Approximation by finite elements, existence and uniqueness for a model of stratified thermoviscoplastic materials

Abstract

abstract In the present paper we consider for a < x < b, 0 < t < T, the system of partial differential equations $$ \begin{array}{l} \displaystyle{\rho(x) {\partial v \over \partial t} - {\partial \over \partial x} \left(\mu(x,\theta) {\partial v \over \partial x}\right) = f,} \displaystyle{c(x,\theta) {\partial\theta \over \partial t} = \mu(x,\theta) \left({\partial v \over \partial x}\right)^2}, \end{array} $$ completed by boundary conditions on v and by initial conditions on v and θ. The unknowns are the velocity v and the temperature θ, while the coefficients ρ, μ and c are Caratheodory functions which satisfy $$ 0 < c_1 \leq \mu(x,s) \leq c_2, \quad {\partial\mu \over \partial s}(x,s) \leq 0, $$ $$ 0 < c_3 \leq c(x,s) \leq c_4,\quad 0 < c_5 \leq \rho(x) \leq c_6. $$ This one dimensional system is a model for the behaviour of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening and temperature dependent rate of plastic work converted into heat. Under the above hypotheses we prove the existence of a solution by proving the convergence of a finite element approximation. Assuming further that μ is Lipschitz continuous in s, we prove the uniqueness of the solution, as well as its continuous dependence with respect to the data. We also prove its regularity when suitable hypotheses are made on the data. These results ensure the existence and uniqueness of one solution of the system in a class where the velocity v, the temperature θ and the stress $\sigma = \mu(x,\theta) \displaystyle{\partial v \over \partial x}$ belong to L∞((0,T) × (a,b)). Keywords: Thermoviscoplastic materials, nonhomogeneous materials, thermal softening, existence, uniqueness, Galerkin’s method Mathematics Subject Classification (2000): 74H20, 74H25, 65M60, 35D05, 35D10, 35R05, 74C10, 74F05, 35Q72, 35M20 This is a “Springer Open Choice” article. Unrestricted non-commercial use, distribution, and reproduction in any medium is permitted, provided the original author and source are credited.