Self-similarity mechanisms of damage growth in solids experiencing quasi-brittle fracture

A phenomenological model and kinetic equations for two independent order parameters (defect density tensor and structure scaling parameter) have been constructed in the framework of the recently developed theory describing the behavior of solids with mesoscopic defects. The analysis of self-similar solutions of the constitutive relations has shown that there are two bifurcation points, one of which corresponds to a transition from plastic to quasi-brittle behavior. As it has been shown previously, in the vicinity of the bifurcation points the kinetic equation for the parameter of defect density has self-similar solutions of a singular type (blow-up regimes), i.e. the parameter of defect density approaches infinity in finite time. The method of averaging is used to analyze qualitatively the kinetic equation for the damage parameter, to determine the types of equilibrium points and characteristic patterns of the system behavior and to reveal the dependence of the amplitude and half-width coordinate of a solitary localized structure on the value of the applied stress and initial location of the system.