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

${{\frac {\partial f}{\partial T}}=f_{T}=\left({\cfrac {1}{T_{m}-T_{0}}}\right)\left[\sigma _{0}+B\alpha ^{n}\right]~.}$

Nonlinear finite elements/Homework11/Solutions/Problem 1