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# 9781118105733.ch2

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```An Introduction to Mathematical Modeling: A Course in Mechanics
by J. Tinsley Oden
CHAPTER 2
MASS AND MOMENTUM
A common dictionary definition of mass is as follows:
Mass The property of a body that is a measure of the amount of
material it contains and causes it to have weight in a gravitational
field.
In continuum mechanics, the mass of a body is continuously distributed over its volume and is an integral of a density field ρ : Ω0 —>■ K +
called the mass density. The total mass M(B) of a body is independent
of the motion φ, but the mass density ρ can, of course, change as the
volume of the body changes while in motion. Symbolically,
M{B)=
ί ράχ,
(2.1)
where dx = volume element in the current configuration Ω( of the body.
Given two motions φ and ψ (see Fig. 2.1), let ρφ and ρψ denote
the mass densities in the configurations ν?(Ω0) and ψ(Ωο), respectively.
An Introduction to Mathematical Modeling: A Course in Mechanics, First Edition. By J. Tinsley Oden
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.
25
26
CHAPTER 2. MASS AND MOMENTUM
Figure 2.1: Two motions ψ and ψ.
Since the total mass is independent of the motion,
M(B) = /
•/^(ίίο)
ρφάχ =
Α/>(Ωο)
ρψάχ.
(2.2)
This fact represents the principle of conservation of mass. The mass of a
body В is thus an invariant property (measuring the amount of material
in B)\ the weight of В is defined as gM(B) where g is a constant gravity
field. Thus, a body may weigh differently in different gravity fields (e.g.,
the earth's gravity as opposed to that on the moon), but its mass is the
same.
2.1
Local Forms of the Principle of Conservation
of Mass
Let £o = 6o(X) be the mass density of a body in its reference configuration and let ρ = ρ(χ, t) be the mass density in the current configuration
üt. Then
I
ß0{X)dX=
[ 6{x)dx
Jnt
2.1 LOCAL FORMS OF THE PRINCIPLE OF CONSERVATION OF MASS
27
(where the dependence of ρ on t has been suppressed). But dx =
detF(X)dX,so
/ [ρο(Χ) - ρ(φ(Χ)) det F(X)} dX = 0,
and therefore
Qo(X) =
(23)
Q(x)detF(X).
This is the material description (or the Lagrangian formulation) of the
principle of conservation of mass. To obtain the spatial description (or
Eulerian formulation), we observe that the invariance of total mass can
be expressed as
— / g(x,t)dx
dtJnt
= Q.
Changing to the material coordinates gives
0=—
/
g{x,t)detF(X,t)dX
= / (дШ¥
+
gdetF)dX,
where (·) = d{-)/dt. Recalling that det F = det F div v, we have
0=/
det F (ρ div i; + - ^ + w-grade) dX
= [ (-£ + άίν(ρυ)) detF dX
=
iMt+άivi^υ)">dX,
from which we conclude
dt
+ div(^v) = 0.
(2.4)
28
CHAPTER 2. MASS AND MOMENTUM
2.2
Momentum
The momentum of a material body is a property the body has by virtue of
its mass and its velocity. Given a motion φ of a body В of mass density
ρ, the linear momentum I (B, t) of В at time ί and the angular momentum
H(B, t) of В at time ί about the origin 0 of the spatial coordinate system
are defined by
I(B,t) = [ ρν dx,
JUt
H(B,t)=
f x x ρν dx.
Jnt
(2.5)
Again, dx (= άχιάχ2άχ3) is the volume element in Qt.
The rates of change of momenta (both / and H) are of fundamental
importance. To calculate rates, first notice that for any smooth field
w = w(x,t),
AC
А
Г
— / wgdx = —
dt Jat
dt JUo
= / -jrQodx
Jn0 dt
w((p(X,t),t)g{x,t)detF(X,t)dX
= /
-jrQdx.
Jat dt
(2.6)
Thus,
(2.7)
```
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