Mixed boundary value problem for quasi-orthotropic elastic plane

Abstract

The quasi-orthotropic elastic plane in which the characteristic roots of the fundamental differential equation for the orthotropic elastic plane are doubled is investigated for a mixed boundary value problem. One of the main purposes of the present paper is to derive a general solution of a mixed boundary value problem. For the associated stress analysis, a rational mapping function is used and a closed-form stress function is obtained. The stress analysis is hence rigorous for the rational mapping function, which gives a comparative arbitrary configuration. The quasi-orthotropic elastic plane problem includes that of an isotropic elastic plane. As a demonstration of the stress analysis, a half plane with an oblique edge rigid line inclusion in which the rim of the half plane and the principal axis for the orthotropic elasticity coincides is analyzed. A uniform tension is applied over the half plane, and the rigid line inclusion is expressed as a fixed boundary. The homogenous solution of the stress function can be obtained without any integration. Stress distributions and stress intensity factors at the tip of the rigid line inclusion are investigated for Poisson’s ratio. The relationship between the quasi-orthotropic elastic plane and the isotropic elastic plane with respect to stress intensity factors is derived. Stress functions in which the rim of the half plane and the principal axis of the orthotropic elasticity do not coincide are also described.