Homogenization of Scalar Wave Equations with Hysteresis
- Franců, Jan
- Krejčí, Pavel
2010 Mathematics Subject Classification
- 35B27 73B27 73E50 47H30
- Prandtl-Ishlinskii operator, hysteresis, homogenization, nonlinear scalar wave equation
The paper deals with a scalar wave equation of the form ρutt = (𝓕[ux])x + ƒ, where 𝓕 is a Prandtl-Ishlinskii operator and ρ, ƒ are given functions. This equation describes longitudinal vibrations of an elastoplastic rod. The mass density ρ and the Prandtl-Ishlinskii distribution function η 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 ρε and ηε, where the spatial period ε tends to 0. We identify the homogenized limits ρ* and η* and prove the convergence of solutions uε to the solution u* of the homogenized equation.
- Contin. Mech. Thermodyn., 11 (1999), pp. 371-391