Homogenization Methods and Multiscale Modeling: Nonlinear Problems

Geers, M.G.D.; Kouznetsova, V Varvara; Matouš, Karel; Yvonnet, Julien

doi:10.1002/9781119176817.ecm107

Cited by 57 publications

(61 citation statements)

References 186 publications

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“…However the non-existence of macroscopic constitutive behaviour for global–local analysis in contrast to separated macro-micro constitutive relations for asymptotic homogenisation can be seen as the main difference. A detailed discussion regarding different numerical homogenisation approaches can be found in Geers et al. (2017).…”

Section: Introductionmentioning

confidence: 99%

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

Bhattacharyya

Dureisseix

Faverjon

2020

International Journal of Damage Mechanics

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This article deals with damage computation of heterogeneous structures containing locally periodic micro-structures. Such heterogeneous structure is extremely expensive to simulate using classical finite element methods, as the level of discretisation required to capture the micro-structural effects is too fine. The simulation time becomes even higher when dealing with highly non-linear material behaviour, e.g. damage, plasticity and such others. Therefore, a multi-scale strategy is proposed here that facilitates the simulation of non-linear heterogeneous material behaviour in a manner that is computationally feasible. Based on the asymptotic homogenisation theory, this multi-scale technique explores the micro–macro behaviour for elasto-(visco)plasticity coupled with damage. The theory inherently segregates the heterogeneous continua into a macroscopic homogeneous structure and an underlying heterogeneous microscopic periodic unit cell. Several heterogeneous structures have been simulated using the multi-scale method along with a one-dimensional verification with respect to a reference solution. Additionally, a reduced order modelling is used to prevent large memory requirement for storing micro-structural quantities of interest.

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Section: Introductionmentioning

confidence: 99%

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

Bhattacharyya

Dureisseix

Faverjon

2020

International Journal of Damage Mechanics

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“…In this context, computational homogenization methods for composite materials with periodic microstructures (unit cells) have been drawing intense research interest because of their potential capabilities for linking the resin's thermal, mechanical and nonmechanical characteristics with the macroscopic material behavior. In general settings, early work typified by Devries et al 9 and Guedes and Kikuchi 10 is known as an alternative to micromechanics for heterogeneous media, and recent developments are found in the review article by Geers et al, 11 the textbook by Yvonnet 12 and references therein. The application of the computational homogenization must be the most promising way to incorporate various microscopic information such as stress, temperature and DOC states into the evaluation of the macroscopic material behavior of FRP subjected to curing by identifying the unit cell of FRP as a representative volume element (RVE); see, for example, Agius et al 13 and Hirsekorn et al 14 for RVE‐based approaches.…”

Section: Introductionmentioning

confidence: 99%

“…Despite the outstanding technological advances made during the 1990s and 2000s, considerable attention is still being paid to extensive applications of the computational homogenization in practice 15 . However, since the computational homogenization is not a methodology to provide analytical expressions of nonlinear or inelastic macroscopic constitutive laws of FRPs, either the coupling (or built‐in) or decoupling (or detached) scheme has to be adopted 11 . Because the micro–macro coupling scheme, which was originally presented by Terada and Kikuchi 16,17 and is often called FE 2 method, 18 does not require explicit function forms of macroscopic constitutive laws, it is still an important subject for study in the literature.…”

Section: Introductionmentioning

confidence: 99%

A decoupling scheme for two‐scale finite thermoviscoelasticity with thermal and cure‐induced deformations

Saito

Yamanaka

Matsubara

et al. 2020

Numerical Meth Engineering

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This article proposes a decoupling scheme for two-scale analysis of fiber-reinforced plastics (FRP) exhibiting finite thermoviscoelasticity in consideration of the dependences of resin's mechanical and nonmechanical deformation characteristics on the degree of cure (DOC) and ambient temperature. To characterize the macroscopic material behavior, numerical material tests are carried out on a unit cell composed of a polymer resin matrix and carbon fibers. The generalized Maxwell model (GMM) is employed for resin' material behavior, while its orthotropic version is assumed for FRP. The evolution of DOC is reflected in the evaluation of the nonmechanical deformation by cure shrinkage in addition to thermal expansion/contraction. The key ingredient of this study is the novel strategy for identifying the macroscopic coefficients of these nonmechanical deformations, both of which must be separately defined in the equilibrium and nonequilibrium elements of the orthotropic GMM. In addition, a modification is originally made on the evolution equations of the nonequilibrium stresses in the GMM. The verification analyses are carried out to confirm the adequateness of the proposed identification methods and followed by numerical examples of two-scale analysis to demonstrate the capability of simulating the macro-and microscopic thermomechanical responses of FRP subjected to curing.

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“…Among noteworthy reduced order methods are the Voronoi cell method, the spectral method, the network approximation method, the fast Fourier transforms, the mesh‐free reproducing kernel particle method, the finite‐volume direct averaging micromechanics, the transformation field analysis, the methods of cells or its generalization, methods based on control theory including balanced truncation, the optimal Hankel norm approximation, the proper orthogonal decomposition, data‐driven–based reduced order methods, the reduced order homogenization methods for two scales and more than two‐scales, and the nonuniform transformation field methods . For a recent comprehensive review of various homogenization‐like method, we refer to the works of Fish and Geers et al…”

Section: Introductionmentioning

confidence: 99%

A second‐order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials

Fish

Yang

Yuan

2019

Numerical Meth Engineering

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Summary An efficient second‐order reduced asymptotic homogenization approach is developed for nonlinear heterogeneous media with large periodic microstructure. The two salient features of the proposed approach are (i) an asymptotic higher‐order nonlinear homogenization that does not require higher‐order continuity of the coarse‐scale solution and (ii) an efficient model reduction scheme for solving higher‐order nonlinear unit cell problems at a fraction of computational cost in comparison to the direct computational homogenization. The former is a consequence of a sequential solution of increasing order solutions, which permits evaluation of higher‐order coarse‐scale derivatives by postprocessing from the zeroth‐order solution. The efficiency and accuracy of the formulation in comparison to the classical zeroth‐order homogenization and direct numerical simulations are assessed on hyperelastic and elastoplastic periodic structures.

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Homogenization Methods and Multiscale Modeling: Nonlinear Problems

Cited by 57 publications

References 186 publications

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

A decoupling scheme for two‐scale finite thermoviscoelasticity with thermal and cure‐induced deformations

A second‐order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials

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