CONTINUUM MECHANICS for ENGINEERS

1 Continuum Theory
1.1 The Continuum Concept
The atomic/molecular composition of matter is well established. On a small
enough scale, for instance, a body of aluminum is really a collection of
discrete aluminum atoms stacked on one another in a particular repetitive
lattice. On an even smaller scale, the atoms consist of a core of protons and
neutrons around which electrons orbit. Thus, matter is not continuous. At
the same time, the physical space in which we live is truly a continuum, for
mathematics teaches us that between any two points in space we can always
find another point, regardless of how close together we choose the original
pair. Clearly then, although we may speak of a material body as “occupying”
a region of physical space, it is evident that the body does not totally “fill”
the space it occupies. However, if we accept the continuum concept of matter,
we agree to ignore the discrete composition of material bodies, and to assume
that the substance of such bodies is distributed uniformly throughout, and
completely fills the space it occupies. In keeping with this continuum
model, we assert that matter may be divided indefinitely into smaller and
smaller portions, each of which retains all of the physical properties of the
parent body. Accordingly, we are able to ascribe field quantities such as
density and velocity to each and every point of the region of space which
the body occupies.
The continuum model for material bodies is important to engineers for
two very good reasons. On the scale by which we consider bodies of steel,
aluminum, concrete, etc., the characteristic dimensions are extremely large
compared to molecular distances so that the continuum model provides a
very useful and reliable representation. Additionally, our knowledge of the
mechanical behavior of materials is based almost entirely upon experimental
data gathered by tests on relatively large specimens.