Proof:
Let
Ω
0
{\displaystyle \Omega _{0}}
be reference configuration of the region
Ω
(
t
)
{\displaystyle \Omega (t)}
. Let
the motion and the deformation gradient be given by
x
=
φ
(
X
,
t
)
;
⟹
F
(
X
,
t
)
=
∇
∘
φ
.
{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)~;\qquad \implies \qquad {\boldsymbol {F}}(\mathbf {X} ,t)={\boldsymbol {\nabla }}_{\circ }{\boldsymbol {\varphi }}~.}
Let
J
(
X
,
t
)
=
det
[
F
(
X
,
t
)
]
{\displaystyle J(\mathbf {X} ,t)=\det[{\boldsymbol {F}}(\mathbf {X} ,t)]}
.
Then, integrals in the current and the reference configurations are
related by
∫
Ω
(
t
)
f
(
x
,
t
)
dV
=
∫
Ω
0
f
[
φ
(
X
,
t
)
,
t
]
J
(
X
,
t
)
dV
0
=
∫
Ω
0
f
^
(
X
,
t
)
J
(
X
,
t
)
dV
0
.
{\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}~.}
The time derivative of an integral over a volume is defined as
d
d
t
(
∫
Ω
(
t
)
f
(
x
,
t
)
dV
)
=
lim
Δ
t
→
0
1
Δ
t
(
∫
Ω
(
t
+
Δ
t
)
f
(
x
,
t
+
Δ
t
)
dV
−
∫
Ω
(
t
)
f
(
x
,
t
)
dV
)
.
{\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)~.}
Converting into integrals over the reference configuration, we get
d
d
t
(
∫
Ω
(
t
)
f
(
x
,
t
)
dV
)
=
lim
Δ
t
→
0
1
Δ
t
(
∫
Ω
0
f
^
(
X
,
t
+
Δ
t
)
J
(
X
,
t
+
Δ
t
)
dV
0
−
∫
Ω
0
f
^
(
X
,
t
)
J
(
X
,
t
)
dV
0
)
.
{\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)~.}
Since
Ω
0
{\displaystyle \Omega _{0}}
is independent of time, we have
d
d
t
(
∫
Ω
(
t
)
f
(
x
,
t
)
dV
)
=
∫
Ω
0
[
lim
Δ
t
→
0
f
^
(
X
,
t
+
Δ
t
)
J
(
X
,
t
+
Δ
t
)
−
f
^
(
X
,
t
)
J
(
X
,
t
)
Δ
t
]
dV
0
=
∫
Ω
0
∂
∂
t
[
f
^
(
X
,
t
)
J
(
X
,
t
)
]
dV
0
=
∫
Ω
0
(
∂
∂
t
[
f
^
(
X
,
t
)
]
J
(
X
,
t
)
+
f
^
(
X
,
t
)
∂
∂
t
[
J
(
X
,
t
)
]
)
dV
0
{\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}}}
Now, the time derivative of
det
F
{\displaystyle \det {\boldsymbol {F}}}
is given by
(see Gurtin: 1981, p. 77)
∂
J
(
X
,
t
)
∂
t
=
∂
∂
t
(
det
F
)
=
(
det
F
)
(
∇
⋅
v
)
=
J
(
X
,
t
)
∇
⋅
v
(
φ
(
X
,
t
)
,
t
)
=
J
(
X
,
t
)
∇
⋅
v
(
x
,
t
)
.
{\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)~.}
Therefore,
d
d
t
(
∫
Ω
(
t
)
f
(
x
,
t
)
dV
)
=
∫
Ω
0
(
∂
∂
t
[
f
^
(
X
,
t
)
]
J
(
X
,
t
)
+
f
^
(
X
,
t
)
J
(
X
,
t
)
∇
⋅
v
(
x
,
t
)
)
dV
0
=
∫
Ω
0
(
∂
∂
t
[
f
^
(
X
,
t
)
]
+
f
^
(
X
,
t
)
∇
⋅
v
(
x
,
t
)
)
J
(
X
,
t
)
dV
0
=
∫
Ω
(
t
)
(
f
˙
(
x
,
t
)
+
f
(
x
,
t
)
∇
⋅
v
(
x
,
t
)
)
dV
{\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}}}
where
f
˙
{\displaystyle {\dot {\mathbf {f} }}}
is the material time derivative of
f
{\displaystyle \mathbf {f} }
. Now,
the material derivative is given by
f
˙
(
x
,
t
)
=
∂
f
(
x
,
t
)
∂
t
+
[
∇
f
(
x
,
t
)
]
⋅
v
(
x
,
t
)
.
{\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)~.}
Therefore,
d
d
t
(
∫
Ω
(
t
)
f
(
x
,
t
)
dV
)
=
∫
Ω
(
t
)
(
∂
f
(
x
,
t
)
∂
t
+
[
∇
f
(
x
,
t
)
]
⋅
v
(
x
,
t
)
+
f
(
x
,
t
)
∇
⋅
v
(
x
,
t
)
)
dV
{\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}}}
or,
d
d
t
(
∫
Ω
(
t
)
f
dV
)
=
∫
Ω
(
t
)
(
∂
f
∂
t
+
∇
f
⋅
v
+
f
∇
⋅
v
)
dV
.
{\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}}~.}
Using the identity
∇
⋅
(
v
⊗
w
)
=
v
(
∇
⋅
w
)
+
∇
v
⋅
w
{\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {v} \otimes \mathbf {w} )=\mathbf {v} ({\boldsymbol {\nabla }}\cdot \mathbf {w} )+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {w} }
we then have
d
d
t
(
∫
Ω
(
t
)
f
dV
)
=
∫
Ω
(
t
)
(
∂
f
∂
t
+
∇
⋅
(
f
⊗
v
)
)
dV
.
{\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}}~.}
Using the divergence theorem and the identity
(
a
⊗
b
)
⋅
n
=
(
b
⋅
n
)
a
{\displaystyle (\mathbf {a} \otimes \mathbf {b} )\cdot \mathbf {n} =(\mathbf {b} \cdot \mathbf {n} )\mathbf {a} }
we have
d
d
t
(
∫
Ω
(
t
)
f
dV
)
=
∫
Ω
(
t
)
∂
f
∂
t
dV
+
∫
∂
Ω
(
t
)
(
f
⊗
v
)
⋅
n
dA
=
∫
Ω
(
t
)
∂
f
∂
t
dV
+
∫
∂
Ω
(
t
)
(
v
⋅
n
)
f
dA
.
{\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}}~.}}
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.