A View of the Material at the Atomic Scale:
We
know that every physical object is made up of molecules, atoms and even smaller
particles. These particles are not continuously distributed over the object.
Microscopic observations reveal that there are gaps (empty spaces) between
particles. Consider an atomic structure of a metal in which the atoms are
separated by inter-atomic distance of the order of
. The nucleus of the atom where most of the mass (neutron and protons) are
concentrated are at least three order lower, thus leaving a vast empty space
where the electrons revolve. In essence the physical space occupied by materials
is very very small. However, this effect is never felt in the everyday
experience of dealing with materials. For all practical purposes, we ignore that
the material is a continuously occupied by matter.
Micro,
Meso and Mesoscopic Scales of the Materials.
Though
there are many possible scales description of materials in terms of
characteristic lengths is very useful. For that pupose if we analyze the problem
at the scale of micrometers
or less
then the descriptions refers to microscopic
scale of the materials. Though in the realms of nuclear physics a scale of
, sometimes referred to as nanoscopic scale, is used in the study of mechanics
of continuous media we will still refer to them as microscopic description.
Understanding the effect of point (vacancy, interstitials), and line (edge or
screw dislocations) defects on the field falls under this category. In the
mesoscopic analysis, we are interested in scales between
. In this scale, we can analyze the effect of individual grains, void, cavities,
cracks and grain boundaries. In the macroscopic scale,
we include the study of structures anywhere between electronic devices, to
automobiles to large space shuttles.
Physical
Scale of the Problem
Every
physical problem in nature, based on mechanics or otherwise has a length scale
associated with it. All of those problems are described by a set of governing
field equations be it be based on mechanics, thermodynamics, magnetic or
electrical fields. With each of the specific problem, there is a characteristic
length scale. For example if one were to study the effect of cracks on the
failure strength of the material, the size of the crack is the characteristic
length. In this case it ranges from a few tenths of a mm (
) to a few mm. If we are to study the effect of the deflection of a large bridge
under dynamic loading, then a few mm is the characteristic length. Even for the
same physical problem, the length scale varies depending on what is the specific
isssue we are analyzing. If we like to study the viscous drag of air on an
airplane, we will focus on the boundary layer which is a less than a mm. On the
other hand, if we like to evaluate the lift of the same plane, the projected
area is the critical parameter leading to a characteristic length a few
decimeters.
Validity of Continuum Assumptions.
In order to validate the assumptions of continuity, we need to compare the
characteristic length of the problem with that of the discontinuity in the
material. For the sake of simplicity, we can assume that the assumptions of
continuity is valid if the material discontinuity is at least two orders of
magnitude lower than that of the characteristic length of the problem. For
example the atomic level discontinuity can be ignored in fracture mechanics
problem since the latter has a characteristic length scale of 1000
compared
to the atomic discontinuity of
.
In
general, mechanics of continuum medium attempts to relate the deformation of a
body from an undeformed to deformed state under the action of all external and
internal forces. The assumption of continuity of material particles that make up
of the body implies that there is a one to one correspondence between the
original and current configurations. That is for every particle
in the original configurations has a corresponding
in
the deformed configuration and there is one and only particle that has a
correspondence in both the states. Thus the deformation
has
one to one correspondence such that the inverse
exists and is unique. Also both the functions can have derivatives of any given
order because of the continuity assumptions. This assumption is important in the
definition of deformation gradient and strain quantities.