Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

Visintin, Augusto

doi:10.1017/s0308210506000709

Cited by 27 publications

(27 citation statements)

References 82 publications

(118 reference statements)

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“…see [8], [9]. Although these classical laws find an indirect confirmation in the everyday practice of engineering, (apparently) they do not stem from any underlying lower-scale model.…”

Section: Constitutive Behaviour Let Us Assume That a Mappingmentioning

confidence: 93%

Rheological models vs. homogenization

Visintin

2011

GAMM-Mitteilungen

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We show that the outcome of the theory of homogenization is at variance with a classical engineering technique, that is widely used for constructing rheological models of mechanical materials. Analogical Models. The constitutive behavior of several materials is often represented via analogical models, These models are discrete, and are built up by arranging few elements, that are representative of the basic behavior of the material, viz., elasticity, viscosity and plasticity. For univariate models (just in this case!), one may also use the pictorial image of parallel and series arrangements.These constructions are ubiquitous in engineering, see e.g. . In continuum mechanics they are usually named rheological models; analogous representations are used in electromagnetism, under the label of circuital models; in magnetism they are called magnetic circuits, and so on. As apparently these models are not derived from any representation of the mesoscopic structure of the material, here we address the following question: may the use of analogical models for continuous systems be justified by homogenizing an underlying lower-scale structure?(1)We anticipate that we shall answer in the negative, and will derive alternative models by means of the theory of homogenization. Although analogical models typically represent linear laws, here we are rather concerned with nonlinear relations: this will enhance the generality, without affecting the basic features of the question. Set-Up.In view of dealing with a periodic material, we might tile the Euclidean space R 3 by 3-dimensional cells [0, 1[ 3 . Equivalently, we shall assume the unit torus of R 3 , that we denote by Y, as a reference cell. As the period is typically very small, this normalization will accordingly need the use of a mesoscopic length-scale, when coupled with the macroscopic *

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“…see [8], [9]. Although these classical laws find an indirect confirmation in the everyday practice of engineering, (apparently) they do not stem from any underlying lower-scale model.…”

Section: Constitutive Behaviour Let Us Assume That a Mappingmentioning

confidence: 93%

Rheological models vs. homogenization

Visintin

2011

GAMM-Mitteilungen

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“…In the homogenization results [29] and [32] of Visintin, the maximal monotone operator g is a subdifferential and kinematic hardening is used. Formally, g = g( · ; x, y) = ∂Ψ( · ; x, y) is assumed to be independent of x, but a remark notes that an additional x-dependence can also be treated.…”

Section: Further Comparison With the Literaturementioning

confidence: 99%

Homogenization of plasticity equations with two-scale convergence methods

Schweizer

Veneroni

2014

Applicable Analysis

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We investigate the deformation of heterogeneous plastic materials. The model uses internal variables and kinematic hardening, elastic and plastic strain are used in an infinitesimal strain theory. For periodic material properties with periodicity length scale η > 0, we obtain the limiting system as η → 0. The limiting two-scale plasticity model coincides with well-known effective models. Our direct approach relies on abstract tools from two-scale convergence (regarding convex functionals and monotone operators) and on higher order estimates for solution sequences.

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“…It will actually turn out to be equivalent to the first concept, but it can be verified more easily for weak limits. We emphasize that this use of an energy inequality was extended into a theory of energetic solutions in [20], and it was used in the analysis of elastoplasticity problems, e.g., in [31,32,22,33,34] in order to characterize weak variational solutions.…”

Section: Solution Conceptsmentioning

confidence: 99%

Periodic Homogenization of the Prandtl–reuss Model With Hardening

Schweizer

Veneroni

2010

J. Multiscale Modelling

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We study the n-dimensional wave equation with an elasto-plastic nonlinear stress-strain relation. We investigate the case of heterogeneous materials, i.e. x-dependent parameters that are periodic at the scale η > 0. We study the limit η → 0 and derive the plasticity equations for the homogenized material. We prove the well-posedness for the original and the effective system with a finite-element approximation. The approximate solutions are also used in the homogenization proof which is based on oscillating test function and an adapted version of the div-curl Lemma.MSC: 74Q10, 35L70, 74D10.

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Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

Cited by 27 publications

References 82 publications

Rheological models vs. homogenization

Rheological models vs. homogenization

Homogenization of plasticity equations with two-scale convergence methods

Periodic Homogenization of the Prandtl–reuss Model With Hardening

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