en>SMesser: /* Variational basis */ spelling fix

2014-01-16T23:44:38Z

Variational basis: spelling fix

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In [[continuum mechanics]], '''objective stress rates''' are time [[derivative]]s of [[stress (physics)|stress]] that do not depend on the [[frame of reference]].<ref>Gurtin et al. (2010). p. 151,242.</ref> Many [[constitutive equation]]s are designed in the form of a relation between a stress-rate and a [[finite strain theory|strain-rate]] (or the [[finite strain theory|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 [[objectivity (frame invariance)|frame indifferent (objective)]]. If the [[stress measures|stress]] and strain measures are [[Continuum mechanics#Lagrangian escription|material]] quantities then objectivity is automatically satisfied. However, if the quantities are [[Continuum mechanics#Eulerian description|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 derivative]]s. Some of the widely used objective stress rates are:
# the ''' Truesdell''' rate of the [[Cauchy stress tensor]],
# the ''' Green-Naghdi''' rate of the Cauchy stress, and
# the ''' Jaumann''' rate of the Cauchy stress.

The adjacent figure shows the performance of various objective rates in a [[pure shear]] test where the material model is [[hypoelastic material|hypoelastic]] with constant [[elastic moduli]]. The ratio of the [[shear stress]] to the [[displacement field (mechanics)|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 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 (<math>\boldsymbol{Q}</math>), the [[Cauchy stress tensor]] <math>\boldsymbol{\sigma}</math> [[Cauchy_stress_tensor#Transformation rule of the stress tensor|transform]]s as
:<math>
\boldsymbol{\sigma}_r = \boldsymbol{Q}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{Q}^T ~;~~ \boldsymbol{Q}\cdot\boldsymbol{Q}^T = \boldsymbol{\mathit{1}}
</math>
Since <math>\boldsymbol{\sigma}</math> is a spatial quantity and the transformation follows the rules of [[covariant transformation|tensor transformation]]s, <math>\boldsymbol{\sigma}</math> is objective. However,
:<math>
\cfrac{d}{dt}(\boldsymbol{\sigma}_r) = \dot{\boldsymbol{\sigma}}_r = \dot{\boldsymbol{Q}}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{Q}^T +
\boldsymbol{Q}\cdot\dot{\boldsymbol{\sigma}}\cdot\boldsymbol{Q}^T + \boldsymbol{Q}\cdot\boldsymbol{\sigma}\cdot\dot{\boldsymbol{Q}}^T \ne \boldsymbol{Q}\cdot\dot{\boldsymbol{\sigma}}\cdot\boldsymbol{Q}^T \,.
</math>
Therefore the stress rate is ''' not objective''' unless the rate of rotation is zero, i.e. <math>\boldsymbol{Q}</math> is constant.

==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 <math>\boldsymbol{\sigma}</math> or the push-forward of <math>\boldsymbol{S}</math> as
:<math>
\boldsymbol{S} = J~\phi^{*}[\boldsymbol{\sigma}] ~;~~ \boldsymbol{\sigma} = J^{-1}~\phi_{*}[\boldsymbol{S}]
</math>

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
:<math>
\overset{\circ}{\boldsymbol{\sigma}} = J^{-1}~\phi_{*}[\dot{\boldsymbol{S}}]
</math>
Expanded out, this means that
:<math>
\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}]
</math>
where the Kirchhoff stress <math>\boldsymbol{\tau} = J~\boldsymbol{\sigma}</math> and the ''' Lie derivative''' of
the Kirchhoff stress is
:<math>
\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 ~.
</math>
This expression can be simplified to the well known expression for the
Truesdell rate of the Cauchy stress
{| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:80%"
| colspan="2" style="width:80%; horizontal-align:right; vertical-align:top; border:1px solid Sienna; background-color:White;" |
<div style="border-bottom:1px solid Sienna; background-color:Wheat; padding:0.2em 0.5em 0.2em 0.5em; font-size:100%; font-weight:bold;">
Truesdell rate of the Cauchy stress
</div>
<div style="padding:2em 5em 0em 3em;">
:<math>
\overset{\circ}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{l}\cdot\boldsymbol{\sigma} - \boldsymbol{\sigma}\cdot\boldsymbol{l}^T +
\text{tr}(\boldsymbol{l})~\boldsymbol{\sigma}
</math>

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Proof:
|-
|
We start with
:<math>
\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 ~.
</math>
Expanding the derivative inside the square brackets, we get
:<math>
\begin{align}
\overset{\circ}{\boldsymbol{\sigma}} & =
J^{-1}~\boldsymbol{F}\cdot(\dot{J}~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T})\cdot\boldsymbol{F}^T +
J^{-1}~\boldsymbol{F}\cdot(J~\dot{\boldsymbol{F}^{-1}}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T})\cdot\boldsymbol{F}^T \\
& +
J^{-1}~\boldsymbol{F}\cdot(J~\boldsymbol{F}^{-1}\cdot\dot{\boldsymbol{\sigma}}\cdot\boldsymbol{F}^{-T})\cdot\boldsymbol{F}^T +
J^{-1}~\boldsymbol{F}\cdot(J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\dot{\boldsymbol{F}^{-T}})\cdot\boldsymbol{F}^T
\end{align}
</math>
or,
:<math>
\overset{\circ}{\boldsymbol{\sigma}} = J^{-1}~\dot{J}~\boldsymbol{\sigma} +
\boldsymbol{F}\cdot\dot{\boldsymbol{F}^{-1}}\cdot\boldsymbol{\sigma} + \dot{\boldsymbol{\sigma}} +
\boldsymbol{\sigma}\cdot\dot{\boldsymbol{F}^{-T}}\cdot\boldsymbol{F}^T
</math>
Now,
:<math>
\boldsymbol{F}\cdot\boldsymbol{F}^{-1} = \boldsymbol{\mathit{1}}
</math>
Therefore,
:<math>
\cfrac{d}{dt}\left(\boldsymbol{F}\cdot\boldsymbol{F}^{-1}\right) = 0
\quad \implies \quad
\dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1} + \boldsymbol{F}\cdot\dot{\boldsymbol{F}^{-1}} = 0
</math>
or,
:<math>
\dot{\boldsymbol{F}^{-1}} = - \boldsymbol{F}^{-1}\cdot\boldsymbol{l} \quad \implies \quad
\dot{\boldsymbol{F}^{-T}} = - \boldsymbol{l}^T\cdot\boldsymbol{F}^{-T}
</math>
where the velocity gradient <math>\boldsymbol{l} = \dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1}</math>.

Also, the rate of change of volume is given by
:<math>
\dot{J} = J~\text{tr}(\boldsymbol{d}) = J~\text{tr}(\boldsymbol{l})
</math>
where <math>\boldsymbol{d}</math> is the rate of deformation tensor.

Therefore,
:<math>
\overset{\circ}{\boldsymbol{\sigma}} = J^{-1}~J~\text{tr}(\boldsymbol{l})~\boldsymbol{\sigma} -
\boldsymbol{F}\cdot\boldsymbol{F}^{-1}\cdot\boldsymbol{l}\cdot\boldsymbol{\sigma} + \dot{\boldsymbol{\sigma}} -
\boldsymbol{\sigma}\cdot\boldsymbol{l}^T\cdot\boldsymbol{F}^{-T}\cdot\boldsymbol{F}^T
</math>
or,
:<math>
\overset{\circ}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{l}\cdot\boldsymbol{\sigma} - \boldsymbol{\sigma}\cdot\boldsymbol{l}^T +
\text{tr}(\boldsymbol{l})~\boldsymbol{\sigma}
</math>
|}
|}
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
:<math>
\boldsymbol{S} = \phi^{*}[\boldsymbol{\tau}] ~;~~ \boldsymbol{\tau} = \phi_{*}[\boldsymbol{S}]
</math>
and defining
:<math>
\overset{\circ}{\boldsymbol{\tau}} = \phi_{*}[\dot{\boldsymbol{S}}]
</math>
Expanded out, this means that
:<math>
\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}]
</math>
Therefore, ''' the Lie derivative of <math>\boldsymbol{\tau}</math> 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
{| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:80%"
| colspan="2" style="width:80%; horizontal-align:right; vertical-align:top; border:1px solid Sienna; background-color:White;" |
<div style="border-bottom:1px solid Sienna; background-color:Wheat; padding:0.2em 0.5em 0.2em 0.5em; font-size:100%; font-weight:bold;">
Truesdell rate of the Kirchhoff stress
</div>
<div style="padding:2em 5em 0em 3em;">
:<math>
\overset{\circ}{\boldsymbol{\tau}} = \dot{\boldsymbol{\tau}} - \boldsymbol{l}\cdot\boldsymbol{\tau} - \boldsymbol{\tau}\cdot\boldsymbol{l}^T
</math>
|}

== 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
:<math>
\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 ~.
</math>
From the polar decomposition theorem we have
:<math>
\boldsymbol{F} = \boldsymbol{R}\cdot\boldsymbol{U}
</math>
where <math>\boldsymbol{R}</math> is the orthogonal rotation tensor (<math>\boldsymbol{R}^{-1} = \boldsymbol{R}^T</math>)
and <math>\boldsymbol{U}</math> is the symmetric, positive definite, right stretch.

If we assume that <math>\boldsymbol{U} = \boldsymbol{\mathit{1}}</math> we get <math>\boldsymbol{F} = \boldsymbol{R}</math>. Also since there is no
stretch <math>J = 1</math> and we have <math>\boldsymbol{\tau} = \boldsymbol{\sigma}</math>. 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
:<math>
\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
</math>
We can show that this expression can be simplified to the
commonly used form of the ''' Green-Naghdi''' rate
{| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:80%"
| colspan="2" style="width:80%; horizontal-align:right; vertical-align:top; border:1px solid Sienna; background-color:White;" |
<div style="border-bottom:1px solid Sienna; background-color:Wheat; padding:0.2em 0.5em 0.2em 0.5em; font-size:100%; font-weight:bold;">
Green-Naghdi rate of the Cauchy stress
</div>
<div style="padding:2em 5em 0em 3em;">
:<math>
\overset{\square}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol{\Omega}
- \boldsymbol{\Omega}\cdot\boldsymbol{\sigma}
</math>
where <math>\boldsymbol{\Omega} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T</math>.
:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Proof:
|-
|Expanding out the derivative
:<math>
\overset{\circ}{\boldsymbol{\sigma}} = \boldsymbol{R}\cdot\dot{\boldsymbol{R}^T}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{R}\cdot\boldsymbol{R}^T +
\boldsymbol{R}\cdot\boldsymbol{R}^T\cdot\dot{\boldsymbol{\sigma}}\cdot\boldsymbol{R}\cdot\boldsymbol{R}^T +
\boldsymbol{R}\cdot\boldsymbol{R}^T\cdot\boldsymbol{\sigma}\cdot\dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T
</math>
or,
:<math>
\overset{\circ}{\boldsymbol{\sigma}} = \boldsymbol{R}\cdot\dot{\boldsymbol{R}^T}\cdot\boldsymbol{\sigma} +
\dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T
</math>
Now,
:<math>
\boldsymbol{R}\cdot\boldsymbol{R}^T = \boldsymbol{\mathit{1}} \quad \implies \quad
\dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T = - \boldsymbol{R}\cdot\dot{\boldsymbol{R}^T}
</math>
Therefore,
:<math>
\overset{\circ}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T
- \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T\cdot\boldsymbol{\sigma}
</math>
If we define the angular velocity as
:<math>
\boldsymbol{\Omega} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T
</math>
we get the commonly used form of the ''' Green-Naghdi''' rate
:<math>
\overset{\square}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol{\Omega}
- \boldsymbol{\Omega}\cdot\boldsymbol{\sigma}
</math>
|}
|}

The Green-Naghdi rate of the Kirchhoff stress also has the form since the
stretch is not taken into consideration, i.e.,
:<math>
\overset{\square}{\boldsymbol{\tau}} = \dot{\boldsymbol{\tau}} + \boldsymbol{\tau}\cdot\boldsymbol{\Omega}
- \boldsymbol{\Omega}\cdot\boldsymbol{\tau}
</math>

== Jaumann rate of the Cauchy stress ==
The Jaumann rate of the Cauchy stress is a further specialization of the
Lie derivative (Truesdell rate). This rate has the form
{| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:80%"
| colspan="2" style="width:80%; horizontal-align:right; vertical-align:top; border:1px solid Sienna; background-color:White;" |
<div style="border-bottom:1px solid Sienna; background-color:Wheat; padding:0.2em 0.5em 0.2em 0.5em; font-size:100%; font-weight:bold;">
Jaumann rate of the Cauchy stress
</div>
<div style="padding:2em 5em 0em 3em;">
:<math>
\overset{\triangle}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol{w} - \boldsymbol{w}\cdot\boldsymbol{\sigma}
</math>
where <math>\boldsymbol{w}</math> is the spin tensor.
|}

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 <math>\boldsymbol{w}</math> (the skew part of the velocity gradient)
can be expressed as
:<math>
\boldsymbol{w} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T + \frac{1}{2}~\boldsymbol{R}\cdot(\dot{\boldsymbol{U}}\cdot\boldsymbol{U}^{-1} -
\boldsymbol{U}^{-1}\cdot\dot{\boldsymbol{U}})\cdot\boldsymbol{R}^T
</math>
Thus for pure rigid body motion
:<math>
\boldsymbol{w} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T = \boldsymbol{\Omega}
</math>
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
:<math>
\boldsymbol{U} = \left[\begin{array}{ccc}
\lambda_{X}\\
& \lambda_{Y}\\
& & \lambda_{Z}\end{array}\right]
</math>
we have
:<math>
\dot{\boldsymbol{U}} = \left[\begin{array}{ccc}
\dot{\lambda}_{X}\\
& \dot{\lambda}_{Y}\\
& & \dot{\lambda}_{Z}\end{array}\right]
</math>
Also,
:<math>
\boldsymbol{U}^{-1} = \left[\begin{array}{ccc}
1/\lambda_{X}\\
& 1/\lambda_{Y}\\
& & 1/\lambda_{Z}\end{array}\right]
</math>of the Cauchy stress
Therefore,
:<math>
\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}
</math>
This once again gives
:<math>
\boldsymbol{w} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T = \boldsymbol{\Omega}
</math>
In general, if we approximate
:<math>
\boldsymbol{w} \approx \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T
</math>
the Green-Naghdi rate becomes the Jaumann rate of the Cauchy stress
:<math>
\overset{\triangle}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol{w}
- \boldsymbol{w}\cdot\boldsymbol{\sigma}
</math>

== Other objective stress rates ==
There can be an infinite variety of objective stress rates. One of these
is the ''' Oldroyd stress rate'''
:<math>
\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
</math>
In simpler form, the Oldroyd rate is given by
:<math>
\overset{\triangledown}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{l}\cdot\boldsymbol{\sigma} - \boldsymbol{\sigma}\cdot\boldsymbol{l}^T
</math>

If the current configuration is assumed to be the reference configuration then
the pull back and push forward operations can be conducted using <math>\boldsymbol{F}^T</math> and
<math>\boldsymbol{F}^{-T}</math> respectively. The Lie derivative of the Cauchy stress is then
called the ''' convective stress rate'''
:<math>
\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}
</math>
In simpler form, the convective rate is given by
:<math>
\overset{\diamond}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} + \boldsymbol{l}\cdot\boldsymbol{\sigma} + \boldsymbol{\sigma}\cdot\boldsymbol{l}^T
</math>

== See also ==
* [[Cauchy stress tensor]]
* [[Stress measures]]
* [[Objectivity (frame invariance)]]
* [[Principle of material objectivity]]
* [[Hypoelastic material]]

== Notes ==
{{Reflist}}

== References ==
* {{Citation|last1=Gurtin|first1= Morton E.|first2=Eliot|last2=Fried|first3=Lallit|last3= Anand|title= The mechanics and thermodynamics of continua|publisher= Cambridge University Press|year= 2010}}

[[Category:Continuum mechanics]]

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en>SMesser: /* Variational basis */ spelling fix