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

Abstract

The focus of this study is the review of a class of solutions associated with the Williams (ASME J. Appl. Mech., 1952, 19, 526–528) eigenexpansion of the stress state in a composite wedge, that are not variable-separable. These ‘wedge paradox’ solutions, which cannot be expressed as a single function of the radial coordinate multiplied by a single function of the angular coordinate, are readily obtained in this linear analysis by standard mathematical procedures associated with multiple roots from a constant coefficient, linear differential equation. The stress state resulting from these ‘non-separable’ solutions is not self-similar, in that the angular dependence of the stresses is a function of the radial coordinate. Such behavior will complicate both stress analysis, and the application of a linear 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 stress function in polar coordinates are obtained, including four solutions associated with non-separable stresses with terms proportional to r − ω ln( r) for ω=0, 1 and 2, as well as the well-known, non-self-similar eigensolution corresponding to complex eigenvalues. In the second part of the paper, non-separable Airy stress solutions are obtained associated with ω=0, 1 and 2, and values of ω that correspond to the transition from real to complex eigenvalues. After providing the form of these non-separable solutions, examples are given that show how frequently they occur, how the solutions are obtained, the behavior of the associated coefficients or stress intensity factors at and near the special circumstances where they occur, and another look at both the concentrated force problem and the Sternberg–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 problem and solution procedure, these linear results are important to consider when solving the related nonlinear problem where standard superposition procedures for multiple roots cannot be applied.