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:
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 (formula_1), the Cauchy stress tensor formula_2 transforms as
Since formula_2 is a spatial quantity and the transformation follows the rules of tensor transformations, formula_2 is objective. However,
Therefore, the stress rate is not objective unless the rate of rotation is zero, i.e. formula_1 is constant.
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 formula_8. 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 formula_9. Consequently, it is necessary to introduce the so-called objective stress rate formula_10, or the corresponding increment formula_11. The objectivity is necessary for formula_10 to be functionally related to the element deformation. It means that formula_10 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).
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 formula_2 or the push-forward of formula_15 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 formula_19 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 formula_24 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 (or the Truesdell rate of the 
Cauchy stress). Recall that the Truesdell rate of the Cauchy stress is
given by
From the polar decomposition theorem we have
where formula_27 is the orthogonal rotation tensor (formula_28)
and formula_29 is the symmetric, positive definite, right stretch.
If we assume that formula_30 we get formula_31. Also since there is no
stretch formula_32 and we have formula_33. 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 (Truesdell rate). This rate has the form
The Zaremba-Jaumann rate is used widely in computations primarily for two reasons
Recall that the spin tensor formula_36 (the skew part of the velocity gradient)
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 formula_48 and
formula_49 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 (or linearized) strain increment tensor 
where formula_53 is the displacement increment of the continuum points. The time derivative 
is the strain rate tensor (also called the velocity strain) and formula_55 is the material point velocity or displacement rate. For finite strains, measures from the Seth–Hill family (also called Doyle–Ericksen tensors) can be used:
where formula_57 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 (or true) stress formula_59 and let formula_2 be the Cauchy stress in the final configuration. Let formula_61 be the work done (per unit initial volume) by the internal forces during an incremental deformation from this initial state. Then the variation formula_62 corresponds to the variation in the work done due to a variation in the displacement formula_63. The displacement variation has to satisfy the displacement boundary conditions.
Let formula_64 be an objective stress tensor in the initial configuration. Define the stress increment with respect to the initial configuration as formula_65. Alternatively, if formula_66 is the unsymmetric first Piola–Kirchhoff stress referred to the initial configuration, the increment in stress can be expressed as formula_67.
Variation of work done.
Then the variation in work done can be expressed as
where the finite strain measure formula_69 is energy conjugate to the stress measure formula_70. Expanded out, 
The objectivity of stress tensor formula_64 is ensured by its transformation as a second-order tensor under coordinate rotations (which causes the principal stresses to be independent from coordinate rotations) and by the correctness of formula_73 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 formula_78, we have 
We can also write the above equation as
Time derivatives.
The Cauchy stress and the first Piola-Kirchhoff stress are related by (see Stress measures)
For small incremental deformations, 
Therefore,
Substituting formula_84,
For small increments of stress formula_86 relative to the initial stress formula_59, the above reduces to
From equations (1) and (3) we have
Recall that formula_15 is an increment of the stress tensor measure formula_64.
Defining the stress rate
and noting that
we can write equation (4) as
Taking the limit at formula_95, and noting that formula_96 at this limit, one gets the following expression for the objective stress rate associated with the strain measure formula_69:
Here formula_99 = material rate of Cauchy stress (i.e., the rate in Lagrangian coordinates of the initial stressed state).
Work-conjugate stress rates.
A rate for which there exists no legitimate finite strain tensor formula_69 associated according to Eq. (6) is energetically inconsistent, i.e., its use violates energy balance (i.e., the first law of thermodynamics).
Evaluating Eq. (6) for general formula_101 and for formula_102, one gets a general expression for the objective stress rate:
where formula_104 is the objective stress rate associated with the Green-Lagrangian strain (formula_102).
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%).
Non work-conjugate stress rates.
Other rates, used in most commercial codes, which are not work-conjugate to any finite strain tensor are:
Objective rates and Lie derivatives.
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. 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 formula_111 are the tangential moduli (components of a 4th-order tensor) associated with strain tensor formula_112. They are different for different choices of formula_101, and are related as follows:
From the fact that Eq. (7) must hold true for any velocity gradient formula_115, it follows that:
where formula_117 are the tangential moduli associated with the Green–Lagrangian strain (formula_102), taken as a reference, formula_8 = current Cauchy stress, and formula_120 = Kronecker delta (or unit tensor).
Eq. (8) can be used to convert one objective stress rate to another. Since formula_121, the transformation
can further correct for the absence of the term formula_123 (note that the term formula_124 does not allow interchanging subscripts formula_125 with formula_126, which means that its absence breaks the major symmetry of the tangential moduli tensor formula_127).
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. (8) means that the nonlinear dependence of formula_128 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 formula_101, for which the moduli can be considered constant.