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

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 , to the ‘non-separable’ form,

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. 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 and 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.