Asymptotics of Arbitrary Order for a Thin Elastic Clamped Plate, I. Optimal Error Estimates

This paper is the rst of a series of two, where we study the asymp-totics 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 " denotes the small parameter in the geometry, we rst construct the algorithm for an innnite \even" Ansatz involving only even powers of " , which is a natural extension of the usual Kirchhoo-Love Ansatz. The boundary conditions of the clamped plate being only satissed 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 suucient conditions for the exponential decay of such terms. In order to fullll 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 non-negative powers of " , the in-plane space variables x , the transverse scaled variable x 3 and the quickly varying variable r=" where r is the distance to the clamped part of the boundary.