Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 55

consider a domain of the object in which the integrand of ( 3.4 ) is always positive
(or negative). Let this domain be an object. For the object chosen in this way the
integral on the left-hand side of ( 3.4 ) cannot be zero. This conclusion contradicts
( 3.4 ) because ( 3.4 ) must be zero. It may therefore be concluded that the require-
ment that the integral ( 3.4 ) be zero for an object and all sub-objects that can
be formed from it means that the integrand must be zero everywhere in the object.
Note that a very important transition has occurred in the argument that obtains ( 3.5 )
from ( 3.4 ). Integral statements such as ( 3.4 ) are global statements because they
apply to an entire object. However the requirement ( 3.5 ) is a local, pointwise
condition valid at the typical point (place) in the object. Thus the transition
( 3.4 ) ( 3.5 ) is from the global to the local or from the object to the point

(or particle) in the object. Note also that the converse proof ( 3.5 ) and ( 3.4 ) is trivial.
Note that ( 3.5 ) may be combined with (2.30), the expression decomposing the
material time derivative into the sum of a local rate of change and a convective rate
of change, to obtain this alternate local statement of mass conservation:

!

@r
@ t þrðrvÞ¼ 0 :

(3.6)

Another consequence of the conservation of mass is a simple formula for the
material time derivative of an integral of the form

ð

K ¼ k ð x ; t Þrð x ; t Þ d v ; (3.7)

O

where k ( x , t ) is a physical quantity (temperature, momentum, etc.) of arbitrary
scalar, vector or tensor character and K is the value of the density times quantity
k ( x , t ) integrated over the entire object O . Since, by ( 3.2 ), the material time rate of
change of

r ( x , t )d v is zero, it follows that

ð

K k

¼ ðx; t Þrðx; t Þ d v : (3.8)

O

Problems

3.2.1. For the first six motions of the form (2.10) given in Problem 2.1.1, namely
2.1.1(a) through 2.1.1 (f), determine the ratio of the time rate of change of
density

r
r

.
3.2.2. In this section it was shown that one could derive the local statement of mass
conservation ( 3.5 ) from the global statement of mass conservation, DM/Dt
¼

r to the instantaneous density r ,

0. Reverse the direction of this derivation, derive the global statement of
mass conservation DM/Dt 0 from the fact that the local statement ( 3.5 )or

( 3.6 ) is true at all points in an object O .

¼