Representative Volume Element Boundary Research Articles

Second-order multi-scale modelling of natural and architected materials in the presence of voids: Formulation and numerical implementation

This contribution proposes a second-order computational homogenisation formulation for natural and architected materials in the presence of voids. The macro-scale is described by a second gradient continuum theory in the finite strain regime, and the micro-scale is modelled by the concept of representative volume element (RVE) within the classical first-order continuum mechanics. The Method of Multi-scale Virtual Power is employed to link the two scales, ensuring a variationally consistent scale transition. Unlike the multi-scale model proposed by Rodrigues Lopes and Andrade Pires (2022), the developed formulation allows the consideration of voids in the RVE domain and boundary, which is critical for modelling cellular materials, lattice structures, and metamaterials, among others. This is achieved by defining the kinematical quantities only in the solid domain of the RVE and postulating a new homogenisation relation for the second-order gradient. The kinematic constraints are imposed on the RVE by the Lagrange multiplier method and particularised for minimal (lower bound), periodic and direct (upper bound) conditions. It is demonstrated that the homogenised macroscopic stress tensors can be expressed in terms of the Lagrange multipliers. The finite element method is adopted for the numerical solution of the micro and macro equilibrium. The Newton–Raphson scheme is employed to solve the non-linear systems of equations at both scales and the consistent macroscopic tangents required for the FE2 framework are derived. Several numerical examples of porous solids, lattice structures and metamaterials illustrate the consistency and applicability of the formulation for two and three-dimensional problems.

Solid Truss to Shell Numerical Homogenization of Prefabricated Composite Slabs.

The need for quick and easy deflection calculations of various prefabricated slabs causes simplified procedures and numerical tools to be used more often. Modelling of full 3D finite element (FE) geometry of such plates is not only uneconomical but often requires the use of complex software and advanced numerical knowledge. Therefore, numerical homogenization is an excellent tool, which can be easily employed to simplify a model, especially when accurate modelling is not necessary. Homogenization allows for simplifying a computational model and replacing a complicated composite structure with a homogeneous plate. Here, a numerical homogenization method based on strain energy equivalence is derived. Based on the method proposed, the structure of the prefabricated concrete slabs reinforced with steel spatial trusses is homogenized to a single plate element with an effective stiffness. There is a complete equivalence between the full 3D FE model built with solid elements combined with truss structural elements and the simplified homogenized plate FE model. The method allows for the correct homogenization of any complex composite structures made of both solid and structural elements, without the need to perform advanced numerical analyses. The only requirement is a correctly formulated stiffness matrix of a representative volume element (RVE) and appropriate formulation of the transformation between kinematic constrains on the RVE boundary and generalized strains.

Effective hydro-mechanical material properties and constitutive behaviors of meso-structured RVE of saturated granular media

In the framework of the second-order concurrent computation homogenization method for saturated granular material, meso-structured RVE (Representative Volume Element) is modeled as a compact assembly of saturated discrete particles. The boundary conditions imposed on the RVE are comprised of the two parts: (1) the boundary displacements and pressures prescribed by downscaled macroscopic strain variables and pore pressure, and (2) constrained fine scale fluctuations of mesoscopic hydro-mechanical variables expressed in terms of periodic boundary conditions.The present paper focuses on the estimation of effective hydro-mechanical material properties and constitutive behaviors of meso-structured RVE of saturated discrete particle assembly via a proposed homogenization procedure. To achieve this target, a novel scheme for determining and imposing periodic boundary conditions on peripheral particles contacting with the RVE boundary of saturated discrete particle assembly is proposed.Numerical results demonstrate performances and capabilities of the proposed scheme in estimating effective hydro-mechanical non-linear material properties and predicting hydro-mechanical constitutive behaviors of saturated granular materials evolving with loading histories. They also demonstrate the significance and necessity of imposing periodic boundary conditions on the RVE in proper prediction of effective hydro-mechanical material properties and constitutive behaviors.

On boundary conditions and constraints for representative volume elements of a two‐scale shell formulation

AbstractIn this contribution a coupled two‐scale shell model with application of the FE2‐method based on the algorithm in [1] is presented. The representative volume element (RVE) is meshed with solid elements. In the RVE boundary effects can be reduced or even avoided using periodic boundary conditions. However, this requires proper constraints to prevent rigid body movements. For this purpose, the theory in [2] for beams, which considers the shape of normal stresses through the cross section, is transferred to the shell problem. Furthermore, the RVE reacts to shear deformations not only with shear stresses but also with bending stresses. This leads to a vanishing shear stiffness with an increase of the freely selectable in‐plane size. This dependence of the shear stiffness is eliminated by a further constraint on the normal stresses. Both constraints are incorporated using the method of Lagrange multipliers.

A novel scheme for imposing periodic boundary conditions on RVE in second-order computational homogenization for granular material

Purpose This paper aims to present a novel scheme for imposing periodic boundary conditions with downscaled macroscopic strain measures of gradient Cosserat continuum on the representative volume element (RVE) of discrete particle assembly in the frame of the second-order computational homogenization methods for granular materials. Design/methodology/approach The proposed scheme is based on the generalized Hill’s lemma of gradient Cosserat continuum and the incremental non-linear constitutive relation condensed to the peripheral particles of the RVE of discrete particle assembly. The generalized Hill’s lemma conducts to downscale the macroscopic strain or stress measures and to impose the periodic boundary conditions on the RVE boundary so that the Hill-Mandel energy equivalence condition is ensured. Because of the incremental non-linear constitutive relation condensed to the peripheral particles of the RVE, the periodic boundary displacement and traction constraints together with the downscaled macroscopic strains and strain gradients, micro-rotations and curvatures are imposed in the point-wise sense without the need of introducing the Lagrange multipliers for enforcing the periodic boundary displacement and traction constraints in a weak sense. Findings Numerical results demonstrate that the applicability and effectiveness of the proposed scheme in imposing the periodic boundary conditions on the RVE. The results of the RVE subjected to the periodic boundary conditions together with the displacement boundary conditions in the second-order computational homogenization for granular materials provide the desired estimations, which lie between the upper and the lower bounds provided by the displacement and the traction boundary conditions imposed on the RVE respectively. Research limitations/implications Each grain in the particulate system under consideration is assumed to be rigid and circular. Practical implications The proposed scheme for imposing periodic boundary conditions on the RVE can be adopted solely for estimating the effective mechanical properties of granular materials and/or integrated into the frame of the second-order computational homogenization method with a nested finite element method-discrete element method solution procedure for granular materials. It will tend to provide, at least theoretically, more reasonable results for effective material properties and solutions of a macroscopic boundary value problem simulated by the computational homogenization method. Originality/value This paper presents a novel scheme for imposing periodic boundary conditions with downscaled macroscopic strain measures of gradient Cosserat continuum on the RVE of discrete particle assembly for granular materials without need of introducing Lagrange multipliers for enforcing periodic boundary conditions in a weak (integration) sense.

Prediction of the Stress–Strain Behavior of Open-Cell Aluminum Foam under Compressive Loading and the Effects of Various RVE Boundary Conditions

The prediction of the mechanical behavior of metallic foams with realistic microstructure and the effects of various boundary conditions on the mechanical behavior is an important and challenging issue in modeling representative volume elements (RVEs). A numerical investigation is conducted to determine the effects of various boundary conditions and cell wall cross sections on the compressive mechanical properties of aluminum foam, including the stiffness, plateau stress and onset strain of densification. The open-cell AA6101-T6 aluminum foam Duocel is used in the analyses in this study. Geometrical characteristics including the cell size, foam relative density, and cross-sectional shape and thickness of the cell walls are extracted from images of the foam. Then, the obtained foam microstructure is analyzed as a 2D model. The ligaments are modeled as shear deformable beams with elastic-plastic material behavior. To prevent interpenetration of the nodes and walls inside the cells with large deformations, self-contact-type frictionless interaction is stipulated between the internal surfaces. Sensitivity analyses are performed using several boundary conditions and cells wall cross-sectional shapes. The predicted results from the finite element analyses are compared with the experimental results. Finally, the most appropriate boundary conditions, leading to more consistent results with the experimental data, are introduced.

Influence of the size and boundary conditions on the predicted effective strengths of particulate reinforced metal matrix composites (PRMMCs)

Understanding how the load bearing capacity of particulate reinforced metal matrix composite (PRMMC) is related to the underlying microstructure is essential for developing new PRMMC materials. In a recent study, we presented a numerical technique which combines the homogenization technique, direct method, and statistical analyses. By using this approach, ultimate strength and endurance limit of the PRMMCs can be predicted from the composite microstructure in the mesoscale. Because PRMMCs are typical random heterogeneous materials, the size of RVE models, as well as boundary conditions, have been acknowledged to have nontrivial influence over the predicted effective behavior. In this paper, the objective is to understand how the strength predicted from the direct method is influenced by the RVE size and boundary condition. Because in our previous study it has already been shown that homogenization results are insensitive to boundary conditions applied on the RVE, the focus of the present paper is laid on the influence of the embedded cell model. In order to capture the influence of RVE size and embedded cell configuration, an exemplary PRMMC material, WC-20 Wt% Co, was selected. A great number of representative volume elements (RVE) models were built as artificial microstructures of this material. By analyzing the effective behaviors, especially the strength, predicted from these samples from a statistical point of view, the influence of the boundary condition and the size effect were revealed. On that basis, this study proposes a feasible solution to these issues.

Localization aligned weakly periodic boundary conditions

SummaryWhen computing the homogenized response of a representative volume element (RVE), a popular choice is to impose periodic boundary conditions on the RVE. Despite their popularity, it is well known that standard periodic boundary conditions lead to inaccurate results if cracks or localization bands in the RVE are not aligned with the periodicity directions. A previously proposed remedy is to use modified strong periodic boundary conditions that are aligned with the dominating localization direction in the RVE. In the present work, we show that alignment of periodic boundary conditions can also conveniently be performed on weak form. Starting from a previously proposed format for weak micro‐periodicity that does not require a periodic mesh, we show that aligned weakly periodic boundary conditions may be constructed by only modifying the mapping (mirror function) between the associated parts of the RVE boundary. In particular, we propose a modified mirror function that allows alignment with an identified localization direction. This modified mirror function corresponds to a shifted stacking of RVEs, and thereby ensures compatibility of the dominating discontinuity over the RVE boundaries. The proposed method leads to more accurate results compared to using unaligned periodic boundary conditions, as demonstrated by the numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

Three-dimensional nonlinear micro/meso-mechanical response of the fibre-reinforced polymer composites

A three-dimensional multi-scale computational homogenisation framework is developed for the prediction of nonlinear micro/meso-mechanical response of the fibre-reinforced polymer (FRP) composites. Two dominant damage mechanisms, i.e. matrix elasto-plastic response and fibre–matrix decohesion are considered and modelled using a non-associative pressure dependent paraboloidal yield criterion and cohesive interface elements respectively. A linear-elastic transversely isotropic material model is used to model yarns/fibres within the representative volume element (RVE). A unified approach is used to impose the RVE boundary conditions, which allows convenient switching between linear displacement, uniform traction and periodic boundary conditions. The computational model is implemented within the framework of the hierarchic finite element, which permits the use of arbitrary orders of approximation. Furthermore, the computational framework is designed to take advantage of distributed memory high-performance computing. The accuracy and performance of the computational framework are demonstrated with a variety of numerical examples, including unidirectional FRP composite, a composite comprising a multi-fibre and multi-layer RVE, with randomly generated fibres, and a single layered plain weave textile composite. Results are validated against the reference experimental/numerical results from the literature. The computational framework is also used to study the effect of matrix and fibre–matrix interfaces properties on the homogenised stress–strain responses.

Multi-scale computational homogenisation to predict the long-term durability of composite structures

A coupled hygro-thermo-mechanical computational model is proposed for fibre reinforced polymers, formulated within the framework of Computational Homogenisation (CH). At each macrostructure Gauss point, constitutive matrices for thermal, moisture transport and mechanical responses are calculated from CH of the underlying representative volume element (RVE). A degradation model, developed from experimental data relating evolution of mechanical properties over time for a given exposure temperature and moisture concentration is also developed and incorporated in the proposed computational model. A unified approach is used to impose the RVE boundary conditions, which allows convenient switching between linear Dirichlet, uniform Neumann and periodic boundary conditions. A plain weave textile composite RVE consisting of yarns embedded in a matrix is considered in this case. Matrix and yarns are considered as isotropic and transversely isotropic materials respectively. Furthermore, the computational framework utilises hierarchic basis functions and designed to take advantage of distributed memory high performance computing.

Reduced order modeling strategies for computational multiscale fracture

The paper proposes some new computational strategies for affordably solving multiscale fracture problems through a FE2 approach. To take into account the mechanical effects induced by fracture at the microstructure level the Representative Volume Element (RVE), assumed constituted by an elastic matrix and inclusions, is endowed with a large set of cohesive softening bands providing a good representation of the possible microstructure crack paths. The RVE response is then homogenized in accordance with a model previously developed by the authors and upscaled to the macro-scale level as a continuum stress–strain constitutive equation, which is then used in a conventional framework of a finite element modeling of propagating fracture.For reduced order modeling (ROM) purposes, the RVE boundary value problem is first formulated in displacement fluctuations and used, via the Proper Orthogonal Decomposition (POD), to find a low-dimension space for solving the reduced problem. A domain separation strategy is proposed as a first technique for model order reduction: unconventionally, the low-dimension space is spanned by a basis in terms of fluctuating strains, as primitive kinematic variables, instead of the conventional formulation in terms of displacement fluctuations. The RVE spatial domain is then decomposed into a regular domain (made of the matrix and the inclusions) and a singular domain (constituted by cohesive bands), the required RVE boundary conditions are rephrased in terms of strains and imposed via Lagrange multipliers in the corresponding variational problem. Specific low-dimensional strain basis is then derived, independently for each domain, via the POD of the corresponding strain snapshots.Next step consists of developing a hyper-reduced model (HPROM). It is based on a second proposed technique, the Reduced Optimal Quadrature (ROQ) which, again unconventionally, is determined through optimization of the numerical integration of the primitive saddle-point problem arising from the RVE problem, rather than its derived variational equations, and substitutes the conventional Gauss quadrature. The ROQ utilizes a very reduced number of, optimally placed, sampling points, the corresponding weights and placements being evaluated through a greedy algorithm. The resulting low-dimensional and reduced-quadrature variational problem translates into very relevant savings on the computational cost and high computational speed-ups.Particular attention is additionally given to numerical tests and performance evaluations of the new hyper-reduced methodology, by “a-priori” and “a-posteriori” error assessments. Moreover, for the purposes of validation of the present techniques, a real structural problem exhibiting propagating fracture at two-scales is modeled on the basis of the strain injection-based multiscale approach previously developed by the authors. The performance of the proposed strategy, in terms of speed-up vs. error, is deeply analyzed and reported.

The method of multiscale virtual power for the derivation of a second order mechanical model

A multi-scale model, based on the concept of Representative Volume Element (RVE), is proposed linking a classical continuum at RVE level to a macro-scale strain-gradient theory. The multi-scale model accounts for the effect of body forces and inertia phenomena occurring at the micro-scale. The Method of Multiscale Virtual Power recently proposed by the authors drives the construction of the model. In this context, the coupling between the macro- and micro-scale kinematical descriptors is defined by means of kinematical insertion and homogenisation operators, carefully postulated to ensure kinematical conservation in the scale transition. Micro-scale equilibrium equations as well as formulae for the homogenised (macro-scale) force- and stress-like quantities are naturally derived from the Principle of Multiscale Virtual Power – a variational extension of the Hill-Mandel Principle that enforces the balance of the virtual powers of both scales. As an additional contribution, further insight into the theory is gained with the enforcement of the RVE kinematical constraints by means of Lagrange multipliers. This approach unveils the reactive nature of homogenised force- and stress-like quantities and allows the characterisation of the homogenised stress-like quantities exclusively in terms of RVE boundary data in a straightforward manner.

Multiscale modeling of continuous fiber and fabric reinforced rubber components

AbstractFabric or continuous fiber reinforced rubber components (e.g. tires, air springs, industrial hoses, conveyor belts or membranes) are underlying high deformations in application and show a complex, nonlinear material behavior. A particular challenge depicts the simulation of these composites. In this contribution we show the identification of the stress and strain distributions by using an uncoupled multiscale modeling method, see [1]. Within this method, two representation levels are described: One, the meso level, where all constituents of the composite are shown in a discrete manner by a representative volume element (RVE) and secondly, the macro level, where the structural behavior of the component is defined by a smeared anisotropic hyperelastic constitutive law. Uncoupled means that the RVE does not drive the macroscopic material behavior directly as in a coupled approach, where a RVE boundary value problem has to be solved at every integration point of the macro level. Thus an uncoupled approach leads to a tremendous reduction in numerical effort because the boundary value problem of a RVE just has to be solved at a point of interest, see [1]. However, the uncoupled scale transition has to fulfill the HILL–MANDEL condition of energetic equivalence of both scales. We show the calibration of material parameters for a given constitutive model for fiber reinforced rubber by fitting experimental data on the macro level. Additionally, we demonstrate the determination of effective properties of the yarns. Finally, we compare the energies of both scales in terms of compliance with the HILL–MANDEL condition by using the example of a biaxial loaded sample and discuss the consequences for the mesoscopic level. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Micro–macro homogenization of granular materials based on the average-field theory of Cosserat continuum

This paper performs further study on the micro–macro homogenization approach of granular materials (Li et al., 2010) based on the advancement of Hill’s lemma for Cosserat continuum (Liu, 2013). Firstly, the average couple stress tensor, expressed as the volume integration of quantities over the representative volume element (RVE) in the average-field theory of Cosserat continuum, is further deduced and expressed in terms of discrete quantities on the discrete particle assembly RVE of granular materials. The expression is also discussed and compared with other typical definitions of the effective couple stress tensor for granular materials in the literature. Then, rate forms of micromechanically based constitutive models consistent with different types of RVE boundary conditions are derived and discussed. Since the presented micro–macro homogenization approach is used, not only the micro–macro energy equivalence is satisfied, but also the microstructure and its evolution can be taken into account in the constitutive formulation with no need of specifying macroscopic phenomenological constitutive model.

Generalized boundary conditions on representative volume elements and their use in determining the effective material properties

• The proof of the Hill–Mandel-condition is given more detailed in the Appendix . • Cubical RVE are considered also. • Many minor changes have been incorporated. When determining an effective stress–strain law by means of the representative volume element (RVE) method, one needs to subject the RVE to the effective strains by appropriate boundary conditions (BC). Usually, classical BC that prescribe a homogeneous stress or strain field at the boundary or a periodic unit cell are used. In this work, we discuss generalized BC, which involve the partitioning of the RVE boundary into n parts. It is demonstrated that the classical BC are contained as special cases, and that the Hill–Mandel-condition holds for all partitionings. By a more or less fine surface partitioning, the generalized BC allow for a smooth scaling between the extremal cases of homogeneous stress or homogeneous strain BC. Further, by an irregular surface partitioning, one can obtain stochastic BC with an elastic stiffness close to the periodic/antipodal BC, but with a higher resistance against localization. This has been demonstrated by examining a softening example material. A test of plausibility for a RVE is to apply it to a homogeneous microstructure. Then, the microscale material law should be conducted directly to the macroscale. In case of softening microscale materials, this test works only for homogeneous strain BC. For homogeneous stress- and periodic/antipodal BC, localization occurs, accompanied by a drastic deviation from the expected stress–strain curve. From the generalization, one can derive stochastic BC that combine the moderate elastic stiffness of periodic BC with the high resistance against localization of homogeneous strain BC.

Micromechanical Modeling of Damage and Failure in Dual Phase Steels

Dual phase (DP) steels consisting of two phases, ferrite and dispersed martensite, offer an attractive combination of strength and stretchability, which is a result of the strong distinctions of these constituents in mechanical properties. However, the damage behavior in DP steels exhibits a rather complex scenario: voids are generated by the debonding of the hard phase from the matrix and the inner cracking of the hard phase in addition to by inclusions. The target of this study is to describe the initiation and evolution of damage in DP steel and develop a microstructure-based model which is capable of reflecting the underlying damage mechanisms. Both uniaxial and biaxial tensile tests are performed and the subsequent metallographic investigations are executed to reveal the mechanisms of damage initiation and evolution under different stress state condition and attention will be paid on the influence of various microstructural features on the initiation of damage. In finite element (FE) simulations, the microstructural features are taken into account by the representative volume elements (RVE). Different treatments of the constitutive behaviour of each constituent including isotropic hardening rule and crystallographically dependent configuration with crystal plasticity finite element method are investigated. Several numerical aspects are also discussed, such as RVE size, mesh size, element type, and boundary connections. In the end, the study is attempting to achieve a quantitative assessment of the cold formability of the investigated steel in a microscopic level based on microstructure information of material as well as to understand the damage mechanisms under different stress states condition which cause the macroscopic failure during plastic deformation.

Hill’s lemma for the average-field theory of Cosserat continuum

There are two typical definitions for the macroscopic average couple stress tensor in the literature, which brings confusion and difficulty to the establishment of Hill’s lemma for heterogeneous Cosserat continuum. Besides, some boundary conditions on the representative volume element (RVE) commonly used in the homogenization method cannot be properly determined by the existing version of Hill’s lemma of Cosserat continuum. To deal with these issues, Hill’s lemma for micro-macro homogenization modeling of Cosserat continuum is further investigated in the frame of the average-field theory in this paper. An intermediate form of Hill’s lemma is constructed in which the micromechanically based definition of the average couple stress tensor is not specified. By substituting two existing definitions of the average couple stress tensor into the presented intermediate form, not only the previous version of Hill’s lemma is derived, but also a new version of Hill’s lemma is obtained. According to the new version of Hill’s lemma, more versatile RVE boundary conditions in the strong form can be properly given, and the periodic RVE boundary conditions in the weak form are also constructed.

Novel boundary conditions for strain localization analyses in microstructural volume elements

SUMMARYMulti‐scale modeling frequently relies on microstructural representative volume elements (RVEs) on which macroscopic deformation is imposed through kinematical boundary conditions. A particular choice of these boundary conditions may influence the obtained effective properties. For strain localization and damage analyses, the RVE is pushed beyond the limits of its representative character, and the applied boundary conditions have a significant impact on the onset and the type of macroscopic material instability to be predicted.In this article, we propose a new type of boundary conditions for microstructural volume elements, called percolation‐path‐aligned boundary conditions. Intrinsically, these boundary conditions capture the constraining effect of the material surrounding the RVE upon developing localization bands. The alignment with evolving localization bands allows the highly strained band to cross the RVE and fully develop with minimal interference of the applied boundary conditions. For an illustration of the performance of the newly proposed boundary conditions, macroscopic deformation has been imposed on a voided elasto‐plastic RVE using different types of boundary conditions. It is observed that the new RVE boundary conditions provide a good estimate for the effective stiffness, are not susceptible to spurious localization, and permit the development of a full strain localization band up to failure. Copyright © 2011 John Wiley & Sons, Ltd.

On the equivalence between spatial and material volume averaging of stress in large strain multi-scale solid constitutive models

This paper discusses some equivalence relationships for large strain multi-scale solid constitutive models based on the volume averaging of the microscopic stress and deformation gradient fields over a representative volume element (RVE). The analysis is carried out within a purely kinematically-based variational framework. Sufficient conditions are established under which the volume average of the microscopic first Piola–Kirchhoff stress over the reference (or material) configuration of the RVE is mechanically equivalent to the average of the microscopic Cauchy stress field over the deformed (or spatial) configuration. The Taylor (or homogeneous deformation) assumption, the affine boundary condition (or linear RVE boundary displacements) as well as the periodic boundary displacement fluctuations constraint – all widely adopted in multi-scale constitutive modelling – satisfy the sufficient conditions. For these classes of models, the choice of the material or spatial configuration for stress volume averaging is immaterial. Some averaging equivalence relations involving the homogenised Kirchhoff stress are also briefly discussed.

Scale transition and enforcement of RVE boundary conditions in second‐order computational homogenization

AbstractFormulation of the scale transition equations coupling the microscopic and macroscopic variables in the second‐order computational homogenization of heterogeneous materials and the enforcement of generalized boundary conditions for the representative volume element (RVE) are considered. The proposed formulation builds on current approaches by allowing any type of RVE boundary conditions (e.g. displacement, traction, periodic) and arbitrary shapes of RVE to be applied in a unified manner. The formulation offers a useful geometric interpretation for the assumptions associated with the microstructural displacement fluctuation field within the RVE, which is here extended to second‐order computational homogenization. A unified approach to the enforcement of the boundary conditions has been undertaken using multiple constraint projection matrices. The results of an illustrative shear layer model problem indicate that the displacement and traction RVE boundary conditions provide the upper and lower bounds of the response determined via second‐order computational homogenization, and the solution associated with the periodic RVE boundary conditions lies between them. Copyright © 2007 John Wiley & Sons, Ltd.