STRESS I
Geologists are not using forces. Instead
they use stresses! Why stress?
Consider two blocks of different size
that are subjected to an equal force F~ .
Intuitively, the smaller block is going to
deform a lot more than the larger block.
For this reason it makes sense to work
with stresses rather than with forces.
STRESS = FORCE / AREA
Like force, the stress is a vector. The
units of stress (in mks) are Pascal,
where:
Pascal = Newton / m2
A stress of 1Pa is very small! For
example, the load due to 1m of water is
about 104 Pa (why?). For this reason,
geologists use MPa, which is 106
Pascals.
Other stress units are bars and atm:
1MPa = 106 Pa = 10bars = 9.8692atm
Shear Versus Normal Stress
In general, a stress acting on a plane
may be expressed as a sum of shear and
normal stresses.
Normal Stress is the component of
stress acting perpendicular to the plane
in question.
Shear Stress is the component of stress
acting parallel to the plane.
Consider a small cubic element of rock
extracted from the earth. The stresses
acting on this element may be
visualized as follows:
The first index indicates the plane in
question (recall that a plane is
indicated by the direction of its
outward normal).
The second index indicates the
direction of the stress.
Sign Convention
Unfortunately, the sign convention
adopted by geologists is different than
that adopted by engineers.
In Engineering, the stress is positive if
it acts in the positive direction on the
positive plane. In other words, the
stress is positive in tension, and
negative in compression.
In Geology, the stress is positive if it
acts in the negative direction on the
positive plane. In other words, the
stress is negative in tension, and
positive in compression. Why is that?
This is because the stresses in the earth
are compressive (although locally,
tensional stresses are also possible).
Stress Tensor
It is convenient to ”pack” the stress
components into a tensor form.
In 3D:

σ11 σ12 σ13





σij = σ21 σ22 σ23 


σ31 σ32 σ33
In 2D, that is when
σ13 = σ31 = σ23 = σ32 = σ33 = 0:

σij = 
σ11 σ12
σ21 σ22


The Symmetry of the Stress Tensor
The torque has to be zero, otherwise
the block rotates. In the example
below, the condition of zero torque may
be written as:
−(σxy δy)δx/2 + (σyx δx)δy/2 = 0
And since δxδy 6= 0, we get that
σxy = σyx , and the number of
independent stress components is equal
to 3.
Similarly, in 3D we get that: σxy = σyx ,
σxz = σzx , and σyz = σzy , and the
number of independent stress
components is equal to 6.
Cauchy’s Formula
Consider a small cubic element of rock
extracted from the earth. Now imagine
a plane boundary with outward normal,
n, and area, δA cutting through this
element so it is reduced to a triangular
element with sides 1 and 2.
The force components are:
f1x = −σxx δAnx
f1y = −σxy δAnx
f2x = −σyx δAny
f2y = −σyy δAny
Summing the force components and
setting these sums to zero we have:
X
fx = tx δA − σxx δAnx − σyx δAny = 0
X
fy = ty δA − σxy δAnx − σyy δAny = 0
Rearranging:
tx = σxx nx + σyx ny
ty = σxy nx + σyy ny
This is equivalent to:
tj = σij ni
where t is the traction acting on n .
Principal Stresses
We have learned that the stress tensor
is symmetric. A property of symmetric
matrices is that they may be
diagonalized. The transformation from
the non-diagonal to the diagonal tensor
requires transformation of the
coordinate system. The axes of the new
coordinate system are the principal
axes, and the diagonal elements of the
tensor are referred to as the principal
stresses.

σ1 0
0





∗
σij =  0 σ2 0 


0 0 σ3
Next we will see that the shear stress
along the principal axes is zero.
The Shear and Normal Stress on Any
Plane
Adding vectors in directions parallel
and normal to the plane in question:
FN = F1 cos θ + F3 sin θ
FS = F1 sin θ − F3 cos θ
Expressing forces in terms of stresses
leads to:
2
2
σN = σ1 cos θ + σ3 sin θ
σS = (σ1 − σ3 ) sin θ cos θ
Substituting these identities:
sin2 θ = (1 − cos 2θ)/2
sin θ cos θ = sin 2θ/2
gives:
σ1 + σ3 σ1 − σ3
+
cos 2θ
σN =
2
2
σ1 − σ3
σS =
sin 2θ
2
The above equation defines a circle
with a center on the horizontal axes at
(σ1 + σ3 )/2, and a radius that is equal
to (σ1 − σ3 )/2