curved and layered structures

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2016 · VOLUME 3 e-ISSN 2353-7396

CURVED AND LAYERED STRUCTURES

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engineering, structural stability, structural dynamics, structural stability/reliability, experimental modeling and smart structures. Therefore, the Journal accepts both theoretical and applied contributions in all subfields of structural mechanics as long as they contribute in a broad sense to the core theme. Journal Editor Francesco Tornabene, University of Bologna, Italy Associate Editors Lorenzo Dozio, Polytechnic of Milan, Italy Elena Ferretti, University of Bologna, Italy Alfred G. Striz, Oklahoma University, USA 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 Moshe Eisenberger, Technion, Israel Luciano Feo, University of Salerno, Italy Fernando Fraternali, University of Salerno, Italy Seyed M. Hashemi, Ryerson University, Canada Alexander L. Kalamkarov, Dalhousie University, Canada Gennady M. Kulikov, Tambov State Technical University, Russia Kim Meow Liew, City University of Hong Kong, China Parviz Malekzadeh, Persian Gulf University, Iran Federico M. Mazzolani, University of Naples "Federico II", 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

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Curved and Layer. Struct. 2016; 3 (1):1–16

Research Article

Open Access

G. M. Kulikov*, A. A. Mamontov, S. V. Plotnikova, and S. A. Mamontov

Exact geometry solid-shell element based on a sampling surfaces technique for 3D stress analysis of doubly-curved composite shells DOI 10.1515/cls-2016-0001 Received Aug 1, 2015; accepted Sep 16, 2015

1 Introduction

Abstract: A hybrid-mixed ANS four-node shell element by using the sampling surfaces (SaS) technique is developed. The SaS formulation is based on choosing inside the nth layer I n not equally spaced SaS parallel to the middle surface of the shell in order to introduce the displacements of these surfaces as basic shell variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree I n − 1 in the thickness direction for each layer permits the presentation of the layered shell formulation in a very compact form. The SaS are located inside each layer at Chebyshev polynomial nodes that allows one to minimize uniformly the error due to the Lagrange interpolation. To implement the efficient analytical integration throughout the element, the enhanced ANS method is employed. The proposed hybrid-mixed four-node shell element is based on the Hu-Washizu variational equation and exhibits a superior performance in the case of coarse meshes. It could be useful for the 3D stress analysis of thick and thin doubly-curved shells since the SaS formulation gives the possibility to obtain numerical solutions with a prescribed accuracy, which asymptotically approach the exact solutions of elasticity as the number of SaS tends to infinity.

A conventional way of developing the higher-order shell formulation consists in the expansion of displacements into power series with respect to the transverse coordinate, which is referred to the direction normal to the middle surface. For the approximate presentation of the displacement field, it is possible to use finite segments of power series because the principal purpose of the shell theory consists in the derivation of approximate solutions of elasticity. The idea of this approach can be traced back to Cauchy [1] and Poisson [2]. However, its implementation for thick shells is not straightforward since it is necessary to retain the large number of terms in corresponding expansions to obtain the comprehensive results. An alternative way of developing the shell theory is to choose inside each layer a set of not equally spaced sampling surfaces (SaS) Ω(n)1 , Ω(n)2 ,. . ., Ω(n)I n parallel to the middle surface in order to introduce the displacement vectors u(n)1 , u(n)2 , u(n)I n of these surfaces as basic shell variables, where I n is the total number of SaS of the nth layer (I n ≥ 3) and n = 1,2,. . .,N, where N is the number of layers. Such choice of displacements with the consequent use of the Lagrange polynomials of degree I n − 1 in the thickness direction for each layer allows the presentation of governing equations of the layered shell formulation in a very compact form. The SaS concept was proposed recently by the authors to evaluate analytically the 3D stress state in rectangular plates and cylindrical and spherical shells [3– 5]. It should be noticed that the SaS shell formulation with equally spaced SaS does not work properly with the Lagrange polynomials of high degree because of the Runge’s phenomenon [6]. This phenomenon can yield the wild oscillation at the edges of the interval when the user deals with any specific functions. If the number of equispaced nodes is increased then the oscillations become even larger. However, the use of the Chebyshev polynomial nodes [7] inside the shell body can help to improve significantly the behavior of the Lagrange polynomials of

Keywords: Sampling surfaces method; Doubly-curved composite shell; Hybrid-mixed four-node solid-shell element; 3D stress analysis

*Corresponding Author: G. M. Kulikov: Department of Applied Mathematics and Mechanics, Tambov State Technical University, Sovetskaya Street, 106, Tambov 392000, Russia; E-mail: [email protected] A. A. Mamontov, S. V. Plotnikova, S. A. Mamontov: Department of Applied Mathematics and Mechanics, Tambov State Technical University, Sovetskaya Street, 106, Tambov 392000, Russia

© 2016 G. M. Kulikov et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

2 | G. M. Kulikov et al. high degree because such a choice permits one to minimize uniformly the error due to the Lagrange interpolation. This fact gives an opportunity to derive displacements and stresses with a prescribed accuracy employing the sufficiently large number of SaS. It means in turn that the solutions based on the SaS technique asymptotically approach the 3D exact solutions of elasticity as the number of SaS goes to infinity. Note also that the origins of the SaS concept can be found in contributions [8, 9] in which three, four and five equally spaced SaS are employed. The SaS formulation with the arbitrary number of equispaced SaS is considered in [10]. The more general approach with the SaS located at the Chebyshev polynomial nodes was developed later [3–5]. In contribution [11], both SaS formulations are compared with each other. It is well-known that the low-order finite elements are sensitive to shear locking. This is because of incorrect shear modes, which infect the pure bending element behavior. In regards to exact geometry or geometrically exact (GeX) four-node shell elements, they reproduce additionally membrane locking. The abbreviation “GeX” reflects the fact that the parametrization of the middle surface is known a priori and, therefore, the coefficients of the first and second fundamental forms are taken exactly at element nodes [12, 13]. The feature of GeX solid-shell elements [12, 13] is that they are based on strain-displacement relationships of the simplest SaS formulation with I n = 3, which precisely represent all rigid-body motions of the shell in any convected curvilinear coordinates. This fact is of great importance since one may read in paper [14] that “shell theory is an absolute academic exercise” due to “the difficulties of representing the rigid body modes in shell finite element formulations”. To avoid shear and membrane locking and have no spurious zero energy modes, the hybrid-mixed finite element method can be applied. This method pioneered by Pian [15] utilizes the robust interpolation of displacements on the element boundary to provide displacement compatibility between elements, whereas the internal stresses are assumed so as to satisfy the differential equilibrium equations. The Pian’s hybrid stress finite element was originally based upon the principle of the stationary complementary energy. Alternatively, the assumed stress finite element was proposed by applying the Hellinger-Reissner variational principle that simplifies the evaluation of the element stiffness matrix [16]. Then, the assumed strain [17] and assumed stress-strain [18] finite elements were developed. The former is based on the modified HellingerReissner functional in which displacements and strains are utilized as basic shell variables, whereas the latter departs from the Hu-Washizu functional depending on dis-

placements, strains and stresses. However, here we do not use these terms because a more general term hybrid-mixed finite element covers all hybrid and mixed finite elements, which are "formulated by multivariable variational functional, yet the resulting matrix equations consist of only the nodal values of displacements as unknowns” [19]. In the present paper, the hybrid-mixed GeX four-node solid-shell element formulation is proposed in which the SaS are located inside each layer at Chebyshev polynomial nodes [5]. To circumvent locking phenomena, the assumed interpolations of displacement-independent strains and stress resultants are invoked and utilized together with displacement and displacement-dependent strain interpolations into the Hu-Washizu variational equation. Such an approach exhibits an excellent performance in the case of coarse mesh configurations and has computational advantages compared to conventional isoparametric hybridmixed solid-shell element formulations [20–23] because it reduces the computational cost of numerical integration in the evaluation of the element stiffness matrix. This is due to the facts that all element matrices require only direct substitutions, i.e., no expensive numerical matrix inversion is needed. It is impossible in the framework of the isoparametric hybrid-mixed shell element formulation. Second, the GeX four-node solid-shell element formulation is based on the effective analytical integration throughout the finite element by the use of the enhanced ANS method [24, 25]. The latter has a great meaning for the numerical modelling of thick doubly-curved shells with variable curvatures. The thin doubly-curved shells based on strong and weak formulations are widely discussed in the literature. The static and dynamic analyses of doubly-curved shells by using the classic Kirchhoff-Love and TimoshenkoMindlin-type theories can be found in books [26–30]. The state-of-the-art development of the problem is analyzed in review articles [31, 32]. The higher-order doubly-curved shell formulation based on the Carrera’s equivalent single layer theory [33] accounting for thickness stretching has been proposed in [34, 35]. Both free vibration and static problems are discussed with a particular emphasis on the stress recovery procedure. The authors report that their procedure leads to stable, accurate and reliable results for the moderately thick and thin doubly-curved shells with variable principal curvatures. However, for the analysis of thick doubly-curved shell structures instead of the postprocessing stress recovery technique a more general approach based on the 3D constitutive equations should be applied. Such a question is discussed here in detail.

cribe all SaS of the nth layer and run from 1 to I n ; Latin tensorial indices i, j, k , l Geometry solid-shell element based on analysis of composite shells

ge from 1 to 3; Greek indices  ,  range from 1 to 2.

| 3

n g(n)i are the base vectors of SaS of the nth layer given by i n n n g(n)i = R(n)i = A α c(n)i eα , α ,α α

Figure 1: Geometry layered shell. Figure of1:the Geometry of the

g3(n)i n = e3 ,

(5)

n n n where c(n)i = c(θ(n)i ) = 1 + k α θ(n)i are the components of α 3 3 the shifter tensor at SaS. Here and in the following developments, (. . .),i stands for the partial derivatives with respect to coordinates θ i ; the index m n identifies the belonging of any quantity to the inner SaS of the nth layer and runs from 2 to I n − 1, whereas the indices i n , j n , k n describe all SaS of the nth layer and run from 1 to I n ; Latin tensorial indices i, j, k, l range from 1 to 3; Greek indices α, β range from 1 to 2. Remark 1: As follows from (3), the transverse coordinates of inner SaS coincide with the nodes of Chebyshev polynomials [7]. This fact has a great meaning for a convergence of the SaS method [3, 11].

layered shell.

Remark 1: As follows from (3), the transverse coordinates of inner SaS coincide

h the

2 Kinematic description of nodes of undeformed Chebyshev polynomials shell[7]. This fact

has a great meaning for a

vergence of Consider the SaS method 11].of the thickness h. Let the middle a layered[3, shell

surface Ω be described by orthogonal curvilinear coordinates θ1 and θ2 , which are referred to the lines of princurvaturesof of deformed its surface. The coordinate θ3 is oriKinematiccipal description shell ented along the unit vector e3 (θ1 , θ2 ) normal to the middleofsurface (see Figure osition vector the deformed shell1).is Introduce written asthe following notations: eα (θ1 , θ2 ) are the orthonormal base vectors of the middle surface; A α (θ1 , θR are of the first 2 ) R the  u coefficients , fundamental form; k α (θ1 , θ2 ) are the principal curvatures of the middle surface; c α = 1 + k α θ3 are the components of the shifter tensor; r = r(θ1 , θ2 ) is the position vector of any point of the middle surface; ai (θ1 , θ2 ) are the base vectors of the middle surface given by aα = r,α = A α eα ,

a3 = e3 ;

3 Kinematic description of deformed shell A position vector of the deformed shell is written as ¯ = R + u, R

(6)

where u is the displacement vector, which is measured in accordance with the total Lagrangian formulation from the (6) initial configuration to the current configuration directly. In particular, the position vectors of SaS of the nth layer are ¯ (n)i n = R(n)i n + u(n)i n , R (7) n u(n)i n = u(θ(n)i ), 3

(1)

(8)

(n)i n

θ[n−1] and θ[n] 3 are the transverse coordinates of layer inter3 faces Ω[n−1] and Ω[n] ; h n = θ3[n] − θ3[n−1] is the thickness of n the nth layer; θ(n)i are the transverse coordinates of SaS of 3 (n)i n the nth layer Ω expressed as θ(n)1 = θ[n−1] , 3 3

n θ(n)m 3

θ3(n)I n = θ3[n] ,

(︂ )︂ )︁ 1 1 (︁ [n−1] 2m n − 3 = θ3 + θ[n] − h cos π ; n 3 2 2 2(I n − 2)

(2)

g3 = R,3 = e3 ;

In particular, the base vectors of deformed SaS of the nth layer (see Figure 2) are n n n n ¯ (n)i + u(n)i g¯ (n)i =R = g(n)i α ,α α ,α ,

(3)

R = r + θ3 e3 is the position vector of any point in the shell body; R(n)i n = r + θ3(n)i n e3 are the position vectors of SaS of the nth layer; gi are the base vectors in the shell body defined as gα = R,α = A α c α eα ,

where u (θ1 , θ2 ) are the displacement vectors of SaS of the nth layer Ω(n)i n . The base vectors in the current shell configuration are defined as ¯ ,i = gi + u,i . g¯ i = R (9)

(4)

g¯ 3(n)i n

=

n g¯ 3 (θ(n)i ) 3

= e3 + β

n β(n)i n = u,3 (θ(n)i ), 3

(n)i n

(n)i n

(10) ,

(11)

where β (θ1 , θ2 ) are the values of the derivative of the displacement vector with respect to the thickness coordinate at SaS of the nth layer.

 u,3 (3



),

(11)

where ( n)in 1,2  are the values of the derivative of the displacement vector with

4 | G. M. Kulikov et al. respect to the thickness coordinate at SaS of the nth layer. 1 (n)i n n u − B α u(n)i for β ≠ α, α A α β,α 1 (n)i n 1 n = u − k α u(n)i , Bα = A for α A α 3,α A α A β α,β

n λ(n)i βα = n λ(n)i 3α

β ≠ α.

Substitution of (16) and (17) into strain-displacement relationships (14) yields the component form of these relationships n 2ε(n)i αβ =

1 n c(n)i β

n λ(n)i αβ +

1

(n)i n n 2ε(n)i + α3 = β α

Figure 2: Initial and current configurations of the shell.

Figure 2: Initial and current configurations of the shell.

The Green-Lagrange strain tensor in an orthogonal curvilinear coordinate system [12] can be written as 1 (¯ g · g¯ − gi · gj ), Ai Aj ci cj i j

2ε ij =

1 n (n)i n A i A j c(n)i cj i

n n n n (¯ g(n)i · g¯ (n)i − g(n)i · g(n)i ), i j i j

n where ε(n)i (θ1 , θ2 ) are the strains of SaS of the nth layer. ij Substituting (5) and (10) into strain-displacement relationships (13) and discarding the non-linear terms, one derives n 2ε(n)i αβ

=

1

n u(n)i ,α n A α c(n)i α

(n)i n n 2ε(n)i · eα + α3 = β

· eβ + 1

n A α c(n)i α

1

n u(n)i ,β n A β c(n)i β n u(n)i · e3 , ,α

· eα ,

(n)i n ε33 = β(n)i n · e3 .

n β(n)i ei . i

(16)

i

Using (15) and formulas for the derivatives of unit vectors ei with respect to orthogonal curvilinear coordinates [25], we have 1 (n)i n ∑︁ (n)i n u = λ iα ei , (17) A α ,α i

where n λ(n)i αα =

1 (n)i n n n u + B α u(n)i + k α u(n)i for 3 β A α α,α

β ≠ α,

(n)i n n ε(n)i . 33 = β 3

Up to this moment, no assumptions concerning displacement and strain fields have been made. We start now with the first fundamental assumption of the proposed higher order layer-wise shell theory. Let us assume that the displacements are distributed through the thickness of the nth layer as follows: ∑︁ (n)i (n)i u(n) L n ui n , θ[n−1] ≤ θ3 ≤ θ[n] (20) 3 3 , i = in

where L(n)i n (θ3 ) are the Lagrange polynomials of degree I n − 1 expressed as ∏︁ θ3 − θ(n)j n 3

L(n)i n =

j n ≠i n

i

β(n)i n =

n λ(n)i 3α ,

(19)

4 Displacement and strain distributions in thickness direction

(14)

Next, we represent the displacement vectors u(n)i n and β(n)i n in the middle surface frame ei as follows: ∑︁ (n)i u(n)i n = u i n ei , (15) ∑︁

n λ(n)i βα ,

(12)

where A3 = 1 and c3 = 1. In particular, the Green-Lagrange strain tensor at SaS is (︁ )︁ n n 2ε(n)i = 2ε ij θ(n)i (13) 3 ij =

n c(n)i α

1 n c(n)i α

(18)

n n θ(n)i − θ(n)j 3 3

.

The use of (11), (16) and (20) yields ∑︁ (n)j (n)i (n)j n β(n)i = M n (θ3 n )u i n , i

(21)

(22)

jn n where M (n)j n = L(n)j ,3 are the derivatives of Lagrange polynomials. The values of these derivatives at SaS are calculated as

)︁ (︁ n M (n)j n θ(n)i = 3

M

(n)i n

(︁

1

∏︁

n n θ(n)i − θ(n)k 3 3

n n θ(n)j − θ(n)i 3 3

k n ≠i n , j n

n n θ(n)j − θ(n)k 3 3

(23)

for j n ≠ i n , )︁ ∑︁ (n)j (︁ (n)i )︁ n =− M n θ3 n . θ(n)i 3 j n ≠i n

n Thus, the key functions β(n)i of the proposed layer-wise i shell theory are represented according to (22) as a linear n combination of displacements of SaS of the nth layer u(n)j . i

Geometry solid-shell element based on analysis of composite shells

| 5

The following step consists in a choice of the consistent approximation of strains through the thickness of the nth layer. It is apparent that the strain distribution should be chosen similar to the displacement distribution (20), that is, ∑︁ (n)i (n)i ε(n) L n ε ij n , θ[n−1] ≤ θ3 ≤ θ[n] (24) 3 3 . ij = in n Remark 2: The functions β(n)1 , β(n)2 ,. . ., β(n)I are linearly dependent, that is, there exist numbers α(n)1 , i i i α(n)2 , . . ., α(n)I n , which are not all zero, such that ∑︁ (n)i (n)i α n β i n = 0. (25)

in

The proof of this proposition follows from a homogeneous system of linear equations ⎡ (n)1 (n)1 ⎤ ⎡ (n)1 ⎤ M (θ3 ) M (n)2 (θ3(n)1 ) ... M (n)I n (θ(n)1 α 3 ) ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ (n)1 (n)2 (n)2 (n)2 (n)I n (n)2 ⎥ ⎢ (n)2 ⎥ ⎢ M (θ3 ) M (θ3 ) ... M (θ3 ) ⎥ ⎢ α ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= ⎢ ⎥ ⎢ ⎥ ... ⎢ ⎥ ⎢ ... ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ (n)1 (n)I n ⎥ ⎢ (n)I n ⎥ (n)I (n)I (n)2 (n)I n n n ⎣M (θ3 ) M (θ3 ) ... M ⎦ (θ3 )⎦ ⎣α

⎡ ⎤ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢...⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0⎦

(26)

and the identity ∑︁

M (n)i n (θ3 ) = 0.

(27)

in

5 Hu-Washizu mixed variational equation To develop the assumed stress-strain finite element formulation, we invoke the Hu-Washizu variational principle in which displacements, strains and stresses are utilized as independent variables [36]. It can be written as follows: δJHW = 0,

(28)

[n]

J HW

]︂ ∫︁ ∫︁ ∑︁ ∫︁θ3 [︂ 1 (n) (n) (n) (n) (n) e ij C ijkl e kl − σ(n) (e − ε ) A1 A2 c1 c2 dθ1 dθ2 dθ3 − W , = ij ij ij 2 n

Ω

∫︁ ∫︁ W=

(︀

(29)

θ[n−1] 3

)︀ c+1 c+2 p+i u+i − c−1 c−2 p−i u−i A1 A2 dθ1 dθ2 + W Σ ,

Ω (n) (n) where σ(n) ij are the stresses of the nth layer; e ij are the displacement-independent strains of the nth layer; C ijkl are the elastic constants of the nth layer; u−i and u+i are the displacements of bottom and top surfaces; p−i and p+i are the tractions acting on the bottom and top surfaces; c−α = 1− k α h/2 and c+α = 1+ k α h/2 are components of the shifter tensor of bottom and top surfaces; W Σ is the work done by external loads applied to the edge surface Σ. Here and in the following developments, the summation on repeated Latin indices is implied. Following the SaS technique, we introduce the third assumption of the proposed hybrid-stress solid-shell element formulation. Let the displacement-independent strains be distributed through the thickness similar to displacement and displacement-dependent strain distributions (20) and (24), that is, ∑︁ (n)i (n)i e(n) L n e ij n , θ[n−1] ≤ θ3 ≤ θ[n] (30) 3 3 , ij = in

6 | G. M. Kulikov et al. n where e(n)i are the displacement-independent strains of SaS of the nth layer. ij Substituting strain distributions (24) and (30) in (28) and (29), and introducing stress resultants [n]

H ij(n)i n =

∫︁θ3

(n)i n σ(n) c1 c2 dθ3 , ij L

(31)

θ[n−1] 3

the following variational equation is obtained: ⎡ ⎛ ⎞ ∫︁ ∫︁ ∑︁ ∑︁ (︁ )︁T (︁ )︁ (︁ )︁T ∑︁ (n)i j (n) (n)j (n)i (n)i ⎣ δ e n ⎝H n − Λ n n C e n ⎠ + δ H(n)i n e(n)i n − ε(n)i n Ω

n

in

(32)

jn

]︂ ∫︁ ∫︁ (︁ )︁T (︀ + + + + )︀ −δ ε(n)i n H(n)i n A1 A2 dθ1 dθ2 + c1 c2 p i δu i − c−1 c−2 p−i δu−i A1 A2 dθ1 dθ2 + δW Σ = 0, Ω

where [︁ ]︁T (n)i n n n n n n , ε(n)i n = ε11 ε(n)i ε(n)i 2ε(n)i 2ε(n)i 2ε(n)i 22 33 12 13 23 [︁ ]︁T (n)i n n n n n n , e(n)i n = e11 e(n)i e(n)i 2e(n)i 2e(n)i 2e(n)i 22 33 12 13 23 [︁ ]︁T (n)i n (n)i n (n)i n (n)i n (n)i n (n)i n H(n)i n = H11 , H22 H33 H12 H13 H23 ⎤ ⎡ (n) 0 0 C(n) C(n) C1111 C(n) 1112 1133 1122 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (n) 0 0 ⎥ C(n) C(n) ⎢C2211 C(n) 2212 2233 2222 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (n) (n) (n) (n) ⎢C 0 0 ⎥ ⎥ ⎢ 3311 C3322 C3333 C3312 ⎥ ⎢ ⎥ ⎢ (n) C = ⎢ (n) ⎥, ⎢C1211 C(n) 0 0 ⎥ C(n) C(n) 1212 1233 1222 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ 0 (n) (n) ⎥ 0 0 0 C1313 C1323 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (n) ⎦ (n) ⎣ 0 0 0 0 C2313 C2323

(33)

[n]

Λ(n)i n j n =

∫︁θ3

L(n)i n L(n)j n c1 c2 dθ3 .

θ[n−1] 3

6 Hybrid-mixed GeX solid-shell element formulation The finite element formulation is based on the simple and efficient interpolation of shells via curved GeX four-node solid-shell elements ∑︁ n n N r u(n)i (34) u(n)i = i ir , r

1 (1 + n1r ξ1 ) (1 + n2r ξ2 ) , 4 ⎧ ⎧ ⎪ 1 for r = 1, 4 ⎪ ⎪ ⎪ ⎨ ⎨ 1 for r = 1, 2 , = , n2r = ⎪ ⎪ ⎪ ⎪ ⎩−1 for r = 3, 4 ⎩−1 for r = 2, 3

Nr =

n1r

(35)

rvilinear coordinates (see Figure 3); 2  are the lengths of the element in 1,2  -

ace; the nodal index r runs from 1 to 4. The surface traction vector is also assumed to Geometry solid-shell element based on analysis of composite shells

| 7

ry bilinearly throughout the element. Ur =

[︂(︁ )︁ (︁ )︁T )︁T (︁ )︁T (︁ )︁T (︁ T 1 −1 u[1] u(2)2 . . . u(1)I u[0] u(1)2 r r r r r

)︁T )︁T (︁ )︁T (︁ (︁ N−1 −1 u[N−1] u(N)2 . . . u(N−1)I r r r )︁T (︁ )︁T ]︂T (︁ [N] N −1 u , . . . u(N)I r r [︁ ]︁T [m] [m] u[m] = u[m] u u (m = 0, 1, . . . , N ) , r 1r 2r 3r ]︁T [︁ n n n n = u(n)m u(n)m u(n)m u(n)m (m n = 2, . . . , I n − 1) , r 3r 2r 1r

Remark 3: The main idea of such approach can be traced back to the ANS method [37–39] developed by many scientists for the isoparametric displacement-based and hybrid-mixed finite element formulations [22, 23, 28, 40, 41]. In contrast with above formulations, we treat the term “ANS” in a broader sense. In the proposed GeX four-node solid-shell element formulation, all components of the strain tensor are assumed to vary bilinearly inside the biunit square. This implies that instead of the expected nonlinear interpolation due to variable curvatures in straindisplacement relationships (18) and (19) the more suitable Figure 3: Biunit square in (ξ1 , ξ2 )-space mapped into the middle bilinear ANS interpolation is utilized. surface of thein GeXfour-node solid-shell element in (x1 , x2 , x3 )Figure 3: Biunit square the GeX4: In order to circumvent curvature thick1 ,  2  -space mapped into the middle surface ofRemark space. ness locking for the isoparametric four-node solid-shell elfour-node solid-shell element in x1, x2 , x3  -space. ement, Betsch and Stein [42] proposed to apply the bilinear where N r (ξ1 , ξ2 ) are the bilinear shape functions of the el- interpolation (36) only for the transverse normal strain. It n ement; u(n)i are the displacements of SaS Ω(n)i n at element is apparent that curvature thickness locking is not related ir nodes; ξ α = (θ α − d α ) /ℓα are the normalized curvilinear co- to the curved GeX four-node solid-shell element because ordinates (see Figure 3); 2ℓα are the lengths of the element it can handle the arbitrary geometry of surfaces properly. in (θ1 , θ2 )-space; the nodal index r runs from 1 to 4. The We advocate the use of the enhanced ANS method for all surface traction vector is also assumed to vary bilinearly components of the strain tensor to implement the efficient throughout the element. analytical integration throughout the element. To implement analytical integration throughout the From the computational point of view it is convenient element, we employ the enhanced ANS interpolation [24, to represent the ANS interpolation (36) as follows: 25] ∑︁ r r (n)i ε(n)i n = (39) (ξ 1 ) 1 (ξ2 ) 2 ε r1 r2n , ∑︁ (n)i n (n)i n r1 , r2 ε = Nr εr , (36) r

[︁

n n ε(n)i = ε(n)i r 11r

n ε(n)i 22r

(n)i n ε33r

(n)i n 2ε12r

(n)i n 2ε13r

(n)i n 2ε23r

]︁T

(40)

where

n where ε(n)i ijr are the strains of SaS of the nth layer at element nodes. These strains can be evaluated as n n ε(n)i = B(n)i U. r r

(n)i n n ε(n)i r1 r2 = Br1 r2 U,

,

[︁ (n)i n n ε(n)i r1 r2 = ε 11r1 r2

(37)

n B(n)i r

Here, - are the constant throughout the element ma∑︀ trices of order 6×12NSaS , where NSaS = I n − N + 1 is the n

total number of SaS; U is the element displacement vector defined as [︁ ]︁T U = U1T U2T U3T U4T , (38)

n B(n)i 00 n B(n)i 01 n B(n)i 10 n B(n)i 11

n ε(n)i 22r1 r2

n n 2ε(n)i 2ε(n)i 13r1 r2 12r1 r2 1 (︁ (n)i n n B1 + B(n)i = 2 4 1 (︁ (n)i n n = B1 + B(n)i 2 4 1 (︁ (n)i n n = B1 − B(n)i 2 4 (︁ 1 n n = B(n)i − B(n)i 1 2 4

n ε(n)i 33r1 r2 n 2ε(n)i 23r1 r2

(41) ]︁T

,

n n + B(n)i + B(n)i 3 4

)︁

,

n n − B(n)i − B(n)i 3 4

)︁

,

n n − B(n)i + B(n)i 3 4

)︁

,

)︁

n n + B(n)i − B(n)i . 3 4

8 | G. M. Kulikov et al. Here and below, the indices r1 and r2 run from 0 to 1. To overcome shear and membrane locking and have no spurious zero energy modes, the robust displacementindependent strain and stress resultant interpolations are utilized ∑︁ r r (n)i (42) e(n)i n = (ξ1 ) 1 (ξ2 ) 2 Qr1 r2 er1 r2n , r1 + r2