Objective stress rate

In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate (or the rate of deformation tensor). The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame-indifferent (objective). If the stress and strain measures are material quantities then objectivity is automatically satisfied. However, if the quantities are spatial, then the objectivity of the stress-rate is not guaranteed even if the strain-rate is objective.

There are numerous objective stress rates in continuum mechanics – all of which can be shown to be special forms of Lie derivatives. Some of the widely used objective stress rates are:

1. the Truesdell rate of the Cauchy stress tensor,
2. the Green–Naghdi rate of the Cauchy stress, and
3. the Zaremba-Jaumann rate of the Cauchy stress.

The adjacent figure shows the performance of various objective rates in a simple shear test where the material model is hypoelastic with constant elastic moduli. The ratio of the shear stress to the displacement is plotted as a function of time. The same moduli are used with the three objective stress rates. Clearly there are spurious oscillations observed for the Zaremba-Jaumann stress rate. This is not because one rate is better than another but because it is a misuse of material models to use the same constants with different objective rates. For this reason, a recent trend has been to avoid objective stress rates altogether where possible.