Homogenization of stratified thermoviscoplastic materials

In the present paper we study the homogenization of the system of partial differential equations \[ \begin {array}{l} \displaystyle {\rho ^\varepsilon (x) {\partial v^\varepsilon \over \partial t} - {\partial \over \partial x} \left (\mu ^\varepsilon (x,\theta ^\varepsilon ) {\partial v^\varepsilon \over \partial x}\right ) = f,} \displaystyle {c^\varepsilon (x,\theta ^\varepsilon ) {\partial \theta ^\varepsilon \over \partial t} = \mu ^\varepsilon (x,\theta ^\varepsilon ) \left ({\partial v^\varepsilon \over \partial x}\right )^2,} \end {array} \] 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 \begin{gather*} 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 s} (x,s) \leq 0, \quad |c^\varepsilon (x,s) - c^\varepsilon (x,s’)|\leq \omega (|s - s’|).} \end{gather*} 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.