Reynolds transport theorem
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Proof:
Let
be reference configuration of the region
. Let
the motion and the deformation gradient be given by
![{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)~;\qquad \implies \qquad {\boldsymbol {F}}(\mathbf {X} ,t)={\boldsymbol {\nabla }}_{\circ }{\boldsymbol {\varphi }}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf32c1ad430eabf08a7928386255aecfc48723d5)
Let
.
Then, integrals in the current and the reference configurations are
related by
![{\displaystyle \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}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a49b5532c9a31af8181b530f93ce95d8880ab0d)
The time derivative of an integral over a volume is defined as
![{\displaystyle {\cfrac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\lim _{\Delta t\rightarrow 0}{\cfrac {1}{\Delta t}}\left(\int _{\Omega (t+\Delta t)}\mathbf {f} (\mathbf {x} ,t+\Delta t)~{\text{dV}}-\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd77b4a4db7c7cc09322a67e4f42131a43e83cbf)
Converting into integrals over the reference configuration, we get
![{\displaystyle {\cfrac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\lim _{\Delta t\rightarrow 0}{\cfrac {1}{\Delta t}}\left(\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t+\Delta t)~J(\mathbf {X} ,t+\Delta t)~{\text{dV}}_{0}-\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\text{dV}}_{0}\right)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/890ff6c7e57700a2973f7e5d6bf09257982cbb6e)
Since
is independent of time, we have
![{\displaystyle {\begin{aligned}{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6bc22c2caea1738fd010bd66624188d03c7c3b)
Now, the time derivative of
is given by
(see Gurtin: 1981, p. 77)
![{\displaystyle {\frac {\partial J(\mathbf {X} ,t)}{\partial t}}={\frac {\partial }{\partial t}}(\det {\boldsymbol {F}})=(\det {\boldsymbol {F}})({\boldsymbol {\nabla }}\cdot \mathbf {v} )=J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} ({\boldsymbol {\varphi }}(\mathbf {X} ,t),t)=J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1395327382631ddbc205ca8cb788cbdab87e3538)
Therefore,
![{\displaystyle {\begin{aligned}{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d84e6b053ff103823871017d8ecfef2700c97ae6)
where
is the material time derivative of
. Now,
the material derivative is given by
![{\displaystyle {\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)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0527b00dc98359f37547adb6ed77d26e9b65d5de)
Therefore,
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d94c20b05aaa00649e1b2a73cdfdbb7f5eedff)
or,
![{\displaystyle {\cfrac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {f} \cdot \mathbf {v} +\mathbf {f} ~{\boldsymbol {\nabla }}\cdot \mathbf {v} \right)~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/644121a19bb0a06facb4ceb35a2768d765de8497)
Using the identity
![{\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {v} \otimes \mathbf {w} )=\mathbf {v} ({\boldsymbol {\nabla }}\cdot \mathbf {w} )+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {w} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e5e488b8fa40a45b6b74d23559e62ab61f48ff)
we then have
![{\displaystyle {\cfrac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\cdot (\mathbf {f} \otimes \mathbf {v} )\right)~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8365e705c40b1ced27c5c720a70ff03d87724b)
Using the divergence theorem and the identity
we have
![{\displaystyle {{\cfrac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {f} \otimes \mathbf {v} )\cdot \mathbf {n} ~{\text{dA}}=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {v} \cdot \mathbf {n} )\mathbf {f} ~{\text{dA}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecf5ac5d673494cfb67949da324ae01ec6a82ff6)
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.