Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 8

Problem 1: Part 8: Consistency condition - 2

The yield stress $\sigma_y$ is given by the Johnson-Cook model

$\sigma_y(\alpha,T) = \left[\sigma_0 + B \alpha^n\right] \left[1 - \left(\cfrac{T - T_0}{T_m -T_0}\right)\right]$

where $\sigma_0$ is the initial yield stress, $B, n$ are constants, $T_0$ is a reference temperature, and $T_m$ is the melt temperature. Derive expressions for $\partial f/\partial \alpha$, and $\partial f/ \partial T$ for the von Mises yield condition with the Johnson-Cook flow stress model.

The yield function is

$f = \sqrt{\cfrac{3}{2}}~\sqrt{\mathbf{s}:\mathbf{s}} - \sigma_y= \sqrt{\cfrac{3}{2}}~\sqrt{\mathbf{s}:\mathbf{s}} - \left[\sigma_0 + B \alpha^n\right] \left[1 - \left(\cfrac{T - T_0}{T_m -T_0}\right)\right]$

Therefore,

${ \frac{\partial f}{\partial \alpha} = f_{\alpha} = - n~B~\alpha^{n-1} \left[1 - \left(\cfrac{T - T_0}{T_m -T_0}\right)\right] }$

and

${ \frac{\partial f}{\partial T} = f_T = \left(\cfrac{1}{T_m - T_0}\right) \left[\sigma_0 + B \alpha^n\right] ~. }$