Page 16 - Introduction to Continuum Mechanics

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Introduction

1.1 CONTINUUM THEORY

Matter is formed of molecules which in turn consist of atoms and sub-atomic particles. Thus
matter is not continuous. However, there are many aspects of everyday experience regarding
the behaviors of materials, such as the deflection of a structure under loads, the rate of
discharge of water in a pipe under a pressure gradient or the drag force experienced by a body
moving in the air etc., which can be described and predicted with theories that pay no attention
to the molecular structure of materials. The theory which aims at describing relationships
between gross phenomena, neglecting the structure of material on a smaller scale, is known
as continuum theory. The continuum theory regards matter as indefinitely divisible. Thus,
within this theory, one accepts the idea of an infinitesimal volume of materials referred to as
a particle in the continuum, and in every neighborhood of a particle there are always neighbor
particles. Whether the continuum theory is justified or not depends on the given situation; for
example, while the continuum approach adequately describes the behavior of real materials
in many circumstances, it does not yield results that are in accord with experimental observa-
tions in the propagation of waves of extremely small wavelength. On the other hand, a rarefied
gas may be adequately described by a continuum in certain circumstances. At any case, it is
misleading to justify the continuum approach on the basis of the number of molecules in a
given volume. After all, an infinitesimal volume in the limit contains no molecules at all.
Neither is it necessary to infer that quantities occurring in continuum theory must be inter-
preted as certain particular statistical averages. In fact, it has been known that the same
continuum equation can be arrived at by different hypothesis about the molecular structure
and definitions of gross variables. While molecular-statistical theory, whenever available, does
enhance the understanding of the continuum theory, the point to be made is simply that
whether the continuum theory is justified in a given situation is a matter of experimental test,
not of philosophy. Suffice it to say that more than a hundred years of experience have justified
such a theory in a wide variety of situations.

1.2 Contents of Continuum Mechanics

Continuum mechanics studies the response of materials to different loading conditions. Its
subject matter can be divided into two main parts: (1) general principles common to all media,

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