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

Charalambakis, Nicolas; Murat, François

doi:10.1007/s11587-006-0011-0

Cited by 9 publications

(17 citation statements)

References 10 publications

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“…Remark 2.2 (Neumann and mixed boundary conditions). As pointed out in [8], a result similar to the result of Theorem 2.1 still holds true when the Dirichlet boundary conditions (1.5) on v ε are replaced either by the Neumann boundary conditions (1.6) on σ ε or by the mixed boundary conditions (1.7) on v ε and σ ε .…”

Section: 2supporting

confidence: 60%

“…Theorems 2.1 and 3.1 of our paper [8] prove the following result of existence, uniqueness and local Lipschitz continuity with respect to the data for the case of Dirichlet boundary conditions (1.5) (see also [7] for another proof of the existence result). …”

Section: 2mentioning

confidence: 75%

“…Theorem 2.4 is similar to Theorem 3.2 of [8], except for the fact that the regularity is now local in time (i.e. in (δ, T − δ) and not in (0, T )) as well as in space (i.e.…”

Section: Proposition 23 (L 6 Regularity Of σ)mentioning

confidence: 83%

“…In contrast, we claim that the temperatures θ ε do oscillate in space, which implies, in view of (2.42) and (2.43), that the strain rates ∂v ε ∂x and the temperature increases ∂θ ε ∂t also oscillate in space. Indeed in the proofs below (see Sections 3, 4 and 5) (and already in the proofs in [7] and [8]), very important quantities, herein named the transformed temperatures τ ε (see (3.19) below), naturally appear. These transformed temperatures τ ε are proved to be bounded in W 1,1 (Q) (see (3.34) below), and therefore do not oscillate.…”

Section: Remark 28 (Oscillation and Nonoscillation Of The Unknowns)mentioning

confidence: 98%

“…As far as existence of a solution is concerned, we just quote the pioneering work of Dafermos and Hsiao [9], the works of Tzavaras [16] and [17], the recent paper [5] which is concerned with the numerical analysis of the problem, and our papers [7] and [8] (the second one presents existence and uniqueness results of the weak solution of (1.1)-(1.3), as well as its approximation by finite elements).…”

Section: ë òóý º ëø ò ö × ùø ñ ð øó ñ ¸ α < X < B¸0 < T < T¸ømentioning

confidence: 99%

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Homogenization of stratified thermoviscoplastic materials

Charalambakis

Murat

2006

Quart. Appl. Math.

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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.

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Section: 2supporting

confidence: 60%

Section: 2mentioning

confidence: 75%

“…Theorem 2.4 is similar to Theorem 3.2 of [8], except for the fact that the regularity is now local in time (i.e. in (δ, T − δ) and not in (0, T )) as well as in space (i.e.…”

Section: Proposition 23 (L 6 Regularity Of σ)mentioning

confidence: 83%

Section: Remark 28 (Oscillation and Nonoscillation Of The Unknowns)mentioning

confidence: 98%

Section: ë òóý º ëø ò ö × ùø ñ ð øó ñ ¸ α < X < B¸0 < T < T¸ømentioning

confidence: 99%

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Homogenization of stratified thermoviscoplastic materials

Charalambakis

Murat

2006

Quart. Appl. Math.

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Mathematical homogenization of inelastic dissipative materials: a survey and recent progress

Charalambakis

Chatzigeorgiou

Chemisky

et al. 2017

Continuum Mech. Thermodyn.

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Biaxial Loading of Continuously Graded Thermoviscoplastic Materials

2006

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In this paper we study the problem of biaxial loading of a sheet made of a continuously graded thermoviscoplastic material. The material phases are supposed to exhibit thermal softening, strain-rate sensitivity and strain hardening, continuously varying along all directions. First we formulate the plane stress problem of a non-homogeneous material and study the behavior of temperature, strain and strain-rate related to inhomogeneities of thermomechanical parameters and geometrical defects. Next we present the "effective" instability analysis of Dudzinski and Molinari (Int J Solids Struc 1991), adapted to the non-homogeneous case, to define the critical conditions and select the localization modes by studying the overall strains for which a certain level of instability growth is developed. Finally, we present the numerical simulation of the fully non-linear dynamical problem. Several aspects of the deformation process and the related role of non-homogeneities are analyzed: onset of strain and temperature localization, ductility, contours of temperature increase as detectors of instability, interplay with initial defects, multiple necking, decrease of thinning-rate, variation of the multiple necking due to boundary conditions.

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

Cited by 9 publications

References 10 publications

Homogenization of stratified thermoviscoplastic materials

Homogenization of stratified thermoviscoplastic materials

Mathematical homogenization of inelastic dissipative materials: a survey and recent progress

Biaxial Loading of Continuously Graded Thermoviscoplastic Materials

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