An investigation on static and dynamic criteria of constitutive stability

The stability characterization of the constitutive incremental behavior of a material in a state of large static deformation is studied. This aspect is of notable interest especially for advanced composite materials which are usually characterized by high anisotropy, soft components and plasticity or damage. Incremental linear and piecewise-linear constitutive relations are considered. Constitutive ( also referred as material) stability may be defined by means of static and dynamic methodologies, producing in different conditions. According to a static criterion, constitutive stability corresponds to the positive definiteness of the scalar product of a strain rate and the conjugated stress rate. On the other hand, dynamic conditions of constitutive stability are usually formulated in the context of propagation of infinitesimal plane or acceleration waves in an infinite body. These conditions are similar to those excluding the emergence of localized instability modes of the shear band type. Interrelations between static and dynamic constitutive stability conditions are established. A static constitutive stability condition corresponding to the Biot strain-rate is obtained and the differences with other constitutive stability conditions are pointed out. The analysis is illustrated firstly by means of a one-dimensional example: subsequently results are generalized to the three-dimensional continuum. Finally, applications are proposed for problems of uniform strain state and representative material models. The paper shows that the relations between the static and dynamic constitutive stability conditions are governed by the combination of two factors: the stress state, present in the examined equilibrium configuration, and the algebraic form of the admissible incremental displacement gradient.