Modeling of composite plates with an arbitrary hole location using the variable separation method

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Modeling of composite plates with an arbitrary hole location using the variable separation method

Philippe Vidal, L. Gallimard, Olivier Polit

To cite this version:

Philippe Vidal, L. Gallimard, Olivier Polit. Modeling of composite plates with an arbitrary hole

location using the variable separation method. Computers and Structures, Elsevier, 2017, 192, pp.157

- 170. �10.1016/j.compstruc.2017.07.020�. �hal-01676226�

Modeling of composite plates with an arbitrary hole location using the variable separation method

P. Vidal ^⇑ , L. Gallimard, O. Polit

UPL, Univ Paris Nanterre, LEME – Laboratoire Energétique, Mécanique, Electromagnétisme, 92410 Ville d’Avray, France

1. Introduction

Composite structures are widely used in the weight-sensitive industrial applications due to their excellent mechanical proper- ties, especially their high specific stiffness and strength. For many cases of engineering application, the interlaminar stresses are very important for prediction of delamination which is one of the criti- cal failure types of laminated systems. It is also well known that the presence of interlaminar stress components is particularly important in the regions close to free and loaded edges like bolted joints. Thus, a particular attention have to be paid to evaluate pre- cisely their influence on local stress fields in each layer, particularly at the interface between layers.

The pioneering work on straight free-edge problems can be found in the review articles [1,2]. However, due to the complicated geometry and three-dimensional nature of the stresses, laminated composite plates with curved free-edges received relatively less attention. In the following, the references are limited to the works involving the modeling of laminated plate with a hole. Classically, 3D Finite Element (FE) approaches are used [3–7], but the compu- tational cost can increase dramatically as a refined mesh is needed near the hole. Thus, the coupling with 2D approach far from the hole can be carried out [8], but the two regions have to be con- nected by dedicated formulations. To overcome that, 2D approaches have been developed and can be classified as follows:

the Equivalent Single Layer Models (ESL): the Reissner-Mindlin (FSDT, [9,10]) and higher-order models (TSDT, [11]) have been addressed. But, only global quantities can be obtained, and the integration of the equilibrium equations is needed to estimate the transverse shear stresses with accuracy. Indeed, transverse shear and normal stress continuity conditions at the interfaces between layers are violated for all of them.

the Layer-Wise Models (LW): It aims at overcoming the restric- tion of the ESL concerning the discontinuity of out-of-plane stresses at the interface between adjacent layers. These models are carried out in [11–13] with a high-order expansion of the displacements (at least three). Unfortunately, the number of unknowns depends on the number of layers.

It should be mentioned alternative approaches to overcome high computational cost. For that, the number of unknowns can be reduced by introducing the continuity conditions at the inter- face layers (on the transverse shear stresses). The so-called Zig- Zag models can be deduced as in [14]. Note also the use of hybrid stress elements based on a mixed form of Hellinger-Reissner prin- ciple in [15]. Both ESL and LW approaches are developed.

In this work, a promising alternative approach in the field of the reduced-order modeling based on the separation of variables [16]

is performed to overcome these drawbacks and give the solution for any location of the hole. Note that the present study is focused on geometrical modification while preserving the topology of the geometry. One the one hand, it is based on the spatial coordinates separation proposed in [17] and also in [18], and the suitable order

http://dx.doi.org/10.1016/j.compstruc.2017.07.020

0045-7949/Ó2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.

E-mail address:philippe.vidal@parisnanterre.fr(P. Vidal).

expansion given in [19–22] for the free-edge effects. On the other hand, a mapping transformation is used for the parameterization of the geometry to refer to a fixed configuration. It has been already carried out in the framework of the reduced basis in [23].

It has been also addressed in [24] with triangular domains for ther- mal problems and recently in [25] applied to non-straight lines for axisymmetric structures made of isotropic material. These two lat- ter references are related to the proper generalized decomposition method. The present work aims at extending theses approaches to composite plates with a hole which can be considered as a serious challenging benchmark for the designer of layered structures.

Thus, a representative test is addressed to assess the reliability and the effectiveness of the present method. The mapping transfor- mation allows us to avoid complex integrations such as in [26].

Another way to solve this problem could be to use the fictitious domain with a indicator function for the hole as in [27]. Neverthe- less, we are particularly interested in the transverse stresses near the hole. The representation of this indicator could influence the accuracy of the results in this zone of interest. Thus, this method is not chosen and we will ensure that the quality of the mesh remains good in this zone.

In our approach, the displacements are written under the form of a sum of products of bidimensional polynomials of (x, y), unidi- mensional polynomials of z and two unidimensional functions of the position of the hole X

T;

Y

T

. A piecewise fourth-order Lagrange polynomial of z is chosen. A linear interpolation is used for the func- tions of X

T

and Y

T

. As far as the variation with respect to the in- plane coordinates is concerned, a 2D eight-node quadrilateral FE is employed. The deduced non-linear problem implies the resolu- tion of four linear problems alternatively. This process yields to a 2D and three 1D problems. Moreover, the explicit solution with respect to the hole position allows us to build a virtual chart in a straightforward manner avoiding the use of numerous expensive LW computations. This could be used to determine the influence of the hole position on the strength failure as in [28]. It could be also seen as a first step towards geometric optimization for laminates.

We now outline the remainder of this article. First, the mechanical formulation is recalled. The parameterization of the hole position requires a mapping transformation which is described. Then, the iterative algorithm to solve the non-linear problem introduced by the variables separation is detailed. The FE discretization is also described. Numerical evaluations are subsequently presented for one-layer and four-layer plate. The behavior of the method is first presented and illustrated. Then, it is assessed to capture local effects and 3D state of the stress, and in particular the transverse stresses near the curved edge, by comparing with reference model available in literature. Results of a LW model issued from Carrera’s Unified Formulation [29] are used. Interesting features near the curved free edge can be emphasized even with involving very localized phenomenon.

2. Reference problem description: the governing equations Let us consider a composite structure occupying the domain

V ¼

X X

z

with X

¼ ½a=

2

;

a

=

2 ½b

=

2

;

b

=

2 and X

z¼ ½h=

2

;

h

=

2 in a Cartesian coordinate

ðx;

y

;

zÞ. The plate is defined by an arbitrary region X in the

ðx;

yÞ plane, located at the midplane for z

¼

0, and by a constant thickness h. See Fig. 1.

2.1. Constitutive relation

The plate can be made of NC perfectly bonded orthotropic lay- ers. The constitutive equations for a layer k can be written as

r

^ðkÞ

^{ðzÞ ¼ C}

^ðkÞ

e ^{ðzÞ for z 2 ½z}

k

; z

kþ1

ð1Þ

where we denote the stress vector r , the strain vector e ^{. z}

k

is the z-coordinate of the bottom surface of the layer k.

We have

C

^ðkÞ

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

ð2Þ

where C

^ðkÞ_ij

is the three-dimensional stiffness coefficients of the layer

ðkÞ.

2.2. The weak form of the boundary value problem

The plate is submitted to a surface force density t defined over a subset C

N

of the boundary and a body force density b defined in

V.

We assume that a prescribed displacement u

¼

u

_d

is imposed on C

D¼@V

C

N

.

Using the above matrix notations and for admissible displace- ment

d

u

2d

U, the variational principle is given by:

find u

2

U such that:

Z

V

e ^ðduÞ

^T

r ^{dV þ Z}

V

du

^T

bdV þ Z

CN

du

^T

td C ^{¼ 0} 8 du 2 dU

ð3Þ

where U is the space of admissible displacements, i.e. U

¼ fu2 ðH¹ðVÞÞ³=

u

¼

u

d

on C

Dg. We have also d

U

¼ fu2 ðH¹ðVÞÞ³=

u

¼

0 on C

^Dg.

3. Application of the separated representation to plate with a hole

In this section, we introduce the application of the variables separation for composite plate analysis with an arbitrary hole loca- tion. The coordinates of the hole center are considered as parame- ters of the solution. The main idea consists in using a mapping to refer to a fixed configuration. It has been already used in [25] for parameterized geometry of axisymmetric structures made of iso- tropic material, and also in [30] for cylindrical shell applications.

Moreover, the separated spatial representation [18] is considered with a suitable degree of interpolation of the transverse function for such structures as in [19].

Fig. 1.The laminated plate and coordinate system.

Thus, the classical mechanical problem defined in Section 2 is considered as a parameterized problem where the coordinates of the hole center

ðXT;

Y

TÞ

is in a bounded interval

IT¼ ITX ITY¼

½X^m_T;

X

^M_T ½Y^m_T;

Y

^M_T. The superscripts^m

and

^M

stand for minimum and maximum, respectively. The solution of this problem for a point M of the structure depends on the values of these coordinates and is denoted uðx

;

y

;

z

;

X

T;

Y

TÞ.

3.1. The parameterized problem

The displacement solution u is constructed as the sum of N products of separated functions (N

2N^þ

is the order of the representation)

uðx; y; z; X

T

; Y

T

Þ ¼ X

^N

i¼1

g

ⁱ_X

ðX

T

Þg

ⁱ_Y

ðY

T

Þ f

ⁱ

ðzÞ v

ⁱ

^{ðx; yÞ ð4Þ}

where g

ⁱ_XðXTÞ;

g

ⁱ_YðYTÞ;

f

ⁱðzÞ

and v

^iðx;

yÞ are unknown functions which must be computed during the resolution process.

g

ⁱ_XðXTÞ;

g

ⁱ_YðYTÞ;

f

ⁱðzÞ

and v

^iðx;

yÞ are defined on

ITX;ITY;

X

z

and X respectively. As this latter depends on the coordinates

ðXT;

Y

TÞ, it

will be denoted X

ðXT;

Y

TÞ

(see Fig. 2). The ‘‘” operator in Eq. (4) is Hadamard’s element-wise product. We have:

f

ⁱ

v

ⁱ

^¼ v

ⁱ

^f

ⁱ

^¼

f

ⁱ₁

ðzÞ v

ⁱ1

ðx; yÞ f

ⁱ₂

ðzÞ v

ⁱ2

ðx; yÞ f

ⁱ₃

ðzÞ v

ⁱ3

ðx; yÞ 2

66 4

3

77 5 with v

ⁱ

^¼ v

ⁱ1

ðx; yÞ

v

ⁱ2

ðx; yÞ

v

ⁱ3

ðx; yÞ 2 64

3 75 f

ⁱ

¼

f

ⁱ₁

ðzÞ f

ⁱ₂

ðzÞ f

ⁱ₃

ðzÞ 2 66 4

3 77 5

ð5Þ

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 u

2 U^ext

(U

^ext¼ fu2 ðH¹ðV ITÞÞ³=

u

¼

u

_d

on C

^{D ITg)}

such that

aðu; duÞ ¼ bðduÞ 8 du 2 dU

^ext

ð6Þ

with aðu; duÞ ¼

Z

XðXT;Y_TÞXzIT

e ^ðduÞ

^T

^C e ^{ðuÞ d} X d X

z

dX

T

dY

T

bðduÞ ¼ Z

CNIT

du

^T

td C ^dX

T

dY

T

ð7Þ

where X

ðXT;

Y

TÞ

X

z IT

is the integration space associated with the geometric space domain X

ðXT;

Y

TÞ

X

z

and the parameters domain of the hole center coordinates

IT

. For the present work, C

^N

is the edge of the plate located at y

¼

b

=

2. At this stage, it should

be noted that the integration domain X depends on the coordinates of the hole center

ðXT;

Y

TÞ.

3.2. Introduction of a mapping transformation

As it is highlighted in the previous section, the integration domain in Eq. (7) depends on the parameters X

T;

Y

T

involved in the unknown function u. To overcome this issue, a mapping trans- formation is carried out as for optimization problems or shell anal- ysis. So, a transformation

TXTYTð

X

0Þ

between the domain X

^ðXT;

Y

TÞ

and the reference fixed domain X

0

(cf. Fig. 2) is defined. The Jaco- bian of this transformation appears in the integration, and the strain has to be updated following this choice. The expression of the left hand side of Eq. (6) can be written as

aðu; duÞ ¼ Z

XðXT;YTÞXzIT

e ^ðduÞ

^T

^C e ^ðuÞd X d X

z

dX

T

dY

T

¼ Z

X0XzIT

e ^ðduðT

^XTYT

ð X

0

ÞÞ

^T

C e ^{ðuðT ð} X

0

ÞÞÞ J

_XTYT

ð X

0

Þ d X

0

d X

z

dX

T

dY

T

ð8Þ

where J

_X

TY_Tð

X

0Þ

is the modulus of the determinant of the geometric transformation Jacobian matrix.

For our application, it should be noted that the geometry will be divided into NbArea areas. A domain decomposition X

^p₀

is defined from the initial one X

0

. In each subdomain, the geometrical trans- formation remains affine (from X

^pXY

to X

^p₀

, see Fig. 3). Indeed, the integral in Eq. (8) must be split into integrals over each variable, so the expression of J

_XTYTð

X

0Þ

must be separated as well. It is an important feature to keep the separability of the variable as it is also noticed in [25]. Moreover, the area around the hole (red part in Fig. 3) is chosen to be fixed to keep the quality of the mesh in this area where out-of-plane stresses occur.

Thus, to express the strain in the reference configuration X

0

, it is needed to introduce the Jacobian matrix of the geometric transfor- mation. The coordinates associated to the reference frame X

^p₀

and the configuration X

^pXY

will be denoted x

;

y and X

;

Y, respectively.

This matrix can be written as J

_X^p

0X^p_XY

¼

@X

@x @X

@y

@Y@x @Y

@y

" #

ð9Þ

and the inverse of the Jacobian matrix is introduced:

J

_X^p

0X^p_XY

h i

1

¼ j

^p₁₁

j

^p₁₂

j

^p₂₁

j

^p₂₂

" #

ð10Þ

Fig. 2.Mapping transformation. Fig. 3.Division intoNbAreaareas.

Note that the Jacobian matrix is constant. And, it is diagonal in our case due to the particular choice of the sub-domain X

^p₀

(the ele- ments remain rectangular), so j

^p₁₂¼

j

^p₂₁¼

0, for p

2 f1;. . .;

NbAreag.

In each sub-domain X

^p0

, the strain can be expressed with respect to the reference frame with the introduction of the Jacobian of the transformation following

e ^{ðuÞ ¼ X}

^N

i¼1

g

ⁱ_X

ðX

T

Þg

ⁱ_Y

ðY

T

Þ

j

^p₁₁

f

ⁱ₁

v

ⁱ1;1

j

^p₂₂

f

ⁱ₂

v

ⁱ2;2

ðf

ⁱ₃

Þ

⁰

v

ⁱ3

ðf

ⁱ₂

Þ

⁰

v

ⁱ2

þ j

^p₂₂

f

ⁱ₃

v

ⁱ3;2

ðf

ⁱ₁

Þ

⁰

v

ⁱ1

þ j

^p₁₁

f

ⁱ₃

v

ⁱ3;1

j

^p₂₂

f

ⁱ₁

v

ⁱ1;2

þ j

^p₁₁

f

ⁱ₂

v

ⁱ2;1

2 66 66 66 66 66 66 4

3 77 77 77 77 77 77 5

ð11Þ

where the prime stands for the classical derivative f

⁰_i¼^df_dzⁱ

, and

ðÞ_;a

for the partial derivative. The dependance with respect to the space coordinates is omitted.

3.3. Resolution of the parameterized problem

The expression of the strain Eq. (11) introduced in the problem Eq. (6) drives to a non-linear parameterized problem that is solved by an iterative process. First, we assume that a sum of m

<

N prod- ucts of separated functions have been already computed. There- fore, the trial function for the iteration m

þ

1 is written as u

^mþ1

ðx; y; z; X

T

; Y

T

Þ ¼ u

^m

ðx; y; z; X

T

; Y

T

Þ þ g

_X

ðX

T

Þg

_Y

ðY

T

ÞfðzÞ v ^{ðx; yÞ}

ð12Þ where g

_X;

g

_Y;

f and v are the functions to be computed, and u

^m

is the associated known sets at iteration m defined by

u

^m

ðx; y; z; X

T

; Y

T

Þ ¼ X

ⁿ

i¼1

g

ⁱ_X

ðX

T

Þg

ⁱ_Y

ðY

T

Þf

ⁱ

ðzÞ v

ⁱ

^{ðx; yÞ ð13Þ}

The problem to be solved given in Eq. (7) can be written as aðg

X

g

Y

f v ^{; duÞ ¼ bðduÞ aðu}

^m

^{; duÞ ð14Þ}

The test function becomes

dðg

X

g

Y

f v ^{Þ ¼ dg}

^X

^g

^Y

^f v ^{þ g}

^X

^dg

^Y

^f v ^{þ g}

^X

^g

^Y

^df v

þ g

_X

g

_Y

f d v ^ð15Þ

Introducing the test function defined by Eq. (15) and the trial function defined by Eq. (12) into the weak form Eq. (14), the four following equations to be solved can be deduced:

for the test functions

dgX

,

aðg

Y

f v ^g

^X

^{; g}

^Y

^f v ^dg

^X

^{Þ ¼ bðg}

^Y

^f v ^dg

^X

^Þ

aðu

^m

; g

Y

f v ^dg

^X

^{Þ 8 dg}

^X

^ð16Þ

for the test functions

d

g

_Y

,

aðg

X

f v ^g

^Y

^{; g}

^X

^f v ^dg

^Y

^{Þ ¼ bðg}

^X

^f v ^dg

^Y

^Þ

aðu

^m

; g

X

f v ^dg

^Y

^{Þ 8 dg}

^Y

^ð17Þ

for the test function

d

f

aðg

_X

g

_Y

v ^{f; g}

X

g

_Y

v ^{dfÞ ¼ bðg}

X

g

_Y

v ^dfÞ

aðu

^m

; g

X

g

Y

v ^{dfÞ 8 df ð18Þ}

for the test function

d

v

aðg

X

g

Y

f v ^{; g}

^X

^g

^Y

^{f d} v ^{Þ ¼ bðg}

^X

^g

^Y

^{f d} v ^Þ

aðu

^m

; g

_X

g

_Y

f d v ^{Þ 8 d} v ^ð19Þ

From Eqs. (16)–(19), a coupled non-linear problem is derived. A fixed point method is chosen to solve it. Starting from an initial function

~

g

^ð0Þ_X ;~

g

^ð0Þ_Y ;~

f

^ð0Þ;

v

~^ð0Þ

^{, we construct a sequence}

~

g

^ðkÞ_X ;~

g

^ðkÞ_Y ;~

f

^ðkÞ;

v

~^ðkÞ

which satisfy Eqs. (16)–(19) respectively. For each problem, only one unknown 1D or 2D function has to be found, the three other ones are assumed to be known (from the previous step of the fixed point strategy). So, the approach leads to the process given in Algorithm 1. The fixed point algorithm is stopped when the distance between two consecutive terms are sufficiently small (Cf. [19]).

Algorithm 1. Present algorithm

for m

¼

1 to N

max

do Initialize

~

g

^ð0Þ_Y ;~

f

^ð0Þ;

v

~^ð0Þ

for k

¼

1 to k

max

do

Compute

~

g

^ðkÞ_X

from Eq. (16), g

~^ðk1Þ_Y ;~

f

^ðk1Þ;

v

~^ðk1Þ

^being known

Compute

~

g

^ðkÞ_Y

from Eq. (17),

~

g

^ðkÞ_X ;~

f

^ðk1Þ;

v

~^ðk1Þ

being known Compute

~

f

^ðkÞ

from Eq. (18) (linear equation on X

z

),

~

g

^ðkÞ_X ;~

g

^ðkÞ_Y ;

v

~^ðk1Þ

being known

Compute

~

v

^ðkÞ

^{from Eq. (19)} (linear equation on X ^),

~

g

^ðkÞ_X ;~

g

^ðkÞ_Y ;~

f

^ðkÞ

being known Check for convergence end for

Set g

^ðmþ1Þ_X ¼~

g

^ðkÞ_X ;

g

^ðmþ1Þ_Y ¼

g

~^ðkÞ_Y ;

f

^ðmþ1Þ¼~

f

^ðkÞ;

v

^ðmþ1Þ¼

v

^~ðkÞ

Set u

^mþ1¼

u

^mþ

g

^mþ1_X

g

^mþ1_Y

f

^mþ1

v

^mþ1

Check for convergence end for

3.4. Finite element discretization

It should be noted that the initial weakform Eq. (6), Eq. (7) involves multidimensional integrals. As the displacement is assumed to be written under a separated representation (one 2D and three 1D functions), it is needed to separate the calculation of these integrals to achieve an efficient process in terms of compu- tational cost. The three following sections are devoted to show how the expression of the weakform (Eqs. (16)–(19)) can be deduced under this suitable form. Then, the FE discretization can be introduced.

To build the plate finite element approximation, a classical finite element approximation in X

0

and X

z

for

ð

v

^;

^fÞ is introduced.

The elementary vector of degrees of freedom (dof) associated with one element X

e

of the mesh in X

0

is denoted q

^v_e

. The elementary vector of dofs associated with one element X

ze

of the mesh in X

z

is denoted q

e^f

. The displacement fields and the strain field are determined from the values of q

^v_e

and q

e^f

by

v

^e

^{¼ N}

^xy

^q

^ve

; E

^e_v

¼ B

xy

q

^v_e

; f

e

¼ N

z

q

e^f

; E

^ef

¼ B

z

q

e^f

ð20Þ where

E

^e_v^T

¼ ½ v

¹

v

1;1

v

1;2

v

²

v

2;1

v

2;2

v

³

v

3;1

v

3;2

E

^ef

T

¼ f

₁

f

⁰₁

f

₂

f

⁰₂

f

₃

f

⁰₃

The matrices N

xy;

B

xy;

N

z

, B

z

contain the interpolation functions,

their derivatives and the jacobian components.

As far as the functions g

_X;

g

_Y

are concerned, a piecewise linear interpolation is considered in

IT

g

X

ðX

T

Þ ¼ N

g_X

ðX

T

Þq

^gX

; g

Y

ðY

T

Þ ¼ N

g_Y

ðY

T

Þq

^gY

ð21Þ where N

g_X

and N

g_Y

are the matrices of the interpolation functions, q

^gX;

q

^gY

are the associated degrees of freedom.

3.5. Finite element problem to be solved on X

0

For the sake of simplicity, the functions

~

f

^ðkÞ;~

g

^ðkÞ_X ;~

g

^ðkÞ_Y

which are assumed to be known, will be denoted

~

f

;~

g

X;~

g

Y

, respectively. And the function v

~^ðkÞ

to be computed will be denoted v . The strains in Eq. (19) are defined as

e ^{ð ~ f} v ^{Þ ¼ R}

^z

^{ð ~ f ÞE}

^Xv^XY

¼ R

z

ð ~ f ÞT

_e

E

v

ð22Þ with

R

z

ð ~ fÞ ¼

0 ~ f

1

0 0 0 0 0 0 0 0 0 0 0 0 ~ f

2

0 0 0 0 0 0 0 0 0 ~ f

⁰₃

0 0 0 0 0 ~ f

⁰₂

0 0 0 0 ~ f

3

~ f

⁰₁

0 0 0 0 0 0 ~ f

3

0 0 0 ~ f

1

0 ~ f

2

0 0 0 0 2

66 66 66 66 66 4

3 77 77 77 77 77 5

ð23Þ

T_e

contains the components of the inverse of the jacobian matrix, namely j

^p_ii;

i

¼

1

;

2. It can be deduced from Eq. (11). It is not detailed here as it is only used for convenience in the expres- sion of the strain and one integral. In practice, the problem has to be written in a more suitable way as it is described below.

The variational problem defined on X

0

from Eq. (19) is X

NbArea

p¼1

Z

X^p₀

h dE

^T_v

k

^p_zXY

ð ~ f ; ~ g

X

; ~ g

Y

ÞE

_v

i d X

^p0

¼

^NbAreaPres

X

p¼1

Z

C^pN

t

^p_X

TYT

d v

^T

^t

^z

^{ð ~ fÞdS}

^p

^NbArea

X

p¼1

Z

X^p₀

h dE

^T_v

r

^pzXY

ð ~ f; ~ g

X

; ~ g

Y

; u

^m

Þ i

d X

^p0

ð24Þ

where

k

^p_zXY

ð ~ f ; ~ g

X

; ~ g

Y

Þ ¼ X

16i63 16j63 i6j

k

z

ð ~ fÞ U

ij

Z

XT

~ g

²_X

T

^p_Xij

dX

T

Z

YT

~

g

²_Y

T

^p_Yij

dY

T

ð25Þ

and t

z

ð ~ f Þ ¼

Z

Xz

~ f tdz

r

^pzXY

ð ~ f; ~ g

X

; ~ g

Y

; u

^m

Þ ¼ Z

XT

Z

YT

Z

Xz

g ~

X

g ~

Y

T

^T_e

R

z

ð ~ f Þ

^T

C e ^ðu

^m

^Þ

h i

dz dX

T

dY

T

t

^p_XTYT

¼ Z

XT

~ g

X

T

^p_X11

dX

T

Z

YT

~ g

Y

dY

T

ð26Þ k

zð~

f

Þ

and the 9 9 matrix U

_ij

are defined by

k

z

ð ~ fÞ ¼ Z

Xz

R

z

ð ~ f Þ

^T

CR

z

ð ~ fÞdz

U

ij

¼

H

ij

H

ij

H

ij

H

ij

H

ij

H

ij

H

ij

H

ij

H

ij

2 64

3

75 with ðh

ij

Þ

_kl

¼ 1 if k ¼ i; l ¼ j or k ¼ j; l ¼ i

¼ 0; otherwise 8 >

<

> :

ð27Þ

The expression of T

^p_Xij

and T

^p_Yij

are given in Appendix A.

It should be noted that the expression of k

^p_zXYð~

f

;

g

~_X;~

g

_YÞ

intro- duced in Eq. (25) is split into 6 matrices. It allows us to obtain a separated representation which is particularly suitable to the pre- sent formulation.

Note that r

^pzXY

is also expressed under a separated form to keep the computational efficiency of the method. For this purpose, the same principle as the computation of k

^p_zXY

is performed. It is detailed in Appendix B.

The introduction of the finite element approximation Eq. (20) in the variational Eq. (24) leads to the linear system

K

zXY

ð ~ f; ~ g

X

; ~ g

Y

Þq

^v

¼ R

t

ð ~ f; ~ g

X

; g ~

Y

Þ R

v

ð ~ f ; g ~

X

; ~ g

Y

; u

^m

Þ ð28Þ where

q

^v

is the vector of the nodal displacements, associated with the finite element mesh in X

0

,

K

zXYð~

f

;~

g

X;

g

~YÞ

is the mechanical stiffness matrix obtained by summing the elements’ stiffness matrices K

^e_zXYð~

f

;~

g

X;

g

~YÞ ¼R

Xe

B

^T_xy

k

^p_zXYð~

f

;

g

~X;

g

~YÞBxy

h i

d X

e

Rtð~

f

;~

g

X;

g

~YÞ Rvð~

f

;

g

~X;

g

~Y;

u

^mÞ

is the equilibrium mechanical residual obtained by summing the elements’ load vectors

R

Ce

N

^T_xy

t

zð~

f

Þd

C

e

and

R

Xe

B

^T_xy

r

^pzXYð~

f

;

g

~X;

g

~Y;

u

^mÞd

X

e

(associated to the known terms)

3.6. Finite element problem to be solved on X

z

As in the previous section, the known functions v

~^ðk1Þ;~

^g

^ðkÞX ;~

g

^ðkÞ_Y

will be denoted v

~;~

g

X;~

g

Y

, and the functions

~

f

^ðkÞ

to be computed will be denoted f. The strain in Eq. (18) is defined as

e ^ð~ v ^{f Þ ¼ R}

^Xxy^XY

ð~ v ^ÞE

^f

^{¼ T}

R

R

^Xxy⁰

ð~ v ^ÞE

^f

^ð29Þ

where

T_R

is the transformation of the derivatives from X

^p0

to X

^pXY

, and

R

^Xxy⁰

ð~ v ^{Þ ¼}

v ~

¹;1

0 0 0 0 0 0 0 v ~

²;2

0 0 0 0 0 0 0 0 v ~

³

0 0 0 v ~

²

v ^~

³;2

0 0 v ~

¹

^{0 0} v ^~

³;1

0

v ~

¹;2

0 v ~

²;1

0 0 0 2

66 66 66 66 4

3 77 77 77 77 5

ð30Þ

As in the previous section,

T_R

is only introduced by convenience.

The variational problem defined on X

z

from Eq. (18) is Z

Xz

dE

^Tf

k

xyXY

ð~ v ^{; g ~}

^X

^{; ~ g}

^Y

^ÞE

^f

^{dz ¼ Z}

Xz

df

^T

t

xyXY

ð~ v ^{; ~ g}

^X

^{; g ~}

^Y

^Þdz

Z

Xz

dE

^Tf

r

^xyXY

^ð~ v ^{; g ~}

^X

^{; ~ g}

^Y

^{; u}

^m

^{Þdz ð31Þ}

where k

xyXYð~

v

;

g

~X;

g

~YÞ

can be expressed under the following sepa- rated form:

k

xyXY

ð~ v ^{; ~ g}

^X

^{; ~ g}

^Y

^{Þ ¼ X}

16i63 16j63 i6j

X

NbArea

p¼1

k

^p_xyij

ð~ v ^{Þ Z}

XT

~ g

²_X

T

^p_Xij

dX

T

Z

YT

~

g

²_Y

T

^p_Yij

dY

T

ð32Þ

k

^p_{xy ij}ð~

v

^Þ

is obtained by integrating the functions

~

v

^i;

ⁱ

^f1;

3g over

each area defined in Fig. 3 and depends on the three-dimensional

stiffness coefficients.

We also define:

t

xyXY

ð~ v ^{; ~ g}

^X

^{; g ~}

^Y

^{Þ ¼}

^NbArea

^X

p¼1

t

^p_XTYT

Z

C^pN

v ~ ^tdx

^p

r

^xyXY

^ð~ v ^{; ~ g}

^X

^{; g ~}

^Y

^{; u}

^m

^{Þ ¼}

^NbArea

^X

p¼1

Z

X^p₀

Z

IT

g ~

X

~ g

Y

R

^Xxy⁰

ð~ v ^Þ

^T

^T

^TR

^ð C e ^ðu

^m

^{Þ Þ}

h i

dX

T

dY

T

d X

^p0

ð33Þ The term r

^xyXYð~

v

^;~

^g

^X;~

^g

^Y;

^u

^mÞ

is given under a short form for con- venience reason, but it can also be written under a separated rep- resentation following the same decomposition as in Eq. (32).

The introduction of the finite element discretization Eq. (20) in the variational Eq. (31) leads to the linear system

K

xyXY

ð~ v ^{; ~ g}

^X

^{; ~ g}

^Y

^Þq

^f

^{¼ R}

^ft

^ð~ v ^{; ~ g}

^X

^{; g ~}

^Y

^{Þ R}

^f

^ð~ v ^{; g ~}

^X

^{; ~ g}

^Y

^{; u}

^m

^{Þ ð34Þ}

where

q

^f

is the vector of degree of freedom associated with the F.E.

approximations in X

z

.

K

xyXYð~

v

;~

g

X;~

g

YÞ

is obtained by summing the elements’ stiffness matrices:

K

^e_xyXY

ð~ v ^{; ~ g}

^X

^{; ~ g}

^Y

^{Þ ¼ Z}

Xze

B

^T_z

k

xyXY

ð~ v ^{; ~ g}

^X

^{; ~ g}

^Y

^ÞB

^z

h i

dz

e

ð35Þ

Rftð~

v

;~

g

X;~

g

YÞ Rfð~

v

;~

g

X;~

g

Y;

u

^mÞ

is a equilibrium residual, it is obtained by the summation of the elements’ residual vectors coming from the right hand side of Eq. (33).

3.7. Finite element problem to be solved on

IT

In this part, we will focus on the computation of the unknown function

~

g

^ðkÞ_X

, denoted g

_X

, the other ones

~

v

^ðk1Þ;~

f

^ðk1Þ;

g

~^ðk1Þ_Y

being known. They are denoted v

~;~

f and

~

g

_Y

, respectively.

The problem Eq. (16) can be solved in a straightforward manner following:

K

xyzY

ð~ v ^{; ~ f ; ~ g}

^Y

^{Þ g}

X

¼ R

g_Xt

ð~ v ^{; ~ f; ~ g}

^Y

^{Þ R}

^gX

ð~ v ^{; ~ f; ~ g}

^Y

^{; u}

^m

^{Þ ð36Þ}

where we have the stiffness matrix

K

xyzY

ð~ v ^{; ~ f ; ~ g}

^Y

^{Þ ¼}

^NbArea

^X

p¼1

X

16i63 16j63 i6j

E

^p_zXij

ð ~ f ; v ~ ^{Þ Z}

YT

~ g

²_Y

T

^p_Yij

dY

T

T

^p_Xij

XT

ð37Þ

with

E

^p_zXij

ð ~ f; ~ v ^{Þ ¼ Z}

X^p₀

E ~

^T_v

k

z

ð ~ f Þ U

ij

h i

~ E

_v

d X

^p0

ð38Þ

k

zð~

fÞ and U

ij

are given in Eq. (27).

the equilibrium residual

R

g_Xt

ð~ v ^{; ~ f ; g ~}

^Y

^{Þ ¼}

^NbArea

^X

p¼1

Z

C^pN

v ~

^T

^dx

^p

^T

^pX11

XT

" #

t

z

ð ~ f Þ Z

Y_T

g ~

Y

dY

T

ð39Þ

R

gX

ð~ v ^{; ~ f ; g ~}

^Y

^{; u}

^m

^Þ

¼

^NbArea

X

p¼1

Z

YT

Z

X^p₀

Z

Xz

~

g

Y

~ E

^T_v

T

^T_e

R

z

ð ~ f Þ

^T

C e ^ðu

^m

^Þ

h i

dz d X

^p0

dY

T

ð40Þ

RgXð~

v

;~

f

;~

g

Y;

u

^mÞ

is built from the known function u

^m

. It is com- puted in the same way as K

_xyzY

to deduce a separated representation.

The computation of the unknown function

~

g

^ðkÞ_Y

is given in Appendix C. We can notice that the computational cost of this problem is very low once the problem associated to

~

g

^ðkÞ_X

is solved.

In fact, the computational cost comes mainly from the evaluation of the 6 NbArea scalar values E

^p_zXijð~

f

;

v

~^Þ

which have been already computed and stored in the

~

g

^ðkÞ_X

problem.

4. Numerical results

This section is devoted to the assessment of the present approach for static response of composite plates with a hole. The example involved in this work is a challenging test in the modeling of such structures. The separated representation has already shown interesting features in the framework of free-edge effects (see [22]) and also for parametric geometry [25]. Thus, it is inter- esting to extend the method to a variable geometry for anisotropic structure where high stress gradient occurs.

In the numerical examples, an eight-node quadrilateral FE based on the classical Serendipity interpolation functions is used for the unknowns depending on the in-plane coordinates. For the unknowns depending on the z-coordinate, the displacement is described by a fourth-order interpolation as it is justified in [19].

A Gaussian numerical integration with 3 3 points is used to eval- uate the elementary matrices. As far as the integration with respect to the transverse coordinate is concerned, an analytical integration is performed.

The results are compared with a fourth-order LayerWise model (denoted LD4) referring to the systematic work of Carrera and his

‘‘Carrera’s Unified Formulation” (CUF), (see [31,32]). It can be con- sidered as reference results. Firstly, the behavior of the method is illustrated on a one layer plate. A convergence study is carried out. Then, the capabilities of the approach are assessed on a 4- layer cross-ply structure and in particular, the accuracy of the out-of-plane stresses occurring in the vicinity of the hole is shown.

Note that the use of subdomains with a fixed area around the hole allows us to have a simple translation in the geometrical transformation described in Section 3.2. So, the size of the ele- ments remains unchanged near the zone of interest and no addi- tional error is introduced by the mapping transformation in this region, as the mesh will not be distorted. However, the usual errors hold.

4.1. Description of the test case

A plate with a circular hole subjected to a uniaxial tension in the longitudinal direction is considered. The angle

h

is defined from the x axis (see Fig. 4). The test is described below:

geometry: rectangular composite cross-ply plate

½0

or

½0=

90

=

90

=

0

the radius of the hole is R

¼

3 m. All layers have the same thickness. h

¼

2 m, a

¼

20 m, b

¼

40 m.

boundary conditions: clamped on one side (y

¼ b=

2) and subjected to a constant pressure q

_yðx;

b

=

2Þ ¼ q

₀

material properties: E

L ¼

25 GPa

;

E

T¼

1 GPa

;

G

LT¼

0

:

2 GPa

;

G

TT¼

0

:

5 GPa

;

m

^LT¼

m

^TT¼

⁰

:

25 where L refers to the fiber direction, T refers to the transverse direction.

mesh: the whole plate is meshed. N

theta¼

10 elements on a

quarter of the hole. The size of the central part is a

hole¼

10 m

(Fig. 3).

number of dofs: mesh 1: one-layer case Ndof

xy¼

5616 and Ndof

z¼

12 NC

þ

3

¼

15mesh 2: our-layer case Ndof

xy¼

16

;

056 and Ndof

z¼

12 NC

þ

3

¼

51

reference values: LD4 (one-layer case NdofLD4

xy¼

28

;

080;

four-layer case NdofLD4

xy¼

272

;

952)

If not mentioned, the dimensional quantities are expressed in SI units.

4.2. The one-layer case

The approach is first assessed on a one-layered plate where the coordinates of the hole center belongs to the interval

IT¼ ½4;

4 ½12

;

12. It allows us to consider a wide range of variation for the position of the hole. A convergence study is per- formed with respect to the number of elements associated to the functions g

_X

and g

_Y

, denoted NDF

XYT

. For a fixed value of X

T;

Y

T

, a local error indicator between a reference solution u

^ref

and a sepa- rated variables solution u

ⁿ

of order n is introduced as

ⁿ

^{¼ 100}

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a

err

ðu

ⁿ

u

^ref

; u

ⁿ

u

^ref

Þ

a

err

ðu

^ref

; u

^ref

Þ s

with a

err

ðu; v ^{Þ ¼ Z}

XXz

e ^ð v ^Þ

^T

^C e ^ðuÞd X d X

z

ð41Þ

The variation of this error indicator with respect to the number of the iterations is represented on Fig. 5 for

ðXT¼ 4;

Y

T¼ 12Þ;ðXT¼

0

;

Y

T¼

0Þ

;ðXT¼

4

;

Y

T¼

12Þ and for dif- ferent values of NDF

XYT

. It corresponds to the bounds of the interval for the variation of the position of the hole center. The trend of these three configurations is rather similar. Thus, the influence of NDF

XYT

is not significant for this range. Moreover, we notice that the configuration with the hole at the center of the plate (namely X

0

) drives to the lowest error rate. In the following, 65 terms are built to obtain the solution.

For further investigations, the contribution of each 4-uplets in the total strain energy D ^{EðmÞ ¼}

^aerrðum;um_aerr^Þaðu^errN;u^ðuNm1Þ ^;um1Þ

is shown in Fig. 6 for the three locations of the hole. The main contributions are brought by the first 4-uplets, in particular for

ðXT¼

0

;

Y

T¼

0Þ where 99% of the energy is included in the first 5 couples.

Nevertheless, depending on the configuration, some modes with a high number can have significant contributions. For

ðXT¼ 4;

Y

_T¼ 12Þ, 46 couples are required to obtain 95% of the

total energy (the 47th couple involves 5%). Thus, each term of

the solution will contribute in a different way for a given configu- ration as wide modification of the geometry of the plate could occur. To illustrate these discrepancies, a local error rate on some maximum values for u

3;

r

²²

^and r

³³

is showed in Fig. 7. It can be inferred from this figure that the accuracy of the results decreases at the bounds of the study domain for X

T;

Y

T

. Nevertheless, the error rate remains less than 2% for r

22

(corresponds to the direc- tion of the traction). For the transverse normal stress, this error is about 15% as it is the most difficult quantity to improve, but it is localized near the bounds of the domain

½4;

4 ½12

;

12.

To illustrate the previous results, the 24 first normalized func- tions v

ⁱ1ðx;

yÞ

;

v

ⁱ2ðx;

yÞ

;

v

ⁱ3ðx;

yÞ, the 32 first normalized f

ⁱ₁ðzÞ;

f

ⁱ₂ðzÞ;

f

ⁱ₃ðzÞ

functions are shown in Figs. 8–13. For the func- tions v

ⁱ2ðx;

yÞ in the direction of the traction, the first 5 modes are global. It corresponds to the main contribution to the total strain energy. The corresponding correction in the transverse direction x is localized in an area near the hole over the whole width of the plate. Due to the type of solicitation, the associated transverse functions (cf. Figs. 11–13) are classically constant through the thickness for f

ⁱ₁;

f

ⁱ₂

and linear for f

ⁱ₃

(Poisson effect).

We also notice local modes. For instance, corrections all around of the circumference of the hole or in the vicinity of this one are obtained with mode 9 and 20 for both v

¹

^and v

²

. The distribution of the associated f

ⁱ_jðzÞ

functions become more complex. Thus, the expression of the solution requires high-order z-expansion terms due to the presence of the hole.

The functions g

ⁱ_XðXTÞ

and g

ⁱ_YðYTÞ

associated to the location of the hole are also given in Fig. 14. They are not similar as the influence of the hole position is not the same in the two directions. We also notice a higher gradient of this function near the bound of the interval, especially for g

ⁱ_XðXTÞ.

To assess the capability of the method, the distribution of r

²²

from the hole to the free edge at the middle of the layer is shown in Fig. 15(a). It is compared with the results from LD4 model for three different locations of the hole. The results are in very good agreement with the reference solution. The high stress gradient near the hole is well captured regardless the configuration. Finally, the approach allows us to compute with accuracy for a low compu- tational cost the distribution of the maximal value of r

²²

^with

respect to the hole location as an explicit solution is obtained in this work. It is represented in Fig. 15(b). It can be noticed that

Fig. 4.Plate under uniaxial tension.

Fig. 5.Energy error for different number of elements associated togXðXTÞand YðYTÞ XT2 ½4;4 and YT2 ½12;12 ðXT¼ 4;YT¼ 12Þ;ðXT¼0;YT¼0Þ;

ðXT¼4;YT¼12Þ.

0 10 20 30 40 50 60 70 -20

0 20 40 60 80 100

10 20 30 40 50 60

-0.1 -0.05 0 0.05 0.1 0.15 0.2

0 10 20 30 40 50 60 70

-20 0 20 40 60 80 100

10 20 30 40 50

-0.5 0 0.5 1

0 10 20 30 40 50 60 70

-20 0 20 40 60 80 100

10 20 30 40 50 60

0 1 2 3 4 5

Fig. 6.Contribution of each couple to the total strain energy –XT2 ½4;4andYT2 ½12;12– 65 couples.

10 20 -10 0

-4 -20 0 -2 2 -5 -4 -3 -2 -1 0 1 2

4 20

0 10 -20 -10

-4 -2 0 2 -2.5

-2 -1.5 0.5 0 -0.5 -1

4

10 20 -10 0

-20 -4 -2 0 2 10

5 0 -5 -10 -15 -204

Fig. 7.Local error on the maximum value with respect toXTandYTat the edge of the hole –XT2 ½4;4andYT2 ½12;12.

Fig. 8.Distribution of

v

^i1ðx;yÞ.

Fig. 9.Distribution of

v

^i2ðx;yÞ.

the stress concentration is more sensitive to the hole location in the width direction of the plate, as expected. In a classical approach, numerous computations are required to build this type of curve, as one point corresponds to one F.E. analysis.

4.3. Four-layer plate

In this section, a four-layer

½0=

90

=

90

=

0

plate is considered with

IT¼ ½4:

5

;

4

:

5 ½13

;

13. The choice of this new domain

Fig. 10.Distribution of

v

ⁱ3ðx;yÞ.

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

Fig. 11.Distribution offⁱ1ðx;yÞ.

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

-1 0 1

z

-1 0 1

Fig. 12.Distribution offⁱ₂ðx;yÞ.