Continuum mechanics/Reynolds transport theorem
From Wikiversity
[edit] Reynolds transport theoremLet Ω(t) be a region in Euclidean space with boundary This relation is also known as the Reynold's Transport Theorem and is a generalization of the Leibniz rule. Content of example. |
Proof:
Let Ω0 be reference configuration of the region Ω(t). Let the motion and the deformation gradient be given by
Let
. Then, integrals in the current and the reference configurations are related by
The time derivative of an integral over a volume is defined as
Converting into integrals over the reference configuration, we get
Since Ω0 is independent of time, we have
Now, the time derivative of
is given by (see Gurtin: 1981, p. 77)
Therefore,
where
is the material time derivative of
. Now, the material derivative is given by
Therefore,
or,
Using the identity
we then have
Using the divergence theorem and the identity
we have
References
- M.E. Gurtin. An Introduction to Continuum Mechanics. Academic Press, New York, 1981.
- T. Belytschko, W. K. Liu, and B. Moran. Nonlinear Finite Elements for Continua and Structures. John Wiley and Sons, Ltd., New York, 2000.
. Let
be the positions of points in the region and let
be the velocity field in the region. Let
be the
be a vector field in the region (it may also be a scalar field). Show that

![\int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV} =
\int_{\Omega_0} \mathbf{f}[\boldsymbol{\varphi}(\mathbf{X},t),t]~J(\mathbf{X},t)~\text{dV}_0 =
\int_{\Omega_0} \hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)~\text{dV}_0 ~.](http://upload.wikimedia.org/math/6/b/7/6b7ed02827498b2dbe7342fcb16aa68a.png)


![\begin{align}
\cfrac{d}{dt}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) & =
\int_{\Omega_0} \left[\lim_{\Delta t \rightarrow 0} \cfrac{
\hat{\mathbf{f}}(\mathbf{X},t+\Delta t)~J(\mathbf{X},t+\Delta t) -
\hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)}{\Delta t} \right]~\text{dV}_0 \\
& = \int_{\Omega_0} \frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)]~\text{dV}_0 \\
& = \int_{\Omega_0} \left(
\frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]~J(\mathbf{X},t)+
\hat{\mathbf{f}}(\mathbf{X},t)~\frac{\partial }{\partial t}[J(\mathbf{X},t)]\right) ~\text{dV}_0
\end{align}](http://upload.wikimedia.org/math/3/b/d/3bde5047de442befcd9dc185f08c26f0.png)

![\begin{align}
\cfrac{d}{dt}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) & =
\int_{\Omega_0} \left(
\frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]~J(\mathbf{X},t)+
\hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right) ~\text{dV}_0 \\
& =
\int_{\Omega_0}
\left(\frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]+
\hat{\mathbf{f}}(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~J(\mathbf{X},t) ~\text{dV}_0 \\
& =
\int_{\Omega(t)}
\left(\dot{\mathbf{f}}(\mathbf{x},t)+
\mathbf{f}(\mathbf{x},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~\text{dV}
\end{align}](http://upload.wikimedia.org/math/b/4/e/b4e7aac62c78aa61f7811b42e283b1d0.png)
![\dot{\mathbf{f}}(\mathbf{x},t) =
\frac{\partial \mathbf{f}(\mathbf{x},t)}{\partial t} + [\boldsymbol{\nabla} \mathbf{f}(\mathbf{x},t)]\cdot\mathbf{v}(\mathbf{x},t) ~.](http://upload.wikimedia.org/math/5/c/0/5c05c282a558a9ce7dd4f17ec271c626.png)
![\cfrac{d}{dt}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) =
\int_{\Omega(t)}
\left(
\frac{\partial \mathbf{f}(\mathbf{x},t)}{\partial t} + [\boldsymbol{\nabla} \mathbf{f}(\mathbf{x},t)]\cdot\mathbf{v}(\mathbf{x},t) +
\mathbf{f}(\mathbf{x},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~\text{dV}](http://upload.wikimedia.org/math/f/1/a/f1a4f88166e8f0c07a1bc8d93affef7b.png)



