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. The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame-indifferent. 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.

Non-objectivity of the time derivative of Cauchy stress

Under rigid body rotations, the Cauchy stress tensor transforms as
Since is a spatial quantity and the transformation follows the rules of tensor transformations, is objective. However,
Therefore, the stress rate is not objective unless the rate of rotation is zero, i.e. is constant.
For a physical understanding of the above, consider the situation shown in Figure 1. In the figure the components of the Cauchy stress tensor are denoted by the symbols. This tensor, which describes the forces on a small material element imagined to be cut out from the material as currently deformed, is not objective at large deformations because it varies with rigid body rotations of the material. The material points must be characterized by their initial Lagrangian coordinates. Consequently, it is necessary to introduce the so-called objective stress rate, or the corresponding increment. The objectivity is necessary for to be functionally related to the element deformation. It means that 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.

Truesdell stress rate of the Cauchy stress

The relation between the Cauchy stress and the 2nd P-K stress is called
the Piola transformation. This transformation can be
written in terms of the pull-back of or the push-forward of as
The Truesdell rate of the Cauchy stress is the Piola transformation of the material time derivative of the 2nd P-K stress. We thus define
Expanded out, this means that
where the Kirchhoff stress and the Lie derivative of
the Kirchhoff stress is
This expression can be simplified to the well known expression for the Truesdell rate of the Cauchy stress
It can be shown that the Truesdell rate is objective.

Truesdell rate of the Kirchhoff stress

The Truesdell rate of the Kirchhoff stress can be obtained by noting that
and defining
Expanded out, this means that
Therefore, the Lie derivative of is the same as the Truesdell rate of the Kirchhoff stress.
Following the same process as for the Cauchy stress above, we can show that

Green-Naghdi rate of the Cauchy stress

This is a special form of the Lie derivative. Recall that the Truesdell rate of the Cauchy stress is
given by
From the polar decomposition theorem we have
where is the orthogonal rotation tensor
and is the symmetric, positive definite, right stretch.
If we assume that we get. Also since there is no
stretch and we have. 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. Therefore,
We can show that this expression can be simplified to the
commonly used form of the Green-Naghdi rate
The Green–Naghdi rate of the Kirchhoff stress also has the form since the stretch is not taken into consideration, i.e.,

Zaremba-Jaumann rate of the Cauchy stress

The Zaremba-Jaumann rate of the Cauchy stress is a further specialization of the
Lie derivative. This rate has the form
The Zaremba-Jaumann rate is used widely in computations primarily for two reasons
  1. it is relatively easy to implement.
  2. it leads to symmetric tangent moduli.
Recall that the spin tensor
can be expressed as
Thus for pure rigid body motion
Alternatively, we can consider the case of proportional loading when
the principal directions of strain remain constant. An example of this
situation is the axial loading of a cylindrical bar. In that situation,
since
we have
Also,
Therefore,
This once again gives
In general, if we approximate
the Green–Naghdi rate becomes the Zaremba-Jaumann rate of the Cauchy stress

Other objective stress rates

There can be an infinite variety of objective stress rates. One of these
is the Oldroyd stress rate
In simpler form, the Oldroyd rate is given by
If the current configuration is assumed to be the reference configuration then
the pull back and push forward operations can be conducted using and
respectively. The Lie derivative of the Cauchy stress is then
called the convective stress rate
In simpler form, the convective rate is given by

Objective stress rates in finite strain inelasticity

Many materials undergo inelastic deformations caused by plasticity and damage. These material behaviors cannot be described in terms of a potential. It is also often the case that no memory of the initial virgin state exists, particularly when large deformations are involved. The constitutive relation is typically defined in incremental form in such cases to make the computation of stresses and deformations easier.

The incremental loading procedure

For a small enough load step, the material deformation can be characterized by the small strain increment tensor
where is the displacement increment of the continuum points. The time derivative
is the strain rate tensor and is the material point velocity or displacement rate. For finite strains, measures from the Seth–Hill family can be used:
where is the right stretch. A second-order approximation of these tensors is

Energy-consistent objective stress rates

Consider a material element of unit initial volume, starting from an initial state under initial Cauchy stress and let be the Cauchy stress in the final configuration. Let be the work done by the internal forces during an incremental deformation from this initial state. Then the variation corresponds to the variation in the work done due to a variation in the displacement. The displacement variation has to satisfy the displacement boundary conditions.
Let be an objective stress tensor in the initial configuration. Define the stress increment with respect to the initial configuration as. Alternatively, if is the unsymmetric first Piola–Kirchhoff stress referred to the initial configuration, the increment in stress can be expressed as.

Variation of work done

Then the variation in work done can be expressed as
where the finite strain measure is energy conjugate to the stress measure. Expanded out,
The objectivity of stress tensor is ensured by its transformation as a second-order tensor under coordinate rotations and by the correctness of as a second-order energy expression.
From the symmetry of the Cauchy stress, we have
For small variations in strain, using the approximation
and the expansions
we get the equation
Imposing the variational condition that the resulting equation must be valid for any strain gradient, we have
We can also write the above equation as

Time derivatives

The Cauchy stress and the first Piola-Kirchhoff stress are related by
For small incremental deformations,
Therefore,
Substituting,
For small increments of stress relative to the initial stress, the above reduces to
From equations and we have
Recall that is an increment of the stress tensor measure.
Defining the stress rate
and noting that
we can write equation as
Taking the limit at, and noting that at this limit, one gets the following expression for the objective stress rate associated with the strain measure :
Here = material rate of Cauchy stress.

Work-conjugate stress rates

A rate for which there exists no legitimate finite strain tensor associated according to Eq. is energetically inconsistent, i.e., its use violates energy balance.
Evaluating Eq. for general and for, one gets a general expression for the objective stress rate:
where is the objective stress rate associated with the Green-Lagrangian strain.
In particular,
.

Non work-conjugate stress rates

Other rates, used in most commercial codes, which are not work-conjugate to any finite strain tensor are:
The objective stress rates could also be regarded as the Lie derivatives of various types of stress tensor and their linear combinations. The Lie derivative does not include the concept of work-conjugacy.

Tangential stiffness moduli and their transformations to achieve energy consistency

The tangential stress-strain relation has generally the form
where are the tangential moduli associated with strain tensor. They are different for different choices of, and are related as follows:
From the fact that Eq. must hold true for any velocity gradient, it follows that:
where are the tangential moduli associated with the Green–Lagrangian strain, taken as a reference, = current Cauchy stress, and = Kronecker delta.
Eq. can be used to convert one objective stress rate to another. Since, the transformation
can further correct for the absence of the term .
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. What Eq. means that the nonlinear dependence of on the stress must be different for different objective stress rates. Yet none of them is fundamentally preferable, except if there exists one stress rate, one, for which the moduli can be considered constant.