

A Consistent Thermodynamic Theory of Inelasticity Resulting in Gradients in Dislocation Density with Naturally Evolving Length Scales
Abstract
A structure for an internal state viable description of inelastic deformation of crystals is developed. The deformation gradient is multiplicatively decomposed into an elastic deformation resulting from externally applied loads and the deformations associated with each density of defects to be included as internal state variables. The deformations associated with these defects or foreign atoms such as diffusing species, may further be decomposed into elastic and plastic parts depending upon the structure of the defect. In many cases, either the elastic or plastic part of a particular deformation gradient will be negligible, depending upon the situation. The appropriate elastic strain-like variable associated with the defects is included in the free energy resulting in conjugate thermodynamic forces (internal stresses) that must be included in the dissipation inequality. In addition, these forces (stresses) are required to satisfy micro or meso scale linear and angular balance laws. All transport equations (e.g. heat conduction or diffusing species) are derived from a combination of the energy balance and these force balance laws. This is in contrast to classic state variable theories in which only temporal evolution equations were specified for the internal state variables. Constraint equations are required for the extra kinematic degrees of freedom that are introduced. These are based upon the physics of the associated defect density/state variable an example given by the flow rule or plastic velocity gradient based upon the Orowan equation relating plastic strain rate to mobile dislocation density and velocity. Examples are given for theories ranging from simple statistically stored dislocations, coupled transport theories for hydrogen and micropolar type theories for asymmetric defects