Asymptotic Solution for a Visco-Elastic Thin Plate: Quasistatic and Dynamic Cases

Abstract

The Kelvin–Voigt model for a thin stratified two-dimensional visco-elastic strip is analyzed both in the quasistatic and in the dynamic cases. The Neumann boundary conditions on the upper and the lower parts of the boundary and periodicity conditions with respect to the longitudinal variable are stated. A complete asymptotic expansion of the solution is constructed in both cases, by using the dimension reduction combined with a homogenization technique. The error between the exact solution and the asymptotic one is evaluated in each case and the obtained results fully justify the asymptotic construction. The results were partially (quasistatic case) announced in the short note in C.R. Acad. Sci. Paris; the present article contains the complete proofs and generalizations in the dynamic case.

Keywords

• Kelvin–Voigt visco-elasticity
• thin plate
• laminate
• asymptotic expansion
• dimension reduction
• homogenization