An ellipsoidal crack and needle in an anisotropic elastic medium: PMM vol. 37, n≗3, 1973, pp. 524–531

Abstract

We consider an ellipsoidal crack and needle (of small, but finite thickness) in an anisotropic elastic medium and a homogeneous external field. We obtain and investigate the explicit expressions for the stresses on their surfaces. We show that by decreasing the thickness of the needle the stresses corresponding to any load tend to a finite limit, i.e. they do not contain singularities, while for the ellipsoidal crack a singularity arises if the external field contains a component normal to the plane of the crack. In the case of extension the maximum stress is always attained at the edge of the crack while in the case of pure shear, as a rule, in its small neighborhood. In the latter case there is a sharp peak of stresse and on the edge itself all the components of the stress tensor may vanish. This points to the necessity of investigating the stresses on the entire surface of the crack and not only in its characteristic points. By an ellipsoidal crack (needle) we will understand an ellipsoidal cavity having one small (large) dimension in comparison with the other two dimensions. This allows us, at the computation of the stresses at the surface of the cavity, to restrict ourselves to the principal term of the expansion with respect to a corresponding small but finite parameter. The limiting case when the parameter tend to zero corresponds to the elliptic crack. In this case, only the stresses in the neighborhood of the crack or the limiting values of the nonsingular components of the stress tensor on the crack itself are meaningful. In most cases (see, for example [1] where there are other references) the ellip tic cracks have been studied. The results for an ellipsoidal crack in an isotropic medium can be obtained by a limiting process from the known solutions for the ellipsoidal cavity constructed in [2–4] and which has been done in [5, 6]. How ever, the components of the stress tensor have been studied not on the entire sur face of the crack but only at its edge. The stresses at the vertices of a spheroid; crack and needle in a transversely isotropic medium have been obtained in [7] for external fields which do not have singularities. As opposed to the mentioned papers, here we consider an arbitrary anisotropic medium and we investigate the complete state of stress on the entire surface of an ellipsoidal crack and needle. Such an investigation turns out to be essential since in some cases there is an abrupt increase of stresses near the edge of the crack, although on the edge itself they are equal to zero. In this paper we make use of the general solution of the stress concentration problem on the surface of an ellipsoidal cavity in an anisotropic medium which has been obtained in [8]. In Sect. 1 we give some formulas from [8] which are necessary in what follows and we introduce parameters which are convenient for the limitimg processes. In Sects. 2 and 3 we obtain expressions for the stresses on the surface of the ellipsoidal needle and crack. In Sects 4 and 5 we give a complete study of the stress concentrations at the surface of an ellipsoidal crack in an isotropic and also in an orthotropic medium for all cases of external homogeneous fields.