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] ~. } $$