Asymptotics of arbitrary order for a thin elastic clamped plate,I. Optimal error estimates

Abstract

This paper is the first of a series of two, where we study the asymptotics of the displacement in a thin clamped plate made of a rigid “monoclinic” material, as the thickness of the plate tends to 0. The combination of a polynomial Ansatz (outer expansion) and of a boundary layer Ansatz (inner expansion) yields a complete multi-scale asymptotics of the displacement and leads to optimal error estimates in energy norm. We investigate the polynomial Ansatz in Part I, and the boundary layer Ansatz in Part II.

If ε denotes the small parameter in the geometry, we first construct the algorithm for an infinite “even” Ansatz involving only even powers of ε, which is a natural extension of the usual Kirchhoff-Love Ansatz. The boundary conditions of the clamped plate being only satisfied at the order 0, we try to compensate for them by boundary layer terms: we rely on a result proved in Part II giving necessary and sufficient conditions for the exponential decay of such terms. In order to fulfill these conditions, the constructive algorithm for the boundary layer terms has to be combined with an “odd” polynomial Ansatz. The outcome is a two-scale asymptotics involving all nonnegative powers of ε, the in-plane space variables x_α, the transverse scaled variable x₃ and the quickly varying variable r/ε where r is the distance to the clamped part of the boundary.