Comparison between Static and Dynamic Criteria of Material Stability

A theoretical investigation on material stability characterisation for incrementally non-linear solids at finite strains is developed. Material stability may be defined by means of static and dynamic approaches leading to different conditions. In the framework of static material stability conditions, the opportunity of defining an infinitesimal stable material behaviour as the positiveness of the scalar product of the Biot strain-rate and the conjugate stress-rate, is taken into consideration. Dynamic conditions of material stability may be formulated in the context of propagation of plane infinitesimal wave (or acceleration waves) in an infinite body, and may lead to conditions similar to those obtained in the context of strain localisation in shear bands. Static and dynamic conditions are thus examined on the basis of the static material stability criterion defined by means of the Biot strain measure. The main objective of the present work is to investigate the interrelations between different material stability criteria. The analysis is illustrated at first by means of a one-dimensional example showing the main features of the problem, subsequently results are obtained for the three-dimensional continuum. Finally, applications are proposed for problems of uniform strain state. Results emphasize the role of the stress state, acting in the examined equilibrium configuration, in the relation between the static and the dynamic material stability condition