Elastic shadow flow and its theoretical implications for inelastic solids

Abstract

Microstructural considerations have led to a constitutive format for the modeling of inelastic solids which does not make use of a fixed material reference in the usual sense. A principal motivation for this Eulerian-type' theoretical structure stems from a belief that large deformation inelastic flow phenomena (such as that encountered in certain metal forming processes) will one day prove amenable to numerical techniques of the sort common to the field of fluid mechanics. Unfortunately, virtually all established theoretical results of a general nature have been developed in a Lagrangian or referential context, and are not readily adapted to this new format. Here, this difficulty is partially overcome through the introduction of an instantaneous local reference corresponding to a material element's so called 'shadow' (elastically unstretched) configuration. A related state variable transformation and the definition of a shadow frame time derivative (analogous to the corotational time derivative) are then shown to facilitate a far more tractable theoretical reformulation. Throughout, three equivalent variants of this general theory are presented, one expressed in terms of a gradient type 'cell placement tensor', a second involving symmetric elastic stretch and orthogonal cell orientation tensors, and a third in which the aforementioned elastic stretch is replaced by the elastic (natural) log-strain tensor. In each case, the theoretical simplifications appropriate for various types of materials are enumerated. It is noteworthy that the latter two forms are instantly specialized for the most common class of structurally isotropic materials by merely dropping dependence on the cell orientation tensor. In all forms, basic thermodynamic considerations (second law) result in general expressions for true stress response in terms of energy derivatives, the rate of mechanical dissipation, and to necessary conditions (in the form of constitutive inequalities) for Il'iushin stability. Finally, the specific circumstances under which these inequalities reduce to the familiar Drucker-like' forms are identified.