Energy profile and bifurcation analysis in softening plasticity

Bifurcations of solutions and energy profile in softening plasticity are discussed in this paper. The localized and non-localized solutions are obtained for a simple softening bar; the second-order derivatives of the incremental energy are evaluated. The second-order derivatives along the fundamental path demonstrate a discontinuity at the bifurcation point; the eigen-analysis of the tangential stiffness matrix fails to identify the post-bifurcation paths. The energy variation near the bifurcation point is investigated; the relationship between the stationary points of the energy profile and post-bifurcation solutions is established. Beyond the bifurcation point, the single stable loading path splits into several post-bifurcation paths and the incremental energy exhibits several competing minima. Among the multiple post-bifurcation equilibrium states, the localized solutions correspond to the minimum points of the energy profile, while the non-localized solutions correspond to the saddles and local maximum points. To determine the real post-bifurcation path, the concept of minimization of the second-order energy is used as the criterion for the bifurcation analysis involved in softening plasticity. As an application, a lattice model of a beam is analyzed and damage localization is obtained.