Homogenization Methods and Multiscale Modeling: Nonlinear Problems

Materials with network-like microstructure, including polymers, are the backbone for many natural and human-made materials such as gels, biological tissues, metamaterials, and rubbers. Fracture processes in these networked materials are intrinsically multiscale, and it is computationally prohibitive to adopt a fully discrete approach for large scale systems. To overcome such a challenge, we introduce an adaptive numerical algorithm for modeling fracture in this class of materials, with a primary application to polymer networks, using an extended version of the Quasi-Continuum method that accounts for both material and geometric nonlinearities. In regions of high interest, for example near crack tips, explicit representation of the local topology is retained where each polymer chain is idealized using the worm like chain model. Away from these imperfections, the degrees of freedom are limited to a fraction of the network nodes and the network structure is computationally homogenized, using the micromacro energy consistency condition, to yield an anisotropic material tensor consistent with the underlying network structure. A nonlinear finite element framework including both material and geometric nonlinearities is used to solve the system where dynamic adaptivity allows transition between the continuum and discrete scales. The method enables accurate modelling of crack propagation without a priori constraint on the fracture energy while maintaining the influence of large-scale elastic loading in the bulk. We demonstrate the accuracy and efficiency of the method by applying it to study the fracture in different examples of network structures. We further use the method to investigate the effects of network topology and disorder on its fracture characteristics. We discuss the implications of our method for multiscale analysis of fracture in networked material as they arise in different applications in biology and engineering.

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“…In terms of macro deformation gradient tensor and the first Piola-Kirchhoff stress tensor, the condition reads [29]:…”

Section: Micro-macro Energy Consistency Conditionmentioning

confidence: 99%

An adaptive quasicontinuum approach for modeling fracture in networked materials: Application to modeling of polymer networks

Ghareeb

Elbanna

2020

Journal of the Mechanics and Physics of Solids

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“…Depending on the application, hybrid one/two‐way coupled models can be constructed (Carr et al., ). Various multiscale models have been constructed over the years such as adaptive model refinement (Garcia, Bell, Crutchfield, & Alder, ) Car–Parrinello method (Car & Parrinello, ), multigrid (Brandt, ), quasi continuum method (Tadmor, Ortiz, & Phillips, ), HMM (Weinan & Engquist, ; Weinan et al., ), equation‐free framework (Kevrekidis et al., ), distributed microstructure model (DMM) (Showalter, ), extended distributed microstructure model (EDMM) (Carr, Perré, & Turner, ) and computational homogenization (Geers, Kouznetsova, Matouš, & Yvonnet, ). However, these approaches are generally valid for specific applications.…”

Section: Current Status Of Multiscale Modelingmentioning

confidence: 99%

Multiscale Modeling for Food Drying: State of the Art

Welsh

Simpson

Khan

et al. 2018

Comp Rev Food Sci Food Safe

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Plant-based food materials are mostly porous in nature and heterogeneous in structure with huge diversity in cellular orientation. Different cellular environments of plant-based food materials, such as intercellular, intracellular, and cell wall environments, hold different proportions of water with different characteristics. Due to this structural heterogeneity, it is very difficult to understand the drying process and associated morphological changes during drying. Transport processes and morphological changes that take place during drying are mainly governed by the characteristics of and the changes in the cells. Therefore, to predict the actual heat and mass transfer process that occurs in the drying process and associated morphological changes, development of multiscale modeling is crucial. Multiscale modeling is a powerful approach with the ability to incorporate this cellular structural heterogeneity with microscale heat and mass transfer during drying. However, due to the huge complexity involved in developing such a model for plant-based food materials, the studies regarding this issue are very limited. Therefore, we aim in this article to develop a critical conceptual understanding of multiscale modeling frameworks for heterogeneous food materials through an extensive literature review. We present a critical review on the multiscale model formulation and solution techniques with their spatial and temporal coupling options. Food structure, scale definition, and the current status of multiscale modeling are also presented, along with other key factors that are critical to understanding and developing an accurate multiscale framework. We conclude by presenting the main challenges for developing an accurate multiscale modeling framework for food drying.

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“…Among noteworthy reduced order methods are the Voronoi cell method, 62,63 the spectral method, 64 the network approximation method, 65 the fast Fourier transforms, 66,67 the mesh-free reproducing kernel particle method, 68,69 the finite-volume direct averaging micromechanics, 70 the transformation field analysis, 71,72 the methods of cells 64 or its generalization, 73 methods based on control theory including balanced truncation, 74,75 the optimal Hankel norm approximation, 76 the proper orthogonal decomposition, 77,78 data-driven-based reduced order methods, [79][80][81][82] the reduced order homogenization methods for two scales 83,84 and more than two-scales, 85,86 and the nonuniform transformation field methods. [87][88][89] For a recent comprehensive review of various homogenization-like method, we refer to the works of Fish 90 and Geers et al 91 The primary objective of this manuscript is to develop an efficient computational framework for analyzing nonlinear periodic materials with large microstructure that combines nonlinear higher-order asymptotic homogenization methods that do not require higher-order continuity of the coarse-scale solution with an efficient model reduction scheme. We will refer to the proposed formulation as the second-order reduced asymptotic homogenization (SRAH).…”

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 91 publications

References 206 publications

An adaptive quasicontinuum approach for modeling fracture in networked materials: Application to modeling of polymer networks

An adaptive quasicontinuum approach for modeling fracture in networked materials: Application to modeling of polymer networks

Multiscale Modeling for Food Drying: State of the Art

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

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