Generalized Unified Formulation Shell Element for

AIAA 2015-0195 AIAA SciTech 5-9 January 2015, Kissimmee, Florida 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

Generalized Unified Formulation Shell Element for Functionally Graded Variable-Stiffness Composite Laminates and Aeroelastic Applications Luciano Demasi,∗ Yonas Ashenafi,† Rauno Cavallaro, ‡ Enrico Santarpia

§

Composite materials have been extensively used in engineering thanks to their lightweight, superior mechanical performances and possibility to tailor the structural behavior, increasing the available design space. Variable Angle Tow (VAT) structures exploits this advantage by adopting a curvilinear patterns for the fibers constituting the lamina. This work, for the first time, extends the Generalized Unified Formulation (GUF) to the case of fourth-order triangular shell elements and VAT composites. Functionally graded material properties in both the thickness and in-plane directions are also possible. The finite element has been formulated with layers of variable thickness with respect to the in-plane coordinates. GUF is a very versatile tool for the analysis of Variable Stiffness Composite Laminates (VSCLs): it is possible to select generic element coordinate systems and define different types of axiomatic descriptions (Equivalent Single Layer, Layer Wise, and Zig-Zag enhanced formulations) and orders of the thickness expansions. Each displacement is independently treated from the others. All the infinite number of theories that can be generated with GUF are obtained by expanding six theory-invariant kernels (formally identical for all the elements), allowing a very general implementation. Finally, the possibility of tailoring the theory/order to increase the accuracy in desired directions makes the GUF VAT capability a very powerful tool for the design of aerospace structures.

I.

Introduction

OMPOSITE structures have been used in several aerospace engineering applications due to their C lightweight and great strength. Moreover, composites offer the possibility of customizing the mechanical behavior with freedom in the choice of materials for matrix and fibers, number of layers, and stacking sequence. Application of such design freedom could be in the aeroelastic tailoring (see ref. [1]) inducing for example a bending-torsion coupling that could reduce the likelihood of having instabilities such as divergence. Traditional straight fiber composites, Constant Stiffness Composite Laminates (CSCL), can be further enhanced by allowing the fiber orientation to change in space. This increases the design space towards more efficient structures. For example, aircraft fuselage presents some regions dominated by bending adjacent to areas in which shear deformation mainly affects the response (see [2]). In that case, the fibers would best be aligned to a certain direction to optimize the bending response; however, the fibers should be placed at 45 degrees to have a more efficient design in respect to the shear. A method to achieve this variability in stiffness properties and tailor the behavior of the structure is to use Variable Stiffness Composite Laminates (VSCLs) obtained with curvilinear fibers’ paths (another term used in this work is Variable Angle Tow (VAT) composites). ∗ Associate

Professor, Department of Aerospace Engineering, San Diego State University, Lifelong AIAA Member Student, Department of Aerospace Engineering, San Diego State University, also Stress Engineer, UTC Aerospace Systems, San Diego, California ‡ PhD Candidate, Department of Aerospace Engineering, San Diego State University and Department of Structural Engineering, University of California San Diego, AIAA Member § PhD Candidate, Department of Aerospace Engineering, San Diego State University and Department of Structural Engineering, University of California San Diego, AIAA Member † Master

1 of 44 American Institute of Aeronautics and Astronautics Copyright © 2015 by Luciano Demasi, Yonas Ashenafi, Rauno Cavallaro, Enrico Santarpia. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

VAT composites have been explored for the last few decades. Refs. [3] showed that with curvilinear fibers it was possible to improve stress concentration around holes. This was accomplished by arranging the fibers in the direction of critical load paths. Ref. [4] analyzed the postbuckling progressive damage behavior of Variable Stiffness Composite Laminates. The importance of taking into account the residual thermal stress resulting from the curing process was emphasized. It was also showed that VSCLs demonstrate capacity for load redistribution: the curvilinear fiber panels can redirect load fluxes from the central regions to their stiffer edges. The buckling load is then increased as a consequence. Tow-steered laminate exhibiting bistability have been studied for a trailing edge flap application or where it is necessary to sustain significant changes in shape without the need for a continuous power supply. Ref. [5] discussed how to accurately predict the cured shapes of tow-steered laminates that are intended to be bistable. Ref. [6] introduced a fiber placement technique (Continuous Tow Shearing) for the manufacturing of VAT composites with tailored fiber paths. Prebuckling analysis of anisotropic VAT plates using Airy’s stress function was presented in ref. [7]. Ref. [8] addressed the problem of impact and compression after impact of VSCLs with emphasis on the interaction between fiber orientations, matric-crack and delaminations. The simulations were carried out using an explicit finite element analysis. Aeroelastic behavior of a rectangular unswept composite wing combined with modified strip theory aerodynamics was studied in ref. [9]. Flutter, divergence, and gust loads were investigated, showning that the speed of instability occurrence could be influenced, both positively and negatively, by changing the fiber angle along the wingspan. It was also observed that VAT laminates allowed improved design compared to traditional unidirectional composite laminates. Ref. [10] introduced a semi-analytical formulation based on a variational approach and Rayleigh-Ritz method to solve the postbuckling problem of VAT plates. The advantage of using variable stiffness for enhanced postbuckling performance of composite laminates was demonstrated. Ref. [11] studied the problem of tailoring the in-plane tow path of VAT composite plates for improved postbuckling resistance. Ref. [12] investigated the structural performance of axially compressed tow-steered shells. Both experimental and computational (finite element analysis) approaches were adopted. Prebuckling stiffness and buckling loads were estimated. An optimization strategy (based on a genetic algorithm) for the design of postbuckling behavior of VAT composite laminates under axial compression was shown in ref. [13]. Ref. [2] introduced a Third Order Shear Deformation Theory (p-version Finite Element approach) with geometric nonlinear effects. Studies on the curvilinear fibers showed the possibility of reducing deflections and stresses under some static loadings. VSCLs led to changes in the stresses altering the position of maximum stress at the plate. It was concluded that this modification could be exploited to improve damage resistance in particular applications. In Ref. [14] VSCLs were analyzed with a p-version Layer Wise finite element approach. Unsymmetric laminates with curvilinear fibers were analyzed with just one element. Moderately large deflection model (Von Karman strain-displacement relationships) was assumed. The Layer Wise displacement field assumed linear variation in the thickness for the in-plane displacements and constant transverse displacement. Different behavior was found in several unsymmetric laminates: a plate with relatively large displacement was actually stiffer in the nonlinear regime, showing the importance of taking into account geometric nonlinearities. Ref. [15] presented a Third-Order Shear Deformation Theory for VSCL rectangular plates. Geometric nonlinearities and damage under various static and dynamic loads (uniform, localized, sinusoidal, and impact) were analyzed. Refs. [16] developed a Layer Wise model in which a First Order Shear Deformation Theory was adopted for each layer. Plates with classical straight-fiber and curvilinear layers were investigated. The curvilinear fibers were used between plies with straight fibers. It was concluded that one can still take advantage of the variable stiffness plates (e.g., redistribute the applied load in the plane [4]) by mixing constant stiffness with variable stiffness plies along the thickness. Ref. [17] discussed manufacturing characteristics of Variable Angle Tow structures with particular attention on layup accuracy, and thickness variation. An experimental approach (layup tests) was adopted. Ref. [18] highlighted the advantages of having fiber-reinforcement following curvilinear paths in space.

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The initial post-buckling response of variable-stiffness cylindrical panels was presented. The model aimed to get an efficient tool for optimization studies of variable-stiffness panels where stability represents a constraint. Ref. [19] introduced a First Order Shear Deformation Theory for buckling analysis of thick stiffened Variable Angle Tow panels. Ref. [20] investigated the stability of VAT panels. It was observed that the Variable Angle Tows laminates manufactured with the Continuous Tow Shearing technique produce laminates with a flat profile on one side and a curved profile on the other. Variable thickness was then modeled. An equivalent single layer model, taking into account shear effects, was adopted. The 3D structure was simulated either as a cylindrical shell or a flat plate. It was argued that the buckling event of the variable angle tow with variable thickness was characterized more by a “shell-like” behavior rather than a “plate-like” behavior. From the above discussion of literature, there is a vast body of work on curvilinear fibers showing the design advantages that can be exploited in increase of buckling load and reduction of stresses. Several models have been introduced with aim to efficient optimization of VAT panels/structures. With the present effort, the authors introduce the Generalized Unified Formulation (GUF) [21, 22, 23, 24] for VAT structures for the first time. With this computational framework it is possible to define an independent local element coordinate system and have different axiomatic models for the different displacement variables. This allows the user to have the design freedom and computational efficiency where required by the problem under investigation. To provide a general background on the theoretical models developed for the analysis of composites (and in some cases for VAT structures as previously discussed), a brief overview on the various axiomatic formulations is provided next.

II. A.

Axiomatic Formulations for the Analysis of Composite Structures

Background and Literature Study

It is well know that composites do not have the same behavior in all the directions. This is actually exploited in the practice but also constitute a difficulty when a good model is sought. Classical Plate Theory (CPT) [25] is generally sufficient for metallic thin panels. When some of the assumptions made in its formulation are removed, the resulting theories can more efficiently capture behaviors typical of composite laminates (for example the Zig-Zag form of the displacements, see ref. [24]). Thus, people formulated First Order Shear Deformation Theory (FSDT) [26, 27]. FSDT was improved even more with the introduction of the so-called Higher Order Shear Deformation Theories (HSDTs) [28, 29, 30, 31, 32, 33, 34]. Some researchers modified HSDTs by adding the transverse strain effects (more details can be found in ref. [24]). This was achieved by adding additional terms in the thickness expansion of the transverse displacement uz . It is known that the interlaminar equilibrium of the transverse stresses and anisotropy of the mechanical properties along the thickness determines a discontinuity of the first displacement derivatives with respect to the thickness coordinate z (“Zig Zag form of the displacements”, see refs. [35, 36, 37]). Following the historical reconstruction provided in ref. [38] and the discussions found in ref. [24], the Zig Zag theories can be subdivided into 3 major groups: • Lekhnitskii Multilayered Theory (LMT) • Ambartsumian Multilayered Theory (AMT) • Reissner Multilayered Theory (RMT) Particularly relevant is the contribution provided by Murakami who proposed in ref. [39] to take into account the Zig Zag effects by enhancing the corresponding displacement variable with a Zig Zag function denoted here as Murakami’s Zig Zag Function (MZZF). Numerous applications [40, 41, 42, 43, 44, 45, 46, 47] of the concept of enhancing the displacement field with MZZF have been presented. This enhancement provides a significant improvement of the accuracy with a marginal increment of the computational cost with respect to the inexpensive (but less accurate) classical methods. Recently researchers [48, 49, 50, 51, 52] adopted Zig Zag models to solve various problems involving bending analysis of functionally graded sandwich structures, laminated beams, and buckling calculations. The effectiveness of taking into account the displacements’ slope discontinuity at the interfaces with Zig Zag models was proven.

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For a detailed quasi-3D type of investigation a Layer Wise [53, 54, 55, 56, 57, 58, 59] description represents a valuable alternative to the computationally demanding Finite Element approaches based on solid elements. The Layer Wise and Equivalent Single Layer approaches can be unified with the adoption of the Compact Notations (CNs). Examples of Compact Notations are represented by Carrera’s Unified Formulation (CUF) [60] and its generalization represented by the Generalized Unified Formulation (GUF) [21, 23, 61, 62, 43, 63, 22, 24] which will be extensively discussed later in this work. This research extends the Generalized Unified Formulation to the case of Variable Angle Tow structures and fourth order triangular shell element. The different types of theories are now classified and discussed (for more details see ref. [24]). B.

Definition of Equivalent Single Layer (ESL) Theories

Definition: A theory in which all its unknown functions do not depend on the considered layer is called Equivalent Single Layer Theory. Examples of ESL theories are Classical Plate Theory, First Order Shear Deformation Theory, Higher k Order Shear Deformation Theories, and Zig Zag Theories obtained by adding MZZF (−1) ζk . 1.

Definition of Zig Zag Theories (ZZTs)

The Zig Zag theories are a special case of Equivalent Single Layer Theories. In particular the following definition is used: A Zig Zag Theory (ZZT) is obtained from an Equivalent Single Layer Theory (called baseline theory) by enhancing all displacements ux , uy , and uz with Murakami’s Zig Zag Function. The baseline theory presents a zero transverse normal strain zz . ZZT is also an Equivalent Single Layer Theory. 2.

Advanced Zig Zag Theories (AZZTs)

Definition: An Advanced Zig Zag Theory (AZZT) is obtained from an Equivalent Single Layer Theory (called baseline theory) by enhancing all displacements ux , uy , and uz with Murakami’s Zig Zag Function. The baseline theory presents a non-zero transverse normal strain zz . AZZT is also an Equivalent Single Layer Theory. C.

Layer Wise (LW) Theories

Definition: A theory in which all its unknown functions do depend on the considered layer is called Layer Wise theory. D.

Partially Zig Zag Higher Order Shear Deformation Theories (PZZHSDTs)

Definition: A partially Zig Zag Higher Order Shear Deformation Theory (PZZHSDT) is obtained from an Equivalent Single Layer Theory (called baseline theory) by enhancing some (but not all) displacements with Murakami’s Zig Zag Function. The baseline theory presents a zero transverse normal strain zz . PZZHSDT is also an Equivalent Single Layer Theory. E.

Partially Zig Zag Advanced Higher Order Shear Deformation Theories (PZZAHSDTs)

Definition: A partially Zig Zag Advanced Higher Order Shear Deformation Theory (PZZAHSDT) is obtained from an Equivalent Single Layer Theory (called baseline theory) by enhancing some (but not all) displacements with Murakami’s Zig Zag Function. The baseline theory presents a non-zero transverse normal strain zz . PZZAHSDT is also an Equivalent Single Layer Theory.

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F.

Partially Layer Wise Higher Order Shear Deformation Theories (PLHSDTs)

Definition: Partially Layer Wise Higher Order Shear Deformation Theory (PLHSDT) presents some quantities described in a Layer Wise sense and other quantities described in an ESL sense. PLHSDT presents a zero transverse normal strain zz and non-zero transverse shear strains. PLHSDT is neither an ESL theory nor a LW theory since both types of representations are present. G.

Partially Layer Wise Advanced Higher Order Shear Deformation Theories (PLAHSDTs)

Definition: Partially Layer Wise Advanced Higher Order Shear Deformation Theory (PLAHSDT) presents some quantities described in a Layer Wise sense and other quantities described in an ESL sense. PLAHSDT presents a non-zero transverse normal strain zz and non-zero transverse shear strains. PLAHSDT is neither an ESL theory nor a LW theory since both types of representations are present. H.

Partially Layer Wise Zig Zag and Higher Order Shear Deformation Theories (PLZZHSDTs)

The Partially Layer Wise Zig Zag and Higher Order Shear Deformation Theories (PLZZHSDTs) have the following properties: • One displacement variable presents a LW description. • One displacement variable presents an ESL description and is enhanced with Murakami’s Zig Zag Function. • One displacement variable presents an ESL description but is not enhanced with Murakami’s Zig Zag Function. • The transverse shear strains are not zero • The transverse normal strain zz is equal to zero I.

Partially Layer Wise Advanced Zig Zag and Higher Order Shear Deformation Theories (PLAZZHSDTs)

Partially Layer Wise Advanced Zig Zag and Higher Order Shear Deformation Theories (PLAZZHSDTs) are very similar to the (PLZZHSDTs) previously discussed. The difference is in the transverse normal strain zz which is not zero. In general, a PLAZZHSDT has the following properties: • One displacement variable is described with a LW expansion. • One displacement variable presents an ESL description and is enhanced with Murakami’s Zig Zag Function. • One displacement variable is axiomatically expanded with an ESL approach and is not enhanced with Murakami’s Zig Zag Function. • The transverse shear strains are not zero. • The transverse normal strain zz is a non-zero quantity. With the Generalized Unified Formulation all these types of theories are described within the same theoretical/computational framework. This is discussed next.

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III.

Notations, Coordinate Systems at Element Level, and Transformations

Consider a portion of the structure (triangular element). With reference to Figure 1, all the edges in the undeformed continuum are assumed perpendicular to the reference plane identified as shown in Figure 1. The corners nodes 1, 2, and 3 are identified by a local numbering freely selected by the user. The notation

actual

FE model

x, y: element reference plane

x, y: element reference plane

3

edges not necessarily perpendicular to x, y plane

1

3 1

2

2

edges are perpendicular to the x, y plane

Figure 1. Selection of the element reference plane.

and coordinate systems (see Figure 2) are now introduced. The global axes are X, Y, Z. The corresponding unit vectors are indicated with e1 , e2 , and e3 . The auxiliary unit vectors e1 , e2 , and e3 are referred to an ˜ ˜ coordinate system ˜ ˜ x , y ˜ , and z (see Figure 2). This˜ intermediate intermediate local coordinate system loc loc loc has the xloc connecting nodes 1 and 2 of the element. The final unit vectors of the local coordinate system at element level are indicated with i1 , i2 , and i3 ˜ ˜The local ˜ respectively. x, y, and z are the actual local coordinates of the element under consideration. coordinate system is selected by the user. Thus, the angle ψ (see Figure 2) is assigned and known. Note that all the layers may have variables thicknesses. However, the plate assumption will be considered. Thus, all the coordinate systems at layer levels will be considered to be on parallel planes. That is, all the layers will have parallel reference planes. This is consistent with the assumption of flat element approximating a curved surface (see Figure 3). The angle ψ will then be the same for all the layers. Notice that the origin of the element coordinate system does not have to be necessarily on node 1. It can be anywhere even outside the element. However, the node 1 is selected for simplicity. After some simple algebra involving the rotation

zloc Z

z element reference plane

e3

e2 1

e3

2

e1

b

e1

e2

Actual local coordinate system. Note that the angle y is known

i3

yloc

1

3

i1

Y

x

y

i2 3

f 2

x loc

X Figure 2. Global and local coordinate systems, for a triangular element

of coordinate systems, it is possible to relate the unit vectors of the local system with the ones of the global coordinate system: i1 = a11 e1 + a12 e2 + a13 e3 ˜ ˜ ˜ ˜ (1) i2 = a21 e1 + a22 e2 + a23 e3 ˜ ˜ ˜ ˜ i3 = a31 e1 + a32 e2 + a33 e3 ˜ ˜ ˜ ˜

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reference plane (layer 2)

3 top

y

3

2 top

h2 /2 h 2 /2 h 2

3 bot

2

1

h /2

h

h /2

1

top

f x

1 bot

reference plane (layer 1)

2 bot reference plane (element)

Figure 3. Layer and element reference planes are parallel. Note that the layer thickness is not constant.

where a11 = cos ψ A1 − sin ψ B1

a21 = sin ψ A1 + cos ψ B1

a31 = C1

a12 = cos ψ A2 − sin ψ B2

a22 = sin ψ A2 + cos ψ B2

a32 = C2

a13 = cos ψ A3 − sin ψ B3

a23 = sin ψ A3 + cos ψ B3

a33 = C3

(2)

Equation 2 contains the quantities defined below (Xi , Yi , and Zi are the global coordinates of the corner nodes 1,2, and 3, respectively): A1 = √ A2 = √ A3 = √

(X2 −X1 ) (X2 −X1 )2 +(Y2 −Y1 )2 +(Z2 −Z1 )2 (Y2 −Y1 ) (X2 −X1 )2 +(Y2 −Y1 )2 +(Z2 −Z1 )2 (Z2 −Z1 ) (X2 −X1 )2 +(Y2 −Y1 )2 +(Z2 −Z1 )2

E1 = (X3 − X1 ) E1 = (Y3 − Y1 ) (3) E3 = (Z3 − Z1 )

A2 E3 −A3 E2 (A2 E3 −A3 E2 )2 +(A3 E1 −A1 E3 )2 +(A1 E2 −A2 E1 )2

B1 = C2 A3 − C3 A2

A3 E1 −A1 E3 (A2 E3 −A3 E2 )2 +(A3 E1 −A1 E3 )2 +(A1 E2 −A2 E1 )2

B2 = C3 A1 − C1 A3

A1 E2 −A2 E1 (A2 E3 −A3 E2 )2 +(A3 E1 −A1 E3 )2 +(A1 E2 −A2 E1 )2

B3 = C1 A2 − C2 A1

C1 = √ C1 = √ C3 = √

(4)

Equation 1 can be used to express the local coordinates as a function of the global coordinates: x = a11 (X − X1 ) + a12 (Y − Y1 ) + a13 (Z − Z1 ) y = a21 (X − X1 ) + a22 (Y − Y1 ) + a23 (Z − Z1 )

(5)

z = a31 (X − X1 ) + a32 (Y − Y1 ) + a33 (Z − Z1 ) and the inverse of equation 5 is (note that the coordinates of point 1, origin of the local coordinate system, are zero in the local frame): X = a11 x + a21 y + a31 z + X1 Y = a12 x + a22 y + a32 z + Y1

(6)

Z = a13 x + a23 y + a33 z + Z1

IV.

Definition of the Fibers’ Curvilinear Path at Layer and Element Levels

The proposed finite element presents a generic definition of the pattern defining the curvilinear fiber. This is achieved as follows. Consider a user-selected layer coordinate system x b k , yb k and zb k with origin on a point on the layer reference plane (see Figure 3). This coordinate system is adopted to identify the fibers’ paths. Note that 7 of 44 American Institute of Aeronautics and Astronautics

each layer needs to have its own coordinate system so that general Variable Angle Tow multi-layer structures can be easily modeled. Given a triangular finite element, the curvilinear fibers’ paths need to be provided. The user can select some points and curve fitting may be adopted to retrieve an analytical expression. From a practical point of view, one needs to provide the “fundamental curve” which is then replicated with translation in the element coordinate system (see Figure 4). The fundamental curve is conveniently defined

yk

1

yk

line of maximum y k coordinate

1

known user-provided points which identify the fundamental curve

3

3

line of minimum y k coordinate

i k2

i k2

xk

2

2

i k1

yk

1

i k1 fundamental curve

3

i k2

xk

the paths are obtained with translation in the x k direction of the fundamental curve

xk

2

i k1 Figure 4. Finite element and curvilinear fibers.

Rk 1

fundamental curve

B

3

2

Yk

A

in this coordinates system the triangle is included in a square (of dimension equal to 2). The fundamental curve is provided in that plane

Figure 5. Coordinate system in which the fundamental curve is defined. k in a different coordinate system ξ k , η k (see Figure 5) where Legendre polynomials can be defined. Let ybmax k be the largest yb coordinate (it can be negative). In the example of Figure 4 it corresponds to point 1. k k Similarly, it is possible to define ybmin (in the example of Figure 4 it corresponds to point 2), x bmax (in the k example of Figure 4 it corresponds to point 3), and x bmin (in the example of Figure 4 it corresponds to point k k 2). The new coordinate system is selected so that ybmax corresponds to η k = 1, ybmin corresponds to η k = −1,

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k k x bmax corresponds to ξ k = 1, and x bmin corresponds to ξ k = −1. The linear transformation is the following:

ξk = −

k k 2 x bmax +x bmin + k x bk k k k −x bmin bmin x bmax x bmax − x

ηk = −

k k 2 ybmax + ybmin + k yb k k k k ybmax − ybmin ybmax − ybmin

(7)

and the inverse transformation is k k k x bmax +x bmin x bk − x bmin + max ξk 2 2 k k yb k − ybmin yb k + ybmin yb k = max + max ηk 2 2

x bk =

(8)

Let µ k be a parameter used to describe the fundamental curve in the ξ k , η k plane. It is selected to have point A (see Figure 5) when µ k = −1 and point B when µ k = +1. Point A is always on the line of minimum yb k coordinate. The fundamental curve

is defined from the originally given points as a combination of Legendre polynomials Pgk µ k and Phk µ k as follows:



ξ k µ k = a0k P0k µ k + a1k P1k µ k + ... = aqk Pqk µ k



η k µ k = b0k P0k µ k + b1k P1k µ k + ... = bhk Phk µ k

q = 0, 1, ..., Q (9) h = 0, 1, ..., H

(The reader should note that h also indicates the thickness. However, in this context h is used as an index) In the practice the coefficients of the Legendre polynomials are calculated with the collocation method. This is now briefly discussed for the variable ξ k . Let’s assume that Q + 1 is the number of coefficients that need to be determined (for example, if Q = 3 it means
that maximum polynomial included is the cubic one and so 4 coefficients need to be determined). ξ k µ k is then calculated at the zeros of the Legendre polynomial Q + 1.
Each evaluation of ξ k µ k at the ith zero corresponds to an equation. Then a system of equations is determined and the coefficients akq found. After collocation, the coefficients akq and bkh are known. Using equations 8 and 9 the parametric representation of the fundamental curve becomes: k k k
x bk − x bmin + xmin x bmax + max aqk Pqk µ k 2 2 k k k k
+ y b − ybmin y b y b min ybfk = max + max bhk Phk µ k 2 2

x bfk =

(10)

The subscript f has been used to clearly indicate that the quantities are referred to the fundamental one. All the other curves describing the fibers patterns (see Figure 4) are obtained from the fundamental curve by a rigid translation in the x b k direction. Let x bck , ybck be the coordinates of a point on one of these curves. From equation 10 it is immediately deduced: k k


x bk + x bmin x bk − x bmin x bck µ k = x bfk µ k + dck = max + max aqk Pqk µ k + dck 2 2 k k k k


y b + y b y b − y b min min k k ybck µ k = ybfk µ k = max + max bh Ph µ k 2 2

(11)

where dck is a constant and represents the “distance”, in the x b k direction, of the curve from the fundamental one. One important quantity that needs to be derived is the local anglea ϑ k (see Figures 6 and 7) at a given location identified by coordinates x b k , yb k . That point is on a curve which has expression of the type shown in equation 11: a In the literature the angle is expressed as a value between −90 and +90 and this is the form that will be used to present the results. However, in this theoretical formulation the angle is selected to vary from 0 to 360 for convenience.

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yk

1

hk

fundamental curve

3

ik

rk

2

2

xk

i k1 Figure 6. Local fiber orientation angle.

k k
x
bk + x bmin x bk − x bmin x b k µ k = max + max aqk Pqk µ k + d k 2 2 (12) k k k k

− ybmin yb + ybmin yb + max bhk Phk µ k yb k µ k = max 2 2 k In the practical where d is a positive or negative constant which has similar meaning of dck earlier discussed.
problem it is not actually relevant to know the value of d k . The position vector r k µ k (see Figure 6) is:



k
k rk µk = x b k µ k bi1 + yb k µ k bi2 ˜ ˜ ˜ The unit tangent vector v k (see Figure 7) is: ˜

y k µ k bk dr k µ k db x k (µ) b k db i + i ˜ k
dµ dµ k ˜1 dµ k ˜2 v k µ k = k k
= s
2  k k
2 dr µ ˜ ˜ db xk µk db y µ dµ k + dµ k dµ k where the derivatives are calculated from equation 12:
k k
db xk µk −x bmin x bmax = aqk Pq0 k µ k dµ k 2
k
db yk µk yb k − ybmin = max bhk Ph0 k µ k k dµ 2

(13)

(14)

(15)

In reality, the layer coordinate system x b k and yb k is not used to define the fibers’ angles. The element coordinate system x, y needs to be adopted (see Figure 7). A transformation of coordinates is in place: x b k = + (x − xo ) cos ϕ k + (y − yo ) sin ϕ k yb k = − (x − xo ) sin ϕ k + (y − yo ) cos ϕ k

(16)

where xo and yo are the coordinates (x, y coordinate system) of the origin of the cartesian frame x b k , yb k (in the case of Figure 7 xo is positive whereas yo is negative). Another equivalent way of representing equation 16 is the following: x b k = −xo cos ϕ k − yo sin ϕ k + x cos ϕ k + y sin ϕ k = x b1k + x cos ϕ k + y sin ϕ k yb k = +xo sin ϕ k − yo cos ϕ k − x sin ϕ k + y cos ϕ k = yb1k − x sin ϕ k + y cos ϕ k

(17)

The angle ϕ k indicates a counterclockwise (i.e., positive rotation if consistent with the element thickness coordinate z according to the right hand rule) rotation required to make the local coordinates x, y parallel to x b k , yb k . (ϕ k is defined in Figure 7). ϕ k is an input provided for each element. 10 of 44 American Institute of Aeronautics and Astronautics

i2

y v k ÝW k Þ

yk

hk

1 hk

jk

x

rk

i k2

3

i1

2

o

xk

i k1

Figure 7. Element coordinate system and element’s fiber coordinate system.

The inverse of equation 16 is the following:

x= x bk − x b1k cos ϕ k − yb k − yb1k sin ϕ k

y= x bk − x b1k sin ϕ k + yb k − yb1k cos ϕ k

(18)

Equation 16 can be used to obtain the transformation of basis (see Figure 7 for a representation of the unit vectors): bi k = + cos ϕ k i + sin ϕ k i 1 2 1 ˜ ˜ ˜ (19) k bi = − sin ϕ k i + cos ϕ k i 1 2 2 ˜ ˜ ˜ and so it is now possible to calculate the trigonometric functions necessary to rotate the stiffness tensor (see equation 14 and consider equation 19):

db yk µk db xk µk k cos ϕ − sin ϕ k   k

dµ k dµ k k k k cos ϑ µ = v µ • i1 = s 2  k k 2 ˜ ˜ db x k (µ k ) db y (µ ) + dµ k dµ k (20)

k k db xk µk db y µ sin ϕ k + cos ϕ k   k

dµ k dµ k k k k sin ϑ µ = v µ • i2 = s 2  k k 2 ˜ ˜ db x k (µ k ) db y (µ ) + dµ k dµ k As will be shown later, to calculate the stiffness matrix, it is necessary to evaluate the different quantities at the Gauss points. Let xg and yg be the coordinates of one of the  elements’s 
Gauss  points (the
element local coordinate system is considered). The trigonometric functions sin ϑ k µ k and cos ϑ k µ k , expressed in equation 20, need to be found at each Gauss point. To achieve this, the corresponding parameter µ k ≡ µgk has to be identified so that equation 15 and then equation 20 can be used. This is accomplished as follows: • The coordinates of each Gauss point are transformed and expressed in the layer coordinate system

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where the fundamental curve is defined (see equation 16): x bgk = + (xg − xo ) cos ϕ k + (yg − yo ) sin ϕ k

(21)

ybgk = − (xg − xo ) sin ϕ k + (yg − yo ) cos ϕ k

• The corresponding parameter µgk is numerically obtained by solving the following equation (see the second relation of equation 12 written in correspondence of the Gauss point under consideration): ybgk =

k k k
yb k − ybmin ybmax + ybmin + max bhk Phk µgk 2 2

(22)

• With the knowledge of µgk (obtained from relation 22) equations 15 and 20 are fully determined. The numerical solution of equation 22 is now discussed. The residual is defined as R=

k k k
ybmax + ybmin yb k − ybmin + max bhk Phk µ k − ybgk 2 2

(23)

It is zero if the solution of equation 22 is exactly known. The first-order Taylor series of the residual about µgk (µgk = 0 could be a good initial choice considering that the parameter is included between −1 and +1) is:

dR
k k R µ ≈ R µg + µ k − µgk (24) dµ µ k =µ k g

The new guess for the value of

µgk

is obtained by setting the residual equal to zero: k µg

=

µgk

R µgk − dR dµ k k µ

or k µg


(25)

=µgk

k k k
+ ybmin yb k − ybmin ybmax + max bhk Phk µgk − ybgk 2 2 = µgk − k k
− ybmin ybmax bhk Ph0 k µgk 2

(26)

k

The procedure is repeated by updating the starting point and setting µgk = µg . At the end of the iterative process the value of the parameter µgk corresponding to the Gauss point is determined. Thus, the sine and cosine of the local fiber angle ϑ k (see Figure 7) can be determined from equation 20.

V.

Formulation of In-plane and Out-of-plane Functionally Graded Properties

The present shell formulation is made of several triangular plate elements. Thus, within the generic finite element, the integrals are calculated assuming a plate geometry and neglecting the curvature effects (if the curvature is important in the structure under investigation, then more elements are required for computational accuracy). The geometric interfaces between layers are assumed to be a function provided by the user and can be non-planar surfaces. We know discuss the case of a generic layer k. It should be noted that the coordinate system x b k , yb k is not the local coordinate system x, y used to define the finite element matrices. This is done to allow a very general placement of the fiber patterns. The choice of the local coordinate system x, y is related to the orders of expansion, type of theory etc. which are used in the Generalized Unified Formulation approach. This approach provides great versatility and generality of the finite element implementation because the user can decide to increase the accuracy of the description in any desired direction (for example the local y direction in one element and the local x direction in another element).

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A.

Element’s Variable Thickness

Modeling the variability of thickness may be relevant in Variable Angle Tow Composite Laminates (see Figs. 1 and 4 of ref. [20]). How this is accomplished in the present computational architecture is discussed next. The bottom surface of layer k is indicated as zbotk . The top surface is indicated as ztopk .These surfaces are both functions of x, y if the element thickness is not constant. zbotk and ztopk are measured from the reference coordinate system of the entire plate (not the layer’s one: see Figure 3 for a pictorial representation of the element reference plane). These functions are provided in the coordinate system reported in Figure 8. The transformation is easily achieved with the following equations:

y

R

note that the axes are parallel

xmax xmin

1 3

2

Y

ymax

ymin

x in this coordinates system the triangle is included in a square (of dimension equal to 2). Figure 8. Element coordinate system used to define the thickness variation along the plate. Note that a single coordinate system is used for the entire plate.

xmax + xmin 2 + x xmax − xmin xmax − xmin ymax + ymin 2 η=− + y ymax − ymin ymax − ymin

ξ=−

(27)

the inverse transformation is

xmax − xmin xmax + xmin + ξ 2 2 (28) ymax + ymin ymax − ymin y= + η 2 2 The layer interfaces are expressed as a function of products of Legendre polynomials of both coordinates ξ and η: ztopk (ξ, η) = aτztop Pτztop (ξ) · asztop Psztop (η) k k k k (29) zbotk (ξ, η) = aτzbot Pτzbot (ξ) · aszbot Pszbot (η) x=

k

where

k

ztopk

τztopk = 0, 1, ..., Nξ

zbotk

τzbotk = 0, 1, ..., Nξ

k

k

ztopk

sztopk = 0, 1, ..., Nη

zbotk

(30)

szbotk = 0, 1, ..., Nη

ztop

Nξ k is the degree of the Legendre polynomial used to express the functional dependance of ztopk with ztop respect to ξ, Nη k is the degree of the Legendre polynomial used to express the functional dependance zbot of ztopk with respect to η; Nξ k is the degree of the Legendre polynomial used to express the functional zbot dependance of zbotk with respect to ξ, Nη k is the degree of the Legendre polynomial used to express the functional dependance of zbotk with respect to η. Due to the mapping (relations 27 and 28) equation 29 can be formally written in the physical coordinate system: ztopk (x, y) = aτztop Pτztop (x) · asztop Psztop (y) k

k

k

k

zbotk (x, y) = aτzbot Pτzbot (x) · aszbot Pszbot (y) k

k

k

k

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(31)

Since the geometry of the multilayer structure made of layers with variable thickness is known, the coefficients of the expansion in equation 31 are provided by the user (preprocessing phase). B.

Element’s Variable Material Properties in both In-plane and Out-of-plane Directions

y yk

xmk ymk 1

hk

3

x 2

o

xk

Figure 9. Material coordinate system at a given location on a generic curvilinear fiber. k k which changes , and zm The material is assumed orthotropic in its material coordinate system xkm , ym point by point according to Figure 9. At any point the material coordinate system has one axis (xkm ) tangent k to the curve representing the curvilinear path at that point, another axis (ym ) is perpendicular to the first one and parallel to the reference plane. The third axis zm follows the right hand rule and is directed along the z direction. k k k The quantities that need to be point-by-point defined are the Young moduli E11 , E22 , and E33 , the shear k k k k k k k moduli G12 , G13 , and G23 , and Poisson’s ratios υ12 , υ13 , and υ23 . For example, E11 is the elastic modulus in the xkm direction. Using the same logic that led to equation 31 and adopting Legendre polynomials in the
thickness direction it is possible to write the explicit form of Gk12 x, y, z k as follows (details for the other material properties are reported in equation 78 in Appendix A):


Gk12 x, y, z k = aτGk PτGk (x) · asGk PsGk (y) · arGk PrGk 12

12

12

12

12

zk




(32)

12

where, for example, it is Gk 12

τGk12 = 0, 1, ..., Nξ Gk

and Nξ 12 indicates the order of the Legendre polynomial. Note that the functionally graded properties have to be provided layer by layer. k k In the material coordinate system xkm , ym , and zm , Hooke’s coefficients are calculated with the formulas: k k k k υ31 k υ k + υ23 1 − υ23 υ32 k k E11 C12 = 21 E11 k k ∆ ∆ k k υk υk + υk k υ k + υ12 υ31 k k = 21 32 k 31 E11 C23 = 32 E22 ∆ ∆k k k = Gk23 C55 = Gk13 C66 = Gk12

k k 1 − υ13 υ31 k E22 ∆k k k 1 − υ12 υ21 k = E33 ∆k

k C11 =

k C22 =

k C13

k C33

k C44

(33)

k k k k k k k k k ∆k = 1 − υ23 υ32 − υ12 υ21 − υ13 υ31 − 2υ21 υ32 υ13 k k E Ek k E k k k k k where υ32 = 33 υ23 υ21 = 22 υ12 υ31 = 33 υ k k k 13 E22 E11 E11

All quantities used in equation 33 are functions of x, y, z. All the functionally graded properties are referred to the plate coordinate system (but defined at layer level).

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Following the procedure that led to equation 26, the stiffness tensor can be rotated at any given location x, y as follows (notice that in the thickness direction within a layer the local fiber orientation does not change): k

C 66 = C k


4

k C66 − 2 Ck


2

Sk


2

k C66 + Sk


4

k C66 + Ck


2

Sk


2

k C11 − 2 Ck


2

Sk


2

k C12 + Ck


2

Sk


2

k C22

where C k = cos ϑk

S k = sin ϑk

(34) Note that if a different layer is considered a different value for the angle ϑk is obtained for the same position in the x, y plane (see equation 20). All the remaining coefficients are reported in equation 79 in Appendix A.

VI.

The Generalized Unified Formulation (GUF)

With GUF the variables are expanded in the thickness direction with an axiomatic approach. The following features are typical of the Generalized Unified Formulation for a displacement-based computational framework (i.e., the Principal of Virtual Displacements, PVD, is used to derive the governing equations): • Given an element coordinate system x, y, z, a different type of representation is possible for each displacement component. For example, the displacement ux in the x direction may be described with an Equivalent Single Layer formulation; the displacement uy in the y direction my be axiomatically expanded with an Equivalent Single Layer approach but including Zig-Zag effects via Murakami’s Zig-Zag Function (MZZF); the displacement uz may be simulated with a Layer Wise (LW) theory. • Different orders of expansions can be used for the different displacements. For example ux may have a cubic expansion whereas uz could have a parabolic dependence on the thickness coordinate. More precise descriptions are now reported. A.

Designation of the Axiomatic Models

The theories are differentiated from each other by the following descriptors: • Variational statement: the governing equations can be variationally derived from a displacement-based statement such as the Principle of Virtual Displacements (PVD). This is the approach followed in the present work. • Type of axiomatic model : the different displacement variables can have an ESL description (enhanced or not with the Zig Zag function) or a Layer Wise representation. • Order of expansion: the different displacement variables can have an independently selected order of expansion in the thickness direction. These descriptors are shown in Figure 10. Figures 11 and 12 shows some theories and relative notations in the GUF framework. B.

GUF Formalism

The Finite Element implementation within the Generalized Unified Formulation framework is now discussed. Consider the displacement variable ux . Assume that it is represented with a parabolic Equivalent Single Layer discretization. This could be the case of theories EEE P VD222 , ELZ P VD224 , EZZ P VD232 , and ELL P VD212 . Within GUF formalism the displacement ux (which is a function of the thickness coordinate z and the in-plane coordinates x, y) is written as follows: X x ux = xFαux uxαux ≡ Fαux uxαux = xFt uxt + xF2 ux2 + xFb uxb αux =t,l,b (35) = ux0 (x, y) + zφ1ux (x, y) + z 2 φ2ux (x, y)

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“PVD” means that the infinite number of theories that can be generated with the Generalized Unified Formulation are based on the Principle of Virtual Displacements (PVD). Other variational statements can be used.

Dux is a descriptor adopted for the displacement variable u x . An Equivalent Single Layer description is used for u x . Dux = E No Zig-Zag form is enforced a priori for u x . Dux = Z

An Equivalent Single Layer description is used for u x . The Zig-Zag form is enforced a priori for u x .

Dux = L

A Layer Wise description is used for u x .

Duy is a descriptor adopted for the displacement variable u y (see detailed explanation of the descriptor D u x ).

N uy is the order of expansion (thickness direction) used for the u y displacement N uz is the order of expansion (thickness direction) used for the u z displacement

Dux Duy Duz

PVD N

ux

N uy N uz

N ux is the order of expansion (thickness direction) used for the u x displacement

Duz is a descriptor adopted for the displacement variable u z (see detailed explanation of the descriptor D u x).

Figure 10. Axiomatic models obtained from the Generalized Unified Formulation (GUF): definition of the acronyms and descriptors.

The unknown functions ux0 (x, y), φ1ux (x, y), and φ2ux (x, y) in equation 35 need to be discretized according to the Finite Element Method. Consider, for example, the function φ2ux (x, y) and its FE representation: φ2ux (x, y) = xNi (x, y) Φ2ux i

i = 1, 2, ..., Nn

(36)

Nn is the number of nodes (for examples, in the case of quadrilateral element with four nodes, Q4, it is Nn = 4; in the case of parabolic triangular element it is Nn = 6, and for a quartic triangular element it is Nn = 15). xNi (x, y) represents the generic known shape function used to reconstruct the function φ2ux (x, y) at element level. Note that the superscript x is used in the symbolic representation of the shape functions to emphasize that one term, φ2ux (x, y), of the expansion of the displacement in the x direction, ux , is considered. In this implementation of the Generalized Unified Formulation the shape functions for all the displacements and geometry are selected to be the same. The methodology that led to the writing of equation 36 for the Finite Element representation of φ2ux (x, y) can be adopted for the remaining variables [ux0 (x, y) and φ1ux (x, y)] used in the expansion of the displacement ux (see equation 35). Expressions similar to equation 36 are then obtained: ux0 (x, y)

= xNi (x, y) Ux0 i

i = 1, 2, ..., Nn

φ1ux (x, y)

= xNi (x, y) Φ1ux i

i = 1, 2, ..., Nn

(37)

where Ux0 i , and Φ1ux i are the nodal values. Now equations 36 and 37 are written in terms of GUF notation

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Example: Z EE PVD132 GUF representation u x = FJu x u Jux u y = yFJuy u Juy u z = z FJu z u Juz x

J u x = t, l, b; l = 2, . . . , N u x +1 J u y = t, m, b; m = 2, . . . , N u y J u z = t, n, b; n = 2, . . . , N u z

u x = xFJu x u Jux u y = yFJuy u Juy u z = z FJu z u Juz

J u x = t, l, b; l = 2, . . . , N u x +1 J u y = t, m, b; m = 2, . . . , N u y J u z = t, n, b; n = 2, . . . , N u z +1

ESL + Zig-Zag ESL ESL

Expansion (  u x,  u y ,and  u z) Final Partially Zig-Zag theory u x = xFt u x t + xF2 u x 2 + xFb u x b u y = yFt u y t + yF2 u y 2 + yF3 u y 3 + yFb u y b u z = zFt u z t + zF2 u z 2 + zFb u z b

Example: Z EZ PVD122 GUF representation

u x: N u x = 1 u y: N uy = 3 uz : N uz = 2

u x: N u x = 1 u y: N uy = 2 uz : N uz = 2

u x = u x 0 + zd 1 ux + Ý?1Þk Q k u x Z

Zig Zag term

u y = u y 0 + zd 1 uy + z d 2 uy + z d 3 uy u z = u z 0 + zd 1 u z + z 2 d 2 u z 2

3

ESL + Zig-Zag ESL ESL + Zig-Zag

Expansion (  u x,  u y ,and  u z) Final Partially Zig-Zag theory u x = xFt u x t + xF2 u x 2 + xFb u x b u y = yFt u y t + yF2 u y 2 + yFb u y b u z = zFt u z t + zF2 u z 2 + zF3 u z 3 + z Fb u z b

u x = u x 0 + zd 1 u x + Ý?1Þk Q k u x Z

Zig-Zag term

u y = u y 0 + zd 1 uy + z d 2 uy 2

u z = u z 0 + zd 1 u z + z 2 d 2 u z + Ý?1Þk Q k u z Z

Zig-Zag term

Figure 11. Examples of possible theories and GUF representation.

(see Equation 35) and the final expansion for the displacement ux becomes: ux = ux0 + zφ1ux + z 2 φ2ux = xFt uxt + xF2 ux2 + xFb uxb = xFt xNi x U ti + xF2 xNi x U 2i + xFb xNi x U bi

(38)

= xNi ( xFt x U ti + xF2 x U 2i + xFb x U bi ) = xNi xFαux x Uαux i = xFαux xNi x Uαux i where the indices of the summations of equation 38 vary according to the formula αux = t, l, b;

l = 2, ..., Nux

i = 1, 2, ..., Nn

(39)

In the case of parabolic expansion for the displacement ux and cubic triangular finite element, it is Nux = 2 and Nn = 10. However, it is important to notice that the last term on the RHS of equation 38 is formally invariant with respect to both the order of expansion (Nux can assume any integer value) and the type of element (Nn is another free parameter which changes if another type of finite element is used). Using the same derivations for all the displacements it is possible to obtain the GUF Finite Element representation of the displacements for a generic PVD-based Higher Order Shear Deformation theory: ux = xFαux xNi x Uαux i

αux = t, l, b;

l = 2, ..., Nux ;

i = 1, 2, ..., Nn

uy = xFαuy yNi y Uαuy i

αuy = t, m, b;

m = 2, ..., Nuy ;

i = 1, 2, ..., Nn

αuz = t, n, b;

n = 2, ..., Nuz ;

i = 1, 2, ..., Nn

x

z

z

uz = Fαuz Ni Uαuz i

(40)

If a generic displacement variable (for example uy ) presents a Layer Wise description, it is sufficient to add the superscript k in the definitions reported in equation 40. Thus, in the GUF formalism the superscript k is always added so that the layer stiffness matrix generation is formally invariant with respect to the type of representation (i.e., ESL with ot without Zig Zag term or LW). The final expression includes all types of

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Example: Z EE PVD132 GUF representation u x: N u x = 1 u y: N uy = 3 uz : N uz = 2

ESL + Zig-Zag ESL ESL

u x = xFJu x u Jux u y = yFJu y u Juy u z = z FJu z u Juz

J u x = t, l, b; l = 2, . . . , N u x +1 J u y = t, m, b; m = 2, . . . , N u y J u z = t, n, b; n = 2, . . . , N u z

Expansion (  u x,  u y ,and  u z) Final Partially Zig-Zag theory u x = xFt u x t + xF2 u x 2 + xFb u x b u y = yFt u y t + yF2 u y 2 + yF3 u y 3 + yFb u y b u z = zFt u z t + zF2 u z 2 + zFb u z b

u x = u x 0 + zd 1 u x + Ý?1Þk Q k u x Z

Zig Zag term

u y = u y 0 + zd 1 u y + z 2 d 2 u y + z 3 d 3 u y u z = u z 0 + zd 1 u z + z 2 d 2 u z

Example: Z EZ PVD122 GUF representation u x: N u x = 1 u y: N uy = 2 uz : N uz = 2

ESL + Zig-Zag ESL ESL + Zig-Zag

u x = xFJu x u Jux u y = yFJu y u Juy u z = z FJu z u Juz

J u x = t, l, b; l = 2, . . . , N u x +1 J u y = t, m, b; m = 2, . . . , N u y J u z = t, n, b; n = 2, . . . , N u z +1

Expansion (  u x,  u y ,and  u z) Final Partially Zig-Zag theory u x = xFt u x t + xF2 u x 2 + xFb u x b u y = yFt u y t + yF2 u y 2 + yFb u y b u z = zFt u z t + zF2 u z 2 + zF3 u z 3 + z Fb u z b

u x = u x 0 + zd 1 u x + Ý?1Þk Q k u x Z

Zig-Zag term

u y = u y 0 + zd 1 u y + z 2 d 2 u y u z = u z 0 + zd 1 u z + z 2 d 2 u z + Ý?1Þk Q k u z Z

Zig-Zag term

Figure 12. Examples of possible theories and GUF representation.

axiomatic theories previously listed and read as follows: ukx = xFαux xNi x Uαkux i

αux = t, l, b;

l = 2, ..., Nux ;

i = 1, 2, ..., Nn

uky = yFαuy yNi y Uαkuy i

αuy = t, m, b;

m = 2, ..., Nuy ;

i = 1, 2, ..., Nn

ukz

αuz = t, n, b;

n = 2, ..., Nuz ;

i = 1, 2, ..., Nn

z

z

= Fαuz Ni

z

Uαkuz i

(41)

The orders of expansions for a generic variable are assumed to be the same for all layers when a Layer Wise approach is adopted. Moreover, the same types of functions are used in the thickness expansions and this means that no superscript k is necessary for example for the term xFαux . Equation 41 is used to express the derivatives of displacements so that strains and their virtual variations can be determined. For example the derivative of the displacement uky with respect to y is the following:  ∂uky ∂ y ∂ yNi y k = Fαuy yNi y Uαkuy i = yFαuy Uαuy i = yFαuy yNi,y y Uαkuy i ∂y ∂y ∂y

(42)

Similarly, it is possible to deduce that ∂ukx = xFαux xNi,x x Uαkux i ∂x

∂ukx = xFαuz ,z xNi x Uαkux i ∂z

∂ukz = zFαuz zNi,x z Uαkuz i ∂x

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(43)

k Using equations 42 and 43 the virtual strains εkyy , εkxx , and γzx can be deduced:

δεkyy = yFαuy yNi,y δ y Uαkuy i δεkxx = xFαux xNi,x δ x Uαkux i

(44)

k δγzx = xFαuz ,z xNi δ x Uαkux i + zFαuz zNi,x δ z Uαkuz i

The other virtual strains can be obtained with a similar procedure. C.

Governing Equations within GUF Formalism

The governing equations are obtained by using the Principle of Virtual Displacements. The internal virtual work δWIk at layer level involves the virtual variations of the strains and the stresses: Z
k k k k k k k k k k δkxx σxx + δkyy σyy + δγxy σxy + δγzx σzx + δγzy σzy + δkzz σzz dV k (45) δWI = Vk

where V k indicates the volume (at element level) of layer k. Using the geometric relations to express the strains as function of the displacements’ derivatives (see for example the derivations that led to equation 44) and adopting Hooke’s law to express the stresses as a function of the strains the internal virtual work at layer level can be demonstrated to be k αu βuy ij y

kα

βux ij x

kα

βuz ij z

Uβkuz j + δ y Uαkuy i Kuy uxy

k αu βuy ij z

Uβkuy j + δ y Uαkuy i Kuy uzy

kα

βux ij x

Uβkux j + δ z Uαkuz i Kuz uyz

kα

βuz ij y

δWIk = δ x Uαkux i Kux uuxx + δ x Uαkux i Kux uuzx + δ y Uαkuy i Kuy uyy + δ z Uαkuz i Kuz uuxz + δ z Uαkuz i Kuz uuzz

Uβkux j + δ x Uαkux i Kux uyy

k αu βux ij x

Uβkuy j

Uβkux j

k αu βuz ij z

Uβkuz j

k αu βuy ij x

Uβkuy j

(46)

Uβkuz j

k αu βu ij

where, for example, the term Kux uyx y is one of the kernels of the Generalized Unified Formulation and is used to generate a portion of the structural stiffness matrix by expanding the indices (see Figure 13). Equating the internal virtual work and the external virtual work leads to the following set of equations: kα

δ x Uαkux i Kux uuxx

βux ij x

k αu βuy ij y

Uβkux j + δ x Uαkux i Kux uyx

kα

Uβkuy j + δ x Uαkux i Kux uuzx

βuz ij z

Uβkuz j = δ x Uαkux i xPαkux i (47)

δ

y

k αu βu ij Uαkuy i Kuy uxy x x Uβkux j

+δ

y

k αu βu ij Uαkuy i Kuy uyy y y Uβkuy j

+δ

y

k αu βu ij Uαkuy i Kuy uzy z z Uβkuz j

=δ

y

Uαkuy i yPαkuy i

(48) Uβkuz j = δ z Uαkuz i zPαkuz i (49) After expansion of the indices and assembling in the thickness direction (see Figure 13) equation 47 becomes:
δ x UT Kux ux x U + Kux uy y U + Kux uz z U − x P = 0 (50) kα

δ z Uαkuz i Kuz uuxz

βux ij x

k αu βuy ij y

Uβkux j + δ z Uαkuz i Kuz uyz

kα

Uβkuy j + δ z Uαkuz i Kuz uuzz

βuz ij z

where the arrays x U, y U, and z U contain the nodal displacements in the x, y, and z directions at element level. x P is an array whose components are the equivalent nodal forces in the x direction at element level. The entries of the array δ x U are the virtual variations of the nodal displacements in the x direction at element level. The superscript T indicates the transpose operator. Equation 50 is satisfied if the following relation holds: Kux ux x U + Kux uy y U + Kux uz z U − x P = 0 (51)

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Expansion at element and layer levels (indices i and j)

k ij

K u xuy expansion indices J ux Kuy

k J ux K uy ij K u xuy

Matrix at layer and nodal levels

k

K uxuy

Assembling in the thickness direction

Matrix at layer and element levels

Kernel (1X1 invariant array)

K uxuy Matrix at multilayer and element levels k αu βuy ij

Figure 13. Definition of one of the invariant kernel: case of Kux uyx

and its expansion.

which represents the first set of governing equations. Repeating the procedure for equations 48 and 49, it is possible to obtain the governing equations at finite element level as follows: KU U · U = P

(52)

where U contains all the arrays of nodal displacements at element level and P contains all the arrays of nodal forces at element level. The explicit expressions for the arrays of unknown nodal displacements and known nodal loads (see equation 52) are:     x x U P     U =  yU  P =  yP  (53) z z U P The finite element stiffness matrix KU U (see equation 52) is:  Kux ux Kux uy   KU U =  KTux uy Kuy uy  KTux uz KTuy uz

Kux uz



  Kuy uz   Kuz uz

(54)

Note that the stiffness matrix of equation 54 is symmetric. The sub-matrices Kux ux , Kuy uy , and Kuz uz are square symmetric matrices (in general of different sizes due to the fact that different representations and orders are possible for the displacements ux , uy , and uz respectively). The partitions Kux uy , Kux uz , and Kuy uz are in general rectangular matrices. D.

Theory-Invariant Arrays: Kernels of the Generalized Unified Formulation

The governing equations (see equation 52) can be solved once KU U is determined. Looking at equation 54, it appears clear that KU U is built from the knowledge of six finite element matrices Kux ux , Kux uy , Kux uz , Kuy uy , Kuy uz , and Kuz uz . This implies that the actual GUF’s kernels required to generate the stiffness 20 of 44 American Institute of Aeronautics and Astronautics

kα

β

ij

k αu βu ij

kα

β

k αu βu ij

ij

k αu βu ij

kα

β

ij

matrix are the following: Kux uuxx ux , Kux uyy y , Kux uuzx uz , Kuy uyy y , Kuy uzy z , and Kuz uuzz uz . These matrices require evaluation of integrals over the volume of the layer (at element level). From a practical point of view the integral is split in an integral over the thickness and one over the element plane (indicated with Ω). Some example of thickness integrals within GUF formalism are presented in Figure 14. ztopk is the top layer z coordinate of the upper layer surface at a given location in the plane of element. zbotk has a similar meaning but is referred to the lower layer surface. After carrying out all integrations,

k 2t Example 1: definition of Z 11 u u x x k 2

t

Z 11 u x u x Ýx, yÞ =

z top k Ýx, yÞ k x

XC 11

x

2

Z 11 u x u x Ýx, yÞ =

Jux = 2 K ux = t

x

Jux = b K ux = 2

z bot k Ýx, yÞ

k b2 Example 2: definition of Z 11 u u

k b

F t dz

x

F2

z top k Ýx, yÞ k x

XC 11

Fb

x

F 2 dz

z bot k Ýx, yÞ

Figure 14. Integrals along the thickness: definitions.

the expressions for the six kernels of the Generalized Unified Formulation can be determined. Their formal k αu βu ij writing is reported in Appendix B. For clarity, Kux uyx y is reported below: Z Z k αu βu ij k αu βu k αu βu = Kux uyx y Z 12 uxx uyy xNi,x yNj,y dxdy + Z 16 uxx uyy xNi,x yNj,x dxdy Ω Ω Z Z k αux βuy x k α u βu y + Z 26 ux uy Ni,y Nj,y dxdy + Z 66 uxx uyy xNi,y yNj,x dxdy (55) Ω Ω Z k αux,z βuy,z x Ni yNj dxdy Z 45 ux uy + Ω

VII.

Interelement Boundary Conditions: the Penalty Method

Within this GUF extension to VAT structures, each element has different kinematics and types of theories referred to different local coordinate system. Thus, the issue of combining (see refs. [64, 65, 66, 67]) different kinematic assumptions arises when the interlement compatibility needs to be imposed. In fact, due to the different discriminations among the adjacent elements, the classic assembling technique, which automatically assures the equality of the global displacements is not the optimal choice. This issue is overcome in this work by adopting the penalty method [68].

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z dk z ck

k di 3

k ci3

Z

k di 1

fd element

d

l

e3 e1

X

k ci1

k di 2

k

fc

i element c 2

c

e2

j

x dk

yc

k

y dk

Y

xc

k

Figure 15. The displacements of nodes j (element c) and l (element d) need to be imposed to be the same.

A.

Displacements of the Linked Nodes

Within GUF formalism, the displacement in the local xk directionb of a generic layer k is written as


ukx xk , y k , z k = xFαkux z k x Ni xk , y k x Uαkux i

(56)

In this formulation it is assumed that two adjacent layers (parts of elements denoted as element c and element d respectively) that have nodes j and l on the “same location” (which will be linked with springs) have the same thickness on that location. The thickness can be variable, but at the finite element nodes is assumed to be the same if the two elements are physically connected (see Figure 15). This is not really a limitation because the variability of the thickness can still be considered at element level. Consequence of this assumption is that a point on layer k of element c will be linked to a point on layer k (note the same identity for the layer) of element d. If the number of layers are different between elements, fictitious mathematical layers can be inserted to make the number of layers and thicknesses the same. The displacement is evaluated in correspondence of a finite element node j of element c. All the element shape functions, except the one corresponding to node j, will be zero and the non-zero shape function takes the unitary value. Thus, it can be inferred (see equation 56) that:
k x k k x k (57) c uxj = c Fα cux zc c Uα cux j where the subscript c has been added to emphasize that finite element c is considered. No summation is implied when the index c is repeated. For the displacements in the other two local directions similar formula is valid:
k y k k y k (58) c uyj = c Fα cuy zc c Uα cuy j k c uzj

=

z k c Fα cuz

zck




z k c Uα cuz j

(59)

The displacements must be understood in local coordinate system relative to element c (see Figure 15). If c akmn indicates the generic entry of the transformation matrix that transforms the components of a vector from global to local coordinate system (at layer level) of element c (similar logic can be used for element d), it is possible to refer equations 57-59 to the global coordinate system by multiplying the array representation of the displacement vector by the transpose of the transformation matrix. This means that the b Note that the in-plane coordinates x, y are the same for all the layers; thus, it is x = xk and y = y k . For clarity of the presentation the superscript k is retained.

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global displacements (indicated with c ukXj , c ukY j , c ukZj , imagined function of the local coordinates xkc , yck , zck ) are the following: k k k k k k k c uXj = c a11 c uxj + c a21 c uyj + c a31 c uzj k c uY j

= c ak12 c ukxj + c ak22 c ukyj + c ak32 c ukzj

k c uZj

= c ak13 c ukxj + c ak23 c ukyj + c ak33 c ukzj

(60)

or k c uXj

= c ak11 xc Fαk cux xc Uαk cux j + c ak21 yc Fαk cuy yc Uαk cuy j + c ak31 zc Fαk cuz zc Uαk cuz j

k c uY j

= c ak12 xc Fαk cux xc Uαk cux j + c ak22 yc Fαk cuy yc Uαk cuy j + c ak32 zc Fαk cuz zc Uαk cuz j

k c uZj

= c ak13 xc Fαk cux xc Uαk cux j + c ak23 yc Fαk cuy yc Uαk cuy j + c ak33 zc Fαk cuz zc Uαk cuz j

(61)

The superscript k is maintained even at global level to emphasize that layer k is considered. Similar method is followed when a point, in correspondence of node l and element d is considered:

B.

k d uXl

= d ak11 xd Fαk

dux

x k d Uα dux l

+ d ak21 yd Fαk

duy

y k d Uα duy l

+ d ak31 zd Fαk

duz

z k d Uα duz l

k d uY l

= d ak12 xd Fαk

dux

x k d Uα dux l

+ d ak22 yd Fαk

duy

y k d Uα duy l

+ d ak32 zd Fαk

duz

z k d Uα duz l

k d uZl

= d ak13 xd Fαk

dux

x k d Uα dux l

+ d ak23 yd Fαk

duy

y k d Uα duy l

+ d ak33 zd Fαk

duz

z k d Uα duz l

(62)

Compatibility Enforced by Springs

k k The potential energy Uk of spring (of stiffness values KX , KYk , and KZ respectively), written considering the displacements expressed in global coordinates (see ref. [69]), is:

 Uk =



k c uXj

 1  k  c uY j 2  k c uZj



k d uXl

     −  d ukY l   k d uZl

T     

k KX

   0  0

 

0

0

KYk

0

0

k KZ

k c uXj

    k   c uY j   k c uZj



k d uXl



     −  d ukY l   k d uZl

   



(63)

Note that in equation 63 the indices j and l are not understood as variables: they indicate the nodes connected via springs as depicted in Figure 15. Using equations 61 and 62 (which report the global displacements of the nodes that need to have the same displacements imposed via penalty method), equation 63 is rewritten as (no summation on repeated indices ll and jj is implied) 2Uk = x k c Uα cux j x k c Uα cux j

where

kα

β cux jj

· K cux ccuuxx ·

kα

k α u β u jl K cux cduxx d x

β cux jj

=

K cux ccuuxx

· xc Uβk cux j + xd Uαk · xd Uβk

l dux

k k k c a11 c a11 KX

+ xd Uαk

kα

l dux l dux

u

β cux lj

· K duxdcuxx ·

k α u β u ll K duxdduxx d x

· xd Uβk · xd Uβk

k + c ak12 c ak12 KYk + c ak13 c ak13 KZ

l dux

l dux




+ + other terms

x k x k c Fαux c Fβux

kα

u

β cux lj

k k = − c ak11 d ak11 KX − c ak12 d ak12 KYk − c ak13 d ak13 KZ

kα

u

β dux jl

kα

u

β dux ll


x k x k k k = − c ak11 d ak11 KX − c ak12 d ak12 KYk − c ak13 d ak13 KZ c Fαux d Fβux
k k x k x k = d ak11 d ak11 KX + d ak12 d ak12 KYk + d ak13 d ak13 KZ d Fαux d Fβux

K duxdcuxx

K cux cduxx

K duxdduxx

(64)




x k x k d Fαux c Fβux

More details, omitted here for brevity, are reported in Appendix C. This formulation has the drawback that the springs impose the compatibility of the displacements only for a single point along the thickness of layer k. To have the compatibility enforced in a weak form in the thickness direction, one may use a distribution of springs.

23 of 44 American Institute of Aeronautics and Astronautics

Let the stiffnesses per unit of length of thickness of element c (or d, being the thickness of each layer the k same according to the assumption earlier introduced) of the springs be indicated with SX , SYk , and SZk (they are assumed constant and not a function of the layer’s thickness coordinate). Let hk be the layer thickness (the same for both elements). The potential energy is the following:

Z Uk =

1 2 hk



k c uXj

  k  c uY j  k c uZj





k d uXl

     −  d ukY l   k d uZl

T     

k SX

   0  0

0

0

SYk

0

0

SZk

 

k c uXj

    k   c uY j   k c uZj





k d uXl

     −  d ukY l   k d uZl

    dz 

(65)

(observe that it always is dz = dz k ). Equation 65 can be written in a very similar to equation 64 form. However, the stiffness terms are now different. For example, it is ztopk k α u β u jl K cuz cduzz d z

=

k − c ak31 d ak31 SX

−

k k k c a32 d a32 SY

−

k k k c a33 d a33 SZ




Z

z k c Fαuz

zk




z k d Fβuz


z k dz

(66)

zbotk

The formulation reported in equation 65 corresponds to the actual implementation of the present GUF-based capability. Note that the integral of equation 66 can be solved numerically. There are some relatively difficult theoretical derivations to enforce the interelement compatibility and the boundary condition (connection of the structure to the ground). This is explained next with particular focus on the thickness assembling of the matrices. C.

Thickness Assembling of the Springs’ Contributions

First the assembling of the finite element stiffness matrices is discussed. Then how the springs modify the resulting matrix is present. 1.

Thickness Assembling of the Finite Element Stiffness Matrix

For simplicity of the discussion assume that both elements c and d have 3 nodes only and are made of two layers. Suppose that for element c the adopted theory is EZL P V D211 :   u = ux0 + zφ1ux + z 2 φ2ux   x   k uy = uy0 + zφ1uy + (−1) ζk uyZ (67)   k k k k    uk = P0 + P1 uk + P0 − P1 uk z zt zb 2 2 Thus, it is clear that the number of terms (DOFs in the thickness direction) is the following: c Nux = 3, k k c Nuy = 3, and c Nuz = 2. P0 and P1 are the constant and linear Legendre polynomials. Suppose that for element d a different theory is adopted. For example assume that LZL P V D111 is used:  P0k − P1k k P0k + P1k k k   u + uxb u =  x x t  2 2   k (68) uy = uy0 + zφ1uy + (−1) ζk uyZ    k k k k    uk = P0 + P1 uk + P0 − P1 uk z zt zb 2 2 Thus, it is clear that d Nux = 2, d Nuy = 3, and d Nuz = 2. It should be emphasized that within GUF formalism the ESL theories do not have index k (except for the Zig-Zag term). However, when the matrix at layer level is derived the index k is considered for consistency of the notation. The assembling in the thickness direction will take care of the ESL or LW description for the different quantities. 24 of 44 American Institute of Aeronautics and Astronautics

In this example:

Jcu x = t,l ,b l = 2 Kcuy = t, m,b m = 2

EZ L PVD211 c u x = 3 and c uy = 3

ö

k tt 31

k 31 K cu x cuy

k t2 31

K cu x cuy

=

k tb 31

K cu x cuy

k 2t 31

K cu x cuy

k 2b31

k 22 31

K cu x cuy

K cu x cuy

K cu x cuy

k bt 31

k b2 31

K cu x cuy

k bb31

K cu x cuy

K cu x cuy

Matrix at layer and nodal levels (in this example nodes 3 and 1)

Kcku xt2cu31 y =

X X

X

X

k t2

Z 26cu x cuy cx N 3,y cy N 1 ,y dxcdyc +

Ic

+

k t2

Z 16cu x cuy cx N 3,x cyN 1,x dxcdyc

Ic

Ic

+

X

k t2

Z 12cu x cuy cx N 3,x cy N 1 ,y dxcdyc +

k t2

Z 66cu x cuy cx N 3,y cyN 1,x dxcdyc

z top k

Ic

k t2 Z 66cu x cuy

kt 2 Z 45,zcu x,zcuy cx N 3 cyN 1 dxcdyc

k k = XC 66 cx Ft cy F2 dz

k

z bot k

Ic

kernel evaluated for Jcu x = t and Kcuy = 2

example of integral along the thickness

Figure 16. Finite element implementation of the Generalized Unified Formulation: expansion of the thickness indices to obtain the stiffness matrix at layer, and nodal levels. Example for matrix Kkc u31 . x c uy

Matrix at layer and element levels

k

K u x uy c

c

c

=

c

c

k 21

c

c

c

k 22

c

c

k 31

k 23

c

c

c

k 32

k 33

K u x uy Ku x uy Ku x uy c

c

k tt 31

k 31

c

K u x uy Ku x uy K u x uy c

K cu x cuy

k 13

k 12

k 11

K u x uy K u x uy K u x uy

=

K cu x cuy

k 2t 31

K cu x cuy

k bt 31

K cu x cuy

c

k t2 31

K cu x cuy

k 22 31

K cu x cuy

k b2 31

K cu x cuy

c

c

k tb 31

K cu x cuy

k 2b31

K cu x cuy

k bb31

K cu x cuy

In this example: EZ L PVD211 c u x = 3 and c uy = 3

Matrix at layer and nodal levels

ö

(i

= 3 j = 1)

Figure 17. Finite element implementation of the Generalized Unified Formulation: expansion of the finite element indices i and j to obtain the matrix at layer and element levels. Example for matrix Kkc ux c uy (see also Figure 16).

25 of 44 American Institute of Aeronautics and Astronautics

Focus is now on the assembling of matrix Kkux uy . Figure 16 shows one of the terms of the matrix at layer and nodal levels. The subscript c is added for clarity. Figure 17 shows the matrix at layer and element levels, whereas Figure 18 presents the matrix after the assembling in the thickness direction is completed. It should be observed that both displacements ux and uy of element c have an ESL description. Thus, all the terms of the stiffness matrix need to be added as shown in Figure 18. Things would be different if one variable (or both) had a LW description. In that case only some elements would be added in correspondence of the degrees of freedom at the interface between layers and Figure 18 would be modified. 1 11

2 11

c

K u x uy c

c

=

1 12

2 12

2 13

1 13

K u x uy + K u x uy K u x uy + K u x uy K u x uy +K u x uy c

c

c

c

1 21

2 21

c

c

c

1 22

2 22

c

c

c

2 23

c

1 23

K u x uy + K u x uy K u x uy + K u x uy K u x uy + K u x uy c

c

c

c

c

1 31

2 31

c

c

c

c

c

1 32

2 32

2 33

c

c

1 33

K u x uy + K u x uy K u x uy + K u x uy K u x uy + K u x uy c

c

c

c

c

c

c

c

c

c

c

c

Figure 18. Finite Element implementation of the Generalized Unified Formulation: example of element matrix Kc ux c uy .

The other finite element matrices are built with similar logic (details are omitted for brevity). 2.

Spring Contributions

Assume that the two elements are located in space as in Figure 19. The goal is to impose the compatibility between the displacements of node 3 and node 4 and between 2 and 6. Suppose for simplicity that the k structure is made of two layers, and that the values of SX , SYk , SZk are assigned and known (in this implementation all the spring stiffness densities are taken to be 103 times the highest entry of the stiffness matrix, numerically obtained). The final set of equations that needs to be solved is depicted in Figure 20. This

Z 4

element d

3

e3

5 element c

2 6

1

Y e2

X

e1

goal: impose the equality of the displacements of nodes 2-6 and 3-4

Figure 19. Two triangles c and d.

formulation has the following properties: • All the different entries of the displacement vector have to be understood in local coordinate systems of the respective finite elements and not in the global coordinate system. • The loads are also provided in local element coordinate systems. • The conceptual structure made of two elements and shown in the example of Figure 19, needs to be constrained to the ground. This will be discussed later. Figures 21 and 22 show how the boundary conditions via springs are imposed. The reader should focus the attention on the thickness integrals. The particular assembling in the thickness direction implies that at the nodes where the springs are used there is the interlaminar compatibility of displacements referred to different coordinate systems (in fact, the two elements have different coordinate systems). This is the case, for example, when matrix K26 is c uz d uy built. That particular assembling in the thickness direction is consistent with the compatibility condition

26 of 44 American Institute of Aeronautics and Astronautics

contributions of the springs

12

11

13

K u x ux K u x ux K u x ux c

c

c

c

22

21

c

c

23

K u x ux K u x ux K u x ux c

c

c

=

c

32

31

K u x ux

c

c

33

K u x ux K u x ux K u x ux c

c

c

c

42

41

c

c

43

0 u x ux 0 u x ux K u x ux d

c

c

d

52

51

d

c

53

0 u x ux 0 u x ux 0 u x ux d

c

c

d

61

d

62

c

63

0 u x ux K u x ux 0 u x ux d

c

d

c

d

c

15

14

16

0 u x ux 0 u x ux 0u x ux d

c

d

c

25

24

d

c

1

26

0 u x ux 0 u x ux K u x ux

R u x ux

35

2

d

c

d

c

34

36

R u x ux

K u x ux 0 u x ux 0 u x ux d

c

44

Ku d

x d ux

54

Ku d

x d ux

64

Ku d

x d ux

d

c

c

45

Ku d

55

Ku d

d

d

x

x d ux

R

56

x d ux

65

Ku

Ku

Ku d

=

Ku d

3

R u x ux c

c

4

R u x ux c

d

x d ux

5

R u x ux

66

x d ux

c

c

d

46

x d ux

c

c

d

c

c

d

x d ux

6

R u x ux d

K u xux

K u xuy

K u xuz

K u yux

K u yuy

K u yuz

K u z ux

K u z uy

K u z uz

x

y

z

x

U

U

=

U

y

z

c

R

R

R

Figure 20. Two triangles c and d. Final system of equations that needs to be solved

at the interface. The fact that the displacements are referred to different coordinate systems is not then an issue: the equality of displacements should be regarded as a property valid for any selected direction at a given point. If only a finite number of springs is used, the procedure is very similar. For example, assume that instead of having a compatibility enforced in weak form along the thickness (procedure adopted in this example) the displacement on a given z is imposed to be the same (as when the user wants to impose the translational displacements but leave free or uncostrained the relative rotations). In that case, the user should: • Build the layer matrix relative to the spring (see Figures 21 and 22 but consider that the thickness integral is not calculated now for the springs; instead, the thickness functions are calculated only at a specific given location). • Set the springs stiffness matrices for the other layers to be null entities. • Perform the assembling in the thickness direction as explained in Figures 21 and 22. The structure has to be restrained to the ground. How this is ontained in this formulation is now discussed. Suppose that node 1 of element c is grounded (i.e., the displacements of node 1 have to be imposed to be zero). This is achieved as shown in Figure 23. As previously discussed, it is also possible to ground only a specific point (or a finite number of points) in the thickness direction (for a given finite element node). The procedure is identical to the one described in Figure 23. However, there is no integral (weak imposition of the compatibility condition) and only the contribution of the layer specifically involved by the spring connection is a non-zero quantity. Note that in the interpretation of Figures 22 and 23, the superscript “2” indicating the identity of layer 2 should not be confused with the exponent 2. 27 of 44 American Institute of Aeronautics and Astronautics

z top 2

26 K2c u2b = Ý?c a 112 d a312 S X2 ? c a122 d a 322 SY2 ? c a 132 d a332 S Z2 Þ x d uz

Xz

= Ý?c a 112 d a312 S X2 ? c a122 d a 322 SY2 ? c a 132 d a332 S Z2 Þ

Xz

2 z top 2 ?z bot 2

where Q 2 =

2 tt 26

=

26

Kcu x d uz

2 2t 26

2 bt 26

12

11

c

c

c

c

13

c

22

21

c

23

K u x uz K u x uz K u x uz c

c

c

=

c

32

31

K u x uz

c

c

33

K u x uz K u x uz K u x uz c

c

c

c

42

41

c

c

43

0 u x uz 0 u x uz K u x uz d

c

c

d

d

52

51

c

53

0 u x uz 0 u x uz 0 u x uz d

c

61

c

d

d

62

c

63

0 u x uz K u x uz 0 u x uz d

c

d

c

d

c

1 tt 26

2 2b26

1 2t 26

2 bb26

1 bt 26

15

14

bot 2

z6

1?Q 2 2

dz

1 tb26

Kcu x d uz 1 2b26

K cu x d uz 1 bb26

Kcu x d uz + Kcu x d uz

K cu x d uz

K u x uz K u x uz K u x uz

2 tb26

Kcu x d uz + K cu x d uz

K cu x d uz

z top 2

x 2 z 2 c F2 d Fb dz

z top 2 +z bot 2 z top 2 ?z bot 2

z?

Kcu x d uz + Kcu x d uz

Kcu x d uz

bot 2

Kcu x d uz

16

0 u x uz 0 u x uz 0 u x uz d

c

d

c

25

24

d

c

26

0 u x uz 0 u x uz K u x uz d

c

34

d

c

35

c

d

36

K u x uz 0 u x uz 0 u x uz d

c

44

Ku d

x d uz

54

Ku d

x d uz

64

Ku d

x d uz

c

d

45

Ku d

x d uz

55

Ku d

x d uz

65

Ku d

x d uz

c

d

46

Ku d

x d uz

56

Ku d

x d uz

66

Ku d

x d uz

Figure 21. Imposition of the compatibility of displacements. Case of matrix K26 . c ux d uz

VIII.

Implemented Triangular Elements and Relative Shape Functions

GUF is a very general technique: it can be applied to different types of axiomatic approaches such as the global-local model [70, 71] or different finite element formulations (see refs. [72, 73]). In the present work GUF finite element implementation includes linear, parabolic, cubic, and quartic triangular elements. The results of the present work have all been obtained with quartic triangular element, which for the investigated cases showed excellent numerical performances. The geometry of a triangular element could be specified by the location of its three corner nodes on the Cartesian x, y plane. The coordinates of the corners are given as xi , yi where i = 1, 2, 3. Alternatively, the corner points may be defined in terms of a parametric coordinate system (triangular coordinate system) A1 , A2 , A3 (see Figure 24). Quantities which are closely related to the element geometry are naturally expressed in area coordinates, and properties such as displacements, strains, and stresses are often expressed in the cartesian x, y coordinate system. It is then required to express the area coordinates as a function of the cartesian coordinates and viceversa.

28 of 44 American Institute of Aeronautics and Astronautics

z top 2

2 2 2 2 2 2 2 2 2 K2c utbz d26 u y = Ý?c a 31 d a 21 S X ? c a 32 d a 22 S Y ? c a 33 d a 23 S Z Þ

Xz

= Ý?c a 312 d a 212 S X2 ? c a 322 d a 222 SY2 ? c a 332 d a 223 S Z2 Þ

Xz

where Q 2 =

2 tt 26

2 t2 26

K uz c

d

uy

=

2 bt 26

2 b2 26

1 tt 26

6 Ý?1Þ Q dz 2

2

z top 2 +z bot 2 z top 2 ?z bot 2

2 bb 26

+

1 tb 26

1 t2 26

Kcu z d uy Kcu z d uy 1 bt 26

1+Q 2 2

Kcu z d uy

+

+

bot 2

Kcu z d uy

Kcu z d uy

Kcu z d uy

z top 2

z 2 y 2 c Ft d Fb dz

2 tb26

Kcu z d uy Kcu z d uy

26

z?

2 z top 2 ?z bot 2

bot 2

Kcu z d uy

1 bb 26

1 b2 26

Kcu z d uy K cu z d uy K cu z d uy

12

11

13

K u z uy K u z uy K u z uy c

c

c

c

c

22

21

c

23

K u z uy K u z uy K u z uy c

c

c

K u z uy

=

c

c

32

31

c

33

K u z uy K u z uy K u z uy c

c

c

c

42

41

c

c

43

0 u z uy 0 u z uy K u z uy d

c

c

d

d

52

51

c

53

0 u z uy 0 u z uy 0 u z uy d

c

61

c

d

d

62

c

63

0 u z uy K u z uy 0 u z uy d

c

d

c

d

c

15

14

0 uz

d

c

uy

c

d

uy

c

d

z d uy

Ku

54

Ku d

z d uy

64

Ku d

c

0 uz

44

Ku

0 uz

uy

d

16

uy

d

z d uy

c

d

uy uy

z d uy

55

Ku d

z d uy

65

Ku d

c

d

uy

K uz c

d

uy

36

45

d

0uz

26

35

34

K uz

d

c

25

24

0 uz

0 uz

z d uy

0 uz c

d

uy

46

Ku d

z d uy

56

Ku d

z d uy

66

Ku d

z d uy

Figure 22. Imposition of the compatibility of displacements. Case of matrix K26 . c uz d uy

A.

Linear and Higher-Order Triangular Element

For a triangular element whose shape functions are polynomials of pth degree, the number Nn of nodes is given by the following formula: (p + 1)(p + 2) Nn = (69) 2 so that a linear element (p = 1) has 3 nodes, a parabolic (p = 2) element has 6 nodes, a cubic triangle (p = 3) has 10 nodes, and a quartic triangle (p = 4) has 15 nodes (see Figure 25). The shape functions are now reported. 1.

Linear Triangular Element N1 = A1

2.

N2 = A2

N3 = A3

(70)

Quadratic Triangular Element N1 = A1 (2A1 − 1)

N2 = A2 (2A2 − 1)

N3 = A3 (2A3 − 1)

N4 = 4A1 A2

N5 = 4A2 A3

N6 = 4A3 A1

29 of 44 American Institute of Aeronautics and Astronautics

(71)

z top 2

2 2 2 2 2 2 2 2 2 K2c utbz c 11 u y = Ý c a 21 c a 31 S X + c a 22 c a 32 S Y + c a 23 c a 33 S Z Þ

Xz

2 2 2 = Ýc a 212 c a 312 S X + c a 222 c a 322 SY + c a 232 c a 332 S Z Þ

Xz

where Q2 =

2 tt 11

2 t2 11

K uz c

c

uy

=

2 bt 11

2 b2 11

1 tt 11

6 Ý?1Þ Q dz 2

2

2 bb 11

+

1 t2 11

1 tb 11

Kcu z c uy Kcu z c uy 1 bt 11

1+Q 2 2

Kcu z c uy

+

+

bot 2

Kcu z c uy

Kcu z c uy

Kcu z c uy

z top 2

2 tb11

Kcu z c uy Kcu z c uy

11

z 2 y 2 c Ft c Fb dz

z top 2 +z bot 2 z top 2 ?z bot 2

z?

2 z top 2 ?z bot 2

bot 2

Kcu z c uy

1 bb 11

1 b2 11

Kcu z c uy K cu z c uy K cu z c uy

12

11

13

K u z uy K u z uy K u z uy c

c

c

c

c

22

21

c

23

K u z uy K u z uy K u z uy c

c

c

=

c

32

31

K u z uy

c

c

33

K u z uy K u z uy K u z uy c

c

c

c

42

41

c

c

43

0 u z uy 0 u z uy K u z uy d

c

c

d

d

52

51

c

53

0 u z uy 0 u z uy 0 u z uy d

c

61

c

d

d

62

c

63

0 u z uy K u z uy 0 u z uy d

c

d

c

d

c

15

14

0 uz

d

c

uy

0 uz

uy

0 uz

c

d

c

0 uz

z d uy

Ku

44

Ku d

54

Ku d

z d uy

64

Ku d

c

uy

d

0uz

uy

K uz

d

z d uy

c

d

uy

z d uy

55

Ku d

z d uy

65

Ku d

d

uy

c

d

uy

36

45

d

c

26

35

34

K uz

16

uy

25

24

0 uz

d

c

z d uy

0 uz c

d

uy

46

Ku d

z d uy

56

Ku d

z d uy

66

Ku d

z d uy

Figure 23. Imposition of the ground conditions for node 1 of element c. Details relative to matrix K11 . c uz d uy

30 of 44 American Institute of Aeronautics and Astronautics

A2 = 0

A2 =

3

examples

1 2

A2 =

A3 = 1

A2 =

3 4

A3 = A3 =

1 4

point 1 Ý1, 0, 0Þ point 2 Ý0, 1, 0Þ point 3 Ý0, 0, 1Þ point A Ý 14 , 12 , 14 Þ

3 4

A2 = 1

A

1 2

2 A3 =

1 4

A3 = 0

1 A1 = 1

A1 =

A1 =

3 4

A1 =

1 2

1 4

A1 = 0

Figure 24. Area Coordinates 3

3

6 5

1

1 4 Quadratic Triangle

2

Linear Triangle

2

3

3 10

8 7 9

11 12

6

13

1

1 5 Cubic Triangle

9 8

10

4

15

14 4

5 6 Quartic Triangle

2

7

2

Figure 25. Linear and higher order triangles: local node numbering.

3.

Cubic Triangular Element N1 =

1 2 A1

(3A1 − 1) (3A1 − 2)

N2 =

1 2 A2

N3 =

1 2 A3

(3A3 − 1) (3A3 − 2)

N4 =

9 2 A1 A2

(3A1 − 1)

N5 =

9 2 A1 A2

(3A2 − 1)

N6 =

9 2 A2 A3

(3A2 − 1)

N7 =

9 2 A2 A3

(3A3 − 1)

N8 =

9 2 A3 A1

(3A3 − 1)

N9 =

9 2 A3 A1

(3A1 − 1)

N10 = 27A1 A2 A3

(3A2 − 1) (3A2 − 2)

31 of 44 American Institute of Aeronautics and Astronautics

(72)

4.

Quartic Triangular Element N1 =

1 3 A1

(4A1 − 3) (2A1 − 1) (4A1 − 1)

N2 =

1 3 A2

N3 =

1 3 A3

(4A3 − 3) (2A3 − 1) (4A3 − 1)

N4 =

16 3 A1 A2

(2A1 − 1) (4A1 − 1)

N6 =

16 3 A1 A2

(2A2 − 1) (4A2 − 1)

N5 = 4A1 A2 (4A1 − 1) (4A2 − 1)

(4A2 − 3) (2A2 − 1) (4A2 − 1)

N7 =

16 3 A3 A2

(2A2 − 1) (4A2 − 1)

N8 = 4A3 A2 (4A3 − 1) (4A2 − 1)

N9 =

16 3 A3 A2

(4A3 − 1) (2A3 − 1)

N10 =

16 3 A1 A3

(2A3 − 1) (4A3 − 1)

N11 = 4A1 A3 (4A1 − 1) (4A3 − 1)

N12 =

16 3 A1 A3

(4A1 − 1) (2A1 − 1)

N13 = 32A1 A2 A3 (4A1 − 1)

N14 = 32A1 A2 A3 (4A2 − 1)

(73)

N15 = 32A1 A2 A3 (4A3 − 1) B.

Numerical Integration and Differentiation

With some algebra [74] it is possible to obtain the expressions for the derivatives of the shape functions in local cartesian coordinate system:   ∂Nj        ∂A1  ∂Nj   J Jy31 Jy12  ∂Nj   ∂x     = 1  y23  (74)  ∂A2   ∂Nj  2J   Jx32 Jx13 Jx21   ∂y  ∂Nj  ∂A3 where J=

1 (Jx21 Jy31 − Jy12 Jx13 ) 2

Jx32 = Jx3 − Jx2 Jx2 = xi

∂Ni ∂A2

Jy23 = Jy2 − Jy3

Jx13 = Jx1 − Jx3 Jx3 = xi

∂Ni ∂A3

Jy31 = Jy3 − Jy1

Jx21 = Jx2 − Jx1

Jy1 = yi

∂Ni ∂A1

Jy2 = yi

Jx1 = xi ∂Ni ∂A2

Jy12 = Jy1 − Jy2 ∂Ni ∂A1

Jy3 = yi

(75) ∂Ni ∂A3

In the expresssion of the kernels several integrals over the volume of the element need to be evaluated R k αu βu (see for example the integral I = Z 12 uxx uyy xNi,x yNj,y dxdy reported in equation 55). This is numerically Ω

accomplished as follows: NΩ

I≈

h

g NG X X

k

W g · W G · C 12 Ag , ζkG




x

Fαkux ζkG




y


Fβkuy ζkG J · xNi,x (Ag ) · yNj,y (Ag ) · J (Ag )

(76)

g=1 G=1

The index g indicates the generic “in-plane” Gauss point (see the reference plane in Figure 1). J is the related Jacobian and W g is the generic Gauss weight. Ag is a compact representation to express that all the area coordinates are evaluated at the Gauss points. The index G indicates the generic Gauss point in the thickness direction. J is the Jacobian that appears in the thickness integral. ζkG is the dimensionless thickness coordinate used to calculate the integrals in the h thickness direction. NgΩ and NG are the Gauss points used for the plane and thickness directions, respectively. The number of Gauss points in the plane of element may change according to the case that needs to be investigated. Higher variability of the element’s material properties requires a larger number of Gauss points. The current implementation adoptes the Gauss points reported in ref. [75] and specifically provided for triangular elements. Table 1 shows an example of Gauss points and corresponding weights.

32 of 44 American Institute of Aeronautics and Astronautics

p

8

NgΩ

Wg

Ag1

Ag2

Ag3

16

0.144315607677787 0.095091634267285 0.103217370534718 0.032458497623198 0.027230314174435

0.333333333333333 0.081414823414554 0.658861384496480 0.898905543365938 0.008394777409958

0.333333333333333 0.459292588292723 0.170569307751760 0.050547228317031 0.263112829634638

0.333333333333333 0.459292588292723 0.170569307751760 0.050547228317031 0.728492392955404

Table 1. In-plane weights and evaluation Gauss points for integration. p indicates the complete polynomial exactly integrated.

IX.

Results

The global coordinate system X, Y, Z is located at the center of the plate (see Figure 26). The edges have length a and b (in the X and Y directions respectively). They are selected to be the same. In this work the loading condition is represented by a transverse constant distributed load. Two-layer and three-layer structures are examined. In the material coordinate system (which is variable in the space being the fibers’ patterns curvilinear) the properties are assumed orthotropic and reported in Figure 26. Called T0k and T1k the angles the fibers form with respect to the X axis at X = 0 (center) and X = a/2 (edge) respectively, the fibers’ angle is selected to change with the in-plane coordinates as follows:
2 T1k − T0k k |X| + T0k (77) ϑ (X) = a Several cases are analyzed and combined in this work. They are described in Figs. 27 and 28 where the

Y

clamped

clamped

clamped

X

h = 0. 01 m a = b = 1m E 11 = 40 GPa E 22 = 1 GPa E 33 = 1 GPa G 12 = 0. 5 GPa G 13 = 0. 5 GPa G 23 = 0. 5 GPa c 12 = 0. 25 c 13 = 0. 25 c 23 = 0. 25

clamped

Figure 26. Geometry, material properties of the layers, and mesh used for this test case (quartic triangular elements).

fibers’ patterns are shown and the corresponding values for the parameters T0 and T1 (see equation 77) provided. Note that in Figs. 27 and 28 the superscript k, indicating the layer ID, is not used because the cases presented can be adopted to any layer in the investigations carried out in this work. Table 2 presents the normalized central displacement for a two-layer structure loaded with a constant distributed load equal to 10 kN/m2 (see ref. [14]). The first layer has thickness h/2 and angles T01 and T11 corresponding to case 4 (see Figure 27). The second layer has variable parameters used to describe the curvilinear fibers (cases 1 − 8 of Figure 27). In Table 3 the first layer has thickness h/2 and angles T01 and T11 corresponding to case 13 (see Figure 28). The second layer has variable parameters used to describe the curvilinear fibers (cases 9 div 16 of Figure 28). Table 4 presents the normalized central displacement for a three-layer structure (see ref. [14]). All layers have thickness equal to h/3. In the first layer the angles T01 33 of 44 American Institute of Aeronautics and Astronautics

T 0 = 10 E T 1 = ?10 E

Y

Y

Y

Y

X

X

X

T 0 = 10 E T 1 = 20 E

T 0 = 10 E T 1 = 10 E

T 0 = 10 E T 1 = 0 E

X

case 1

case 2

case 3

case 4

T 0 = 10 E T 1 = 30 E

T 0 = 10 E T 1 = 40 E

T 0 = 10 E T 1 = 50 E

T 0 = 10 E T 1 = 60 E

Y

Y

X

case 5

Y

Y

X

case 6

X

X

case 8

case 7

Figure 27. Different patterns for the curvilinear fibers.

T 0 = 60 E T 1 = 10 E

case 9

T 0 = 0 E T 1 = 10 E

Y

X

T 0 = ?10 E T 1 = 10 E Y

Y

X

case 14

X

case 12

case 11

T 0 = 10 E T 1 = 10 E

Y

Y

X

X

case 10

T 0 = 20 E T 1 = 10 E

T 0 = 30 E T 1 = 10 E

Y

Y

X

case 13

T 0 = 40 E T 1 = 10 E

T 0 = 50 E T 1 = 10 E

Y

X

X

case 15

Figure 28. Different patterns for the curvilinear fibers.

34 of 44 American Institute of Aeronautics and Astronautics

case 16

T01 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦

Layer 1 T11 Case 20◦ 4 20◦ 4 ◦ 20 4 20◦ 4 ◦ 20 4 20◦ 4 ◦ 20 4 20◦ 4

T02 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦

Layer 2 T12 Case ◦ −10 1 0◦ 2 ◦ 10 3 20◦ 4 ◦ 30 5 40◦ 6 ◦ 50 7 60◦ 8

uz /h Present Ref [14] 1.409 (1.405) 1.210 (1.204) 1.032 (1.019) 0.993 (0.978) 1.131 (1.125) 1.353 (1.360) 1.577 (1.589) 1.766 (1.773)

Table 2. Normalized central deflection uz /h of a two-layer unsymmetric clamped Variable Stiffness Composite Laminate under a transverse distributed load equal to 10 kN/m2 . The present results have been obtained using the LLL P VD111 theory.

T01 20◦ 20◦ 20◦ 20◦ 20◦ 20◦ 20◦ 20◦

Layer 1 T11 Case 10◦ 13 10◦ 13 ◦ 10 13 10◦ 13 ◦ 10 13 10◦ 13 ◦ 10 13 10◦ 13

T02 −10◦ 0◦ 10◦ 20◦ 30◦ 40◦ 50◦ 60◦

Layer 2 T12 Case 10◦ 16 10◦ 15 ◦ 10 14 10◦ 13 ◦ 10 12 10◦ 11 ◦ 10 10 10◦ 9

uz /h Present Ref [14] 1.234 (1.210) 1.106 (1.084) 0.992 (0.973) 0.972 (0.956) 1.065 (1.048) 1.207 (1.187) 1.340 (1.319) 1.445 (1.425)

Table 3. Normalized central deflection uz /h of a two-layer unsymmetric clamped Variable Stiffness Composite Laminate under a transverse distributed load equal to 10 kN/m2 . The present results have been obtained using the LLL P VD111 theory.

T01 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦

Layer 1 T11 Case 20◦ 4 ◦ 20 4 20◦ 4 ◦ 20 4 20◦ 4 20◦ 4 ◦ 20 4 20◦ 4

T02 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦

Layer 2 T12 Case 30◦ 5 ◦ 30 5 30◦ 5 ◦ 30 5 30◦ 5 30◦ 5 ◦ 30 5 30◦ 5

T03 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦ 10◦

Layer 3 T13 Case ◦ −10 1 ◦ 0 2 10◦ 3 ◦ 20 4 30◦ 5 40◦ 6 ◦ 50 7 60◦ 8

uz /h Present Ref [14] 2.469 (2.425) 2.213 (2.177) 2.014 (1.983) 1.998 (1.971 ) 2.257 (2.232) 2.694 (2.670) 3.105 (3.071) 3.386 (3.331)

Table 4. Normalized central deflection uz /h of a three-layer unsymmetric clamped Variable Stiffness Composite Laminate under a transverse distributed load equal to 20 kN/m2 . The present results have been obtained using the LLL P VD111 theory.

35 of 44 American Institute of Aeronautics and Astronautics

and T11 correspond to case 4 (see Figure 27). The second layer has angle parameters corresponding to case 5 (see Figure 27). The third layer presents variable angle parameters used to describe the curvilinear fibers (cases 1 − 8 of Figure 27). There is an excellent correlation with published data. With the generality of the GUF-based approach several studies regarding the displacements and stresses distributions will be analyzed by changing theories and orders of expansions in the different directions. Being the fibers curvilinear, it is anticipated that for Variable Stiffness Composite Laminates GUF will provide a useful investigation tool since each direction is independently handled and this can be freely changed by the user. Finally, the above discussed extension of the Generalize Unified Formulation framework to analyze Variable Angle Tows (or Variable Stiffness Composite Laminate) is summarized in Figure 29. Variable Angle Tows and Functionally Graded Materials can be analyzed. Pure structural and aeroelastic problems can be addressed.

z2

Z

z1

Generalized Unified Formulation (GUF): The coordinate systems are freely selected

y2

x2 y1

Y X

Different kinematics for each direction can be selected Example: parabolic Equivalent Single Layer expansion in the x1direction and cubic Layerwise model in the y2 direction

x1

Figure 29. Main properties of the Generalized Unified Formulation for Variable Stiffness Composite Laminates.

X.

Conclusions

This work presented for the first time the Generalized Unified Formulation extended to the case of Variable Angle Tow composite structures. Functionally graded properties in both the thickness and in-plane direction have also been theoretically formulated. The numerical implementation was based on a quartic triangular shell element with user-defined curvilinear fiber directions. Main features of the proposed computational framework for Variable Stiffness Composite Laminates are the following: • An Equivalent Single Layer or Layer Wise description for a displacement in any element-wise direction can be selected. This feature allows the user to concentrate the computational effort where it is necessary (case dependent property). Optimization and reliability analysis are then natural applications. • The thickness axiomatic expansion of a displacement along a generic direction is completely independent of the choice made for another direction. For example, a cubic Higher Order Theory can be used for a local in-plane displacement and a linear Layer Wise approach can be adopted for a local transverse displacement variable. • Equivalent Single Layer expansions can be enhanced with Murakami’s Zig Zag functions. • Spatially variable thickness and material properties are included in the formulation. • Several different finite elements can be adopted with no additional programming effort thanks to the theory-invariant kernels of the Generalized Unified Formulation from which an infinite number of theories can be generated.

36 of 44 American Institute of Aeronautics and Astronautics

• The interelement displacement compatibility is imposed with the penalty method (weak form). This has proven to be efficient and provides to the user great versatility since the finite elements are independently modeled (different theories for user-selected directions). • The interlaminar displacement compatibility is automatically enforced via thickness assembling of the finite element matrices. • No shear correction factors are required and the transverse strain effects can be retained. The results showed excellent correlation with available data on Variable Stiffness Composite Laminates regarding two- and three-layer unsymmetric multilayer structures. Future works will assess the performance of a large number of theories which will exploit the generality of the Generalized Unified Formulation. Moreover, mixed variational approaches will also be investigated for an a-priori interlaminar transverse stress equilibrium enforcement.

XI.

Acknowledgment

This paper is dedicated to the memory of Todd O. Williams (Los Alamos National Lab, Theoretical Division) for his contributions to the theoretical modeling of structures.

37 of 44 American Institute of Aeronautics and Astronautics

Appendix A: Functionally Graded Properties and Hooke’s Coefficients This appendix presents the explicit writing (see discussion that led to equation 32) of the material properties as a function of the thickness and in-plane element’s coordinates.

k E11 x, y, z k = aτEk PτEk (x) · asEk PsEk (y) · arEk PrEk z k 11 11 11 11 11 11

k E22 x, y, z k = aτEk PτEk (x) · asEk PsEk (y) · arEk PrEk z k 22 22 22 22 22 22

k k E33 x, y, z = aτEk PτEk (x) · asEk PsEk (y) · arEk PrEk z k 33 33 33 33 33 33

k k G12 x, y, z = aτGk PτGk (x) · asGk PsGk (y) · arGk PrGk z k 12 12 12 12 12 12

k k G13 x, y, z = aτGk PτGk (x) · asGk PsGk (y) · arGk PrGk z k 13 13 13 13 13 13

k k G23 x, y, z = aτGk PτGk (x) · asGk PsGk (y) · arGk PrGk z k 23 23 23 23 23 23

k k υ12 x, y, z = aτυk Pτυk (x) · asυk Psυk (y) · arυk Prυk z k 12 12 12 12 12 12

k k υ13 x, y, z = aτυk Pτυk (x) · asυk Psυk (y) · arυk Prυk z k 13 13 13 13 13 13

k υ23 x, y, z k = aτυk Pτυk (x) · asυk Psυk (y) · arυk Prυk z k 23

23

23

23

23

(78)

23

Hooke’s coefficients obtained after considering the rotation of the coordinate system are now reported (see the discussion that led to equation 34): k


2 k
2 k
4 k
2 k
2 k k C11 + 2 Ck S C12 + S k C22 + 4 Ck S C66






2 k
2 k 4 k 4 k 2 2 k 2 2 k = C k C12 + S k C12 + Ck S k C11 + Ck S k C22 − 4 Ck S C66





3 3 3 3 k 3 k 3 k k k k = C k S k C11 − C k S k C12 + C k S k C12 − C k S k C22 + 2C k S k C66 − 2 C k S k C66
2 k
2 k = C k C13 + S k C23
4 k
2 k
2 k
4 k
2 k
2 k = C k C22 + 2 Ck S C12 + S k C11 + 4 Ck S C66




3
3 k 3 3 3 3 k k k k k = C S k C11 + C k S k C12 − C k S k C12 − C k S k C22 + 2 C k S k C66 − 2C k S k C66
2 k
2 k = C k C23 + S k C13
4 k
2 k
2 k
4 k
2 k
2 k
2 k
2 k
2 k
2 k = C k C66 − 2 Ck S C66 + S k C66 + Ck S C11 − 2 C k S C12 + C k S C22

C 11 = C k k

C 12 k

C 16 k

C 13 k

C 22 k

C 26 k

C 23 k

C 66


4

k

k k − C k S k C23 C 36 = C k S k C13
2 k
2 k k C 55 = C k C55 + S k C44
2 k
2 k k + S k C55 C 44 = C k C44 k

k k C 45 = −C k S k C44 + C k S k C55 k

k C 33 = C33

C k = cos ϑk

where S k = sin ϑk (79)

38 of 44 American Institute of Aeronautics and Astronautics

Appendix B: Kernels of the Generalized Unified Formulation The kernels of the Generalized Unified Formulation discussed when equation 55 was derived, are reported below. Z Z k αux βux x k α u βu k αux βux ij x = Z 11 ux ux Ni,x Nj,x dxdy + Z 16 uxx uxx xNi,x xNj,y dxdy Kux ux Ω Ω Z Z k αux βux x k αu βu + Z 16 ux ux Ni,y xNj,x dxdy + Z 66 uxx uxx xNi,y xNj,y dxdy (80) Ω Ω Z k αux,z βux,z x + Ni xNj dxdy Z 55 ux ux Ω

k αu βuy ij

Kux uyx

Z

k αu βu

= Ω

Z + Ω

Z + Ω

kα

Kux uuzx

βuz ij

Z Z + Ω

Z = Ω

Z + Ω

Z + Ω

k αu βuz ij

Kuy uzy

Z Ω

+ Ω

kα

βuz ij

Z = Ω

Z + Ω

Z + Ω

Z Ω

Z

k αu βu

Z 16 uxx uyy xNi,x yNj,x dxdy k α u βu

+ Ω

Z 66 uxx uyy xNi,y yNj,x dxdy

(81)

k αux βuy Z 45 ux,zuy ,z xNi yNj dxdy

Z 13 ux uz

Ni,x zNj dxdy +

k α ux

Ni zNj,x dxdy +

βuz x

Z 55 ux,zuz

Z

k αux βuz,z x

Ω

Z Ω

k α u βu Z 22 uyy uyy yNi,y yNj,y dxdy k α u βu Z 26 uyy uyy yNi,x yNj,y dxdy

Z

Z 36 ux uz

Ni,y zNj dxdy

k α ux

Ni zNj,y dxdy

βuz x

Z 45 ux,zuz

(82)

k αu βu

+ Ω

Z + Ω

Z 26 uyy uyy yNi,y yNj,x dxdy k α u βu

Z 66 uyy uyy yNi,x yNj,x dxdy

(83)

k αuy βuy Z 44 uy,zuy ,z yNi yNj dxdy

k αuy βuz,z y

= Z

Kuz uuzz

k αu βu Z 26 uxx uyy xNi,y yNj,y dxdy

k αux βuz,z x

= Ω

k αu βu ij Kuy uyy y

Z 12 uxx uyy xNi,x yNj,y dxdy +

Z 23 uy uz

Ni,y zNj dxdy +

k αuy βuz,z y

Ω

k αuy βuz Z 45 uy,zuz yNi zNj,x dxdy

Z +

k αu βu

Z 55 uzz uzz zNi,x zNj,x dxdy + k αu βu

Z 45 uzz uzz zNi,y zNj,x dxdy + k αuz

Z

Ω

Z Ω

Z Ω

Z 36 uy uz

Ni,x zNj dxdy

k α uy

Ni zNj,y dxdy

βuz y

Z 44 uy,zuz

k αu βu

Z 45 uzz uzz zNi,x zNj,y dxdy k α u βu

Z 44 uzz uzz zNi,y zNj,y dxdy

βuz,z z

Z 33 uz,zuz

(84)

Ni zNj dxdy

39 of 44 American Institute of Aeronautics and Astronautics

(85)

Appendix C: Springs’ Potential Energy and Constants Complete expression of the spring’s potential energy (see equation 64 and related discussion) is now reported. 2Uk = kα

kα

u

β cux jj

· xc Uβk cux j + zc Uαk cuz j · K cuz ccuuzx

kα

u

β cuy jj

· yc Uβk cuy j + zc Uαk cuz j · K cuz ccuzy

kα

u

β cuz jj

· zc Uβk cuz j + zc Uαk cuz j · K cuz ccuuzz

kα

u

β cuy lj

· yd Uβk

x k c Uα cux j

· K cux ccuuxx

β cux jj

· xc Uβk cux j + yc Uαk cuy j · K cuy ccuyx

x k c Uα cux j

· K cux ccuxy

kα

β cuy jj

· yc Uβk cuy j + yc Uαk cuy j · K cuy ccuyy

x k c Uα cux j

· K cux ccuuxz

kα

β cuz jj

· zc Uβk cuz j + yc Uαk cuy j · K cuy ccuyz

x k d Uα dux l

· K duxdcuxx

β cux lj

· xd Uβk

y k d Uα duy l

· K duy dcuyx

z k d Uα duz l

· K duz dcuzx

x k c Uα cux j

· K cux cduxx

· xd Uβk

l dux

x k c Uα cux j

· K cux cduxy

kα

· yd Uβk

l duy

x k c Uα cux j

· K cux cduxz

kα

· zd Uβk

l duz

x k d Uα dux l

· K duxdduxx

x k d Uα dux l

· K duxdduxy

x k d Uα dux l

· K duxdduxz

kα

u

kα

u

kα

u

kα

kα

kα

kα

u

u

u

u

u

u

u

β cux lj

· xd Uβk

β cux lj

· xd Uβk

β dux jl

β duy jl

β duz jl

β dux ll

β duy ll β duz ll

· K duxdcuxy

+

y k d Uα duy l

· K duy dcuyy

· yd Uβk

+

z k d Uα duz l

· K duz dcuzy

kα

· yd Uβk

+

y k c Uα cuy j

· K cuy cduyx

· xd Uβk

+

y k c Uα cuy j

· K cuy cduyy

kα

· yd Uβk

l duy

+

y k c Uα cuy j

· K cuy cduyz

kα

· zd Uβk

l duz

+

y k d Uα duy l

· K duy dduyx

kα

· xd Uβk

+

y k d Uα duy l

· K duy dduyy

+

y k d Uα duy l

· K duy dduyz

l dux

· yd Uβk

l duy

· zd Uβk

l duz

kα

kα

kα

kα

β cuy lj

u

u

β cuy lj

β dux jl

u

u

u

u

β duy jl

β duz jl

β dux ll β duy ll

u

u

β duz ll

u

kα

x k d Uα dux l

l dux

l dux

kα

+

l dux

· xd Uβk

kα

l duy

l duy

l duy

· yd Uβk · zd Uβk

l duy

l duz

· xc Uβk cux j +

β cuy jj

· yc Uβk cuy j +

β cuz jj

· zc Uβk cuz j +

β cuz lj

· zd Uβk

u

+

x k d Uα dux l

· K duxdcuxz

+

y k d Uα duy l

· K duy dcuyz

+

z k d Uα duz l

· K duz dcuzz

· zd Uβk

kα

· xd Uβk

l dux

l dux

l dux

kα

β cux jj

kα

kα

β cuz lj

u

u

u

β cuz lj

β dux jl

+ zc Uαk cuz j · K cuz cduzx kα

β duy jl

· zd Uβk

l duz

l duz

l duz

+

z k c Uα cuz j

· K cuz cduzy

· yd Uβk

l duy

+

z k c Uα cuz j

· K cuz cduzz

kα

· zd Uβk

l duz

+

z k d Uα duz l

· K duz dduzx

· xd Uβk

l dux

kα

· yd Uβk

l duy

· zd Uβk

l duz

+ zd Uαk

+

kα

u

u

u

u

β duz jl

β dux ll β duy ll

l duz

· K duz dduzy

z k d Uα duz l

· K duz dduzz

kα

u

β duz ll

+ + +

+ + +

+ +

(86) where

kα

β cux jj

=

k k k c a11 c a11 KX

k + c ak12 c ak12 KYk + c ak13 c ak13 KZ




x k x k c Fαux c Fβux

K cuy ccuyx

=

k k k c a11 c a21 KX

k + c ak12 c ak22 KYk + c ak13 c ak23 KZ




y k x k c Fαuy c Fβux

kα

=

k k k c a11 c a31 KX

k + c ak12 c ak32 KYk + c ak13 c ak33 KZ




z k x k c Fαuz c Fβux

=

k k k c a11 c a21 KX

k + c ak12 c ak22 KYk + c ak13 c ak23 KZ




x k y k c Fαux c Fβuy

=

k k k c a21 c a21 KX

k + c ak22 c ak22 KYk + c ak23 c ak23 KZ




y k y k c Fαuy c Fβuy

=

k k k c a21 c a31 KX

k + c ak22 c ak32 KYk + c ak23 c ak33 KZ




z k y k c Fαuz c Fβuy

=

k k k c a11 c a31 KX

k + c ak12 c ak32 KYk + c ak13 c ak33 KZ




x k z k c Fαux c Fβuz

K cuy ccuyz

=

k k k c a21 c a31 KX

k + c ak22 c ak32 KYk + c ak23 c ak33 KZ




y k z k c Fαuy c Fβuz

kα

=

k k k c a31 c a31 KX

k + c ak32 c ak32 KYk + c ak33 c ak33 KZ




z k z k c Fαuz c Fβuz

K cux ccuuxx kα

β cux jj

u

β cux jj

K cuz ccuuzx kα

u

β cuy jj

kα

u

β cuy jj

kα

u

β cuy jj

K cux ccuxy

K cuy ccuyy

K cuz ccuzy kα

β cuz jj

K cux ccuuxz kα

β cuz jj

u

β cuz jj

K cuz ccuuzz

kα

u

β cux lj

k k = − c ak11 d ak11 KX − c ak12 d ak12 KYk − c ak13 d ak13 KZ




x k x k d Fαux c Fβux

kα

u

β cuy lj

k k = − c ak21 d ak11 KX − c ak22 d ak12 KYk − c ak23 d ak13 KZ




x k y k d Fαux c Fβuy

K duxdcuxx K duxdcuxy

40 of 44 American Institute of Aeronautics and Astronautics

kα

u

β cuz lj

k k = − c ak31 d ak11 KX − c ak32 d ak12 KYk − c ak33 d ak13 KZ




x k z k d Fαux c Fβuz

kα

u

β cux lj

k k = − c ak11 d ak21 KX − c ak12 d ak22 KYk − c ak13 d ak23 KZ




y k x k d Fαuy c Fβux

K duy dcuyy

kα

u

β cuy lj

k k − c ak22 d ak22 KYk − c ak23 d ak23 KZ = − c ak21 d ak21 KX




y k y k d Fαuy c Fβuy

kα

u

β cuz lj

k k = − c ak31 d ak21 KX − c ak32 d ak22 KYk − c ak33 d ak23 KZ




y k z k d Fαuy c Fβuz

kα

u

β cux lj

k k = − c ak11 d ak31 KX − c ak12 d ak32 KYk − c ak13 d ak33 KZ




z k x k d Fαuz c Fβux

K duz dcuzy

kα

u

β cuy lj

k k = − c ak21 d ak31 KX − c ak22 d ak32 KYk − c ak23 d ak33 KZ




z k y k d Fαuz c Fβuy

kα

u

β cuz lj

k k = − c ak31 d ak31 KX − c ak32 d ak32 KYk − c ak33 d ak33 KZ




z k z k d Fαuz c Fβuz

K duxdcuxz

K duy dcuyx

K duy dcuyz

K duz dcuzx

K duz dcuzz

kα

u

β dux jl

k k = − c ak11 d ak11 KX − c ak12 d ak12 KYk − c ak13 d ak13 KZ




x k x k c Fαux d Fβux

K cuy cduyx

kα

u

β dux jl

k k = − c ak21 d ak11 KX − c ak22 d ak12 KYk − c ak23 d ak13 KZ




y k x k c Fαuy d Fβux

kα

u

β dux jl

k k = − c ak31 d ak11 KX − c ak32 d ak12 KYk − c ak33 d ak13 KZ




x k z k c Fαuz d Fβux

kα

u

β duy jl

k k = − c ak11 d ak21 KX − c ak12 d ak22 KYk − c ak13 d ak23 KZ




y k x k c Fαux d Fβuy

K cuy cduyy

kα

u

β duy jl

k k = − c ak21 d ak21 KX − c ak22 d ak22 KYk − c ak23 d ak23 KZ




y k y k c Fαuy d Fβuy

kα

u

β duy jl

k k = − c ak31 d ak21 KX − c ak32 d ak22 KYk − c ak33 d ak23 KZ




y k z k c Fαuz d Fβuy

K cux cduxz

kα

u

β duz jl

k k = − c ak11 d ak31 KX − c ak12 d ak32 KYk − c ak13 d ak33 KZ




x k z k c Fαux d Fβuz

kα

u

β duz jl

k k = − c ak21 d ak31 KX − c ak22 d ak32 KYk − c ak23 d ak33 KZ




y k z k c Fαuy d Fβuz

kα

u

β duz jl

k k = − c ak31 d ak31 KX − c ak32 d ak32 KYk − c ak33 d ak33 KZ




z k z k c Fαuz d Fβuz

K cux cduxx

K cuz cduzx K cux cduxy

K cuz cduzy

K cuy cduyz

K cuz cduzz

kα

u

β dux ll

kα

u

β dux ll

kα

u

β dux ll

kα

u

β duy ll

kα

u

β duy ll

kα

u

β duy ll

kα

u

β duz ll

kα

u

β duz ll

kα

u

β duz ll

K duxdduxx

K duy dduyx

K duz dduzx K duxdduxy

K duy dduyy

K duz dduzy K duxdduxz

K duy dduyz

K duz dduzz

=

k k k d a11 d a11 KX

k + d ak12 d ak12 KYk + d ak13 d ak13 KZ




x k x k d Fαux d Fβux

=

k k k d a11 d a21 KX

k + d ak12 d ak22 KYk + d ak13 d ak23 KZ




y k x k d Fαuy d Fβux

=

k k k d a11 d a31 KX

k + d ak12 d ak32 KYk + d ak13 d ak33 KZ




x k z k d Fαuz d Fβux

=

k k k d a11 d a21 KX

k + d ak12 d ak22 KYk + d ak13 d ak23 KZ




y k x k d Fαux d Fβuy

=

k k k d a21 d a21 KX

k + d ak22 d ak22 KYk + d ak23 d ak23 KZ




y k y k d Fαuy d Fβuy

=

k k k d a21 d a31 KX

k + d ak22 d ak32 KYk + d ak23 d ak33 KZ




y k z k d Fαuz d Fβuy

=

k k k d a11 d a31 KX

k + d ak12 d ak32 KYk + d ak13 d ak33 KZ




x k z k d Fαux d Fβuz

=

k k k d a21 d a31 KX

k + d ak22 d ak32 KYk + d ak23 d ak33 KZ




y k z k d Fαuy d Fβuz

=

k k k d a31 d a31 KX

k + d ak32 d ak32 KYk + d ak33 d ak33 KZ




z k z k d Fαuz d Fβuz

41 of 44 American Institute of Aeronautics and Astronautics

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