Instabilities in deformation processes of solids

Concepts related to the loss of stability in quasistatic deformation of solids are summarized from the point of view of the discretized representation by finite elements and the numerical following up of the process. Two classes of material models are considered: incrementally nonlinear materials as elastic–plastic and quasi-brittle damaging ones and nonlinear viscous solids. In either case the issues of interest regard the uniqueness of the solution, the stability of the state and of the deformation path, but the respective approaches depend on the material model. The various situations that can arise at a state marked as critical for the case of continuing loading in the incrementally nonlinear solid are pointed out and their significance for the deformation path is discussed. Stability of viscous deformation is examined by means of the temporal evolution of perturbations in the actual reference process. Reflection on the numerical stability of the time integration of viscous deformation is included.