On the finite deformation of a sector plate

Abstract

In this paper an investigation is given into the behaviour of the stress singularity which occurs in the linear theory of elasticity at the deformation of a sector plate, if finite deformations are considered. It is assumed that for very small deformations Hooke's law is valid, and only in the neighbourhood of the singularity Hooke's law has to be extended. This extension is not unique. It is shown that for two different strain-energy functions, which have the same asymptotic expansion for infinitesimal deformations, the behaviour of the solutions is quite different. One of the strain-energy functions leads to a bounded solution, while the solution, obtained from the other one becomes singular for the case of contraction. As it cannot be expected that it will be possible to decide on an experimental basis about the right extension, an assertion about the difference in smoothness of solutions to problems in linear and non-linear theory cannot be given. An open question is raised: whether or not the requirement of regularity for this kind of problems in non-linear theory can be posed as a restriction on the admissible energy functions.