Analysis of the influence of incremental material response on the structural stability

An unstable material behavior, which occurrs when some of the tangential moduli attain negative values, is usually associated with failure processes such as fracture, instabilities, and localization. An accurate prediction of these phenomena requires a realistic modelling of nonlinear effects in material behaviour. Therefore, in this work some relations between incremental behavior of a finitely deformed material and the stability of uniform equilibrium states for homogeneous bodies are studied. A class of materials is considered, for which a positive or negative ( also referred as unstable or softening) incremental response is defined as the presence or absence of positive definiteness of the scalar product of a strain rate and the conjugated stress-rate, respectively. A novel characterization of these material states is provided by considering the stability of equilibrium configurations for a material element under deformation-dependent surface forces. As a consequence, new methods to detect material instabilities are provided, and necessary and sufficient conditions are established, which is useful to analyze the influence of a positive incremental material response on structural stability. The circumstances for which a softening material state may be attained while the equilibrium configuration is still stable, are also examined. Results are devoted to boundary conditions leading to uniform and non-uniform instability modes. A collection of representative examples characterized by prescribed loading or boundary displacements, is proposed. It serves to clarify the main aspects of the above general considerations.