Homogenization of scalar wave equations with hysteresis (Q1976243)

The paper deals with a scalar wave equation of the form \(\rho u_{tt}= ({\mathcal F}[u_x])_x+f\), where \({\mathcal F}\) is a Prandtl-Ishlinskii operator and \(\rho,f\) are given functions. This equation describes longitudinal vibrations of an elastoplastic rod. The mass density \(\rho\) and the Prandtl-Ishlinskii distribution function \(\eta\) are allowed to depend on the space variable \(x\). We prove existence, uniqueness and regularity of solution to a corresponding initial-boundary value problem. The system is then homogenized by considering a sequence of equations of the above type with spatially periodic data \(\rho^\varepsilon\) and \(\eta^\varepsilon\), where the spatial period \(\varepsilon\) tends to 0. We identify the homogenized limits \(\rho^*\) and \(\eta^*\), and prove convergence of solutions \(u^\varepsilon\) to the solution \(u^*\) of the homogenized equation.