A Dynamical Systems Approach to Bifurcation and Instability of Constitutive Relations for Brittle Damage Materials

A methodology based on stability theory of dynamical systems is proposed to study the stability of the constitutive relations for brittle damage materials. The bifurcation conditions and the subsequent stability issues of the equilibrium paths are examined in the framework of bifurcation and stability theories of nonlinear dynamical systems. This framework also demonstrates that stability is not only a constitutive property, it also depends on the imposed traction and displacement boundary conditions. Moreover, the loss of uniqueness and stability of the primary equilibrium path does not necessarily imply failure in the dynamical sense since there may exist a stable secondary equilibrium path. Stability criteria for both stress-controlled and displacement-controlled conditions are formulated and an equivalent stability condition for the corresponding statical system is also presented.