Page 450 - Introduction to Continuum Mechanics

434 integral Formulation of General Principles

2
with fixed values of x^ and x^ \ is an integral over a fixed control volume ; it gives the total
mass at time t within the spatially fixed cylindrical volume of constant cross-sectional area ,4
and bounded by the end facesx = jr ' and* = JT .
Let A^ and A^ be the material coordinates for the particles which, at time t are at jr '
(2)
1
(1)
(2)
and* respectively, i.e.,.x = jc^ *, 0 and* = x(X^\ t}, then the integral

with its integration limits functions of time, (in accordance with the motion of the material
particles which at time t are at JT ' andjr '), is an integral over a material volume; it gives the
total mass at time / , of that part of material which is instantaneously (at time t) coincidental
with that inside the fixed boundary surface considered in Eq. (7.3.3). Obviously, at time t, both
integrals, i.e., Eqs. (7.3.3) and (7.3.4), have the same value. At other times, say at t+dt,
however, they have different values. Indeed,

is different from

We note that dm /dt in Eq. (7.3.5) gives the rate at which mass is increasing inside the fixed
1
control volume bounded by the cylindrical lateral surface and the end faces x = jr ) and
x = jr \ whereas 3M /dt in Eq. (7.3.6) gives the rate of increase of the mass of that part of
material which at time t is coincidental with that in the fixed control volume. They should
obviously be different. In fact, the principle of conservation of mass demands that the mass
within a material volume should remain a constant, whereas the mass within the control volume
in general changes with time.
The above one dimensional example serves to illustrate the two types of volume integrals
which we shall employ in the following sections. We shall use V c to indicate a fixed control
volume and V m to indicate a material volume. That is, for any tensor T (including a scalar)
the integral

is over the fixed control volume V c and the rate of change of this integral is denoted by