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

where

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