Homogenization of layered elastoplastic composites: Theoretical results

International Journal of Non-Linear Mechanics 47 (2012) 367–376

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Homogenization of layered elastoplastic composites: Theoretical results Q.-C. He a,c,n, Z.-Q. Feng b,d a

Southwest Jiaotong University, School of Mechanical Engineering, Chengdu 610031, China Southwest Jiaotong University, School of Applied Mechanics and Engineering, Chengdu 610031, China Universite´ Paris-Est, Laboratoire Mode´lisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 Boulevard Descartes, 77454 Marne-la-Valle´e Cedex 2, France d Universite´ d’Evry-Val d’Essonne, Laboratoire de Me´canique d’Evry, 40 rue du Pelvoux, 91020 Evry, France b c

a r t i c l e i n f o

abstract

Available online 14 September 2011

Exact closed-form solutions are derived that completely characterize the effective behavior of a composite material made of elastic-perfectly plastic parallel plane layers perfectly bonded together. The derivation is framed within a rigorous theory of homogenization for elastoplastic composites, and based on the fundamental fact that the in-plane part of the strain tensor and the out-of-plane part of the stress tensor are uniform throughout the composite provided no free-edge effects occur. The obtained expressions are coordinate-free and valid in the general anisotropic case. As an example, a layered composite material with isotropic constituents is examined in detail. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Layered material Composite Elastoplasticity Homogenization Effective properties

1. Introduction The problem of predicting the effective behavior of layered composite materials is important for two reasons. From a practical point of view, some natural and man-made composites, such as stratified geological strata and laminates, can be considered as belonging to this category of materials when free-edge effects are negligible. From a theoretical standpoint, layered composites have a simple microstructure prone to getting exact closed-form solutions for their effective properties, which, once available, may serve as benchmarks for approximate analytical or numerical prediction methods. Researchers in the fields of materials science, solid mechanics and geophysics have long been interested in the homogenization of layered composite materials. However, their efforts have been mainly directed to solving this problem for linear phenomena (see, e.g., [1–10]). Recently, in the setting of non-linear phenomena, a few researchers have also investigated the problem of homogenization of simply and sequentially layered composite materials (see, e.g., [11–14]). The present work is concerned with the homogenization of composite materials made of elastic-perfectly plastic multiphase plane layers perfectly bonded together. This problem was investigated by several authors [15–19]. Nevertheless, the investigations of these authors are based on rather restrictive hypotheses and the results obtained are not complete. In the work of Sawicki [15], the constituents of the composite are required to be isotropic and n Corresponding author at: Universite´ Paris-Est, Laboratoire Mode´lisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 Boulevard Descartes, 77454 Marne-la-Valle´e Cedex 2, France. Tel.: þ33 1 60 95 77 86; fax: þ33 1 60 95 77 99. E-mail address: [email protected] (Q.-C. He).

0020-7462/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2011.09.018

the plastic strain-induced residual stress and the yield surfaces subsequent to the initial one are not determined. The work presented by de Buhan [16] and de Buhan and Salenc- on [17] is framed within a yield design homogenization theory. This theory, intended for assessing the ultimate strength of materials or structures, cannot describe the macroscopic elastoplastic behavior of a layered composite before its ultimate failure. However, the initial elastic range of a number of metal matrix composites is small in comparison with their ultimate strength. Therefore, there is a need for characterizing their macroscopic behavior between the first yielding and ultimate failure. In the works of Elomri and Sidoroff [18] and Elomri et al. [19], the constituents of the composite are assumed to have the same isotropic elastic properties. This elastic homogeneity leads them to define the macroscopic plastic strain as the unweighted volume average of its microscopic counterpart. As has been pointed out by Hill [20], such a definition is physically unrealistic in the general case where the elastic properties of the constituents of a composite are different. The objective of this work is twofold. First, it aims at deriving exact closed-form expressions for a complete characterization of the effective behavior of elastic-perfectly plastic layered composites in the general anisotropic case. Second, it has the purpose of presenting a novel application of a rigorous theory of homogenization for elastoplastic composites, essentially due to Mandel [21,22] and Hill [20,23,24] and then developed by Suquet [25] among others. The results obtained in this work substantially complete the previous relevant ones and are directly usable for layered metal composites which arise in several technologies including coating. These results also allow us to gain insight into the effective behavior of more complicated inhomogeneous elastoplastic materials.

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Q.-C. He, Z.-Q. Feng / International Journal of Non-Linear Mechanics 47 (2012) 367–376

The paper is organized as follows. In Section 2, the aforementioned homogenization theory is reviewed and structured in such a way to provide a framework and guide for us to subsequently make a mathematically consistent and physically sound analysis of the effective behavior of a layered elastoplastic composite. In Section 3, we begin with justifying that the in-plane part of the strain and the out-of-plane part of the stress are uniform throughout a layered composite even in the elastoplastic case. On the basis of this fact, the stress and strain localization tensors and the plastic strain-induced residual stress are determined. The expressions for the macroscopic measures of elastic strain, plastic strain and plastic dissipation are given. The macroscopic elastic domain is specified. The effective elastoplastic behavior of the composite as a generalized standard material is characterized. In Section 4, the results of Section 3 are applied to a composite with isotropic layers complying with von Mises’ criterion. Some remarks are given in Section 5. Notation: In order to present the essential structure of subsequent formulae in a clear manner, coordinate-free notation is employed as much as possible. A bold-face Latin lowercase letter, say a, and a bold-face Latin capital letter, say A, will denote a vector and second-order tensor, respectively. An outline Latin capital letter, say A, refers to a fourth-order tensor. If their Cartesian components are noted as ai, Aij and Aijkl, the main coordinate-free symbols used in this paper are related to the corresponding index symbols in the following way: a  b ¼ ai bi ; A : B ¼ Aij Bij ;

ðAbÞi ¼ Aij bj ;

ðABÞij ¼ Aik Bkj

ðABÞij ¼ Aijkl Bkl ;

ðABÞijkl ¼ Aijmn Bmnkl ; ðABÞijkl ¼ Aik Bjl ;

ðABBÞik ¼ Aijkl Bjl

ða  bÞij ¼ ai bj ;

ðA  BÞijkl ¼ Aij Bkl

ðABÞijkl ¼ Ail Bjk

where the convention of summation over repeated indices is adopted. In fact, the above coordinate-free symbols are widely used, except ABB, AB and AB. Designating by I the identity tensor on the vector space V, then the identity tensor I on the second-order symmetric tensor space Sym admits the expression:

I ¼ I I ¼

1 ðIIþ IIÞ 2

where use is made of the notation two second-order tensor A and B.

taken as being admissible. Within the framework of small deformation and to within a rigid-body transformation, the periodicity implies that a displacement field u of the composite is such that [26]: ~ uðxÞ ¼ Ex þ uðxÞ

~ where uðxÞ is periodic on @O. Here, E is a constant symmetric second-order tensor and the periodicity of u~ on @O means that u~ takes the same value when evaluated at any two opposed points of @O. Next, the perfect bonding of the phases at the interfaces has the consequence that u over O is continuous across G. In this work, it is assumed that u is continuous over O , and continuously ðrÞ differentiable over each O so that u is continuously differentiable almost everywhere over O . Under this condition and in view of (1), the (infinitesimal) strain tensor E is almost everywhere defined by ~ E ¼ E þ E;

for any

~ E~ ¼ 12ðu~  r þ r  uÞ

ð2Þ

~ ij ¼ @u~ j =@xi . Owing to the where ðu~  rÞij ¼ @u~ i =@xj and ðr  uÞ ~ both E and E~ are periodic on @O. Moreover, periodicity of u, denoting by n a unit normal vector field on G, the continuity of u across G implies that ~ ðIn  nÞdEcðIn  nÞ ¼ ðIn  nÞdEcðIn  nÞ ¼ 0

P ¼ PðNÞ ¼ ðINÞ ðINÞ ¼ IN II N þN  N

Consider a composite consisting of qð Z 2Þ phases that are periodically distributed and firmly bonded at the interfaces. For such a composite, there exists a unit cell which, by definition, is a representative volume element (RVE) containing all information necessary to completely characterize the material by its periodicity. For a given periodic composite, the shape of the unit cell is not unique and may, without loss of generality, be supposed a parallelepiped. Let O be the closed domain of a three-dimensional Euclidean space occupied by the chosen parallelepiped unit cell, and let ðrÞ O be the corresponding closed sub-domain occupied by phase r. We denote the interior and boundary of O by O and @O and those of ðrÞ O by OðrÞ and @OðrÞ , and assume that OðrÞ \ OðsÞ ¼ | unless r¼s. The phase interfaces inside O will collectively be designated by G. 2.1. Admissible strain and stress fields: the averaging theorem Due to the periodicity and perfect interface bonding of the considered composite, only a class of displacement fields can be

ð3Þ

ð4Þ

~ ¼ 0. In the following, we can compactly express (3) as PdEc ¼ PdEc a displacement field u over O is said to be admissible if u is continuous over O , continuously differentiable on O \G and can be represented in the form (1); a strain field E over O \G is qualified as admissible if it is derivable from an admissible displacement field. Let S be a (Cauchy) stress tensor field possible for the composite under consideration, and assume S to be differentiable over O\G. In the absence of body forces, S must satisfy the equation of static equilibrium: div S ¼ 0

over O\G

ð5Þ

The perfect bonding at G implies that the surface tractions are continuous across G: on

G

ð6Þ

In addition, the periodicity of the composite has the consequence that Sm is anti-periodic on @O

2. A theoretical framework for the homogenization of elastoplastic composites

on G

where dc represents the jump across G. Setting N ¼ n  n and introducing the interior projection operator P of Hill [27] defined by

dScn ¼ 0 A B ¼ 12ðAB þ ABÞ

ð1Þ

ð7Þ

where m denotes the outward unit normal vector field on @O and the anti-periodicity of Sm on @O means that the traction vectors Sm acting on any two opposed surfaces of the boundary @O have the same magnitude but two opposed directions. With the help of the exterior projection operator P? of Hill [27] defined by

P? ¼ P? ðNÞ ¼ IPðNÞ ¼ N I þ I NN  N

ð8Þ ?

the continuity condition (6) amounts to writing P dSc ¼ 0. In the sequel, a stress field S over O \G is said to be admissible if it satisfies (5)–(7). In micromechanics, the foregoing strain and stress tensors, E and S, are taken to be microscopic and the macroscopic strain and stress tensors, E and S, are defined as Z Z 1 1 E ¼ /ES ¼ EðxÞ dv; S ¼ /SS ¼ SðxÞ dv ð9Þ volðOÞ O volðOÞ O Hereafter, a letter with (without) an over bar refers to a macroscopic (microscopic) quantity and /  S represents the volume average of a microscopic quantity over O. With the definitions of (9), the classical virtual work principle applied to the unit cell O yields the averaging theorem [22,23,28]: for any admissible strain

Q.-C. He, Z.-Q. Feng / International Journal of Non-Linear Mechanics 47 (2012) 367–376

S ¼ LðEEp Þ

field E and any admissible stress field S: /S : ES ¼ /SS : /ES ¼ S : E

ð10Þ

This far-reaching theorem, independent of any constitutive laws, will repeatedly be used, in particular in defining macroscopic elastic and plastic quantities.

div S ¼ 0

over

over

369

O,

O\G;

P? dSc ¼ 0 on G,

u~ is periodic and Sm is anti-periodic on @O:

 Problem (PS): given a macroscopic stress tensor S and a plastic 2.2. Microscopic constitutive laws The constituents of the composite under consideration are assumed to be elastic-perfectly plastic. Precisely, we first admit that each microscopic strain field E over O \G can be decomposed into a microscopic elastic and plastic strain field over O \G: EðxÞ ¼ Ee ðxÞ þ Ep ðxÞ

or

Ee ðxÞ ¼ MSðxÞ

ð12Þ

Here, the stiffness tensor L and the compliance tensor M are assumed to have the classical minor and major symmetries, i.e. L ¼ ðIIÞL ¼ LT and M ¼ ðIIÞM ¼ MT , and to be positive-definite and related by M ¼ L1 . The elastic energy density is then given by wðx,E,Ep Þ ¼ 12ðEEp Þ : LðEEp Þ

ð13Þ

With the expressions (11) and (13), (12) can be written as S ¼ @w=@E. It is worth noting that Ee , Ep , L, M and w are well defined only over O \G. Next, let KðxÞ be the set of all stress tensors that a generic point xA O \G can have with no plastic strain. The elastic domain and yield surface of this point correspond to the interior and boundary of KðxÞ, respectively. In this work, KðxÞ is assumed to be convex and characterized by a yield function yðx,SÞ as follows: KðxÞ ¼ fS9yðx,SÞ r0g

and

½SðxÞS0  : E_ p Z 0

for all

S0 A KðxÞ

ð15Þ

Accordingly, the plastic dissipation power density is given by

Cðx, E_ p Þ ¼ sup fS : E_ p g

ð16Þ

S A KðxÞ

In the case when yðx,SÞ is differentiable with respect to S, (15) is equivalent to @yðx,SÞ ; E_ p ðxÞ ¼ L @S

S ¼ LðEEp Þ

over

L Z0;

y r0;

Ly ¼ 0

ð17Þ

where L is the Lagrange multiplier field over O \G. 2.3. Associated elastic homogenization problem It has been realized from the works of Mandel [21,22] and Hill [20,23,24] that the solution of the elastic boundary value problem associated with the homogenization of an elastoplastic composite in the absence of plastic evolution plays a fundamental role in determining both its effective elastic and plastic properties. For the composite considered in this work and according to whether a macroscopic strain or a macroscopic stress is prescribed, the aforementioned elastic boundary value problem is now formulated as:

 Problem (PE): given a macroscopic strain tensor E and a plastic ~ continstrain field Ep over O \G, find a displacement field u, uous over O and twice continuously differentiable over O \G, such that ~ E ¼ E þ 12ðu~  r þ r  uÞ

div S ¼ 0

over

O \G,

over

O,

O,

over O\G;

P? dSc ¼ 0 on G,

S ¼ /SS, u~ is periodic and Sm is anti  periodic on @O: In the foregoing formulation, Ep behaves as an eigenstrain. To within a rigid-body displacement, either (PE) or (PS) has a unique solution provided an appropriate functional space (for instance (H1 ðOÞÞ3 ) is chosen for u~ and the classical assumptions are made on the symmetry, coercivity and boundedness of L [25]. Since the problems (PE) and (PS) are linear, their respective stress and strain solutions can formally be written as Z EðxÞ ¼ AðxÞE þ E0 ðxÞ; E0 ðxÞ ¼ FnEp ðxÞ ¼ Fðx,x0 ÞEp ðx0 Þ dv O

SðxÞ ¼ BðxÞS þ Sr ðxÞ;

Z Sr ðxÞ ¼ GnEp ðxÞ ¼  Gðx,x0 ÞEp ðx0 Þ dv O

ð18Þ

ð14Þ

Denoting the plastic strain rate by E_ p and adopting the normalityflow rule, we have SðxÞ A KðxÞ

~ E ¼ /ES þ 12ðu~  r þ r  uÞ

ð11Þ

and that the microscopic stress field S is related to the microscopic elastic strain field Ee by Hooke’s law: SðxÞ ¼ LEe ðxÞ

~ continstrain field Ep over O \G, find a displacement field u, uous over O and twice continuously differentiable over O \G, such that

The fourth-order tensor A (or B) is determined by solving the problem (PE) (or (PS)) with Ep ¼ 0, while Fðx,x0 Þ (or Gðx,x0 ÞÞ is the Green function obtained by solving the problem (PE) (or (PS)) with E ¼ 0 (or S ¼ 0). Physically, E0 is the compatible total strain induced by an eigenstrain Ep , and Sr the residual stress field produced by Ep . Mathematically, E0 and Sr are two second-order tensor-valued linear functionals of Ep . When Ep ¼ 0, (11) and (18) imply that E ¼ Ee , E ¼ /Ee S, E0 ¼ 0 and Sr ¼ 0. In this case, it is natural to define the macroscopic elastic strain E e as equal to E. Correspondingly, (18) reduces to Ee ðxÞ ¼ AðxÞE e ;

SðxÞ ¼ BðxÞS

ð19Þ

Below, A and B will be called the elastic strain and stress localization tensors, respectively. In general, AT a A and BT a B, but /AS ¼ /BS ¼ I. It is important to remark that, for any macroscopic strain tensor F and any macroscopic stress tensor T, the microscopic strain field AF and the microscopic stress field BT are individually admissible. Using this fact and applying the averaging theorem, it can be shown that A and B have the useful property that /BT ES ¼ /ES;

/AT SS ¼ /SS

ð20Þ

for any admissible microscopic strain E and stress S [20,24]. By definition, the effective stiffness and compliance tensors, L and M , are such that S ¼ LEe;

Ee ¼ MS

ð21Þ

Combining (12), (19) and (21), we obtain the classical formulae:

L ¼ /LAS;

M ¼ /MBS

ð22Þ

With the help of (20) and the averaging theorem, it can be proved that the equalities L ¼ LT , M ¼ MT , and M ¼ L1 imply the equalities

370

Q.-C. He, Z.-Q. Feng / International Journal of Non-Linear Mechanics 47 (2012) 367–376 T

T

1

L ¼ L , M ¼ M , and M ¼ L . Further, it follows from (12), (19) and (21) that A, B, L and M are interconnected by AM ¼ MB;

BL ¼ LA

ð23Þ

Thus, A and B can be derived from each other. In this sense, (PE) and (PS) are equivalent. 2.4. Definitions of macroscopic elastic and plastic quantities When the plastic strain Ep is not everywhere equal to zero in

O , care is demanded in the transition from the microscopic elastic (or plastic) strain to the macroscopic elastic (or plastic) strain, even though (9) continues making sense for the macroscopic total strain and stress [20,22,25]. Starting from (91) and then using (201) and (11), we can write E ¼ /BT Ee Sþ /BT Ep S. Accounting for (122), (231), (201) and (92), we see that /BT Ee S ¼ M S. Comparing the latter with (212), it now becomes physically sound to define the macroscopic elastic and plastic strains as follows: E e ¼ /BT Ee S;

E p ¼ EE e ¼ /BT Ep S

ð24Þ

Note that /BT Ee S a/Ee S and /BT Ep S a/Ep S, because neither Ee nor Ep is generally an admissible strain field. For this reason, it makes no physical sense to define E e and E p as the unweighted volume averages of their microscopic counterparts. However, as additive quantities, the macroscopic elastic energy and plastic dissipation power, w and C , can simply be defined as the volume averages of its microscopic counterparts (13) and (16). After some computations (Appendix A), we obtain their expressions: w ¼ 12 ðEE p Þ : L ðEE p Þ þ 12/GnEp : Ep S

ð25Þ

C ¼ S : E_ p /GnEp : E_ p S

ð26Þ

The first term in (25) is the macroscopic elastic energy associated with the macroscopic elastic strain, and is hence recoverable upon macroscopic unloading; the second term represents the stored energy due to the residual stress induced by the incompatibility of the microscopic plastic strain field and is not recoverable upon macroscopic unloading. We see from (25) and (26) that the microscopic plastic strain field cannot generally be eliminated from the expressions of w and C and may be viewed as equivalent to an infinite number of internal variables.

the intersection of all sets obtained by the foregoing two operations. Recalling that the convexity of a set is conserved under an affine transformation and that the intersection of two convex sets gives a convex set, we deduce that the hypothesis that KðxÞ at any x A O \G is convex implies that KðEp Þ is convex. It is worth noting that this preservation of convexity in the transition from microscopic to macroscopic scales is closely related to the linearity of the problem (PS). Once the convex macroscopic elastic domain is determined, the question arises whether the normality-flow rule holds at the macroscopic level. In fact, Mandel [21,22] and Hill [20,23] have independently shown that the microscopic flow normality implies the macroscopic flow normality. Precisely, (15) results in S A KðEp Þ

and

½SS  : E_ p Z0 0

for all

0

S A KðEp Þ

ð29Þ

A proof for this important result is provided in Appendix B. 2.6. Characterization of the effective behavior of elastoplastic composites as generalized standard materials It is structurally useful to recast the equations governing the macroscopic behavior of the composite within the framework of generalized standard materials [25,29]. As has been seen previously, the microscopic plastic strain field Ep plays the role of a parametric field at the macroscopic level and cannot rigorously be replaced by the macroscopic plastic strain tensor. Hence, strictly speaking, the field Ep must be taken as a state variable. If the macroscopic strain tensor is chosen as another state variable, then fE,Ep g completely characterize the isothermal state of O . By definition, the associated thermodynamic forces are @w=@E and @w=@Ep and their expressions in terms of E and Ep are the state laws. With the function (25) for w and the definition (24)2 for E p , they are given by @w @E

¼ L ðE/BT Ep SÞ;



@w ¼ BL ðE/BT Ep SÞGnEp @Ep

ð30Þ

Comparing (301) and (302) with (211) and (18), we can identify @w=@E and @w=@Ep with the macroscopic stress tensor S and microscopic stress field S: S¼

@w ¼ L ðE/BT Ep SÞ; @E

S¼

@w ¼ BL ðE/BT Ep SÞGnEp @Ep ð31Þ

2.5. Macroscopic elastic domain and normality-flow rule Consider the case where the plastic strain Ep is not zero everywhere over O \G. Then, it is physically meaningful to define the corresponding macroscopic elastic domain and yield surface as the interior and boundary of the set K of all macroscopic stresses that O can have without changing the given plastic strain field. According to this definition and accounting for (15) together with (14) and (18), K is determined by KðEp Þ ¼ fS9yðx, BSGnEp Þ r0

for all x A O \Gg

or equivalently by \ KðEp Þ ¼ B1 ½KðxÞ þ GnEp ðxÞ

ð27Þ

ð28Þ

The complementary or evolution law is given by specifying the convex domain KðxÞ or the yield function yðx,SÞ as in (14) and by adopting the normality-flow rule (15). In conclusion, within the framework of generalized standard materials, the effective behavior of the considered composite is entirely characterized when its elastic domain is prescribed at each point xA O \G and the macroscopic elastic energy w is determined. In the general case, this implies an infinite number of internal variables and thus analytical or numerical approximation has to be made for the problem to be solvable [25]. However, as will be shown in the next section, in the case of layered composites, the number of internal variables reduces to being finite and the theory of this section is directly and exactly applicable.

x A O \G

Here, it is to be understood that B1 ½K þ GnEp  ¼ fS9BS A ðK þ GnEp Þg. From (28), we see that the macroscopic elastic domain KðEp Þ

3. Exact closed-form solution for the homogenization of layered elastoplastic composites

for a given plastic strain field Ep is obtained from the microscopic elastic domains after undergoing three geometrical transformations. First, the microscopic elastic domain KðxÞ at x A O \G is translated by GnEp ðxÞ; this gives rise to a kinematic hardening. Second, each set KðxÞ þ GnEp ðxÞ undergoes a linear transformation B1 which is equivalent to a rotation and a non-uniform dilation. Third, KðEp Þ is

The composite considered in this section consists of qð Z2Þ parallel plane layers which are perpendicular to and periodically distributed along the axis defined by the unit vector n. Each layer is assumed to be homogeneous and infinite in the plane transverse to n, called the layer plane, and to have a finite thickness,

Q.-C. He, Z.-Q. Feng / International Journal of Non-Linear Mechanics 47 (2012) 367–376

371

following way. First, when the layers are elastoplastic, the microscopic stress–strain relation is in general incrementally linear: dS ¼ TdE

Fig. 1. A typical unit cell for a layered multiphase composite.

say hi for layer i. A typical unit cell O for this material is shown in Fig. 1, where h denotes the total thickness of O along n. Then, the volume average operation /  S is simplified into /  S ¼ cð1Þ /  S1 þcð2Þ /  S2 þ    þ cðqÞ /  Sq ;

cðiÞ ¼ hi =h

where cðiÞ represents the volume fraction of phase i and /  Si the phase volume average. Now, we proceed to apply the theory of homogenization presented in the last section to the layered composite described above. It will tacitly be assumed that all the hypotheses made there hold in the following. 3.1. Fundamental considerations The basic characteristic of the layered composite under consideration is that it is only unidirectionally heterogeneous, i.e. homogeneous in the layer plane and heterogeneous along the direction normal to the layer plane. In the case where the constituent layers are all linearly elastic, the problem of predicting its effective behavior has analytically been solved [1–5,7]. The method of solution is based on the fundamental fact that the in-plane part of the strain and the out-of-plane part of the stress are uniform throughout O provided no free-edge effects occur. This holds even when the layers are heterogeneous along the axis normal to them. However, if the layers are individually homogeneous, then the strain and stress are additionally uniform in each layer. In virtue of the projection operators P and P? defined in (4) and (8), the in-plane part of the strain and the out-of-plane part of the stress are given by PE and P? S. Explicitly, relative to a three-dimensional orthonormal basis fe1 ,e2 ,e3 g with e3 ¼ n, the matrices of PE and P? S take the forms: 2 3 2 3 0 0 S13 E11 E12 0 6 6 7 0 S23 7 ½PE ¼ 4 E21 E22 0 5; ½P? S ¼ 4 0 5 S31 S32 S33 0 0 0 The assumption that the composite is infinite in the layer plane has indeed the purpose of eliminating free-edge effects. Practically, for a layered composite of finite geometry in the layer plane, the prevention of free-edge effects can be accomplished by imposing the following mixed uniform boundary conditions: uðxÞ ¼ ðPEÞx

on @OL ;

Sn ¼ ðP? SÞn

on @OT

where E and S are the macroscopic strain and stress tensors and @OL is the lateral boundary perpendicular to the transverse boundary @OT (Fig. 1). The key to solving our problem of predicting the effective behavior of a layered elastoplastic composite is the observation that the foregoing fundamental fact is still valid. This can be justified in the

dE ¼ T1 dS

or

where T and T1 are the microscopic tangent stiffness and compliance tensors and depend on the deformation history. Then, the incremental boundary value problem of the homogenization of a layered composite has exactly the same structure as in the linearly elastic case. Consequently, we can invoke the arguments used in the linearly elastic case to infer that the in-plane part of an incremental strain field and the out-of-plane part of the associated incremental stress field are uniform throughout O . Finally, after a time integration, the conclusion is reached that the fundamental fact still holds. Moreover, if the layers are individually homogeneous, the strain and stress are uniform in every layer. In the latter case, the elastic and plastic parts of the strain are, respectively, uniform in every layer. Nevertheless, the in-plane part of the elastic or plastic strain is generally not uniform throughout O . 3.2. Strain and stress localization tensors: plastic strain-induced residual stress Bearing in mind the theoretical framework of Section 2, the main step towards the homogenization of an elastoplastic composite turns out to be the determination of the stress (or strain) localization tensor and the residual stress due to the plastic strain. This amounts to solving the boundary value problem (PS) or (PE) formulated in Section 2.3 and hence numerical methods are in general needed. However, for a layered composite, the fundamental fact pointed out in Section 3.1 allows to reduce the solution of the boundary value problem to that of an algebraic one. By virtue of the fundamental fact, the in-plane part PE of the microscopic strain E and the out-of-plane part P? S of the microscopic stress S are, respectively, uniform throughout O . Then, it follows that PE ¼ /PES ¼ P/ES ¼ PE and P? S ¼ /P? SS ¼ P? /SS ¼ P? S, i.e.

PE ¼ PE

ð32Þ

P? S ¼ P? S

ð33Þ ?

Using the property P þ P ¼ I from (8), we can recast (32) and (33) as E ¼ E P? ðEEÞ

ð34Þ

S ¼ S PðSSÞ

ð35Þ

On the other hand, when a microscopic plastic strain field Ep is given, the stress and strain are by hypothesis related by (12) with (11), i.e. S ¼ LðEEp Þ

or

E ¼ MS þ Ep

ð36Þ

Introducing (34) into (361) and then using (33), yields

P? LP? ðEEÞ ¼ P? ðSLE þ LE p Þ

ð37Þ

Dually, substituting (35) into (36)2 and invoking (32), comes

PMPðSSÞ ¼ PðEMSE p Þ

ð38Þ

Due to the supposed positive definiteness of L and M, it makes sense to define the inverses

C ¼ ðP? LP? Þ1 ;

D ¼ ðPMPÞ1

ð39Þ

such that CðP? LP? Þ ¼ ðP? LP? ÞC ¼ P? and DðPMPÞ ¼ ðPMPÞ D ¼ P [7]. As a result of these definitions, we have P? C ¼ CP? ¼ C and PD ¼ DP ¼ D. Then, using (39) and accounting for (32) and (33), we can group (37) and (38) into E ¼ ðICLÞE þ CS þ CLEp

ð40Þ

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Q.-C. He, Z.-Q. Feng / International Journal of Non-Linear Mechanics 47 (2012) 367–376

S ¼ DE þðIDMÞSDEp

ð41Þ

These two dual relations must be equivalent under (36). It is thus deduced that L, M, C and D are interconnected by

LCþ DM ¼ CL þ MD ¼ I

we find (Appendix C)   1 L5 K¼ Iþ N L2 L2 þ L5

ð42Þ /KS

1

Therefore, (40) and (41) can further be written in the compact forms: E ¼ MDE þ CS þ CLEp

ð43Þ

S ¼ DE þ LCSDEp

ð44Þ

Averaging (40) and (41) and using (32) and (33), we have

P? S ¼ /CS1 /CLSE/CS1 /CLEp S

where /CS1 and /DS1 are understood in the sense that /CS1 /CS ¼ /CS/CS1 ¼ P? and /DS1 /DS ¼ /DS /DS1 ¼ P. Substituting (45) and (46) into (43) and (44), respectively, we obtain E ¼ ðMD þ C/CS

  1 L5 P? þ NN L2 L2 þ L5

ð60Þ

D¼

  1 M3 Pþ ðINÞ  ðINÞ M1 M1 þ 2M3

ð61Þ

/CS1 ¼ ð46Þ

/CLSÞE þ CLEp C/CS

1

/CLEp S

ð59Þ

C¼

ð45Þ

PE ¼ /DS1 /DMSS þ/DS1 /DEp S

1

! /L5 =ðL22 þL2 L5 ÞS 1 Iþ N ¼ /1=L2 S /1=L2 S/L5 =ðL22 þL2 L5 ÞS

ð58Þ

! /L5 =ðL22 þL2 L5 ÞS 1 N  N P? þ /1=L2 S /1=L2 S/L5 =ðL22 þL2 L5 ÞS

1 /DS1 ¼ /1=M1 S " # /M3 =ðM12 þ 2M1 M3 ÞS ðINÞ  ðINÞ Pþ /1=M1 S2/M3 =ðM12 þ 2M1 M3 ÞS

ð62Þ

ð63Þ

ð47Þ

S ¼ ðLC þ D/DS1 /DMSÞSDEp þ D/DS1 /DEp S

ð48Þ

Comparing (47) and (48) with (18), respectively, we identify the strain and stress localization tensors:

A ¼ MD þ C/CS1 /CLS

ð49Þ

B ¼ LCþ D/DS1 /DMS

ð50Þ

and the compatible strain and residual stress fields due to the plastic strain field:

These formulae generalize the relevant ones given by Hill [27] for the isotropic case. 3.3. Expressions of macroscopic measures and effective elastic tensors We are now in a position to explicitly define the macroscopic elastic and plastic strain tensors of a layered composite in terms of its microscopic counterparts. Introducing the expression (50) of B into the definitions (24) of E e and E p , we obtain

E0 ¼ FnEp ¼ CLEp C/CS1 /CLEp S

ð51Þ

E e ¼ /BT Ee S ¼ /CLEe S þ /MDS/DS1 /DEe S

ð64Þ

Sr ¼ GnEp ¼ DEp þ D/DS1 /DEp S

ð52Þ

E p ¼ /BT Ep S ¼ /CLEp Sþ /MDS/DS1 /DEp S

ð65Þ

It is useful to note that E0 is an out-of-plane tensor, i.e. P? E0 ¼ E0 , while Sr is an in-plane tensor, i.e. PSr ¼ Sr . Application of the formulae (49)–(52) mainly resides in calculating the tensors C and D, defined by (39), and their volume average inverses /CS1 and /DS1 . It is known (see, e.g., [27]) that the calculation of C involves the inverse of the Christoffel (or acoustic) tensor LBN. More precisely, C is given by

C ¼ 12ðK N þ N KÞ;

K ¼ ðLBNÞ1

ð53Þ

Once C is available, D is given by (42) as

D ¼ LLCL

ð54Þ

Concerning /CS

1

, an explicit expression was derived by Norris [7]:

/CS1 ¼ 12ðJ N þ N JÞ;

J ¼ ð2INÞ/KS1 ð2INÞ

ð55Þ

As expected, E e a /Ee S and E p a /Ep S unless all the layers are elastically homogeneous so that /CLEe S þ/MDS/DS1 /DEe S ¼ ðCL þ MDÞ/Ee S ¼ /Ee S ð66Þ /CLEp Sþ /MDS/DS1 /DEp S ¼ ðCL þ MDÞ/Ep S ¼ /Ep S ð67Þ where (42)2 is used in the second equalities. Next, the expressions for the effective stiffness and compliance tensors are given by inserting (49) and (50) into (22):

L ¼ /LAS ¼ /DS þ /LCS/CS1 /CLS

ð68Þ

M ¼ /MBS ¼ /CS þ/MDS/DS1 /DMS

ð69Þ

By contrast, no similar expression seems to exist for /DS . In the general anisotropic case, K and /KS1 are complicated functions of N and so are C, D, /CS1 and /DS1 . However, when the layers are all transversely isotropic about the axis normal to them, the expressions for these tensors are quite simple. Consider the elastic stiffness and compliance tensors which are transversely isotropic about n and given by

These two concise coordinate-free formulae, which have already been obtained by Norris [7], summarize all the results precedently reported in the literature on the effective behavior of linearly elastic layered composites. Substituting (52) into (25) and (26), we obtain the expressions for the macroscopic elastic energy and plastic dissipation power of a layered elastoplastic composite:

L ¼ L1 P þL2 P? þ L3 ðINÞ  ðINÞ

w ¼ 12 ðEE p Þ : L ðEE p Þ þ 12 /Ep : DEp S12/DEp S/DS1 /DEp S

1

þL4 ½N  ðINÞ þ ðINÞ  N þ L5 N  N

ð56Þ

M ¼ M1 P þM2 P? þM3 ðINÞ  ðINÞ þM4 ½N  ðINÞ þðINÞ  N þ M5 N  N

ð70Þ

C ¼ S : E_ p E p Þ/Ep : DE_ p S þ /DEp S/DS1 /DE_ p S ð57Þ

where P, P? ,ðINÞ  ðINÞ,N  ðINÞ þ ðINÞ  N and N  N are a complete set of transversely isotropic tensor generators [30]. Then,

ð71Þ

In (70), the stored energy w 0 ðEp Þ ¼ 12 /Ep : DEp S12/DEp S/DS1 /DEp S

ð72Þ

Q.-C. He, Z.-Q. Feng / International Journal of Non-Linear Mechanics 47 (2012) 367–376

depends only on the in-plane plastic strain PEp because PD ¼ DP ¼ D. This indicates that he hardening of the considered composite is due to the in-plane microscopic plastic strain. It is also worth noting that the microscopic strain field Ep cannot be eliminated from w 0 , even when all the layers are elastically homogeneous. Indeed, in the latter case, E p ¼ /Ep S but w 0 ðEp Þ ¼ ðD :: /Ep  Ep SE p : D1 E p Þ=2 with D :: /Ep  Ep S ¼ ðDÞijkl /Ep  Ep Sijkl . 3.4. Macroscopic elastic domain and plastic strain flow When the microscopic elastic domain at each point x A O \G is specified, the macroscopic elastic domain of the layered composite with any given microscopic plastic strain field can now be determined. Introducing the expression (50) of the stress localization tensor and the expression (52) of the residual stress field into (27), we obtain the characterization of the macroscopic elastic domain: ^ KðEp Þ ¼ fS9yðx,S; Ep Þ r0

for all x A O \Gg

ð73Þ

^ Ep Þ is the yield function defined by where yðx,S; ^ Ep Þ ¼ y½x,ðLCþ D/DS1 /DMSÞSDEp þ D/DS1 /DEp S yðx,S; ð74Þ Noting that DEp ¼ DðPEp Þ, we see that only the in-plane part PEp of the plastic strain field contributes to the evolution of the macroscopic elastic domain. The macroscopic behavior of a layered elastoplastic composite as a generalized standard material is completely described by (70) and (74) together with the normality-flow rule. To obtain explicit expressions for the plastic flow, assume that y(x, S) is differentiable with respect to S at each point x so that (17) applies. Define ^ the tensor Gðx,S; Ep Þ as follows: ^ Ep Þ ¼ G½x, BSDEp þ D/DS1 /DEp S Gðx,S; Gðx,SÞ ¼

@yðx,SÞ @S

ð75Þ ð76Þ

Then (17) becomes ^ E_ p ðxÞ ¼ LGðx,S; Ep Þ;

Next, the microscopic and macroscopic plastic strain flows are given by ðiÞ ðiÞ E_ p ¼ Hij1 s_ j G^

y^ r 0;

Ly^ ¼ 0

ð77Þ

ð84Þ

m X ðiÞ E_ p ¼ /BT E_ p S ¼ cðiÞ Li ðCðiÞ LðiÞ þ/MDS/DS1 DðiÞ ÞG^

ð85Þ

i¼1 ðiÞ Numerical methods are needed for the time integration of E_ p _ and E p .

4. An illustrative example The results obtained in the previous section are valid for the most general anisotropic case and most of them do not require the homogeneity along the axis normal to the layer plane. In this section, we consider a simple case of practical interest, i.e. a composite with isotropic elastic-perfectly plastic layers which are individually homogeneous and comply with von Mises’ yield criterion. This model idealizes some problems arising in coating engineering. By hypothesis, the elastic behavior of layers is described by the isotropic stiffness or compliance tensor:

L ¼ 2mI I þ lI  I;

M¼

1 l II I I 2m 2mð3l þ 2mÞ

ð86Þ

where l and m are the Lame´ elastic coefficients which are uniform in every layer. The plastic behavior of layers is characterized by von Mises’ criterion: rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð87Þ yðx,SÞ ¼ S : HS y0 ðxÞ r0 3 where H ¼ I II  I=3 and y0 ðxÞ is the tensile yield stress which is assumed to be uniform in every layer. To calculate the tensors C and D defined by (39), we apply the formulae (60)–(63) to (86). For this, it suffices to set L1 ¼ L2 ¼ 2m, L3 ¼ L4 ¼ L5 ¼ l,M1 ¼ M2 ¼ 1=ð2mÞ and M3 ¼ M4 ¼ M5 ¼ l=½2mð3l þ 2mÞ in these formulae. First, (60) and (61) give

C¼ L Z0;

373

1 ðP? ZN  NÞ 2m

D ¼ 2m½P þ ZðINÞ  ðINÞ

ð88Þ ð89Þ

When the condition y^ ¼ 0 is satisfied at some x A O \G, the corresponding Lagrange multiplier L is determined by the consistency condition y^_ ¼ 0, i.e.

where Z ¼ l=ðl þ 2mÞ ¼ n=ð1nÞ with n being the Poisson ratio. Then,

^ G^ : D/DS1 /LDGS ^ ¼ G^ : BS_ LG^ : DG

/CS ¼ 12 /m1 SP? 12/m1 ZSN  N

ð90Þ

/DS ¼ 2/mSP þ 2/mZSðINÞ  ðINÞ

ð91Þ

ð78Þ

Further, assume that the layers are individually homogeneous. In this case, the microscopic stress and plastic strain fields are uniform in each layer and the Lagrange multiplier field L can be replaced by a finite number of multipliers. Thus, when mð r qÞ phases are plastified, (78) can be written in the matrix form: Hij Lj ¼ s_ i

ði,j ¼ 1; 2, . . . ,mÞ

ð79Þ

where Hij and s_ i are defined by ðiÞ ðiÞ ðiÞ ðiÞ Hii ¼ G^ : DðiÞ G^ cðiÞ G^ : DðiÞ /DS1 DðiÞ G^

Hij ¼ cðjÞ G^

ðiÞ

: DðiÞ /DS1 DðjÞ G^

ðiÞ _ s_ i ¼ G^ : BðiÞ S

ðjÞ

ði ajÞ

/DS1 ¼

  1 /mZS ðINÞ  ðINÞ P 2/mS /mS þ 2/mZS

ð92Þ

ð93Þ

ð81Þ

To our knowledge, these two expressions have not been presented in the literature. Next, by using the formula (50) and after some computations, the stress localization tensor has the following expression:

ð82Þ

B ¼ P? þ b1 P þ b2 ðINÞ  ðINÞ þ b3 ðINÞ  N

ð80Þ

Here, (i) refers to phase i, c(i) denotes the volume fraction of phase i, and the summation convention does not apply to repeated indices. As the layers are elastic-perfectly plastic, the matrix Hij can be inverted. Then, it follows from (79) that

Li ¼ Hij1 s_ j

Applying (62) and (63), we obtain   2 /m1 ZS ? N  N P þ /CS1 ¼ /m1 S /m1 S/m1 ZS

ð83Þ

ð94Þ

with

b1 ¼ b3 ¼

m /mS

;

b2 ¼

mZ/mS/mZSm /mSð/mSþ 2/mZSÞ

/mSðZ/ZSÞ þ 2Zð/mZS/mS/ZSÞ /mS þ2/mZS

ð95Þ

ð96Þ

374

Q.-C. He, Z.-Q. Feng / International Journal of Non-Linear Mechanics 47 (2012) 367–376

To give a physical interpretation to the coefficients b1 , b2 and b3 , let e1 ,e2 ,e3 be a three-dimensional orthonormal basis with e3 ¼ n and consider the following particular macroscopic stress tensors: S 1 ¼ e1  e2 þ e2  e1 ;

S 2 ¼ IN;

ð106Þ

S2 ¼ BS 2 ¼ ðb1 þ 2b2 ÞS 2 ;

S3 ¼ BS 3 ¼ S 3 þ b3 ðINÞ

Thus, we see that b1 represents the magnitude of the microscopic in-plane shear stress produced by a unit in-plane macroscopic shear stress, b1 þ2b2 the one of the microscopic in-plane isotropic stress induced by a unit macroscopic in-plane isotropic stress, b3 the one of the microscopic in-plane isotropic stress generated by a unit macroscopic simple traction stress along the axis normal to the layer plane. In the particular case where the phases of the composite have the same Z, i.e. the same Poisson ratio, it is immediate from (59) and (96) that b2 ¼ b3 ¼ 0 and b1 ¼ m=/mS ¼ E=/ES with E being the Young modulus. The effective elastic stiffness and compliance tensors are obtained by applying the formulae (68) and (69):

L ¼ L 1 P þ L 2 P? þL 3 ðINÞ  ðINÞ þL 4 ½N  ðINÞ þ ðINÞ  N þL 5 N  N L 1 ¼ 2/mS;

L2 ¼

2 ; /m1 S

ð97Þ

L 3 ¼ 2/mZSþ

2/ZS2 /m1 S/m1 ZS ð98Þ

L4 ¼

2/ZS ; /m1 S/m1 ZS

L5 ¼

E p ¼ /P? Ep S þ /b1 PEp S þ/b2 trðPEp ÞSðINÞ þ /b3 trðPEp ÞSN

S3 ¼ N

In the absence of plastic strain, the microscopic stresses induced by these tensors are S1 ¼ BS 1 ¼ b1 S 1 ;

macroscopic plastic strain tensor is related to the microscopic plastic strain field by

2/m1 ZS /m1 Sð/m1 S/m1 ZSÞ

ð99Þ

M ¼ M 1 P þ M 2 P? þ M 3 ðINÞ  ðINÞ þ M 4 ½N  ðINÞ þ ðINÞ  N þM 5 N  N

ð100Þ

We observe that, if fe1 ,e2 ,ng is a three-dimensional orthonormal basis, then e1  E p n ¼ /e1  E p nS and e2  E p n ¼ /e2  E p nS, but n  E p n a /n  E p nS because the macroscopic normal plastic strain may result from the isotropic part of the microscopic plastic strain field. With (106), we have the macroscopic elastic strain E e ¼ EE p . The corresponding macroscopic elastic energy is calculated by applying (70): w ¼ 12ðEE p Þ : L ðEE p Þ þw 0 ðEp Þ where the stored energy w ðEp Þ is given by w 0 ðEp Þ ¼ 2/mPEp : PEp S

/m1 S ; M2 ¼ 2

þ

2/mZS/m trðPEp ÞS2 4/mS/mZ trðPEp ÞS2 /mSð/mSþ 2/mZSÞ

with ^ ¼ BT HB ¼ P? þ b2 P1ðb2 2b2 2b b ÞðINÞ  ðINÞ H 1 2 1 2 3 1 13ðb1 þ 2b2 b1 b3 2b2 b3 Þ½N  ðINÞ þ ðINÞ  N

M4 ¼

/ZS ; 2ð/mS þ 2/mZSÞ

M5 ¼

/ZS /m ZS  2 /mS þ 2/mZS 1

ð102Þ

As expected, both L and M are transversely isotropic with respect to n. The expressions (97)–(99) for L are known in different equivalent forms in the literature [1–4], but the expressions (100)–(102) for M have been omitted. The residual stress for a given plastic strain field is determined by inserting the expressions of D and D1 into (52):  ð/mSZ/mZSÞ/m trðPEp ÞS 2m /mPEp S2mPEp þ 2m Sr ¼ /mSð/mS þ 2/mZSÞ /mS  ð1þ 2ZÞ/mZ trðPEp ÞS Z trðPEp Þ ðINÞ þ ð103Þ /mS þ2/mZS This stress field is in-plane and depends only on the in-plane plastic strain field PEp . In the particular case where all of the layers have the same elastic properties but different yield stresses, (103) reduces to Sr ¼ 2mð/PEp SPEp Þ þ 2mZ½/trðPEp ÞStrðPEp ÞðINÞ

2

S0r ¼ HSr ð101Þ

ð104Þ

which is generally non-zero. This shows that the residual stress field may originate from elastic or/and plastic inhomogeneity. By (18), the microscopic stress field is given by S ¼ BS þ Sr ¼ P? S þ b1 PS þ½b2 trðPSÞ þ b3 trðP? SÞðINÞ þ Sr ð105Þ It is easy to see that (105) together with (103) verifies the condition that P? S ¼ P? S. According to the definition (24)2, the

ð108Þ

and depends only on the in-plane part PEp of the plastic strain field. Expressed in terms of the macroscopic stress tensor S and microscopic plastic strain field Ep , the von Mises’ criterion (87) takes the form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi ^ S þ 2S0 : BS þ S0 : S0  2y0 ðxÞ r 0 ^ yðx,S,E Þ ¼ S:H ð109Þ p r r r 3

13ð1 þ 4b2 4b2 6b3 ÞN  N

/mZS M3 ¼ 2/mSð/mS þ 2/mZSÞ

2

2 /mPEp S /mS

: /mPEp S þ 2/mZ½trðPEp Þ2 S

2

1 ; M1 ¼ 2/mS

ð107Þ

0

ð110Þ ð111Þ

The macroscopic elastic domain is then characterized by ^ KðEp Þ ¼ fS9yðx,S; Ep Þ r 0

for all xA O g

ð112Þ

It is seen from (109) and (112) that, even though the layered composite in question is the simplest one, its macroscopic hardening, i.e. the dependence of K on Ep , remains rather complicated under a general loading. Of particular interest is the initial case when the plastic strain field Ep is absent and hence the residual stress field calculated by (103) is zero. Correspondingly, (109) reduces to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi ^ S  2y0 ðxÞ r 0 ^y 0 ðx,SÞ ¼ yðx,S,0Þ ^ ¼ S:H ð113Þ 3 The initial macroscopic elastic domain K 0 is then given by K 0 ¼ fS9y^ 0 ðx,SÞ r0 for all x A O g:

ð114Þ

This domain is transversely isotropic with respect to the direction ^ normal to the layer plane since so is the fourth-order tensor H given by (110). However, this domain is no longer independent of ^ in general. The shape of ^ aH hydrostatic pressures, because HHH KðEp Þ is different from K 0 . The relation of KðEp Þ to K 0 cannot be described neither by an isotropic nor a kinematic hardening. On the other hand, KðEp Þ is not transversely isotropic but monoclinic with respect to the layer plane. Moreover, as Sr is an in-plane stress, it follows from (109) that Ep affects the shape of K in a inplane way. In this sense, the hardening of the considered layered composite is said to be two-dimensional. It finally remains to specify the expression of E_ p . Using the von Mises criterion (87) and the normality-flow rule, E_ p is given by

Q.-C. He, Z.-Q. Feng / International Journal of Non-Linear Mechanics 47 (2012) 367–376

Accounting for (11) and (12) and using (18), we can further write (A.1) as

(77) together with

HBS þ S0r

G^ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ S þ 2S0 : BS þ S0 : S0 S:H r r r

ð115Þ

2w ¼ /ðBS þ Sr Þ : MðBS þSr ÞS ¼ /BS : MBSS þ2/Sr : MBSSþ /Sr : MSr S

The Lagrange multiplier in (77) can be calculated by applying formulae (79)–(83). However, a detailed quantitative analysis could not be done without having recourse to numerical methods. This will be presented in another paper dedicated to numerical simulations and examples. ˜eda [31] and Ponte In the papers of deBotton and Ponte Castan ˜eda and deBotton [32], a composite with ideally plastic rigid Castan layers complying with the von Mises criterion was considered as a limiting case of a relevant power-law layered composite. By applying a variational procedure, they obtained an exact solution for the ultimate yield domain of the composite. In particular, they observed that this domain cannot be described as a level set of a quadratic yield function. In the present paper, the macroscopic elastic domain is shown to be characterized by Eq. (112) and can be interpreted as the intersection of a certain number of domains described by different quadratic functions of form (109). Clearly, this macroscopic elastic domain cannot in general be expressed as a level set of a quadratic function. Thus, we see that our result (112) is compatible with the ˜eda [31] and Ponte Castaobservation of deBotton and Ponte Castan ˜eda and deBotton [32]. In the forthcoming paper dedicated to n numerical simulations and examples, we shall make use of their exact result about the ultimate yield domain for comparison.

ðA:2Þ

Note that MBS is an admissible strain field and Sr is an admissible stress field such that /Sr S ¼ 0. Applying the averaging theorem yields /BS : MBSS ¼ /BSS : /MBSS ¼ /BSS : /MBSS ¼ S : M S

ðA:3Þ /Sr : MBSS ¼ /Sr S : /MBSS ¼ 0

ðA:4Þ

Then, the macroscopic elastic energy density in terms of S and Sr is given by w ¼ 12 S : M S þ 12/Sr : MSr S

ðA:5Þ

Recall that Sr is the stress solution of the problem (PS) defined in Section 2.3 and subjected to the condition /SS ¼ 0. Therefore, MSr þ Ep is an admissible strain field. Thus, /Sr : MSr S ¼ /Sr : ðMSr þ Ep ÞS/Sr : Ep S ¼ /Sr : Ep S

ðA:6Þ

Using (A.6), S ¼ L ðEE p Þ and Sr ¼ GnEp in (A.5), we obtain w ¼ 12 ðEE p Þ : L ðEE p Þ þ 12/GnEp : Ep S

ðA:7Þ

The macroscopic plastic dissipation power density C is, by definition, the volume average of its microscopic counterpart (16): _ ¼ S : E_ w _ _ wS _ _ ¼ /SS : /ES C ¼ /CS ¼ /S : E_ p S ¼ /S : E w

5. Final remarks

ðA:8Þ

Exact closed-form solutions have been derived for the effective behavior of a composite consisting of anisotropic elastic-perfectly plastic plane layers. These results can directly be extended to the case where the constituent layers themselves exhibit plastic hardening. In such a case, the macroscopic hardening of the composite results both from the microscopic hardening and plastic straininduced residual stress. Another extension can be directed toward composites with curvilinear layers whose properties are uniform along the corresponding curvilinear coordinates. For example, such a composite occurs when bonding concentric cylindrical layers which are cylindrically anisotropic. It should be emphasized that the closed-form solutions presented in this paper are exact if and only if no free-edge effects take place. As discussed at the beginning of section 3, this requirement is satisfied if the assumption is made that each layer is infinite. For a layered composite of finite boundaries, it amounts to imposing the uniform stress condition on the boundaries parallel to the layer plane, and the uniform strain condition on the boundaries perpendicular to the layer plane. In practice, these boundary conditions are rarely met. However, in the elastic case, the work by Pipes and Pagano [33] has illustrated that the stress field of a layered composite only suffers disturbances in boundary regions of a dimension comparable to its thickness. Thus, the basic role played by the stress localization tensor in the elastoplastic case gives strong evidence that this conclusion would remain valid and that the results obtained in this paper would be accurate for any layered composite whose thickness is much smaller than its plane dimensions but is much larger than the maximum layer thickness.

Using the expression (A.5) of w and the fact that MSr þ Ep is an admissible strain field, we have _ ¼ S : E_ p /Sr : MS_ r S ¼ S : E_ p /Sr : ðMS_ r þ E_ p ÞS þ /Sr : E_ p S S : E_ w

ðA:9Þ _ _ Applying the average theorem, /Sr : ðMS r þ E p ÞS ¼ /Sr S : /MS_ r þ E_ p S ¼ 0. Finally,

C ¼ S : E_ p þ /Sr : E_ p S ¼ S : E_ p /GnEp : E_ p S

ðA:10Þ

Appendix B. Proof of (29) 0

Let S and S be any two macroscopic stress tensors belonging to KðEp Þ. The associated microscopic stress fields are given by S ¼ BS þ Sr ;

0

S0 ¼ BS þ Sr

ðB:1Þ 0

The residual stress fields corresponding to S and S are identical to 0 each other, since both S and S belong to the same domain KðEp Þ. Using the definition (24)2 of E p , we have 0 0 0 0 ðSS Þ : E_ p ¼ ðSS Þ : /BT E_ p S ¼ /ðSS Þ : BT E_ p S ¼ /BðSS Þ : E_ p S

ðB:2Þ Accounting for (B.1), we can further write 0 ðSS Þ : E_ p ¼ /ðSS0 Þ : E_ p S

ðB:3Þ

Therefore, if the microscopic plastic normality-flow rule (15) applies, then 0 ðSS Þ : E_ p Z0

ðB:4Þ

In short, (15) implies (29).

Appendix A. Derivation of (25) and (26) By definition, the macroscopic elastic energy density is the volume average of its microscopic counterpart (13): w ¼ /wS ¼ 12/ðEEp Þ : LðEEp ÞS

375

ðA:1Þ

Appendix C. Derivation of (58)–(63) Let U be a tensor representing a linear transformation from a vector or tensor space V into itself, and let a, a and b be a scalar

376

Q.-C. He, Z.-Q. Feng / International Journal of Non-Linear Mechanics 47 (2012) 367–376

and two elements of V, respectively. Then, ðU þ aa  bÞ

1

¼U

1



a 1 þ aðb  U1 aÞ

1

ðU

aÞ  ðU

T

bÞ

ðC:1Þ

provided U is invertible and 1 þ aðb  U1 aÞ a0. This algebraic formula is the key to the derivation of formulae (58)–(63). We also need the following formulae: ðA  BÞBC ¼ ACBT ;

ðABÞBC ¼ ðB : CÞA;

[8] [9] [10]

[11]

ðABÞBC ¼ ACT B

ðC:2Þ

[12]

When L is given by (56), then with the help of (C.2) we obtain

LBN ¼ L2 Iþ L5 N

ðC:3Þ

By definition, K ¼ ðLBNÞ1 . Recalling that N ¼ n  n and using (C.1), we have   1 L5 I N ðC:4Þ K¼ L2 L2 þ L5 As N is constant for a composite with plane layers, it follows from (C.4) that /KS ¼ /1=L2 SI/L5 =ðL2 þ L5 ÞSN

ðC:5Þ

Applying (C.1) yields ! /L5 =ðL22 þ L2 L5 ÞS 1 Iþ N /KS1 ¼ /1=L2 S /1=L2 S/L5 =ðL22 þ L2 L5 ÞS

[14]

[15] [16]

[17]

[18]

ðC:6Þ

With the expressions (56) and (57) for L and M, we have ?

[13]

[19]

[20]

P LP? ¼ L2 P? þL5 N  N

ðC:7Þ

PMP ¼ M1 P þM3 ðINÞ  ðINÞ

ðC:8Þ

[21]

ðC:9Þ

[22]

To use (C.1) to calculate C, we take the symmetric second-order tensor subspace fS9P? S ¼ Sg for V, and P? for the identity on V. Then, the expressions for C and /CS1 can be obtained in the same way as K and /KS1 . To apply (C.1) to the calculation of D and /DS1 , we take the subspace fE9PE ¼ Sg for V, and P for the identity on V.

[23]

References

[26]

?

? 1

The tensor C ¼ ðP LP Þ ?

?

?

?

is defined in the sense that

CðP LP Þ ¼ ðP LP ÞC ¼ P?

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