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TECHNISCHE MECHANIK, 32, 2-5, (2012), 251 264

submitted: September 29, 2011

Numerical Multiscale Modelling of Sandwich Plates

C. Helen, S. Diebels

Sandwich plates present a complicated material behaviour, strongly depending on the considered materials and on

the layer composition. Thereore, it is an increasing interest to eature a model o their mechanical behaviour. A

multiscale model is developed in the paper. In the scope o this project, some layers show non-linear material be-

haviour at large strains and, as a consequence, the classical plate theory cannot be considered. At this stage, only

linear elastic material behaviour and small deormations are taken into account, in order to validate the presented

model; but to guaranty the possibility to consider non-linear material behaviour, a numerical homogenisation is

chosen explicitly taking into account the stacking order and the material behaviour o the individual layers.

A numerical homogenisation, or so-called FE2, consists o a Finite Element computation on a macroscale -here a

plate which contains the plate kinematics and balance equations; but instead o applying the constitutive equations

on this scale, the deormations are projected on a mesoscale where the mesostructure is ully resolved and another

Finite Element computation is perormed on this level. In this paper, a plate theory ollowing Mindlin concept with

fve degrees o reedom is considered on the macroscale, and a three dimensional boundary value problem is solved

in the mesoscale resolving the stacking order o the sandwich. Whereas the macroscale problem is implemented

in a non-commercial FORTRAN code, the mesoscale is modelled using the commercial sotware ABAQUS R andan UMAT SUBROUTINE. In order to fnd an analytical tangent or the global iterations, a Multi-Level Newton

Algorithm is applied, which enables a aster computation.

1 Introduction

Composites plates are nowadays widely used because o their outstanding mechanical properties, especially or

transport industries like airplane and automotive industries. However, they are quite difcult to test experimen-

tally, hence there exists an increasing interest or modelling. To model the mechanical behaviour o sandwich

plates, one can basically use the classical plate theory (or one o its derivatives), or any homogenisation strategy.

Since some o the considered layers o the sandwich plate are metallic and show elasto-plastic material behaviour,

the plate theory is not sufcient. To the knowlegde o the authors, no correct developments o the plate theory have

been made that take into account non-linear material behaviour, c. Altenbach (1988); Altenbach et al. (2010).

Indeed the plate theory is unable to take into account non-linear material behaviour, like or instance plasticity.

However, in this paper, only elastic material behaviour and small deormations are considered, in order to validate

the presented model. But because o the need in a urther work to consider elasto-plastic material behaviour, a

multi-scale approach and more precisely a numerical homogenisation is used in the scope o this work, in order to

predict the eective material behaviour o the sandwich plate taking into account an arbitrary stacking order and

non-linear material behaviour o the layers.

In the considered numerical multi-scale method, the macroscale is computed as a plate and thereore contains

the plate kinematics. For a better estimation o the shear stresses, a plate theory ollowing Mindlin concept is

considered, which has fve degrees o reedom, i. e. three translations o a point located on the midplane and two

rotations o the cross-sections. The transer o the deormations rom the macroscale to the mesoscale is made by a

projection o the macroscopic deormation on the boundaries o a three dimensional volume on the mesoscale. In

this context, the volume takes the whole plate thickness into account and - as a consequence or a sandwich plate

- the dierent layers and their stacking order. On this level, a representative volume element (RVE) is defned,

and a mesoscopic boundary value problem is solved. Forces and moments are then calculated as resultants o themesoscale computation and transerred back to the macroscale.

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The frst multi-scale modelling was done by Castaneda (2002a,b) and Suquet (1985). Further developments in the

scope o a numerical homogenisation were presented, e.g., by Schroder et al. (1999), Feyel and Chaboche (2000);

Feyel (2001, 2003) and Kouznetsova (2002); Kouznetsova et al. (2004). Recently, a second order or Cosserat

theory is used, in order to improve the technique by considering complex heterogeneities, c. Forest (1998, 2002);

Forest and Trinh (2011), Kouznetsova et al. (2001), and Larsson and Diebels (2007) and Janicke and Diebels

(2010a,b); Janicke et al. (2009). For the specifc case o plates, an analytical homogenisation was frst used by

Laschet et al. (1989) and Hohe (2003). Later on, a numerical homogenisation was also studied or this kind omaterials, c. Geers et al. (2007); Coenen et al. (2010); Landervik and Larsson (2008).

Within the ramework o this paper, an FE2 approach is applied and an analytical tangent is defned with the

help o a Multi-Level Newton Algorithm, c. Ellsiepen and Hartmann (2001); Hartmann (2005); Hartmann et al.

(2008). The procedure is the same as the global-local iteration in the case o constitutive equations o evolutionary

type (viscoelasticity, elasto-plasticity or viscoplasticity). The same way o thinking can be applied or a numerical

homogenisation, in order to identiy an accurate tangent stiness. Whereas most o the authors working on FE2

are using a numerical tangent, what considerably slows down the computations, we will use an analytical tangent

or sake o efciency.

In the next chapter, the principle o a FE2 is shown in details: the plate kinematics on the macroscale is given,

ollowed by the possible projections rules. Then the boundary value problem, solved on the mesoscale, is exposed.Lastly, the transer o the stress and moment resultants rom the mesoscale to the macroscale is explained. In the

ourth chapter, a Multi-Level-Newton Algorithm is presented, in order to defne the analytical tangent. In the fth

chapter, the results or a single layer plate, as well as or a sandwich plate are presented in case o traction, shear

and bending tests.

2 Principle of a Numerical Homogenisation of Composites Plates

In this section, the homogenisation method used or the sandwich plate is introduced. The components o the

macroscale are written in capital letters or the displacements or with the index (.)M or the stress resultants,whereas the components o the mesoscale are defned with lower case letters or the displacements or with the

index (.)m. The principle o a numerical homogenisation or plates is explained in Figure 1 and can be divided inour steps.

Macroscale

Plate

Mesoscale

3-D RVE

1. Plate kinematics

2. Projection o the deormations

on the boundary o the RVE

3. Solution o the BVP

4. Meso-macro transition

UM =

U0 +x3 1V0 +x3 2

W0

um = GradUM X + u

Balance equation Constitutive equation

divPm = 0 Sm = f(Fm)

NM = h Sijm dx3MM =

hSijm dx3

QM =hS3im dx3

NM

Figure 1: Principle o a numerical homogenisation o plates

1. The frst step is related to the macroscale. In the scope o this work, a plate theory with fve degrees oreedom is used. The plate kinematics are defned and the equilibrium equations are solved using a FEM

discretisation o the plate.

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2. The projection o the deormations rom the macroscale on the boundary o the representative volume el-

ement on the mesoscale consists o the second step defning an additional boundary value problem on the

mesoscale. At each integration point o the macroscale an individual RVE is attached.

3. Then the boundary value problem on the mesoscale is solved with the balance equations and the constitutive

equations. An advantage o this technique is the possibility to consider any three dimensional constitutive

law without any urther transormations o it, and to take into account an arbitrary stacking order o thesandwich plate.

4. The last step is related to the determination o the stress and moment resultants or the macroscale, which

are computed rom the local stress distribution obtained rom the boundary value problem on the mesoscale.

In the next subsections, the our steps will be explained in detail.

2.1 Macroscale: Plate Kinematics

In the context o this work, a classical plate theory o Mindlin type is used, c. Altenbach et al. (1996); Reddy

(1997). It contains fve degrees o reedom: U0, V0,W0 are the three translational degrees o reedom, whereas1, 2 are the two rotational degrees o reedom. The 5-degrees o reedom plate theory corresponds to a Cosserattheory or plates, as illustrated in Altenbach et al. (2010).

As a result, the local displacement feld over the plates thickness is expressed as

U = U0 +x3 1,

V = V0 +x3 2,

W = W0, (1)

where x3 is the thickness (or out-o-plane) coordinate and is by convention defned as zero in the mid plane. Forclarity, the frst order components are written in gray.

The in-plane deormations are also defned as

11 = 011 + x3 11 =

U0x1

+ x31x1

, (2)

22 = 022 + x3 22 =

V0x2

+ x32x2

, (3)

where by defnition the coordinates (x1, x2) are the in-plane coordinates. Thus, the in-plane shear deormation isdefned as

12 = 012 + x3 12 =

1

2

U0x2

+V0x1

+ x3

1

2

1x2

+2x1

, (4)

whereas the out-o-plane shear deormations

3 =

13 = 013 =

12

W0x1

+ 1

,

23 = 023 =

12

W0x2

+ 2

,

(5)

are not dependent on the coordinate in transverse direction. In the ramework o the plate theory with fve degrees

o reedom, the deormation in thickness direction 33 is neglected in relation with 11 and 22, c. Altenbach et al.(1996).

We also defne the stress resultants on the macroscale NM, i. e. the normal orce NM, moment MM, and shear

orce QM as the integration o the stress distribution and its frst moment across the plate thickness as

NM = [NM,MM,QM]T =

h/2h/2

[Pijm,Pijm x3,P

3im]

T dx3, (i, j) = (1, 2) (6)

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where Pm is the mesoscopic 1st Piola-Kirchho stress tensor. The fve equilibrium equations are thus defned on

the macroscale as, c. Reddy (1990, 1997), when no external load is applied

G =

N11,1 + N12,2N12,1 + N22,2Q1,1 + Q2,2

M11,1 + M12,2 Q1M12,1 + M22,2 Q2

= 0. (7)

For the use o a FE2 method or plates, kinematic relations or the plate have to be defned. The major dierence

with the classical plate theory is that the constitutive law is not modifed according to the kinematic assumptions

like or the plate theory, but on each Gauss point o the macroscopic FE discretisation, the deormations are

projected on a three dimensional mesoscale volume. In the mesoscale, a separate Finite Element computation o

the mesostructure problem is perormed; and the results are transerred back to the macroscale in terms o the

stress resultants. The constitutive law is then defned within the mesoscale problem, i. e., in a local 3-dimensionalproblem. As a consequence, two major advantages occur: frstly, the constitutive law has not to be changed in order

to pass the plate kinematics; secondly, any constitutive law can be used, even non-linear ones. On the one hand, the

major drawback o this kind o theory lies in the high computational costs, which, however, can be dramatically

reduced due to parallelisation.

2.2 Projection Rules

There are dierent possibilities to project the deormations on the mesoscale, Kouznetsova (2002); Coenen et al.

(2010):

the Taylor or Voigt assumption, in which a constant deormation is projected to the mesoscale,

the Sachs or Reuss assumption, in which a constant stress is projected to the mesoscale.

The Taylor assumption leads to an overestimation o the results in terms o the eective stiness, whereas theSachs assumption leads to an underestimation o the results. Both assumptions cannot be accurately used or sand-

wich structures, since neither the stresses nor the strains are homogeneous in a sandwich composite.

Another possibility - which is preerred in most o the cases because o its efciency - is to project the deor-

mations on the boundaries o the RVE taking into account an additional uctuation feld. We choose a projection

on the nodes o the element, and the mesoscopic deormations u are defned as

u = GradU X+ u (8)where u is chosen as periodic on opposite boundaries, written as

x+ x

=FM

(X+ X

),(9)

where the index ()+ accounts or the ront panel o the RVE and the index () or the opposite side. The deorma-tions are assumed to be periodic, since the mesoscopic uctuations o two parallel sides o an unit cell are identical.

The macroscopic deormation gradient is defned as the volume average o the mesoscopic deormation gradient

FM =1

V0

V0

Fm dV0. (10)

2.3 RVE

In the mesoscale, a 3-dimensional Representative Volume Element (RVE) is defned, which takes into account thestacking order o the dierent layers o the sandwich plate. Since the order o magnitude in the thickness direction

is not much smaller in the mesoscale than in the macroscale, no homogenisation can be done in this direction. For

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this reason, the numerical homogenisation is only done in two directions. In the thickness direction, no homogeni-

sation is possible and a ull resolution o the materials occurs. Furthermore, we choose a plate theory o Mindlin

type which means that no thickness change occurs. In the present paper, a sandwich plate is studied, i. e. the

structure is composed o three layers o dierent materials; whereas the core material is thick and has a low elastic

modulus, the top panels are thin and sti. It is to dierentiate with laminates, which contains dierent layers o

the same thickness and materials, but with dierent fbers orientations. However, the presented FE2 method has

the advantage to consider laminates plates as well as sandwich plates.

On the mesoscale, the balance equation between the internal and external workV0

PTm : Fm dV0

V0

t x dV0 = 0 (11)

as well as the local constitutive equation

Pm = F(Fm) (12)

apply, wherePm is the mesoscopic 1st Piola-Kirchho stress tensor andFm the mesoscopic deormation gradient.

In contrast to the classical plate theory, even non-linear constitutive equations can be considered, like or example

or elasto-plasticity. Furthermore, the constitutive equations do not need any urther transormations, like it is the

case or a 5-degrees o reedom plate theory.

2.4 Macro-Meso Transition

The Hill-Mandel condition, c. Larsson and Diebels (2007) or a classical numerical homogenisation, postulates

the equivalence o the macroscopic and the mesoscopic stress power

1

V0

V0

Pm : FTm dV0 = PM : F

TM. (13)

In order to ft into the plate kinematics o the macroscale, the Hill-Mandel condition is modifed

1

S0

V0

Pm : FTm dV0 =

S

[U0,1 N11 + (U0,2 + V0,1)N12 + V0,2 N22

+ W0,1 Q1 + W0,2 Q2 + 1,1 M11 + (1,2 + 2,1)M12h

+ 2,2 M22 + 1 Q1 + 2 Q2] dS. (14)

However, the stress resultants are obtained rom

NN =4C1:

0+4C2: +

3C4 3,

MM =4C2:

0+4C3: +

3C5

3

,

QM =3C4:

0+3C5: +

2C6 3, (15)

where NN is the second order orce tensor, MM is the second order moment tensor and QM is the shear stress

vector. Thej

Ci are the tangent stiness, which are tensors o order j, used or an analytical tangent within the FE2

approach; the determination o them will be shown in the next chapter.

3 Multi-Level Newton Algorithm (MLNA)

3.1 Principle of the MLNA

The difculty applying an FE2 model consists o the determination o an accurate tangent stiness or the macroscale

(that is the plate) rom the mesoscale (the 3-dimensional problem). To solve this issue, we will use a Multi-Level

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Newton Algorithm, frstly used by Rabat et al. Rabbat et al. (1979) or electrical networks, later on by Ellsiepen and

Ellsiepen and Diebels (2001) or the solution o mechanical problems when the constitutive law is o evolutionary

type. For these issues, the principle o a MLNA is to separate the two systems o equations in one global system,

which contains the equilibrium equations, and one local system, which consists o the evolution equations. The

same kind o procedure can be used or FE2, c. Hartmann et al. (2008), in which the macroscale is represented by

the global level and the mesoscale by the local level.

The discretised system o the equilibrium equationsG (NM), see equation (7), represents the global level, whereasthe local level consists o the 3-dimensional boundary value problem o the mesoscale defned at each macroscopicintegration point. In the local level, the problem will be subdivided in two parts: one on the boundary l o the RVE

- which is obtained by the projection - and the other inside the RVE l, which corresponds to the local equilibrium

l =

ll

=

u GradUM X

div u,u

= 0. (16)

Thus, the displacements o the local problem are divided into an unknown and a known (prescribed) part, as

represented in fgure 2. u is the unknown part, inside the RVE and u is the prescribed part, on the boundary o

the RVE Hartmann et al. (2008).

u

u

Figure 2: Schematic representation o the displacement on the local level: on the boundary (u) and inside the

element (u)

Then a discretisation is made by multiplication with a test unction and integration over the volume. The equilib-

rium equations o the macroscale (7) become

G =

S

[U0,1 N11 + U0,2 N12] dxdy

s

P1 U0 dsS

[V0,1 N12 + V0,2 N22] dxdy

s

P2 U0 dsS

[W0,1 Q1 + W0,2 Q2 W0 q+N1] dxdy

s

Q U0 ds

S[1,1 M11 + 1,2 M12 + 1 Q1] dxdy s

T1 U0 dsS [2,1 M12 + 2,2 M22 + 2 Q2] dxdy

s

T2 U0 ds

= 0, (17)

where, c. Reddy (1997)

P1 = N11 n1 + N12 n2,

P1 = N12 n1 + N22 n2,

N1 = W0,1 (N11 W0,1 + N12 W0,2) + W0,2 (N12 W0,1 + N22 W0,2),

Q = (Q1 + N11 W0,1 + N12 W0,2)n1 + (Q2 + N11 W0,1 + N22 W0,2)n2,

T1 = M11 n1 + M12 n2,

T2 = M12 n1 + M22 n2, (18)

and n is defned as the unit vector normal and n = [n1, n2]T. A discretisation o the equilibrium equation o the

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mesoscale (16 a) is based on the weak orm o the equilibrium conditionB0

u : dB0

B0

u t dB0 = 0

Ater a discretization, it ollows or the deormations on the local level

=nk=1

Bk uk +

nk=1

Bk uk , (19)

where uk is the nodal displacement at node k and the Bk are the derivatives o the shape unctions. The virtualdeormations are defned as

=nk=1

Bk uk , (20)

since the virtual displacement on the boundary vanishes

u = 0. (21)

Subsequently or the macro-meso problem, it returns to solve the ollowing 2-levels systemG

l

=

RURu

= 0, (22)

composed o the global level (the plate), and the local level, containing the 3-dimensional Finite Element compu-tation o the RVE. In order to solve this system, a Newton iteration is required

G

U

G

u

l

U

l

u

dU

du

=

RU

Ru

. (23)

The partitioning o the Jacobian is with respect to the MLNA. The system (22) is solved in a Newton iteration in 3steps, c. Hartmann et al. (2008):

Step 1: In the frst step, the local level is considered, thereore the global displacement is assumed to beknown (dU = 0). The second line o the system (23) can be simplifed as

l

u du = Ru, (24)

which leads to the determination o du.

Step 2: In the second step, an update o the local variable u is done, and we now return to the global level.We assume that an equilibrium is ound on the local level, and thus the residuum is set to zero

Ru = 0. (25)

By considering the second line o the system (23), we can determine a relation between the local increment

du and the global increment dU

l

U dU+

l

u du = 0, (26)

which leads to a direct relation between the global increments and the local displacements

du =

l

u

1

l

U

dU. (27)

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Step 3: In the last step, the frst line o the system (23) is solved on the global level taking into account (27)

G

U dU+

G

u du =

G

U+

G

u

du

dU

dU = RU, (28)

where some simplifcations can be done: the global level does not depend directly on the macroscopic

increments, that leads to

G

U= 0. (29)

Hence, the equation (28) reduces toG

u

du

dU

dU = RU. (30)

From this equation, one can identiy the tangent stiness K as K dU = RU

K =

G

u

du

dU

, (31)

and according to the chain rule

K =G

NMNMu

du

dU, (32)

taking into account the explicit dependences oG on the macroscopic stress resultants NM.

3.2 Application to FE2

The macroscopic tangent stiness is decomposed according to the chain rule, in order to use the dependence o the

mesoscopic stresses on the mesoscopic deormations

NMM

=NMm

mm

mM

(33)

where M is defned as containing the whole macroscopic deormations and m as the mesoscopic deormations.

From the defnitions o the macroscopic stress resultants as a unction o the mesoscopic stresses, rom the classical

plate theory

NTM = [Nij ,Mij ,Qi]T =

h/2h/2

[ijm, ijm x3,

3im]

T dx3, (i, j) = (1, 2),

we can fnd a direct relation between the macroscopic orces and the mesoscopic stresses, in the linear case

Nij =h/2h/2

ijm dx3 =h/2h/2

k,l

Cijkl mkl dx3 =p,k,l

Cpijkl

pmkl h, (34)

where the macroscopic stress resultant is equal to the sum over the elements o the mesoscopic stresses multiplied

by the thickness o the plate. C is defned as the elastic tensor o the mesoscopic model, = C : andC = l/.p,k,l is defned as the sum over the indices p, k, l; k and l are the sum index and k, l = 1, 3; p is the index over

the thickness o the RVE. The same principle applies or the moment

Mij =

h/2h/2

ijm x3 dx3 =p,k,l

Cpijkl

pmkl x3 h, (35)

and the shear stresses

Qi =

h/2h/2

3im dx3 =p,k,l

Cp3ikl

pmkl h. (36)

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The relation between the macroscopic and mesoscopic deormation is also needed

M =

0M + x3 M ij = 11, 12, 223 ij = 13, 23

=

1

V0

V0

mdV0. (37)

Thus, we obtain the tangent stiness

NM =

NMQ

= CM : M = CM :

0

3

, (38)

with CM

CM =

p C

pijkl h

p C

pijkl hx3 0

p Cpijkl hx3

p C

pijkl hx

23 0

0 0

p Cp3j3l h

. (39)

The use o an analytical tangent enables a aster computation as or a numerical tangent. However, a numerical

homogenisation remains a computationally expensive technique, which is one o its major drawbacks. It is to

mention that this drawback can be minimised by parallelisation o the local solutions.

4 Results

In order to evaluate the proposed multi-scale scheme, a tension, shear, and bending test o a single layer plate, as

well o a sandwich plate are proposed. The sandwich plate is composed o three elastic layers, the bottom and top

layers are supposed to be sti; whereas the core material is soter. The restriction to elastic material behaviour is

due to simplicity. Non-linear behaviour can be easily included by choice o another UMAT interace.

4.1 Tension Test

u ul

2 4 6 8 1000

6000

4000

2000

Deormation (%)

NormalforceN11

(MPa/m)

plate theory with 5 DOF

FE2 based on 5 DOF plate th.

3-D solution

2 4 6 8 1000

6000

4000

2000

Deormation (%)

plate theory with 5 DOF

FE2 based on 5 DOF plate th.

3-D solution

Figure 3: Tension test or a 3 layers material with a zero Poissons ratio on the let and a non-zero Poissons ratioon the right

In order to validate our model, a sandwich plate made o 3 elastic layers is computed or a tension test. The FE2

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approach is compared to a ully resolved 3-dimensional FE computation and to the solution o a ully macroscopicsolution. In the scope o this academic example, the Poissons ratio will vary between 0 and 0.4, in order totest the proposed model. As observed in fgure 3, i the Poissons ratios are set to zero or both materials, the

results o the reerence (here the plate theory or the 3-dimensional problem) and the proposed FE2 are similar.However, a discrepancy o 15% occurs i the Poissons ratio o the core material is set to 0.4 and the one o thepanels is set to 0.3. The reason or this are the dierent assumptions done in the multi-scale modelling: on the

macroscale, we consider a plate ollowing Mindlin concept, i. e., a plate without thickness change based on a 2-dimensional constitutive law, whereas the mesoscale is a 3-dimensional problem, with a thickness change and witha 3-dimensional constitutive law. It ollows an obvious contradiction between the assumptions on the macroscaleand on the mesoscale. As a consequence, a plate theory with thickness change would avoid this contradiction and

could bring better results. For these reasons, or a non-zero Poissons ratio, the results can not be the same or a

tension test. In the next part, a shear test and later a bending test will be computed and the results compared with

the plate theory.

4.2 Shear Test

A simple shear test is computed, as shown in Figure 4, where the results or the proposed multi-scale model, or a

plate theory and or a 3-dimensional computation are the same, or the single material, as well as or the sandwichstructure. It makes sense, because no transverse deormation occurs or this kind o test.

ShearforceN12

(MPa/m)

40000

30000

20000

10000

0= 0 = 0 = 0 = 0

1 Layer 3 Layers

plate

FE23D

Figure 4: Shear orce or a shear test or dierent cases

4.3 Bending Test

Lastly, a bending test o a single material as well as o a sandwich structure is computed, as represented in Figure 5

or the composite. The distribution o the bending moment is shown in fgure 6 or a single layer. The classical

x

u

Figure 5: Schematic representation o the computed bending test

plate theory ollowing Mindlin concept with fve degrees o reedom is represented in red, whereas the three other

curves give the results or the numerical homogenisation, or a mesoscale mesh o 23, 33, 43 and 103 elementsrespectively. For a material with a Poissons ratio o 0, the convergence is very good; however, or a Poissons

ratio dierent o 0, the values are converging, but to another value. In the stage o this work, the numerical ho-mogenisation is based on a plate theory which does not take into account any thickness change. For these reasons,we assume that the presented multi-scale model is partly taking the transverse deormations into account (not rom

the macroscale but rom the 3-dimensional problem o the mesoscale), and that this element could explain the

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observed discrepancy.

M11

x

20

15

10

10

5

00

2 4 6 8

plate th. 5 DOF

FE2

: 23

el.

FE2: 33 el.

FE2: 43 el.

FE2: 103 el.M11

x

20

15

10

10

5

00

2 4 6 8

plate th. 5 DOF

FE2

: 23

el.

FE2: 43 el.

FE2: 103 el.

FE2: 33 el.

Figure 6: Moment or a single layer, with a Poissons ratio o 0 (let) and or a Poissons ratio o 0.35 (right)

A similar bending test is applied or a sandwich structure composed o three layers, as represented in Figure 5; the

moment distribution is given in Figure 7. For a Poissons ratio equal to 0 or both materials, we observe that theerror is not to large (between 3% and 8%), but the results are not converging to the results given by the plate theory.The same eect can be observed when the Poissons ratio is not set to 0, however the errors are larger (rom 6% to11%). At this stage, we ormulate the assumption that the considered plate theory on the macroscale is not takinginto account any thickness change, and that a plate theory with thickness change could give an accurate result.

M11

x0

02

2

4

4

6

6

8

8

10

10

12

14

16 plate th. 5 DOF

FE2: 23 el.

FE2: 42 el.

FE2: 103 el.

FE2: 33 el.

M11

x

20

15

10

10

5

00

2 4 6 8

plate th. 5 DOF

FE2: 23 el.

FE2: 43 el.

FE2: 103 el.

FE2: 33 el.

Figure 7: Moment or a sandwich structure, with a Poissons ratio o 0 or both materials (let) and or a Poissonsratio average o0.35 (right)

In Figure 8, the shear stress in the middle o the plate or the single layer, as well as or the considered composite,

is represented. For the single material, as well as or the sandwich structure, the dierence between the plate

theory and the presented model are larger or non-zero Poissons ratio materials. For a single layer with a zero

Poissons ratio, the error is about 3%. On the contrary, it reaches 20% or a single layer with a Poissons ratio o

0.35. For the sandwich structure, the error lies between 11% and 15%. Furthermore, we see clearly that the resultso the FE2 model are not converging to the results o the plate theory. For the plate theory o the reerence as well

as or the plate theory o the macroscale, a shear correction coefcient o 5/6 (Reissner) is applied; however, weare not awaiting a better result i this actor is changed. At the stage o this work, we give the assumption that

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2

1.2

Q1

nelement

1.4

1.6

1.8

0 23 43 63 83 103

plate th. 5 DOF

FE2

plate th. 5 DOF

FE2

= 0

= 0

2

1.2

Q1

nelement

1.4

1.6

1.8

0 23 43 63 83 103

plate th. 5 DOF

FE2= 0

plate th. 5 DOF

FE2= 0

Figure 8: Shear stress or a bending test or a single material (let) and or a sandwich structure (right)

the observed discrepancies derive rom the considered plate theory without thickness change. For these reasons,

urther comparison with classical examples, like or instance the Pagano solution, c. Pagano (1969) and later de-

velopped by Meenen and Altenbach (1998) about the boundary conditions, computed or an anisotropic elastic

material behavior, will be developped in a urther work dealing with anisotropic plates.

5 Conclusions and Outlook

This paper has proposed a numerical homogenisation method, so-called FE2, o composites plates. The macroscale

is defned as a Finite Element computation o a plate which contains fve degrees o reedom. From each Gauss

point, the deormations are projected on a 3-dimensional RVE on the mesoscale, where another Finite Elementcomputation is done. The RVE takes the whole thickness o the plate into account and thereore the numerical

homogenisation is only made in the two directions parallel to the mid plane, but not in the thickness direction. In

the scope o this work, periodic boundary conditions are chosen and a boundary value problem is computed on the

mesoscale. In order to identiy an analytical tangent, a Multi-Level Newton Algorithm is used and applied to the

proposed FE2 model: the global level consists o the equilibrium equations, whereas the local level deals with the

mesoscale. The plate kinematics are used to project the macroscopic deormation to the mesoscale.

The proposed model in computed within tension, shear and bending tests o a single material and o a sand-

wich structure. For simplicity, the considered materials are elastic, in order to enable a better verifcation o the

model. In a uture work, elasto-plastic material behaviour as well as anisotropy will be taken into account.

For the tension test and or the bending test, a discrepancy between the classical plate theory and the proposed

multi-scale model occurs or materials with a non-zero Poissons ratio. We give the assumption that it is caused bythe act that the considered plate theory, or the reerence calculations as well as or the numerical homogenisation,

does not allow or any thickness change and thereore is unable to consider this kind o eects. As a solution, we

have proposed to use a plate theory with thickness change, or instance a plate theory with 7 degrees o reedom;this aspect will be developed in a orthcoming work.

Acknowledgment

The authors thank the German Science Foundation DFG or their fnancial support under the grant Di 430/11-1.

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Address: Dipl.-Ing. Cecile Helen and Pro. Dr.-Ing. Stean Diebels, Chair o Applied Mechanics, Saarland

University, Campus A 4.2, Zi. 1.08, D-66123, Saarbrucken, Germany.

email: c.helfen@mx.uni-saarland.de;s.diebels@mx.uni-saarland.de

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