Multi-scale asymptotic expansion for a singular problem of a free plate with thin stiffener

Abstract

In this paper, we consider a partially clamped plate which is stiffened on a portion of its free boundary. Our aim is to build an asymptotic expansion of the displacement, solution of the Kirchhoff-Love model, with respect to the thickness of the stiffener. Due to the mixed boundary conditions, singularities appear, obstructing the construction of the terms of the asymptotic expansion in the same way as if the plate was surrounded by the stiffener on its whole boundary. Using a splitting into regular and singular parts, we are able to formulate an asymptotic expansion involving profiles which allow to take into account the singularities.

Impedance operator of a curved thin layer in linear elasticity with voids

“…The concept of impedance boundary condition is widely used in numerous studies, mainly in electro-magnetics and mechanics; see, for instance, earlier studies [4,5] for the Helmholtz equation in acoustics, prior studies [6,7] for Maxwell equations, and earlier research [8,9] in structure mechanics; see also previous research [10,11]. In the framework of linear elasticity with voids, impedance is a measure of how a material or structure resists deformation or responds to the application of external forces and moments of forces.…”

“…Then the equilibrium Equations ( 7) and ( 9), the boundary conditions on Γ 𝛿 + (10), and the transmission conditions on Σ, (11)…”