Grain Size

The GrainSize class manages the laws estimating the stress based on the size of grains and the laws governing the evolution of their size. When this size is taken into account, both laws must be defined (the stress due to grain size is taken into account in the plastic criterion).

Perzyna model

Stress:

$$\sigma_d =K\,d^p\,\left (\dot{\bar{\varepsilon}}^{pl} \right )^m\,\left (\bar{\varepsilon}^{pl} \right )^n$$

Evolution of grain size :

$$\dot{d}=\dfrac{\alpha_1}{d^Q}+\dfrac{\alpha_2 \dot{\bar{\varepsilon}}^{pl}}{d^R}$$

Prandtl model

Stress:

$$\sigma_{d}= \dfrac{1}{\beta}\, \mbox{arcsinh} \left(\dfrac{\dot{\bar{\varepsilon}}^{pl}d^p}{A}\right)$$

Evolution of grain size :

$$\dot{d}= \dfrac{\alpha_1}{d^Q} + \dfrac{\alpha_2\dot{\bar{\varepsilon}}^{pl}}{d^R}$$