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})~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a1a57334406446db9e517e10040b7d8fc3168c)
- In classical (small strain) rate-independent plasticity we start off with an additive decomposition :of the strain tensor

- Assuming linear elasticity, we have the following elastic stress-strain law

- Let us assume that the
theory applies during plastic deformation of the material. :Hence, the material obeys an associated flow rule

- where
is the plastic flow rate,
is the yield function,
is the temperature, and
is an internal variable.