Nonlinear symmetric bending of circular elastic plates

Abstract

Communicated by V. G. HartIn recent years, there has been considerable interest in the refining of thin plate theories. In this paper, the method of matching asymptotic expansions is used to obtain one such refinement which is believed to be an improvement on several previous results. Previous authors (Habip (1967), Widera (1969) attempted such refinements within the framework of a partially nonlinear theory of elasticity whereas in the present work all terms neglected by these authors have been retained.The forms of the solutions for the displacements and stresses in the interior of the plate are given in equations (3.7) to (3.12) while those for the boundary layer are given in (4.40) to (4.45). On comparing both sets of solutions it will be observed that the order of magnitude of the stresses increases near the edge of the plate. This effect is due to the occurrence of a boundary layer depending on a dimensionless thickness parameter e defined in §1. Schematic diagrams illustrating the change in the order of the stresses appear at the end of the paper.The assumptions made on the magnitude of the deformation are given in §2. Formulation of the problemWe consider a thin isotropic ideally elastic circular plate of radius R o , thickness 2H, with the origin of the Langrangian coordinates at the centre on the middle surface. The plate is initially flat. We shall use cylindrical polar coordinates R, 0, Z. Thusfor the plate.In deformations symmetrical about the origin, the two non-zero displacements are independent of 0 . Thus the material point with coordinates R, 0, Z before the deformation has coordinates R + U(R, Z), 0, Z + W(R, Z) after the defor-481 use, available at https://www.cambridge.org/core/terms. https://doi