# Yield And Von Mises Stress

When a material is loaded the stress experienced by the material can be considered as two components a mean hydrostatic stress and a deviatoric stress.

The mean stress is defined as

$\sigma_{p} = \frac{\sigma_{x} + \sigma_{y} + \sigma_{z}}{3}$

The deviarotic stress is often represented by the von Mises stress is

$\sigma_{vM} = \frac{1}{\sqrt{2}}\left\lbrack \left( \sigma_{x} - \sigma_{y} \right)^{2} + \left( \sigma_{y} - \sigma_{z} \right)^{2} + \left( \sigma_{z} - \sigma_{x} \right)^{2} + 6\left( {\tau_{xy}}^{2} + {\tau_{yz}}^{2} + {\tau_{zx}}^{2} \right) \right\rbrack^{\frac{1}{2}}$

Yield depends on the type of material and the material model which represents the behaviour of that material. For the most commonly used structural materials, steel is not sensitive to pressure or mean hydrostatic stress while for concrete failure depends on the mean hydrostatic stress

For a material like steel the normal yeild condition is when

$\sigma_{vM} = \sigma_{yield}$

The yield and ultimate stress and the hardening parameters are not used in linear analysis. For nonlinear analysis in GSA, only the yield stress is used and ultimate stress and hardening parameters are ignored.

In general, material yielding follows the line defined by the hardening modulus. Either isotropic hardening $(\beta = 1)$ or kinematic hardening $(\beta = 0)$ can be defined.

If the hardening parameter, $\beta$ is 1 (isotropic hardening as shown above) the yield stress retains its maximum value on reversal of stress. A value of 0 corresponds to kinematic hardening, where the diameter of the yield surface remains constant so on reversal of stress the material yield when the stress reversal is twice the original yield stress.

For material like concrete a more advance material yield condition is required.