*under construction*

S = lim F/A

A=>0

Like force and velocity, stress is a vector quantity. The stress
vector can be resolved into a component perpendicular to the surface (the
normal stress), and a component parallel to the surface (the tangential,
or * shear * stress)¹.

The stress field about any point in the body may be isotropic (also
called hydrostatic) or anisotropic (non-hydrostatic). If we imagine a
simple, infinitesimal cube, with its sides oriented orthogonally to the
three cartesian coordinates, we can define a normal stress, S1, S2, S3
perpendicular to each pair of sides of the cube. These stresses are
defined so that S1>S2>S3. Note that stress has units of Pascals =
N/m², (where N = newton). The Pascal is the SI unit of pressure; a
Newton is
defined as the force necessary to accelerate a one kilogram mass by one
meter per second, * per* second. Twiss and Moores¹ note that
the weight (which of course is a force = mg, where g = acceleration due
to gravity) of an apple is about one newton! A pressure of 1 bar
(nominally, atmospheric pressure at sea level) is the equivalent of
100,000 apples per square meter.

Generally, we might ask what the state
of stress might be for * any arbitrarily defined plane * that passes
through the point in question (say, a bedding plane). To answer this type
of question, the concept of a tensor was introduced. We use a second rank
tensor to determine the magnitude and orientation of a stress vector
acting on any given plane, and the tensor can be represented by a 3 x 3
matrix. a Vector is a first rank tensor, and the three components of a
vector can be represented by a column matrix. A tensor of rank zero is a
scalar quantity (such as speed, or temperature).

Because the "stress tensor" is discussed in structural geology,
geophysics, fluid dynamics etc., I will say something about this
sometimes mysterious, but in fact, rather simple concept.
When the stress tensor matrix, which mathematically desribes the state of
stress around a point, is multiplied (dot product) by the column vector
which represents the unit vector that is normal to the surface of
interest, the product is *another* vector which represets the stress
vector acting on that surface. Putting it poetically, the state of stress
around a point, and hence the matrix itself, holds the "potential" for a
stress vector that may act on any surface that may pass through the point
in question; however, the stress is only "**realized**" when one
plane (defined by the surface normal) is identifed, by writing the column
vector (the three numbers in the column matrix (vector) are the magnitudes
of the x,y,z components of the surface normal, written
in a column). The general form of a stress tensor is given
by:

|s11 s21 s31|

|s12 s22 s32| = S(m,n)

|s13 s23 s33|

of which an example is

| 2 1 -3|

| 1 1 2|

|-3 2 1|

Now, of these, which numbers represent the normal stresses and which
represent the shear stresses?

If this stress tensor is dotted against a surface normal, such
as:

n = (3/7)i + (6/7)j + (2/7)k

where i,j,k are unit vectors along the x,y, and z axes, then, e.g., the x-component of the stress vector is given by:

S(x1) = (3/7)s_{11} + (6/7)s_{21} +
(2/7)s_{31}.

and s_{11} = 2, s_{21} = 1 and
s_{31} = -3. The other two components of the stress vector are
defined
similarly, yielding:

S = S_{mn}N_{n} = (6/7)i + (13/7)j + (5/7)k.

The indices (m,n) after S are the subscripts of the elements of the stress tensor, and the i,j,k in this equation are unit vectors.

Anytime the stress is nonhydrostatic in a rock system (i.e., most of the time), then, strictly, there is no "pressure". Further, there is no unambiguous "Gibbs Free Energy", since G = U + PV -TS. Helmholtz Free Energy can be defined unambigously under non-hydrstatic conditions, however. The stress that defines the thermodynamic properties of solids under nonhydrostatic conditions is the microscopic stress, and I agree with Dahlen (AJS, 1992, pp184-198) who states: "In fact, there is no global equilbrium condition that depends only on the macroscopic principle stresses in addition to the fluid pressures, and the temperature, T. Local equilbrium on the scale of individual solid grains is the only equilbrium possible..."Hence, there is a limit to how precisely geological thermodynamics can be related to

pure shear (s1=-s3; s2=0)

deviatoric stress = matrix, not a number

differential stress = S_{1} - S_{3}, a positive scalar quantity

effective stress

Mohr circle: center of circle = mean stress; radius of circle (height
of circle) = max poss shear stress;

Prof Lori Kennedy at UBC has discussion of Stress/Mohr circles that you might want to look at.

FOOTNOTE 1
Technically, these are tractions, not stresses; when we pair tractions
up on either side of a body, so as to reduce both the acceleration of
the center of mass and the angular acceleration about the the center of
mass to zero, the tractions on opposite sides of the body become equal
and opposite. These paired tractions are, technically, the
*stresses*. This is discussed nicely in Twiss and Moore's
book,Structural Geology (W.H. Freeman & Co., 1992) section 8.2. I see
no point in introducing this technicality in a thermodynamics forum, but
add it here for completeness.
Some notes, and other topics to be dealt with presently:

principal stresses

scalar invariants of 2D and 3D stress matrix; mean normal stress is
an inv.

uniaxial compression

uniaxial tension