On bifurcation and instability due to softening

The paper focuses on strict definitions of bifurcation and instability applicable to problems involving dynamic phenomena in rock (seismic events, rockbursts, earthquakes). It is noted that in these problems softening is the only physical process, which may cause instability and bifurcation when geometrical non-linearity is inessential or neglected. Discrete models with one and two degrees of freedom (DOF) are considered to clarify the matter. Strict definitions and conditions of instability and bifurcation are given for systems with finite number of DOF. Comments are made on cases when the effects of instability and bifurcation on the level of microstructure manifest themselves macroscopically as instability of crack propagation and bifurcation of its trajectory.