Homogenized elastoplastic properties for a partially cohesive composite material

Z. angew. Math. Phys. 49 (1998) 568–5890044-2275/98/040568-22 $ 1.50+0.20/0c© 1998 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Homogenized elastoplastic properties for a partiallycohesive composite material

Erick Pruchnicki

Abstract. The two-scale asymptotic process is used to investigate microscopic and macroscopicmechanical behaviors of an elastic-perfectly plastic composite material when a tangential slip isallowed on the interface. For multilayered media the microstress and strain tensors are constantin each constituent. This result allows explicit expressions for the macrofree energy to be derivedfrom the macrostiffness tensor, macroelastic domain and hardening rule. Finally a numericalillustration of the macroscopic behavior is presented.

Mathematics Subject Classification (1991). 35C, 35Q, 73E, 73K, 73T.

Keywords. Homogenization, elastoplastic, composite, interface, slip.

1. Introduction

Composite materials provide a good weight/strength ratio and high performancestructures. Consequently the use of composite materials is becoming increasing-ly important in the realization of structures of extreme performance. However,analysis of such structures using finite element methods would be very expen-sive, since discretization of the body is important in order to represent detailedmicrostructure of the material. As a consequence a structural analysis is feasi-ble if the composite can be replaced with an equivalent homogeneous material.The determination of these macroscopic properties is called the homogenizationprocess.

This paper is concerned with the overall behavior of linear elastic and nonlinearplastic composites under the small deformation hypothesis. A brief review of someof the relevant contributors is also given.

For the most complicated case of random (aperiodic) inhomogeneous microstruc-tures, there are two different approaches to the problem. On the one hand, ap-proximate methods have been developed. The most popular of these methods arethe “self consistent” scheme developed initially by Hill [1] and Kroner [2]. On theother hand, variational principles are employed to obtain mathematical lower andupper bounds for effective properties of linear and nonlinear composites Hashin

Vol. 49 (1998) Elastoplastic properties for a composite material 569

and Shtrikman [3], Willis [4]. These bounds have been generalized to include dif-ferent kinds of boundary conditions on the representative volume element of theheterogeneous material Huet [5], Hazanov and Huet [6].

For the simplest possible microstructures, namely the periodic ones, the repre-sentative volume element (called in this case the unit cell) is the smallest elementwhich generates by periodicity the entire structure of the composite. A rigorousmathematic approach of the homogenization theory deals with the limit behaviorof composite materials when their microstructure becomes increasingly thinner(Bakhvalov and Panasenko [7]). The prediction of the homogenized behavior ofcomposite materials with elastoplastic properties has been investigated by a num-ber of authors (Hill [8], Agarwal et al [9], Nemat-Nasser and Taya [10], Aboudi[11], Guedes and Kikuchi [12], Jansson [13], Shkoller and Hegemier [14], Ghosh etal [15], Pruchnicki [16, 17, 18]).

Interfaces between the constituents are probably among the most importantcomponent of a composite material, because the load transfer between fiber andmatrix strongly depends on the degree of contact and the transfer of load at theinterface. In order to take into account these effects, some homogenized models arepresented in which mechanical models of interfacial zones are incorporated (Aboudi[19], Aboudi and Pindera [20]). In another type of modelling the thickness of theinterface is considered as a small parameter. Besides introducing some assumptionabout the dependence of the mechanical constants of the interface on its thickness,the interfacial effect can be taken into account in the asymptotic process Lene andLeguillon [21, 22], Licht [23], Leguillon [24, 25]. Thus Licht and Michaille [26],Geymonat et al [27] has studied the behavior of a bounded joint. For the jointedmaterial, de Buhan and Maghous [28] has pointed out that the strains inside thejoint material can be large. Furthermore, the question of average relations inmaterials with defects has been discussed in Suquet [29].

In the present study, the heuristic asymptotic expansion method is employedto justify an elastoplastic homogenized model for composite materials when atangential slip is allowed on the interface between the constituents and the case ofsmall deformations and displacements has been considered.

In the particular case of a multilayered structure, it was established that boththe microstress and strain tensors are constant in each constituent (Salamon [30],Sawicki [31]). This result permits the formulation of an explicit macroconstitutivelaw. In this work, we consider a possible sliding at interface described by lin-ear elastic and nonlinear plastic law, extending the framework of Pruchnicki andShahrour [32, 33]. First the relevant state variables are identified and then themacrofree energy is computed and used to derive expressions for the macroconsti-tutive law. Finally, the elastic domain and hardening rule are described. Then anumerical application illustrates this theory.

570 E. Pruchnicki ZAMP

2. Setting of the problem

This section establishes notation and precisely describes the problem we consider.

2.1. Notation

We shall write x = (x1, x2, x3) for vectors in R3 and x = (x1, x2) for vectors inR2. I denotes the unit matrix. δ is the Kronecker delta. The superscript refers tothe ith constituent and the particular superscript p is associated to plastic entities.We shall adopt the following notations:|Y | stands for the volume of the domain Y , parentheses <> implies averaging

over the volume Y , [v](= v1−v2) denotes the jump of the vector v through surfaceΓ, 〈r(y)〉Γ represents 1

|Y |Γ

∫rdΓ, divx (resp. gradx) refers to the divergence (resp.

gradient) with respect to the local variable x, the rate of function r with respectto time is denoted by r, “·” and “:” signify respectively the simple and the doublecontractions symbol, ⊗

srepresents the symmetrical tensorial product, BT is the

transpose of the matrix B.

2.2. Geometry of the domain

Let us consider a macroscopic volume Ω with a smooth boundary ∂Ω of compositematerial referred to the Cartesian system of axes (0, x1, x2, x3) associated withmacroscopic space variable (x). The microstructure of composite material is con-stituted by spatially repeated unit cells (Y ) related to microscopic scale (y). Eachperiod Y is composed of two constituents (the inclusion Y 1 and the matrix Y 2)(Fig. 1) whose volume fractions are w1 and w2 respectively. Γ is the interfacebetween the two constituents. In real heterogeneous materials the dimensions ofthe unit cell are typically very small compared with the dimensions of the body.The ratio of these microscopic and macroscopic scales is represented by a verysmall positive number ε (Fig. 1). Thus the relation between the global scale xfor the body and the local one y for the unit cell can then be written as y = x

ε .Throughout the paper, we denote the association of a function with the two lengthscales by using the subscript ε.

2.3. Constitutive law

The stress tensor is associated with any displacement uε:

exε(x, t) =12(gradx(uε(x, t)) + gradx(uε(x, t))T

)(1)

in which t represents the time.The isotropic constituents of the composite material are assumed to obey an

associated elastic-perfectly plastic law. The linear behavior is characterized by the


Figure 1.Macroscopic and microscopic scales.

following free energy:

Wε(x, t) =12aε(x) :

(exε(x, t)− epε(x, t)

):(exε(x, t) − epε(x, t)

)where aε is the stiffness tensor (the inverse compliance tensor is denoted by Aε)and epε is the plastic strain.

The evolution of plasticity in each constituent is governed by a convex hyper-surface in the space of stresses.

fc(σε, y, t) = 0

in which σε represents the Cauchy stress tensor.At the interface the contact holds and the tangential slip is modelled by non

associated elastic-perfectly plastic law. In accordance with, the elastic stiffnessof the interface (kIε) is assumed to be inversely proportional to the scale ratio ε(Lene and Leguillon [21, 22], Murakami and Hegemier [34], Shkoller et al [35]).The corresponding free energy of the linear elastic law is given as:

WIε(x, t) = kIε(

[uTε(x, t)]− [upTε(x, t)])·(

[uTε(x, t)] − [upTε(x, t)])

where kIε is equal to kIε and kI is a positive slip coefficient, uTε represents the

tangential displacement along the interface defined by uTε = uε − (uε · n)n, n isthe inward unit normal on the interface Γ (Fig. 1).


Three different orders of the elastic stiffness of the interface (kIε = kI , kIε = kIε ,

kIε = kIε2

) have been considered by Caillerie and Tollenaere [36] in order to studythe local and global behaviors of woven materials.

The discontinuity of plastic sliding through the interface is classically character-ized by a convex hypersurface (fI(Tε, y, t) = 0) and a plastic potential gI(TTε , y, t).Tε(σε.n) denotes the stress vector along the interface Γ and TTε is its tangentialpart (σε.n− (n.σε.n)n).

2.4. Loading

The loading is defined in the following way: the body is subjected to a systemfor body forces f , a prescribed stress vector T (x, t) on a portion of the boundary∂ΩT and a prescribed displacement field (u(x, t)) on the remaining part of theboundary (noted ∂Ωu). We set nT is the unit outward normal along the boundary∂ΩT .

3. The family of Pε problems

3.1. Formulation of these problems

The Cauchy stress tensor, the strain tensor and the displacement field dependon the scale ratio ε, they satisfy at time instant t the following evolution staticproblem:

divx σε(x, t) + f(x) = 0 in Ω (2)

σε =∂Wε(exε − epε)

∂exεin Ω (3)

TTε =∂WIε

([uTε]− [upTε]

)∂[uTε]

on Γ (4)

fc(σε) < 0 ⇒ epε = 0 in Ω

fc(σε) = 0 ⇒

∣∣∣∣∣∣ ∗fc(σε) = 0 epε = λε∂fc(σε)∂σε

λε ≥ 0

∗ fc(σε) < 0 epε = 0in Ω (5)

fI(Tε) < 0 ⇒[upTε

]= 0 on Γ

fI(Tε) = 0 ⇒

∣∣∣∣∣∣∣∗fI(Tε) = 0

[upTε

]= λIε

∂gI(TTε)∂(TTε)

λIε ≥ 0

∗ fI(Tε) < 0[upTε

]= 0

on Γ (6)

σε.nT = T on ∂ΩT(7)uε = u on ∂Ωu(8)


in which λε and λIε denote respectively the plastic consistency parameter in ma-terials and at interface.

3.2. Resolution using two-scale asymptotic process

Assuming a solution of the field variables could be found that is a function of thespace variables x and y and the scaling ratio ε. Hence, we seek a solution forthe problem Pε in the form of the following asymptotic expansion (Bakhvalov andPanasenko [7])

uε(x, t) = u(x, t) + εu1(x, y, t) + ε2u2(x, y, t) +O(ε3) (9)

σε(x, t) = σ(x, y, t) + ε(x, y, t) +O(ε2) (10)

where the functions u1, u2, σ, σ1 are periodic in y.By using the definition for the strain tensor (1) together with differentiation

with respect to the two length scales(∂∂x

(Ψ(x, xε

)))=(∂∂x + ε−1 ∂

∂y

) (Ψ(x, y)

)and the asymptotic expansion of the displacement field (9), we obtain the corre-sponding one of the strain tensor:

exε(x, t) = ey(v(x, y, t)

)+ ε(ex(u1(x, y, t)

)+ ey

(u2(x, y, t)

))+O(ε2) (11)

by settingv = E · y + u1 (12)

in which E is defined by ex(u) and ey(v) is the strain tensor calculated accordingto the variable y.

By inserting the asymptotic expansion of the Cauchy stress tensor (10) intothe equilibrium equation (2) and identifying each coefficient up to zeroth order,we obtain:

Order-1 divy(σ) = 0 (13)Zeroth-order divx(σ) + divy(σ1) + f = 0. (14)

By considering the asymptotic expansion (10) and the continuity of the yieldfunctions (fc(σε) and fI(Tε)), we see that:

fc(σ) +O(ε) ≤ 0fI(T ) +O(ε) ≤ 0

where T = σ · n and O(ε) is an infinitely small in ε.When ε is small enough. We require:

fc(σ) ≤ 0 (15)fI(T ) ≤ 0. (16)


As a consequence the rate of plastic state(ep,[vpT])

is associated with the firstterms of the asymptotic expansion of the stress state (σε, TTε). By consideringregular points of the yield surface, we see that the corresponding plastic ratedifferent from zero is given by:

fc(σ) = 0 and fc(σ) = 0 ⇒ ep = λ∂fc(σ)∂σ

λ ≥ 0 (17)

fI(T ) = 0 and fI(T ) = 0 ⇒[vpT]

= λI∂gI(TT )∂(TT )

λI ≥ 0. (18)

4. Effective behaviour

In this section, we establish problems for defining ey(v), σ, ep and [vpT ], we canobserve that these mechanical quantities satisfy the original problem Pε up toinfinitely small of the order of ε.

4.1. Microscopic aspect

The microscopic problem arises by considering the conditions (15) and (16), byinserting the asymptotic expansions (9), (10), (11) and (13) and the formulas (17)and (18) and identifying the various ε powers into the equations of the problemPε thus we see that the microscopic problem Pm is expressed as follows:

divy(σ) = 0 in Y

σ =∂W (ey(v)− ep)

∂ey(v)in Y (19)

TT =∂WI([vT ]− [vpT ])

∂[vT ]on Γ (20)

fc(σ) < 0 ⇒ ep = 0 in Y

fc(σ) = 0 ⇒

∣∣∣∣∣∣ ∗fc(σ) = 0 ep = λ∂fc(σ)∂σ

λ ≥ 0

∗ fc(σ) < 0 ep = 0in Y

fI(T ) < 0 ⇒ [vpT ] = 0 on Γ

fI(T ) = 0 ⇒

∣∣∣∣∣∣ ∗fI(T ) = 0 [vpT ] = λI∂gI(TT )∂(TT )

λI ≥ 0

∗ fI(T ) < 0 [vpT ] = 0on Γ (21)

〈v〉 = 0 (22)

v is periodic in y.


The problem Pm is settled in terms of the variable y and x plays the role of aparameter. For a given plastic state in the unit cell at the time instant t(εp, [vpT ]),this problem has a unique solution up to a constant vector (Lene and Leguillon[22]). The additional condition (22) allows us to avoid indetermination due tofields of rigid translation. Thanks to the linearity of this problem with respect toE, εp and [vpT ], its solution v can be split up into two terms:

v = Ekhvkh + L

(εp, [vpT ]

)where vkh is the solution of the problem Pm when E is equal to the unit tensorEkh (of which components are defined by Ekhij = 1

2(δki δhj + δhi δ

kj )) and when the

plastic state does not exists (εp = 0, [vpT ] = 0). L(εp, [vpT ]

)is the solution of the

problem Pm when the strain tensor E vanishes and the plastic state at time t isconsidered (εp 6= 0, [vpT ] 6= 0). Besides, L is a integro-differential operator of whichthe expression is given in terms of the Green function of the elastic problem Pm,but here we shall not need its exact expression.

4.2. Macroscopic aspect

4.2.1. Preliminary. The macrostress can be obtained by performing the integra-tion over the unit cell

Σ = 〈σ〉. (23)

It is an easy matter to realize that E is also the macrostrain tensor. Let usestablish an expression of the macrostrain tensor from the microdisplacement v.Firstly, by considering the relation defining the microdisplacement field v (12), weeasily establish:

ey(v) = E + ey(u1).

Secondly, by averaging this relation over the unit cell, applying the divergencetheorem, the periodicity conditions of the field displacement u1 on the oppositefaces of the unit cell and considering that the microdisplacement fields u1 and vhas the same discontinuity condition through the interface, we obtain:

E =1|Y |

∫∂Y

v⊗snd(∂Y ). (24)

4.2.2. Macroscopic elastic constitutive law. We shall establish the general ex-pression of the macrocompliance tensor. For a given macrostress tensor Σ, themicromechanical quantities are determined in elastic phase (ep = 0, [vpT ] = 0)by searching the periodic microdisplacement field v solution of the problem Pmeformed by equations (13), (19), (20) and (23). As a consequence of the linearityof this problem, the solution is a linear combination of particular solutions asso-ciated with the unit macrostress tensor Σkh (of which components are defined byΣkhij = 1

2(δki δ

hj + δhi δ

kj

))

v = Σkhvkh.


As the compliance homogenized tensor relates the macrostrain and stress ten-sors, its expression is given by:

Ahomijkh = Ekhij =

∫∂Y

(vkh⊗Sn)ijd(∂Y ) (25)

in which the macrostrain tensor Ekhij is associated to the microdisplacement vkh.For the future developments, let us introduce the stress localization tensor (CΣ)

which enables the microstress tensor to be obtained from the macrostress tensor:

σ = CkhΣ : Σkh (26)

withCkhΣ = σkh. (27)

The definition of the stress localization tensor (27) is introduced by Suquet[37]. Moreover by averaging this formula over the unit cell, we immediately arriveat the following property:⟨(

CkhΣ)ij

⟩=

12

(δki δhj + δhi δ

kj ). (28)

4.2.3. Macroscopic strain. Let us consider the microscopic constitutive relation(19) in the reverse form then we multiply both sides of this formula by the stresslocalization tensor and finally by averaging over the unit cell, we obtain:

< CkhΣ : A : σ > + < CkhΣ : ep >=< CkhΣ : ey(v) > . (29)

In the first step, we shall modify the first term of the equation (29). Wemultiply the microconstitutive relation (19) of the system Pme (when Σ = Σkh)by the compliance tensor and considering the definition of the stress localizationtensor (27), we establish:

CkhΣ : A = ey(vkh). (30)

By multiplying (30) by the microstrain tensor and integrating over the unitcell, we see that:

< CkhΣ : A : σ >=< ey(vkh) : σ > . (31)

By integrating by parts and using the equilibrium equation (13), we see thatthe second side of the equation (31) can be rewritten:

< ey(vkh) : σ >=1|Y |

∫Y

divy(vkh · σ)dY. (32)

By applying the divergence theorem to second side of the equation (32) andconsidering the decomposition of the displacement field vkh(= Ekh · y+ukh1 ) (12),we see that:

< ey(vkh) : σ >= Ekh : Σ− < TT ·[vkhT]>Γ .


By taking into account the expression of the compliance homogenized tensor(25) and the constitutive elastic behavior of the interface (20), we arrive at:

< ey(vkh) : σ >= AhomkhlmΣlm− < T khT ·

([vT ]− [vpT ]

)>Γ . (33)

In the second step, we modify the second side of relation (29). By applying thedivergence theorem, using the decomposition of the displacement field v (12) andnoticing that the stress localization tensor satisfies the equilibrium equation (13)and the property given by formula (28), we get:

< CkhΣ : ey(v) >= Ekh− < T khT · [vT ] >Γ . (34)

By substituting equations (33) and (34) into formula (29), we finally arrive at:

AhomkhlmΣlm+ < CkhΣ : ep > + < T khT · [vpT ] >Γ= Ekh. (35)

We recognize in Ahom : Σ the elastic part of the macrostrain and therefore theplastic part of the macrostrain is given by:

Epkh =< CkhΣ : ep > + < T khT · [vpT ] >Γ . (36)

We note that the macroscopic strain is not the average of its microscopic ana-logue. In the case of perfect bonding between the constituents, the macroplasticstrain is only defined by the first term of the preceding relation (Suquet [37]).

4.2.4. Macroscopic formulation of the boundary evolution problem. The first stepis to determine the macroequilibrium equation. By taking the average over theunit cell of the equilibrium equation at zeroth order (14) yields:

divx Σ+ < divy(σ1) + f = 0 in Ω.

By applying the divergence theorem on the second term of the previous relationtogether with the periodicity in y of the microstrain tensor, this relation becomes:

divx Σ + f = 0 in Ω. (37)

With the help of equations (35) and (36), we obtain the macroscopic elasticlaw:

Ahom : Σ +Ep = E in Ω. (38)

The second step of the analysis is the determination of the macroscopic bound-ary condtions. By taking the mean value of the Neumann boundary condition (7)at zeroth order we lead to the corresponding macroscopic condition

Σ · nT = T on ∂ΩT . (39)


As at zeroth order the displacement field is independent of the microscopicscale y, the zero displacement boundary condition (8) can be verified accurately:

u = u on ∂Ωu. (40)

As a consequence, the homogenized problem is defined by the system of equa-tions (37)–(40) in which ep and [vpT ] appear as a microscopic internal variablesdetermined by resolution up to time instant t of the microscopic problem Pm.This result has been established for the first time by Suquet [38] when the inter-face between the constituents is perfect.

5. Microscopic and macroscopic constitutive laws for the multi-layered medium

5.1. Preliminary

The unit cell of a multilayered medium is illustrated in Fig. 2. The interfaceis divided into two parts Γ1 and Γ2 having respectively inward unit normal n1and n2. Following a procedure similar to the one described by Pruchnicki andShahrour [31], we show that the microstrain and stress tensors are constant ineach constituent. Therefore the sliding along interfaces Γ1 (VΓ1m = [vm]Γ1 form = 1 or 3) and Γ2 are also uniform. As a consequence of this result, the averageof a quantity (r) over the unit cell is given by: 〈r〉 =

∑2i=1 w

iri. In order to derivethe macroscopic law, we first have to identify the relevant state variables. Theuniformity of the stress and strain tensors in each constituent implies that thedistribution of the microplastic strain tensor is also uniform. This result meansthat the behavior of elastoplastic multilayered media can be fully described usingthe state variables (E,Hα, α = 1, 2, V pΓ1

), where Hα stands for the plastic straintensor in the αth constituent and V pΓ1

(= [vpm]Γ1 for m = 1 or 3) is the plasticsliding along Γ1.

5.2. Localization procedure

The computation of the macrofree energy requires the determination of the mi-crostress and strain tensors in terms of the state variables. The elastic constitutivelaw in each constituent (19) can be rewritten:

σi = ai : (ey(vi)−Hi) (41)

in which the repeated index i does not imply summation.The elastic constitutive law of the interface Γ1 (20) becomes:

σm2 = kI(VΓ1m − VpΓ1m

) m = 1 or 3. (42)


Figure 2.Unit cell of a stratified composite.

The continuity of the microstress and strain vectors at the interface of the twocontituents leads to the following conditions:

eylm(v1) = eylm(v2) for (1,m) = (1, 1), (1, 3) and (3.3), (43)

σ1ij = σ2

ij for (i, j) = (1, 2), (2, 2) and (2.3). (44)

By combining the constitutive elastic behavior (41) the plastic evolution law(21) and the continuity of the microstress vector at the interface (44), we establishthat:

VΓ2 = −VΓ1

V pΓ2= −V pΓ1

.(45)

By considering the expression of the macroscopic strain tensor given by relation(24), the formulas linking the sliding between both the interfaces Γ1 and Γ2 (45)and applying the divergence theorem, we arriwe at:∑

k

wkekyij = Eij − (VΓ1 i(n1)j + VΓ1 j

(n1)i). (46)

Equations (41)–(44) and (46) allow the microstress and strain tensors to beexpressed in terms of the state variables.

ey(vi) = CiE : E +∑j

Cij : Hj + CIi · V pΓ1

σi = SiE : E −∑j

Sij : Hj + SI · V pΓ1

(47)


withSiE = ai : CiE , S

ij = ai : (Iδij − Cij) and SI = GiCIi.

Gi denotes shear modulus of the ith constituent and CiE is the strain concentrationtensor in the elastic domain and the expressions of tensors CiE , Cij and CIi aregiven in Appendix (formulas (A1)–(A3)).

5.3. Determination of the macrofree energy

The macrofree energy is equal to the average of the microfree one (Hill and Rice[39]). Its expression is given by:

W =12< σ : ey(v) > −1

2< σ : ep > +

12< T (n) · (VΓ − V pΓ ) >Γ . (48)

The first term of equation (48) is computed by integrating by parts and applyingthe divergence theorem.

< σ : e(u) >= Σ : E − 2σi2VΓ1 i. (49)

The expression of the macrostress tensor in terms of the state variables can bededuced from equations (23) and (47).

Σ = ahom : E −∑i

∑j

Bij : Hj + SI · V pΓ1(50)

whereahom =

∑i

BiE BiE = wiSiE Bij = wiSij.

ahom stands for the elastic macrostiffness tensor whose expression is given in Ap-pendix (formula (A4)).

By noticing that∑j

BiTj = BiE , equation (50) becomes:

Σ = ahom : E −∑j

BjET

: Hj + SI · V pΓ1. (51)

For the calculation of the second term of the macrofree energy, which stands forthe plastic strain effect, we use the expression of the microstress tensor given inequation (47):

< σ : ep >=∑i

BiE : E : Hi −∑i

∑j

Bij : HjHi + SI · V pΓ1:∑i

wi : Hi. (52)


The last term of macrofree energy (48) corresponds to the interface effect:

< T (n) · (VΓ − V pΓ ) >Γ= 2σi2(VΓ1i − VpΓ1i

). (53)

Finally, equations (48)–(49) and (51)–(53) allow the macrofree energy to be com-puted in terms of the state variables. Its expression is given by:

W =12

(ahom : E : E − 2

∑i

BiE : E : Hi +∑i

∑j

Bij : Hj : Hi

+ 2SI · V pΓ1:(E −

∑i

wiHi)

+4

ΓI(V pΓ1

)T · V pΓ1

) (54)

with

ΓI =∑i

wi

Gi+

2kI.

5.4. Derivation of the macroconstitutive law

The relationship between the macrostress tensor and the state variables is obtainedfrom the derivative of macrofree energy (W) with respect to the macrostrain tensor(E):

Σ =∂W∂E

= ahom : (E −Ep), (55)

where the macroplastic strain tensor Ep can be expressed as follows:

Ep =∑i

Ri − 1ΓISI · V pΓ1

(56)

in whichRi = Ahom : BiTE Ahom = (ahom)−1.

The analytical expressions of tensor Ri is given in Appendix (formula (A5)).Expression given in equation (56) show that the macroscopic strain is equal

to the average of the microplastic strain in the case of a multilayered mediumwith homogeneous elastic properties (a1 = a2) and perfect bounding at interface( 1kI

= 0 and UpΓ1= 0).

The thermodynamical forces associated to the internal parameters (Hα, α =1, 2) are determined from the partial derivative of the macrofree energy: Pα =∂W∂Hα . This quantity can be computed using equation (54):

Pα = wασα. (57)


The thermodynamical force associated to plastic sliding at interface is deter-mined from the partial derivative of the macrofree energy with respect to theplastic discontinuity at interface.

(PI)m = 2Σm2 = − ∂W∂V pΓ1m

for m = 1 or 3. (58)

The expressions (57) and (58) show that the thermodynamical forces (Pα, α =1, 2, PI) stand for the internal forces in each constituent and on interface. Theexpression of the microstress tensor in terms of the variables (Σ, Hα = 1, 2) canbe obtained from equations (47) and (55).

σi = CiΣ : Σ + σir (59)

with

CiΣ =1wiRi

Tσir =

∑j

U ij : Hj U ij = CiΣ : BjTE − Sij

where the stress localization tensor (CiΣ) and the residual stress tensor (σir) areonly induced by plastic strain. These tensors are independent of the plastic slidingat interface. The reason is that plastic sliding at interface only depends on themacrostress Σm2 (m = 1 or 3). The expression of tensor U ij is given in Appendix(formula (A6)). Finally, equations (57) and (58) allow us to express thermody-namical forces (Pα, α = 1, 2; PI) in terms of the state variables.

5.5. Macroelastic domain

Equation (59) shows that the macroelastic domain is defined by:

DE =

Σ, f ic(CiΣ : Σ + σir

)≤ 0, i = 1, 2, fI(Σm2) ≤ 0, m = 1, 2, 3

. (60)

We notice that the macroelastic domain results from the intersection of thecorresponding domain for perfect interface (DEP = Σ, f ic(CiΣ : Σ + σir) ≤ 0, i =1, 2) and the one of the interface (DI = fI(Σm2) ≤ 0, m = 1, 2, 3).

Setting the tensor σir equal to zero in equation (60), we obtain the followingexpression for the inital yield surface (Hα = 0, α = 1, 2, V pΓ1

= 0).

DEI = Σ, F ic(Σ) ≤ 0, i = 1, 2, fI(Σm2) ≤ 0, m = 1, 2, 3

with F ic (Σ) = f ic(CiΣ : Σ).


5.6. Hardening

The expression of the microstress tensor in equation (59) can be rewritten asfollows:

σi = CiΣ : (Σ + ti), (61)

where ti = (CiΣ)−1 : σir =∑j

V ij : Hj .

The expression of tensor V ij is given in Appendix (formula (A7)).Computation of tensor (ti) shows that:

tikh = 0 for (k, h) = (2, 2), (1, 2) and (2, 3).

By using equations (60) and (61), the following expression for the macroelasticdomain is obtained:

DE = DEP ∩DIDEP =

Σ, F ic(Σ11 + ti11, Σ22 Σ33 + ti33, Σ23,Σ13 + ti13, Σ12) ≤ 0, i = 1, 2

.

(62)This expression shows that during loading the macroyield surface (DE) un-

dergoes a kinematic hardening due to the microplastification in each constituentand independent of the plastic sliding at interface beause the elastic domain ofthe interface does not undergo hardening. Hardening parameters are given byT i = (−ti11, 0,−ti33, 0,−ti13, 0) for i = 1 or 2.

5.7. Numerical application

5.7.1. Geometrical and mechanical properties of the composite material. Considera bilaminated composite, the volume ratio of each constituent is equal to 50%. Theelastic properties of the individual phases (reinforcement and matrix) used in thecalculations are given in Table 1.

Table 1. Constituents elastic constants for reinforcement and matrix.

E (MPa) νYoung’s modulus Poisson’s ratio

Reinforcement 4 · 105 0.2

Matrix 7.5 · 104 0.3

Both the two constituents obeys a classical von Mises condition, the yieldcriterion of which takes the form:

f ic(σ) =[1

2((σ11−σ22)2 +(σ11−σ33)2 +(σ22−σ33)2)+3(σ2

12 +σ213 +σ2

23)] 1

2 − σi


Figure 3.Elastic slip coefficient of the homogenized shear modulus in plane parallel to the stratification.

where σi represents uniaxial tensile-compressive elastic limit of the reinforcement(i = 1) and the matrix (i = 2). The corresponding numerical values are respec-tively taken to be equal to 3400 MPa and 450 MPa.

Besides the interface’s elastic limit is supposed to be governed by a tensioncriterion (de Buhan and Talierco [40]), which limits the normal component of thestress vector (here Σ22) acting on the interface to a prescribed value:

fI(Σ22) = Σ22 − σint ≤ 0.

The numerical value of the positive parameter σint is taken to be equal to 40 MPa.5.7.2. Elastic analysis In Table 2, we give the component of the homogenizedelastic stiffness independent of the elastic coefficient of the interface (see expression(A4) into the Appendix).

Table 2. Components of the homogenized elastic stiffness independent of the elasticity of theinterface.

unit a 1111 a 2222 a 1122 a 1133 a 1212

GPa 268 165 56 73 98

In constrast, the shear modulus in the plane parallel to the stratification is ahyperbolic function of the slip coefficient (kI) of the interface. This moduli (a1313)is drawn for various values of the slip coefficient of the interface in Fig. 3. Thehomogenized stiffness goes to zero as the stiffness of the interface vanishes. In ad-dition, the homogenized shear modulus converges on the corresponding coefficientof the perfect interfacial composite (here equal to 49 GPa) as the value of the slipcoefficient tends towards infinity (perfect interface).


Figure 4.Yield and failure stresses in simple tension for both perfect and imperfect interfaces.

5.7.3. Inelastic analysis in uniaxial tension. The bilaminated composite wasloaded to simple macrotension (Σ1) in a plane perpendicular to the stratification.Let αΣ (= (y1,Σ1)) denote the angle between y1-axis and the macrotension (Σ1).In Fig. 4, we give the yield and failure macrostresses when interface is perfect(Pruchnicki [16]). In view of interfacial considerations about the yield criterion andthe uniformity of the stress vector, the failure of the homogenized material arisesby perfect plastification of the interface (Fig 4). We notice that the interface hasno influence on the failure in direction parallel to the stratification. Neverthelessthe interfacial influence upon the failure increases with the angle between tensileand stratification.

For the following development, we recall a result established in section 5.6 (for-mula 62), the macroscopic elastic domain is the intersection between the globalelastic domain for perfect bonding at interface with the yield domain of the in-terface. When direction of tension path is approximately inferior to 12, it canbe observed in Fig. 4 that the initial elastic phase is followed by a plastic phase,due to non linear hardening in each constituent, up to perfect plastification of theinterface at failure. We also notice that the hardening phase decreases when thedirection of tension moves away to the y1-axis. When tension path with the y1-axisis approximately superior to 12, the failure in the homogenized material occurs byperfect interfacial plastification at failure without hardening in each constituent.

For perfect interface, we see that the macroresponse is composed of the follow-ing phases:

(i) The first phase corresponds to the elastic domain.(ii) The second phase corresponds to plastification in each constituent (except


for angle αΣ contained in 30 to 45).(iii) The third phase corresponds to perfect plastification at failure.

6. Conclusion

The process using double scale asymptotic developments appears as an heuristicmethod giving the elastoplastic homogenized behavior associated to compositematerial. The plastic behavior is governed by the microscopic problem posed onthe unit cell. The general form of the macrocompliance tensor and macroplasticstrain are expressed. The macroscopic boundary value problem is given. Theapplication of this method on a multilayered media shows that the microstressand strain tensors are uniform in each constituent and the sliding is constant alongthe interface. The macroconstitutive law can then be derived from the macrofreeenergy. The initial yield surface is the intersection of the corresponding one ofeach constituent and also of the interface. The hardening law depends on themicroplastic state in each constituent but it is not affected by the microplasticsliding at interface. A numerical illustration has shown that the yield and failuredomains of the homogenized material depend strongly on the elastic domain ofthe interface.

Appendix

We set in preliminary

αi =νi

1− νi βi =(1 + νi)(1− 2νi)

(1− νi)EY i γi =1Gi

ci =1− (αi)2

βidi =

αi(1− αi)βi

f i =EY i

(1− νi)Λ =

∑i

wiαi ∆ =∑i

wiβi C =∑i

wici

D =∑i

widi F =∑i

wif i M = 2∑i

wi

γi

Ca = cicj + didj Cb = cjdi + cidj

Let us express the following tensors:

CiE =

1 0 0 0 0 0

βi Λ∆ − αi

βi

∆ βi Λ∆ − αi 0 0 0

0 0 1 0 0 00 0 0 γi

ΓI 0 00 0 0 0 1 00 0 0 0 0 γi

ΓI

(A1)


Cij =

0 0 0 0 0 0

αiδij − βi wjαj

∆ δij − wjβi

∆ αiδij − βi wjαj

∆ 0 0 00 0 0 0 0 00 0 0 δij − wjγi

ΓI 0 00 0 0 0 0 00 0 0 0 0 δij − wjγi

ΓI

(A2)

CIi = −2γi

ΓI

0 00 00 01 00 00 1

(A3)

ahom =

C + Λ2

∆Λ∆ D + Λ2

∆ 0 0 0Λ∆

1∆

Λ∆ 0 0 0

D + Λ2

∆Λ∆ C + Λ2

∆ 0 0 00 0 0 1

Γ10 0

0 0 0 0 M2 0

0 0 0 0 0 1Γ1

(A4)

Ri =

wi(Cci−Ddi)MF 0 wi(Cdi−Dci)

MF 0 0 0

wi(αi − ΛgiF ) wi wi(αi − Λgi

F ) 0 0 0wi(Cdi−Dci)

MF 0 wi(Cci−Ddi)MF 0 0 0

0 0 0 wi 0 00 0 0 0 2wi

Mγi 00 0 0 0 0 wi

(A5)

U ij =

wj(CCa−DCb)MF − ciδij 0 wj(CCb−DCa)

MF − diδij 0 0 00 0 0 0 0 0

wj(CCb−DCa)MF − diδij 0 wj(CCa−DCb)

MF − ciδij 0 0 00 0 0 0 0 00 0 0 0 2wi

Mγiγj −δijγi 0

0 0 0 0 0 0

(A6)

V ij =

wjcj − Cδij 0 wjdj −Dδij 0 0 0

0 0 0 0 0 0wjdj −Dδij 0 wjcj − Cδij 0 0 0

0 0 0 0 0 00 0 0 0 wj

γj −Mδij

2 00 0 0 0 0 0

(A7)


References

[1] R. Hill, A self-consistent mechanics of composite materials, J. Mech. Phys. Solids 13(1965), 213–222.

[2] E. Kroner, Elasitc moduli of perfectly disordered composite materials, J. Mech. Phys.Solids 15 (1967), 319–340.

[3] Z. Hashin and S. Shtrikman, On some variational principles in anisotropic and non homo-geneous elasticity, J. Mech. Phys. Solids 10 (1962), 335–342.

[4] J. R. Willis, Variational and related methods for the overall properties of composites, In:Advances in Applied Mechanics, Academic Press, New York 1981, 1–78.

[5] C. Huet, Application of variational concepts to size effects in elastic heterogeneous bodies,J. Mech. Phys. Solids 38 (1990), 813–841.

[6] S. Hazanov and C. Huet, Order relationships for boundary conditions effect in heterogeneousbodies smaller than the representative volume, J. Mech. Phys. Solids 12 (1994), 1995–2011.

[7] N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging Processes in PeriodicMedia, Kluwer, Dordrecht 1989.

[8] R. Hill, The essential structure of constitutive laws for metal composites and polycrystals,J. Mech. Phys. Solids 15 (1967), 79–95.

[9] B. D. Agarwal, J. M. Lifshitz and L. J. Broutman, Elastic-plastic element analysis of shortfibre composites. Fibre Science and Technology , Applied Sciences Publishers LTD, England1974, 45–62.

[10] S. Nemat-Nasser and M. Taya, On effective moduli of an elastic body containing periodicallydistributed voids. Q. Appl. Math. 39 (1981), 43–59.

[11] J. Aboudi, Elastoplasticity theory for composite materials. SM Archives, Oxford UniversityPress 1986, 141–183.

[12] J. M. Guedes and N. Kikuchi, Preprocessing and postprocessing for materials based on thehomogenization method with adaptive finite element methods, Comp. Meth. Appl. Mech.Engng. 83 (1991), 143–198.

[13] S. Jansson, Homogenized non linear constitutive properties and local stress concentrationsfor composites with periodic internal structure. Int. J. Solids Structures 29 (1992), 2181–2200.

[14] S. Shkoller and G. Hegemier, Homogenization of plain weave composites using two-scaleconvergence. Int. J. Solids Structures 32 (1995), 783–794.

[15] S. Ghosh, K. Lee and S. Moorthy, Multiple scale analysis of heterogeneous elastic struc-tures using homogenization theory and voronoi cell finite element method, Int. J. SolidsStructures 32 (1995), 27–62.

[16] E. Pruchnicki, Homogenized nonlinear constitutive law using Fourier series expansion, Int.J. Solids Structures 35 (1998), 1895–1913.

[17] E, Pruchnicki, Hyperelastic homogenized law for reinforced elastometer at finite strain withedge effects, to appear to Acta Mechanica.

[18] E. Pruchnicki, Overall properties of thin hyperelastic plate at finite strain with edge effectsusing asymptotic method, to appear to Int. J. Engng. Sci.

[19] J. Aboudi, Damage in composites-modeling of imperfect bonding, Compos. Sci. Tech. 28(1987), 103–128.

[20] J. Aboudi and M.-J. Pindera, Thermoelastic theory for the response of materials function-ally graded in two directions, Int. J. Solids Structures 33 (1996), 931–966.

[21] F. Lene and D. Leguillon, Etude de l’influence d’un glissement entre les constituants d’unmateriau composite sur ses coefficients effectifs, J. Mec. 20 (1981), 509–536.

[22] F. Lene and D. Leguillon, Homogenized consitutive law for a partially cohesive compositematerial, Int. J. Solids Structures 18 (1982), 443–358.

[23] C. Licht, Comportement asymptotique d’une bande dissipative mince de faible rigidite, C.


R. Acad. Sci. Paris I 317 (1989), 429–433.[24] D. Leguillon, Un exemple d’interaction singularite-couche limite pour la modelisation de la

fracture dans les composites. C. R. Acad. Sci. Paris II 319 (1994), 161–166.[25] D. Leguillon, Influence de l’epaisseur d’une interface sur les parametres de rupture. C. R.

Acad. Sci. Paris II 322 (1996), 533–539.[26] C. Licht and G. Michaille, Une modelisation du comportement d’un joint colle elastique,

C. R. Acad. Sci. Paris I 322 (1996), 295–300.[27] G. Geymonat, F. Krasucki and S. Lenci, Analyse asymptotique du comportement d’un

assemblage colle, C. R. Acad. Sci. Paris I 322 (1996), 1107–1112.[28] P. de Buhan and S. Maghous, Comportement elastique non lineaire macroscopique d’un

materiau comportant un reseau de joints, C. R. Acad. Sci. Paris II 324 (1997), 209–218.[29] P. Suquet, Approach by homogenization of some linear and nonlinear problems in solid

mechanics, in: Plastic behavior of anisotropic solids, J. P. Boehler, ed., CNRS, InternationalColoquium 319, 1985, 77–117.

[30] M. D. G. Salamon, Elastic moduli of a stratified rock mass. Int. J. Rock Mech. 5 (1968),519–527.

[31] A. Sawicki, Yield condition for layered composites. Int. J. Solids Structures 17 (1981),969–979.

[32] E. Pruchnicki and I. Shahrour, Loi d’evolution homogeneisee du materiau multicouche aconstituants elastoplastiques parfaits, C. R. Acad. Sci. Paris II 315 (1992), 137–142.

[33] E. Pruchnicki and I. Shahrour, A macroscopic elastoplastic constitutive law for multilayeredmedia: Application to reinforced earth material, Int. J. Numer. Anal. Meth. Geomech.18 (1994), 507–518.

[34] H. Murakami and G. A. Hegemier, Development of a nonlinear continuum model for wavepropagation in jointed media: theory for single joint set, Mech. Mater. 8 (1989), 199–218.

[35] S. Shkoller, A. Maewal and G. A. Hegemier, A dispersive continuum model of jointed media,Q. Appl. Math. 3 (1994), 481–498.

[36] D. Caillerie and H. Tollenaere, Modeles bidimensionnels de tisses, C. R. Acad. Sci. ParisI 318 (1994), 87–92.

[37] P. Suquet, Local and global aspects in the mathematical theory of plasticity. Plasticitytoday , Branchi and Sawczuk, ed., Elsevier, Amsterdam 1984, 279–308.

[38] P. Suquet, Plasticite et homogeneisation, These d’etat Paris VI. Paris 1982.[39] R. Hill and J. R. Rice, Elastic potentials and the structure of inelastic constitutive laws,

SIAM J. Appl. Math. 25 (1973), 448–461.[40] P. de Buhan and A. Talierco, A homogenization approach to the yield strength of composite

materials. Eur. J. Mech. A/Solids 10 (1991), 129–154.

Erick PruchnickiLaboratoire de Mecanique de Lille CNRS URA1441 Ecole Centrale de LilleB. P. 48, F-59651 Villeneuve d’Ascq CedexFrance(Fax +33 03 2033 5499)

(Received: May 5, 1997; revised: February 14, 1998)

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Homogenized elastoplastic properties for a partially cohesive composite material

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