# 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
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