Asymptotics of arbitrary order for a thin elastic clamped plate,II. Analysis of the boundary layer terms

Abstract

This paper is the last of a series of two, where we study the asymptotics of the displacement in a thin clamped plate as its thickness tends to $0$ . In Part I, relying on the structure at infinity of the solutions of certain model problems posed on unbounded domains, we proved that 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 optimal estimates in energy norm. The “profiles” for the boundary layer terms are solutions of such model problems. In this paper, adapting Saint‐Venant’s principle to our framework, we prove the results which we used in Part I.

Investigating more precisely the structure of the boundary layer terms, we go further in the analysis performed in Part I: the introduction of edge layer terms along the intersections of the clamped face with the top and the bottom of the plate, respectively, allows estimates in higher order norms. These edge layer terms are constructed with the help of stable asymptotics, and are the singular parts of the boundary layer terms. As a by‐product of all these investigations, we obtain expansions and estimates for the stress tensor in various anisotropic norms, and also estimates in $L^\infty$ ‐norm for the displacement field.