The nine stress components are shown acting on
the surfaces of this infinitesimal element in Figure 12-7
*
b
*
and
*
c
*
. The components
*
xx
*
,
*
yy
*
,
and
*
zz
*
are
the normal stresses, so called because they act, respectively, in directions
normal to the
*
x
*
,
*
y,
*
and
*
z
*
surfaces
of the cube. The components
*
xy
*
and
*
xz
*
,
for example, are shear stresses that act on the
*
x
*
face
and whose directions of action are parallel to the
*
y
*
and
*
z
*
axes, respectively. The sign of any one of
these components is defined as positive if the signs of its surface normal and
its stress direction are the same, and as negative if they are different. Thus
the components shown in Figure 12-7
*
b
*
are all positive because they
are acting on the positive faces of the cube and their directions are also
positive. The components shown in Figure 12-7
*
c
*
are all
negative because they are acting on the positive faces of the cube and their
directions are negative. This sign convention makes tensile normal stresses
positive and compressive normal stresses negative.

For
the 2-D case, only one face of the stress cube may be drawn. If the
*
x
*
and
*
y
*
directions are retained and
*
z
*
eliminated,
we look normal to the
*
xy
*
plane of the cube of Figure 12-7 and see the
stresses shown in Figure 12-8, acting on the unseen faces of the cube. The
reader should confirm that the stress components shown in Figure 12-8 are all
positive by the sign convention stated above.

Note
that the definition of the dual subscript notation given above is consistent
when applied to the normal stresses. For example, the normal stress
*
xx
*
acts
on the
*
x
*
face and is also in the
*
x
*
direction.
Since the subscripts are simply repeated for normal stresses, it is common to
eliminate one of them and refer to the normal components simply as
*
x
*
,
*
y
*
,
and
*
z
*
.
Both subscripts are needed to define the shear-stress components and they will
be retained. It can also be shown[2] that the stress tensor is symmetric, which
means that

This
reduces the number of stress components to be calculated.

**
**

**
12.7 STRAIN
**

Stress
and strain are linearly related by Hooke’s law in the elastic region of most
engineering materials as discussed in Chapter 2. Strain is also a second-order
tensor and can be expressed for the 3-D case as

and
for the 2-D case as

where
represents
either a normal or a shear strain, the two being differentiated by their
subscripts. We will also simplify the repeated subscripts for normal strains to
*
x
*
,
*
y
*
,
and
*
z
*
for
convenience while retaining the dual subscripts to identify shear strains. The
same symmetric relationships shown for shear stress components in equation 12.3
also apply to the strain components.

**
**

**
12.8 PRINCIPAL STRESSES
**

The
axis systems taken in Figures 12-7 and 12-8 are arbitrary and are usually
chosen for convenience in computing the applied stresses. For any particular
combination of applied stresses, there will be a continuous distribution of the
stress field around any point analyzed. The normal and shear stresses at the
point will vary with direction in any coordinate system chosen. There will
always be planes on which the shear-stress components are zero. The normal
stresses acting on these planes are called the principal stresses. The planes
on which these principal stresses act are called the
**
principal planes
**
. The directions of the surface normals to the
principal planes are called the
**
principal
axes
**
, and the normal stresses
acting in those directions are the
**
principal
normal stresses
**
. There will
also be another set of mutually perpendicular axes along which the shear
stresses will be maximal. The
**
principal
shear stresses
**
act on a set of
planes that are at 45
°
angles to the planes of the principal normal
stresses. The principal planes and principal stresses for the 2-D case of
Figure 12-8 are shown in Figure 12-9.

**
FIGURE 12-9
**

Principal stresses on a two-dimensional
stress element

Since,
from an engineering standpoint, we are most concerned with designing our
machine parts so that they will not fail, and since failure will occur if the
stress at any point exceeds some safe value, we need to find the largest
stresses (both normal and shear) that occur anywhere in the continuum of
material that makes up our machine part. We may be less concerned with the
directions of those stresses than with their magnitudes as long as the material
can be considered to be at least macroscopically isotropic, thus having
strength properties that are uniform in all directions. Most metals and many
other engineering materials meet these criteria, although wood and composite
materials are notable exceptions.

The
expression relating the applied stresses to the principal stresses is

where
is
the principal stress magnitude and
*
nx
*
,
*
ny
*
, and
*
nz
*
are the direction cosines of
the unit vector
**
n
**
, which is normal to the principal plane:

For
the solution of equation 12.5a to exist, the determinant of the coefficient
matrix must be zero. Expanding this determinant and setting it to zero, we
obtain

where

Equation
12.5c is a cubic polynomial in
. The coefficients
*
C
*
0,
*
C
*
1, and
*
C
*
2 are called the tensor invariants because they
have the same values regardless of the initial choice of
*
xyz
*
axes
in which the applied stresses were measured or calculated. The units of
*
C
*
2
are psi (MPa), of
*
C
*
1 psi2 (MPa2), and of
*
C
*
0
psi3 (MPa3). The three principal (normal) stresses
1,
2,
3 are the three roots of this cubic polynomial
The roots of this polynomial are always real[2] and are usually ordered such
that
1 >
2 >
3. If needed, the directions of the principal stress
vectors can be found by substituting each root of equation 12.5c into 12.5a and
solving for
*
nx
*
,
*
ny
*
, and
*
nz
*
for each of the three
principal stresses. The directions of the three principal stresses are mutually
orthogonal.