On the solutions of clamped Reissner-Mindlin plates under transverse loads

The governing equations in the Reissner—Mindlin theory may be written in a form such that a small parameter $\epsilon$ is involved. This parameter $\epsilon$ depends on a combination of the shear modulus and the plate thickness. The governing equations are singularly perturbed with respect to $\epsilon$. However, as $\epsilon \to 0$ one does recover the biharmonic equation of the classical plate theory. In a previous work of ours [1] the behavior of solutions for clamped Reissner—Mindlin plates as $\epsilon \to 0$ was studied and it was shown there that these solutions tend continuously, in various functional norms, to their corresponding solutions in the classical plate theory. This paper deals with two specific questions concerning the detailed dependence of these solutions on $\epsilon$ as $\epsilon \to 0$. We shall show the nonexistence of regular asymptotic expansions of the solutions in integral powers of $\epsilon$ for general clamped Reissner—Mindlin plates. We shall also construct an exact solution for a circular plate which exhibits dependence on fractional powers of $\epsilon$. This latter solution shows a boundary layer phenomenon, decaying away from the boundary, often encountered in singular perturbation problems.