Here is the angle between
the two unit vectors which, in this case, must be
/2. A similar derivation for the other two pairs of
principal stresses shows that they are mutually
orthogonal.
If the Cartesian coordinate system is chosen to
coincide with the axes of principal stress at a
point then all the off-diagonal terms in the matrix
of stress components (6.35) are zero and the diagonal terms are the principal stresses:
(6.64)
΄
1
0
0
0
2
0
0
0
3
΅
n(1) · n(2) ϭ cos ϭ 0.
The stress state
(MPa) is illustrated in Fig. 6.20a as a traction ellipsoid. In this
illustration we have chosen xy ϭ yz ϭ zx ϭ 0, so
the principal axes of stress are aligned with the
coordinate axes, and
If the three principal stresses were equal
the state of stress would be isotropic
and the traction ellipsoid would degenerate to a
sphere. In this case any orthogonal set of axes may
serve as the principal axes. If two of the principal
stresses were equal the traction vectors would
trace out an ellipsoid with the one unequal axis
being the axis of revolution. In the plane perpendicular to the axis of revolution any two orthogonal axes may serve as the principal axes and only
the principal axis parallel to the axis of revolution
is unique. Because the three principal stresses are
unequal in this example the traction vectors trace
out an ellipsoid with three unequal axes which
are coincident with the coordinate axes (Fig.
( 1 ϭ 2 ϭ 3 )
xx ϭ 1 , yy ϭ 2 , zz ϭ 3 .
1 ϭ 1, 2 ϭ
1
2 , 3 ϭ
1
4
218
FORCE, TRACTION, AND STRESS
Fig 6.20 (a) Traction ellipsoid for state of stress with
principal axes parallel to coordinate axes. (b) Traction
ellipsoid for state of stress defined in (6.68) with principal
axes not parallel to coordinate axes.
1
0
1
0
z
y
x
(a)
2
1
0
1
2
1
0
1
z
y
x
0.5
0.5
0.5
0.5
1.5
1.5
(b)
0.5
0
0.5
0.2
0.1
0
0.1
0.2
0.5
0.5
0.5
0.5
Fig 6.19 (a) Geometric relations among the Cartesian
coordinates and the unit normal vectors directed parallel to
the axes of principal stresses. (b) Volume element on which
principal stresses act.
n(1)
x
y
z
(a)
(x, 1)
(z, 1)
(y, 1)
O
n(2)
n(3)
s 2
s 3
s 1
z
y
x
(b)
the two unit vectors which, in this case, must be
/2. A similar derivation for the other two pairs of
principal stresses shows that they are mutually
orthogonal.
If the Cartesian coordinate system is chosen to
coincide with the axes of principal stress at a
point then all the off-diagonal terms in the matrix
of stress components (6.35) are zero and the diagonal terms are the principal stresses:
(6.64)
΄
1
0
0
0
2
0
0
0
3
΅
n(1) · n(2) ϭ cos ϭ 0.
The stress state
(MPa) is illustrated in Fig. 6.20a as a traction ellipsoid. In this
illustration we have chosen xy ϭ yz ϭ zx ϭ 0, so
the principal axes of stress are aligned with the
coordinate axes, and
If the three principal stresses were equal
the state of stress would be isotropic
and the traction ellipsoid would degenerate to a
sphere. In this case any orthogonal set of axes may
serve as the principal axes. If two of the principal
stresses were equal the traction vectors would
trace out an ellipsoid with the one unequal axis
being the axis of revolution. In the plane perpendicular to the axis of revolution any two orthogonal axes may serve as the principal axes and only
the principal axis parallel to the axis of revolution
is unique. Because the three principal stresses are
unequal in this example the traction vectors trace
out an ellipsoid with three unequal axes which
are coincident with the coordinate axes (Fig.
( 1 ϭ 2 ϭ 3 )
xx ϭ 1 , yy ϭ 2 , zz ϭ 3 .
1 ϭ 1, 2 ϭ
1
2 , 3 ϭ
1
4
218
FORCE, TRACTION, AND STRESS
Fig 6.20 (a) Traction ellipsoid for state of stress with
principal axes parallel to coordinate axes. (b) Traction
ellipsoid for state of stress defined in (6.68) with principal
axes not parallel to coordinate axes.
1
0
1
0
z
y
x
(a)
2
1
0
1
2
1
0
1
z
y
x
0.5
0.5
0.5
0.5
1.5
1.5
(b)
0.5
0
0.5
0.2
0.1
0
0.1
0.2
0.5
0.5
0.5
0.5
Fig 6.19 (a) Geometric relations among the Cartesian
coordinates and the unit normal vectors directed parallel to
the axes of principal stresses. (b) Volume element on which
principal stresses act.
n(1)
x
y
z
(a)
(x, 1)
(z, 1)
(y, 1)
O
n(2)
n(3)
s 2
s 3
s 1
z
y
x
(b)