Multiple root solutions, wedge paradoxes and singular stress states that are not variable-separable

The focus of this study, is the review of a class of solutions associated w ith the Williams (ASME J. Appl. Mech., 1952, 19, 526-528) eigenexpansion of the stress stare in a composite wedge, that are nor variable-separable. Th ese 'wedge paradox' solutions, which cannot be expressed as a single functi on of the radial coordinate multiplied by a single function of the angular coordinate, are readily obtained in this linear analysis by standard mathem atical procedures associated with multiple roots from a constant coefficien t, linear differential equation. The stress state resulting from these 'non -separable' solutions is not self-similar, in that the angular dependence o f the stresses is a function of the radial coordinate. Such behavior will c omplicate both stress analysis, and the application of a linens solution to the failure analysis of an inherently nonlinear problem. In the first part of the paper all appropriate variable-separable solutions of the Airy stre ss function in polar coordinates ape obtained including four solutions asso ciated with non-separable stresses with terms proportional to r(-omega)ln ( r) for omega = 0, 1 and 2, as well as the well-known, non-self-similar eige nsolution corresponding to complex eigenvalues. In the second part of the p aper, non-separable Airy stress solutions are obtained associated with omeg a = 0, 1 and 2, and values of omega that correspond to the transition from real to complex eigenvalues. After providing the form of these non-separabl e solutions, examples are given that show how frequently they occur, how th e solutions are obtained, the behavior of the associated coefficients or st ress intensity factors at and near the special circumstances where they occ ur, and another look at both the concentrated force problem and the Sternbe rg-Koiter (ASME J. Appl. Mech., 1958, 4, 575-581) problem of a wedge with a concentrated couple. In addition to providing a thorough review of the pro blem and solution procedure, these linear results are important to consider when solving the related nonlinear problem where standard superposition pr ocedures for multiple roots cannot be applied. (C) 1998 Elsevier Science Lt d. All rights reserved.