Homogenization of stratified thermoviscoplastic materials

Abstract

In the present paper we study the homogenization of the system of partial differential equations \[ ρ ε ( x ) ∂ v ε ∂ t − ∂ ∂ x ( μ ε ( x , θ ε ) ∂ v ε ∂ x ) = f , c ε ( x , θ ε ) ∂ θ ε ∂ t = μ ε ( x , θ ε ) ( ∂ v ε ∂ x ) 2 , \begin {array}{l} \displaystyle {\rho ^\varepsilon (x) {\partial v^\varepsilon \over \partial t} - {\partial \over \partial x} \left (\mu ^\varepsilon (x,\theta ^\varepsilon ) {\partial v^\varepsilon \over \partial x}\right ) = f,} \displaystyle {c^\varepsilon (x,\theta ^\varepsilon ) {\partial \theta ^\varepsilon \over \partial t} = \mu ^\varepsilon (x,\theta ^\varepsilon ) \left ({\partial v^\varepsilon \over \partial x}\right )^2,} \end {array} \] posed in a > x > b a > x > b , 0 > t > T 0 > t > T , completed by boundary conditions on v ε v^\varepsilon and by initial conditions on v ε v^\varepsilon and θ ε \theta ^\varepsilon . The unknowns are the velocity v ε v^\varepsilon and the temperature θ ε \theta ^\varepsilon , while the coefficients ρ ε \rho ^\varepsilon , μ ε \mu ^\varepsilon and c ε c^\varepsilon are data which are assumed to satisfy 0 > c 1 ≤ μ ε ( x , s ) ≤ c 2 , 0 > c 3 ≤ c ε ( x , s ) ≤ c 4 , 0 > c 5 ≤ ρ ε ( x ) ≤ c 6 , − c 7 ≤ ∂ μ ε ∂ s ( x , s ) ≤ 0 , | c ε ( x , s ) − c ε ( x , s ′ ) | ≤ ω ( | s − s ′ | ) . \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 ε ′ \varepsilon ’ for which the velocity v ε ′ v^{\varepsilon ’} and the temperature θ ε ′ \theta ^{\varepsilon ’} converge to some homogenized velocity v 0 v^0 and some homogenized temperature θ 0 \theta ^0 which solve a system similar to the system solved by v ε v^\varepsilon and θ ε \theta ^\varepsilon , for coefficients ρ 0 \rho ^0 , μ 0 \mu ^0 and c 0 c^0 which satisfy hypotheses similar to the hypotheses satisfied by ρ ε \rho ^\varepsilon , μ ε \mu ^\varepsilon and c ε c^\varepsilon . These homogenized coefficients ρ 0 \rho ^0 , μ 0 \mu ^0 and c 0 c^0 are given by some explicit (even if sophisticated) formulas. In particular, the homogenized heat coefficient c 0 c^0 in general depends on the temperature even if the heterogeneous heat coefficients c ε c^\varepsilon do not depend on it.