Mechanical Modeling of Material Behavior - Continuum Mechanics of Anisotropic Materials - page 2

volumes are selected to satisfy the criteria of the modeler, which are generally to
define the problem by representing the unknown quantities and the known
quantities in a way that the conservation principles can be applied to obtain an
equation(s) representing the unknown quantities in terms of the known quantities.
The construction of free object diagrams or control volumes is an artisan-like skill
because there is not a unique way to construct them; the modeler must have insight
into the problem. Many such physically correct diagrams can be drawn but it is
likely that only a few will yield the modeler the relationship between quantities that
he or she sought. This creative aspect of the application of conservation principles
of mechanics is prominent because of the great diversity of the situations to which
they are applied. These applications range in size from the Nano scale to the Macro
scale, from a portion of a protein to a molecule to bridges, airplane structures, and
the structure of the universe.
Both free object diagrams and control volumes are drawings made to simplify
the application of conservation principles to a particular physical situation. A
conservation principle can often be written in the form of an accounting statement:

½ The time rate of change of a quantity in a system

¼½ the amount of the quantity coming into the system per unit time

½ the amount of the quantity leaving the system per unit time

þ½ the amount of the quantity produced within the system per unit time

½ the amount of the quantity consumed within the system per unit time :

Thus the application of the conservation of mass (or momentum) of a fluid
employs a drawing to balance the net flow or change in the quantity, much like a
financial account for an organization is balanced.
Note that the word “body” used in modeling nonbiological applications of
Newton's laws, for example “free body diagrams” and “rigid body motion,” is
replaced in this work by the word “object”; thus reference is made here to “free
object diagrams” and “rigid object motions.” The reason for this shift in terminol-
ogy is to avoid the use of the same word for two different meanings in the same
phrase, like a “free body diagram” of a body or a “rigid body motion” of a body.

1.3 Models and the Real Physical World

Models have been found to be very effective tools for the analysis of physical
problems. The basic elements of these models are Euclidean or classical geometry
and the concept of time. The concept of time is intuitive while our mathematical
model of time, the real line, that is to say the line representing all real numbers, is
abstract. The connection between the mathematical abstraction and our intuitive
perception of time is a philosophical matter that should be accepted by the reader;