Homogenization of stratified thermoviscoplastic materials

Abstract

Abstract. In the present paper we study the homogenization of the system of partial differential equations, posed in a < x < b, 0 < t < T , completed by boundary conditions on v ε and by initial conditions on v ε and θ ε . The unknowns are the velocity v ε and the temperature θ ε , while the coefficients ρ ε , µ ε and c ε are data which are assumed to satisfy. This sequence of one-dimensional systems is a model for the homogenization of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening and a temperature-dependent rate of plastic work converted into heat.Under the above hypotheses we prove that this system is stable by homogenization. More precisely one can extract a subsequence ε for which the velocity v ε and the temperature θ ε converge to some homogenized velocity v 0 and some homogenized temperature θ 0 which solve a system similar to the system solved by v ε and θ ε , for coefficients ρ 0 , µ 0 and c 0 which satisfy hypotheses similar to the hypotheses satisfied by ρ ε , µ ε and c ε . These homogenized coefficients ρ 0 , µ 0 and c 0 are given by some explicit (even if sophisticated) formulas. In particular, the homogenized heat coefficient c 0 in general depends on the temperature even if the heterogeneous heat coefficients c ε do not depend on it.Résumé. Dans cet article, nousétudions l'homogénéisation du système d'équations aux dérivées partielles Cette suite de problèmes unidimensionnels modélise l'homogénéisation de matériaux thermoviscoplastiques hétérogènes dont la résistance diminue avec la température et dont le taux de travail plastique converti en chaleur dépend de la température. Sous les hypothèses ci-dessus, nous démontrons que ce système est stable par homogénéisation. Plus précisément, on peut extraire une sous-suite ε pour laquelle la vitesse v ε et la température θ ε convergent vers une vitesse homogénéisée v 0 et une température homogénéisée θ 0 qui sont solution d'un système similaireà celui dont v ε et θ ε sont solution, pour des coefficients ρ 0 , µ 0 et c 0 qui satisfont des hypothèses analoguesà celles satisfaites par ρ ε , µ ε et c ε . Les coefficients homogénéisés ρ 0 , µ 0 et c 0 sont donnés par des formules explicites (même si elles sont assez compliquées). En particulier le coefficient thermique homogénéisé c 0 dépend en général de la température, même si les coefficients thermiques hétérogènes c ε n'en dépendent pas.

Approximation by finite elements, existence and uniqueness for a model of stratified thermoviscoplastic materials

“…However recent experimental work of Hodowany et al [5] based on a Kolski bar technique, and theoretical studies by Rosakis et al [8] suggest that β can depend, among others, on the temperature. We actually prove in our paper [3] that homogenization of nonhomogeneous, stratified, thermoviscoplastic materials where c depends only on x produces an homogenized material where c in general does depend on the temperature.…”

“…In our paper [3], we investigate the asymptotic behaviour of a thermoviscoplastic material made of numerous layers of different phases of the type considered here, each of those layers having a small thickness, i.e. the homogenization of such materials, and we prove that system (1.1)-(1.5) is stable by homogenization.…”

Stability by Homogenization of Thermoviscoplastic Problems

“…Lemma 3.8 in Charalambakis and Murat (2006b)) asserts that one can extract a subsequence ε ′ and that there exist two functions Y 0 (x, r) and π 0 (x, r): Ω × R → R (which are also Lipschitz continuous in r uniformly in x and measurable in x and bounded for every r ∈ R fixed), such that for every r ∈ R fixed…”

“…with thermal softening, the dynamical problem (1.1)-(1.4) with Dirichlet or Neumann or mixed boundary conditions has been studied by the authors (Charalambakis and Murat (1989), Charalambakis and Murat (2006a)) and its homogenization has been presented in Charalambakis and Murat (2006b).…”

“…We then define Recall that in our papers Charalambakis and Murat (1989), Charalambakis and Murat (2006a) and Charalambakis and Murat (2006b) we considered the dynamical problem associated with (2.1)-(2.8) in the special case where n ε (x) = 1 (linear setting) and where ψ ε (x, θ, γ) = µ ε (x, θ) with ∂µ ε ∂θ (x, θ) ≤ 0 (thermoviscous problem with thermal softening). In this case…”