Thermally induced logarithmic stress singularities in a composite wedge and other anomalies

Abstract Wedge paradoxes, which were first studied by Sternberg and Koiter (Sternberg E, Koiter WT. The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity. ASME Journal of Applied Mechanics 1958;4:575–81), occur due to multiple roots in the Williams (Williams ML. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. ASME Journal of Applied Mechanics 1952;19:526–28) eigenfunction expansion. The consequence of such a paradox is a change in behavior of the stresses from σ ij r, θ =r −ω h 1 ij θ , to the ‘non-separable’ form, σ ij r, θ =r −ω − ln r h 1 ij θ +h 3 ij θ . The focus of this study is the problem of thermally induced logarithmic stress singularities in a composite wedge associated with ω=0. Both double and triple root examples are presented which lead to ln r and ln 2 r behavior in the stresses, respectively. This behavior is primarily associated with incompressible materials for the clamped–clamped single material case, and for the full range of Poisson’s ratio for the clamped-free case. The study also includes non-separable eigenfunctions that occur when complex conjugate roots transition to double real roots. Perhaps the most interesting result is that for the clamped–clamped wedge with Poisson’s ratio equal to 1/2, the hydrostatic stress has a logarithmic singularity proportional to the thermal strain for all wedge angles. This result can be extended to conclude that for a confined, incompressible or nearly incompressible material with a relatively sharp corner, and subject to some expansion or contraction phenomena, high hydrostatic stresses can result.