令
Ω
0
{\displaystyle \Omega _{0}}
Ω
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t
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{\displaystyle \Omega (t)}
x
=
φ
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t
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;
⟹
F
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=
∇
∘
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{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)~;\qquad \implies \qquad {\boldsymbol {F}}(\mathbf {X} ,t)={\boldsymbol {\nabla }}_{\circ }{\boldsymbol {\varphi }}~.}
令
J
(
X
,
t
)
=
det
[
F
(
X
,
t
)
]
{\displaystyle J(\mathbf {X} ,t)=\det[{\boldsymbol {F}}(\mathbf {X} ,t)]}
∫
Ω
(
t
)
f
(
x
,
t
)
dV
=
∫
Ω
0
f
[
φ
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X
,
t
)
,
t
]
J
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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}~.}
That this derivation is for a material element is implicit in the time constancy of the reference configuration: it is constant in material coordinates. 针对体积积分的微分定义为
d
d
t
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∫
Ω
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t
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f
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x
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t
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dV
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=
lim
Δ
t
→
0
1
Δ
t
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∫
Ω
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t
+
Δ
t
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f
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x
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Δ
t
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dV
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∫
Ω
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f
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x
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dV
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.
{\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\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)~.}
将上式转换为对参考组态的积分，可得
d
d
t
(
∫
Ω
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t
)
f
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x
,
t
)
dV
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=
lim
Δ
t
→
0
1
Δ
t
(
∫
Ω
0
f
^
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X
,
t
+
Δ
t
)
J
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X
,
t
+
Δ
t
)
dV
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−
∫
Ω
0
f
^
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X
,
t
)
J
(
X
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t
)
dV
0
)
.
{\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\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)~.}
因为
Ω
0
{\displaystyle \Omega _{0}}
d
d
t
(
∫
Ω
(
t
)
f
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x
,
t
)
dV
)
=
∫
Ω
0
[
lim
Δ
t
→
0
f
^
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t
+
Δ
t
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J
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+
Δ
t
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−
f
^
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X
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J
(
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Δ
t
]
dV
0
=
∫
Ω
0
∂
∂
t
[
f
^
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X
,
t
)
J
(
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]
dV
0
=
∫
Ω
0
(
∂
∂
t
[
f
^
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t
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]
J
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+
f
^
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∂
∂
t
[
J
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]
)
dV
0
{\displaystyle {\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\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}}}
现在，
det
F
{\displaystyle \det {\boldsymbol {F}}}
[ 6]
∂
J
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∂
t
=
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∂
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det
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=
(
det
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∇
⋅
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J
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⋅
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{\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)~.}
因此
d
d
t
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Ω
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t
)
f
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x
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t
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dV
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=
∫
Ω
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t
[
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^
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J
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^
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J
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v
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dV
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=
∫
Ω
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∂
t
[
f
^
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]
+
f
^
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∇
⋅
v
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J
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dV
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=
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Ω
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(
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f
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∇
⋅
v
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x
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t
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)
dV
{\displaystyle {\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\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}}}
其中
f
˙
{\displaystyle {\dot {\mathbf {f} }}}
f
{\displaystyle \mathbf {f} }
材料导数
f
˙
(
x
,
t
)
=
∂
f
(
x
,
t
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∂
t
+
[
∇
f
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x
,
t
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]
⋅
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)~.}
因此
d
d
t
(
∫
Ω
(
t
)
f
(
x
,
t
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dV
)
=
∫
Ω
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t
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(
∂
f
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∂
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+
[
∇
f
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+
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dV
{\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\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}}}
或者
d
d
t
(
∫
Ω
(
t
)
f
dV
)
=
∫
Ω
(
t
)
(
∂
f
∂
t
+
∇
f
⋅
v
+
f
∇
⋅
v
)
dV
.
{\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\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}}~.}
利用以下的恒等式
∇
⋅
(
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} }
可得
d
d
t
(
∫
Ω
(
t
)
f
dV
)
=
∫
Ω
(
t
)
(
∂
f
∂
t
+
∇
⋅
(
f
⊗
v
)
)
dV
.
{\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\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}}~.}
利用高斯散度定理 及恒等式
(
a
⊗
b
)
⋅
n
=
(
b
⋅
n
)
a
{\displaystyle (\mathbf {a} \otimes \mathbf {b} )\cdot \mathbf {n} =(\mathbf {b} \cdot \mathbf {n} )\mathbf {a} }
d
d
t
(
∫
Ω
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f
dV
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∂
f
∂
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dV
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∂
Ω
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(
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⋅
n
dA
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∂
f
∂
t
dV
+
∫
∂
Ω
(
t
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(
v
⋅
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f
dA
◻
{\displaystyle {{\cfrac {\mathrm {d} }{\mathrm {d} t}}\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}}\qquad \square }}
![{\displaystyle J(\mathbf {X} ,t)=\det[{\boldsymbol {F}}(\mathbf {X} ,t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/488e6f753749ea03647f6dfbdcdbd6cf515575bc)
![{\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)
![{\displaystyle {\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\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/dccf7b716b8f34767f136efbceb2832faa96de7e)
![{\displaystyle {\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\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/8fb723181cce1321f237c980778b89bc72717f7d)
![{\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)
![{\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\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/7ccadc7aec02eeff04b0876b7d6c97d41f277add)
