This paper examines the composite, two-dimensional, linear elastic wedge for singular stresses at its vertex. A full range of wedge boundary and matching conditions is considered. Using separation of variables on the Airy stress function, the usual determinant conditions for singularities of the form O(r -x) as r --~ 0 are established and further conditions are derived for singularities of the form O(r -x In r) as r ~ 0.The order of the determinant involved in these conditions depends upon the number of materials comprising the wedge. Two systematic methods of expanding the determinant for the N-material wedge are presented. RESUMECe papier examine le coin compos~, lin~aire et 61astique, en deux dimensions, pour d6terminer les contraintes singuli~res ~ son sommet. On va consid6rer la rang6e totale des conditions aux limites du coin, et les conditions correspondantes dans le coin. On se sert de la s6paration des variables de la fonction de contrainte d'Airy, pour d6terminer les conditions usuelles sur le d~terminant pour les singularit6s de la forme O(r -x) quand r ~ 0, et on d~rive des conditions additionnelles pour les singularit6s de la forme O(r -x In r) quand r--~ 0. L'ordre du d6terminant impliqu~ dans ces conditions d6pend du nombre des mat6riaux dans le coin. D'abord on propose deux m6thodes syst6matiques de d~velopper le d6terminant du coin de N-mat~riaux.

This review article ͑Part II͒ is a sequel to an earlier one ͑Part I͒ that dealt with means of removal and interpretation of stress singularities in elasticity, as well as their asymptotic and numerical analysis. It reviews contributions to the literature that have actually effected asymptotic identifications of possible stress singularities for specific configurations. For the most part, attention is focused on 2D elastostatic configurations with constituent materials being homogeneous and isotropic. For such configurations, the following types of stress singularity are identified: power singularities with both real and complex exponents, logarithmic intensification of power singularities with real exponents, pure logarithmic singularities, and log-squared singularities. These identifications are reviewed for the in-plane loading of angular elastic plates comprised of a single material in Section 2, and for such plates comprised of multiple materials in Section 3. In Section 4, singularity identifications are examined for the out-of-plane shear of elastic wedges comprised of single and multiple materials, and for the out-of-plane bending of elastic plates within the context of classical and higher-order theory. A review of stress singularities identified for other geometries is given in Section 5, axisymmetric and 3D configurations being considered. A limited examination of the stress singularities identified for other field equations is given as well in Section 5. The paper closes with an overview of the status of singularity identification within elasticity. This Part II of the review has 227 references.

Information on the singular behavior at the vertex of a bi-material wedge is the objective of this paper. A summary of the necessary conditions, which depend heavily on the associated eigenvalue equation, for stress singularities of O(r -~ In r) as r --~ 0 or O(r x) as r -~ 0 is stated. The eigenvalue equations arising from a wide range of boundary and interface conditions are then provided. Bi-material wedge problems that have been subjected to singularity analyses of some generality in the literature are briefly reviewed.

This review article has two parts, published in separate issues of this journal, which consider the stress singularities that occur in linear elastostatics. In the present Part I, after a brief review of the singularities that attend concentrated loads, attention is focused on the singularities that occur away from such loading, and primarily on 2D configurations. A number of examples of these singularities are given in the Introduction. For all of these examples, it is absolutely essential that the presence of singularities at least be recognized if the stress fields are to be used in attempts to ensure structural integrity. Given an appreciation of a stress singularity’s occurrence, there are two options open to the stress analyst if the stress analysis is to actually be used. First, to try and improve the modeling so that the singularity is removed and physically sensible stresses result. Second, to try and interpret singularities that persist in a physically meaningful way. Section 2 of the paper reviews avenues available for the removal of stress singularities. At this time, further research is needed to effect the removal of all singularities. Section 3 of the paper reviews possible interpretations of singularities. At this time, interpretations using the singularity coefficient, or stress intensity factor, would appear to be the best available. To implement an approach using stress intensity factors in a general context, two types of companion analysis are usually required: analytical asymptotics to characterize local singular fields; and numerical analysis to capture participation in global configurations. Section 4 of the paper reviews both types of analysis. At this time, methods for both are fairly well developed. Studies in the literature which actually effect asymptotic analyses of specific singular configurations will be considered in Part II of this review article. The present Part I has 182 references.

SUMMARYA set of path independent integrals is constructed for the calculation of the generalized stress intensity factors occurring in elastic plates having sharp re-entrant corners or notches with stress-free faces and subjected to Mode I, I1 or 111 type loading. The Mode I integral is then demonstrated to enjoy a reasonable degree of numerical path independence in a finite element analysis of a test problem having an exact solution. Finally, this integral is used on the same problem in conjunction with a regularizing, finite element, procedure or superposition method. The results indicate that sufficiently accurate estimates of these stress intensity factors for engineering purposes can be achieved with little computational effort.