Standardized complex and logarithmic eigensolutions for n-material wedges and junctions

Abstract

The Airy stress eigenfunction expansion of Williams [1] has been used to obtain simple expressions for the angular variations of the stress and displacement fields for n-material wedges and junctions subjected to inplane loading. This formulation applies to real and complex roots, as well as the special transition case giving rise to r ^−ω singular behavior. The asymptotic behavior of the general problem is similar to that of the bi-material interface crack. In the case of real roots, the stress and displacement expressions can be determined to within a multiplicative real constant (amplification), while for the complex case, the fields are determined to within a multiplicative complex constant (amplification plus rotation). Because of the rotation in the complex case, there are an infinite number of equivalent ways to express the angular variations (eigenfunctions) of the stress and displacement fields. Therefore, the fields are standardized in terms of ‘generalized stress intensity factors’ that are consistent with the bi-material interface crack and the homogeneous crack problems. As in the bi-material crack problem, for the complex case there are two stress intensity factors for each admissible order of the stress singularity. For specific n-material wedges and junctions, a small variation of material properties and/or geometry can change the eigenvalues from a pair of complex conjugate roots to two distinct real roots or vice-versa. An r ^−ω singularity associated with a nonseparable solution in υ and θ exists at this point of bifurcation. Such behavior requires an adjustment in the standard eigenfunction approach to insure bounded stress intensity factors. The proper form of the solution is given both at and near this special material combination, and the smooth transition of the eigenfunctions as the roots change from real to complex is demonstrated in the results. Additional eigenfunction results are provided for particular cases of 2 and 3-material wedges and junctions.