Section 3.1
Solid Mechanics Part III
Kelly
317
3.1 Conservation of Mass
3.1.1
Mass and Density
Mass
is a nonnegative scalar measure of a body’s tendency to resist a change in motion.
Consider a small volume element
v
Δ
whose mass is
m
Δ
.
Define the average
density
of
this volume element by the ratio
v
m
Δ
Δ
=
AVE
ρ
(3.1.1)
If
p
is some point within the volume element, then define the
spatial mass density
at
p
to
be the limiting value of this ratio as the volume shrinks down to the point,
v
m
t
v
Δ
Δ
=
→
Δ
0
lim
)
,
(
x
ρ
Spatial Density
(3.1.2)
In a real material, the incremental volume element
v
Δ
must not actually get too small
since then the limit
ρ
would depend on the atomistic structure of the material; the
volume is only allowed to decrease to some minimum value which contains a large
number of molecules.
The spatial mass density is a representative average obtained by
having
v
Δ
large compared to the atomic scale, but small compared to a typical length
scale of the problem under consideration.
The density, as with displacement, velocity, and other quantities, is defined for
specific
particles
of a continuum, and is a continuous function of coordinates and time,
)
,
(
t
x
ρ
ρ
=
.
However, the mass is not defined this way – one writes for the mass of an
infinitesimal volume of material – a
mass element
,
dv
t
dm
)
,
(
x
ρ
=
(3.1.3)
or, for the mass of a volume
v
of material at time
t
,
(
)
∫
=
v
dv
t
m
,
x
ρ
(3.1.4)
3.1.2
Conservation of Mass
The law of conservation of mass states that mass can neither be created nor destroyed.
Consider a collection of matter located somewhere in space.
This quantity of matter with
welldefined boundaries is termed a
system
.
The law of conservation of mass then
implies that the mass of this given system remains constant,
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Section 3.1
Solid Mechanics Part III
Kelly
318
0
=
Dt
Dm
Conservation of Mass
(3.1.5)
The volume occupied by the matter may be changing and the density of the matter within
the system may be changing, but the mass remains constant.
Considering a differential mass element at position
X
in the reference configuration and
at
x
in the current configuration, Eqn. 3.1.5 can be rewritten as
)
,
(
)
(
t
dm
dm
x
X
=
(3.1.6)
The conservation of mass equation can be expressed in terms of densities.
First,
introduce
0
ρ
, the
reference mass density
(or simply the
density
), defined through
V
m
V
Δ
Δ
=
→
Δ
0
0
lim
)
(
X
ρ
Density
(3.1.7)
Note that the density
0
ρ
and the spatial mass density
ρ
are
not
the same quantities
1
.
Thus the
local
(or
differential
)
form
of the conservation of mass can be expressed as (see
Fig. 3.1.1)
const
)
,
(
)
(
0
=
=
=
dv
t
dV
dm
x
X
ρ
ρ
(3.1.8)
Figure 3.1.1: Conservation of Mass for a deforming mass element
Integration over a finite region of material gives the
global
(or
integral
)
form
,
const
)
,
(
)
(
0
=
=
=
∫
∫
v
V
dv
t
dV
m
x
X
ρ
ρ
(3.1.9)
or
0
)
,
(
=
=
=
∫
v
dv
t
dt
d
dt
dm
m
x
ρ
&
(3.1.10)
1
they not only are functions of different variables, but also have different values; they are not different
representations of the same thing, as were, for example, the velocities
v
and
V
.
One could introduce a
material mass density,
)
),
,
(
(
)
,
(
t
t
X
x
t
X
ρ
=
Ρ
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