A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals

Rosakis, Phoebus; Rosakis, A. J.; Ravichandran, G.; Hodowany, J.

doi:10.1016/s0022-5096(99)00048-4

Cited by 451 publications

(284 citation statements)

References 25 publications

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“…even if c ε (x, s) = c ε (x), the homogenized heat coefficient c 0 (x, s) does depend in general on s (see Theorem 6.1, where explicit formulas are given in the case of a material made of layers of some given homogeneous phases). This mathematical result is in accordance with recent experiments and theoretical mechanical studies based on a temperaturedependent fraction of plastic work converted into heating (see [10], [12] and [13]). Indeed the heat coefficient c is given by c = β ρh , where h is the specific heat coefficient and β the rate of plastic work converted into heating, a quantity which is related to the rearrangement of crystals during deformation.…”

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

confidence: 92%

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

confidence: 92%

Homogenization of stratified thermoviscoplastic materials

Charalambakis

Murat

2006

Quart. Appl. Math.

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“…The theoretical basis for the dependence of J3 on strain rate can be found in Rosakis et 0/. [6] where the experimental results are compared with the results of a thermodynamic theoretical model.…”

Section: A-titaniummentioning

confidence: 99%

On the Conversion of Plastic Work into Heat During High-Strain-Rate Deformation

Ravichandran¹,

Rosakis²,

Hodowany³

et al. 2002

AIP Conference Proceedings

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Abstract. Heat generation in metals during high-strain-rate plastic deformation was investigated. Experiments were designed to measure the partition of plastic work into heat and stored energy during dynamic deformations under adiabatic conditions. A Kolsky pressure bar was used to determine mechanical properties at high strain rates while a servo-hydraulic material testing system was used at low strain rates. For dynamic loading, in-situ temperature changes were measured using a high-speed infrared detector. The dependence of the fraction of plastic work converted to heat on strain and strain rate was determined for an aluminum 2024-T3 alloy and a-titanium. The flow stress and the fraction of plastic work converted to heat for 2024-T3 aluminum alloy were found to be a function of strain but not of the strain rate while they were found to be strongly dependent on strain rate for a-titanium.

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“…In almost all works concerned with this problem under adiabatic conditions, β is supposed to be a constant which is equal to 0.9. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Charalambakis

Murat

2006

Ricerche mat.

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In the present paper we consider for a < x < b, 0 < t < T , the system of partial differential equationscompleted by boundary conditions on v and by initial conditions on v and θ . The unknowns are the velocity v and the temperature θ , while the coefficients ρ, µ and c are Carathéodory functions which satisfyCommunicated by the Editor-in-Chief This one dimensional system is a model for the behaviour of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening and temperature dependent rate of plastic work converted into heat. Under the above hypotheses we prove the existence of a solution by proving the convergence of a finite element approximation. Assuming further that µ is Lipschitz continuous in s, we prove the uniqueness of the solution, as well as its continuous dependence with respect to the data. We also prove its regularity when suitable hypotheses are made on the data. These results ensure the existence and uniqueness of one solution of the system in a class where the velocity v, the temperature θ and the stress σ = µ(x, θ ) ∂ v ∂ x belong to L ∞ ((0, T ) × (a, b)).

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A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals

Cited by 451 publications

References 25 publications

Homogenization of stratified thermoviscoplastic materials

Homogenization of stratified thermoviscoplastic materials

On the Conversion of Plastic Work into Heat During High-Strain-Rate Deformation

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

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