State variables and kinematics

We start by defining degrees of freedom in the system. Let be the reference configuration and be the current configuration at time . Let be the deformation map from to . The deformation gradient is denoted as , where the operator denotes differentiation with respect to . The deformations due to thermal expansion and plastic flow are modeled using the multiplicative decomposition

where , , and are referred to as the eigen, elastic, and plastic deformation gradients, respectively. For convenience, is defined as the mechanical deformation gradient.

In general, there exist constraints for the plastic flow and the effective plastic strain rate of the following form:

For example, the Prandtl-Reuss flow rule requires the plastic flow to be purely isochoric and normalized so that the plastic deformation is uniaxial:

In contrast, Tresca's flow rule is more restrictive in the sense that the plastic flow may only occur along the crystal slip system, i.e. :

In the phase-field approach to fracture, crack surfaces are regularized and modeled using a phase-field . Both the plastic flow and the crack evolution are considered irreversible, i.e.

The set of state variables are .