Stress systems in aeolotropic plates. I

Abstract
1. The distribution of stress in sheets of uniform aeolotropic material has been discussed by Michell (1900 a ). In particular, he gave the solution of the elastic equations for an isolated force acting at the edge of an aeolotropic plate and at the vertex of an angle cut from such a plate. In these cases the solution is very simple because the stress turns out to be purely radial. When expressed in polar co-ordinates the stress components rθ and θθ are zero, and this is true whether the material is isotropic or aeolotropic. Several examples of the stress systems due to loads distributed along the edges of aeolotropic plates and disks have recently been published by Wolf (1935), Okubo (1937, 1939) and Sen (1939). The method of solution which is used in these examples differs from that used by Michell in the paper referred to above but is related to that used by him in a subsequent paper (1900 b ), in which he shows that the differential equations in three dimensions for stress and strain in an aeolotropic material possessing elastic symmetry equivalent to that of a crystal of the hexagonal system, can be expressed in the form (∂ 2 /∂ x 2 + ∂ 2 /∂ z 2 + k 1 ∂ 2 /∂ y 2 ) V 1 = 0, (∂ 2 /∂ x 2 + ∂ 2 /∂ z 2 + k 2 ∂ 2 /∂ y 2 ) V 2 = 0, where k 1 and k 2 are functions of the elastic constants, and V 1 and V 2 are functions of the components of strain, and the body is elastically symmetrical around lines parallel to the axis of y . A particular case which is included in Michell’s analysis is that of an aeolotropic sheet parallel to the plane z = 0 when it is subjected to a system of generalized plane stress.

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