Nonlinear finite elements/Homework 11/Solutions/Problem 1/Part 11

Problem 1: Part 11: Discrete Kuhn-Tucker

The Kuhn-Tucker loading-unloading conditions are

{\dot {\gamma }}\geq 0~,~~f({\boldsymbol {\sigma }},\alpha ,T)\leq 0~;~~{\dot {\gamma }}f({\boldsymbol {\sigma }},\alpha ,T)=0~.

Write down a discrete form of the Kuhn-Tucker conditions.

The discrete form of the Kuhn-Tucker conditions is

{\begin{aligned}\Delta \gamma &\geq 0\\f({\boldsymbol {\sigma }}_{n+1},\alpha _{n+1},T_{n+1})&\leq 0\\\Delta \gamma ~f({\boldsymbol {\sigma }}_{n+1},\alpha _{n+1},T_{n+1})&=0\end{aligned}}