). Both one-layered and multilayered configurations are analyzed. The order of expansion employed for the exponential matrix is N=3. P=100 fictitious layers have been used for the investigation of one-, two- and four-layered structures. P=102 fictitious layers have been used for the investigation of three-layered structures. An appropriate convergence study has already been proposed in [5], [6] and [7]. The first assessment considers a simply supported one-layered square plate (a=b=10 m) with thickness h=1 m. The only embedded layer is the isotropic aluminium alloy with Young modulus E=70 GPa, Poisson ratio ν=0.3 and mass density ρ=2702 kg/m3 . The first six circular frequen√︁ 2

¯ = ω ah Eρ for cies are given in non-dimensional form ω half-wave numbers m=n=1. Table 1 compares the present 3D solution with 3D solutions by Vel and Batra [22] and Srinivas et al. [12]. Compared results are identical for each proposed vibration mode.


67

Table 1: First assessment for the √︁ 3D exact solution, simply supported plate made of aluminium alloy with thickness ratio a/h=10. First six 2

¯ = ω ah circular frequencies ω

ρ E

for half-wave numbers m=n=1.

Mode 3D [22] 3D [12] Present 3D

I 5.7769 5.7769 5.7769

II 27.554 27.554 27.554

III 46.503 46.503 46.502

The second assessment shows the free frequencies for a simply supported one-layered cylinder with radii of curvature R α =1 m and R β = ∞, and circular dimension a = 2π R α . The material is isotropic (the same aluminium alloy proposed in the first assessment). In Table 2, the proposed thickness values are h=0.12 m and h=0.18 m, and the b dimension values can be 2m or 1 m. Table 2 shows the first √︁ cir¯ = ω πh Gρ for cular frequency in non-dimensional form ω longitudinal half-wave number n=1 and several m circular half-wave numbers. The present 3D solution is coincident with 3D solutions by Armenakas et al. [29] and Bhimaraddi [30] for each proposed h/R α and b/R α ratios. Table 2: Second assessment for the 3D exact solution, simply supported cylinder made of aluminium alloy with√︁different thickness ¯ = ω πh ratios h/R α . First circular frequency ω m values.

m

2

3D [29, 30] Present 3D

0.03730 0.03730


0.05853 0.05853

m

2


0.05652 0.05652


0.09402 0.09402

ρ G

for n=1 and several

h/R α = 0.12 4 6 b/R α = 2 0.02359 0.02462 0.02359 0.02462 b/R α = 1 0.04978 0.04789 0.04978 0.04789 h/R α = 0.18 4 6 b/R α = 2 0.03929 0.04996 0.03929 0.04996 b/R α = 1 0.08545 0.09093 0.08545 0.09093

8 0.03686 0.03686 0.05545 0.05545 8 0.07821 0.07821 0.11205 0.11205

The third assessment gives several frequencies for a simply supported multilayered composite cylindrical panel with radii of curvature R α =10m and R β = ∞, and dimensions a=b=5m. The thickness is h=0.5m (the shell is moderately thick with a thickness ratio R α /h=20). The composite layers have properties E1 = 25E0 , E2 = E3 = E0 ,

IV 196.77 196.77 196.77

V 201.34 201.34 201.34

VI 357.42 357.42 357.42

G12 = G13 = 0.5E0 , G23 = 0.2E0 , ν12 = ν13 = ν23 = 0.25 3 and ρ = 1500kg/m . Non-dimensional circular frequen√︁ ρ ¯ = ωR α E0 for several physical layers N L and lamcies ω ination sequence (0∘ /90∘ /0∘ /90∘ / . . .) are given in Table 3. The imposed half-wave numbers m and n are indicated in the table. For m=n=1 the first three modes are shown, only the first mode is given for the other combinations of half-wave numbers m and n. The present 3D solution is coincident with the 3D solution by Huang [25] for each proposed half-wave number, vibration mode, and number of physical layers N L embedded in the multilayered composite structure. The last assessment considers several frequencies for a simply supported multilayered composite spherical panel with radii of curvature R α =10m and R β =10m, dimensions a=b=2m and thickness value h=0.2m (the shell is moderately thin with a thickness ratio R α /h=50). The composite layers have the same properties already seen for the third assessment. Non-dimensional circular frequen√︁ ¯ = ωR α Eρ0 for several physical layers N L and lamcies ω ination sequence (0∘ /90∘ /0∘ /90∘ / . . .) are given in Table 4. The present 3D solution gives the same results of the 3D solution by Huang [25] for each proposed half-wave number and for each number of physical layers N L embedded in the multilayered composite structure. The first vibration mode is investigated for each combination of half-wave numbers m and n. The proposed 3D solution has successfully been validated for each geometry (plate, cylinder, cylindrical and spherical shell panels), lamination sequence, embedded material, thickness ratio, vibration mode and imposed half-wave numbers. Therefore, it can be used with confidence to validate the FE models and also to make the comparisons between the exact 3D solutions and the computational 2D models.

3 Finite element model The 2D finite element results proposed in this paper have been obtained by means of the FE commercial code known

68 | Salvatore Brischetto and Roberto Torre Table 3: Third assessment √︁ for the 3D exact solution, simply supported composite cylindrical panel with thickness ratio R α /h = 20. Circular ¯ = ωR α Eρ for several physical layers N L , lamination sequence (0∘ /90∘ /0∘ /90∘ / . . .) and half-wave numbers m and n. frequencies ω 0

m,n 1,1 1,1 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3

mode I II III I I I I I I I I

NL = 2 3D [25] Present 3D 1.8971 1.8971 18.813 18.813 20.169 20.169 4.4492 4.4492 7.8195 7.8195 4.3485 4.3485 6.0384 6.0384 8.8895 8.8895 7.7503 7.7503 8.9012 8.9012 11.103 11.103

NL = 4 3D [25] Present 3D 2.3415 2.3415 21.545 21.545 22.902 22.902 4.9620 4.9620 8.0752 8.0753 4.8493 4.8493 6.5486 6.5486 9.1438 9.1438 7.9573 7.9573 9.1290 9.1290 11.164 11.164

N L = 10 3D [25] Present 3D 2.4930 2.4930 22.387 22.387 23.694 23.694 5.3017 5.3017 8.5254 8.5253 5.1853 5.1853 6.9739 6.9739 9.6347 9.6346 8.3952 8.3950 9.6122 9.6120 11.686 11.686

Table 4: Fourth assessment for the 3D exact solution, simply supported composite spherical panel with thickness ratio R α /h = 50. Circular √︁ ¯ = ωR α Eρ for several physical layers N L , lamination sequence (0∘ /90∘ /0∘ /90∘ / . . .) and half-wave numbers m and n. frequencies ω 0

m,n 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3

mode I I I I I I I I I

NL = 2 3D [25] Present 3D 4.6238 4.6240 10.753 10.753 19.130 19.130 10.864 10.864 14.909 14.909 21.961 21.961 19.315 19.315 22.053 22.053 27.483 27.483

NL = 4 3D [25] Present 3D 5.8070 5.8070 12.134 12.134 19.846 19.845 12.188 12.188 16.298 16.298 22.719 22.719 19.932 19.931 22.757 22.757 27.790 27.790

as MSC Nastran & Patran [8]. Only simple geometries are analyzed in this paper (plates and cylinders). For these structures a maximum number of 5000 elements is sufficient for a correct convergence in the case of free vibration analysis (as it will be demonstrated in the section about the validation of the FE model). The 2D element employed in the free vibration analysis is the SHELL QUAD4 element of Nastran, it has four nodes for each element that are collocated in the four corners. The kinematic model used by Nastran in its 2D FEs is based on the Reissner-Mindlin hypotheses (equivalent single layer approach and constant transverse displacement in the z direction).

3.1 Validation of the 2D FE model The FE model will be validated only for plates and cylinders because in Section 4 comparisons will be made only

N L = 10 3D [25] Present 3D 6.2293 6.2293 13.050 13.050 21.042 21.042 13.076 13.076 17.432 17.432 24.027 24.027 21.082 21.081 24.045 24.045 29.189 29.189

for these two geometries. This choice is due to the fact that we want to compare several laminations, materials and modes without lose in clarity and conciseness. The investigation for cylindrical and spherical panels could be the topic of a future work. The first assessment for the FE model considers a simply supported one-layered square plate (dimensions a=b=1 m) with thickness ratios a/h=1000 and a/h=100. The layer is made of an isotropic aluminium alloy with Young modulus E=73 GPa, Poisson ratio ν=0.3 and mass density ρ =2800 kg/m3 . The second assessment for the FE model analyzes a simply supported two-layered cylinder (radii of curvature R α =10m and R β =∞, and dimensions a = 2πR α and b=20m) with thickness ratios R α /h=1000 and R α /h=100. The isotropic layer at the bottom is the same already seen for the first assessment about the onelayered isotropic plate. The isotropic layer at the top is a titanium alloy with Young modulus E=114 GPa, Poisson


5

f[Hz]

4.95

2D FE 3D exact

4.9

4.85

4.8 0

1000

2000 3000 number of elements

4000

5000

14

13.5

f[Hz]

ratio ν=0.3 and mass density ρ = 2768kg/m3 . The third assessment for the FE model proposes the same geometry of the cylinder described in the second assessment. In this third case, the structure is simply supported and it is multilayered embedding three composite layers with lamination sequence 90∘ /0∘ /90∘ . The composite material has Young modulii E1 =132.38 GPa and E2 =E3 =10.756 GPa, shear modulii G12 =G13 =5.6537 GPa and G23 =3.603 GPa, Poisson ratios ν12 =ν13 =0.24 and ν23 =0.49, and mass density ρ = 1600 kg/m3 . In each table and figure, the first two frequencies obtained via the FE code are shown. The frequency values are given in Hz. The results for the first assessment are shown in Table 5. For both thickness ratios (a/h=1000 and a/h=100), the first frequency is obtained with imposed half-wave numbers m=n=1, and the second frequency is obtained with m=1 and n=2. FE results with a rare mesh are more rigid than 3D results, therefore frequency values are bigger. These frequency values decrease and they are coincident with the 3D exact results when the mesh size increases. For a 70×70 mesh (that means 4900 elements), the FE value is very close to the 3D exact value for both frequencies (first and second) and for both thickness ratios (a/h=1000 and a/h=100). The error in percentage ∆(%) is always less than 0.1%. Results of Table 5 are also shown in graphical form in Figures 4 and 5 for a/h=1000 and a/h=100, respectively. The top image is for the first frequency and the bottom image is for the second frequency. The results for the second assessment are shown in Table 6. The structure is two-layered with a transverse anisotropy, the FE results converge to the 3D exact solution when the mesh size increases. For the cylinder with R α /h=1000, the first frequency is obtained with circular half-wave number m=18 and longitudinal half-wave number n=1, the second frequency is obtained with m=20 and n=1. For the thicker cylinder (R α /h=100), the first frequency is for m=10 and n=1, and the second frequency is for m=12 and n=1. For a rare mesh, the error given by the 2D FE model is large, this error is almost zero for a refinement of the mesh (127 × 38 that means 4826 elements). These results are confirmed in graphical form in Figures 6 and 7 for thickness ratios R α /h=1000 and R α /h=100, respectively. In this assessment, 2D frequencies are sometimes smaller than 3D frequencies. However, these differences are negligible (always less than | 0.1 | %). The third assessment is a composite three-layered cylinder, it is shown in Table 7. The first frequency is obtained with m=22 and n=1 for thickness ratio R α /h=1000, and with m=12 and n=1 for R α /h=100. The second frequency is obtained with m=24 and n=1 for thickness ratio R α /h=1000, and with m=10 and n=1 for R α /h=100. 2D

69

2D FE 3D exact

13

12.5

12 0

1000


4000

5000

Figure 4: First assessment for the 2D FE solution, simply supported plate made of aluminium alloy with thickness ratio a/h=1000. Frequency f[Hz] vs. number of elements for the mode (1,1) (at the top) and for the mode (1,2) (at the bottom).

FE results with a rare mesh give bigger frequencies than 3D exact results. FE results converge to the 3D solution when the mesh increases. There is a very small error (always ∆(%) ≤ 0.18%) when the mesh is 127 × 38. All these results are confirmed in graphical form in Figures 8 and 9. The 2D FE model has been validated for both geometries (plates and cylinders) and for several lamination sequences (one-layered, two-layered and three-layered structures embedding isotropic or orthotropic materials). The FE model correctly converges with a 70 × 70 mesh for the plate geometry, and with a 127 × 38 mesh for the cylinder. Such values will always be used in Section 4 for the detailed comparison between the 3D exact solution and the 2D computational model.

70 | Salvatore Brischetto and Roberto Torre Table 5: First assessment for the 2D FE solution, simply supported plate made of aluminium alloy with thickness ratios a/h=1000 and a/h=100. First two frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

Mesh 5×5 10×10 20×20 30×30 40×40 50×50 60×60 70×70

Num.El. 25 100 400 900 1600 2500 3600 4900

Mesh 5×5 10×10 20×20 30×30 40×40 50×50 60×60 70×70

Num.El. 25 100 400 900 1600 2500 3600 4900

a/h=1000 Mode I (1,1) 2D FE 3D ∆(%) 4.982 4.854 2.64 4.873 4.854 0.39 4.857 4.854 0.06 4.855 4.854 0.02 4.855 4.854 0.02 4.855 4.854 0.02 4.855 4.854 0.02 4.854 4.854 0.00 a/h=100 Mode I (1,1) 2D FE 3D ∆(%) 49.70 48.52 2.43 48.67 48.52 0.31 48.55 48.52 0.06 48.54 48.52 0.04 48.53 48.52 0.02 48.53 48.52 0.02 48.53 48.52 0.02 48.53 48.52 0.02

4 Results This section proposes a detailed comparison between the 3D exact model discussed in this paper and 2D FE models obtained via the code MSC Nastran & Patran [8]. As demonstrated in the sections about the validation of the models, all the 3D exact results use an order of expansion N=3 for the exponential matrix, and P=100 fictitious layers for one-layered, two-layered and four-layered structures or P=102 fictitious layers for three-layered geometries. The 2D FE results use the SHELL QUAD4 element of Nastran with a 70 × 70 mesh for all the plate geometries and a 127 × 38 mesh for all the cylinder geometries. The comparisons will be made only for plates and cylinders in order to focus our attention to several laminations and materials. In this way, we are able to contain the length of the paper and we do not lose in clarity. For shell geometries, frequencies with w≠0 are obtained twice by Nastran (for each couple of (m,n)) because the section of the cylinder is symmetric. However, these two vibration modes are equal and we will write only one frequency in the table. Further geometries, such as cylindrical and spherical shell panels, that have already

Mode I (1,2) 2D FE 3D 13.77 12.13 12.43 12.13 12.20 12.13 12.16 12.13 12.15 12.13 12.14 12.13 12.14 12.13 12.14 12.13

∆(%) 13.5 2.47 0.58 0.25 0.16 0.08 0.08 0.08

Mode I (1,2) 2D FE 3D 136.8 121.2 123.9 121.2 121.8 121.2 121.5 121.2 121.4 121.2 121.4 121.2 121.3 121.2 121.3 121.2

∆(%) 12.9 2.23 0.49 0.25 0.16 0.16 0.08 0.08

been validated in Section 2.1 via the 3D exact model, could be the topic of a future comparison work.

4.1 Comparison between the two models The first geometry considered in this investigation is a simply supported square plate with dimensions a=b=1 m. Thickness values are h=0.1 m, 0.05 m, 0.01 m and 0.001 m that mean thickness ratios a/h=10, 20, 100 and 1000, respectively. The second geometry is a simply supported cylinder with radii of curvature R α =10m and R β =∞. The dimensions are a=2πR α and b=20 m. The thickness values are h=0.01 m, 0.1 m, 1 m and 2 m that mean thickness ratios R α /h=1000, 100, 10 and 5, respectively. Both geometries will be considered as isotropic one-layered (h1 = h), isotropic two-layered (h1 = h2 = h/2), isotropic threelayered (h1 = h2 = h3 = h/3), three-layered composite cross-ply 90∘ /0∘ /90∘ (h1 = h2 = h3 = h/3), fourlayered composite cross-ply 90∘ /0∘ /90∘ /0∘ (h1 = h2 = h3 = h4 = h/4) and three-layered sandwich (skins with h1 = h3 = 0.2h and core with h2 = 0.6h). The onelayered structures embed an aluminium alloy with Young modulus E=73 GPa, Poisson ratio ν=0.3 and mass density

Exact 3D solutions and finite element 2D models for free vibration analysis of plates and cylinders

| 71

Table 6: Second assessment for the 2D FE solution, simply supported cylinder made of aluminium alloy and titanium alloy with thickness ratios R α /h=1000 and R α /h=100. First two frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

Mesh 20×6 40×12 60×18 70×21 90×27 110×33 120×36 127×38

Num.El. 120 480 1080 1470 2430 3630 4320 4826

Mesh 20×6 40×12 60×18 70×21 90×27 110×33 120×36 127×38

Num.El. 120 480 1080 1470 2430 3630 4320 4826

R α /h=1000 Mode I (18,1) 2D FE 3D ∆(%) 4.207 3.518 19.6 3.547 3.518 0.82 3.523 3.518 0.14 3.521 3.518 0.08 3.519 3.518 0.03 3.518 3.518 0.00 3.518 3.518 0.00 3.518 3.518 0.00 R α /h=100 Mode I (10,1) 2D FE 3D ∆(%) 11.01 10.76 2.32 10.76 10.76 0.00 10.75 10.76 -0.09 10.75 10.76 -0.09 10.75 10.76 -0.09 10.76 10.76 0.00 10.76 10.76 0.00 10.76 10.76 0.00

ρ =2800 kg/m3 . The two-layered structures have the bottom layer in aluminum alloy and the top layer in titanium alloy (E=114 GPa, ν = 0.3 and ρ =2768 kg/m3 ). The threelayered isotropic structures have the bottom layer in aluminum alloy, the mid layer in titanium alloy and the top layer in steel (E=210 GPa, ν = 0.3 and ρ =7850 kg/m3 ). The composite material for the three-layered and four-layered cross-ply structures has Young modulii E1 =132.38 GPa and E2 =E3 =10.756 GPa, shear modulii G12 =G13 =5.6537 GPa and G23 =3.603 GPa, Poisson ratios ν12 =ν13 =0.24 and ν23 =0.49, and mass density ρ =1600 kg/m3 . Sandwich configurations have the skins in aluminium alloy (see one-layered, two-layered and three-layered isotropic structures) and the core is in PVC (E=0.18 GPa, ν=0.37 and ρ =50 kg/m3 ). For all the benchmarks, the comparison is proposed calculating the first ten frequencies via the 2D FE code. From the visualization of these ten vibrations modes, it is possible to understand the half-wave numbers in the α and β directions. Therefore, these half-wave numbers have been used to calculate the same ten frequencies via the 3D exact model. There are some frequencies missed by the FE code, but they have not been investigated via the 3D exact model because this is not the main aim of the paper. The

Mode I (20,1) 2D FE 3D ∆(%) 4.366 3.540 23.3 3.573 3.540 0.93 3.542 3.540 0.06 3.540 3.540 0.00 3.538 3.540 -0.06 3.538 3.540 -0.06 3.538 3.540 -0.06 3.538 3.540 -0.06 Mode I (12,1) 2D FE 3D ∆(%) 12.15 11.65 4.11 11.61 11.65 -0.34 11.62 11.65 -0.26 11.62 11.65 -0.26 11.63 11.65 -0.17 11.64 11.65 -0.08 11.64 11.65 -0.08 11.64 11.65 -0.08

main aim of the paper is to understand the differences between the 2D FE and the 3D exact model for the first ten frequencies given by the 2D FE code. It is also important to understand what are the features that influence these differences (geometry of the structures, materials, lamination sequences, thickness ratios, order of frequencies, vibration modes). The first benchmark considers a one-layered isotropic plate (see Table 8 and Figure 10). For plate geometry the first ten frequencies for thin structures (a/h=1000 and a/h=100) are obtained with all the possible combinations of half-wave numbers in α and β directions from 1 to 4. The 3D model obtains such frequencies as the first mode for each couple of (m,n). The FE code works very well for thin plates (a/h=1000 and 100), where the error is meaningless (always less than 0.3%) for all the first ten frequencies. In the case of thick plates (a/h=20 and 10), the FE code works well only for low frequencies and it gives significant errors for higher order frequencies (reaching an error of 2.31% for the tenth frequency of the a/h=10 plate). For thick plates, the FE code gives some modes that have the transverse displacement w=0 (they are in-plane modes). The 3D model also gives such values (mode II for

72 | Salvatore Brischetto and Roberto Torre Table 7: Third assessment for the 2D FE solution, simply supported composite cylinder 90∘ /0∘ /90∘ with thickness ratios R α /h=1000 and R α /h=100. First two frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

Mesh 20×6 40×12 60×18 70×21 90×27 110×33 120×36 127×38

Num.El. 120 480 1080 1470 2430 3630 4320 4826

Mesh 20×6 40×12 60×18 70×21 90×27 110×33 120×36 127×38

Num.El. 120 480 1080 1470 2430 3630 4320 4826

R α /h=1000 Mode I (22,1) 2D FE 3D ∆(%) 4.224 2.743 54.0 2.829 2.743 3.13 2.771 2.743 1.02 2.763 2.743 0.73 2.754 2.743 0.40 2.750 2.743 0.25 2.749 2.743 0.22 2.748 2.743 0.18 R α /h=100 Mode I (12,1) 2D FE 3D ∆(%) 8.271 7.717 7.18 7.783 7.717 0.85 7.740 7.717 0.30 7.734 7.717 0.22 7.727 7.717 0.13 7.724 7.717 0.09 7.723 7.717 0.08 7.722 7.717 0.06

(0,1), (1,0) and (1,1)) even if it also gives other frequencies that have not been calculated by the 2D FE code. For inplane modes (w=0), the error obtained with the FE code is almost zero because the 3D effects are not important in vibration modes with w=0. Left part of Figure 10 shows the first five vibration modes with transverse displacement w≠0 obtained via the FE code. These modes are important to understand the half-wave numbers (m,n) to use for the 3D exact investigation. The 3D exact model gives the vibration modes in terms of displacement components u, v and w in the thickness direction z (right side of Figure 10) because the behavior in α and β directions is already known (via the imposed half-wave numbers (m,n) obtained from the FE analysis). The second benchmark is a a one-layered isotropic cylinder embedding the same material already seen for the benchmark 1. Frequencies and vibration modes are shown in Table 9. Even if the employed material is the same, the behavior is completely different due to the coupling caused by the radius of curvature R α . For each thickness ratio the lowest frequency is obtained for different couples of half-wave numbers (m,n), these values decrease when the thickness of the shell increases. For such a struc-

Mode I (24,1) 2D FE 3D ∆(%) 4.514 2.754 63.9 2.830 2.754 2.76 2.774 2.754 0.73 2.767 2.754 0.47 2.760 2.754 0.22 2.757 2.754 0.11 2.756 2.754 0.07 2.755 2.754 0.04 Mode I (10,1) 2D FE 3D ∆(%) 8.653 8.151 6.16 8.250 8.151 1.21 8.193 8.151 0.51 8.182 8.151 0.38 8.170 8.151 0.23 8.163 8.151 0.15 8.162 8.151 0.13 8.161 8.151 0.12

ture the vibration behavior is not a priori predictable because there is not a regular sequence of (m,n) that gives the first ten frequencies for thin structures (R α /h=1000 and R α /h=100). In this case, it is fundamental to obtain the first ten frequencies via the FE code because the mode visualization of Nastran allows to understand what are the half-wave numbers (m,n) to use in the 3D exact model. The cylinder has a symmetric geometry, for this reason for each couple of (m,n) the frequencies with w≠0 are twice given. However, only one value is written in the tables. For thin cylinders (R α /h=1000 and 100) the 2D FE results are coincident with the 3D exact results (the error ∆(%) is always negligible). For thick cylinders (R α /h=10 and R α /h=5), there are some FE frequencies that are quite different from the corresponding 3D frequencies (see for example the tenth frequency for R α /h = 10). These differences are not a priori predictable because the frequencies do not monotonously increase with the increasing of halfwave numbers (m,n). For thick cylinders, the 2D FE code gives some modes with w=0 that are also given by the 3D model. The third benchmark considers a two-layered isotropic plate, see Table 10. In the plate case there is not


50

| 73

4.4

49.8 4.2

49.6

2 FE 3D exact

2D FE 3D exact

49.4

f[Hz]

f[Hz]

4 49.2 49

3.8 48.8 48.6

3.6

48.4 48.2

3.4 0

1000


4000

5000

0

138

1000


4000

5000

4.4 2D FE 3D exact

136

4.2

134

2D FE 3D exact

132 f[Hz]

f[Hz]

4 130 128

3.8 126 124

3.6

122 120

3.4 0

1000


4000

5000

Figure 5: First assessment for the 2D FE solution, simply supported plate made of aluminium alloy with thickness ratio a/h=100. Frequency f[Hz] vs. number of elements for the mode (1,1) (at the top) and for the mode (1,2) (at the bottom).

any coupling due to the curvature, therefore the frequencies have the same behavior already seen for the benchmark 1 (they monotonously increase with the increasing of the half-wave numbers (m,n)). Therefore, the vibration behavior can be predicted for these structures. Errors for thin plates (a/h=1000 and 100) are small, but these errors for thick plates (a/h=20 and 10) are bigger even when low frequencies are investigated. For thick plates, there are some in-plane modes with w=0 that are coincident with the values given by the 3D model. However, some modes given by the 3D code have tragically been lost by the FE code (e.g., see mode I for m=0 and n=1 or m=1 and n=0 in the case of a/h=20 and a/h=10 plates). The fourth benchmark analyzes the free vibrations for a simply supported two-layered isotropic cylinder. Results are shown in Table 11 and in Figure 11. The radius of curvature R α gives a coupling between the displacement components. In this way, the vibration behavior is not a priori

0

1000


4000

5000

Figure 6: Second assessment for the 2D FE solution, simply supported multilayered cylinder made of aluminium alloy and titanium alloy with thickness ratio R α /h=1000. Frequency f[Hz] vs. number of elements for the mode (18,1) (at the top) and for the mode (20,1) (at the bottom).

predictable, as already seen for the second benchmark for the one-layered cylinder. The considerations are the same seen for the one-layered cylinder even if some errors given by the 2D FE model are bigger because there is a transverse anisotropy due the presence of these two different isotropic layers (see Table 11 for further details). Nastran gives some modes with w=0 for thick shells (R α /h=10 and R α /h=5). The 3D model gives such results but it also gives further frequencies that are not obtained by the 2D FE code (e.g., the mode I with m=0 and n=1 for R α /h=5). Figure 11 shows the first five frequencies with w≠0 obtained via Nastran (on the left side) and the corresponding first five frequencies with w≠0 obtained via the 3D exact model (on the right side). 3D modes are plotted only through the thickness direction because the behavior in α and β directions is known (imposed half-wave numbers (m,n)). However, all the first ten vibration modes given in the tables have been

74 | Salvatore Brischetto and Roberto Torre

11.1 4.2 11

2 FE 3D exact

4

2D FE 3D exact

3.8 3.6 f[Hz]

f[Hz]

10.9

10.8

3.4 3.2 3

10.7

2.8 2.6

10.6 0

1000


4000

5000

0

1000


4000

5000

12.4 4.5 12.2

2D FE 3D exact

2D FE 3D exact

4

f[Hz]

f[Hz]

12 3.5

11.8

3

11.6

11.4

2.5 0

1000


4000

5000

Figure 7: Second assessment for the 2D FE solution, simply supported multilayered cylinder made of aluminium alloy and titanium alloy with thickness ratio R α /h=100. Frequency f[Hz] vs. number of elements for the mode (10,1) (at the top) and for the mode (12,1) (at the bottom).

plotted via Nastran in order to make easier the comparison with the 3D exact results. The fifth benchmark about a three-layered isotropic plate is discussed in Table 12 and in Figure 12. From the Table 12, it is clear how the behavior is similar to that already seen for the one-layered and two-layered plates even if the transverse anisotropy is different from the one-layered and two-layered cases. For the sake of clarity, the first five vibration modes with w≠0 are shown in Figure 12 for the 2D FE analysis (left side) and the 3D exact model (right side). The behavior is the same already seen for the first benchmark, the introduction of further layers does not change the vibration modes in terms of half-wave numbers (m,n). For thick plates, the 2D FE code gives some in-plane modes (w=0) that are also calculated by the 3D model. However, the 3D model also gives further frequencies (e.g., for m=1

0

1000


4000

5000

Figure 8: Third assessment for the 2D FE solution, simply supported composite cylinder 90∘ /0∘ /90∘ with thickness ratio R α /h=1000. Frequency f[Hz] vs. number of elements for the mode (22,1) (at the top) and for the mode (24,1) (at the bottom).

and n=0, or m=0 and n=1) that have not been calculated by the 2D FE code. The sixth benchmark proposes a three-layered isotropic cylinder (see Table 13), the behavior is the same already seen for the one-layered cylinder in benchmark two and for the two-layered cylinder in benchmark four. The behavior of the vibration modes obtained with the imposed half-wave numbers (m,n) is not a priori predictable. The presence of three different isotropic layers gives a bigger transverse anisotropy. Therefore, FE results give bigger errors, in particular for thicker structures and for some vibration modes. In the case of thick cylinder, there are some modes with transverse displacement w=0 that are not present for thin geometries. The seventh benchmark considers a three-layered composite cross-ply 90∘ /0∘ /90∘ plate. The frequency results for this symmetric laminated plate are given in Table 14. Even if a plate geometry is considered, the half-


8.3 8.2

2D FE 3D exact

f[Hz]

8.1 8 7.9 7.8 7.7 7.6 0

1000


4000

5000

8.8 8.7

2D FE 3D exact

8.6

f[Hz]

8.5 8.4 8.3 8.2 8.1 8 0

1000


4000

5000

Figure 9: Third assessment for the 2D FE solution, simply supported composite cylinder 90∘ /0∘ /90∘ with thickness ratio R α /h=100. Frequency f[Hz] vs. number of elements for the mode (12,1) (at the top) and for the mode (10,1) (at the bottom).

wave number (m,n) for the first ten frequencies are different with respect to the three-layered isotropic plate already proposed. This feature is due to the fact that there is an orthotropy. For this reason, the imposition of a certain value for half-wave number m in x direction is different from the n value in y direction. Therefore, the first ten frequencies for thin plates are not given by the first ten possible combinations of m and n from 1 to 4. For thick plates (a/h=20 and a/h=10), the FE code gives some in-plane modes with w=0. The 3D model also obtains such modes, but it also gives further vibration modes (e.g., the first ones for (0,1) and (0,2)) that are not given by the FE code. The transverse anisotropy in the thickness direction is smaller than the isotropic cases that have three layers embedding different materials. For this reason, the errors given by the 2D FE model are smaller than errors seen in benchmarks three and five. Bigger errors are given for thick plate (a/h=20 and

| 75

10) but these errors are negligible for the in-plane modes with w=0 that have not any 3D effects. The eight benchmark analyzes a three-layered composite cross-ply 90∘ /0∘ /90∘ cylinder (see Table 15 and Figure 13). The errors given by the 2D FE code are smaller than those seen for benchmarks four and six. In fact, in the case of two-layered and three-layered isotropic cylinders the transverse anisotropy is bigger. The 2D FE code seems to give acceptable errors for all the thickness ratios and for each frequency (from the first to the tenth). The first five modes with transverse displacement w≠0 for this cylinder are shown in Figure 13 for both 2D FE and 3D exact analyses. There are not big differences with respect to the modes already plotted for the two-layered isotropic cylinder. The half-wave numbers (m,n) to calculate the first five modes with w≠0 change with respect to the isotropic case, but the general behavior remains the same. Some vibration modes with w=0 are calculated by the FE code in the case of thick cylinder. For thickness ratio R α /h=5, the FE code gives a mode with w=0 for m=4 and n=0. The 3D model also gives such a value but it also proposes a lower vibration mode for m=4 and n=0 that has not been calculated by the 3D FE code. The four-layered composite cross-ply 90∘ /0∘ /90∘ /0∘ plate for the benchmark nine is analyzed in Table 16, and the first five vibration modes with transverse displacement w≠0 are given in Figure 14. In this case the lamination is not symmetric and the bottom 0∘ layer is able to compensate the behavior of the top 90∘ layer. For this reason, even if an orthotropic material is employed, results for half-wave numbers (1,2) are equal to those for (2,1), those for (1,3) are equal to those for (3,1), and so on. FE results are quite correct for each vibration mode. For thick plates (a/h=20 and a/h=10), the FE code gives some inplane modes (w=0) for half-wave numbers (0,1) and (1,0), and (0,2) and (2,0). The 3D method also gives such frequencies but it also gives further modes (the first I for (1,0), (0,1), (2,0) and (0,2)) that are not obtained by the FE code. Figure 14 about vibration modes with w≠0 confirm that the behavior is a priori predictable. The modes through the thickness show a transverse anisotropy (zigzag form of the displacements). However, this transverse anisotropy is small. The tenth benchmark is about the four-layered composite cross-ply 90∘ /0∘ /90∘ /0∘ cylinder. Results shown in Table 17 confirm the same behavior already seen for the other cylinder configurations. Moreover, the errors obtained via the 2D FE analysis are smaller with respect to the two-layered and three-layered isotropic cases because there is a smaller transverse anisotropy. For the thick cases with R α /h=10 or 5, there are some vibration modes with w=0 for half-wave numbers (2,0) and (0,1). For R α /h=5


Figure 10: First benchmark, simply supported plate made of aluminium alloy with thickness ratio a/h=10. First five frequencies (with transverse displacement w≠0) via 2D FE solution (on the left) and via 3D exact solution (on the right).

shell, the FE code gives a frequency with w=0 for halfwave numbers (4,0). This frequency is also obtained by the

3D method. The 3D solution also gives another lower frequency for (4,0).


| 77

Figure 11: Fourth benchmark, simply supported cylinder made of aluminium alloy and titanium alloy with thickness ratio R α /h=10. First five frequencies (with transverse displacement w≠0) via 2D FE solution (on the left) and via 3D exact solution (on the right).

The eleventh benchmark considers a square sandwich plate with isotropic aluminium skins and an isotropic core in PVC. In this case the transverse anisotropy is very big because the elastic and mechanical properties of the core

are completely different from those of the skins. This feature is confirmed by the results given in Table 18. The errors given by the 2D FE code are acceptable only for thin plates (a/h=1000 or 100) but they are too large for thick


Figure 12: Fifth benchmark, simply supported plate made of aluminium alloy, titanium alloy and steel with thickness ratio a/h=10. First five frequencies (with transverse displacement w≠0) via 2D FE solution (on the left) and via 3D exact solution (on the right).

plates (a/h=20 or 10) for each frequency investigated (from the first to the tenth). The simple 2D kinematic model em-

ployed in the MSC Nastran code is not able to investigate thick and moderately thin sandwich structures with a big


| 79

Figure 13: Eight benchmark, simply supported composite cylinder 90∘ /0∘ /90∘ with thickness ratio R α /h=10. First five frequencies (with transverse displacement w≠0) via 2D FE solution (on the left) and via 3D exact solution (on the right).

transverse anisotropy. The 3D element of Nastran could give better results but is is not used in this work because the main aim is the comparison between 3D exact models and 2D FE models. This 2D element of Nastran gives wrong

frequencies that are smaller than 3D results, and this feature demonstrates how the code does not work in a correct way for such a benchmark. The plate is isotropic in the plane and for this reason the vibration behavior is a priori


Figure 14: Ninth benchmark, simply supported composite plate 90∘ /0∘ /90∘ /0∘ with thickness ratio a/h=10. First five frequencies (with transverse displacement w≠0) via 2D FE solution (on the left) and via 3D exact solution (on the right).

predictable. In this benchmark, the FE code does not give any in-plane frequency (see all the thickness ratios a/h).

The same considerations seen in Table 18 for the sandwich plate are confirmed in Table 19 for the sand-


81

Figure 15: Twelfth benchmark, simply supported sandwich cylinder embedding isotropic skins and PVC core with thickness ratio R α /h=10. First five frequencies (with transverse displacement w≠0) via 2D FE solution (on the left) and via 3D exact solution (on the right).

wich cylinder. FE results are acceptable for thin shells (R α /h=1000 and 100) but they are completely wrong for thick shells (R α /h=10 and 5). This feature is due to the big transverse anisotropy of this configuration. This trans-

verse anisotropy is confirmed by Figure 15 where the first five frequencies with w≠0 obtained by the 3D exact model exhibit an important zigzag behavior for displacement components through the thickness. The cylinder is


5 Conclusions This paper has proposed an exact three-dimensional model for the free vibration analysis of one-layered and multilayered plates, cylinders, cylindrical and spherical shell panels. Comparisons with a commercial finite element code have been proposed (for the cases of plates and cylinders) in order to explain the method used for such a comparison and to see the possible differences between an exact 3D solution and a numerical 2D solution. The exact 3D solution gives infinite vibration modes (for all the possible combinations of half-wave numbers (m,n)). A 2D FE code gives a finite number of vibration

250

n=1 n=2 n=3

f[Hz]

200

150

100

50

0 1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

6

7

8

9

10

m

2500

n=1 n=2 n=3

f[Hz]

2000

1500

1000

500

0 1

2

3

4

5 m

14000 n=1 n=2 n=3

12000 10000

f[Hz]

isotropic but its behavior in terms of vibration modes and half-wave numbers is not easily predictable because of the coupling given by the radius of curvature R α . In this last case there are not in-plane frequencies given by the FE code for thick cylinders. The FE results are completely wrong for thick shells as demonstrated by the values of frequency that are smaller than 3D values. Figures 16-19 show how the first (I) 3D frequency values change with the half-wave numbers (m,n) in the case of simply supported one-layered and two-layered isotropic plates and cylinders. Figure 16 gives frequencies for onelayered isotropic plates with thickness ratios a/h equal 1000, 100 and 10. For n=1, frequency increases with the increasing of half-wave number m. The same behavior is confirmed for the curves related to n=2 and n=3. When n increases the curves move to higher values of frequency. Figure 17 shows frequencies for one-layered isotropic cylinders with thickness ratios R α /h equal 1000, 100 and 10. The coupling between displacement components due to the curvature R α gives the minimum value of frequency for a circumferential half-wave number different from one (e.g., m=18 for longitudinal half-wave number n equals 1 and thickness ratio R α /h = 1000). When the longitudinal half-wave number n increases, the curves move to higher values of frequency and the minimum in frequency moves to higher values of m. When the thickness ratio of the cylinder decreases (thicker shells), the values of circumferential half-wave number m that give the minimum of frequency also decrease. For example, for longitudinal half-wave number n=1, there is a minimum in frequency for m=18 for thickness ratio R α /h = 1000, m=10 for R α /h = 100, and m=6 for R α /h = 10. The behavior of the twolayered structures (Figures 18-19) is the same already seen for the corresponding one-layered structures (Figures 1617), only the numerical values are different.

8000 6000 4000 2000 0 1

2

3

4

5 m

Figure 16: First benchmark, simply supported plate made of aluminium alloy with thickness ratios a/h=1000 (top), a/h=100 (middle) and a/h=10 (bottom). First (I) 3D frequencies versus half-wave numbers m (from 1 to 10) and n (from 1 to 3).

modes because it uses a finite number of degrees of freedom in the plane and in the thickness direction. A possible method to make a 3D versus 2D comparison is to calculate the frequencies via the 2D FE code and then to evaluate the 3D exact frequencies by means of the appropriate halfwave numbers (obtained via a correct visualization of the vibration modes via FE). It is obvious that the 3D analysis could give some frequencies that are missed by the 2D FE code, but this is not the aim of the paper. The paper tries to explain what could be the limitations of a commercial


83

350

16 14

n=1 n=2 n=3

12

n=1 n=2 n=3

300 250

f[Hz]

f[Hz]

10 8

200 150

6 100

4

50

2

0

0 10

15

20

25

30

35

40

1

45

2

3

4

5

6

7

8

9

10

6

7

8

9

10

6

7

8

9

10

m

m 3500

60 n=1 n=2 n=3

50

n=1 n=2 n=3

3000 2500

f[Hz]

f[Hz]

40

30

2000 1500

20 1000 10

500 0

0 5

10

15

20

25

1

30

2

3

4

5 m

m 12000 n=1 n=2 n=3

200

n=1 n=2 n=3

10000

8000 f[Hz]

f[Hz]

150 6000

100 4000 50 2000

0

0 0

5

10 m

15

20

Figure 17: Second benchmark, simply supported cylinder made of aluminium alloy with thickness ratios R α /h=1000 (top), R α /h=100 (middle) and R α /h=10 (bottom). First (I) 3D frequencies versus halfwave numbers m (values around that for minimum frequency) and n (from 1 to 3).

2D FE code. A typical 2D FE code uses a Reissner-Mindlin model for the approximation of displacement components through the thickness direction. Results in this paper show how this model employed by commercial FE codes could give errors for thick and moderately thick structures, complicated lamination sequences, higher order frequencies and particular vibration modes. In these cases, the use of 3D finite elements or refined 2D finite elements is mandatory.

1

2

3

4

5 m

Figure 18: Third benchmark, simply supported plate embedding aluminium alloy and titanium alloy with thickness ratios a/h=1000 (top), a/h=100 (middle) and a/h=10 (bottom). First (I) 3D frequencies versus half-wave numbers m (from 1 to 10) and n (from 1 to 3).

The behavior of frequency values and vibration modes versus imposed half-wave numbers has been investigated via the 3D exact model. The behavior is simple and easily predictable for plate structures because the increasing of m and/or n values gives bigger frequency values. In the case of cylinder geometry there is a coupling between the displacement components due to the curvature. For this reason, when longitudinal half-wave number n is imposed, the minimum of frequency is obtained for a value of circumferential half-wave number m differ-

84 | Salvatore Brischetto and Roberto Torre Table 8: First benchmark, simply supported plate made of aluminium alloy with thickness ratios a/h=1000, a/h=100, a/h=20 and a/h=10. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

18 16

n=1 n=2 n=3

14

f[Hz]

12 10

2D FE

8 6 4 2 0 10

15

20

25

30

35

40

45

m 70 n=1 n=2 n=3

60

f[Hz]

50 40

4.854 12.14 12.14 19.42 24.30 24.30 31.57 31.57 41.36 41.36

30 20 10 0 5

10

15

20

25

30

m 300 n=1 n=2 n=3

250

f[Hz]

200

150

100

50

0 0

5

10 m

15

20

Figure 19: Fourth benchmark, simply supported cylinder embedding aluminium alloy and titanium alloy with thickness ratios R α /h=1000 (top), R α /h=100 (middle) and R α /h=10 (bottom). First (I) 3D frequencies versus half-wave numbers m (values around that for minimum frequency) and n (from 1 to 3).

ent from 1. Such values decrease when the thickness ratio decreases (thicker cylinders). When the longitudinal halfwave number n increases the curves frequency versus m move to higher values of frequencies and the minimum in frequency moves to higher values of the circumferential half-wave number m. These last considerations are very similar for one-layered and multilayered structures.

48.53 121.3 121.3 194.0 242.7 242.7 315.1 315.1 412.6 412.6 241.0 596.6 596.6 944.7 1174 1174 1511 1511 1584 1584 472.3 1137 1137 1583 1583 1754 2146 2146 2239 2701 2701

3D

Mode a/h=1000 4.854 I 12.13 I 12.13 I 19.42 I 24.27 I 24.27 I 31.55 I 31.55 I 41.26 I 41.26 I a/h=100 48.52 I 121.2 I 121.2 I 193.9 I 242.3 I 242.3 I 314.8 I 314.8 I 411.4 I 411.4 I a/h=20 240.6 I 594.0 I 594.0 I 938.9 I 1164 I 1164 I 1496 I 1496 I 1583 II(w=0) 1583 II(w=0) a/h=10 469.5 I 1122 I 1122 I 1583 II(w=0) 1583 II(w=0) 1724 I 2102 I 2102 I 2239 II(w=0) 2640 I 2640 I

m,n

∆(%)

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

0.00 0.08 0.08 0.00 0.12 0.12 0.06 0.06 0.24 0.24

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

0.02 0.08 0.08 0.05 0.16 0.16 0.09 0.09 0.29 0.29

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 0,1 1,0

0.17 0.44 0.44 0.62 0.86 0.86 1.00 1.00 0.06 0.06

1,1 1,2 2,1 0,1 1,0 2,2 1,3 3,1 1,1 2,3 3,2

0.60 1.34 1.34 0.00 0.00 1.74 2.09 2.09 0.00 2.31 2.31


Table 9: Second benchmark, simply supported cylinder made of aluminium alloy with thickness ratios R α /h=1000, R α /h=100, R α /h=10 and R α /h=5. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

2D FE 3.123 3.153 3.392 3.405 3.813 4.047 4.335 4.942 5.219 5.620 9.557 10.41 11.28 12.84 16.20 16.68 19.41 20.23 20.34 20.65 28.74 30.21 41.45 49.07 50.41 57.94 61.37 63.20 64.22 76.03 35.84 47.58 49.89 50.42 75.03 79.40 79.41 79.87 85.97 86.05

3D

Mode R α /h=1000 3.123 I 3.154 I 3.391 I 3.407 I 3.816 I 4.045 I 4.339 I 4.947 I 5.216 I 5.625 I R α /h=100 9.558 I 10.41 I 11.28 I 12.85 I 16.22 I 16.68 I 19.45 I 20.26 I 20.34 I 20.73 I R α /h=10 28.72 I 30.19 I 41.42 I 49.08 I 50.42 I(w=0) 57.69 I 61.20 I 62.90 I 64.06 I 75.72 I R α /h=5 35.85 I 47.66 I 49.91 I 50.48 I(w=0) 74.93 I 78.47 I 79.16 I(w=0) 79.63 II 85.35 I 84.74 I

85

Table 10: Third benchmark, simply supported plate made of aluminium alloy and titanium alloy with thickness ratios a/h=1000, a/h=100, a/h=20 and a/h=10. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

m,n

∆(%)

2D FE

18,1 20,1 16,1 22,1 24,1 14,1 26,1 28,1 12,1 30,1

0.00 -0.03 0.03 -0.06 -0.08 0.05 -0.09 -0.10 0.06 -0.09

5.409 13.53 13.53 21.64 27.08 27.08 35.18 35.18 46.09 46.09

10,1 12,1 8,1 14,1 16,1 6,1 14,2 18,1 12,2 16,2

-0.01 0.00 0.00 -0.08 -0.12 0.00 -0.21 -0.15 0.00 -0.39

54.08 135.2 135.2 216.2 270.4 270.4 351.2 351.2 459.8 459.8

6,1 4,1 8,1 2,1 2,0 6,2 10,1 4,2 8,2 2,2

0.07 0.07 0.07 -0.02 -0.02 0.43 0.28 0.48 0.25 0.41

268.6 664.8 664.8 1053 1309 1309 1684 1684 1797 1797

4,1 6,1 2,1 2,0 8,1 4,2 0,1 0,1 6,2 2,2

-0.03 -0.17 -0.04 -0.12 0.13 1.18 0.32 0.30 0.73 1.55

526.3 1266 1266 1797 1797 1955 2391 2391 2541 3010 3010

3D

Mode a/h=1000 5.409 I 13.52 I 13.52 I 21.64 I 27.05 I 27.05 I 35.16 I 35.16 I 45.98 I 45.98 I a/h=100 54.07 I 135.1 I 135.1 I 216.1 I 270.0 I 270.0 I 350.8 I 350.8 I 458.4 I 458.4 I a/h=20 268.1 I 661.9 I 661.9 I 1046 I 1298 I 1298 I 1668 I 1668 I 1797 II(w=0) 1797 II(w=0) a/h=10 523.1 I 1250 I 1250 I 1797 II(w=0) 1797 II(w=0) 1922 I 2344 I 2344 I 2540 II(w=0) 2945 I 2945 I

m,n

∆(%)

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

0.00 0.07 0.07 0.00 0.11 0.11 0.06 0.06 0.24 0.24

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

0.02 0.07 0.07 0.05 0.15 0.15 0.11 0.11 0.30 0.30

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 0,1 1,0

0.19 0.44 0.44 0.67 0.85 0.85 0.96 0.96 0.00 0.00

1,1 1,2 2,1 0,1 1,0 2,2 1,3 3,1 1,1 2,3 3,2

0.61 1.28 1.28 0.00 0.00 1.72 2.00 2.00 0.04 2.21 2.21

86 | Salvatore Brischetto and Roberto Torre Table 11: Fourth benchmark, simply supported cylinder made of aluminium alloy and titanium alloy with thickness ratios R α /h=1000, R α /h=100, R α /h = 10 and R α /h=5. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

2D FE 3.518 3.538 3.810 3.835 4.259 4.585 4.837 5.511 5.920 6.265 10.76 11.64 12.78 14.31 18.05 18.93 21.81 22.53 22.96 23.12 32.07 34.21 45.89 55.81 57.05 65.01 67.80 71.37 71.52 86.03 40.33 52.56 56.83 56.91 82.46 89.08 90.19 91.56 95.68 96.93

3D

Mode R α /h=1000 3.518 I 3.540 I 3.812 I 3.834 I 4.263 I 4.583 I 4.841 I 5.516 I 5.917 I 6.271 I R α /h=100 10.76 I 11.65 I 12.77 I 14.33 I 18.07 I 18.92 I 21.86 I 22.56 I 22.97 I 23.20 I R α /h=10 32.11 I 34.21 I 45.94 I 55.77 I 57.07 I(w=0) 64.74 I 67.78 I 70.94 I 71.41 I 85.43 I R α /h=5 40.39 I 52.79 I 56.76 I 56.99 I(w=0) 82.61 I 87.89 I 89.32 I 91.32 II(w=0) 95.04 I 95.00 I

m,n

Table 12: Fifth benchmark, simply supported plate made of aluminium alloy, titanium alloy and steel with thickness ratios a/h=1000, a/h=100, a/h=20 and a/h=10. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

∆(%) 2D FE

18,1 20,1 22,1 16,1 24,1 14,1 26,1 28,1 12,1 30,1

0.00 -0.06 -0.05 0.03 -0.09 0.04 -0.08 -0.09 0.05 -0.10

10,1 12,1 8,1 14,1 16,1 6,1 14,2 18,1 12,2 16,2

0.00 -0.09 0.08 -0.14 -0.11 0.05 -0.23 -0.13 -0.04 -0.34

6,1 4,1 8,1 2,1 2,0 6,2 10,1 4,2 8,2 2,2

-0.12 0.00 -0.11 0.07 -0.03 0.42 0.03 0.61 0.15 0.70

4,1 6,1 2,1 2,0 8,1 4,2 0,1 0,1 6,2 2,2

-0.15 -0.44 0.12 -0.14 -0.18 1.35 0.97 0.26 0.67 2.03

4.914 12.29 12.29 19.66 24.60 24.60 31.96 31.96 41.86 41.86 49.12 122.8 122.8 196.3 245.6 245.6 318.9 318.9 417.6 417.6 243.9 603.5 603.5 955.4 1187 1187 1527 1527 1687 1687 477.6 1148 1148 1685 1685 1771 2165 2165 2724 2724

3D

Mode a/h=1000 4.913 I 12.28 I 12.28 I 19.65 I 24.57 I 24.57 I 31.94 I 31.94 I 41.76 I 41.76 I a/h=100 49.12 I 122.7 I 122.7 I 196.3 I 245.2 I 245.2 I 318.6 I 318.6 I 416.4 I 416.4 I a/h=20 243.6 I 601.3 I 601.3 I 950.5 I 1179 I 1179 I 1515 I 1515 I 1687 II(w=0) 1687 II(w=0) a/h=10 475.2 I 1136 I 1136 I 1687 II(w=0) 1687 II(w=0) 1745 I 2129 I 2129 I 2673 I 2673 I

m,n

∆(%)

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

0.02 0.08 0.08 0.05 0.12 0.12 0.06 0.06 0.24 0.24

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

0.00 0.08 0.08 0.00 0.16 0.16 0.09 0.09 0.29 0.29

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 0,1 1,0

0.12 0.37 0.37 0.51 0.68 0.68 0.79 0.79 0.00 0.00

1,1 1,2 2,1 0,1 1,0 2,2 1,3 3,1 2,3 3,2

0.50 1.06 1.06 -0.12 -0.12 1.49 1.69 1.69 1.91 1.91


Table 13: Sixth benchmark, simply supported cylinder made of aluminium alloy, titanium alloy and steel with thickness ratios R α /h=1000, R α /h=100, R α /h = 10 and R α /h=5. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

Table 14: Seventh benchmark, simply supported composite plate 90∘ /0∘ /90∘ with thickness ratios a/h=1000, a/h=100, a/h=20 and a/h=10. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

2D FE 2D FE 3.256 3.259 3.487 3.575 3.886 4.292 4.404 5.013 5.551 5.695 9.952 10.65 11.94 13.03 16.40 17.76 20.13 20.45 21.18 21.36 29.32 31.98 41.40 52.49 53.38 59.89 60.98 65.08 66.43 80.25 37.33 47.33 53.08 53.51 73.68 81.95 84.12 86.93 87.36 89.80

3D

Mode R α /h=1000 3.257 I 3.259 I 3.489 I 3.574 I 3.889 I 4.290 I 4.408 I 5.018 I 5.548 I 5.701 I R α /h=100 9.962 I 10.67 I 11.94 I 13.06 I 16.43 I 17.76 I 20.18 I 20.50 I 21.27 I 21.38 I R α /h=10 29.47 I 31.99 I 41.68 I 52.17 I 53.07 I(w=0) 59.77 I 61.35 I 65.26 I 66.02 I 80.66 I R α /h=5 37.41 I 47.84 I 52.50 I(w=0) 52.88 I 74.43 I 80.89 I 83.04 I 86.66 I 84.05 II(w=0) 87.80 I

m,n

∆(%)

20,1 18,1 22,1 16,1 24,1 14,1 26,1 28,1 12,1 30,1

-0.03 0.00 -0.06 0.03 -0.08 0.05 -0.09 -0.10 0.05 -0.10

10,1 12,1 8,1 14,1 16,1 6,1 14,2 18,1 16,2 12,2

-0.10 -0.19 0.00 -0.23 -0.18 0.00 -0.25 -0.24 -0.42 -0.09

6,1 4,1 8,1 2,1 2,0 6,2 10,1 8,2 4,2 10,2

-0.51 -0.03 -0.67 0.61 0.58 0.20 -0.60 -0.28 0.62 -0.51

4,1 6,1 2,0 2,1 8,1 4,2 0,1 6,2 0,1 2,2

-0.21 -1.07 1.10 1.19 -1.01 1.31 1.30 0.31 3.94 2.28

4.697 7.898 14.45 16.76 18.79 23.52 24.10 31.63 36.71 37.09 46.90 78.87 144.2 166.6 186.9 234.0 240.2 314.6 365.4 366.1 227.1 382.3 688.9 738.1 832.9 940.0 1051 1119 1407 1443 416.3 702.7 940.0 1145 1222 1315 1686 1880 1882 1959

3D

Mode a/h=1000 4.696 I 7.895 I 14.43 I 16.74 I 18.78 I 23.50 I 24.04 I 31.58 I 36.57 I 37.03 I a/h=100 46.89 I 78.83 I 143.9 I 166.5 I 186.8 I 233.7 I 239.5 I 313.9 I 363.7 I 365.5 I a/h=20 226.7 I 380.9 I 684.1 I 736.1 I 829.1 I 939.9 II(w=0) 1043 I 1107 I 1393 I 1440 I a/h=10 414.5 I 696.3 I 939.9 II(w=0) 1144 I 1205 I 1308 I 1668 I 1880 II(w=0) 1851 I 1977 I

m,n

∆(%)

1,1 2,1 3,1 1,2 2,2 3,2 4,1 4,2 5,1 1,3

0.02 0.04 0.14 0.12 0.05 0.09 0.25 0.16 0.38 0.16

1,1 2,1 3,1 1,2 2,2 3,2 4,1 4,2 5,1 1,3

0.02 0.05 0.21 0.06 0.05 0.13 0.29 0.22 0.47 0.16

1,1 2,1 3,1 1,2 2,2 0,1 3,2 4,1 4,2 1,3

0.18 0.37 0.70 0.27 0.46 0.01 0.77 1.08 1.00 0.21

1,1 2,1 0,1 1,2 3,1 2,2 3,2 0,2 4,1 1,3

0.43 0.92 0.01 0.09 1.41 0.53 1.08 0.00 1.67 -0.91

87

88 | Salvatore Brischetto and Roberto Torre Table 15: Eighth benchmark, simply supported composite cylinder 90∘ /0∘ /90∘ with thickness ratios R α /h=1000, R α /h=100, R α /h = 10 and R α /h=5. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

2D FE 2.748 2.755 2.889 2.894 3.123 3.218 3.434 3.748 3.807 4.231 7.722 8.161 8.407 9.919 9.949 12.01 13.44 14.53 14.58 14.60 20.79 22.34 26.29 29.92 32.61 36.49 47.01 47.27 49.08 49.73 27.54 29.93 32.14 34.47 45.22 47.14 59.84 63.20 65.21 65.99

3D

Mode R α /h=1000 2.743 I 2.754 I 2.887 I 2.887 I 3.122 I 3.209 I 3.435 I 3.738 I 3.809 I 4.233 I R α /h=100 7.717 I 8.151 I 8.406 I 9.923 I 9.938 I 12.02 I 13.43 I 14.54 I 14.57 I 14.62 I R α /h=10 20.81 I 22.38 I 26.27 I 29.93 I(w=0) 32.64 I 36.39 I 46.99 I(w=0) 47.28 I 49.09 I 49.51 I R α /h=5 27.65 I 29.97 I(w=0) 32.17 I 34.56 I 45.11 I 46.99 I(w=0) 59.93 II(w=0) 62.90 I 65.39 I 66.12 I

Table 16: Ninth benchmark, simply supported composite plate 90∘ /0∘ /90∘ /0∘ with thickness ratios a/h=1000, a/h=100, a/h=20 and a/h=10. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

m,n

∆(%)

2D FE

22,1 24,1 26,1 20,1 28,1 18,1 30,1 16,1 32,1 34,1

0.18 0.04 0.07 0.24 0.03 0.28 -0.03 0.27 -0.05 -0.05

4.412 12.24 12.24 17.65 26.24 26.24 30.04 30.04 39.75 46.04

12,1 10,1 14,1 16,1 8,1 18,1 6,1 20,1 14,2 16,2

0.06 0.12 0.01 -0.04 0.11 -0.08 0.07 -0.07 0.07 -0.14

44.07 122.0 122.0 175.8 260.4 260.4 298.1 298.1 393.8 454.3

6,1 4,1 8,1 2,0 2,1 10,1 0,1 6,2 8,2 12,1

-0.10 -0.18 0.08 -0.03 -0.09 0.27 0.04 -0.02 -0.02 0.44

215.1 566.1 566.1 803.0 940.0 940.0 1116 1116 1277 1277

4,1 2,0 6,1 2,1 8,1 0,1 4,0 10,1 4,2 6,2

-0.40 -0.13 -0.09 -0.26 0.24 0.32 -0.15 0.48 -0.27 0.20

401.3 940.0 940.0 947.2 947.2 1305 1655 1655 1880 1880

3D

Mode a/h=1000 4.411 I 12.24 I 12.24 I 17.64 I 26.20 I 26.20 I 30.00 I 30.00 I 39.69 I 45.92 I a/h=100 44.06 I 121.9 I 121.9 I 175.7 I 259.9 I 259.9 I 297.7 I 297.7 I 393.3 I 452.9 I a/h=20 215.0 I 565.1 I 565.1 I 802.5 I 939.9 II(w=0) 939.9 II(w=0) 1112 I 1112 I 1276 I 1276 I a/h=10 401.3 I 939.9 II(w=0) 939.9 II(w=0) 947.9 I 947.9 I 1312 I 1665 I 1665 I 1880 II(w=0) 1880 II(w=0)

m,n

∆(%)

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 3,3 4,1

0.02 0.00 0.00 0.06 0.15 0.15 0.13 0.13 0.15 0.26

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 3,3 4,1

0.02 0.08 0.08 0.06 0.19 0.19 0.13 0.13 0.13 0.31

1,1 1,2 2,1 2,2 0,1 1,0 1,3 3,1 2,3 3,2

0.05 0.18 0.18 0.06 0.01 0.01 0.36 0.36 0.08 0.08

1,1 0,1 1,0 1,2 2,1 2,2 1,3 3,1 0,2 2,0

0.00 0.01 0.01 -0.07 -0.07 -0.53 -0.60 -0.60 0.00 0.00


Table 17: Tenth benchmark, simply supported composite cylinder 90∘ /0∘ /90∘ /0∘ with thickness ratios R α /h=1000, R α /h=100, R α /h = 10 and R α /h=5. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

2D FE 3.558 3.624 3.788 3.924 4.368 4.398 5.001 5.364 5.705 6.492 9.679 10.19 11.26 13.22 14.25 17.24 18.16 18.19 18.35 19.55 21.85 25.37 29.92 32.04 38.95 43.95 46.69 47.01 51.63 57.21 26.73 29.93 32.81 38.93 47.14 58.01 59.57 59.84 62.06 68.88

3D

Mode R α /h=1000 3.552 I 3.619 I 3.780 I 3.920 I 4.358 I 4.395 I 4.999 I 5.353 I 5.704 I 6.491 I R α /h=100 9.665 I 10.17 I 11.24 I 13.20 I 14.24 I 17.22 I 18.15 I 18.18 I 18.33 I 19.54 I R α /h=10 21.78 I 25.20 I 29.93 I(w=0) 32.03 I 38.70 I 43.72 I 46.47 I 46.99 I(w=0) 51.44 I 56.89 I R α /h=5 26.54 I 29.97 I(w=0) 32.78 I 38.63 I 46.99 I(w=0) 57.54 I 59.36 I 59.93 II(w=0) 61.81 I 68.44 I

m,n

Table 18: Eleventh benchmark, simply supported sandwich plate embedding isotropic skins and PVC core with thickness ratios a/h=1000, a/h=100, a/h=20 and a/h=10. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

∆(%) 2D FE

18,1 20,1 16,1 22,1 14,1 24,1 26,1 12,1 28,1 30,1

0.17 0.14 0.21 0.10 0.23 0.07 0.04 0.20 0.02 0.01

10,1 8,1 12,1 6,1 14,1 12,2 14,2 16,1 10,2 4,1

0.14 0.20 0.18 0.15 0.07 0.12 0.05 0.05 0.11 0.05

4,1 6,1 2,0 2,1 8,1 6,2 4,2 0,1 8,2 10,1

0.32 0.67 -0.03 0.03 0.65 0.53 0.47 0.04 0.37 0.56

4,1 2,0 2,1 6,1 0,1 4,2 8,1 4,0 6,2 2,2

0.72 -0.13 0.09 0.78 0.32 0.82 0.35 -0.15 0.40 0.64

6.705 16.75 16.75 26.77 33.47 33.47 43.43 43.43 56.82 56.82 62.79 144.2 144.2 214.4 256.8 256.8 314.7 314.7 384.6 384.6 157.3 267.0 267.0 344.4 387.7 387.7 444.8 444.8 511.5 511.5 172.1 277.8 277.8 353.2 395.7 395.7 451.9 451.9 517.7 517.7

3D

Mode a/h=1000 6.704 I 16.74 I 16.74 I 26.76 I 33.43 I 33.43 I 43.41 I 43.41 I 56.69 I 56.69 I a/h=100 62.79 I 144.2 I 144.2 I 214.7 I 257.3 I 257.3 I 315.9 I 315.9 I 386.4 I 386.4 I a/h=20 161.6 I 287.6 I 287.6 I 387.5 I 448.1 I 448.1 I 534.2 I 534.2 I 643.2 I 643.2 I a/h=10 193.7 I 361.1 I 361.1 I 513.2 I 611.6 I 611.6 I 756.7 I 756.7 I 947.5 I 947.5 I

m,n

∆(%)

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

0.01 0.06 0.06 0.04 0.12 0.12 0.05 0.05 0.23 0.23

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

0.00 0.00 0.00 -0.14 -0.19 -0.19 -0.38 -0.38 -0.47 -0.47

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

-2.66 -7.16 -7.16 -11.1 -13.5 -13.5 -16.7 -16.7 -20.5 -20.5

1,1 1,2 2,1 2,2 1,3 3,1 2,3 3,2 1,4 4,1

-11.1 -23.1 -23.1 -31.2 -35.3 -35.3 -40.3 -40.3 -45.4 -45.4

89

90 | Salvatore Brischetto and Roberto Torre Table 19: Twelfth benchmark, simply supported sandwich cylinder embedding isotropic skins and PVC core with thickness ratios R α /h=1000, R α /h=100, R α /h = 10 and R α /h=5. First ten frequencies in Hz, comparison between the present 3D exact solution and the 2D FE solution via Nastran. The error is calculated as ∆(%) = FE−3D × 100. 3D

2D FE 3.642 3.687 3.919 4.171 4.420 5.075 5.233 5.846 6.709 7.122 10.68 11.68 12.18 14.94 16.63 18.32 20.96 21.40 22.04 22.37 18.30 19.63 20.47 23.85 27.60 28.41 30.71 31.26 31.48 32.15 18.65 19.91 20.81 24.15 27.87 28.57 30.96 31.50 31.72 32.39

3D

Mode m,n R α /h=1000 3.644 I 18,1 3.687 I 16,1 3.922 I 20,1 4.170 I 14,1 4.425 I 22,1 5.082 I 24,1 5.230 I 12,1 5.855 I 26,1 6.721 I 28,1 7.119 I 10,1 R α /h=100 10.69 I 10,1 11.68 I 8,1 12.19 I 12,1 14.98 I 14,1 16.62 I 6,1 18.40 I 16,1 21.03 I 14,2 21.41 I 12,2 22.17 I 18,1 22.51 I 16,2 R α /h=10 19.79 I 8,1 20.04 I 6,1 23.70 I 10,1 29.48 I 12,1 36.36 I 14,1 28.46 I 4,1 37.10 I 12,2 34.84 I 10,2 44.15 I 16,1 42.10 I 14,2 R α /h=5 24.68 I 8,1 21.93 I 6,1 32.50 I 10,1 42.96 I 12,1 55.27 I 14,1 28.83 I 4,1 52.38 I 12,2 44.69 I 10,2 69.01 I 16,1 63.30 I 14,2

∆(%) -0.05 0.00 -0.08 0.02 -0.11 -0.14 0.06 -0.15 -0.18 0.04 -0.09 0.00 -0.08 -0.27 0.06 -0.43 -0.33 -0.05 -0.59 -0.62 -7.53 -2.05 -13.6 -19.1 -24,1 -0.18 -17.2 -10.3 -28.7 -23.6 -24.4 -9.21 -36.0 -41.4 -49.6 -0.90 -40.9 -29.5 -54.0 -48.8


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Research Article

Open Access

Nicholas Fantuzzi*

New insights into the strong formulation finite element method for solving elastostatic and elastodynamic problems Abstract: This present paper has a complete and homogeneous presentation of plane stress and plane strain problems using the Strong Formulation Finite Element Method (SFEM). In particular, a greater emphasis is given to the numerical implementation of the governing and boundary conditions of the partial differential system of equations. The paper’s focus is on numerical stability and accuracy related to elastostatic and elastodynamic problems. In the engineering literature, results are mainly reported for isotropic and homogeneous structures. In this paper, a composite structure is investigated. The SFEM solution is compared to the ones obtained using commercial finite element codes. Generally, the SFEM observes fast accuracy and all the results are in very good agreement with the ones presented in literature. Keywords: Elastostatic Problem; Elastodynamic Problem; Composite Structure; Strong Formulation Finite Element Method; Differential Quadrature Method. DOI 10.2478/cls-2014-0005 Received September 24, 2014 ; accepted October 30, 2014

1 Introduction The elastostatic and elastodynamic problems [1–3] for engineering applications constitute a widely known approach for the study of structural components, laboratory tests, composite materials, composite structures, and so on. Some of the most studied problems of the present class are two-dimensional (2D) models, which find application in several engineering fields, such as solid mechanics, dynamics of structures, fracture mechanics, wave propagation, seismic stability and rock mechanics. It should be

*Corresponding Author: Nicholas Fantuzzi: DICAM - Department, School of Engineering and Architecture, University of Bologna, Italy; E-mail: [email protected]; http://software. dicam.unibo.it/diqumaspab-project

cited that the mechanics of static and dynamic composite systems has been studied by several researches. Thus, several books have been hitherto published [4–10]. Problems involving the theory of elasticity can be solved analytically only when simple geometries occur. Thus, the most common way of treating these problems is to make numerical models. Generally the elastic behavior of homogeneous and composite solids is studied using the co-called Boundary Value Problems (BVPs). These problems are governed by partial differential equations where the displacement parameters (the unknowns of the model) are the physical displacements in the three-dimensional (3D) space. Mathematically speaking, all the quantities involved in the formulation must be smooth all over the definition domain. However, discontinuities can occur in practical applications. The most common discontinuations are related to the material and/or the geometry. The former is connected to composite materials such as reinforced fibers immersed into a matrix and the latter is directly connected to cracks and slits. When these cases come out, some mathematical tricks have to be introduced. The present approach has its roots in the Differential Quadrature (DQ) method [11, 12], which have the peculiarity of being very accurate using a small number of degrees of freedom when compared to classic numerical approaches, such as the Finite Element Method (FEM). It should be mentioned that DQ method belongs to the big family of the Methods of Weighted Residuals (MWRs) [13], since the functional approximation using weighting coefficients must minimize the error between the given function and its approximation. Furthermore, DQ method can be seen as a general form of the so-called collocation methods or spectral collocation [14, 15]. The original pseudospectral collocation (DQ method) presented by Bellman has been improved by the approach proposed by Quan [16] and by Quan and Chang [17, 18]. All the advancements in the DQ method has been reviewed by Bert and Malik [19]. The DQ method in its original version had some numerical instabilities, when the number of grid points is large, since

© 2014 Nicholas Fantuzzi, licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

94 | Nicholas Fantuzzi the weighting coefficient matrix becomes ill-conditioned due to the chosen basis functions (power basis). Following the ideas by Quan and Chang [17, 18] a generalized version of the DQ method, known as Generalized Differential Quadrature (GDQ) method, had been developed by Shu [20] and by Shu and Richards [21, 22]. Nowadays, the GDQ method is the most widely used in literature due to its stability and reliability when applied to different engineering problems. The DQ method had demonstrated to be a useful tool for solving structural components [23–32]. Subsequently, the GDQ method showed the same properties, in fact several numerical applications related to structural mechanics and solid mechanics can be found in [33–84]. The GDQ method turned out to be really useful for studying structural components and the mechanics of composite structures, due to the easy implementation of the governing system of equations for one-dimensional (1D) and twodimensional (2D) systems. As it is well-known the GDQ method solves the strong formulation of the differential problem. This implies that the boundary conditions must be enforced a posteriori. On the contrary using a weak (variational) formulation they are a priori satisfied. The GDQ solving system contains algebraic equations, which come from the governing equations (domain equations) and from the boundary equations (boundary conditions). Despite what someone might think, strong formulation approaches can be used when irregularities are present. In literature two different approaches were presented. The first one is related to the so-called mesh-less methods in which scattered points are chosen in the physical domain [85–109]. The second one is based on the domain decomposition, such as the one used in the classic FEM. Contrarily to the FEM, domain decomposition methods solve the strong formulation inside each element instead of the weak one. Furthermore, the continuity conditions between two elements must be enforced in order to satisfy the connectivity among them. Hence, the accuracy of this approach depends on the predefined mesh used for the discretization other than the approximation inside each element. Historically speaking, researchers divided a regular domain using regular elements, this approach was termed multi-domain differential quadrature [110–115]. Inserting the mapping technique in the previous works the differential quadrature element method (DQEM) or SFEM can be defined [116–125]. The author presented a preliminary development of the present work in his PhD Thesis [126]. Some other papers followed about the vibration of arbitrary shaped laminated composite plates [127], some comparisons between SFEM and the cell method [128–130]. Moreover, the stress and strain recovery has been published on the static analysis of arbi-

trary shaped plates [131, 132]. A review article about the vibration problem of composite membranes has been presented also [133]. A comparison between the SFEM based on GDQ and RBF methods was presented in [134]. The free vibration behavior of arbitrary shaped functionally graded plates was presented in [135]. A first review about the stability and accuracy for the static and free vibration analysis of SFEM was illustrated in [136]. Nevertheless, a complete survey about SFEM and related methods was recently published in [137]. At a first instance mesh-less methods are better since they do not depend on the mesh used. Mesh-less methods generally use local basis functions, which depend on the distance between the points. In fact they are also called radial basis functions. These local functions depend on a parameter (shape parameter) which influences the accuracy of the solution. On the contrary SFEM, based on DQ method, uses global higher-order basis functions in each element, which have a high accuracy, fast convergence and do not depend on any shape parameter. The present manuscript presents the SFEM for 2D plane stress and strain problems. After a brief introduction on the 2D elasticity, the algebraic equations in extended and in matrix forms are presented. In particular, a wide focus is given to the application of the mapping technique to a single element. Furthermore, some new details about the boundary conditions and inter-element connectivity conditions are given. For the sake of clarity, it is given to the reader a graphical representation of the boundary conditions for a general SFEM mesh. This helps the reader to better understand all the different boundary conditions that should be implemented. Subsequently, a section is dedicated to the assemblage of the solving system in discrete form for both the static and free vibration case. Some numerical tests are performed using some reference solutions in literature and new numerical applications are proposed for further studies on the subject.

2 Preliminary remarks A wide number of applications have been presented throughout several years about structures which are in deformation or tension state. As it is well-known these states are better known as plane strain and plane stress. For instance, if a thin plate is loaded by forces applied on its boundaries, parallel to the plane of the plate, this state is called plane stress, whereas when the dimensions of the same body is very large when compared to the other two dimensions, this is termed plane strain. The plane states are

New insights into the strong formulation finite element method | 95

particular cases of the 3D theory of elasticity, when some assumptions are made. Thus, the three equilibrium equations for the 3D solid are the starting point of the present formulation ∂σ x ∂τ xy ∂τ xz + + + fx = 0 ∂x ∂y ∂z ∂τ xy ∂σ y ∂τ yz + + + fy = 0 ∂x ∂y ∂z ∂τ xz ∂τ yz ∂σ z + + + fz = 0 ∂x ∂y ∂z

(1)

For the plane case, it can be assumed that the stress components do not depend on z (∂/∂z = 0). For this reason, the remaining components do not vary through the thickness and they are functions of the in-plane Cartesian coordinates only σ x (x, y), σ y (x, y), τ xy (x, y). Considering all the previous assumptions the equilibrium equations (1) become ∂σ x ∂τ xy + + fx = 0 ∂x ∂y (2) ∂τ xy ∂σ y + + fy = 0 ∂x ∂y

2.1 Plane strain state Consider a prismatic solid in which the plane state occurs in a plane parallel to the x-y one. Hence, according to the state plane hypotheses, the not zero strain components are ε x (x, y), ε y (x, y), 𝛾xy (x, y), since by definition w = 0, ∂u/∂z = 0 and ∂v/∂z = 0, where u, v, w are the displacements along the three Cartesian axes respectively. The well-known kinematic relations are εx =

∂u x , ∂x

εy =

∂u y , ∂y

𝛾xy =

∂u x ∂u y + ∂y ∂x

(3)

The stress components can be found by introducing the inverse Hooke’s laws σ x = (2G + λ) ε x + λε y , σ y = λε x + (2G + λ) ε y , τ xy = G𝛾xy , σ z = λε x + λε y

(4)

It should be pointed out that in equation (4) the normal stress σ z is not negligible. Thus, a plane strain is not also a plane stress. The elastic constants G, λ in equation ((4)) are the shear modulus and Lamé constant. These constants are related to the better known elastic (Young’s) modulus and Poisson’s ratio with the following relations E=

G (3λ + 2G) , λ+G

ν=

λ 2 (λ + G)

(5)

Introducing the kinematic expressions (3) into the inverse Hooke’s laws (4), it is obtained (excluding the definition

for the normal stress) ∂u y ∂u x +λ , ∂x ∂y ∂u y ∂u x σy = λ + (2G + λ) , ∂x (︂ )︂∂y ∂u x ∂u y + τ xy = G ∂y ∂x σ x = (2G + λ)

(6)

Finally, the governing equations in terms of displacements can be found using (6) into (2) as follows ∂2 u ∂2 u ∂2 v + fx = 0 + G + λ + G ( ) ∂x∂y ∂x2 ∂y2 2 2 2 ∂ v ∂ u ∂ v + G 2 + (2G + λ) 2 + f y = 0 (λ + G) ∂x∂y ∂x ∂y (2G + λ)

(7)

Equations (7) correspond to the elastostatic case for the plane strain case. However, if the elastodynamic problem has to be solved, the inertia forces should be added to the governing equations as ∂2 u ∂2 u ∂2 v ∂2 u + G + λ + G + f = ρ ( ) x ∂x∂y ∂x2 ∂y2 ∂t2 ∂2 v ∂2 u ∂2 v ∂2 v + G 2 + (2G + λ) 2 + f y = ρ 2 (λ + G) ∂x∂y ∂x ∂y ∂t (2G + λ)

(8) where ρ represents the material density. If the free vibration problem is wanted to be studied, the body forces should be set zero f x = f y = 0. In the present paper the free vibration problem is investigated, thus the dynamic solution is found in the form u (x, y, t) = Φ x (x, y) e iωt v (x, y, t) = Φ y (x, y) e iωt

(9)

Substituting equation (9) into equation (8) one yields ∂2 Φ x ∂2 Φ x +G + (λ + G) 2 ∂x ∂y2 2 +ρω Φ x u = 0 ∂2 Φ y ∂2 Φ x +G + (2G + λ) (λ + G) ∂x∂y ∂x2 2 +ρω Φ y v = 0 (2G + λ)

∂2 Φ y + ∂x∂y ∂2 Φ y + ∂y2

(10)

where Φ x , Φ y are the mode shapes related to the in-plane displacements u, v and ω represents the natural circular frequencies of the structure under study.

2.2 Plane stress state In order to solve the plane stress problem, it is possible to follow the same mathematical developments of the strain case above. Thus the same systems of equations (7) and (8) 2Gλ or λ* = can be found substituting λ → λ* where λ* = 2G+λ Eν . In the following all the equations are reported using 1−ν2 G, λ for the strain case, taking into account that the stress case can be found using G, λ* .

96 | Nicholas Fantuzzi

2.3 Boundary conditions In order to solve any partial differential system of equations, boundary conditions must be introduced. According to the theory of elasticity [1–3] the boundary conditions for a 3D solid considering the state plane assumptions are σ x n x + τ xy n y + p x = 0 τ xy n x + σ y n y + p y = 0

(11)

where n x , n y are the direction cosines of the outward unit normal of the current edge, written with respect to the outer Cartesian reference system. However, it happens that the edge is generally oriented, so that a transformation matrix should be applied to the stress components in order to evaluate the normal and tangential stresses of that edge. Furthermore, the same occurs for the in-plane displacement parameters. For these reasons the following relations can be written according to the literature [1–3] un = u nx + v ny ut = v nx − u ny σ n = σ x n2x + σ y n2y + 2τ xy n x n y (︀ )︀ τ nt = (σ y − σ x ) n x n y + τ xy n2x − n2y

3.1 Derivative approximation and mapping (12)

where u n , u t are the normal and tangential displacements and σ n , τ nt are the normal and shear stresses at the edge. For the application of the SFEM all the equations should be written as functions of the displacement parameters as in the following un = σn = τ nt =

u n + v ny , ut = v nx − u ny )︀ (︀ x ∂u (2G + λ) n2x + λn2y ∂u ∂x + 2Gn x n y)︀∂y + (︀ ∂v ∂v +2Gn x n y ∂x + (2G + λ) n2y + λn2x ∂y (︀ 2 )︀ 2 ∂u ∂u −2Gn x n y ∂x + G n x − n y ∂y + (︀ )︀ ∂v ∂v +G n2x − n2y ∂x + 2Gn x n y ∂y

governing equations are discretized inside each element, whereas continuity conditions are used in order to connect all of them. In the present section the discrete form of all these equations are presented in order to give a general overview on the problem. The 2D DQ implementation has been already presented by several authors in literature [33–84]. Summarizing, the DQ method can approximate a derivative along a direction using several grid points located along two directions in the reference coordinate system. However, the present approach is general and the DQ method is applied at the master element level as in the standard FEM where the Gauss quadrature is applied in the master element. For this reason all the following formulae are referred to the local master element reference system ξ -η. At the end it will be shown how to solve the differential problems in Cartesian coordinates (7) and (8) using the following notation.

(13)

As any other collocation method, in order to accurately approximate the derivative, two grid locations have to be set, one along ξ and the other along η. It is recalled that the reference element or master element or parent element belongs to the unit space as already presented in the previous works [126–135]. The number of points, which defines these collocations, is indicated by N and M respectively. The weighting coefficients can be evaluated afterwards using the classic formulae provided by the DQ and the GDQ approaches. In this way two matrices, containing the weighting coefficients of all the points of the master element, are carried out. So that, the following derivatives can be approximated ⃒

∂(n) f (ξ ,η) ⃒ ∂ξ (n) ⃒ξ =ξ i

η=η j

3 Discretized forms It is recalled that the SFEM is based on the DQ method which discretizes the derivative of a function as a weighted linear sum of some functional values. The main advantage of the DQ method is the possibility of having extremely high accuracy. However, its principal drawback is that it cannot deal with arbitrarily shaped domains and cannot have discontinuities. In order to overcome these difficulties a domain decomposition approach is followed. Thus, the whole problem is numerically solved in several domains or elements, which compose the global geometry. Subsequently, all these elements are connected among them in order to achieve the global solution. Hence, the

=

N ∑︀ k=1

ς ξi,k(n) f k,j =

= ς ξi,1(n) f1,j + ς ξi,2(n) f2,j + · · · + ς ξiN(n) f N,j ⃒ M ∑︀ ∂(m) f (ξ ,η) ⃒ = ς η(m) f i,l = j,l ∂η(m) ⃒ξ =ξ i η=η j

l=1

η(m) = ς η(m) · + ς η(m) j,1 f i,1 + ς j,2 f i,2 + · ·(︂ j,M f i,M )︂ ⃒ M N (n+m) ∑︀ ∑︀ ∂ f (ξ ,η) ⃒ = ς ξi,k(n) ς η(m) f k,l = j,l ∂ξ (n) ∂η(m) ⃒ξ =ξ i η=η j k=1 l=1 (︁ )︁ η(m) η(m) = ς ξi,1(n) ς η(m) j,1 f 1,1 + ς j,2 f 1,2 + · · · + ς j,M f 1,M + (︁ )︁ η(m) η(m) +ς ξi,2(n) ς η(m) f + ς f + · · · + ς f + 2,1 2,2 2,M j,1 j,2 j,M

(14)

+ · · · + (︁ )︁ η(m) η(m) (n) , +ς ξi,N ς η(m) j,1 f N,1 + ς j,2 f N,2 + · · · + ς j,M f N,M (︀ )︀ where for the sake of conciseness f ξ i , η j = f i,j . It must be pointed out that i, j define the location of the point at which the derivative is evaluated and k, l are their respec-


tively sum indices. Observing equation (14) the first expression takes all the points which have j fixed, whereas the second one fixes i index. Finally, the mixed derivative comprehends all the points of the element, since k, l appear both in the third of equation (14). In order to be implemented in a computer code the expressions (14) have to be written in matrix form. The easiest and more compact way is to use the Kronecker product ⊗, which is an operation on two matrices of arbitrary size resulting in a block matrix. For instance A ⊗ B of size n × m and p × q respectively, the resulting product gives a block matrix of size n p × m q. Hence, if ς ξ (n) and ς η(n) indicate the matrices of the weighting coefficients along ξ and η, the resulting block matrices for all the points presented in equation (14) are given by

In conclusion π k defines the location of all the points in the master element.

Cξ (n) = I ⊗ ς ξ (n) ,

NM×NM

C

η(m)

NM×NM

M×M

=ς

η(m)

N×N

⊗ I ,

N×N M×M η(m) ξ (n)

Cξη(n+m) = ς NM×NM

M×M

⊗ς

(15)

N×N

where I is the identity matrix. At this point, the derivatives of any order for the master element can be evaluated using expressions (15). In detail, each row of the matrices Cξ (n) , Cη(m) , Cξη(n+m) represents the approximation of the derivative of the generic point ξ i , η j of the grid. The components ξ (n) η(m) ξη(n+m) of the matrices (15) are indicated by C kl , C kl , C kl for k, l = i + (j − 1) N with i = 1, 2, . . . , N and j = 1, 2, . . . , M. It is important to understand the meaning of each row of the matrices (15). With reference to Figure 1 it is clear that the grid point order is taken “by columns” such as (ξ1 , η1 ), (ξ2 , η1 ),. . . ,(ξ N , η1 ),(ξ1 , η2 ),. . . ,(ξ N , η2 ),. . . . . . ,(ξ1 , η M ), . . . ,(ξ N , η M ). In conclusion the whole vector can be seen as M times a vector of length N. For the sake of simplicity another sequence should be defined since the grid location should be set in vector form for computational needs. For this reason the arrow in Figure 1 has been drawn, so that the point coordinates can be grouped in vector form as π = [ ( ξ 1 , η 1 )1 ⏟

( ξ 2 , η 1 )2 . . . ( ξ N , η 1 ) N ⏞ first column (ξ1 , η2 )N+1 . . . (ξ N , η2 )2N . . . . . . ⏟ ⏞ second column T (ξ1 , η M )N·M−N+1 . . . (ξ N , η M )N·M ] ⏟ ⏞ last column

(︀ )︀ πk = ξi , ηj k

for

The approximate derivatives (15) are related to the master element system ξ -η, nevertheless the governing equations are referred to the outer Cartesian one. Thus, the mapping technique should be introduced into equations (7) and (8) in order to investigate structures of arbitrary shape. A general presentation for a given set of order m of shape functions is given by x=

m ∑︁

P i (ξ , η) x i ,

i=1

y=

m ∑︁

P i (ξ , η) y i

(18)

i=1

It has been already presented by previously published articles that, the first and second order derivatives for a standard linear mapping are the following ∂ ∂ ∂ = ξx + ηx ∂x ∂ξ ∂η

(16)

∂2 ∂2 ∂2 ∂2 ∂ ∂ =ξ x2 2 + η2x 2 + 2ξ x η x + ξ xx + η xx 2 ∂η ∂x ∂η ∂ξ ∂ξ∂η ∂ξ ∂ ∂ ∂ = ξy + ηy ∂y ∂ξ ∂η ∂2 ∂2 ∂2 ∂2 ∂ ∂ =ξ y2 2 + η2y 2 + 2ξ y η y + ξ yy + η yy ∂η ∂y2 ∂η ∂ξ ∂ξ∂η ∂ξ

The definition (16) can be shortened as follows i = 1, 2, . . . , N j = 1, 2, . . . , M k = i + ( j − 1) N

Figure 1: Grid point orders for DQ computation: the column order (boxes) and the corresponding vector (arrow).

(17)

∂2 ∂2 ∂2 ∂2 =ξ x ξ y 2 + η x η y 2 + (ξ x η y + ξ y η x ) + ∂x∂y ∂η ∂ξ ∂ξ∂η ∂ ∂ + η xy + ξ xy (19) ∂η ∂ξ

98 | Nicholas Fantuzzi ∂ξ ∂η ∂η where ξ x = ∂ξ ∂x , ξ y = ∂y , η x = ∂x and η y = ∂y . These coordinate derivatives depend on the Jacobian matrix of the transformation and on the shape functions (mapping nodal coordinates) used, as follows yξ yη xη ξx = , ξy = − , ηx = − , det J det J det J (20) xξ ηy = for det J = x ξ y η − x η y ξ det J ∂y ∂y ∂x ∂x where x ξ = ∂ξ , x η = ∂η , y η = ∂η , y ξ = ∂ξ are the derivatives of the Cartesian coordinates of each mapped element using a given mapping of order m (18). Hence, they are easily known. As far as the mapping technique is concerned, 4 node elements (linear), 8 node elements (quadratic), 12 node elements (cubic) have been presented in [126– 137], but the mapping could be general such as the one presented by Zhong and He [138]. Expressions (20) represent the first order derivatives for the mapping transformation. However, the second order must be computed, since they appear in equation (19). So, evaluating the derivatives of (20) accordingly, the following expressions appear (︃ )︃ yξ yη y2η 1 ξ xx = y η y ξη − det Jξ − y ξ y ηη + det Jη det J det J det J2 (︃ )︃ xξ xη x2η 1 x η x ξη − ξ yy = det Jξ − x ξ x ηη + det Jη det J det J det J2 (︂ )︂ yξ xη yη xη 1 ξ xy = −y x + det J + y x − det J η ηη η ξη ξ ξ det J det J det J2 (︃ )︃ y2ξ yξ yη 1 −y η y ξξ + det Jξ + y ξ y ξη − det Jη η xx = det J det J det J2 (︃ )︃ x2ξ xξ xη 1 det Jξ + x ξ x ξη − det Jη η yy = −x η x ξξ + det J det J det J2 (︂ )︂ xξ yη yξ xξ 1 η xy = −y x − det J + y x + det J η η ξ ξη ξ ξξ det J det J det J2 (21)

where det Jξ = x ξ y ξη − y ξ x ξη + y η x ξξ − x η y ξξ and det Jη = −x η y ξη +y η x ξη −y ξ x ηη +x ξ y ηη . As a consequence the derivatives of the Cartesian coordinates take the form ∂2 x ∂2 x ∂2 x , x ηη = , x ξη = , 2 2 ∂η ∂ξ ∂ξ∂η 2 2 2 ∂ y ∂ y ∂ y = ,y = , y ηη = ∂η2 ξη ∂ξ∂η ∂ξ 2

x ξξ = y ξξ

(22)

Summarizing, by simple algebraic manipulations expressions (20) and (21) can be evaluated ones the element type is chosen (4 node, 8 node, 12 node, etc. . . ). Expressions (19) can be evaluated afterwards. The first step is to carry out the derivatives of the Cartesian coordinates with respect to the local master element system. Each point in Cartesian coordinates has its own mapping transformation. Hence, x ξξ , x ξ , x ηη , x η , x ξη , y ξξ , y ξ , y ηη , y η , y ξη are

vectors of dimension N · M ×1, since the whole domain has N · M points of coordinates. Following the nomenclature defined by expressions (16) and (17) it is possible to define the following vectors xξξ , yξξ ,

xξ , yξ ,

xηη , yηη ,

and their components (︀ )︀ (︀ )︀ , (x ηη )k , x x ξξ k , (︀ ξ )︀ k (︀ )︀ y ξ k , (y ηη )k , y ξξ k , k = i + ( j − 1) N for i = 1, 2, . . . , N j = 1, 2, . . . , M

xη , yη ,

xξη yξη

( x η )k , ( y η )k ,

(23)

(︀ )︀ x (︀ ξη)︀ k y ξη k

(24) Now equations (20) and (21) can be evaluated. Thus, the following vectors can be numerically defined ξ xx , η xx , for

ξ x , ξ yy , ξ y , ξ xy η x , η yy , η y , η xy det J = xξ yη − xη yξ det Jξ = xξ yξη − yξ xξη + yη xξξ − xη yξξ det Jη = −xη yξη + yη xξη − yξ xηη + xξ yηη

(25)

or their components in their correspondent form (ξ xx )k , (ξ x )k , (ξ yy )k , (ξ y )k , (ξ xy )k , (η xx )k , (︀(η x ))︀k , (η yy )k , (︀(η y)︀)k , (η xy )k (det J)k = x ξ k (y η )k − (x η )k y ξ k (︀ )︀ (︀ )︀ (︀ )︀ (︀ )︀ (︀ )︀ det Jξ k = x ξ k y ξη k − y ξ k x ξη k + (︀ )︀ (︀ )︀ + (y η )k x ξξ k − (x η )k y ξξ k (︀ )︀ (︀ )︀ (det Jη )k = − (x η )k y ξη k + (y η )k x ξη k + (︀ )︀ (︀ )︀ − y ξ k (x ηη )k + x ξ k (y ηη )k for k = i + (j − 1) N with i = 1, 2, . . . , N and j = 1, 2, . . . , M

(26)

Finally, using equations (25) and the block matrices (15) the map of the Cartesian derivatives with respect to the local reference system of the master element (19) can be carried out. Thus, equations (19) become D x(1) = (ξ x )k C ξkl(1) + (η x )k C η(1) kl kl x(2) D kl = (ξ x )2k C ξkl(2) + (η x )2k C η(2) + kl ξ (1) +2 (ξ x )k (η x )k C ξη(11) + ξ ( xx )k C kl + kl + (η xx )k C η(1) kl y(1) D kl = (ξ y )k C ξkl(1) + (η y )k C η(1) kl y(2) D kl = (ξ y )2k C ξkl(2) + (η y )2k C η(2) + kl ξ (1) +2 (ξ y )k (η y )k C ξη(11) + ξ ( )k C kl + yy kl η(1) + (η yy )k C kl xy(11) D kl = (ξ x )k (ξ y )k C ξkl(2) + (η x )k (η y )k C η(2) + kl (︀ )︀ + (ξ x )k (η y )k + (ξ y )k (η x )k C ξη(11) + kl + (ξ xy )k C ξkl(1) + (η xy )k C η(1) kl for k, l = i + (j − 1) N with i = 1, 2, . . . , N and j = 1, 2, . . . , M

(27)


where the components of the Cartesian derivatives x(1) D kl , D x(2) , D y(1) , D y(2) , D xy(11) can be written in matrix kl kl kl kl form as Dx(1) ,

Dx(2) ,

Dy(1) ,

Dy(2) ,

Dxy(11)

(28)

At this point, it is possible to approximate the Cartesian derivatives using the matrices (28) which have the mapping transformation included. Finally, the mathematical expressions for the outward unit normal vector n are given for the master element. Considering Figure 2 as a reference for the nomenclature of corners (numbers in circles) and edges (numbers in squares), for the first and third edges ξ = ∓1 the outward unit normal vector components are [︃ ]︃ [︃ ]︃ nx yη ξ = √︁ (29) ny x2 + y2 −x η η

η

placement vector U should be firstly defined. It can be di]︁T [︁ vided into U = Ux Uy , where Ux and Uy contain the u and v displacement components of all the grid points, respectively. It is remarked that Ux and Uy have dimension N · M × 1 and they follow the same structure indicated by expression (16). Ultimately, the single components of the displacement vectors can be indicated by (U x )k and (U y )k as suggested in expression (16) by π k . Thus, using the definition of the derivation matrices (27) and the governing equations of the elastostatic problem (7) the following discrete form is exhibited (2G + λ)

N·M ∑︀ l=1

+ (λ + G)

ξ

ξ

N·M ∑︀ l=1

(λ + G)

N·M ∑︀ l=1

whereas for the second and fourth edges η = ±1 they are [︃ ]︃ [︃ ]︃ nx −y ξ η = √︁ (30) ny x2 + y2 x ξ

D x(2) U +G kl ( x )l

D y(2) U + kl ( x )l

l=1

xy(11) D kl ( U y )l + ( F x )k = 0

xy(11) D kl ( U x )l + G

+ (2G + λ)

N·M ∑︀

N·M ∑︀ l=1

N·M ∑︀ l=1

U + D x(2) kl ( y )l

(31)

D y(2) U + ( F y )k = 0 kl ( y )l

for k = i + (j − 1) N with and j = 1, 2, . . . , M

i = 1, 2, . . . , N

where (F x )k , (F y )k are the components of the force vectors Fx , Fy which obviously have the same structures indicated by equation (16). In the present case G and λ are constant terms. Hence, the present implementation allows to consider mechanical properties that vary element by element. It is remarked that equation (31) is written at the master element level, since the mapping technique is embedded in the derivative terms. A compact form of equation (31) is given in the following

Figure 2: Corners and edges enumeration of the master element.

3.2 Discretized form of the governing equations In order to write and understand the discrete forms of the elastostatic (7) and elastodynamic (8) problems the dis-

(2G + λ) Dx(2) Ux + GDy(2) Ux + (λ + G) Dxy(11) Uy + Fx = 0 (λ + G) Dxy(11) Ux + GDx(2) Uy + (2G + λ) Dy(2) Uy + Fy = 0 (32) Rebuilding the displacement vector U equation ((32)) becomes ]︃[︃ ]︃ [︃ Ux (2G + λ) Dx(2) + GDy(2) (λ + G) Dxy(11) + GDx(2) + (2G + λ) Dy(2) Uy (λ + G) Dxy(11) [︃ ]︃ [︃ ]︃ Fx 0 + = (33) Fy 0 It is clear that the algebraic system has dimension (2N · M ) × (2N · M ). Furthermore equation (33) could not be solved directly since the boundary conditions have not been applied yet to the problem. Thus, the lines related to the boundary conditions have to be substituted by the algebraic expressions of the boundary conditions.

100 | Nicholas Fantuzzi Using the same notation as the one presented above, equation (13) can be written in algebraic form as follows ( U n )k = ( n x )k ( U x )k + ( n y )k ( U y )k , (U t )k = ((︁n x )k (U y )k − (n y )k (U x )k

2 (σ n )k = (2G + λ) (n x )k + )︁ N·M ∑︀ x(1) +λ (n y )2k D kl (U x )l + l=1

+2G (n x )k (n y )k

N·M ∑︀ l=1 N·M ∑︀

(37)

D y(1) U + kl ( x )l D x(1) kl

( U y )l + l=1 (︁ )︁ N·M ∑︀ y(1) + (2G + λ) (n y )2k + λ (n x )2k D kl (U y )l +2G (n x )k (n y )k

l=1

(34)

N·M ∑︀

D x(1) U + (τ nt )k = −2G (n x )k (n y )k kl ( x )l l=1 (︁ )︁ N·M ∑︀ y(1) +G (n x )2k − (n y )2k D kl (U x )l + l=1 (︁ )︁ N·M ∑︀ x(1) +G (n x )2k − (n y )2k D kl (U y )l + l=1

+2G (n x )k (n y )k

N·M ∑︀ l=1

D y(1) U kl ( y )l

for k = i + (j − 1) N with and j = 1, 2, . . . , M

i = 1, 2, . . . , N

where (n x )k , (n y )k are the components of the normal vector projected along x and y, respectively. It should be noted that equation (34) does not have to be necessarily evaluated in all the points of the domain, as indicated by the index k. On the contrary, only the boundary points are involved in this process. Nonetheless, in order to use the definitions (28) it is preferable to implement equation (34) skipping the points that are not involved by the boundaries. For this reason a set of algebraic equations are written for the boundary conditions in a similar form to the one presented by equation (33). This set is presented in matrix form below Un = nx Ux + ny Uy Ut = nx Uy − ny Ux (︀ )︀ σ n = (2G + λ) n2x + λn2y Dx(1) Ux + +2Gnx ny Dy(1) Ux + 2Gnx ny Dx(1) Uy + (︀ )︀ + (2G + λ) n2y + λn2x Dy(1) Uy (︀ )︀ τ nt = −2Gnx ny Dx(1) Ux + G n2x − n2y Dy(1) Ux + (︀ )︀ +G n2x − n2y Dx(1) Uy + 2Gnx ny Dy(1) Uy

(35)

Thus the kinematic displacements of (35) can be written as Un =

[︁

Ut =

[︁

nx −ny

ny nx

]︁

[︃

]︁

and the stresses from (35) become

[︃

Ux Uy Ux Uy

σn = [︃ ]︃ [︃ ]︃ (︀ )︀ ... Ux (2G + λ) n2x + λn2y (︀ Dx(1) + 2Gnx ny Dy(1) )︀ . . . 2Gnx ny Dx(1) + (2G + λ) n2y + λn2x Dy(1) Uy [︃ ]︃ [︃ ]︃ (︀ )︀ −2Gnx ny Dx(1) + G n2x − n2y Dy(1) . . . Ux (︀ )︀ τ nt = . . . G n2x − n2y Dx(1) + 2Gnx ny Dy(1) Uy

]︃ ]︃

(36)

For the sake of clarity the points that belong to the boundaries are identified by k = 1, 2, . . . , N (first column) k = N + 1, 2N, 2N + 1, 3N, . . . , (M − 2) N + 1, (M − 1) N (first and last points of the middle columns) and k = (M − 1) N + 1, . . . , MN (last column). These points are part of the boundary matrices which compose the global stiffness matrix. It should be remarked that four of this set of points are the corner points, which need particular conditions in order to have a correct implementation of the method. In order to have a general implementation of the present method it is fundamental to separate the corner points from the boundary points of the edges (as suggested by Francesco Tornabene during a private communication with the author in April 2012). An analytical way of solving the corner point problem has not been given yet, even though several numerical solutions [126–137] have been proposed. Expressions (37) give all the possible combinations in order to have different boundary conditions for the plane problems. For example one can have the clamped ¯ n , Ut = U ¯ t , the free condition σ n = condition Un = U ¯ n , τ nt = τ¯ nt and the mixed condition that can be a symσ metric condition respect to an axis orthogonal to the nor¯ n , τ nt = τ¯ nt . U ¯ n, U ¯ t and σ ¯ n , τ¯ nt are the mal one Un = U applied displacements and forces on the boundaries, respectively. Each element, of the domain decomposition, is identified by a set of domain and boundary equations. Only the domain points are taken from equation (33) and only the boundary points are extracted from equation (37). Moreover, it is more convenient to separate the boundary and domain degrees of freedom from the displacement vectors Ux , Uy . The displacement vector containing the boundary points is identified by Ub so that it has dimension (2N + 2 (M − 2)) n d , where n d = 2 is the number of degrees of freedom (two in-plane displacements). Grouping the displacements of the domain points the vector Ud is defined and it has dimension ((N − 2) (M − 2)) n d . As it was anticipated in the introduction prior versions of the present formulations were called multi-domain differential quadrature, since regular (no mapping) divisions were employed. For these cases only N ≠ M, nevertheless, it is possible to simplify the presentation of the whole theory considering N = M. Hence, the boundary points be-

New insights into the strong formulation finite element method |

101

come (4N − 4) n d and the domain points are (N − 2)2 n d . The general implementation (N ≠ M) is recommended in order to have both techniques within the same code. In conclusion the algebraic equations for a single SFEM element are the following [︃

Kbb Kdb

Kbd Kdd

]︃(e) [︃

Ub Ud

]︃(e)

[︃ +

Fb Fd

]︃(e)

[︃ =

0 0

]︃(e) (38)

where (e) identifies the generic e-th element, Fd is the vector of the domain loads Fx , Fy and Fb is the vector of the boundary loads (that can be displacements or stresses). As far as the free vibration problem is concerned (10), its discrete form can be presented in matrix form (using the previously presented elastostatic problem (33)) as follows [︃ ]︃[︃ ]︃ Ux (2G + λ) Dx(2) + GDy(2) (λ + G) Dxy(11) + Uy GDx(2) (2G + λ) Dy(2) (λ + G) Dxy(11) [︃ ]︃ [︃ ]︃ [︃ ]︃ I 0 Ux 0 2 + ρω = (39) 0 I Uy 0 where Ux , Uy contain the displacement components of the mode shapes according to Φ x , Φ y defined by equation (9) and Iis the identity matrix of dimension (N · M ) × (N · M ). Applying the boundary conditions (37) the final matrix form of the free vibration problem, for a generic e-th element, becomes ⎛[︃ ]︃(e) [︃ ]︃(e) ⎞ [︃ ]︃(e) K K 0 0 Ub bb bd 2 ⎝ ⎠ +ω = Kdb Kdd 0 Mdd Ud [︃ ]︃(e) 0 = 0 (40) that after the assembly section can be clearly solved as an eigenvalue problem.

3.3 Graphical representation of a SFEM mesh Before jumping into the set of equations for the continuity and external boundary conditions, it could be helpful to have a graphical representation of a general SFEM mesh as the one in Figure 3. This figure represents the inter-element edges and the external boundaries with solid lines. In addition Figure 4 presents the nomenclature of the unit vectors for the same mesh of Figure 3. Since quadrilateral elements are used, at least four normal vectors have to be defined. It should be noted that if the edge is curved the outward unit vector is not constant but changes point by

Figure 3: Internal and external boundary conditions for element edges and corners.

Figure 4: Outward unit normal vectors definition for a generic subdivision.

point according to the element geometry as expressed by equations (29) and (30). It is observed that two groups of points occur, the one on the edges (E) and the others at the element corners (C). Kinematic (Dirichlet) boundary conditions are indicated with the type E1 as for the element Ω(1) . Static (Neumann) boundary conditions are termed for type E2. The stress vector referred to the edge 3 of the element Ω(1) can be indicated as σ(1) n3 . The subscript of the normal vector indicates the edge 3 and the superscript stands for the current element (1) that holds that normal vector. It should be emphasized that the corner conditions strongly depend on the conditions of the pair of edges at the corners where they belong. Thus, looking at the corner conditions several configurations can occur. The two corners on element

102 | Nicholas Fantuzzi Ω(1) on the edge 4 have a kinematic (Dirichlet) condition E1 because the clamped boundary condition is stronger than a static (Neumann) one. The other corners indicated by C1 do not have static boundary conditions since two Neumann conditions have to be enforced at the same time at a single point. The compatibility conditions are indicated by E3 along the element edges. For example the edge points of element Ω(1) along 1 are superimposed to the points along 3 of element Ω(2) . Hence, only one group of points is underlined in Figure 3. Nevertheless computationally speaking a double set of equations have to be enforced. Considering the edge 1 of element Ω(1) and the edge 3 of element Ω(2) that face each other, the compatibility conditions have to be enforced (the equations governing this case will be shown in the following). The external and internal corner type conditions are indicated as C2 and C3 in Figure 3. It is recalled that the corners of all the elements concurring at a specific node are superimposed as well as the points on the edges. For instance, the two corners with C2 conditions belong to the two neighbour elements. In the cited cases the C2 conditions have the same form of the E3 ones because only two elements concur at the corner. Nevertheless, the internal corners C3 should have different continuity conditions. The solution for that problem will be described in the following. It must be remarked that all the enforced continuity conditions are continuous with their first derivative at the interfaces, and they can be indicated as C(1) continuous.

3.4 Element connectivity Since a strong formulation is proposed, the boundary conditions are not automatically satisfies such as in FEM. Hence, they must be defined as it was done in the previous section. However, only the external boundary conditions have been introduced. To connect the elements and performing the assemblage of the whole system, the continuity conditions should be enforced between facing elements. In order to perform that, the same definitions given by equations (37) are used. When two elements share the same boundary, two lines of points are superimposed and conditions per physical point can be written (because each grid point has 2 degrees of freedom). The continuity conditions enforce the equality of the displacements and the stresses between these two edges. Just to give an example if the element (e) faces the element (e+1) the following conditions must be written (e+1) U(e) = 0, n − Un (e) (e+1) σn − σn = 0,

(e+1) U(e) =0 t − Ut (e) (e+1) τ nt − τ nt = 0

(41)

The algebraic equations (41) will be part of matrices K(e) , bb (e+1) (e+1,e) K(e,e+1) , K , K whereas the other matrices related bd bb bd to the domain points are unchanged, since no mathematical condition relates the two facing elements. It can be noted that the first line of equations (41) is related to the kinematic (Dirichlet) conditions, thus no derivation is involved. Whereas the second line of equations (41) are the static (Neumann) conditions in which derivatives of the displacement parameters occur as shown by equations (37). The current implementation follows this rule: when the compatibility conditions are written between two elements, identified by (e) and (e+1) , the kinematic conditions are enforced on the boundary points of the element (e) , and the static conditions are set on the boundary points of the element (e+1) . It is recalled that when a derivative is approximated using DQ method, all the points in the derivative direction (or all the domain points in case of the mixed derivative) are involved. Hence, when the kinematic equations are considered on the boundary points of the = 0, since it contains the element (e) the matrix K(e,e+1) bd domains points which are related to the boundary ones. On the contrary for the static equations enforced on the ≠ 0, since it boundary points of the element (e+1) , K(e+1,e) bd contains the algebraic terms of the derivative approximation of the stresses between the elements. As far as the corner point conditions are concerned, their implementation could follow the approaches presented in the past [126– 137]. It is recalled that in order to treat the corners, different conditions should be taken into consideration as a function of the internal or external boundaries involved. The dealing of the corners is an open problem in literature and researches proposed different solutions on the subject. One of the most interesting solutions has been given by Boyd [15]: “. . . the corner singularities will dominate the asymptotic behaviour of the Chebyshev coefficients. . . , and the convergence will be algebraic rather than exponential”. Moreover: “The generic recommendation is to ignore the singularities unless either (i) one has prior knowledge that u (x, y) is discontinuous or has other strongly pathological behaviour or (ii) poor convergence and rapid variation of the numerical solution near the corners suggests a posteriori that the solution is strongly singular”. Following the suggestions given by Boyd, for the study of the mechanics of structural components a finite element without the corner points was implemented (as suggested by Francesco Tornabene during a private communication with the author in June 2013), in order to avoid them in the program. A graphical representation of this implementation is given in Figure 5. In order to proceed with this numerical procedure the weighting coefficients of the boundaries


103

have to be computed separately from the inner points since different discretizations occur. It is remarked that this kind of implementation does not give accurate results for these structural problems and no numerical result is presented in this work. The instability of the corner-less approach is mainly due to the approximation of the mixed derivative, which is mainly involved in the free external conditions and static inter-element connectivity. Thus, the following corner point implementation has been followed. The first configuration is presented in Figure 6a when a corner point of a single element is studied. This corner can have two edges both clamped, both free or just one of them clamped. The symbol EB is used when external boundaries are considered, whereas the internal boundaries are indicated with IB. The represented element is the general element (e) . It is obvious that when at least one of the two edges of element (e) is clamped the corner is fixed too. So only kinematic conditions have to be imposed.

Figure 6: Definition of the external corners conditions for: a) a single element, two facing elements.

means that the algebraic vector contains the normal displacement components of the element (e) with respect to the normal n1 . It is important to note that when both edges have a Dirichlet condition the following relation should be set Figure 5: Grid points involved in the derivative approximation, when the corner points are avoided from the implementation. The boxes on the edges are related to the boundary points which have two points less than inner rounded boxes.

U(e) =0 n(n1 ) (e) Ut(n1 ) = 0

or

U(e) =0 n(n2 ) (e) Ut(n2 ) = 0

(42)

(e) where U(e) n , Ut are the algebraic displacement vectors (36) that contain the normal and tangential components to the edge, respectively. The complete symbol U(e) , U(e) n(n1 ) t(n1 )

U(e) + U(e) =0 n(n1 ) n(n2 )

and

U(e) + U(e) =0 t(n1 ) t(n2 )

(43)

It is important to define equation (43) because physically the corner belongs to two edges. Analogously, when both edges are set free (Neumann condition) a similar expression can be reported σ(e) + σ(e) =0 n(n1 ) n(n2 )

and

τ (e) + τ (e) =0 nt(n1 ) nt(n2 )

(44)

(e) where σ(e) n , τ nt are the algebraic stress vectors (37) with the same meaning of the symbols of the previous definitions. Equations (42)- (44) are extremely important when mixed boundary conditions are set, such as the symmetry. It is

104 | Nicholas Fantuzzi remarked that equations (42)- (44) change when boundary and stress loads are applied to the edge. In particular if a displacement is imposed on the edge with n1 , equation (43) becomes ¯n =U U(e) n(n1 )

and

¯t U(e) =U t(n1 )

Finally, the static (Neumann) conditions are set between

(45)

Similarly if a stress load is applied on the edge with n1 , equation (44) is ¯n σ(e) =σ n(n1 )

and

τ (e) = τ¯ nt nt(n1 )

(46)

Another configuration with two facing elements and an external boundary is depicted in Figure 6b. The corners of the elements (e) and (e+1) both have an external edge with free conditions (Neumann). The facing edge should be used to set the compatibility conditions (41). Since the continuity condition is physically stronger than the Neumann one, equation (41) are also used in these corners. If boundary loads are enforced the continuity conditions continue to be the best choice for having a better accuracy. A more general configuration is presented in Figure 7. At the moment a theoretical counterpart of the equations needed for this implementations has not been found yet. For this reason the following numerical trick is proposed, for a general implementation of this kind of configuration. First of all an internal corner is studied, as in Figure 7a, where only internal boundaries (IBs) are present. Second of all an external corner point occur in Figure 7b. It is obvious that for both corners, continuity conditions (41) must be prescribed, with the only exception of one of EB clamped or when boundary loads are set, so equations (42), (45) or (46) must be used. The present approach for multi-corner configuration sets a static (Neumann) condition and several kinematic (Dirichlet) ones. For instance, five elements concur at the displayed node. First, the code identifies the sequence of elements, e.g. 1, 3, 5, 2, 4 and enforce four kinematic conditions (eight algebraic equations), following the first expressions of equation (41) as (3) e = 1, e + 1 = 3 → U(1) n − Un (1) (3) Ut − Ut = 0 (5) e = 3, e + 1 = 5 → U(3) n − Un (3) (5) Ut − Ut = 0 (2) e = 5, e + 1 = 2 → U(5) n − Un (2) U(5) t − Ut = 0 (4) e = 2, e + 1 = 4 → U(2) n − Un (2) (4) Ut − Ut = 0

= 0, = 0, = 0,

(47)

the last two elements of the group. Following the second expressions of equation (41) they are (1) e = 4, e + 1 = 1 → σ(4) n − σ n = 0,

= 0,

(1) τ (4) nt − τ nt = 0 (49)

Analogously to the kinematic expressions above, equation (49) can be shortened as

Equation (47) can be graphically shortened as e = 1, e + 1 = 3 → U(1,3) e = 3, e + 1 = 5 → U(3,5) e = 5, e + 1 = 2 → U(5,2) e = 2, e + 1 = 4 → U(2,4)

Figure 7: Multiple corner boundary conditions schemes: a) internal corner of five elements with IB conditions; b) external corner of five elements with EB and IB conditions.

e = 4, e + 1 = 1 → σ(4,1) (48)

(50)

In the second configuration of Figure 7b, considering only external free (Neumann) boundary conditions the following implementation is followed. The previous sequence


of the element changes, due to the opening, and for the case depicted in Figure 7b become 1, 4, 2, 5, 3. With the same meaning of the symbols reported in equations (48) and (50) the following conditions are set e = 1, e + 1 = 4 → U(1,4) e = 4, e + 1 = 2 → U(4,2) e = 2, e + 1 = 5 → U(2,5) e = 5, e + 1 = 3 → U(5,3) e = 3, e + 1 = 5 → σ(3,5)

(51)

It should be pointed out that the conditions U(e,e+1) , U(e+1,e) or σ(e,e+1) , σ(e+1,e) are physically the same but numerically different, since they refer to different grid points in the global stiffness matrix. Further details about this aspect will be given in the following subsection. Two other aspects are raised: firstly if the conditions U(1,3) , U(3,1) or σ(1,3) , σ(3,1) were set between the two edges that share external boundaries, inaccurate and unstable numerical solution would occur. Secondly, the corner condition using four static (Neumann) and one kinematic (Dirichlet) equations has been tested, but the accuracy achieved was not sufficient when compared to the present solution. Thus, at the moment, the multi-corner point implementation remains unchanged with respect to the previous published works [126–137].

3.5 Assemblage The final step before the solution is the assembly section. It is a very well-known fact how to assembly a classic FEM algebraic system using C0 boundary conditions. On the contrary it is less common to see a C1 implementation using strong formulation and continuity conditions. The global structure has the same form of equation (38) where all the sub-matrices are located accordingly. For instance if a mesh is made of three elements the following global system occur ⎡ ⎤ K(1) K(1,2) K(1,3) K(1) K(1,2) K(1,3) bb bb bb bd bd bd ⎢ (2,1) ⎥ K(2) K(2,3) K(2,1) K(2) K(2,3) ⎢ Kbb ⎥ bb bb bd bd bd ⎢ (3,1) ⎥ (3,2) (3) (3,1) (3,2) (3) ⎢ K ⎥ K K K K K bb bb bd bd bd ⎢ bb ⎥ (1) ⎢ K(1) ⎥ 0 0 Kdd 0 0 ⎢ db ⎥ ⎢ ⎥ (2) (2) Kdb 0 0 Kdd 0 ⎣ 0 ⎦ (3) (3) 0 0 Kdb 0 0 Kdd ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

U(1) b U(2) b U(3) b U(1) d U(2) d U(3) d

⎤

⎡

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ = −⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎣ ⎦

F(1) b F(2) b F(3) b F(1) d F(2) d F(3) d

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (52)

105

The first thing that can be noted is that the boundary matrices are full matrices, whereas the domain ones are diagonal as expected, since no connection occurs among the inner points of the elements. Moreover, it should be noted that each sub-matrix contains a particular set of algebraic equations. The matrix form (52) is general and comprehend any configuration. however, some matrices could be empty. For instance, if two elements are not connected Kbb = 0 and Kbd = 0. moreover, if the kinematic equations between two elements are set, the matrices Kbd = 0. In order to give a simpler example, element (1) is connected to element (2) and at the same time element (2) is connected to element (3) , but (1) and (3) do not share any boundary. This is the classic case of l-shaped domain. considering these connectivity conditions equation (52) becomes ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

K(1) bb K(2,1) bb 0 K(1) db 0 0

K(1,2) bb K(2) bb K(3,2) bb 0 K(2) db 0

0 K(2,3) bb K(3) bb 0 0 K(3) db

K(1) bd K(2,1) bd 0 K(1) dd 0 0

0 K(2) bd K(3,2) bd 0 K(2) dd 0

0 0 K(3) bd 0 0 K(3) dd

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (53)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

U(1) b U(2) b U(3) b U(1) d U(2) d U(3) d

⎤

⎡

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ = −⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

F(1) b F(2) b F(3) b F(1) d F(2) d F(3) d

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

= 0, since kinematic con= K(1,3) It is noted that K(1,3) bd bb = 0, whereas nectivity is set on the element (1) K(1,2) bd the static conditions are written on the element (2) , thus ≠ 0. Analogously the connectivity is en≠ 0, K(2,1) K(2,1) bd bb forced between elements (2) and (3) . In conclusion equation ((52)) can be rewritten in a more compact form as follows [︃ ]︃ [︃ ]︃ [︃ ]︃ ˜ bb K ˜ bd ˜b ˜b K U F =− (54) ˜ db K ˜ dd ˜d ˜d K U F ˜ bb , K ˜ bd , K ˜ db , K ˜ dd contain the upper-left, upperwhere K right, lower-left and lower-right parts of expression (52) respectively. with the similar meaning of the symbols ˜b, U ˜d, F ˜b , F ˜ d are defined. analogously the global U algebraic system for the free vibration problem is represented by the form (︃[︃ ]︃ [︃ ]︃)︃ [︃ ]︃ [︃ ]︃ ˜ bb K ˜ bd ˜b K 0 0 U 0 2 +ω = ˜ db K ˜ dd ˜ dd ˜d K 0 M U 0 (55) In order to improve the performance of the final code the static condensation can be carried out for both equa-

106 | Nicholas Fantuzzi tion (54) and (55). For the static case expression (54) becomes (︁ )︁ ˜ b = −K ˜ −1 ˜ ˜ ˜ U bb Fb + Kbd Ud (︁ )︁−1 (︁ )︁ (56) ˜d = K ˜ dd − K ˜ db K ˜ −1 ˜ ˜ db K ˜ −1 ˜ ˜ U K bb Kbd bb Fb − Fd And the dynamic case (55) can be rewritten as ˜ b = −K ˜ −1 ˜ ˜ U bb Kbd Ud (︁(︁ )︁ )︁ 2˜ ˜ ˜ ˜ −1 ˜ ˜ Kdd − Kdb K bb Kbd + ω Mdd Ud = 0

(57)

For the first case, the second equation of expression (56) is solved by gaussian elimination and the boundary displacements are retrieved using the first equation of (56). In the latter study, the generalized eigenvalue problem is solved (the second expression of (57)). Once the mode ˜ d are evaluated, the boundary shapes of the inner points U quantities are retrieved using the first equation in (57).

4 Validation studies As most of the numerical methodologies based on domain decomposition or finite elements, the present numerical approach suffers from two main issues: one is due to the accuracy inside each element (related to the derivative approximation), the other one is brought by the mapping technique and the element distortion. For these reasons several validation tests are presented in order to show and demonstrate what has been hitherto cited. It is not the purpose of the present work to investigate the behavior of the technique using several basis functions and point collocations. Thus, the most well-known accurate method has been considered: Lagrange polynomials (known also as Polynomial Differential Quadrature (PDQ)) and Chebyshev-Gauss-Lobatto (C-G-L) grid. For further details about this the reader can refer to the review papers [136, 137]. It is recalled that PDQ has been introduced by Shu [20] and the C-G-L grid has a non-uniform distribution which takes the following form in the master element (︀ N−i )︀ π )︁, ξ i = cos (︁N−1 η j = cos

M−j M−1 π

i = 1, 2, . . . , N ,

j = 1, 2, . . . , M

(58)

All the following computations have been carried out changing the number of grid points N, M inside each element and the number of elements n e used for the mesh subdivision. In particular, emphasis has been put on the use of several boundary conditions, especially the results related to the use of the mixed ones (such as the symmetry), since this has never been applied before [126–137].

4.1 Free in-plane vibrations of a square plate In order to first present the good accuracy of the present methodology the well-known case of the in-plane free vibrations of a square plate is given in the following. Several articles presented this problem in literature in previous published papers [139–141]. Thus, the past solutions are used here as a benchmark for the present code. Furthermore, some observations can be made while modeling this structure. It is a very well-known fact that the easiest boundary condition for the GDQ method is the kinematic (fixed, Dirichlet) one. This is self-explained by the fact that the problem is solved using the partial differential system of equations as a function of the displacements. Thus, using the identity matrix as matrix for the boundary conditions, automatically all the edges are fixed. On the contrary, it is more difficult to enforce the natural (free, Neumann) boundary conditions, since the conditions on the stresses also comprehend the derivatives of the displacements. For this reason the first comparison is given with respect to a fully clamped (C-C-C-C) plate using a single element and a mesh made of four elements. The plate subdivision is performed using regular elements, so that the element dimensions are given by the plate edge divided by the number of element per edge. In other words if a and b identify the two edges of the plate, when four elements are considered each element has dimension a/2 and b/2. It is recalled that using a regular subdivision (a multi-domain technique is under consideration) a different number of points per side can be considered. This is particularly important when a rectangular plate is taken into account (a/b = 2), whereas for a square plate (a/b = 1) it is better to have N = M. A FEM model is also presented using Abaqus, with a regular (100×100 for a/b = 1 and 200×100 for a/b = 2) mesh made of CPS8 elements. The results related to a C-C-C-C plate are shown in Table 1. The natural frequencies are presented in their dimensionless form √︀ as Ω = ωa ρ (1 − ν2 ) /E, where ω indicates the circular frequency. The case of a completely free (F-F-F-F) plate is reported in Table 2, where only Neumann conditions are used. For both cases the same number of grid points is used and it is noted that when N ≠ M in the rectangular plate model, the results are more accurate than the case with N = M. This effect is more noticeable when a ≫ b like 5 or 10 times. It is also remarked that the present solutions agree with the results presented by other authors in literature.


107

Table 1: First ten dimensionless frequencies for C-C-C-C isotropic in-plane square and rectangular plates using different techniques.

a/b = 1 Ω

Ref. [139]

Ref. [140]

ne = 1 N=M=7

ne = 1 N = M = 21

ne = 4 N=M=7

ne = 4 N = M = 11

1 2 3 4 5 6 7 8 9 10 a/b = 2 Ω

3.555 3.555 4.235 5.186 5.859 5.895 – – – –

3.549 3.549 4.221 5.201 5.967 6.000 – – – –

3.55507 3.55507 4.23521 5.19662 5.83147 5.83147 5.86942 6.72321 7.10014 7.10014

3.55519 3.55519 4.23501 5.18570 5.85862 5.89441 5.89441 6.70768 7.11317 7.11317

3.55521 3.55521 4.23587 5.18481 5.85948 5.89867 5.89867 6.70674 7.11554 7.11554

3.55518 3.55518 4.23501 5.18570 5.85862 5.89442 5.89442 6.70767 7.11317 7.11317

Ref. [139]

Ref. [140]

ne = 1 N=M=7

ne = 1 N = 21, M = 11

ne = 4 N = 11, M = 7

ne = 4 N = 21, M = 11

1 2 3 4 5 6 7 8 9 10

4.789 6.379 6.712 7.049 7.608 8.140 – – – –

4.741 6.387 6.682 7.037 7.565 8.128 – – – –

4.78860 6.37838 6.71533 7.05019 7.61663 8.20558 9.33430 9.78770 10.17764 10.94940

4.78902 6.37856 6.71212 7.04875 7.60830 8.14019 8.99796 9.51559 9.71655 10.60077

4.78903 6.37860 6.71238 7.04885 7.60877 8.14008 8.99838 9.51617 9.71624 10.60080

4.78902 6.37856 6.71213 7.04875 7.60831 8.14019 8.99797 9.51560 9.71655 10.60079

4.2 Free in-plane vibrations of a circular plate In the present subsection, the mapping technique is introduced and tested. The reference article is the one by Park [142], where a clamped isotropic circular plate is investigated. The plate has a radius of 0.5 m and it is made of Aluminium with Young’s modulus of 71 GPa, Poisson’s ratio of 0.33 and density 2700 kg/m3 . The problem of annular and circular plates is very well-known in literature [143]. The results are proposed for two different meshes with n e = 4 and n e = 12. The elements have 8 nodes in order to map correctly the curvature. The meshes used are the same as the ones used in the previous works [126, 133, 134]. Table 3 shows the comparisons with the results proposed by Park and also a FEM solution using 9802 CPS8 elements. Good agreement is observed for the first 25 natural frequencies. It is underlined that the

FEM (CPS8) 100 × 100 3.55519 3.55519 4.23501 5.18571 5.85861 5.89442 5.89442 6.70768 7.11318 7.11318 FEM (CPS8) 200 × 100 4.78903 6.37855 6.71213 7.04878 7.60827 8.14022 8.99795 9.51557 9.71655 10.60078

reference articles show only the symmetric frequencies, whereas the present solution and the FEM report all the physical quantities of the structure.

4.3 Static analysis of a thick walled cylinder Consider the standard thick-walled cylinder test presented by MacNeal and Harder [144]. In this example, the structure is under plane strain conditions and simulate a thickwalled cylinder (of infinite length) subjected to an internal pressure p. A 10 degrees segment is modeled with an inner radius Ri = 3 m and an outer radius Ro = 9 m. In the present analysis the radial, tangential and longitudinal stresses are evaluated and compared to an exact solution given in literature. Furthermore, the radial displacement is also presented. The analyses are carried out considering different number of Poisson’s ratios ν =

108 | Nicholas Fantuzzi Table 2: First ten dimensionless frequencies for F-F-F-F isotropic in-plane square and rectangular plates using different techniques.

a/b = 1 Ω

Ref. [139]

Ref. [140]

ne = 1 N=M=7

ne = 1 N = M = 21

ne = 4 N=M=7

ne = 4 N = M = 11

1 2 3 4 5 6 7 8 9 10 a/b = 2 Ω

2.321 2.472 2.472 2.628 2.987 3.452 – – – –

2.321 2.472 2.472 2.628 2.987 3.452 – – – –

2.32451 2.45872 2.48889 2.62683 2.96949 3.45888 3.69459 3.69459 4.21216 4.72090

2.32171 2.46801 2.46801 2.62859 2.98631 3.44945 3.71042 3.71244 4.31437 4.96327

2.31605 2.45043 2.45388 2.62428 2.97801 3.45045 3.68715 3.68902 4.18625 4.80540

2.32145 2.47238 2.47641 2.62825 2.98617 3.44062 3.72062 3.72062 4.32203 4.96643

Ref. [139]

Ref. [140]

ne = 1 N=M=7

ne = 1 N = 21, M = 11

ne = 4 N = 11, M = 7

ne = 4 N = 21, M = 11

1 2 3 4 5 6 7 8 9 10

1.954 2.961 3.267 4.726 4.784 5.205 – – – –

1.938 2.927 3.238 4.702 4.752 5.178 – – – –

1.96633 2.96229 3.30916 4.61846 4.68062 5.24640 5.30536 5.64674 6.02023 6.08108

1.95803 2.95793 3.29986 4.68942 4.76869 5.21348 5.27862 5.39243 6.15741 6.41060

1.95933 2.95905 3.26824 4.72633 4.79928 5.21344 5.25165 5.37669 6.09865 6.43985

1.94801 2.96102 3.26463 4.72400 4.81284 5.19110 5.26765 5.35378 6.15654 6.46616

(0.3, 0.49, 0.499, 0.4999) and the elastic modulus is set equal to E = 1 Pa. The plane strain exact solution can be found in the book by Timoshenko [2]. The equation for the radial displacement is (︂ )︂ d 1 d (59) (r u r ) = 0 dr r dr where u r indicates the radial displacement and r is the radial coordinate. Integrating two times equation (59), the following expression for the radial displacement comes out b ur = a r + (60) r where a, b are two integration constants that can be easily derived as a = (1 − 2ν)

b R2o

b=

−p (1 + ν) (︁ )︁ E R12 − r12 o

(61)

FEM (CPS8) 100 × 100 2.32060 2.47162 2.47162 2.62845 2.98738 3.45224 3.72313 3.72313 4.30307 4.96863 FEM (CPS8) 200 × 100 1.95365 2.96082 3.26705 4.72633 4.78411 5.20445 5.25689 5.36510 6.14655 6.44752

Once the radial displacement is defined the stress quantities can be evaluated afterwards as follows (︂ (︂ )︂)︂ E du r ν du r u r σr = + + 1 + ν dr 1 − 2ν dr r (︂ (︂ )︂)︂ ur ν du r u r E σt = + + (62) 1+ν r 1 − 2ν dr r (︂ (︂ )︂)︂ E ν du r u r σz = + 1 + ν 1 − 2ν dr r It is remarked form equation (62) that σ z ≠ 0 since a plane strain conditions has been considered. The results of the present case are reported in Table 4 for three different meshes that are depicted in Figure 8 using different number of grid points. In each case the number of degrees of freedom is kept relatively high, using polynomials of high degree in order to catch the solution with a small number of finite elements and a small error. Table 4 reports not only the numerical solution but also the percentage of the


109

Table 3: First twenty-five frequencies for C-C-C-C isotropic circular plate using different techniques.

f [Hz] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Ref. [142] 3363.6 – 3836.4 5217.5 – 5380.5 – 6624 6749.3 – 6929 – 7019.3 8093 – 8476.5 – 8530.6 – 9258 – 9328.1 – 9887.7 –

Ref. [142] 3362 – 3835 5219 – 5383 – 6626 6764 – 6939 – 7021 8130 – 8489 – 8557 – 9263 – 9401 – 9925 –

FEM (CPS8) n e = 9802 3361.73 3361.73 3834.85 5219.37 5219.37 5382.86 5382.86 6625.62 6763.75 6763.75 6938.50 6938.50 7021.35 8130.48 8130.48 8489.44 8489.44 8557.33 8557.33 9262.99 9262.99 9401.23 9401.23 9925.01 9925.01

relative error. The numerical solutions are in good agreement with the exact solution provided by hand calculations, for different values of Poisson’s ratio and number of elements. In fact, the error computed is very small in all cases. In order to complete the analysis a convergence test is carried out in Figure 9 where the number of degrees of freedom (dofs) is increasing keeping the same number of elements. The three curves obtained using SFEM are compared to a FEM solution obtained in Straus. The exact solution is used as a reference for computing the error along the vertical axis of the plot. From the double-log plot it can be noticed as expected that the convergence ratio of the SFEM is steeper than the FEM due to the high-order polynomial approximation.

n e = 12 N = M = 7 3363.074 3363.074 3836.446 5221.504 5221.522 5385.225 5385.225 6628.278 6767.236 6767.236 6941.662 6941.838 7024.409 8134.337 8137.422 8496.113 8496.113 8562.647 8562.647 9266.436 9266.436 9409.051 9409.051 9932.331 9935.469

n e = 4 N = M = 15 3363.044 3363.044 3836.351 5221.404 5221.407 5384.962 5384.962 6628.200 6766.389 6766.389 6941.210 6941.214 7024.091 8133.058 8134.240 8492.753 8492.753 8560.662 8560.662 9266.602 9266.602 9404.896 9404.896 9928.872 9928.876

4.4 Sensitivity analysis of a composite thick walled cylinder Simulating the previous example a composite case is deducted. In particular, the aim of this application is to investigate the sensitivity of the present geometry when two different materials are considered. The mesh used in the computations is depicted in Figure 10. The geometry is kept the same as the previous case, whereas the value of the elastic modulus of the inner and outer sheets is variable using 10, 100, 1000. All the quantities are evaluated at the inner radius as it can be deducted from the radial stress that is equal to the external applied load. The results presented in Table 5 are aimed to be used as a reference for further studies on the subject.

Normal stress σ z [Pa]

Circumferential stress σ ϑ [Pa]

Radial stress σ r [Pa]

Radial displacement u r [m]

a 0.065 0.003725 0.00037475 3.74975E-05 0.065 0.003725 0.00037475 3.74975E-05 0.065 0.003725 0.00037475 3.74975E-05 0.065 0.003725 0.00037475 3.74975E-05

ν 0.3 0.49 0.499 0.4999 0.3 0.49 0.499 0.4999 0.3 0.49 0.499 0.4999 0.3 0.49 0.499 0.4999

13.1625 15.08625 15.177375 15.1864875 13.1625 15.08625 15.177375 15.1864875 13.1625 15.08625 15.177375 15.1864875 13.1625 15.08625 15.177375 15.1864875

b 4.5825000000 5.0399250000 5.0602492500 5.0622749925 -1.0000000000 -1.0000000000 -1.0000000000 -1.0000000000 1.2500000000 1.2500000000 1.2500000000 1.2500000000 0.0750000000 0.1225000000 0.1247500000 0.1249750000

exact solution ne = 1 N = M = 41 4.5824855379 5.0399133157 5.0602377434 5.0622631266 -1.0000000000 -1.0000000000 -1.0000000000 -1.0000000009 1.2501853668 1.2501845236 1.2501844623 1.2501842979 0.0750556100 0.1225904166 0.1248420467 0.1250671301

Table 4: Results for a thick-walled isotropic cylinder for several values of the Poisson’s ratio using the meshes of Figure 8.

SFEM ne = 5 N = M = 21 4.5824734104 5.0398991044 5.0602223432 5.0622358712 -1.0000000000 -1.0000000000 -1.0000000000 -1.0000000010 1.2501750962 1.2501576128 1.2501528187 1.2501449947 0.0750525289 0.1225772303 0.1248262565 0.1250474823

n e = 10 N = M = 11 4.5824856751 5.0399082663 5.0602213416 5.0621272942 -0.9999896226 -0.9999896226 -0.9999896226 -0.9999896221 1.2500078161 1.2500122765 1.2500140604 1.2499580798 0.0750054581 0.1225111004 0.1247621944 0.1249592320



Table 5: Results for a thick-walled composite cyilinder for several values of the Poisson’s ratio and the ratio between the core and sheet moduli using 21×21 grid points per element and the meshes of Figure 10.

ν

Radial displacement u r [m]

Radial stress σ r [Pa]

Circumferential stress σ ϑ [Pa]

Normal stress σ z [Pa]

0.3 0.49 0.499 0.4999 0.3 0.49 0.499 0.4999 0.3 0.49 0.499 0.4999 0.3 0.49 0.499 0.4999

E/E c = 10 ne = 3 12.14834 11.91896 11.88706 11.88384 -0.99999 -0.99999 -0.99999 -0.99999 4.02352 4.26979 4.28238 4.28367 0.90706 1.60220 1.63791 1.64151

ne = 5 12.14834 11.91895 11.88705 11.88381 -0.99999 -0.99999 -0.99999 -0.99999 4.02352 4.26979 4.28238 4.28365 0.90706 1.60220 1.63791 1.64150

E/E c = 100 ne = 3 15.48735 13.92165 13.75455 13.73592 -0.99999 -0.99999 -0.99999 -0.99999 5.24712 5.14866 5.11164 5.10707 1.27414 2.03285 2.05171 2.05313

ne = 5 15.48735 13.92165 13.75455 13.73592 -0.99999 -0.99999 -0.99999 -0.99999 5.24712 5.14866 5.11164 5.10707 1.27414 2.03285 2.05171 2.05313

E/E c = 1000 ne = 3 ne = 5 16.21028 16.21028 14.53872 14.53872 14.08313 14.08313 13.96643 13.96643 -0.99999 -0.99999 -0.99999 -0.99999 -0.99999 -0.99999 -0.99999 -0.99999 5.51205 5.51205 5.41946 5.41946 5.25754 5.25754 5.20955 5.20955 1.35362 1.35362 2.16554 2.16554 2.12452 2.12452 2.10436 2.10436

Figure 9: Convergence and stability curves for an isotropic thickwalled cylinder using different meshes.

4.5 2D elastic structure made of two different materials

Figure 8: Isotropic and composite thick-walled cylinder meshes used for the convergence computations.

The present numerical application has been taken from the book by Zong and Zhang [122], where a 2D rectangular 12 m×6 m body has a lateral traction q = 10 Pa. The structure can be studied using a doubly symmetry on the x and y axis and the whole domain can be just divided into two squared elements n e = 2. In this way a composite structure can be studied without considering the mapping technique. The mechanical properties of the half on the left are E1 = 3 · 107 Pa, ν1 = 0.25, whereas the ones of the half on


4.6 Square plate with a square inclusion

Figure 10: Composite thick-walled cylinder meshes used for the convergence computations.

the right are E2 = 3·106 Pa, ν1 = 0.25. The results are compared in terms of stresses σ x and τ xy , at section y = 1.5 m. The SFEM solution is superimposed to the reference ones using N = M = 21 in Figure 11. The black solid line is the solution proposed by Zong and Zhang and the black circles are related to a FEM (Abaqus) solution obtained by Zong and Zhang. Extremely good agreement is observed and this example shows that it is possible to investigate composite structures easily with the present method.

Another application taken from the book by Zong and Zhang [122] considers a square plate with a square inclusion subjected to a horizontal traction q = 100 Pa. The problem has a double symmetry so a quarter of the plate can be studied. For the present case four regular (squared) elements are used n e = 4. The material properties for the inclusion are E1 = 3·106 Pa, ν1 = 0.25 and the ones for the matrix are E1 = 3 · 107 Pa, ν1 = 0.3. The dimension of the quarter of the plate is L = 2 m and the side of the squared inclusion is L/2 = 1m. The results are presented in terms of displacements u = u x and v = u y in Figure 12, where the present solution with a solid line is compared to the solutions proposed by Zong and Zhang [122]. The black solid line is the Abaqus solution, the stars and triangles markers are the solutions proposed by Zong and Zhang [122]. Figure 12 is obtained drawing a section at y = L/4 = 0.5 m. Analogously at the same section the stresses σ x , σ y and τ xy are shown in Figure 13. The present solution is obtained using N = M = 21. A good agreement is observed, even though some differences can be seen at the material discontinuity interface, since a difference between the present solution and the others occurs. Thus, Figure 14 is presented where another FEM solution has been carried out using a very fine mesh and it can be noted that the present results are in very good agreement with this new FEM reference solution. This is due to the fact that Zong and Zhang used a coarse Abaqus mesh for their calculations.

Figure 11: Stress distributions at y = H/2 for a bi-material beam. The present result using N = M = 21 for each element is compared to the same presented by Zong and Zhang [122], who solved the problem using multi-domain DQ and FEM (Abaqus). Figure 12: The present result using N = M = 21 for each element is compared to the same presented by Zong and Zhang [122], who solved the problem using multi-domain DQ and FEM (Abaqus).


Figure 14: The present result using N = M = 21 for each element is compared to the same computed by Abaqus.

4.7 Square plate with a circular inclusion

Figure 13: The present result using N = M = 21 for each element is compared to the same presented by Zong and Zhang [122], who solved the problem using multi-domain DQ and FEM (Abaqus).

Considering the same material data of the previous example a square plate with a circular inclusion is investigated in the following as in Zong and Zhang [122]. The plate is subjected to a vertical traction q = 100 Pa and a double symmetry occurs in this case also. The quarter plate has L = 2.5 m and the inclusion has a radius R = 1m. The mapping technique is compulsory for this case, since distorted elements must be used in order to map the circumference correctly. The plots in Figure 15 use the same symbols of the previous case. For solving this problem four elements are used n e = 4 with N = M = 21. The top-left subfigure

114 | Nicholas Fantuzzi of Figure 15 shows the horizontal displacement u x on the x axis, whereas the others represent the vertical displacement u y and the stresses σ x and σ y on the yaxis. A very good agreement is observed for the present case also.

4.8 Vibrations of a 2D cantilever elastic beam The problems presented in the present section have been investigated in the past [128–130], when the first ten natural frequencies have been also compared to FEM and other methods [145, 146]. The same problem is now used to show the convergence of the technique when different number of points are used in each element. It is recalled that the structure is a 2D cantilever beam of length L = 0.1m and height H = 0.01 m with the following mechanical properties E = 205.939 GPa, ν = 0.3 and ρ = 7845.32 kg/m3 . In this section the convergence and stability behaviour of the technique is investigated using a reference solution obtained through commercial FEM code Abaqus. A very fine reference solution is calculated using n e = 105 regular (squared) CPS8 (8-node biquadric without reduced integration) elements. Three SFEM structures are drawn using n e = 1, n e = 3 and n e = 10. It is noted that the first two meshes are made of distorted elements (rectangular shape), whereas the latest has all regular (squared) elements. This choice has been made, because in this way the effect on the mesh distortion can be studied. In particular the mesh distortion can be overcome by using different number of points along two directions. The convergence error is evaluated in two different ways: using the logarithm of the relative error and the absolute one. The relative error is computed as ⃒ ⃒ ⧸︀ log10 ⃒f1 f1ref − 1⃒, whereas the absolute one as f1 − f1ref . The first measures the difference between the two solutions looking at the absolute value of the significant digits; it is intrinsically dimensionless and lets the user understand the global trend of the solution. The second one just shows the dimensional differences between the two quantities, it is physical so that the user can understand directly the real trend of the solution. It should be noted that the absolute error shows immediately when the trend oscillates towards the minimum error, whereas this aspect is slightly hidden by the relative error due to the imposition of the absolute value. The relative error is a classic way of error measurement in FEM approach, since it is very wellknown that FEM has a ‘convergence from above’ when the mesh is refined. On the contrary in SFEM, due to the fact that the approximation inside each element can change

(as in p-FEM), a ‘convergence from above’ not always occurs. In Figure 16 the abscissa contains the logarithm of the number of degrees of freedom of each problem dofs = 2 (N − 2) (M − 2) n e and the reference value has been taken with respect to a computed eigenvalue of 2.66805 · 107 → f1ref = 822.08573764683945 Hz. It is clear from Figure 16 that when N ≠ M and in particular when N ≫ M the solution tends to diverge. On the contrary the solution is always stable when N = M, but a strong accuracy is reached only when several dofs are considered.

4.9 Cook isotropic beam with two eccentric holes In the present section a new finite element benchmark is proposed, considering the reference problem of Cook’s beam [147–149]. The plan-form of the Cook’s beam has been kept the same, but two holes have been added in two eccentric locations as depicted in Figure 17a . The elastic material data are E = 3 · 107 Pa and ν = 0.3 and the shear stress applied at the right side of the plate is equal to F = 100 Pa. The beam is in plane stress conditions. The mesh used in the computation is presented in Figure 17b . The results are presented in tabular and graphical forms. Table 6 reports the comparison in terms of global displacement and Mises stress at a specific point for the static analysis, and the first ten natural frequency for the modal analysis. The reference point for the static analysis has coordinates (48,52) m. All the results are compared to a FEM solution obtained with a relatively fine mesh. Next to each result the percentage of the relative error is presented. As it is obvious, the error decreases when the number of elements increases. It can be noted that the solution can be considered accurate for N = M = 9 both statically and dynamically. Figure 18 shows a graphical coloured plot of the Mises stress on the whole geometry compared to the same map create with Abaqus software.

4.10 Laminated composite circular arch with circular holes As a final numerical test the free vibration problem of the structure depicted in Figure 19 is described. The arch has an opening angle of 90 degrees and radius R = 2.5 m. The total width of the arch is h = 1 m, with a bottom and top sheets of thickness h s = 0.25 m and a core with h c = 0.5 m. The holes have a diameter of d = 0.25 m and they are centred in the core of the arch. There are three holes, one on

dofs N=M Mises [Pa] Umag [m] f [Hz] 1 2 3 4 5 6 7 8 9 10

err (%) 10.599 4.714

1.852 -0.092 0.275 0.106 -0.316 -0.790 -0.090 -0.175 -0.145 -0.822

5 3.152 546.155

0.0020 0.0050 0.0060 0.0100 0.0126 0.0140 0.0161 0.0180 0.0185 0.0187

756

0.0019 0.0050 0.0060 0.0099 0.0127 0.0141 0.0161 0.0180 0.0185 0.0189

0.224 -0.028 0.096 0.024 -0.036 -0.046 -0.015 0.018 0.005 -0.061

2100 7 err (%) 2.920 2.480 534.781 2.534 0.0019 0.0050 0.0060 0.0099 0.0127 0.0141 0.0161 0.0180 0.0185 0.0189

0.056 0.001 0.019 0.004 -0.005 -0.008 -0.004 0.000 0.002 -0.006

0.0019 0.0050 0.0060 0.0099 0.0127 0.0141 0.0161 0.0180 0.0185 0.0189

0.008 0.000 -0.003 -0.001 -0.001 0.000 -0.001 -0.005 0.000 0.001

SFEM 4116 6804 9 err (%) 11 err (%) 2.883 1.188 2.867 0.623 528.872 1.401 526.252 0.898 0.0019 0.0050 0.0060 0.0099 0.0127 0.0141 0.0161 0.0180 0.0185 0.0189

-0.006 -0.001 -0.009 -0.003 -0.001 0.000 -0.001 -0.007 0.000 0.001

10164 13 err (%) 2.858 0.289 524.868 0.633

0.0019 0.0050 0.0060 0.0099 0.0127 0.0141 0.0161 0.0180 0.0185 0.0189

-0.010 -0.002 -0.012 -0.004 -0.001 0.000 -0.002 -0.007 -0.001 0.000

14196 15 err (%) 2.856 0.215 524.060 0.478

Table 6: First ten frequencies and static Mises stress and magnitude displacement for an isotropic Cook beam with holes, increasing the number of grid points.

0.0019 0.0050 0.0060 0.0099 0.0127 0.0141 0.0161 0.0180 0.0185 0.0189

FEM 42914 n e = 7018 2.850 521.567 New insights into the strong formulation finite element method | 115


Figure 15: The present result using N = M = 21 for each element is compared to the same presented by Zong and Zhang [122], who solved the problem using multi-domain DQ and FEM (Abaqus).

the symmetry axis and two half circles are drawn at the boundary edges. Both the right and left edges are clamped (the circular holes are free). The right and left edges are inclined of 45 degrees with respect to the horizontal axis. The arch is in plane stress condition. The reference numerical solution is carried out using FEM Straus with a relatively fine mesh with 74646 dofs. The SFEM mesh is made of n e = 48 elements according to Figure 19. Four different solutions are calculated. The first one is referred to an isotropic arch with E = 30 MPa, ν = 0.3 and ρ = 1000 kg/m3 . The others consider laminated structures, when the ratios between the sheets and the core is variable. The elastic modulus of the core is kept constant E c = E = 30 MPa and the Young’s modulus of the sheets increases following the ratios E s /E c = 10, 100, 1000. Table 7 reports the results in terms of natural frequencies where the readiness is helped with the relative error between the reference FEM solution and the SFEM one. As expected, the error decreases when the number of grid

points increases. Furthermore the first six mode shapes of the four cases considered are illustrated in Figures 20 – 23. It can be noted that the modes are global, when the structure is isotropic and the ratio between the core and the sheets is small, whereas when the ratio is large the modes concentrate on the soft-core which is the softest part of the structure and the global behaviour is less significant.

5 Closure It can be concluded that the complete set of governing and boundary equations in differential and discrete forms have been presented for elastostatic and elastodynamic problems using the SFEM approach. The present procedure demonstrated to be very accurate to solve both classical and new numerical applications when also composite materials are taken into account. Moreover, some new


Figure 16: Convergence of the first natural frequency of a cantilever elastic beam with n e = 1, n e = 3, n e = 10 and different number of points along the two directions.

numerical benchmarks have been presented for future developments on the same subject. The manuscript deals with the modelling of the present method and the implementation technique of the SFEM. It should be mentioned that the attention is not focused on the stability of the numerical technique when different basis functions and grid distributions are used. Thus, this topic could be the aim of a future paper, where it could be investigated the accuracy and stability of the method when these parameters are changed. The numerical applications provided solutions from the literature and new geometries compared with classic FEM. In all the cases very good agree-

ment is observed and this demonstrates the correctness of the methodology, when compared to other classical approaches and FEM commercial codes. Furthermore, following previously published works, the author would like to deepen the knowledge about the application of elastostatic SFEM problems related to fracture mechanics topics [150–153] in future papers. Acknowledgement: The research topic is one of the subjects of the Centre of Study and Research for the Identification of Materials and Structures (CIMEST)-“M. Capurso” of the University of Bologna (Italy). I am grateful to Erasmo


(a)

(b)

Figure 17: a) Geometry of the Cook’s beam with two eccentric holes (dimensions in meters [m]). b) SFEM mesh used in the computations.

(a)

(b)

Figure 18: a) SFEM Mises stress color map, b) Abaqus FEM Mises color map; both on the undeformed shape.

New insights into the strong formulation finite element method | 119 Table 7: First ten frequencies for a circular composite arch with circular holes, increasing the number of grid points per element.

FEM f [Hz] 1 2 3 4 5 6 7 8 9 10

dofs = = 74646 12.28002 15.35798 23.78326 29.17763 38.81811 40.03110 53.52950 57.36951 60.65324 63.30650

n e = 48 N=M=5 12.28312 15.36383 23.80035 29.18699 38.86387 39.88207 53.36088 57.43483 60.70018 63.21378

1 2 3 4 5 6 7 8 9 10

26.39486 26.93003 50.24074 51.76582 66.56456 74.78007 82.17393 84.01456 90.81161 97.57337

26.36465 26.99436 50.21545 51.77008 66.59229 74.31327 82.13006 83.82778 90.75894 97.25551

1 2 3 4 5 6 7 8 9 10

52.92372 64.73952 92.17044 101.1039 103.1860 108.9535 122.7692 127.7322 128.5981 133.6918

53.09493 64.72375 92.03309 100.9813 103.0722 108.6645 122.4327 127.4390 128.1156 133.3974

1 2 3 4 5 6 7 8 9 10

105.7750 110.1931 116.8438 126.3414 137.8245 150.4189 150.7741 155.8039 172.5842 177.8687

105.7615 110.0682 116.7519 125.9453 137.4203 149.6223 150.2760 155.7738 172.3676 177.4118

SFEM Isotropic arch err (%) n e = 48 err (%) n e = 48 N=M=7 N=M=9 0.025 12.27989 -0.001 12.28021 0.038 15.35464 -0.022 15.35973 0.072 23.78339 0.001 23.78345 0.032 29.17475 -0.010 29.18183 0.118 38.80420 -0.036 38.82353 -0.372 40.02701 -0.010 40.02956 -0.315 53.51688 -0.024 53.53366 0.114 57.36167 -0.014 57.37167 0.077 60.65306 0.000 60.65354 -0.146 63.29492 -0.018 63.31129 Composite arch E/E c = 10 -0.114 26.39481 0.000 26.39486 0.239 26.94819 0.067 26.93976 -0.050 50.27271 0.064 50.25536 0.008 51.77039 0.009 51.76707 0.042 66.58919 0.037 66.58092 -0.624 74.77522 -0.006 74.77924 -0.053 82.17678 0.003 82.17621 -0.222 84.03978 0.030 84.02784 -0.058 90.81334 0.002 90.81344 -0.326 97.57839 0.005 97.57610 Composite arch E/E c = 100 0.323 52.99118 0.127 52.95069 -0.024 64.74849 0.014 64.74164 -0.149 92.22634 0.061 92.19096 -0.121 101.1082 0.004 101.1075 -0.110 103.2130 0.026 103.1994 -0.265 108.9564 0.003 108.9577 -0.274 122.7823 0.011 122.7800 -0.230 127.7277 -0.004 127.7340 -0.375 128.6563 0.045 128.6226 -0.220 133.6987 0.005 133.7026 Composite arch E/E c = 1000 -0.013 105.8164 0.039 105.7919 -0.113 110.1940 0.001 110.1935 -0.079 116.8677 0.020 116.8522 -0.314 126.3211 -0.016 126.3378 -0.293 137.8758 0.037 137.8512 -0.530 150.3782 -0.027 150.4160 -0.330 150.7483 -0.017 150.7715 -0.019 155.8231 0.012 155.8049 -0.126 172.6097 0.015 172.5949 -0.257 177.8791 0.006 177.8712

err (%)

0.002 0.011 0.001 0.014 0.014 -0.004 0.008 0.004 0.000 0.008

n e = 48 N = M = 11 12.27998 15.35916 23.78313 29.18046 38.82181 40.03057 53.53343 57.37036 60.65304 63.31032

0.000 0.036 0.029 0.002 0.025 -0.001 0.003 0.016 0.002 0.003

26.39478 26.93453 50.24771 51.76588 66.57215 74.78089 82.17474 84.02101 90.81228 97.57478

0.000 0.017 0.014 0.000 0.011 0.001 0.001 0.008 0.001 0.001

0.051 0.003 0.022 0.004 0.013 0.004 0.009 0.001 0.019 0.008

52.93544 64.73995 92.17969 101.1055 103.1920 108.9560 122.7749 127.7330 128.6095 133.6980

0.022 0.001 0.010 0.002 0.006 0.002 0.005 0.001 0.009 0.005

0.016 0.000 0.007 -0.003 0.019 -0.002 -0.002 0.001 0.006 0.001

105.7829 110.1935 116.848 126.3409 137.8364 150.4190 150.7744 155.8037 172.5893 177.8700

0.007 0.000 0.004 0.000 0.009 0.000 0.000 0.000 0.003 0.001

err (%)

0.000 0.008 -0.001 0.010 0.010 -0.001 0.007 0.001 0.000 0.006


Figure 19: SFEM mesh of a laminated composite circular arch with circular holes.

Viola and Francesco Tornabene for their encouragement throughout the preparation of this work. A special thanks goes to Francesco Tornabene who pushed me to improve the manuscript day after day.

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Figure 21: First six mode shapes of a composite arch with ratio E s /E c = 10 using N = M = 7 grid per element.

Figure 22: First six mode shapes of a composite arch with ratio E s /E c = 100 using N = M = 7 grid per element.

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2014 · VOLUME 1

OPEN

e-ISSN 2353-7396

EDITORIAL ADVISORY BOARD LIST ASSOCIATE EDITORS JOURNAL EDITOR Lorenzo DOZIO Francesco TORNABENE Polytechnic of Milan, Italy University of Bologna, Italy ASSISTANT EDITORS Nicholas FANTUZZI University of Bologna, Italy Ana M.A. NEVES Universidade do Porto, Portugal

LANGUAGE EDITOR Almut POHL EnginEdit Translations, Italy

Alfred G. STRIZ Oklahoma University, USA Elena FERRETTI University of Bologna, Italy

EDITORIAL ADVISORY BOARD Serge ABRATE Southern Illinois University, USA Sarp ADALI University of Kwazulu-Natal, South Africa Luigi ASCIONE University of Salerno, Italy Ferdinando AURICCHIO University of Pavia, Italy J. Ranjan BANERJEE City University London, UK Romesh C. BATRA Virginia Polytechnic Institute and State University, USA Zdenek P. BAZANT Northwestern University, USA Jeng-Tzong CHEN National Taiwan Ocean University, Taiwan Zhengtao CHEN Univeristy of New Brunswick, Canada Ömer CIVALEK Akdeniz University, Turkey Angelo DI TOMMASO University of Bologna, Italy Moshe EISENBERGER Technion, Israel Luciano FEO University of Salerno, Italy Fernando FRATERNALI University of Salerno, Italy Seyed M. HASHEMI City University of Hong Kong, China Alexander L. KALAMKAROV Dalhousie University, Canada Gennady M. KULIKOV Tambov State Technical University, Russia www.degruyter.com/journals/cls

Kim Meow LIEW City University of Hong Kong, China Parviz MALEKZADEH Persian Gulf University, Iran Federico M. MAZZOLANI University of Naples, Italy Claudio MAZZOTTI University of Bologna, Italy Evgeny V. MOROZOV The University of New South Wales, Australia Roberto NASCIMBENE European Centre for Training and Research in Earthquake Engineering, Italy Eugenio OÑATE Technical University of Catalonia, Spain Wiesław M. OSTACHOWICZ Polish Academy of Sciences, Poland Wojciech PIETRASZKIEWICZ Polish Academy of Sciences, Poland Mohamad S. QATU Central Michigan University, USA Alessandro REALI University of Pavia, Italy Junuthula N. REDDY Texas A&M University, USA Mohammad SHARIYAT Khaje Nasir Toosi University of Technology, Iran Hui-Shen SHEN Shanghai Jiao Tong University, China Abdullah H. SOFIYEV Suleyman Demirel University, Turkey Francesco UBERTINI University of Bologna, Italy Erasmo VIOLA University of Bologna, Italy Antony M. WAAS University of Michigan, USA Ashraf M. ZENKOUR Kafrelsheikh University, Egypt

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