An approach to micro-macro modeling of heterogeneous materials

Computational Mechanics 27 (2001) 37±48 Ó Springer-Verlag 2001

An approach to micro-macro modeling of heterogeneous materials V. Kouznetsova, W. A. M. Brekelmans, F. P. T. Baaijens

37 Abstract A micro-macro strategy suitable for modeling the mechanical response of heterogeneous materials at large deformations and non-linear history dependent material behaviour is presented. When using this micromacro approach within the context of ®nite element implementation there is no need to specify the homogenized constitutive behaviour at the macroscopic integration points. Instead, this behaviour is determined through the detailed modeling of the microstructure. The performance of the method is illustrated by the simulation of pure bending of porous aluminum. The in¯uence of the spatial distribution of heterogeneities on the overall macroscopic behaviour is discussed by comparing the results of micromacro modeling for regular and random structures.

1 Introduction Heterogeneous materials are found in nature as well as man-made situations. Typical examples are metal alloy systems, polymer blends, porous and cracked media, polycrystalline materials and composites. The effective physical behaviour of a heterogeneous structure, to be considered here as a matrix material with separated inclusions, strongly depends on the size, shape, properties and spatial distribution of the second phase. To determine the macroscopic overall characteristics of heterogeneous media is an essential problem in many engineering applications. From the time and cost viewpoints, performing straightforward experimental measurements on a number of material samples, for various phase properties, volume fractions and loading histories is a hardly feasible task. On the other hand, due to the usually enormous difference in length scales involved, it is impossible, for instance, to generate a ®nite element mesh that accurately represents the microstructure and also allows the numerical solution of the macroscopic structural component within a reasonable amount of time on today's computational systems. To overcome this problem several homogenization techniques have been created to obtain a suitable constitutive model to be inserted at the macroscopic level.

Received 9 July 2000

V. Kouznetsova (&), W. A. M. Brekelmans, F. P. T. Baaijens Faculty of Mechanical Engineering, Netherlands Institute for Metals Research, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

The simplest method leading to the homogenized moduli of a heterogeneous material applies the rule of mixtures. The overall property is then calculated as an average over the respective properties of the constituents, weighted with their volume fractions. This approach takes into consideration only one microstructural characteristic, the volume ratio of the heterogeneities and, strictly speaking, is justi®ed only for linear material properties (e.g. elastic moduli). A more sophisticated method is the self-consistent or effective medium approximation, that has been established by Eshelby (1957) and further developed by a number of authors, see, e.g. Hashin (1962), Hashin and Shtrikman (1963), Budiansky (1965), Hill (1965), Christensen and Lo (1979), Hashin (1983). Equivalent material properties are derived as a result of an analytical (or semi-analytical) solution of a boundary value problem for a spherical or ellipsoidal particle of one material in an in®nite matrix of another material. This self-consistent approach gives a reasonable approximation for structures that possess some kind of geometrical regularity, but it fails to describe the behaviour of clustered structures. Moreover, high contrasts between properties of the phases cannot be represented. Another mathematical approach is the asymptotic homogenization theory, documented in Bensoussan et al. (1978) and Sanchez-Palencia (1980). This method applies an asymptotic expansion of displacement and stress ®elds on the ``natural length parameter'', which is the ratio of a characteristic size of the heterogeneities to a measure of the macrostructure, to approximate their respective macroscopic distributions and then utilizes variational principles to create a link between the scales, see, e.g. Tolenado and Murakami (1987), Devries et al. (1989), Guedes and Kikuchi (1990), Hollister and Kikuchi (1992) and Fish et al. (1999). The asymptotic homogenization approach provides effective overall properties as well as local stress and strain values. However, usually the considerations are restricted to very simple microscopic geometries and simple material models, mostly at small strains. Along with the development of computational methods so-called unit cell methods became widely used. These methods are based on the concept of a representative volume element (RVE), originally introduced by Hill (1963). The homogenized material properties are determined by ®tting the results of the detailed modeling of the RVE (typically performed by the ®nite element method) on macroscopic phenomenological equations. This kind of approaches has been used in a great number of different

38

applications. A selection of some examples may be found in Christman et al. (1989), Tvergaard (1990), Bao et al. (1991), Brockenbrough et al. (1991), Nakamura and Suresh (1993), McHugh et al. (1993a), Smit et al. (1999b) and van der Sluis et al. (1999). The unit cell methods allow to account easily for a complex microstructural morphology and enable the investigation of the in¯uence of different geometrical features on the overall response. But, because these approaches formulate the macroscopic constitutive relations based on the behaviour of a single RVE subjected to a given loading history, they are in fact successful only for small deformations and simple material behaviour (e.g. elastic, elasto-plastic), if the structure of the macroscopic constitutive equations can be established a priori. However, using these techniques for large deformations, complex microstructures and/or non-linear history dependent constitutive behaviour usually does not lead to adequate results. For example, McHugh et al. (1993b) have demonstrated that when a composite is characterized by power-law slip system hardening, the power-law hardening behaviour is not preserved at the macroscale. As a possible way of solving this problem a variety of direct micro-macro methods has been proposed. These approaches do not lead to an overall material description valid for the whole macroscopic piece of material, but estimate the relevant stress±strain relationship at a macroscopic point by performing separate calculations on the RVE, assigned to that macroscopic point. The analysis on the RVE level is realized using the ®nite element method, e.g. in Terada and Kikuchi (1995), Smit (1998), Smit et al. (1998), Miehe et al. (1999), Feyel and Chaboche (2000) and the Voronoi cell method in Ghosh et al. (1995, 1996) or the Fourier series approach in Moulinec and Suquet (1998). To bridge the gap between micro- and macrolevel these methods apply the ideas of the asymptotic homogenization theory in Terada and Kikuchi (1995) and Ghosh et al. (1995, 1996) or perform a volume averaging in Smit (1998), Smit et al. (1998), Miehe et al. (1999) and Feyel and Chaboche (2000). Probably one of the most straightforward methods among the various micro-macro approaches mentioned above is the one elaborated by Smit (1998) and Smit et al. (1998). Despite its simplicity the method enables the incorporation of large deformations and rotations (geometrical non-linearity) and arbitrary material behaviour (including history dependence). In the context of this approach to each integration point of the macromesh an RVE is assigned and a separate ®nite element computation is performed. From the macroscopic deformation tensor appropriate boundary conditions are derived to be imposed on the RVE. The macroscopic stress tensor is obtained by averaging the resulting RVE stress ®eld over the volume of the RVE. The consistent macroscopic stiffness matrix at the macroscopic integration point is derived from the total RVE stiffness matrix by reducing the latter to the relation between the forces acting on the vertices of the RVE and displacements of these vertices. When using this multilevel approach there is no need to specify the macroscopic constitutive behaviour, which, in the case of large deformations and complex microstructures, would be a hardly feasible task generally. Instead, the constitutive behaviour

at the macroscopic integration point is deduced from the averaged behaviour of the associated microstructure. In order to validate the method Smit (1998) and Smit et al. (1998) applied the procedure for hypo-elastic material behaviour. In the present work this micro-macro approach is evaluated for time dependent material behaviour, described by the elasto-visco-plastic model, initially proposed by Bodner and Partom (1975). As an example pure bending of porous aluminum is considered. The micro-macro methods, as well as most of other homogenization techniques, are based on the concept of an RVE. The proper choice of the RVE determines the success of the heterogeneous material modeling. As the simplest RVE, a piece (for example a square or cube) of the matrix material containing one single heterogeneity (e.g. inclusion) could be suggested. In fact, this implicitly assumes the regular arrangement of the heterogeneities in the matrix, which contradicts to the observations that almost all materials, at least the ones produced in industrial amounts, have a spatially random (or non-regular) microstructural composition. Examples are precipitates in metal alloys arranged randomly by their nature and arti®cial ®ber reinforced composites, possessing a non-regular distribution of the ®bers due to the production process. At the same time, several experimental evidences exist showing that the spatial variability in the microstructure signi®cantly in¯uences the overall behaviour and particularly the fracture characteristics of composites, as reported by Mackay (1990) and Barsoum et al. (1992). Different authors, e.g. Brockenbrough et al. (1991); Nakamura and Suresh (1993), Ghosh et al. (1996) and Moulinec and Suquet (1998), have performed a comparison of the overall composite responses resulting from modeling of regular and random structures. They have found a signi®cant difference in the plastic regime, while there is almost no deviation in elastic regime. Also it has been shown by Smit et al. (1999b), that the softening behaviour of a regular structure may change to hardening in case of a random structure. Most of these considerations, except for the latter, have been performed for small deformations, very simple elasto-plastic behaviour and stiff inclusions (®bers). In the present paper the overall behaviour of regular and random structures is compared at large deformations, non-linear history dependent material behaviour, for voided material (an appropriate approximation for material with soft inclusions). Apart from the calculations on the RVE level (tensile con®guration), also a full multi-level analysis (pure bending) of both regular and random structures is presented. The paper is organized as follows. In Sect. 2, the micromacro approach is described. Section 3 contains a numerical example, evaluating the applicability of the presented procedure to the simulation of large-strain timedependent bending of porous aluminum. In Sect. 4 the overall response of a perfectly regular structure (cubic stacking of heterogeneities) is compared with the overall response of a random structure for different material models (hyper-elastic, elasto-visco-plastic with hardening and elasto-visco-plastic with softening) and different loading histories (uniaxial extension and pure bending). The paper ends with some concluding remarks.

2 Micro-macro modeling 2.1 Basic hypotheses The material con®guration to be considered is assumed to be macroscopically homogeneous (continuum mechanics theory is suitable to describe the macroscopic behaviour), but microscopically heterogeneous (the morphology consists of distinguishable components as e.g. grains, interfaces, cavities). This is schematically illustrated in Fig. 1. The microscopic length scale is much larger than the molecular dimensions, so that a continuum approach is justi®ed for every constituent, but much smaller than the characteristic length of the macroscopic sample. In this case a hypothesis on periodicity of the microstructure is acceptable. Most of the homogenization approaches make an assumption on global periodicity of the microstructure, suggesting the whole macroscopic specimen consists of spatially repeated unit cells. In the present method a more realistic assumption on local periodicity is proposed, i.e. the microstructure can have different morphologies corresponding to different macroscopic points, while it repeats itself in a small vicinity of each individual macroscopic point. Due to the repetitive character of the microscopic deformation the macroscopic stresses and strains around a certain point P (see Fig. 1) can be estimated by averaging the stresses and strains in a small representative part of the microstructure, attributed to that point. The macroscopic stress±strain relation at point P can be established by applying the local macroscopic strain tensor to a microstructural RVE, belonging to P, and by averaging the resulting non-uniform RVE stress ®eld. The consistent stiffness at a macroscopic point is derived by reducing the total RVE stiffness. In the subsequent sections these issues are discussed in more detail. First the problem on microlevel is de®ned, then the aspects of the coupling between micro- and macrolevel are considered and ®nally the realization of the whole procedure within the ®nite element method framework is outlined. In this contribution the considerations are restricted to two dimensional con®gurations only in order to reduce computational effort. However, all derivations can be easily extended to three dimensions.

Fig. 1. Continuum macrostructure and heterogeneous microstructure associated with point P

2.2 Definition of the problem on the micro-level The physical and geometrical properties of the microstructure are identi®ed by an RVE. An example of a typical two dimensional RVE is depicted in Fig. 2. The RVE must be selected such that the local microscopic material structure can be considered as the RVE surrounded by copies of itself, without overlapping of the RVEs and without voids between the boundaries of the RVEs. The actual choice of the RVE is a rather delicate task. The RVE should be large enough to represent the microstructure, without introducing non-existing properties (e.g. anisotropy) and at the same time it should be small enough to allow ef®cient computational modeling. This issue has been discussed in a number of publications, see for example Huet (1990) and Terada et al. (2000). In this section it is assumed that the appropriate RVE has been already selected. Then the problem on the RVE level can be formulated as a standard problem of a quasi-static continuum solid analysis. Equilibrium equation The RVE is in equilibrium. In absence of body forces the Cauchy stress tensor r should satisfy

r
r ˆ 0 in XRVE ;

…1†

where r
r represents the divergence of r and XRVE is the RVE domain. Constitutive behaviour The mechanical characterizations of the microstructural constituents are described by certain constitutive laws, that specify a time and history dependent stress-strain relationship

rk …t† ˆ F k fF…s†; s 2 ‰0; tŠg;

k ˆ 1; 2; . . . ; N ;

…2†

where N is the number of the microstructural constituents with different material behaviour (e.g. matrix and ®ber), t denotes the current time and F…s† ˆ …r0 y†c represents the deformation gradient tensor at time s in an arbitrary point with actual position vector y…s† while the gradient opera-

Fig. 2. Typical two dimensional representative volume element (RVE)

39

tor r0 is taken with respect to the reference con®guration dient tensor ®eld F…y0 † over the initial volume V0 of the (the superscript c indicates conjugation). RVE and scaling by this volume

40

Boundary conditions Periodic boundary conditions are imposed on the RVE. This choice has been justi®ed by a number of authors, e.g. van der Sluis et al. (2000) and Terada et al. (2000), who showed that periodicity conditions give a more reasonable estimation of the effective moduli, than both uniform displacements or uniform traction boundary conditions. The periodic boundary conditions imply that (i) the shape and orientation in space of two opposite edges is and remains identical during the deformation process and (ii) in order to have stress continuity across the boundaries the stress vectors acting on opposite sides are opposite in direction. The spatial periodicity of XRVE is enforced when the edges C43 and C23 in Fig. 2 are linked to the other (retained or independent) edges C12 and C14 and vertices 1, 2 and 4. This de®nes the kinematical boundary tyings

y43 …s43 † ˆ y12 …s12 †

y1 ‡ y4

for s43 ˆ s12 ;

…3†

y23 …s23 † ˆ y14 …s14 †

y1 ‡ y2

for s23 ˆ s14 ;

…4†

RVE ˆ 1 F V0

Z

F…y0 †dV0 :

…9†

y0 2V0

Stress The averaged RVE stress tensor can be de®ned according to a simple virtual work consideration. The variation of the internal work dWint performed by the RVE averaged Cauchy RVE on arbitrary virtual displacements of stress tensor r the RVE du is required to be equal to the work dWext performed by external loads on the RVE. The internal work performed by the RVE averaged stress can be written as

Z

Z

RVE : RVE : rdu dV ˆ r r

dWint ˆ V

ndu dC ;

…10†

C

where the Gauss±Ostrogradskii divergence theorem has been applied. In Eq. (10) V denotes the current volume of the RVE and n is the unit outward normal vector on the deformed RVE surface C; the gradient operator r is taken with respect to the current con®guration. With incorporation of the periodicity conditions (3) and (4), relation (10) for the internal work is transformed to

where yi is the position vector of angular point i with respect to some ®xed reference system, while yij and sij

denote the position vector and local coordinate (with RVE : n41 …du1 du2 † y1 y4 vertex i as the origin), respectively, of a point on boundary dWint ˆ r


Cij . ‡n12 …du1 du4 † y1 y2 ; …11† The requirement of opposite stress vectors on opposite where nij is the unit outward normal to the straight line boundaries can be written as connecting the vertices i and j in the current con®guration, r
n43 …s43 † ˆ r
n12 …s12 † for s43 ˆ s12 ; …5† and kmk denotes the length of any vector m. Enforcing the applied boundary conditions by means of r
n23 …s23 † ˆ r
n14 …s14 † for s23 ˆ s14 ; …6† tying conditions implies not only that the position of the where nij is the unit outward normal to the edge Cij . tied points is prescribed by the retained points, but also The displacements of the vertices 1, 2 and 4 that loads on the tied points have to be transferred to the ui ˆ u0i ; i ˆ 1; 2; 4 ; …7† retained points. Therefore the virtual work of the external loads is performed only on the retained vertices and on the are prescribed by applying the local macroscopic deforretained parts of the boundary of the constrained RVE mation gradient tensor to the initial positions of these (vertices 1, 2 and 4 and boundaries C12 and C14 in Fig. 2). vertices. This performs the actual transition of the mac- As a consequence of the continuity of the stress across the roscopic deformation ®eld to the microlevel. The micro- boundary of the RVE represented by the relations (5) and macro coupling is discussed in the next section. (6), the distributed external loads pext on the retained parts of the boundary also vanish from the formulation

2.3 Coupling of the macroscopic and microscopic levels

pext 12 …s12 † ˆ 0;

Deformation Let Fmacro be the deformation gradient tensor at some macroscopic point, then this deformation gradient tensor can be directly applied to the RVE associated to this point by prescribing the new positions of the angular points of the periodic RVE according to

yi ˆ Fmacro
y0i ;

i ˆ 1; 2; 4 ;

…8†

where y0i represents the initial position of angular point i. It can easily be proved that prescribing displacements of the RVE vertices by relation (8) is equivalent to the asRVE , where F RVE is the volume aversumption Fmacro ˆ F aged deformation gradient tensor, that follows from a straightforward integration of the RVE deformation gra-

pext 14 …s14 † ˆ 0 :

…12†

Obviously the work of external loads can be written as

dWext ˆ f 1
du1 ‡ f 2
du2 ‡ f 4
du4 ;

…13†

where f i is the external force per unit of current thickness that is applied to angular point i. In both relations (11) and (13) the variation of the displacements of the retained nodes dui can be expressed in terms of the variation of the deformation gradient tensor by application of relation (8)

dui ˆ dFmacro
y0i ;

i ˆ 1; 2; 4 :

…14†

Equating (11) and (13) and taking into account that the equality should be satis®ed for any arbitrary variation of the deformation gradient tensor dFmacro eventually an expression for the averaged stress tensor is obtained

RVE ˆ r

1 1 X ‰f 1 y 1 ‡ f 2 y 2 ‡ f 4 y 4 Š ˆ f iy ; A A iˆ1;2;4 i

…15†

The reduced RVE stiffness matrix SRVE de®ned by (18) can, in the particular case of a periodic RVE with prescribed vertices numbered as 1, 2 and 4, be partitioned as

2 3 where A represents the current area of the RVE. If the S…11† S…12† S…14† forces f i are in rotational equilibrium (that is the case for 4 5 the converged solution at RVE level), then the symmetry of SRVE ˆ S…21† S…22† S…24† ; S…41† S…42† S…44† RVE is guaranteed. Also it can be proved that the above r derivation of the averaged stress based on the virtual en- with the matrices S given by …ij† ergy balance is equivalent to a simple volume averaging of " # 11 12 the RVE stress ®eld r…y† over the current volume V of the S…ij† S ; i; j ˆ 1; 2; 4 : S…ij† ˆ …ij† RVE S21 S22 RVE r

1 ˆ V

…ij†

Z

r…y†dV :

…16†

y2V



Krr Kpr

Krp Kpp



dur dup





0 ˆ dfp



…17†

…21†

Every matrix S…ij† can be considered as the representation of a second-order tensor S…ij† . With this interpretation relation (181 ) is rewritten as

X

Consistent stiffness When the micro-macro approach is realized within the framework of a non-linear ®nite element code, the consistent stiffness matrix at every macroscopic integration point is required. Because there is no speci®c form of the constitutive behaviour on the macrolevel assumed beforehand in the present method, the stiffness matrix has to be determined directly from the numerical relation of the macroscopic stress and the macroscopic deformation at this point. Miehe (1996) has suggested a method based on a forward difference approximation. In the present work another approach is proposed. The consistent macroscopic stiffness matrix at a macroscopic integration point is derived from the total RVE stiffness matrix by reducing the latter to the relation between the forces acting on the retained vertices of the RVE and the displacements of these vertices. For this purpose the total RVE system of equations is rearranged to the form

…ij†

…20†

jˆ1;2;4

S…ij†
duj ˆ df i ;

i ˆ 1; 2; 4 :

…22†

RVE at every integration point, the variation As rmacro ˆ r of the stress tensor drmacro is determined by varying relation (15)


rmacro 1 X dA ‡ df i yi ‡ f i dyi : A A iˆ1;2;4

drmacro ˆ

…23†

The variation dA of the current area of the RVE in the case of two dimensional con®gurations is written as c dA ˆ AFmacro : dFcmacro :

…24†

Substituting (24), (22) and (14) into (23) leads to

 X  1 X y ˆ S…ij† y0j ‡ f i Iy0i A iˆ1;2;4 i jˆ1;2;4  c rmacro Fmacro : dFcmacro ; 

drmacro

…25†

where the expression in square brackets equals the required stiffness tensor 4 Smacro at the macroscopic with Krr , Krp , Kpr , and Kpp partitions of the stiffness matrix integration point; I denotes the unit tensor. while dup and dfp refer to the columns of the iterative 2.4 displacements and external forces of the prescribed reFinite element implementation tained vertices and dur to the columns of the iterative Based on the above developments the micro-macro stratdisplacements of the dependent nodes. In the formulation egy may be described by the following subsequent steps. of (17) it is assumed that for the converged solution the (residual) reaction forces of the dependent nodes can be 1. The macroscopic structure to be analyzed is assumed to neglected. Equation (17) may be rewritten to obtain the be macroscopically homogeneous and is discretised by reduced stiffness matrix SRVE that relates displacement ®nite elements. The external load is applied by an variations to vertex force variations incremental procedure. Increments can be associated with discrete time steps. SRVE dup ˆ dfp with SRVE ˆ Kpp Kpr Krr 1 Krp : …18† 2. To each macroscopic integration point P, a unique discretised periodic RVE is assigned. The geometry of Next, Eq. (181 ) relating displacement variations and force the RVE is based on the microstructural morphology of variations needs to be transformed to arrive at an exthe material under consideration. pression relating the variations of the macroscopic stress 3. Compute for each integration point P the evolution of and deformation according to the local macroscopic deformation gradient tensor 4 c Fmacro (P) from the estimation of the macroscopic nodal Smacro : dFmacro ˆ drmacro ; …19† displacements of the macroscopic element enclosing P RVE and r RVE , where Fmacro and rmacro can be identi®ed by F and apply appropriate displacements to the retained respectively, and where the fourth-order tensor 4 Smacro vertices of the RVE(P) according to relation (8). represents the required consistent stiffness at the macro- 4. Compute the resulting stress and deformation distriscopic integration point level. butions in the RVE(P) by solving the RVE boundary

41

composed of ®ve quadrilateral 8 node plane strain reduced integration elements. The undeformed and deformed geometries of the macromesh are shown in Fig. 3. At the left side the strip is ®xed in axial (horizontal) direction, the displacement in transverse (vertical) direction is left free. At the right side the rotation of the cross section is prescribed. As pure bending is considered the behaviour of the strip is uniform in axial direction and, therefore, a single layer of elements on the macrolevel suf®ces to simulate the situation. In this example two heterogeneous microstructures consisting of a homogeneous matrix material with 12 and 30% volume fraction of voids are studied. To generate a random distribution of cavities in the matrix with a prescribed volume fraction, maximum diameter of holes and minimum distance between two neighboring holes, for a two dimensional RVE, the procedure from Hall (1991) and Smit (1998) has been adopted. The RVEs used in the calculations are presented in Fig. 4.

42

Fig. 3a, b. Undeformed a and deformed b con®gurations of the macromesh

5.

6.

7. 8.

value problem outlined in Sect. 2.2 using a separate ®nite element discretisation. Compute the RVE vertex forces f i ; i ˆ 1; 2; 4 and substitute these into Eq. (15), yielding the RVE averaged RVE . This stress is returned to the macroscopic stress r integration point as a local macroscopic stress rmacro . Based on the stress rmacro , obtained for each macroscopic integration point, the internal nodal forces at the macroscopic level can be calculated. If these forces are in balance with the external load, incremental convergence has been achieved and the next time increment can be evaluated. If there is no convergence the procedure is continued to arrive at an updated estimation of the macroscopic nodal displacements. Calculate for each macroscopic integration point the consistent macroscopic stiffness matrix 4 Smacro according to (25) from the global RVE stiffness matrix. Assemble the macroscopic stiffness matrix and righthand side vector and produce an updated estimation of the macroscopic displacement ®eld.

This algorithm has been implemented into a multi-level ®nite element program. The macroscopic procedure utilizes an Updated Lagrange environment of a ®nite element code in MATLAB (1996). For the microscopic part of the scheme the commercially available ®nite element package MARC (1997) is used.

3 Example of micro-macro modeling 3.1 Micro- and macrogeometries In order to evaluate the presented method pure bending of a rectangular strip under plane strain conditions has been examined. Both the length and the height of the sample Fig. 4a, b. RVEs used in the calculations with 12% voids a and equal 0.2 m, the thickness is taken 1 m. The macromesh is 30% voids b

3.2 Constitutive modeling To demonstrate the performance of the method for the case of non-linear history and time dependent material behaviour the matrix material has been described by the elasto-visco-plastic model initially developed by Bodner and Partom (1975). They proposed a stress dependent viscosity expression, supplemented with a work hardening contribution to describe the strain rate dependent yield and post-yield behaviour of metals (at elevated temperatures). Smit et al. (1999a) have embedded this Bodner± Partom viscosity into the generalized Leonov model, which is a compressible version of the model established by Leonov (1976), proposed by Baaijens (1991). In this model the total deformation is decomposed in an elastic and a plastic contribution. The elastic deformation is coupled to the stress according to an isotropic neoHookean relationship (also see Sect. 4.1). The dissipative plastic deformation rate tensor Dp is related to the deviator of the Cauchy stress tensor rd according to the generalized Fig. 5. Moment±curvature diagram resulting from the micro-macro calculations Newtonian ¯ow rule Dp ˆ

rd 2g

…26†

holes also the response of a homogeneous con®guration (without cavities) is shown. It can be concluded that even with the stress dependent viscosity de®ned as the presence of 12% voids induces a reduction of the  2n ! bending moment of more than 25% in the plastic regime. r 1 Z …27† This implies that a straightforward application of the rule g ˆ p exp 2 r 12C0 of mixtures leads to erroneous results. p Figure 6 shows the contour plots of the effective plastic where r ˆ 3=2…rd : rd † is the equivalent von Mises strain for the case of the RVE with 12% volume fraction stress. The material parameters C0 and n re¯ect the voids at a curvature of 1.25 m 1 and an applied moment smoothness of the elastic-to-plastic transition and the equal to 6:8
105 N m in the deformed macrostructure and strain rate sensitivity, respectively. The state variable Z in three deformed, initially identical RVEs at different controls the hardening as the resistance to plastic ¯ow, locations in the macrostructure. Each hole acts as a plastic which is de®ned by the following evolution equation strain concentrator and causes higher strains in the RVE Z ˆ Z1 ‡ …Z0 Z1 †e mep …28† than occurring in the homogenized macrostructure. In the present calculations the maximum effective plastic strain where the constants Z 0 and Z 1 denote the lower and upper in the macrostructure is about 25%, whereas at RVE level bounds of Z, respectively, and m is a material constant this strain reaches 80%. It is obvious from the deformed controlling the rate of work hardening. The internal geometry of the holes in Fig. 6 that the RVE in the upper variable ep represents the effective plastic strain, which part of the bended strip is subjected to tension and the is de®ned by the following evolution equation p  ep ˆ 2=3…Dp : Dp †. Issues concerning the numerical time integration of this constitutive model are discussed in detail in Smit et al. (1999a). In the present calculations the material parameters for annealed aluminum AA 1050 determined by van der Aa (1999) have been used; elastic parameters: shear modulus G ˆ 2:6
104 MPa, bulk modulus K ˆ 7:8
104 MPa and viscosity parameters: C0 ˆ 108 s 2 , m ˆ 13:8, n ˆ 3:4, Z0 ˆ 81:4 MPa, Z1 ˆ 170 MPa.

3.3 Results Micro-macro calculations for a heterogeneous structure, represented by the RVEs shown in Fig. 4 have been carried out, simulating pure bending at a prescribed moment rate equal to 5
105 N m s 1 . In Fig. 5 the moment±curvature Fig. 6. Contour plots of the effective plastic strain in the diagram resulting from the micro-macro approach is deformed macrostructure and in three deformed RVEs, presented. To give an impression of the in¯uence of the corresponding to different points of the macrostructure

43

containing one single hole (Fig. 7a). To avoid a discussion on the minimum required size (including the number of voids) of a random RVE, 10 different random RVEs have been generated (Fig. 7b). The averaged behaviour of these 10 RVEs is expected to be representative for the real random structure with a given volume fraction of heterogeneities. In the subsequent sections the comparison is performed for three different constitutive models of the matrix material: hyper-elastic, elasto-visco-plastic with hardening and elasto-visco-plastic with intrinsic softening. 4 First the uniaxial extension of a macroscopic sample is Regular versus random representation In the introduction it has been pointed out that the spatial considered. Because in this case the macroscopic deforarrangement of the microstructure may have an important mation ®eld is homogeneous a full micro-macro modeling is not necessary and an analysis of an isolated RVE with impact on the macroscopic response of heterogeneous adequate boundary conditions suf®ces. In Sect. 4.4 the materials. To contribute to the discussion about the inresults of a micro-macro simulation of bending using the ¯uence of the randomness of the microstructure on the random and the regular microstructures are compared. macroscopic behaviour, in this section the overall response of a perfectly regular structure (cubic stacking of heterogeneities) is compared with the overall response of a 4.1 random structure. Elastic behaviour, tension A material with 12% volume fraction of voids is consid- In this section the comparison of the overall behaviour of ered. The regular structure is modeled by a square RVE the regular and random structures is carried out for the RVE in the lower part to compression, while the RVE in the vicinity of the neutral axis is loaded considerably milder than the other RVEs. This con®rms the conclusion that the method realistically describes the evolution of the microstructure. The method may be applied not only as an alternative for the homogenization of the material behaviour, but also to describe mechanical transformations of the microstructure.

44

Fig. 7a, b. RVE with one hole a, representing a regular structure, and 10 random RVEs b

45 Fig. 9a, b. Contour plots of the equivalent von Mises stress (MPa) in the deformed regular a and random b structures for the case of hyper-elastic behaviour of the matrix material

Fig. 8. Tensile stress±strain responses (RVE averages) of the regular and random structures for the case of hyper-elastic behaviour of the matrix material

case of hyper-elastic behaviour of the matrix material. The compressible neo-Hookean model is used, given by the following constitutive equation

s ˆ K…J

d ; 1†I ‡ GB

…29†

where s ˆ Jr denotes Kirchhoff stress tensor, J ˆ det…F†  d the deviator of the isochoric left the volume ratio, B  ˆ J 2=3 B with B ˆ F
Fc , Cauchy Green (Finger) tensor B K the bulk modulus, and G the shear modulus. The material parameters used in the calculations are K ˆ 2667 MPa, G ˆ 889 MPa. Figure 8 shows the stress±strain curves for the RVEs with regular and random void stacks. For small deformations there is almost no difference in the responses originating from the regular and random void distributions. This result is in agreement with the experiences reported in the literature for small deformations, see, e.g. Brockenbrough et al. (1991), Nakamura and Suresh (1993) and Moulinec and Suquet (1998). For large deformations the stiffer behaviour of the regular cubic structure becomes a little more pronounced, however, the deviations remain small. The difference between the response of the regular structure and the response averaged over the random RVEs does not exceed 2%. This small deviation is explained by Fig. 9, presenting the contour plots of the equivalent von Mises stress in the regular RVE and a random RVE for 20% macroscopic strain. The stress ®eld around any hole of the random structure is almost the same as around the hole of the regular structure. If only the averaged elastic constants are of interest, it is concluded that calculations performed on the simplest regular RVE usually provide the answer within an acceptable tolerance.

Fig. 10. Tensile stress±strain responses (RVE averages) of the regular and random structures for the case of elasto-visco-plastic behaviour with hardening of the matrix material

section investigates the responses of the regular and random RVEs under tensile loading when the matrix material exhibits elasto-visco-plastic behaviour with hardening. The constitutive description is performed by the Bodner± Partom model speci®ed in Sect. 3.2. The RVEs are subjected to uniaxial tension at a constant strain rate of 0.5 s 1 . In Fig. 10 the stress±strain curves are presented. In this case the difference between the overall response of the regular structure and the averaged response of the random structures reaches 10%. The fundamental mechanism that governs this deviation is illustrated in Fig. 11, where the distribution of the effective plastic strain in the deformed regular and random RVEs at 15% applied macroscopic strain is presented. In the regular RVE the ligaments yield simultaneously rather than sequentially which is the case for the random RVE. As a result, at the same value of the macroscopic strain the regular RVE is loaded relatively smooth, while some ligaments in the random RVE have 4.2 already accumulated a signi®cant amount of plastic strain. Elasto-visco-plastic behaviour with hardening, tension The in¯uence of the randomness of the microstructure on Consequently, the regular structure with a cubic stacking the macroscopic response becomes more signi®cant when of heterogeneities results in an overestimated overall plastic yielding of one or more constituents occurs. This stiffness, compared to a random con®guration.

46 Fig. 11a, b. Contour plots of the effective plastic strain in the deformed regular a and random b structures for the case of elasto-visco-plastic behaviour with hardening of the matrix material

4.3 Elasto-visco-plastic behaviour with softening, tension The difference in yielding mechanisms for regular and random microstructures outlined in the previous section causes not only a quantitative deviation in the responses of these structures (as illustrated by Fig. 10), but in some cases also the qualitative character changes, as has been shown by Smit et al. (1999b). For example, such a phenomenon can be observed when the matrix material is described by a generalized compressible Leonov model with intrinsic softening and subsequent hardening. The model incorporates a stress dependent Eyring viscosity extended by pressure dependence and intrinsic softening effects. It is beyond the scope of this paper to enter in more details; the interested reader is addressed to the references Baaijens (1991) and Smit et al. (1999a). The resulting stress±strain curves for uniaxial tension of polycarbonate at a constant strain rate of 0.01 s 1 are given in Fig. 12. The overall behaviour of the regular structure in the plastic regime exhibits some initial softening followed by hardening. The response of the regular structure is, in fact, identical to the response of one single ligament, that softens according to the intrinsic material

behaviour. A completely different situation can be observed for the random con®guration. Although some of the random RVEs also demonstrate some softening behaviour, originating from the relatively simple arrangement of the RVEs used in the calculations, the average response of the random RVEs does not show any softening but exhibits hardening. This is caused by the sequential appearance of elastic, softening and hardening zones within the random microstructure. This example illustrates that the overall response of heterogeneous materials, when determined from a modeling by a regular structure, should be interpreted with great care, especially in case of complex material behaviour (e.g. in case of softening followed by hardening or vice versa).

4.4 Elasto-visco-plastic behaviour with hardening, bending The comparison of the overall behaviour of the regular and random microstructures performed in Sects. 4.1±4.3 has been based on the averaged behaviour of a single RVE subjected to a particular loading history (uniaxial extension). The question remains how the randomness of the microstructure does in¯uence the overall behaviour when the macroscopic sample is deformed heterogeneously, so that potentially every material point of the sample is subjected to a different loading history. In order to investigate this item the micro-macro modeling approach described in Sect. 2 is a helpful tool. As an example the in¯uence of the spatial arrangement of the microstructure on the overall moment±curvature response of the voided material under pure bending is studied. The behaviour of the matrix material is described by the elasto-visco-plastic model with hardening outlined in Sect. 3.2. The macrogeometry and the material parameters are the same as these used in Sect. 3. Figure 13 shows the moment±curvature diagram resulted from the full micro-macro analysis of pure bending of the

Fig. 12. Tensile stress±strain responses (RVE averages) of the Fig. 13. Moment±curvature responses of the regular and random regular and random structures for the case of elasto-visco-plastic structures for the case of elasto-visco-plastic behaviour with behaviour with softening of the matrix material hardening of the matrix material

material using the regular and the random microstructures. Again, the regular cubic structure exhibits a stiffer response than the averaged random result, while the maximum deviation is only about 5%, which is considerably less than for the tensile test with the same material behaviour (see Sect. 4.2, Fig. 10). This smaller deviation originates from the fact that in the case of bending all the RVEs assigned to the various macroscopic points over the height of the bended strip are loaded differently, supported by Fig. 6. The RVE at the top of the bended strip experiences tension, so that the observations dealt with in Sect. 4.2 apply. At the same time, there are also RVEs that are stretched less or still in elastic regime, like for example the one in the vicinity of the neutral line, so that in average for the whole bending process the in¯uence of randomness can be expected to be smaller than for the uniaxial extension.

5 Conclusions A micro-macro strategy for the simulation of the mechanical behaviour of heterogeneous materials has been outlined. The performance of the method is illustrated by the modeling of pure bending of porous aluminum. To demonstrate the capability of the procedure to account for large deformations (and rotations) and history dependent material behaviour, the matrix material has been selected to be described by an elasto-visco-plastic Bodner±Partom model in the context of ®nite kinematics. The in¯uence of the spatial distribution of heterogeneities on the overall behaviour of a heterogeneous material has been investigated. It has been shown that for elastic material behaviour this in¯uence is negligible, while in case of more complex material behaviour (e.g. elasto-viscoplastic at large strains) the difference in response of the regular and random microstructure is signi®cant. For some constitutive features, such as elasto-visco-plastic behaviour with intrinsic softening, the overall response of the regular structure exhibits softening, while the averaged response of the random structures demonstrates hardening. The presented micro-macro strategy provides an approach to determine the macroscopic response of heterogeneous materials with accurate account for microstructural characteristics and evolution of the morphology. When using this multi-level strategy there is no need to specify (phenomenologically) the homogenized macroscopic constitutive behaviour, which in the case of large deformations and complex microstructures, would be a hardly feasible task generally. Instead, the constitutive behaviour at macroscopic integration points is determined by averaging the results of a detailed modeling of the microstructure. This enables the straightforward application of the method to geometrically and physically nonlinear problems. The disadvantage of the full micro-macro analysis, using the detailed numerical description of the microstructure corresponding to every macroscopic integration point, is the relatively large computation time. To reduce the computational efforts a so-called ``data base'' approach has been established by Terada and Kikuchi (1995) and further developed by Lee and Ghosh (1999). This method utilizes a data base, containing an averaged constitutive

quanti®cation of the heterogeneous material, derived from calculations on an RVE subjected to various loading conditions, corresponding to different paths in strain (or stress) space. The homogenized responses in between these points are then determined by interpolation. This approach may reduce the calculation time signi®cantly and it is especially effective when only macroscopic variables are of interest. However, as it has been shown in Lee and Ghosh (1999) this approach only gives a proper approximation of the macroscopic variables in case of small deformations and under the assumption of loading path independent material behaviour. Therefore, in spite of its computational expenses, the direct micro-macro strategy as proposed here is de®nitely a valuable tool for general applications purposes.

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