Page 188 - Introduction to Continuum Mechanics

4

Stress

In the previous chapter, we considered the purely kinematical description of the motion of
a continuum without any consideration of the forces that cause the motion and deformation.
In this chapter, we shall consider a means of describing the forces in the interior of a body
idealized as a continuum. It is generally accepted that matter is formed of molecules which in
turn consists of atoms and subatomic particles. Therefore, the internal forces in real matter
are those between the above particles. In the classical continuum theory the internal forces
are introduced through the concept of body forces and surface forces. Body forces are those
that act throughout a volume (e.g., gravity, electrostatic force) by a long-range interaction with
matter or charge at a distance. Surface forces are those that act on a surface (real or imagined)
separating parts of the body. We shall assume that it is adequate to describe the surface force
at a point of a surface through the definition of a stress vector, discussed in Section 4.1, which
pays no attention to the curvature of the surface at the point. Such an assumption is known as
Canchy's stress principle which is one of the basic axioms of classical continuum mechanics.

4.1 Stress Vector

Let us consider a body depicted in Fig. 4.1. Imagine a plane such as S, which passes through
an arbitrary internal point P and which has a unit normal vector n. The plane cuts the body
into two portions. One portion lies on the side of the arrow of n (designated by II in the figure)
and the other portion on the tail of n (designated by I). Considering portion I as a free body,
there will be on plane S a resultant force AF acting on a small area A/l containing P. We
define the stress vector (from II to I) at the point P on the plane S as the limit of the ratio
AF /AA as A^4 -*0. That is, with t,, denoting the stress vector,

If portion II is considered as a free body, then by Newton's law of action and reaction, we shall
have a stress vector (from I to II), t_ n at the same point on the same plane equal and opposite
to that given by Eq. (4.1.1). That is,

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