# Nonlinear finite elements/Homework 11/Solutions

• Problem 1: Small Strain Elastic-Plastic Behavior
Given:
For small strains, the strain tensor is given by
${\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{T}\right]\qquad {\text{or}}\qquad \varepsilon _{ij}={\frac {1}{2}}(u_{i,j}+u_{j,i})~.}$
In classical (small strain) rate-independent plasticity we start off with an additive decomposition :of the strain tensor
${\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}^{e}+{\boldsymbol {\varepsilon }}^{p}\qquad {\text{or}}\qquad \varepsilon _{ij}=\varepsilon _{ij}^{e}+\varepsilon _{ij}^{p}~.}$
Assuming linear elasticity, we have the following elastic stress-strain law
${\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\mathsf {C}}}:{\boldsymbol {\varepsilon }}^{e}\qquad {\text{or}}\qquad \sigma _{ij}=C_{ijkl}\varepsilon _{kl}^{e}~.}$
Let us assume that the ${\displaystyle J_{2}}$ theory applies during plastic deformation of the material. :Hence, the material obeys an associated flow rule
${\displaystyle {\text{(1)}}\qquad {\dot {\boldsymbol {\varepsilon }}}^{p}={\dot {\gamma }}{\frac {\partial f({\boldsymbol {\sigma }},\alpha ,T)}{\partial {\boldsymbol {\sigma }}}}}$
where ${\displaystyle {\dot {\gamma }}}$ is the plastic flow rate, ${\displaystyle f}$ is the yield function, ${\displaystyle T}$ is the temperature, and ${\displaystyle \alpha }$ is an internal variable.