Homogenization of scalar wave equations with hysteresis

Peer-reviewed article not indexed in WoS or Scopus

en

The paper deals with a scalar wave equation of the form $\rho u_{tt} = (F[u_x])_x + f$ where $F$ 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 $\rho^\eps$ and $\eta^\eps$, where the spatial period $\eps$ tends to $0$. We identify the homogenized limits $\rho^*$ and $\eta^*$ and prove the convergence of solutions $u^\e$ to the solution $u^*$ of the homogenized equation.

scalar wave equation, homogenization, hysteresis operator

1999-01-01

0935-1175

CONTINUUM MECHANICS AND THERMODYNAMICS

11

6

371–390

21