Homogenization of nonlinear visco-elastic composites

Visintin, Augusto

doi:10.1016/j.matpur.2008.02.002

Cited by 16 publications

(14 citation statements)

References 65 publications

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“…Stochastic homogenization of subdifferential inclusions was analyzed in [26]. A related homogenization result is contained in [31], but it does not concern plasticity in our sense. The emphasis in [2] is on the interface conditions along (microscopic) interfaces.…”

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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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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“…In [43] the homogenization of the Prager model was also studied for univariate systems via hysteresis operators (in passing we point out that this analysis might easily be extended to the nonlinear Kelvin-Voigt model). The present work is part of research on the two-scale homogenization of nonlinear models of continuum mechanics, electromagnetism and heat conduction (see [102,103,106,107]). …”

Section: Literaturementioning

confidence: 99%

“…In this work we proceeded along the following lines, which can also be applied to the homogenization of other either stationary or evolutive nonlinear phenomena (see [102,103,106,107]). …”

Section: Two-scale and Single-scale Homogenizationmentioning

confidence: 99%

“…It may be of some interest to apply the procedure of two-scale convergence to the corresponding time homogenization. In view of applications, it may also be useful to allow for purely viscous and/or purely elastic parts of the body, by allowing B and/or ϕ to degenerate in subdomains of Ω; a similar issue is addressed in [106].…”

Section: Further Questionsmentioning

confidence: 99%

“…In [106] this notion is also applied to the homogenization of an evolution problem; after that, this is reformulated as a minimization principle. That approach may be applied to the system (1.1), (1.2).…”

Section: Further Questionsmentioning

confidence: 99%

See 2 more Smart Citations

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

Visintin

2008

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This paper deals with processes in nonlinear inelastic materials whose constitutive behaviour is represented by the inclusionhere we denote by σ the stress tensor, by ε the linearized strain tensor, by B(x) the compliance tensor and by ∂ϕ(·, x) the subdifferential of a convex function ϕ (·, x). This relation accounts for elasto-viscoplasticity, including a nonlinear version of the classical Maxwell model of viscoelasticity and the Prandtl-Reuss model of elastoplasticity. The constitutive law is coupled with the equation of continuum dynamics, and well-posedness is proved for an initial-and boundary-value problem. The function ϕ and the tensor B are then assumed to oscillate periodically with respect to x and, as this period vanishes, a two-scale model of the asymptotic behaviour is derived via Nguetseng's notion of two-scale convergence. A fully homogenized single-scale model is also retrieved, and its equivalence with the two-scale problem is proved. This formulation is non-local in time and is at variance with that based on so-called analogical models that rest on a mean-field-type hypothesis.

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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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Homogenization of nonlinear visco-elastic composites

Cited by 16 publications

References 65 publications

Homogenization of plasticity equations with two-scale convergence methods

Homogenization of plasticity equations with two-scale convergence methods

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

Rheological models vs. homogenization

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