Homogenization of scalar wave equations with hysteresis

The paper deals with a scalar wave equation of the form $\rho u_{tt} = ({\cal F}[u_x])_x + f \,,$ where $\cal F$ 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 the convergence of solutions $u^\varepsilon$ to the solution $u^*$ of the homogenized equation.