Summary

• Stress and strain and the relationship between them can be expressed in tensor formalism.
• The stress tensor is symmetric and can be separated into hydrostatic and deviatoric components.
• The stress state can be expressed by a tensor that has only diagonal components – the principal stress tensor. This is achieved by rotating the axes of the stress tensor, so that the axes are parallel to the forces on the body.
• The measured strain tensor can be separated into a symmetric real strain tensor and an antisymmetric rotation tensor. The real strain tensor can then be separated into dilatational (volume expansion) and deviatoric (shape change) components.
• We can define combinations of the three principal stress components that will cause yield – yield criteria. Different criteria are best used for different materials. The best one for metals is the von Mises yield criterion:


$${({\sigma _1} - {\sigma _2})^2} + {({\sigma _2} - {\sigma _3})^2} + {({\sigma _3} - {\sigma _1})^2} = 6{k^2} = 2{Y^2}$$


A mathematically simpler approximation to the von Mises yield criterion is the Tresca yield criterion:


$$\frac{{\left( {{\sigma _1} - {\sigma _3}} \right)}}{2} = k = \frac{Y}{2}$$

• If a yield criterion is plotted in 3D stress space, we have a yield surface.