Explicit solutions for the modeling of laminated composite plates with arbitrary stacking sequences

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Explicit solutions for the modeling of laminated composite plates with arbitrary stacking sequences

Philippe Vidal, L. Gallimard, O. Polit

To cite this version:

Philippe Vidal, L. Gallimard, O. Polit. Explicit solutions for the modeling of laminated composite plates with arbitrary stacking sequences. Composites Part B: Engineering, Elsevier, 2014, 60, pp.697- 706. �10.1016/j.compositesb.2014.01.023�. �hal-01366906�

Explicit solutions for the modeling of laminated composite plates with arbitrary stacking sequences

P. Vidal^⇑, L. Gallimard, O. Polit

Laboratoire Energétique, Mécanique, Electromagnétisme – EA4416, Université Paris Ouest Nanterre-La Défense, 50 rue de Sèvres, 92410 Ville d’Avray, France

a b s t r a c t

In this paper, a method to compute explicit solutions for laminated plate with arbitrary stacking sequences is presented. This technique is based on the construction of an a posteriori Reduced-Order Model using the so-called Proper Generalized Decomposition. The displacement field is approximated as a sum of separated functions of the in-plane coordinatesx; y, the transverse coordinatezand the ori- entation of each plyhi. This choice yields to an iterative process that consists of solving a 2D and some 1D problems successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the in-plane description, classical Finite Element method is used. The functions ofhiare discretized with linear interpolations. Mechanical tests with different numbers of layers are performed to show the accuracy of the method.

1. Introduction

Composite and sandwich structures are widely used in the industrial field due to their excellent mechanical properties, espe- cially their high specific stiffness and strength. In this context, they can be subjected to severe mechanical loads. For laminated composite design, accurate knowledge of displacements and stresses is required. Moreover, the choice of the stacking se- quences has an important influence on the behavior of the struc- tures. The classical way consists in performing different computations with a fixed value of each orientation of the plies.

The present approach based on the Proper Generalized Decompo- sition (PGD) aims at building the explicit solutions with respect to any stacking sequences avoiding the computational cost of numerous computations.

According to published research, various theories in mechanics for the modeling of composite structures have been developed. On the one hand, the Equivalent Single Layer approach (ESL) in which the number of unknowns is independent of the number of layers, is used. But, the transverse shear and normal stresses continuity on the interfaces between layers are often violated. We can distin- guish the classical laminate theory[1](unsuitable for composites and moderately thick plates), the first order shear deformation theory [2], and higher order theories with displacement[3–10]

and mixed[11,12]approaches. On the other hand, the Layerwise

approach (LW) aims at overcoming the restriction of the ESL con- cerning the discontinuity of out-of-plane stresses on the interface layers and taking into account the specificity of layered structure.

But, the number of degrees of freedom (dofs) depends on the num- ber of layers. We can mention the folllowing contributions[13–17]

within a displacement based approach and [18,11,19]within a mixed formulation. As an alternative, refined models have been developed in order to improve the accuracy of ESL models avoiding the additional computational cost of LW approach. So, a family of models, denoted zig-zag models, was derived in [20–22] from the studies described in[23,24]. Note also the refined approach based on the Sinus model[25–27]. This above literature deals with only some aspects of the broad research activity about models for layered structures and corresponding finite element formulations.

An extensive assessment of different approaches has been made in [28–32].

Over the past years, the so-called Proper Generalized Decompo- sition (PGD)[33]has shown interesting features in the reduction model framework. A separated representation of variables called radial approximation was also introduced in the context of the LA- TIN method [34] for reducing computational costs and used to solve parametric problems [35]. For the scope of our study, the PGD has been successfully applied for the modeling of composite beams and plates [36–38]. The principle of the method consists in using separated representation of the unknown fields to build an approximate solution of the partial differential equations. A first attempt to obtain analytical solutions for composite plate based on a Navier-type solution with a separation of variables can also be found in[39].

⇑Corresponding author. Tel.: +33 140974855.

E-mail address:[email protected](P. Vidal).

This work aims at modeling composite plate structures regard- less of the stacking sequences. For this purpose, the present approach is based on the separation representation where the dis- placements are written under the form of a sum of products of (i) bidimensional polynomials of (x,y), (ii) unidimensional polynomi- als ofzand (iii) unidimensional polynomials of the orientations of the plieshi. As in[38], a piecewise fourth-order Lagrange polyno- mial ofzis chosen as it is particularly suitable to model composite structures. As far as the variation with respect to the in-plane coor- dinates is concerned, a 2D eight-node quadrilateral Finite Element (FE) is employed. The functions of the orientations of each ply are piecewise linear. Finally, the deduced non-linear problem implies the resolution ofNCþ2 linear problems alternatively (NC is the number of layers). This process yields to a 2D and 1D problems in which the number of unknowns is smaller than in a Layerwise approach. Finally, the solution depends explicitly on the fibers ori- entation of each ply.

We now outline the remainder of this article. First, the reference mechanical formulation is recalled. Then, the resolution of the parametrized problem by the PGD is described. The particular assumption on the displacements yields a non-linear problem which is solved by an iterative process. Then, the FE approxima- tions are described. Finally, numerical tests are performed. A one-layer case is first considered to assess and illustrate the behavior of the method. The influence of the discretization of the functions depending on the orientation of the plies is shown.

Two-layer and four-layer configurations are also addressed. The accuracy of the results is evaluated by comparison with reference solutions issued from PGD solutions with different fixed stacking sequences using the work developed in[38].

2. Reference problem description 2.1. The governing equations

Let us consider a plate occupying the domain V ¼XXz

bounded by@XwithXz¼ ^h₂;^h₂

in a Cartesian coordinate system ðx;y;zÞ. The plate is defined by an arbitrary regionX, in theðx;yÞ plane, located at the midplanez¼0, and by a constant thickness h. SeeFig. 1.

2.1.1. Constitutive relation

The plate can be made ofNCperfectly bonded orthotropic layers of the same material. Using matrix notation, the three dimensional constitutive law in the material coordinate is given by:

rm¼eCem ð1Þ

where the stress vectorrm, the strain vectoremandCeare written in the material coordinate.eCis defined as:

Ce¼

eC11 Ce12 eC13 0 0 0 Ce22 eC23 0 0 0 eC33 0 0 0 eC44 0 0

sym Ce55 0

Ce66

2 66 66 66 66 64

3 77 77 77 77 75

ð2Þ

The stiffness coefficientseCijare not reported for brevity reason (see[40]for more details). For each layerðkÞ, a rotation matrixR^ðkÞ is defined to obtain the constitutive law in the global coordinate system. So, we have:

r^ðkÞ11

r^ðkÞ22

r^ðkÞ33

r^ðkÞ23

r^ðkÞ₁₃ r^ðkÞ₁₂ 2 66 66 66 66 66 4

3 77 77 77 77 77 5

¼

C^ðkÞ₁₁ C^ðkÞ₁₂ C^ðkÞ₁₃ 0 0 C^ðkÞ₁₆ C^ðkÞ₂₂ C^ðkÞ₂₃ 0 0 C^ðkÞ₂₆ C^ðkÞ₃₃ 0 0 C^ðkÞ₃₆ C^ðkÞ₄₄ C^ðkÞ₄₅ 0 sym C^ðkÞ₅₅ 0 C^ðkÞ₆₆ 2

66 66 66 66 66 4

3 77 77 77 77 77 5

e^ðkÞ₁₁ e^ðkÞ₂₂ e^ðkÞ₃₃ c^ðkÞ₂₃ c^ðkÞ₁₃ c^ðkÞ₁₂ 2 66 66 66 66 66 4

3 77 77 77 77 77 5

i:e:r^ðkÞ¼C^ðkÞe^ðkÞ ð3Þ

withC^ðkÞ¼R^ðkÞeCR^ðkÞT and

R^ðkÞ¼

cos²hk sin²hk 0 0 0 2sinhkcoshk

sin²hk cos²hk 0 0 0 2 sinhkcoshk

0 0 1 0 0 0

0 0 0 coshk sinhk 0

0 0 0 sinhk coshk 0

sinhkcoshk sinhkcoshk 0 0 0 cos²hksin²hk

2 66 66 66 66 64

3 77 77 77 77 75

ð4Þ

We denote the stress vectorr^ðkÞ, the strain vectore^ðkÞ, the fibers orientationhk(Fig. 1) andC^ðkÞ_ij the three-dimensional stiffness coef- ficients of the layerðkÞin the global coordinate system.

2.1.2. The classical weak form of the boundary value problem The plate is submitted to a surface force densitytdefined over a subsetCNof the boundary and a body force densitybdefined inX. We assume that a prescribed displacementu¼udis imposed on CD¼@XCN.

The classical formulation of the elastic problem is recalled: find a displacement field u and a stress field r defined in V which verify:

the kinematic constraints:

u2 U ð5Þ

the equilibrium equations:

r2 Sand8u2 U

Z

V

r^:eðuÞdV þ Z

V

b:udV þ Z

CN

t:udC¼0 ð6Þ the constitutive relation:

r¼CeðuÞ ð7Þ

U ¼ fuju2 ðH¹ðVÞÞ³;u¼u_donCDg is the space in which the displacement field is being sought, S ¼ L²½V³ the space of the stresses, and eðuÞ denotes the linearized strain associated with the displacement.

3. Application of the Proper Generalized Decomposition to plate with any stacking sequences

The Proper Generalized Decomposition (PGD) was introduced in [33]and is based on ana prioriconstruction of separated variables Fig. 1.Geometry of the plate and orientation of the fibers of the laminated plate.

representation of the solution. The following sections are dedicated to the introduction of the PGD to build a parametric solution for composite plate with any stacking sequences. An illustration is pro- posed in[37]. We consider the problem defined by Eqs.(5)–(7)as a parametrized problem where the orientation of each plyhkis in a bounded interval Ik¼ ½h^k_m;h^k_M. The solution of Eqs. (5)–(7)for a pointMof the structure depends on the values of these angles and is denoteduðM;h1;h2;. . .hNCÞ, and the stress is linked with the strain by a parametrized constitutive lawC^ðkÞðhkÞ.

3.1. The parametrized problem

The displacement solution u is constructed as the sum of N products of separated functions (N2N is the order of the representation)

uðx;y;z;h1;h2;. . .hNCÞ ¼X^N

i¼1

gⁱ₁ðh1Þgⁱ₂ðh2Þ. . .gⁱ_NCðhNCÞfⁱðzÞ vⁱðx;yÞ ð8Þ wheregⁱ_kðhkÞ; fⁱðzÞandv^iðx;yÞare unknown functions which must be computed during the resolution process. gⁱ_kðhkÞ; fⁱðzÞ and

v^iðx;yÞare defined onIk; XzandXrespectively. The ‘‘’’ operator in Eq.(8)is Hadamard’s element-wise product. We have:

fⁱv^i¼v^ifi¼

f₁ⁱðzÞvⁱ1ðx;yÞ f₂ⁱðzÞvⁱ2ðx;yÞ f₃ⁱðzÞvⁱ3ðx;yÞ 2

64

3

75withv^i¼

vⁱ1ðx;yÞ

vⁱ2ðx;yÞ

vⁱ3ðx;yÞ 2 64

3 75fⁱ¼

f₁ⁱðzÞ f₂ⁱðzÞ f₃ⁱðzÞ 2 64

3 75

ð9Þ Note that the in-plane coordinates and the transverse coordi- nate are separated in the expression of the displacements as in [38].

For sake of clarity, the body forces are neglected in the develop- ments and the new problem to be solved is written as follows: find u2 U(space of admissible displacements) such that

aðu;uÞ ¼bðuÞ8u2 U ð10Þ

with aðu;uÞ ¼

Z

XXzI1I2...INC

CeðuÞ:eðuÞdXdXzdh1. . .dhNC

bðuÞ ¼ Z

CNI1I2...INC

t:udCdh1. . .dhNC

ð11Þ

whereXXz I1 I2. . . INCis the integration space associated with the geometric space domain XXzand the parameters do- main of the laminae orientation I1 I2. . . INC. For the present work,CN is considered as the top or bottom surfaces of the plate, that isz¼zFwithzF¼ h=2.

The strain derived from Eq.(8)is

eðuÞ ¼X^N

i¼1

gⁱ₁ðh1Þgⁱ₂ðh2Þ. . .gⁱ_NCðhNCÞ

f₁ⁱvⁱ1;1

f₂ⁱvⁱ2;2

f₃^{i 0}vⁱ3

f₂^{i 0}vⁱ2þf₃ⁱvⁱ3;2

f₁^{i 0}vⁱ1þf₃ⁱvⁱ3;1

f₁ⁱvⁱ1;2þf₂ⁱvⁱ2;1

2 66 66 66 66 66 4

3 77 77 77 77 77 5

ð12Þ

where the prime stands for the classical derivativef_i⁰¼^df_dzⁱ , andðÞ_;a for the partial derivative. The dependance with respect to the space coordinates is omitted.

3.2. decomposition of the parametrized problem

We assume that a sum ofn<Nproducts of separated functions have been already computed. Therefore, we have:

uⁿðx;y;z;h1;h2;. . .hNCÞ ¼Xⁿ

i¼1

gⁱ₁ðh1Þgⁱ₂ðh2Þ. . .gⁱ_NCðhNCÞfⁱðzÞ v^iðx;yÞ

ð13Þ and the solution at step nþ1 is sought as the sum of uⁿand a product

g^nþ1₁ ðh1Þg^nþ1₂ ðh2Þ. . .g^nþ1_NC ðhNCÞf^nþ1ðzÞ v^{nþ1ðx;yÞ of unknown}

functions. So,u^nþ1can be written as

u^nþ1ðx;y;z;h1;h2;. . .hNCÞ ¼uⁿðx;y;z;h1;h2;. . .hNCÞ

þg^nþ1₁ ðh1Þg^nþ1₂ ðh2Þ. . .g^nþ1_NCðhNCÞf^nþ1ðzÞ v^nþ1ðx;yÞ ð14Þ

Let us denote byg^nþ1the set of functionsðg^nþ1₁ ;g^nþ1₂ ;. . .g^nþ1_NC Þand by

piðgÞandpðgÞthe two following products

piðgÞ ¼Y^NC

j¼1 j–i

g_jandpðgÞ ¼Y^NC

j¼1

g_j ð15Þ

The test functions are derived from Eq.(14)

u¼X^NC

i¼1

piðg^nþ1Þg_i f^nþ1v^nþ1þpðg^nþ1Þv^nþ1f

þpðg^nþ1Þf^nþ1v ^ð16Þ

Introducing the trial function udefined by Eq. (14)and the test functionsudefined by Eq.(16)into the weak form (Eq.(10)) leads to theNCþ2 following equations, where the unknown functions g^nþ1_i ðhiÞ; g^nþ1; f^nþ1ðzÞ; v^nþ1ðx;yÞare denotedg_i; g; f; vfor sake of clarity:

for the test functionsg_i; i2 f1;. . .;NCg apiðgÞg_ifv^;piðgÞg_ifv

¼bpiðgÞfv^gi

auⁿ;piðgÞg_ifv 8g_i ð17Þ for the test functionf

aðpðgÞvf;pðgÞvfÞ ¼bðpðgÞvfÞ

aðuⁿ;pðgÞv^fÞ 8f ð18Þ for the test functionv

aðpðgÞfv^;pðgÞfvÞ ¼bðpðgÞfvÞ

aðuⁿ;pðgÞfv^Þ 8v ^ð19Þ

As these equations define a coupled non-linear system, a non- linear resolution strategy has to be used. The fixed point method is used. Starting from an initial solutionð~g^ð0Þ;~f^ð0Þ;~v^ð0ÞÞ, we construct a sequence ð~g^ðjÞÞ_jP1;ð~f^ðjÞÞ_jP1 andð~v^ðjÞÞjP1 which satisfy Eqs. (17)–

(19)respectively. For each problem involving one unknown func- tion, the two other ones are supposed to be known and come from the previous step of the fixed point strategy. So, the PGD leads to the algorithm given inTable 1.

In practice, only few iterations of the fixed point algorithm are necessary to reach a good approximation ð~f^ðjÞ;v~^{ðjÞ;~gðjÞÞ of} ðf^nþ1;v^{nþ1;gnþ1Þ(see[41]}for more details).

3.3. discretization

To compute the solution of Eqs.(17)–(19), a discrete represen- tation of the functionsðf;v^;g1;. . .g_NCÞmust be introduced. We use a classical finite element approximation inX, and a polynomial expansion inXz forv ^{and f}respectively. The elementary vector of degrees of freedom (dof) associated with one element Xe of the mesh inXis denotedq^v_e. The vector of dofs associated with the polynomial expansion inXz is denotedq^f. The displacement fields and the strain fields are determined from the values ofq^v_e andq^f by

ve¼Nxyq^v_e; E^e_v¼Bxyq^v_e; f¼Nzq^f andEf ¼Bzq^f ð20Þ The matrices Nxy; Bxy; Nz,Bz contain the interpolation functions, their derivatives and the jacobian components.E^ev andEf are de- fined inAppendices A and B, respectively.

As far as the functionsg₁;. . .;g_NCare concerned, a piecewise lin- ear interpolation is considered inðIkÞ₁6k6NC

g_kðhkÞ ¼NgkðhkÞq^gk ð21Þ

whereNgk is the matrix of the interpolation functions,q^gk are the degree of freedom.

3.3.1. Finite element problem to be solved onIk

For the sake of simplicity, the functions ~f^ðj1Þ; v~^{ðj1Þ and}

~g^ðjÞ_i

16i6NC;i–k which are assumed to be known, will be denoted

~f; v~^andg~nk¼ ðg_iÞ₁6i6NC;i–k, and the function~g^ðjÞ_k to be computed will be denotedg_k.

The variational problem defined onIkfrom Eq.(17)is Z

Ik

g_kkkxyzð~gnk;~f;v~Þg_kdhk¼ Z

Ik

g_ktkxyzð~gnk;~f;~vÞdhk Z

Ik

g_krkxyzð~gnk;~f;~v;uⁿÞdhk ð22Þ

k_kxyzð~g_nk;~f;v~Þ ¼ Z

XXzInk

eð~f~vÞ^TCeð~fv~ÞY^NC i¼1 i–k

~g²_i 2

66 66 4

3 77 77

5dXdzdh₁. . .dh_NC ð23Þ

tkxyzðg~nk;~f;~vÞ ¼ Z

XInk

ð~fv~Þ^TtpkðgÞ~

h i

dXdh1. . .dhNC

z¼zF

ð24Þ rkxyzð~g_nk;~f;v~^;unÞ ¼

Z

XXzInk

eð~fv~Þ^TCeðuⁿÞpkð~gÞ

h i

dXdzdh₁. . .dh_NC ð25Þ

withInk¼ I1 I2. . . Ik1 Ikþ1. . . INC

Note that the domain of integration in Eqs. (23)–(25)can be splitted into the parameters domains and the geometric space domain.

The introduction of the finite element approximation Eq.(21)in the variational Eq.(22)leads to the linear system

Kkxyzð~gnk;~f;v~^Þqgk¼Rg_kð~gnk;~f;v~^;unÞ ð26Þ where

q^gk is the vector of dofs associated with the expansion inIk, Kkxyzð~gnk;~f;v~^Þis the stiffness matrix obtained by

Kkxyzð~gnk;~f;v~^{Þ ¼Z}

Ik

N^T_gkkkxyzð~gnk;~f;v~^ÞNgkdhk

Rgkð~gnk;~f;v~^;unÞis the equilibrium residual obtained by Rg_kð~gnk;~f;v~^{;unÞ ¼Z}

Ik

N^T_gktkxyzð~gnk;~f;v~^Þdhk

Z

Ik

N^T_g

krkxyzð~gnk;~f;~v^;unÞdhk

3.3.2. Finite element problem to be solved onXandXz

The Finite Element problems onXandXzare a straightforward extension of the formulation given in[38] to the parametrized problem involvinghk. However, for the sake of completeness, their formulations are given inAppendices A and B.

4. Numerical results

In this section, an eight-node quadrilateral FE based on the classical Serendipity interpolation functions[10]is used for the un- knowns depending on the in-plane coordinates. A Gaussian numer- ical integration with 3 3 points is used to evaluate the elementary matrices. A fourth-order Layerwise z-expansion is also considered. Concerning the functions associated to the orientations of the plieshk, a piecewise linear function is chosen. An analytical integration of the functions with respect tohkand the transverse coordinate is performed. In the following, the plate is discretized with NxNy elements. The total number of numerical layer is denotedNz. The numbers of dofs are also precised for the three problems associated toðg_jⁿÞ_16j6NC; v^{n and fn}. They are denoted Ndofhj; NdofxyandNdofzrespectively.

First, a one-layer plate is considered to show the accuracy of the method. The influence of the discretization of the functiong₁ðh1Þis studied. Then, the method is assessed on two and four-layer plate configurations. The results are compared with reference results issued from PGD computations without introducing the laminae orientations as parameters. Indeed, the performance of this latter

10⁰ 10¹

10⁻² 10⁻¹ 10⁰

Fig. 2.Local error indicator with respect to the PGD iterations –S= 4 – 1 layer – h12 ½90;90.

Table 1

Proper Generalized Decomposition algorithm.

forn¼1 toNmax

Initialize~f^ð0Þ;v~^{ð0Þand~gð0Þ}i fori2 f1;. . .NCg forj¼1 toj_max

fori¼1 toNC

Compute~g^ðjÞ_i from Eq.(17),~g^ðjÞp (p<iÞ;~g^ðj1Þp (p>iÞ;~f^ðj1Þandv~^ðj1Þ being known

end for

Compute~f^ðjÞfrom Eq.(18),~g^ðjÞand~v^ðj1Þbeing known Computev~^ðjÞfrom Eq.(19),~g^ðjÞand~f^ðjÞbeing known Check for convergence

end for

Setf^nþ1¼~f^ðjÞ;v^nþ1¼v~^ðjÞandg^nþ1¼g~^ðjÞ Setu^nþ1¼uⁿþpðg^nþ1Þf^nþ1v^nþ1

Check for convergence end for

has been shown in [38] and can be considered as a reference solution.

All the following tests involve a simply-supported plate submit- ted to a sinusoidal pressure. They are described below:

geometry:rectangular composite plate withb¼3a. All layers have the same thicknessS¼^a_h¼4.

boundary conditions:cylindrical bending of a plate submitted to a sinusoidal pressureqðx;yÞ ¼q₀sin^p_a^x.

material properties: EL¼25 GPa; ET¼1 GPa; GLT¼0:2 GPa;

GTT ¼0:5 GPa; mLT¼mTT¼0:25 whereLrefers to the fiber direc- tion,Trefers to the transverse direction.

mesh:Nx¼Ny¼16 and the whole plate is meshed.

number of dofs:Ndofxy¼2499 andNdofz¼3ð4NCþ1Þ.

results:The results are made non-dimensional using:

u¼u1ð0;b=2;zÞ ET

hq₀S³; w ¼u3ða=2;b=2;zÞ100ET

S⁴hq₀

raa¼raaða=2;b=2;zÞ

q₀S² ; r12¼r12ð0;0;zÞ q₀S²

r13¼r13ð0;b=2;zÞ

q0S ; rzz¼rzzða=2;b=2;zÞ q0

−1000 0 100 0.5

1

−100 0 100

−5 0 5

−100 0 100

−5 0 5

−1000 0 100 1

2

−100 0 100

0.5 1 1.5

−100 0 100

−2 0 2

−1000 0 100 1

2

−100 0 100

−2 0 2

−1000 0 100 1

2

−100 0 100

−5 0 5

−100 0 100

−5 0 5

−100 0 100

−2 0 2

−100 0 100

−5 0 5

−100 0 100

0.6 0.8 1

−1000 0 100 0.5

1

−100 0 100

−2 0 2

−1000 0 100 0.5

1

−1000 0 100 0.5

1

−100 0 100

0.6 0.8 1

−100 0 100

−2 0 2

−100 0 100

−2 0 2

−1000 0 100 1

2

−100 0 100

−2 0 2

−1000 0 100 1

2

−1000 0 100 1

2

−1000 0 100 1

2

−1000 0 100 1

2

−1000 0 100 1

2

−1000 0 100 0.5

1

−100 0 100

−2 0 2

−100 0 100

−5 0 5

−100 0 100

−2 0 2

Fig. 5.16 First functionsgiðhiÞ–S= 4 – 2 layers –hi2 ½90;90;i¼1;2.

0 20 40 60 80 0

0.05 0.1 0.15 0.2

0 20 40 60 80 0

5 10 15 20

0 20 40 60 80 0.7

0.8 0.9 1

0 20 40 60 80 0.4

0.45 0.5

present ref. [38]

0 20 40 60 80 0

0.05 0.1 0.15 0.2

0 20 40 60 80 0

5 10 15 20

0 20 40 60 80 0.7

0.8 0.9 1

0 20 40 60 80 0.4

0.45 0.5

present ref. [38]

0 20 40 60 80 0

0.05 0.1 0.15 0.2

0 20 40 60 80 0

5 10 15 20

0 20 40 60 80 0.7

0.8 0.9 1

0 20 40 60 80 0.4

0.45 0.5

present ref. [38]

Fig. 4.Distribution of maxðuÞ;maxðwÞ; maxðr11Þ, and maxðr13Þwith respect to the orientationh1–Ndofh1¼30ðleftÞ;50ðmiddleÞ;150ðrightÞ–S= 4 – 1 layer – h12 ½90;90.

0 10 20 30 40 50

0 5 10 15 20 25 30 35 40

0 10 20 30 40 50

0 5 10 15 20 25 30 35 40

0 10 20 30 40 50

0 5 10 15 20 25 30 35 40

Fig. 3.Error on the maximal values ofu; w; r11andr13with respect to the PGD iterations –½0(left),½68(middle),½90(right) –S= 4 – 1 layer –h12 ½90;90.