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 e and by initial conditions on v e and θ e . The unknowns are the velocity v e and the temperature θ e , while the coefficients ρ e , μ e and c e are data which are assumed to satisfy 0 < c 1 ≤ μ e (x,s) ≤ c 2 , 0 < c 3 ≤ c e (x,s) ≤ c 4 , 0 < c 5 ≤ ρ e (x) ≤ c 6 , -c 7 < ∂μ e ∂s(x, S) < 0, |c e (x, s) - c e (x, s')| < ω(|s - s'|). 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 e' for which the velocity v e' and the temperature θ e converge to some homogenized velocity v° and some homogenized temperature θ° which solve a system similar to the system solved by u e and θ e , for coefficients p°, μ° and c° which satisfy hypotheses similar to the hypotheses satisfied by ρ e , μ e and c e . These homogenized coefficients ρ°, μ° and c° are given by some explicit (even if sophisticated) formulas. In particular, the homogenized heat coefficient c° in general depends on the temperature even if the heterogeneous heat coefficients c e do not depend on it.

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