Conditions of damage initiation in bi-material structures are analysed in the paper. The considerations are derived from the knowledge of the stress state caused by the existence of the sharp notch and the bi-material interface. The step change of material properties and the geometrical discontinuity at the interface leads to a singular peak of stress that is similar to the stress singularity near the crack tip in homogeneous material. That fact urges to use generalized linear elastic fracture mechanics for assessment of conditions of crack initiation. In the paper crack initiation direction and quantification of the external load connected with crack initiation are discussed. Within the stability assessment, possible directions of crack initiation should be considered and tested in both material components and at their interface as well. Knowing the critical applied stresses, one can decide among the damage eventualities.