Objective stress rate
From Wikipedia, the free encyclopedia
In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference.[1] 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:
- the Truesdell rate of the Cauchy stress tensor,
- the Green–Naghdi rate of the Cauchy stress, and
- the Zaremba-Jaumann rate of the Cauchy stress. [2]
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.[3] 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.[4] For this reason, a recent trend has been to avoid objective stress rates altogether where possible.[citation needed]