Dual Variables

Abstract

The general theory of material behaviour shows that wide scope exists for formulating material models, neatly contained in a perceivably well-ordered system: the categories of material behaviour — elasticity, viscoelasticity, plasticity and viscoplasticity — demand differently structured functionals and evolution equations combined with an appropriate regard for the properties of material symmetry and kinematic conditions of constraint. One of the most important conclusions resulting from the assumption of objectivity is the fact that the Green strain tensor E (or the right Cauchy-Green tensor C = 1 + 2E) has to be joined to the 2nd Piola-Kirchhoff tensor $\tilde T$ to guarantee invariance when the reference frame changes (or when undergoing of superimposed rigid body motions). Since the material time derivatives of these tensors, in addition to E and $\tilde T$, also remain unaltered during a change of frame, it is in accordance with the assumption of objectivity to use the material time derivatives to establish evolution equations for internal variables, which are either of Green strain or Piola-Kirchhoff stress type. In view of the material description’s independence of the reference system, there should not be any objections, therefore, to using exclusively type E, $\tilde T$, or Ė, $\dot \tilde T$ tensors, in both material functionals and evolution equations.