# Elastic Plastic with Kinematic Hardening Material Model

I have a model which calculates the rate of deformation tensor using the Green-Lagrange strain rate, the material velocity gradient and the gradient of deformation tensor.

$$\bar{\bar{D}} = \bar{\bar{F}}^{-T}\dot{E}{\bar{\bar{F}}}^{-1}$$

I then used a constitutive model to calculate the true stress rate and converted this into the 1st Piola-Kirchoff stress using the Jacobian and gradient of deformation tensor. This worked well with a simple linear elastic constitutive model.

I am now trying to implement an elastic plastic with kinematic hardening constitutive model. I am basing it on the theory found here on page 273 (19.15). My problem is that it requires an effective plastic strain which is calculated as:

$$\epsilon_{eff}^{p} = \int_0^t ({\frac{2}{3}\dot{\epsilon}_{ij}^{p}\dot{\epsilon}_{ij}^{p}})^\frac{1}{2}dt$$

And this plastic strain rate is given as the total strain rate minus the elastic:

$$\dot{\epsilon}_{ij}^{p} = \dot{\epsilon}_{ij} - \dot{\epsilon}_{ij}^{e}$$

This leads me to believe that the strain rates are interchangeable with the rate of deformation tensors in this calculation. If this is true then the total rate of deformation is calculated as above. My question is, in this case, how to I calculate the elastic component of the rate of deformation tensor?