Stress systems in aeolotropic plates. II

Abstract

1. Fundamental equations which can be applied to problems of generalized plane stress in any plane plate of aeolotropic material have been obtained by Huber (1938). When the material has two directions of symmetry at right angles in the plane of the plate the differential equation for the stress function simplifies, and in a previous paper by G. I. Taylor and the present writer (1939) a number of stress functions were obtained which satisfied this equation and also gave single-valued expressions for the mean values of the stresses and displacements. Some of these functions were used to solve the problem of an isolated force in an infinite plate. In the present paper formulae are obtained for generalized plane stress systems in an infinite aeolotropic strip and also in a semi-infinite plate which is bounded by one straight edge. In particular, a solution is given for the general problem of any force acting at any point either within or on the boundary of a strip or semi-infinite plate. The stresses due to any distribution of force over the strip or semi-infinite plate may be deduced by integration. The method of solution is similar to that used by Howland (1929) for stress systems in a strip of isotropic material and Howland’s results may be obtained from our general formulae by a limiting process. When a force acts on the boundary of a semi-infinite plate the stresses may be deduced from our general results. It is, however, easier to evaluate these stresses independently by considering the problem of an isolated force at the vertex of a wedge. This problem was actually solved by Michell (1900) for any aeolotropic plate whose moduluses are not functions of the distance from the vertex of the wedge, but we give a solution here using the methods and notation of this and our previous paper. For the case of a wedge with one straight boundary the results agree with those deduced from the general formulae for a force in a semi-infinite plate.