Homogenization of stratified thermoviscoplastic materials

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 $v^\varepsilon$ 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

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

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.