12.6 STRESS
Stress
is defined as force per unit area with units of psi or MPa. In a part subjected
to some forces, stress is generally distributed as a continuously varying
function within the continuum of material. Every infinitesimal element of the
material can conceivably experience different stresses at the same time. Thus,
we must look at stresses as acting on vanishingly small elements within the
part. These infinitesimal elements are typically modeled as cubes, shown in
Figure 12-7. The stress components are considered to be acting on the faces of
these cubes in two different manners.
Normal
stresses
act perpendicular (i.e., normal) to the face of the cube and tend
to either pull it out (tensile normal stress) or push it in (compressive normal
stress).
Shear stresses
act parallel to the faces of the cubes, in
pairs (couples) on opposite faces, which tends to distort the cube into a
rhomboidal shape. This is analogous to grabbing both pieces of bread of a
peanut butter sandwich and sliding them in opposite directions. The peanut
butter will be sheared as a result. These normal and shear components of stress
acting on an infinitesimal element make up the terms of a
tensor
.*
* For a discussion of tensor notation, see C. R.,
Wylie and L. C. Barrett,
Advanced Engineering Mathematics
, 6th ed., McGraw-Hill, New York, 1995.
Stress
is a tensor of order two† and thus requires nine values or components to
describe it in three dimensions. The 3-D stress tensor can be expressed as the
matrix:
where
each stress component contains three elements, a magnitude (either
or
), the direction of a normal to the reference
surface (first subscript), and a direction of action (second subscript). We
will use
to
refer to normal stresses and
for
shear stresses.
† Equation 12.2a is more correctly a tensor for
rectilinear Cartesian coordinates. The more general tensor notation for
curvilinear coordinate systems will not be used here.
Many
elements in machinery are subjected to three-dimensional stress states and thus
require the stress tensor of equation 12.2a. There are some special cases,
however, which can be treated as two-dimensional stress states. The stress
tensor for 2-D is
Figure
12-7 shows an infinitesimal cube of material taken from within the material
continuum of a part that is subjected to some 3-D stresses. The faces of this
infinitesimal cube are made parallel to a set of
xyz
axes
taken in some convenient orientation. The orientation of each face is defined
by its surface normal vector* as shown in Figure 12-7
a
.
The
x
face has its surface normal parallel to the
x
axis,
etc. Note that there are thus two
x
faces, two
y
faces,
and two
z
faces, one of each being positive and one
negative as defined by the sense of its surface normal vector.
* A surface normal vector is defined as “growing out
of the surface of the solid in a direction normal to that surface.” Its sign is
defined as the sense of this surface normal vector in the local coordinate
system.