# 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

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

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


# Maximum principal stress – Lamé

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

This criteria is applicable to brittle materials.

# Maximum shear stress – Tresca

From Mohr's circle, based on the principal stresses , and , the largest shear stress is

leading to a yield criterion

# Effective stress – von Mises

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

The von Mises yield criterion is then

As with the Tresca criterion a hydrostatic state of stress will not result in yielding