Yield and Failure

Analysis using elastic properties is only applicable when the stress remains below the yield stress. However yield stress is a single value while the stress state is a tensor. The simplest case is for a material in a uni-axial stress state where the material yields when

$$\sigma_{xx} = \sigma_y$$

When there is a general (multi-axial) stress state there are a number of possible yield (or failure) criteria.

Principal stresses

The general stress tensor is

$$\sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \\ \end{bmatrix}$$

The principal stresses are derived by rotating the stress tensor so that the shear stresses are zero resulting in a set of principal stresses

$$\sigma = \begin{bmatrix} \sigma_{I} & 0 & 0 \\ 0 & \sigma_{II} & 0 \\ 0 & 0 & \sigma_{III} \\ \end{bmatrix}$$

where

$$\sigma_I > \sigma_{II} > \sigma_{III}$$

Maximum principal stress – Lamé

The simplest of these if that the material yields when the maximum principal stress reaches the yield stress.

$$\sigma_I = \sigma_y$$

This criteria is applicable to brittle materials.

Maximum shear stress – Tresca

From Mohr's circle, based on the principal stresses $\sigma_I$, $\sigma_{II}$ and $\sigma_{III}$, the largest shear stress is

$$\tau_y = max\left(\dfrac{\sigma_I - \sigma_{II}}{2}, \dfrac{\sigma_{II} - \sigma_{III}}{2}, \dfrac{\sigma_I - \sigma_{III}}{2}\right)$$

leading to a yield criterion

$$max\left((\sigma_I - \sigma_{II}), (\sigma_{II} - \sigma_{III}), (\sigma_I - \sigma_{III})\right) = \sigma_y$$

Effective stress – von Mises

Using the principal stresses an effective (distortional) stress can be defined as

$$\sigma_e = \sqrt{\dfrac{(\sigma_I - \sigma_{II})^2 + (\sigma_{II} - \sigma_{III})^2 + (\sigma_{III} - \sigma_I)^2}{2}}$$

The von Mises yield criterion is then

$$\sqrt{\dfrac{(\sigma_I - \sigma_{II})^2 + (\sigma_{II} - \sigma_{III})^2 + (\sigma_{III} - \sigma_I)^2}{2}} = \sigma_y$$

As with the Tresca criterion a hydrostatic state of stress $\sigma_I = \sigma_{II} = \sigma_{III}$ will not result in yielding