.. _frame_indifference:

##################
Frame Indifference
##################

An important concept in the formulation of constitutive theories in large deformations is frame indifference, alternatively referred to as objectivity. Although somewhat mathematically involved, the concept of objectivity is fairly simple to understand physically.

When we write constitutive laws in their most general form, we seek to express tensorial quantities, such as stress and stress rate, in terms of kinematic tensorial quantities, most commonly strain and strain rate. The basic physical idea behind frame indifference is that this constitutive relationship should be unaffected by any rigid body motions of the material. Mathematically, we evaluate frame indifference by defining an alternative reference frame that is rotating and translating with respect to the coordinate system in which we pose the problem. For our constitutive description to make sense, the tensorial quantities we use (stress, stress rate, strain, and strain rate) should transform according to the laws of tensor calculus when subjected to a change in reference frame. If a given quantity does this, we say it is material frame indifferent, and if it does not, we say it is not properly invariant.

*****************************************
Objective Strain and Strain Rate Measures
*****************************************

Consider a motion, :math:`\varphi(\mathbf{X},t)`.  We imagine ourselves to be viewing this motion from another reference frame, denoted in the following by :math:`*`, which is related to the original spatial frame via

.. math::
   :label: frame_indifference:eq:01

   \mathbf{x}^{*} = \mathbf{c}(t) + \mathbf{Q}(t)\mathbf{x} ,

where :math:`\mathbf{x} = \varphi(\mathbf{X},t)`.  In :eq:`frame_indifference:eq:01`, :math:`\mathbf{c}(t)` and :math:`\mathbf{Q}(t)` are rigid body translation and rotation, respectively, between the original frame and observer :math:`*`. To observer :math:`*`, the motion appears as defined by

.. math::
   :label: frame_indifference:eq:02

   \mathbf{x}^{*} = \varphi^{*} (\mathbf{X},t)
               = \mathbf{c}(t) + \mathbf{Q}(t) \varphi(\mathbf{X},t) .

The time derivative of this motion equation gives the relationship between the deformation gradients in the two frames:

.. math::
   :label: frame_indifference:eq:03

   \mathbf{F}^{*} = \frac{\partial}{\partial \mathbf{X}} \varphi^{*}_t 
               = \mathbf{Q} \frac{\partial}{\partial\mathbf{X}} 
                 \varphi_t (\mathbf{X}) 
               = \mathbf{Q} \mathbf{F} .

The spatial velocity gradient :math:`\mathbf{L}^{*}` is then

.. math::
   :label: frame_indifference:eq:04

   \mathbf{L}^{*} = \nabla^{*}\mathbf{v}^{*} 
               = \dot{\mathbf{F}}^{*} (\mathbf{F}^{*})^{-1} 
               = \frac{d}{dt}(\mathbf{Q}\mathbf{F})(\mathbf{Q}\mathbf{F})^{-1} 
               = \left( \mathbf{Q} \dot{\mathbf{F}} \mathbf{F}^{-1} \mathbf{Q}^T 
                 + \dot{\mathbf{Q}} \mathbf{F} \mathbf{F}^{-1} \mathbf{Q}^T \right) ,

which simplifies to

.. math::
   :label: frame_indifference:eq:05

   \mathbf{L}^{*} = \mathbf{Q} \mathbf{L} \mathbf{Q}^T + \dot{\mathbf{Q}} \mathbf{Q}^T .

For :math:`\mathbf{L} = \nabla \mathbf{v}` to be objective, it would transform according to the laws of tensor transformation between the two frames, i.e., only the first term on the right-hand side of :eq:`frame_indifference:eq:05` would be present. Clearly, :math:`\mathbf{L} = \nabla \mathbf{v}` is *not* objective.

Examining the rate of deformation tensor :math:`\mathbf{D}^{*}`, on the other hand, one finds: 

.. math::
   :label: frame_indifference:eq:06

   \mathbf{D}^{*} = \frac{1}{2} \left( \mathbf{L}^{*} + (\mathbf{L}^{*})^{T} \right) 
               = \frac{1}{2} \left[ \mathbf{Q} \mathbf{L} \mathbf{Q}^{T} 
                 + \dot{\mathbf{Q}} \mathbf{Q}^T 
                 + \mathbf{Q} \mathbf{L}^T \mathbf{Q}^T 
                 + \mathbf{Q} \dot{\mathbf{Q}}^T \right] ,

where

.. math::
   :label: frame_indifference:eq:07

   \dot{\mathbf{Q}} \mathbf{Q}^T + \mathbf{Q} \dot{\mathbf{Q}}^T 
    = \frac{d}{dt} \left[ \mathbf{Q} \mathbf{Q}^T \right] 
    = \frac{d}{dt} \left[ \mathbf{I} \right] = 0 .

Hence, :eq:`frame_indifference:eq:06` simplifies to

.. math::
   :label: frame_indifference:eq:08

   \mathbf{D}^{*} = \frac{1}{2} \mathbf{Q} \left[ \mathbf{L} 
                 + \mathbf{L}^T \right] \mathbf{Q}^T 
               = \mathbf{Q} \mathbf{D} \mathbf{Q}^T ,

which shows us that :math:`\mathbf{D}` is objective.

Therefore we have a tensorial quantity for the spatial rate-of-strain that is objective. The question arises whether corresponding reference measures of rate are objective. It turns out that such material rates are automatically objective, since they do not change when superimposed rotations occur spatially. Consider, for example, the right Cauchy-Green tensor :math:`\mathbf{C}`:

.. math::
   :label: frame_indifference:eq:09

   \mathbf{C}^{*} = (\mathbf{F}^{*})^T (\mathbf{F}^{*}) 
               = \mathbf{F}^T \mathbf{Q}^T \mathbf{Q} \mathbf{F} 
               = \mathbf{C} .

Similarly, the time derivative of :eq:`frame_indifference:eq:09` simplifies to

.. math::
   :label: frame_indifference:eq:10

   \dot{\mathbf{C}^{*}} = \dot{\mathbf{C}} .

************
Stress Rates
************

Turning our attention to stress rates, examine the material time derivative of the Cauchy stress :math:`\mathbf{T}`:

.. math::
   :label: frame_indifference:eq:11

   \dot{\mathbf{T}} = \left[ \frac{d}{dt} (\mathbf{T} \circ \varphi_t) \right] 
                   \bullet \varphi^{-1}_t 
                 = \left( \frac{\partial \mathbf{T}}{\partial t} 
                   + \mathbf{v} \cdot \nabla \mathbf{T} \right) .

:math:`\mathbf{T}` is itself objective by its very definition as a tensorial quantity. Thus, we can write

.. math::
   :label: frame_indifference:eq:12

   \mathbf{T}^{*} =  \mathbf{Q} \mathbf{T} \mathbf{Q}^T .

Computing the material time derivative of :eq:`frame_indifference:eq:12` gives

.. math::
   :label: frame_indifference:eq:13

   \dot{\mathbf{T}}^{*} = \dot{\mathbf{Q}} \mathbf{T} \mathbf{Q}^T 
                       + \mathbf{Q} \dot{\mathbf{T}} \mathbf{Q}^T 
                       + \mathbf{Q} \mathbf{T} \dot{\mathbf{Q}}^T .

Since the first and third terms on the right-hand side of :eq:`frame_indifference:eq:13` do not, in general, cancel, we see that the material time derivative of the Cauchy stress :math:`\mathbf{T}` is *not* objective.

It therefore becomes critical to consider a frame indifferent measure of stress rate when formulating a constitutive description that requires a stress rate.  A multitude of such rates have been contrived; the interested reader is encouraged to consult Reference [:footcite:`marsdenHughes`] for a highly theoretical treatment. For our discussion here, we consider two such rates especially prevalent in the literature: the Jaumann rate and the Green-Naghdi rate. Both rates rely on roughly the same physical idea. Rather than taking the derivative of the Cauchy stress itself, we rotate the object from the spatial frame before taking the time derivative, so that the reference frame in which the time derivative is taken is the same for all frames related by the transformation in :eq:`frame_indifference:eq:01`.

For example, we consider the Jaumann rate of stress, which we denote here as :math:`\hat{\mathbf{T}}`. Its definition is given as

.. math::
   :label: frame_indifference:eq:14

   \hat{\mathbf{T}} = \dot{\mathbf{T}} - \mathbf{W} \mathbf{T} + \mathbf{T} \mathbf{W} ,

where :math:`\mathbf{W}=\mathbf{L}-\mathbf{D}`.  We can verify that this rate of stress is truly objective by considering the object as it would appear to observer :math:`*`:

.. math::
   :label: frame_indifference:eq:15

   \hat{\mathbf{T}}^{*} = \dot{\mathbf{T}}^{*} - \mathbf{W}^{*} \mathbf{T}^{*} 
                       + \mathbf{T}^{*} \mathbf{W}^{*}  .

The quantity :math:`\dot{\mathbf{T}}^{*}` is given by :eq:`frame_indifference:eq:13`, :math:`\mathbf{T}^{*}` is given by :eq:`frame_indifference:eq:12`, and :math:`\mathbf{W}^{*} ` is computed with the aid of :eq:`frame_indifference:eq:05` and :eq:`frame_indifference:eq:08`:

.. math::
   :label: frame_indifference:eq:16

   \mathbf{W}^{*} = \mathbf{L}^{*} - \mathbf{D}^{*} 
               = \mathbf{Q}\mathbf{L} \mathbf{Q}^{T} 
                 + \dot{\mathbf{Q}} \mathbf{Q}^{T} 
                 - \mathbf{Q} \mathbf{D} \mathbf{Q}^{T} .

Substituting these quantities into :eq:`frame_indifference:eq:15`, we find

.. math::
   :label: frame_indifference:eq:17

   \begin{aligned}
   \hat{\mathbf{T}}^{*}  & = \dot{\mathbf{Q}} \mathbf{T} \mathbf{Q}^{T}
                          + \mathbf{Q} \dot{\mathbf{T}} \mathbf{Q}^{T}
                          - \mathbf{Q} \mathbf{T} \dot{\mathbf{Q}}^{T}  \\
                   {} & - \left( \mathbf{Q}\mathbf{L} \mathbf{Q}^{T}
                          + \dot{\mathbf{Q}} \mathbf{Q}^{T}
                          - \mathbf{Q} \mathbf{D} \mathbf{Q}^{T} \right)
                            \mathbf{Q} \mathbf{T} \mathbf{Q}^{T} \\
                   {} & + \mathbf{Q} \mathbf{T} \mathbf{Q}^{T}
                            \left( \mathbf{Q}\mathbf{L} \mathbf{Q}^{T}
                          + \dot{\mathbf{Q}} \mathbf{Q}^{T}
                          - \mathbf{Q} \mathbf{D} \mathbf{Q}^{T} \right) .
   \end{aligned}

Canceling terms and using the fact that :math:`\dot{\mathbf{Q}}\mathbf{Q}^T = -\mathbf{Q} \dot{\mathbf{Q}}^T`, we can simplify :eq:`frame_indifference:eq:17` to

.. math::
   :label: frame_indifference:eq:18

   \hat{\mathbf{T}}^{*} = \mathbf{Q} 
                       \left[ \dot{\mathbf{T}} - \mathbf{W} \mathbf{T} 
                              + \mathbf{T} \mathbf{W} \right] \mathbf{Q}^{T} 
                     = \mathbf{Q} \hat{\mathbf{T}} \mathbf{Q}^T ,

which ensures us that, indeed, :math:`\hat{\mathbf{T}}` is objective.

By considering the Green-Naghdi rate we can gain more insight into how objective rates are defined.  The Green-Naghdi rate of Cauchy stress is defined via

.. math::
   :label: frame_indifference:eq:19

   \tilde{\mathbf{T}} = \mathbf{R} \dot{ \pmb{\mathit{T}} } \mathbf{R}^{T} ,

where :math:`\mathbf{R}` is the rotation tensor from the polar decomposition of :math:`\mathbf{F}`, and :math:`\pmb{\mathit{T}}` is the rotated Cauchy stress defined in :eq:`stress_measure:eq:14`.

We examine how the rotation tensor :math:`\mathbf{R}` transforms. Utilizing :eq:`frame_indifference:eq:03` and the polar decomposition, we get

.. math::
   :label: frame_indifference:eq:20

   \mathbf{F}^{*} = \mathbf{R}^{*} \mathbf{U}^{*} 
               = \mathbf{Q} \mathbf{F} = \mathbf{Q} \mathbf{R} \mathbf{U} .

We now note two things: first, the product :math:`\mathbf{Q}\mathbf{R}` is itself a proper orthogonal tensor; and second, the polar decomposition is unique for a given deformation gradient. Therefore, comparing the second and fourth terms of :eq:`frame_indifference:eq:20`, we must conclude

.. math::
   :label: frame_indifference:eq:21

   \mathbf{U}^{*} = \mathbf{U} ,

and

.. math::
   :label: frame_indifference:eq:22

   \mathbf{R}^{*} = \mathbf{Q} \mathbf{R} .

Using :eq:`frame_indifference:eq:22` and :eq:`frame_indifference:eq:19`, we can compute

.. math::
   :label: frame_indifference:eq:23

   \tilde{\mathbf{T}}^{*} = \mathbf{R}^{*} \dot{\pmb{\mathit{T}}}^{*} {\mathbf{R}^{*}}^T 
                       = \mathbf{Q} \mathbf{R} \dot{\pmb{\mathit{T}}}^{*} 
                         \mathbf{R}^T \mathbf{Q}^T .

Returning to the definition of :math:`\pmb{\mathit{T}}` in :eq:`stress_measure:eq:14` and incorporating  :eq:`frame_indifference:eq:12` and  :eq:`frame_indifference:eq:22`, we can write

.. math::
   :label: frame_indifference:eq:24

   \pmb{\mathit{T}}^{*} = {\mathbf{R}^{*}}^T \mathbf{T}^{*} \mathbf{R}^{*} 
                     = \mathbf{R}^T \mathbf{Q}^T (\mathbf{Q} \mathbf{T} 
                       \mathbf{Q}^T ) \mathbf{Q} \mathbf{R} 
                     = \mathbf{R}^T \mathbf{T} \mathbf{R} 
                     = \pmb{\mathit{T}} .

Therefore, the rotated stress tensor appears exactly the same in both frames of reference. It follows that

.. math::
   :label: frame_indifference:eq:25

   \dot{\pmb{\mathit{T}}}^{*}  = \dot{\pmb{\mathit{T}}} ,

which, when substituted into :eq:`frame_indifference:eq:23`, gives

.. math::
   :label: frame_indifference:eq:26

   \tilde{\mathbf{T}}^{*}  = \mathbf{Q} \mathbf{R} \dot{\pmb{\mathit{T}}} 
                          \mathbf{R}^T \mathbf{Q}^T 
                        = \mathbf{Q} \tilde{\mathbf{T}} \mathbf{Q}^T .

This is recognized as the properly objective transformation of :math:`\tilde{\mathbf{T}}`.

One may note that this result gives considerable insight into how objective rates can be constructed. In the current case, we transform the stress into the rotated configuration before computing its time derivative, and then we transform the result back to the spatial configuration. Since the rotated stress is exactly the same for all reference frames, related by :eq:`stress_measure:eq:01`, taking the time derivative of it and then transforming produces an objective tensor. This idea can be generalized as follows: construction of an objective rate of stress is achieved by considering the time derivative of a stress measure defined in a coordinate system that is rotating about some set of axes.  In fact, one can show that the Jaumann stress rate can be similarly interpreted.

Finally, the Green-Naghdi rate can be manipulated further to a form more closely resembling the form for the Jaumann rate (:eq:`frame_indifference:eq:14`). That is, we can write

.. math::
   :label: frame_indifference:eq:27

   \begin{aligned}
   \tilde{\mathbf{T}}  & = \mathbf{R} \frac{d}{dt} (\mathbf{R}^T \mathbf{T} \mathbf{R})
                        \mathbf{R}^T \\
                 {} & = \mathbf{R} \dot{\mathbf{R}}^T \mathbf{T} + \dot{\mathbf{T}}
                        + \mathbf{T} \dot{\mathbf{R}} \mathbf{R}^T \\
                 {} & = \dot{\mathbf{T}} + \pmb{\mathit{L}}^T \mathbf{T}
                        + \mathbf{T} \pmb{\mathit{L}} \\
                 {} & = \dot{\mathbf{T}} + \mathbf{T} \pmb{\mathit{L}}
                        - \pmb{\mathit{L}} \mathbf{T} ,
   \end{aligned}

where :eq:`deformation_rates:eq:19` is used to define :math:`\pmb{\mathit{L}}`, recalling also that this object is skew.

The reference for Chapter 9 is [:footcite:`marsdenHughes`]

.. raw::
   html

   <hr>

.. footbibliography::
