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

Yotsawat Terapabkajornded (1) , Somsak Orankitjaroen (2, 1) , Christian Licht (1, 2, 3)

1 Mahidol University [Bangkok]
2 Centre of Excellence in Mathematics
3 M3 - Mathématiques et Modélisations en Mécanique

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.

Mots clés: Asymptotic model; Thin visco-elastic plates; Kelvin-Voigt visco-elasticity; Trotter theory

Domaines: Mécanique des solides [physics.class-ph]

Yotsawat Terapabkajornded, Somsak Orankitjaroen, Christian Licht. Asymptotic model of linearly visco-elastic Kelvin–Voigt type plates via Trotter theory. Advances in Difference Equations, 2019, 2019, pp.186. ⟨10.1186/s13662-019-2104-6⟩. ⟨hal-02318639⟩