Application of the Multiscale Fem to the Modeling of Nonlinear Composites With a Random Microstructure

Klinge, Sandra; Hackl, Klaus

doi:10.1615/intjmultcompeng.2012002059

Cited by 30 publications

(21 citation statements)

References 35 publications

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“…Because in general, the non-linear processes are the result of many complex mechanisms at the micro-scale level which depend of the specific analyzed material. Since the pioneering works of Swan [116], Smit et al [110], Michel et al [73], Miehe et al [76] and Nemat-Nasser [82] many developments have been done on this issue for several authors [117,26,59,25,124,90,16,49,58].…”

Section: Review Of Multiscale Methodsmentioning

confidence: 99%

Multiscale Computational Homogenization: Review and Proposal of a New Enhanced-First-Order Method

Otero

Oller

Martı́nez

2016

Arch Computat Methods Eng

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Aquesta és una còpia de la versió author's final draft d'un article publicat a la revista Archives of computational methods in engineering. Abstract The continuous increase of computational capacity has encouraged the extensive use of multiscale techniques to simulate the material behaviour on several fields of knowledge. In solid mechanics, the multiscale approaches which consider the macro-scale deformation gradient to obtain the homogenized material behaviour from the micro-scale are called first-order computational homogenization. Following this idea, the second-order FE2 methods incorporate high-order gradients to improve the simulation accuracy. However, to capture the full advantages of these high-order framework the classical Boundary Value Problem (BVP) at the macro-scale must be upgraded to high-order level, which complicates their numerical solution. With the purpose of obtaining the best of both methods i.e. firstorder and second-order, in this work an enhanced-firstorder computational homogenization is presented. The proposed approach preserves a classical BVP at the macro-scale level but taking into account the high-order gradient of the macro-scale in the micro-scale solution. The developed numerical examples show how the proposed method obtains the expected stress distribution at the micro-scale for states of structural bending loads. Nevertheless, the macro-scale results achieved are the same than the ones obtained with a first-order framework because both approaches share the same macroscale BVP.

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Section: Review Of Multiscale Methodsmentioning

confidence: 99%

Multiscale Computational Homogenization: Review and Proposal of a New Enhanced-First-Order Method

Otero

Oller

Martı́nez

2016

Arch Computat Methods Eng

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“…The solution of this BVP yields the effective values for the current state of deformations. The connection between the two BVPs is achieved through the Hill macrohomogeneity condition which has the following form in the case of finite strains [4,5] P :…”

Section: Homogenization Approach -Multiscale Finite Element Methodsmentioning

confidence: 99%

Viscoelastic effects and shrinkage as accompanying phenomena of the curing of polymers. Single‐ and multiscale effects

Klinge¹,

Bartels

Hackl³

et al. 2012

Proc Appl Math and Mech

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The contribution deals with the modeling of two phenomena that are characteristic of the curing of polymers, namely the increasing viscosity and the volume decrease known as autogeneous shrinkage. Both of these processes are caused by the crosslinking of polymer chains during polymerization. In order to model the viscoelastic effects, the free energy consisting of an equilibrium and a non-equilibrium part is proposed. The former is related to the elastic processes and depends on total deformations. The latter is caused by the viscoelastic effects and only depends on the elastic part of deformations. In order to avoid volume locking effects typical of isochoric materials, both parts of the free energy density are furthermore split into a volumetric and a deviatoric part. A multifield description depending on the displacements, volume change and hydrostatic pressure is introduced as well. Different from the viscous process, the modeling of shrinkage effects does not require a new assumption for the free energy but a split of the total deformation gradient into a shrinkage and a mechanical part. The model suitable for simulating both of the mentioned phenomena is implemented in the single-and multiscale FE program. Mechanical modeling of viscoelastic and shrinkage effectsThe curing of polymers is a complex process accompanied by the change of material properties such as stiffness, viscosity or opacity. Within this contribution, the emphasis will first be placed on the viscous effects for the modeling of which the free energy density is decomposed into an equilibrium and a non-equilibrium partIn addition, a decomposition into a volumetric and deviatoric part is introduced since the polymers commonly show an isochoric material behaviorWithin the proposed formulation, Ψ denotes the strain energy density, u the displacement field, θ the volume change, p is the hydrostatic pressure, J = det F is the Jacobian related to the deformation gradient F, C is the right Cauchy-Green deformation tensor, the index "e" denotes the elastic quantities and the notation "iso" represents the isochoric part of the deformations. The remaining notation is self-explanatory. The multiplicative decomposition of the deformation gradient F = F e · F i is introduced to define the relationships between the elastic and inelastic deformations. In order to complete the model, the equilibrium energy parts are assumed in the form of convolution integrals, which is substantiated by their dependence on the time-sensitive material parameters [1]Here, C and K are deviatoric and volumetric stiffnesses. Different from the equilibrium part, the non-equilibrium part depends on the time-insensitive material parameters such that it can directly be equalized with the non-equilibrium strain energy density Φ neq = Ψ neq . The evolution of the elastic deformations is controlled through the evolution of the viscous deformations determined by the approach of Reese andwhere T denotes the relaxation time [2]. Apart from the viscous effects, the behavior of curing poly...

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“…This idea has been further developed in Refs. [222] and [352][353][354][355][356][357][358][359][360][361][362][363]. Mo€ es et al [364] presented an extended version of the classical finite element method, referred to as XFEM, to solve microproblems involving complex geometries [365].…”

Section: Analysis At the Rve Levelmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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Application of the Multiscale Fem to the Modeling of Nonlinear Composites With a Random Microstructure

Cited by 30 publications

References 35 publications

Multiscale Computational Homogenization: Review and Proposal of a New Enhanced-First-Order Method

Multiscale Computational Homogenization: Review and Proposal of a New Enhanced-First-Order Method

Viscoelastic effects and shrinkage as accompanying phenomena of the curing of polymers. Single‐ and multiscale effects

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

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