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

Wedge paradoxes, which were first studied by Sternbrrg and Koiter (Sternber g 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 b ehavior of the stresses from sigma(ij)(r, theta) = r(-omega)h(ij)(1)(theta) , to the 'non-separable' form, sigma(ij)(r, theta) = r(-omega)[-ln(r)h(ij)( 1)(theta) + h(ij)(3)(theta)]. The focus of this study is the problem of the rmally induced logarithmic stress singularities in a composite wedge associ ated with omega = 0. Both double and triple root examples are presented whi ch lead to ln(r) and ln(2)(r) behavior in the stresses, respectively. This behavior is primarily associated with incompressible materials for the clam ped-clamped single material case, and for the full range of Poisson's ratio for the clamped-free case. The study also includes non-separable eigenfunc tions that occur when complex conjugate roots transition to double real roo ts. Perhaps the most interesting result is that for the clamped-clamped wed ge with Poisson's ratio equal to 1/2, the hydrostatic stress has a logarith mic singularity proportional to the thermal strain for all wedge angles. Th is result can be extended to conclude that for a confined, incompressible o r nearly incompressible material with a relatively sharp corner, and subjec t to some expansion or contraction phenomena, high hydrostatic stresses can result. (C) 2000 Published by Elsevier Science Ltd. All rights reserved.