Some perspectives on the dynamic history of a material element

Some kinematic concepts like MWCRPSH, extensional motion, and persistence of straining tensor are adapted to a dynamic framework. Then, a criterion based on the dynamics of a material is developed to classify motions. This criterion can be applied to second order tensor fields, Γ, chosen to describe the relation between material response, forces and kinematics. For example Γ can be the deviatoric part of the stress tensor or the conformation tensor of a microstructured material. It can also be a kinematic quantity such as the symmetric part of the velocity gradient. An aim of the present criterion is to provide a measure of the local instantaneous tendency of the material to persist on, for example, stressing the same material line or purely stretching a conformation tensor. The present criterion encompasses a large variety of materials including ones with no explicit constitutive equations, since it only needs the fields of tensor Γ and velocity vector to calculate the scalar field of the measurer considered. In order to construct the present criterion some important concepts are introduced. One of these concepts is a decomposition of a tensor into in-phase and out-of-phase   parts with respect to a second tensor. Another one is the TT-natural convected time derivative, DT()Dt. This operator, which can be applied to any second order tensor, is a time derivative measured from a frame attached to the eigenvectors of the symmetric part, TSTS, of a second order tensor TT. Although different kinds of measurers of the dynamic history can be used to calculate the persistence of a dynamic tensor, it is suggested that the one related to the evolution of the natural configuration of the material is special somehow. This conclusion is based on the fact that the evolution considered and the motivation for the study of dynamic persistency of a motion are related to the elastic response of the material.