Objective stress rate
[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).
If the stress and strain measures are material quantities then objectivity is automatically satisfied.
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 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.
Clearly there are spurious oscillations observed for the Zaremba-Jaumann stress rate.
[4] For this reason, a recent trend has been to avoid objective stress rates altogether where possible.
For a physical understanding of the above, consider the situation shown in Figure 1.
In the figure the components of the Cauchy (or true) stress tensor are denoted by the symbols
must be invariant with respect to coordinate transformations, particularly the rigid-body rotations, and must characterize the state of the same material element as it deforms.
The objective stress rate can be derived in two ways: While the former way is instructive and provides useful geometric insight, the latter way is mathematically shorter and has the additional advantage of automatically ensuring energy conservation, i.e., guaranteeing that the second-order work of the stress increment tensor on the strain increment tensor be correct (work conjugacy requirement).
The Truesdell rate of the Kirchhoff stress can be obtained by noting that
This is a special form of the Lie derivative (or the Truesdell rate of the Cauchy stress).
Note that this doesn't mean that there is not stretch in the actual body - this simplification is just for the purposes of defining an objective stress rate.
The Green–Naghdi rate of the Kirchhoff stress also has the form since the stretch is not taken into consideration, i.e.,
The Zaremba-Jaumann rate is used widely in computations primarily for two reasons Recall that the spin tensor
Alternatively, we can consider the case of proportional loading when the principal directions of strain remain constant.
Many materials undergo inelastic deformations caused by plasticity and damage.
It is also often the case that no memory of the initial virgin state exists, particularly when large deformations are involved.
[9] The constitutive relation is typically defined in incremental form in such cases to make the computation of stresses and deformations easier.
For finite strains, measures from the Seth–Hill family (also called Doyle–Ericksen tensors) can be used:
Imposing the variational condition that the resulting equation must be valid for any strain gradient
at this limit, one gets the following expression for the objective stress rate associated with the strain measure
A rate for which there exists no legitimate finite strain tensor
(6) is energetically inconsistent, i.e., its use violates energy balance (i.e., the first law of thermodynamics).
In particular, (Note that m = 2 leads to Engesser's formula for critical load in shear buckling, while m = -2 leads to Haringx's formula which can give critical loads differing by >100%).
Other rates, used in most commercial codes, which are not work-conjugate to any finite strain tensor are:[8] The objective stress rates could also be regarded as the Lie derivatives of various types of stress tensor (i.e., the associated covariant, contravariant and mixed components of Cauchy stress) and their linear combinations.
, which means that its absence breaks the major symmetry of the tangential moduli tensor
Large strain often develops when the material behavior becomes nonlinear, due to plasticity or damage.
Then the primary cause of stress dependence of the tangential moduli is the physical behavior of material.
