Asymptotic model of linearly visco-elastic Kelvin\u2013Voigt type plates via Trotter theory

Yotsawat Terapabkajornded,Christian Licht,Somsak Orankitjaroen

doi:10.1186/s13662-019-2104-6

Abstract

We confirm the study (Licht in C. R., Méc. 341:697–700, 2013) devoted to the quasi-static response for a visco-elastic Kelvin–Voigt plate whose thickness goes to zero. For each thickness parameter, the quasi-static response is given by a system of partial differential equations with initial and boundary conditions. Reformulating scaled systems into a family of evolution equations in Hilbert spaces of possible states with finite energy, we use Trotter theory of convergence of semi-groups of linear operators to identify the asymptotic behavior of the system. The asymptotic model we obtain and the genuine one have the same structure except an occurrence of a new state variable. Eliminating the new state variable from our asymptotic model leads to the asymptotic model in (Licht in C. R., Méc. 341:697–700, 2013) which involves an integro-differential system.

Highlights

1 Introduction In a recent study [2], Licht and Weller promoted an old but not so well-known convergence tool, namely Trotter theory of convergence of semi-groups of linear operators acting on variable Hilbert spaces, in determining the asymptotic modeling in physics of continuous media
One of the models mentioned is a reduction of the dimension problem on thin linear visco-elastic Kelvin–Voigt type plates
Licht [1] studied this problem before in 2013 and derived the asymptotic model with Laplace transform technique. He found that the mechanical behavior of the limit model is no longer of Kelvin–Voigt type, because a term of fading memory appears like in the homogenization problem

Summary

Introduction

In a recent study [2], Licht and Weller promoted an old but not so well-known convergence tool, namely Trotter theory of convergence of semi-groups of linear operators acting on variable Hilbert spaces, in determining the asymptotic modeling in physics of continuous media. One of the models mentioned is a reduction of the dimension problem on thin linear visco-elastic Kelvin–Voigt type plates. With Trotter theory of convergence, Licht and Weller suggested that the mechanical behaviors of limit and genuine models are the same except for the appearance of a new state variable.

Objectives

Results

Conclusion