Mechanical Modeling of Material Behavior - Continuum Mechanics of Anisotropic Materials - page 3

The Mathematical Model
of the Object

The Real Object

A similarly shaped
portion of a three-
dimensional space

A Cartesian Reference
Coordinate System

THE REAL WORLD

EUCLIDEAN 3 SPACE

Fig. 1.1 The Euclidean space model of a real object

this relationship is rigorously discussed in a classical topic by the German-
American mathematician Hermann Weyl ( 1883 -1955) entitled “The Continuum.”
Once one has accepted the real line as a geometric model for time, the next step of
accepting the real line as a geometric model for a one-dimensional structure (such
as a string or a fiber) is not difficult. Real lines are used in forming one-, two-, and
three-dimensional Cartesian reference coordinate systems for one-, two-, and three-
dimensional Euclidean spaces. The real lines used as the reference Cartesian
coordinate systems in two and three dimensions are mutually orthogonal.
In the initial stages of model construction for a material object of any size or shape,
it is necessary to project the object into Euclidean space (Fig. 1.1 ). The portion of the
three-dimensional Euclidean space that the object occupies is often an exact replica of
the space in the real world occupied by the real object. However, the structure of the
real object is not carried over into the model unless the modeler makes provision for it.
For example, if the object is fibrous, the fibers are not represented in the model unless
the modeler explicitly provides for their representation. The advantage of the model is
that all the points in the real object now have Cartesian addresses. The triplet of
Cartesian coordinates ( x 1 , x 2 , x 3 ) locates a point in the Euclidean space occupied by a
particular point in the real object. This permits the measurement of distance between
points in the object, and if the object moves or deforms, the resulting motion can be
quantitatively documented. The volume of the object and its centroid may be deter-
mined by use of the integral calculus. The greatest advantage, however, is the ability to
define functions of physical interest in terms of the reference coordinate system. Thus,
for example, the temperature distribution throughout the object can be specified by a
function ( x 1 , x 2 , x 3 ) defined for all of the Cartesian coordinates ( x 1 , x 2 , x 3 )

located within the object. If the function is smooth, then a derivative of the function
can be taken and the temperature gradient is determined for all of the Cartesian
coordinates ( x 1 , x 2 , x 3 ) located within the object. The point of these remarks is that

y ¼ y