Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 53

the quantity produced within the system. On the minus side are the amount of the
quantity leaving the system and the quantity consumed within the system. The
system will be either a fixed material system consisting always of the same set of
particles or a fixed spatial (continuum) volume through which material is passing.
The quantity will be either mass, linear momentum, angular momentum, or energy.
The focus of this chapter is the development of continuum formulations for the
conservation principles of mass, linear momentum, angular momentum, and
energy. The statement of conservation of mass is usually the statement that mass
cannot be created or destroyed. The conservation of momentum is usually stated in
the form of Newton's second law: the sum of the forces acting on an object is equal
to the product of the mass of the object and the acceleration of the object. The
conservation of angular momentum is the statement that the time rate of change of
angular momentum must equal the sum of the applied moments. The conservation
of energy is the requirement that the time rate of change of the sum of all the kinetic
and internal energies must equal the mechanical power and heat power supplied to
the object.
In the next section the continuum formulation of the conservation of mass is
developed. In the following section the concept of stress is introduced and its
important properties are derived and illustrated. The conservation of momentum,
or the second law of Newton, when expressed in terms of stress, is called the stress
equation of motion . The conservation of angular momentum is employed in the
development of the stress equations of motion to show that the stress tensor is
symmetric. In the last section the continuum formulation of the conservation of
energy is developed.

3.2 The Conservation of Mass

The total mass M at time t of an object O is given by

ð
O rðx;

M ¼ t Þ d v ; (3.1)

where ( x , t ) is the mass density at the place x within the object at the time t . The

statement of mass conservation for the object O is that M does not change with time:

r

ð
O rð

DM
Dt ¼

D
Dt

x ; t Þ d v ¼ 0 : (3.2)

The material time derivative may be interchanged with the integration over the
object O since a fixed material volume is identified as the object,