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

${\sigma }_y(\alpha ,T)=[{\sigma }_0+B{\alpha }^n][1-\left(\frac{T-T_0}{T_m-T_0}\right)]$

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

$f=\sqrt{\frac{3}{2}}\sqrt{𝐬:𝐬}-{\sigma }_y=\sqrt{\frac{3}{2}}\sqrt{𝐬:𝐬}-[{\sigma }_0+B{\alpha }^n][1-\left(\frac{T-T_0}{T_m-T_0}\right)]$

Therefore,

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

and

$\frac{\partial f}{\partial T}=f_T=\left(\frac{1}{T_m-T_0}\right)[{\sigma }_0+B{\alpha }^n].$