Section 3.1
Solid Mechanics Part I
Kelly
29
3.1
Surface and Contact Stress
The concept of the force is fundamental to mechanics and many important problems
can be cast in terms of forces only, for example the problems considered in Chapter 2.
However, more sophisticated problems require that the action of forces be described
in terms of
stress
, that is, force divided by area.
For example, if one hangs an object
from a rope, it is not the weight of the object which determines whether the rope will
break, but the weight divided by the cross-sectional area of the rope, a fact noted by
Galileo in 1638.
3.1.1 Stress Distributions
As an introduction to the idea of stress, consider the situation shown in Fig. 3.1.1a: a
block of mass
m
and cross sectional area
A
sits on a bench.
Following the
methodology of Chapter 2, an analysis of a free-body of the block shows that a force
equal to the weight
mg
acts upward on the block, Fig. 3.1.1b.
Allowing for more
detail now, this force will actually be distributed over the surface of the block, as
indicated in Fig. 3.1.1c.
Defining the stress to be force divided by area, the stress
acting on the block is
A
mg
=
σ
(3.1.1)
The unit of stress is the Pascal (Pa): 1Pa is equivalent to a force of 1 Newton acting
over an area of 1 metre squared.
Typical units used in engineering applications are
the kilopascal, kPa (
Pa
10
3
), the megapascal, MPa (
Pa
10
6
) and the gigapascal
(P
a
10
9
).
Figure 3.1.1: a block resting on a bench; (a) weight of the block, (b) reaction of
the bench on the block, (c) stress distribution acting on the block
The stress distribution of Fig. 3.1.1c acts on the block.
By Newton’s third law, an
equal and opposite stress distribution is exerted by the block on the bench; one says
that the weight force of the block is
transmitted
to the underlying bench.
The stress distribution of Fig. 3.1.1 is
uniform
, i.e. constant everywhere over the
surface.
In more complex and interesting situations in which materials contact, one is
more likely to obtain a
non-uniform
distribution of stress.
For example, consider the
case of a metal ball being pushed into a similarly stiff object by a force
F
, as
mg
(a)
(c)
mg
(b)