Let the undeformed (or reference) configuration of the body be
Ω
0
{\displaystyle \Omega _{0}}
Γ
0
{\displaystyle \Gamma _{0}}
Ω
{\displaystyle \Omega }
Γ
{\displaystyle \Gamma }
φ
(
X
,
t
)
{\displaystyle {\boldsymbol {\varphi }}(\mathbf {X} ,t)}
Figure 1. The motion of a body.
We write
x
=
φ
(
X
,
t
)
{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}({\boldsymbol {X}},t)}
where
x
{\displaystyle \mathbf {x} }
X
{\displaystyle {\boldsymbol {X}}}
t
{\displaystyle t}
In index notation,
x
i
=
φ
i
(
X
j
,
t
)
,
i
,
j
=
1
,
2
,
3.
{\displaystyle x_{i}=\varphi _{i}(X_{j},t)~,\qquad i,j=1,2,3.}
The displacement of a material point is given by
u
(
X
,
t
)
=
φ
(
X
,
t
)
−
φ
(
X
,
0
)
=
φ
(
X
,
t
)
−
X
=
x
−
X
.
{\displaystyle \mathbf {u} ({\boldsymbol {X}},t)={\boldsymbol {\varphi }}({\boldsymbol {X}},t)-{\boldsymbol {\varphi }}({\boldsymbol {X}},0)={\boldsymbol {\varphi }}({\boldsymbol {X}},t)-{\boldsymbol {X}}=\mathbf {x} -{\boldsymbol {X}}~.}
In index notation,
u
i
=
φ
i
(
X
j
,
t
)
−
X
i
=
x
i
−
X
i
.
{\displaystyle u_{i}=\varphi _{i}(X_{j},t)-X_{i}=x_{i}-X_{i}~.}
The velocity is the material time derivative of the motion (i.e., the time derivative with
X
{\displaystyle \mathbf {X} }
total derivative .
v
(
X
,
t
)
=
∂
∂
t
[
φ
(
X
,
t
)
]
.
{\displaystyle \mathbf {v} ({\boldsymbol {X}},t)={\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)\right]~.}
Now,
u
(
X
,
t
)
=
φ
(
X
,
t
)
−
X
.
{\displaystyle \mathbf {u} ({\boldsymbol {X}},t)={\boldsymbol {\varphi }}({\boldsymbol {X}},t)-{\boldsymbol {X}}~.}
Therefore, the material time derivative of
u
{\displaystyle \mathbf {u} }
u
˙
=
∂
∂
t
[
u
(
X
,
t
)
]
=
∂
∂
t
[
φ
(
X
,
t
)
−
X
]
=
∂
∂
t
[
φ
(
X
,
t
)
]
=
v
(
X
,
t
)
.
{\displaystyle {\dot {\mathbf {u} }}={\frac {\partial }{\partial t}}\left[\mathbf {u} ({\boldsymbol {X}},t)\right]={\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)-{\boldsymbol {X}}\right]={\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)\right]=\mathbf {v} ({\boldsymbol {X}},t)~.}
Alternatively, we could have expressed the velocity in terms of the
spatial coordinates
x
{\displaystyle \mathbf {x} }
u
(
x
,
t
)
=
u
(
φ
(
X
,
t
)
,
t
)
.
{\displaystyle \mathbf {u} (\mathbf {x} ,t)=\mathbf {u} ({\boldsymbol {\varphi }}({\boldsymbol {X}},t),t)~.}
Then the material time derivative of
u
(
x
,
t
)
{\displaystyle \mathbf {u} (\mathbf {x} ,t)}
D
D
t
[
u
(
x
,
t
)
]
=
∂
u
∂
t
+
∂
u
∂
x
∂
x
∂
t
=
∂
u
∂
t
+
∂
u
∂
x
∂
∂
t
[
φ
(
X
,
t
)
]
=
v
(
x
,
t
)
+
∂
u
∂
x
v
(
X
,
t
)
.
{\displaystyle {\cfrac {D}{Dt}}\left[\mathbf {u} (\mathbf {x} ,t)\right]={\frac {\partial \mathbf {u} }{\partial t}}+{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {x} }{\partial t}}={\frac {\partial \mathbf {u} }{\partial t}}+{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)\right]=\mathbf {v} (\mathbf {x} ,t)+{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {v} ({\boldsymbol {X}},t)~.}
The acceleration is the material time derivative of the velocity of a
material point.
a
(
X
,
t
)
=
∂
∂
t
[
v
(
X
,
t
)
]
=
v
˙
=
∂
2
∂
t
2
[
u
(
X
,
t
)
]
=
u
¨
.
{\displaystyle \mathbf {a} ({\boldsymbol {X}},t)={\frac {\partial }{\partial t}}\left[\mathbf {v} ({\boldsymbol {X}},t)\right]={\dot {\mathbf {v} }}={\frac {\partial ^{2}}{\partial t^{2}}}\left[\mathbf {u} ({\boldsymbol {X}},t)\right]={\ddot {\mathbf {u} }}~.}
![{\displaystyle \mathbf {v} ({\boldsymbol {X}},t)={\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)\right]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08c84579f64cf9450946bbe227d331985e6cf6a5)
![{\displaystyle {\dot {\mathbf {u} }}={\frac {\partial }{\partial t}}\left[\mathbf {u} ({\boldsymbol {X}},t)\right]={\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)-{\boldsymbol {X}}\right]={\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)\right]=\mathbf {v} ({\boldsymbol {X}},t)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66d27fe88260f0dc0ade3672ff9733b87cd097c6)
![{\displaystyle {\cfrac {D}{Dt}}\left[\mathbf {u} (\mathbf {x} ,t)\right]={\frac {\partial \mathbf {u} }{\partial t}}+{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {x} }{\partial t}}={\frac {\partial \mathbf {u} }{\partial t}}+{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)\right]=\mathbf {v} (\mathbf {x} ,t)+{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {v} ({\boldsymbol {X}},t)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37a74fad482755baa7bc04bd14701eea758b5857)
![{\displaystyle \mathbf {a} ({\boldsymbol {X}},t)={\frac {\partial }{\partial t}}\left[\mathbf {v} ({\boldsymbol {X}},t)\right]={\dot {\mathbf {v} }}={\frac {\partial ^{2}}{\partial t^{2}}}\left[\mathbf {u} ({\boldsymbol {X}},t)\right]={\ddot {\mathbf {u} }}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11acd7a3b3e1a12971161bc6963ebc941ff1a9e)