Ss. Pageau et al., STANDARDIZED COMPLEX AND LOGARITHMIC EIGENSOLUTIONS FOR N-MATERIAL WEDGES AND JUNCTIONS, International journal of fracture, 77(1), 1996, pp. 51-76
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(-omega) ln(r
) singular behavior. The asymptotic behavior of the general problem is
similar to that of the bi-material interface crack. In the case of re
al 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 com
plex constant (amplification plus rotation). Because of the rotation i
n the complex case, there are an infinite number of equivalent ways to
express the angular variations (eigenfunctions) of the stress and dis
placement 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 ma
terial properties and/or geometry can change the eigenvalues from a pa
ir of complex conjugate roots to two distinct real roots or vice-versa
. An r(-omega) ln(r) singularity associated with a nonseparable soluti
on in r and theta exists at this point of bifurcation. Such behavior r
equires 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 smoo
th transition of the eigenfunctions as the roots change from real to c
omplex is demonstrated in the results. Additional eigenfunction result
s are provided for particular cases of 2 and 3-material wedges and jun
ctions.