Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Homogenization of stratified thermoviscoplastic materials

Authors: Nicolas Charalambakis and François Murat
Journal: Quart. Appl. Math. 64 (2006), 359-399
MSC (2000): Primary 74Q15, 74Q10, 35B27; Secondary 74C10, 74F05, 35Q72, 35M20, 35K55
DOI: https://doi.org/10.1090/S0033-569X-06-01017-3
Published electronically: May 22, 2006
MathSciNet review: 2243868
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Abstract | References | Similar Articles | Additional Information

Abstract: In the present paper we study the homogenization of the system of partial differential equations

\begin{displaymath} \begin{array}{l} \displaystyle{\rho^\varepsilon(x) {\partia... ...\partial v^\varepsilon \over \partial x}\right)^2,} \end{array}\end{displaymath}

posed in $ a < x < b$, $ 0 < t < T$, completed by boundary conditions on $ v^\varepsilon$ and by initial conditions on $ v^\varepsilon$ and $ \theta^\varepsilon$. The unknowns are the velocity $ v^\varepsilon$ and the temperature $ \theta^\varepsilon$, while the coefficients $ \rho^\varepsilon$, $ \mu^\varepsilon$ and $ c^\varepsilon$ are data which are assumed to satisfy

$\displaystyle 0 < c_1 \leq \mu^\varepsilon(x,s) \leq c_2, \quad 0 < c_3 \leq c^\varepsilon(x,s) \leq c_4,\quad 0 < c_5 \leq \rho^\varepsilon(x) \leq c_6,$    
$\displaystyle \displaystyle{- c_7 \leq {\partial \mu^\varepsilon \over \partial... ...t c^\varepsilon(x,s) - c^\varepsilon(x,s')\vert\leq \omega(\vert s - s'\vert).}$    

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 $ \varepsilon'$ for which the velocity $ v^{\varepsilon'}$ and the temperature $ \theta^{\varepsilon'}$ converge to some homogenized velocity $ v^0$ and some homogenized temperature $ \theta^0$ which solve a system similar to the system solved by $ v^\varepsilon$ and $ \theta^\varepsilon$, for coefficients $ \rho^0$, $ \mu^0$ and $ c^0$ which satisfy hypotheses similar to the hypotheses satisfied by $ \rho^\varepsilon$, $ \mu^\varepsilon$ and $ c^\varepsilon$. These homogenized coefficients $ \rho^0$, $ \mu^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^\varepsilon$ do not depend on it.

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Additional Information

Nicolas Charalambakis
Affiliation: Department of Civil Engineering, Aristotle University, GR 54124 Thessaloniki, Greece
Email: charalam@civil.auth.gr

François Murat
Affiliation: Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte courrier 187, 75252 Paris Cedex 05, France
Email: murat@ann.jussieu.fr

DOI: https://doi.org/10.1090/S0033-569X-06-01017-3
Keywords: Homogenization, thermoviscoplastic materials
Received by editor(s): December 1, 2005
Published electronically: May 22, 2006
Article copyright: © Copyright 2006 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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