Section 2.4
Solid Mechanics Part III
Kelly
239
2.4 Material Time Derivatives
The motion is now allowed to be a function of time,
( )
t
,
X
χ
x
=
, and attention is given to
time derivatives, both the
material time derivative
and the
local time derivative
.
2.4.1
Velocity & Acceleration
The velocity of a moving particle is the time rate of change of the position of the particle.
From 2.1.3, by definition,
dt
t
d
t
)
,
(
)
,
(
X
χ
X
V
≡
(2.4.1)
In the motion expression
()
t
,
X
χ
x
=
,
X
and
t
are independent variables and so
X
is
independent of time, denoting the particle for which the velocity is being calculated.
The
velocity can thus be written as
t
t
∂
∂
/
)
,
(
X
χ
or, denoting the motion by
)
,
(
t
X
x
, as
dt
t
d
/
)
,
(
X
x
or
t
t
∂
∂
/
)
,
(
X
x
.
The spatial description of the velocity field may be obtained from the material description
by simply replacing
X
with
x
, i.e.
( )
t
t
t
),
,
(
)
,
(
1
x
χ
V
x
v
−
=
(2.4.2)
As with displacements in both descriptions, there is only one velocity,
)
,
(
)
,
(
t
t
x
v
X
V
=
–
they are just given in terms of different coordinates.
The velocity is most often expressed in the spatial description, as
dt
d
t
x
x
x
v
=
=
)
,
(
velocity
(2.4.3)
To be precise, the right hand side here involves
x
which is a function of the material
coordinates, but it is understood that the substitution back to spatial coordinates, as in
2.4.2, is made.