Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 8
Problem 1: Part 8: Consistency condition - 2
editThe yield stress 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20acd46f1b209919acb430425ac1ab381b8e9462)
where is the initial yield stress, are constants, is a reference temperature, and is the melt temperature. Derive expressions for , and 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3425aaf6a2bd98eb2ddac6675411789c03f5bd91)
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]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0034bcce800010bf26cd51d09fb349f8c8832eab)
and
![{\displaystyle {{\frac {\partial f}{\partial T}}=f_{T}=\left({\cfrac {1}{T_{m}-T_{0}}}\right)\left[\sigma _{0}+B\alpha ^{n}\right]~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebde5331943161926373d83520a64d861abebebe)