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§2.3 BASIC EQUATIONS OF CONTINUUM MECHANICS

Continuum mechanics is the study of how materials behave when subjected to external influences.
External influences which affect the properties of a substance are such things as forces, temperature, chemical
reactions, and electric phenomena. Examples of forces are gravitational forces, electromagnetic forces, and
mechanical forces. Solids deform under external forces and so deformations are studied. Fluids move under
external forces and so the velocity of the fluid is studied.
A material is considered to be a continuous media which is a collection of material points interconnected
by internal forces (forces between the atoms making up the material). We concentrate upon the macroscopic
properties rather than the microscopic properties of the material. We treat the material as a body which is
homogeneous and continuous in its makeup.
In this introduction we will only consider solid media and liquid media. In general, most of the ideas
and concepts developed in this section can be applied to any type of material which is assumed to be a
collection of material points held together by some kind of internal forces.
An elastic material is one which deforms under applied forces in such a way that it will return to its
original unloaded state when the applied forces are removed. When a linear relation exists between the
applied forces and material displacements, then the material is called a linear elastic material. In contrast, a
plastic material is one which deforms under applied forces in such a way that it does not return to its original
state after removal of the applied forces. Plastic materials will always exhibit some permanent deformation
after removal of the applied forces. An elastic material is called homogeneous if it has the same properties
throughout. An isotropic material has the same properties, at a point, in all directions about the point.
In this introduction we develop the basic mathematical equations which describe how a continuum
behaves when subjected to external forces. We shall discover that there exists a set of basic equations
associated with all continuous material media. These basic equations are developed for linear elastic materials
and applied to solids and fluids in later sections.

Introduction to Elasticity

Take a rubber band, which has a rectangular cross section, and mark on it a parallelepiped having a
length `, a width w and a height h, as illustrated in the figure 2.3-1.
Now apply a force F to both ends of the parallelepiped cross section on the rubber band and examine
what happens to the parallelepiped. You will see that:
1. ` increases by an amount ∆`.
2. w decreases by an amount ∆w.
3. h decreases by an amount ∆h.
There are many materials which behave in a manner very similar to the rubber band. Most materials,
when subjected to tension forces will break if the change ∆` is only one or two percent of the original length.
The above example introduces us to several concepts which arise in the study of materials when they are
subjected to external forces. The first concept is that of strain which is defined as

change in length
strain = , (dimensionless).
original length