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

Problem 1: Part 8: Consistency condition - 2

The yield stress ${\displaystyle \sigma _{y}}$ is given by the Johnson-Cook model

${\displaystyle \sigma _{y}(\alpha ,T)=\left[\sigma _{0}+B\alpha ^{n}\right]\left[1-\left({\cfrac {T-T_{0}}{T_{m}-T_{0}}}\right)\right]}$

where ${\displaystyle \sigma _{0}}$ is the initial yield stress, ${\displaystyle B,n}$ are constants, ${\displaystyle T_{0}}$ is a reference temperature, and ${\displaystyle T_{m}}$ is the melt temperature. Derive expressions for ${\displaystyle \partial f/\partial \alpha }$, and ${\displaystyle \partial f/\partial T}$ for the von Mises yield condition with the Johnson-Cook flow stress model.

The yield function is

${\displaystyle f={\sqrt {\cfrac {3}{2}}}~{\sqrt {\mathbf {s} :\mathbf {s} }}-\sigma _{y}={\sqrt {\cfrac {3}{2}}}~{\sqrt {\mathbf {s} :\mathbf {s} }}-\left[\sigma _{0}+B\alpha ^{n}\right]\left[1-\left({\cfrac {T-T_{0}}{T_{m}-T_{0}}}\right)\right]}$

Therefore,

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

and

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