State of stress and strain of a small neighbourhood of the apex of a wedge for a physical non-linearity and different boundary conditions

• Published in: Journal of applied mathematics and mechanics Vol. 51; no. 4; pp. 509 - 515
• Main Authors: Aleksandrov, V.M., Grishin, S.A.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 1987
• ISSN: 0021-8928; 0021-8928
• DOI: 10.1016/0021-8928(87)90092-X

Three plane strain problems of a small neighbourhood of the apex of a physically non-linear incompressible wedge are investigated by the Cherepanov-Rice-Hutchinson method using a non-linear differential equation for the Airy stress function: Problem 1 — one face is free, and a smooth contact condition is given on the other; Problem 2 — one face if free, and a flexible inextensible cover plate is glued to the other; Problem 3 — one fact is free and the condition of adhesion to a stiff flat stamp is given on the other. Numerical results are presented that illustrate the influence of the degree of non-linearity of the governing relationships and the wedge aperture angle on the solution. The method is also applied to the stream function which enables us to formulate an analogy between different plane problems and affords the possibility of extending it to the axisymmetric case. In many problems of the mechanics of a deformable solid, the investigation of the asymptotic form of the solution near an angular point of a domain occupied by a body plays a fundamental role. In the elastic case this question has been studied quite broadly and an extensive literature exists. The situation is more complicated if the governing relationships are non-linear. The majority of papers deal only with the case of a crack. This paper attempts to fill this gap somewhat.