Many constitutive equations are given in rate form as the relation between a
stress rate and a strain rate (or the rate of deformation). We would like our
constitutive equations to 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.
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,

or,

Therefore the stress rate is not objective unless the rate of rotation
is zero, i.e.
is constant.
There are numerous objective stress rates in the literature on continuum
mechanics - all of which can be shown to be special forms of Lie derivatives.
However, we will focus on three which are widely used.
- The Truesdell rate
- The Green-Naghdi rate
- The Jaumann rate
Truesdell stress rate of the Cauchy stress[edit | edit source]
The relation between the Cauchy stress and the 2nd P-K stress is called
the Piola transformation. Recall that this transformation can be
written in terms of the pull-back of
or the push-forward of
as
![{\displaystyle {\boldsymbol {S}}=J~\phi ^{*}[{\boldsymbol {\sigma }}]~;~~{\boldsymbol {\sigma }}=J^{-1}~\phi _{*}[{\boldsymbol {S}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32a0fc1c8e641da74680ad94e60d2a320efbf686)
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
![{\displaystyle {\overset {\circ }{\boldsymbol {\sigma }}}=J^{-1}~\phi _{*}[{\dot {\boldsymbol {S}}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ec757173cee76d92f820ffd1576d08c2558ab1)
Expanded out, this means that
![{\displaystyle {\overset {\circ }{\boldsymbol {\sigma }}}=J^{-1}~{\boldsymbol {F}}\cdot {\dot {\boldsymbol {S}}}\cdot {\boldsymbol {F}}^{T}=J^{-1}~{\boldsymbol {F}}\cdot \left[{\cfrac {d}{dt}}\left(J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}\right)\right]\cdot {\boldsymbol {F}}^{T}=J^{-1}~{\mathcal {L}}_{\varphi }[{\boldsymbol {\tau }}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a13cd7fcf157924ff74053f2ce0cae6352ccc020)
where the Kirchhoff stress
and the Lie derivative of
the Kirchhoff stress is
![{\displaystyle {\mathcal {L}}_{\varphi }[{\boldsymbol {\tau }}]={\boldsymbol {F}}\cdot \left[{\cfrac {d}{dt}}\left({\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\tau }}\cdot {\boldsymbol {F}}^{-T}\right)\right]\cdot {\boldsymbol {F}}^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7eac7ab5d3f3c209142d8526c2e1a6492ec84ec3)
This expression can be simplified to the well known expression for the
Truesdell rate of the Cauchy stress
Truesdell rate of the Cauchy stress

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Proof:
We start with
![{\displaystyle {\overset {\circ }{\boldsymbol {\sigma }}}=J^{-1}~{\boldsymbol {F}}\cdot \left[{\cfrac {d}{dt}}\left(J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}\right)\right]\cdot {\boldsymbol {F}}^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06192ed96d64616c6a3960e8a24eae30a96ea4f6)
Expanding the derivative inside the square brackets, we get

or,

Now,

Therefore,

or,

where the velocity gradient
.
Also, the rate of change of volume is given by

where
is the rate of deformation tensor.
Therefore,

or,

You can easily show that the Truesdell rate is objective.
Truesdell rate of the Kirchhoff stress[edit | edit source]
The Truesdell rate of the Kirchhoff stress can be obtained by noting that
![{\displaystyle {\boldsymbol {S}}=\phi ^{*}[{\boldsymbol {\tau }}]~;~~{\boldsymbol {\tau }}=\phi _{*}[{\boldsymbol {S}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf443117690b959c0c840fac140fbf8d385fca5)
and defining
![{\displaystyle {\overset {\circ }{\boldsymbol {\tau }}}=\phi _{*}[{\dot {\boldsymbol {S}}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/287a08eebc89dfc4c2a071c44238b9cc0dfca7f5)
Expanded out, this means that
![{\displaystyle {\overset {\circ }{\boldsymbol {\tau }}}={\boldsymbol {F}}\cdot {\dot {\boldsymbol {S}}}\cdot {\boldsymbol {F}}^{T}={\boldsymbol {F}}\cdot \left[{\cfrac {d}{dt}}\left({\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\tau }}\cdot {\boldsymbol {F}}^{-T}\right)\right]\cdot {\boldsymbol {F}}^{T}={\mathcal {L}}_{\varphi }[{\boldsymbol {\tau }}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e487613ed28659fb19d8619bfe3d7a4aed3f4898)
Therefore, the Lie derivative of
is the same as the Truesdell rate of the Kirchhoff stress.
FFollowing the same process as for the Cauchy stress above, we can show that
Truesdell rate of the Kirchhoff stress

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Green-Naghdi rate of the Cauchy stress[edit | edit source]
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
![{\displaystyle {\overset {\circ }{\boldsymbol {\sigma }}}=J^{-1}~{\boldsymbol {F}}\cdot \left[{\cfrac {d}{dt}}\left(J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}\right)\right]\cdot {\boldsymbol {F}}^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06192ed96d64616c6a3960e8a24eae30a96ea4f6)
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
![{\displaystyle {\overset {\circ }{\boldsymbol {\sigma }}}={\boldsymbol {R}}\cdot \left[{\cfrac {d}{dt}}\left({\boldsymbol {R}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {R}}^{-T}\right)\right]\cdot {\boldsymbol {R}}^{T}={\boldsymbol {R}}\cdot \left[{\cfrac {d}{dt}}\left({\boldsymbol {R}}^{T}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {R}}\right)\right]\cdot {\boldsymbol {R}}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd338b972f8bdf47e0fbdc189b89d75904a6b92)
We can show that this expression can be simplified to the
commonly used form of the Green-Naghdi rate
Green-Naghdi rate of the Cauchy stress

where .
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The Green-Naghdi rate of the Kirchhoff stress also has the form since the
stretch is not taken into consideration, i.e.,

Proof:
Expanding out the derivative

or,

Now,

Therefore,

If we define the angular velocity as

we get the commonly used form of the Green-Naghdi rate

Jaumann rate of the Cauchy stress[edit | edit source]
The Jaumann rate of the Cauchy stress is a further specialization of the
Lie derivative (Truesdell rate). This rate has the form
Jaumann rate of the Cauchy stress

where is the spin tensor.
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The Jaumann rate is used widely in computations primarily for two reasons
- it is relatively easy to implement.
- it leads to symmetric tangent moduli.
Recall that the spin tensor
(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
![{\displaystyle {\boldsymbol {U}}=\left[{\begin{array}{ccc}\lambda _{X}\\&\lambda _{Y}\\&&\lambda _{Z}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f72d68b9b45d52cb594a840dd8025ac6d039c75)
we have
![{\displaystyle {\dot {\boldsymbol {U}}}=\left[{\begin{array}{ccc}{\dot {\lambda }}_{X}\\&{\dot {\lambda }}_{Y}\\&&{\dot {\lambda }}_{Z}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2795874d1c862f28d143b4825378bcddbbade701)
Also,
![{\displaystyle {\boldsymbol {U}}^{-1}=\left[{\begin{array}{ccc}1/\lambda _{X}\\&1/\lambda _{Y}\\&&1/\lambda _{Z}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1cf72694de7fca85cfe38569adfa21988a21ae)
Therefore,
![{\displaystyle {\dot {\boldsymbol {U}}}\cdot {\boldsymbol {U}}^{-1}=\left[{\begin{array}{ccc}{\dot {\lambda }}_{X}/\lambda _{X}\\&{\dot {\lambda }}_{Y}/\lambda _{Y}\\&&{\dot {\lambda }}_{Z}/\lambda _{Z}\end{array}}\right]=U^{-1}{\dot {U}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205c72aaa61ac5292c384112d3e2d77144e868ac)
This once again gives

In general, if we approximate

the Green-Naghdi rate becomes the Jaumann rate of the Cauchy stress

Other objective stress rates[edit | edit source]
There can be an infinite variety of objective stress rates. One of these
is the Oldroyd stress rate
![{\displaystyle {\overset {\triangledown }{\boldsymbol {\sigma }}}={\mathcal {L}}_{\varphi }[{\boldsymbol {\sigma }}]={\boldsymbol {F}}\cdot \left[{\cfrac {d}{dt}}\left({\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}\right)\right]\cdot {\boldsymbol {F}}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8897f9c18e90410fff650086c77e6eb2bbe1c114)
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
![{\displaystyle {\overset {\diamond }{\boldsymbol {\sigma }}}={\boldsymbol {F}}^{-T}\cdot \left[{\cfrac {d}{dt}}\left({\boldsymbol {F}}^{T}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}\right)\right]\cdot {\boldsymbol {F}}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e710a455f6c8bde299314d90d224931ecd61d7b)
In simpler form, the convective rate is given by

Caveat on objective stress rates[edit | edit source]
The following figure shows the performance of various objective rates
in a pure shear test where the material model is a hypoelastic one 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.
Predictions from three objective stress rates under shear
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Clearly there are spurious oscillations observed for the Jaumann stress rate.
This is not because one rate is better than another but because its 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.