Von Mises Stress and Strain

The von Mises Stress provides a measure of the shear, or distortional, stress in the material. This type of stress tends to cause yielding in metals. It is independent of the amount of hydrostatic stress (s1= s2= s3) action on the material.

The Von Mises Stress is identified in terms of the principal stresses as
svm=Ö{1/2[(s1- s2)2+( s1- s3)2+( s2- s3)2]}.

In a state of pure tension, say s11=s and all other stresses are zero, then svm=s. In a state of pure shear, say s12=t and all other stresses are zero, then svm=Ö3 t.

For materials, initial yielding can be expected when svm=sy, where sy is the tensile yield stress, or when svm=Ö3 ty, where ty is the yield stress in shear. For other materials, particularly frictional materials such as soil and concrete, the von Mises Stress may have no value in predicting yield or failure.

The von Mises Strain, or equivalent strain, is identified in terms of the principal strains as:

evm=Ö{2/9[(e1- e2)2+( e1- e3)2+( e2- e3)2]}.

This is intended to be a measure of plastic deformation, which is assumed to incompressible in metals. In this case, e11 +e22+e33=0 , which is also true when Poisson’s ratio is 0.5.

In a state of pure tension, say s11=s and all other stresses are zero, then evm=e11=s/E, provided that Poisson’s ratio is 0.5. In a state of pure shear, say s12=t and all other stresses are zero, then e_vm = Ö(1/3) g₁₂ = Ö(1/3) t/G, since pure shear satisfies the incompressibility requirement.

Note that for incompressible behavior, the von Mises stress and strain are energy conjugates. For example, s_vme_vm =((Ö3 t) (Ö(1/3) g₁₂) = t² / G, as expected.