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30 MPa. It is important to realize that all resulting angles
of these tractions over the corresponding surfaces, except
the two initial directions acting over
A
2 and
A
3, are dif-
ferent from 90
Normal Stress component (
s
n
)
. In other words, in a 2D study of the
stress tensor, for all possible orientations there are only
two directions in which the tractions are located at right
angles over the surface. The rest are inclined with differ-
ent angles over the inclined surfaces. This allows us to
resolve for normal and shear stress components of the
tractions.
As we have seen for the particular case of hydrostatic
stress, all tractions have the same value in all directions,
act perpendicular to any surfaces and there are no shear
stresses. As an example, try substituting equal values of
the vertical and horizontal components into the previous
case, for example 25 MPa. Calculate the value of the trac-
tion acting over any inclined surface (65
Extensional
Compressional
Shear Stress component (
t
)
Left handed (+)
Right handed (-)
Fig. 3.65
Sign conventions for the normal and shear stress
components of the stress.
or any other)
and subsequently, the angle of the selected surface. The
calculated tractions should have a constant value of
25 MPa and the angle over the corresponding surface
should be of 90
(a)
s
zz
s
xx
t
xz
τ
zx
s
zz
. Thus all tractions are normal stresses
and the shear stress values are always zero. Try different
angles for the inclined surfaces, the result should always
be the same.
t
zx
s
xx
s
xx
(b)
s
1
t
xz
t
xz
3.13.3
Stress notation conventions
t
xz
s
3
s
zz
Normal stresses are compressive when the traction compo-
nents converge at the surface and the arrows of the pair of
tractions, defining the surface stress components point to
each other; any two blocks of rock at either side of the sur-
face are being pushed closer together (Fig. 3.65).
Conversely, normal stresses are tensile when the surface
stress traction components diverge from the surface, the
two arrows point away from each other and the blocks of
rock on each side are being pulled apart. Normal stresses
are the components which tend to press or decompress the
masses of rock over a surface. In structural geology, it is
conventional that compressive stresses are considered pos-
itive and tensile stresses negative. Shear stress components
promote sliding of rock masses at each side of the surface.
Shear stresses are right-handed or clockwise when, facing a
surface, the direction is to the right (a ball positioned
between both arrows would rotate clockwise). Shear
stresses are left-handed or anticlockwise when the arrow
points left and a ball would rotate anticlockwise. Right-
handed shear stresses are generally considered negative
and left-handed positive, although sign conventions do
not have general agreement. Our usage is the same as that
used for vorticity (Section 3.8).
s
3
(c)
s
1
≥
s
3
s
1
s
1
0
0
s
3
s
1
s
3
Fig. 3.66
(a) The stress state at a point centered on a cube in 2D is
defined by two normal stresses and two shear stresses; (b) The main
stress directions; (c) The stress ellipse.
3.13.4 The stress ellipse, the stress ellipsoid, and
the principal stresses
The whole family of tractions acting over a certain point
define an ellipse in 2D and an ellipsoid in 3D. In 2D, the
longest and shortest tractions correspond to the axis of the
ellipse which are located at 90
to each other and are called
respectively
3
(Fig. 3.66). These surface stresses
are called the
principal stresses
and their corresponding
directions
principal stress directions
. The principal stresses
1
and
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