# 5.2: Deviatoric (von Mises) and Hydrostatic Stresses and Strains


Plastic deformation of metals is stimulated solely by the deviatoric (shape-changing) component of the stress state, often termed the von Mises stress, and is unaffected by the hydrostatic component. This is consistent with the fact that plastic deformation (of metals) occurs at constant volume. It follows that the material response (stress-strain relationship) should be the same in tension and compression. This is basically correct, although the difference between true and nominal stresses and strains should be noted (see next page), as should the possible effects of necking in tension and of friction (leading to barrelling) in compression - see following pages.

The von Mises stress is given by:

$\sigma_{\mathrm{VM}}=\sqrt{\frac{\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2}-\sigma_{3}\right)^{2}+\left(\sigma_{3}-\sigma_{1}\right)^{2}}{2}}$

where $$\sigma_1$$, $$\sigma_2$$, and $$\sigma_3$$ are the principal stresses (see the TLP on Stress Analysis). It can thus be seen that the von Mises stress is a scalar quantity. The hydrostatic stress can be written

$\sigma_{\mathrm{H}}=\frac{\sigma_{1}+\sigma_{2}+\sigma_{3}}{3}$

This is also a scalar. In the simulation below, the slider bars can be used to change the principal stresses. The von Mises and hydrostatic stresses are then displayed.

Simulation 1: Von Mises and Hydrostatic Stresses

Under simple uniaxial tension or compression, the von Mises stress is equal to the applied stress, while the hydrostatic stress is equal to one third of it. The von Mises stress is always positive, while the hydrostatic stress can be positive or negative. It’s not appropriate to think of the von Mises stress as being “tensile”, as one would if it were a normal stress (with a positive sign). It’s effectively a type of (volume-averaged) shear stress. Shear stresses do not really have a sign, but it’s conventional to treat them as positive, as indeed is done for the von Mises stress.

It’s also possible to identify deviatoric and hydrostatic components of the (plastic) strain state. Analogous equations to those above are used to obtain these values. The von Mises strain is often termed the “equivalent plastic strain”. Again, it always has a positive sign, but this does not mean that it is a “tensile” strain. The hydrostatic plastic strain, on the other hand, always has a value of zero. This follows from the fact that plastic strain does not involve a change in volume. (This is not true of elastic strains, which do in general involve a volume change.)