1. Introduction to Seismology
Geology 5640/6640
Introduction to Seismology
23 Jan 2015
Last time: Complex Numbers; Tensors; Stress
• A complex number z = x + iy = r ei has both real and
imaginary parts. Its conjugate z* = x – iy = r e–i
Tensors are generalizations of scalars and vectors that
can be used to describe spatially-varying properties.
Tensorial order n determines the number of components
of the tensor (= 3n for 3D)
• Stress is a second-order tensor describing force per unit
area acting in three directions acting on each of three
planes…
A stress traction is a vector that acts on one particular
plane:
Read for Mon 26 Jan: S&W (§ )
© A.R. Lowry 2015

2. We denote the stress components acting on the plane with
normal (i.e. a plane on which x = constant) as:

3. ij More generally we can denote: index i denotes index j denotes
direction of normal
to the plane acted
on by the force
index j denotes
direction of the
force
Hence σxx is a normal stress and σxy and σxz are stresses in
the plane (called “shear stresses”). We can do same with
the y direction: σyx, σyy, σyz act on the plane parallel to
xz-plane. For the z-direction, σzx, σzy, σzz act on the plane
parallel to the xy-plane. Thus we can use all nine σij to
represent the internal force distribution at point P for any
plane passing through P.

4. We combine our 9 components of stress defined in these
coordinate planes to represent the stress on any surface
through the medium. The 9 σij define the state of stress
at point P. The full stress tensor is given by :
Recall that strain is the deformation that results from a
given stress. A common PhD exam question is to
derive stress and strain & elucidate their difference,
particularly in relation to your specific discipline…

5. Note that our text introduces these concepts with slightly
different diagrams & notation:

6. (Note that many texts
also use  to denote
stress instead of ).
In order for the body to remain
in static equilibrium, we require
that the stresses balance. This
effectively requires that
ij = ji

7. The stress tensor is symmetric,
with only six independent elements. The diagonal
terms ij for i = j are the normal stresses. These are
defined (in our usage at least) positive outward, implying
a volumetric expansion for strain, and tension at the point.
Hence compression has negative sign.
The off-diagonal terms, ij; i ≠ j, are shear stresses.

8. Units of stress are force per unit area, or kg m-1 s-2.
The mks (SI) unit is Pa; but also commonly expressed
in bars (= 106 dyn cm-2). 1 bar = 105 Pa.
For any state of stress, there exists a set of orthogonal
coordinate axes for which stress is entirely normal (i.e.,
shear stresses are zero). These normal stresses are
called principal stresses, and the coordinate axis
directions are principal axes.
These correspond to the eigenvalues and
eigenvectors of the tensor, and can be found using
linear algebra. Eigenvalues are the set of scalar
values  for which
for some choice of eigenvector . Eigenvectors define
angles for transformation between the axis system of
and the principal axis system.

9. can be rearranged as ,
where is the identity matrix (e.g., for 3D):
This system of equations has a nontrivial (nonzero)
solution only when the determinant is equal to
zero:
which turns out to be a (3D = 3rd-order) polynomial
equation in . Once the roots of the polynomial are
found, the eigenvectors can be solved for each 
by substituting and solving.

10. Example: Let be and . Then
This has polynomial roots  = 1, 2, 3.

11. For our particular application (solving for principal stress
and axes), we can write the problem in indicial
notation as:
where ij denotes the Kronecker delta:
Within the Earth, stress is dominated by pressure due to
the weight of overlying material. We commonly
remove that effect by removing the mean stress
M = Tr[ij]/3 = ii/3
to get a deviatoric stress :