Asymptotic model of linearly visco-elastic Kelvin–Voigt type plates via Trotter theory

Advances in Difference Equations

Yotsawat Terapabkajornded¹  Christian Licht²  Somsak Orankitjaroen³ 

[1] 0000 0004 1937 0490, grid.10223.32, Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand;0000 0004 1937 0490, grid.10223.32, Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand;0000 0001 2097 0141, grid.121334.6, LMGC-UMR 5508, Université de Montpellier-CC048, Montpellier, France;Centre of Excellence in Mathematics, CHE, Bangkok, Thailand;0000 0004 1937 0490, grid.10223.32, Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand;Centre of Excellence in Mathematics, CHE, Bangkok, Thailand;

关键词: Asymptotic model;    Thin visco-elastic plates;    Kelvin–Voigt visco-elasticity;    Trotter theory;    74B99;   

DOI  :  10.1186/s13662-019-2104-6

来源: publisher

【 摘 要 】

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.

【 授权许可】

CC BY