Atomistic simulation of transformation toughening due to martensitic transformation in Ti-V phase particles dispersed in a gamma-TiAl matrix containing cracks requires knowledge of the continuum elastic stress and displacement fields for the problem of a crack touching the gamma-beta interface. Because of the anisotropic characters of the two phases, analytical solutions for these fields are not available and they must be determined numerically. In the present paper a finite element method-based eigenanalysis is developed and subsequently applied to the gamma-beta system to determine the order of the stress singularity and the angular dependences of the stress and displacement fields. These fields are subsequently used to enrich the finite elements surrounding the crack tip and, through the use of the general finite element code ABAQUS, to determine the generalized stress intensity factors and thus the total singular crack-tip stress and displacement fields. It is found that there are two coupled singular terms in the singular stress and displacement fields, and consequently pure (uniaxial) mode I loading gives rise to mixed modes I-II near-crack-tip behaviour.