CONTINUUM MECHANICS for ENGINEERS

3 Stress Principles
3.1 Body and Surface Forces, Mass Density
Stress is a measure of force intensity, either within or on the bounding surface
of a body subjected to loads. It should be noted that in continuum mechanics
a body is considered stress free if the only forces present are those interatomic
forces required to hold the body together. And so it follows that the
stresses that concern us here are those which result from the application of
forces by an external agent.
Two basic types of forces are easily distinguished from one another and
are defined as follows. First, those forces acting on all volume elements, and
distributed throughout the body, are known as body forces. Gravity and inertia
forces are the best-known examples. We designate body forces by the vector
symbol b i (force per unit mass), or by the symbol p i (force per unit volume).
Second, those forces which act upon and are distributed in some fashion
over a surface element of the body, regardless of whether that element is
part of the bounding surface, or an arbitrary element of surface within the
body, are called surface forces. These are denoted by the vector symbol f i, and
have dimensions of force per unit area. Forces which occur on the outer
surfaces of two bodies pressed against one another (contact forces), or those
which result from the transmission of forces across an internal surface are
examples of surface forces.
Next, let us consider a material body B having a volume V enclosed by a
surface S, and occupying a regular region R 0 of physical space. Let P be an
interior point of the body located in the small element of volume ∆V whose
mass is ∆m as indicated in Figure 3.1. Recall that mass is that property of a
material body by virtue of which the body possesses inertia, that is, the
opposition which the body offers to any change in its motion. We define the
average density of this volume element by the ratio
ρ ave
m
=
V
∆
∆
(3.1-1)