Stability against localization of softening into ellipsoids and bands: Parameter study

Strain-localization instabilities due to strain-softening which result from distributed damage such as cracking in heterogeneous brittle materials are analysed. Attention is restricted to the stability problem of equilibrium states. This problem is not equivalent to bifurcation of the equilibrium path, which may occur before stability of equilibrium is lost. The continuum is local but is enhanced by the localization limiter used in the crack band model, consisting of a lower bound on the minimum dimension of the strain-localization region, which is regarded as a material property. Presented are derivations of the critical state conditions for localization of initially uniform strain into ellipsoidal domains within an infinite continuum and into a planar band within a layer of finite thickness. These derivations are simpler than the previous Bažant's derivations of the general stability conditions for these localizations. A numerical parameter study of the critical states is made for a broad range of material properties as well as various initial stress states and relative sizes of the strain-softening region. The material is described by Drucker-Pragcr plasticity with strain-softening that is caused by yield limit degradation. The Hatter the ellipsoidal domain, or the larger the size of the body (layer thickness), the smaller is found to be the strain-softening slope magnitude at which the critical state is reached. A softening Drucker-Prager material is found to be stable against planar-band localizations in infinite continuum for a certain range of softening material parameters.