12—Tensors 298∆A~
∆F~
cutThe stress tensor in matter is defined as follows: If a body has forces on it
(compression or twisting or the like) or even internal defects arising from its formation,
one part of the body will exert a force on another part. This can be made precise
by the following device: Imagine making a cut in the material, then because of the
internal forces, the two parts will tend to move with respect to each other. Apply
enough force to prevent this motion. Call it∆F~. Typically for small cuts∆F~will be
proportional to the area of the cut. The area vector is perpendicular to the cut and of magnitude equal
to the area. For small areas you have differential relationdF~=S
(
dA~
)
. This functionSis called the
stress tensor or pressure tensor. If you did problem8.11you saw a two dimensional special case of this,
though in that case it was isotropic, leading to a scalar for the stress (also called the tension).
There is another second rank tensor called the strain tensor. I described it qualitatively in section
9.2and I’ll simply add here that it is a second rank tensor. When you applystressto a solid body it
will developstrain.This defines a function with a second rank tensor as input and a second rank tensor
as output. It is the elasticity tensor and it has rank four.
So far, all the physically defined tensors except elasticity have been vector-valued functions of
vector variables, and I haven’t used then-linear functional idea directly. However there is a very simple
example of such a tensor:
work =F~.d~
This is a scalar valued function of the two vectorsF~andd~. This is of course true for the scalar product
of any two vectors~aand~b
g
(
~a,~b
)
=~a.~b (12.9)
gis a bilinear functional called the metric tensor. There are many other physically defined tensors that
you will encounter later. In addition I would like to emphasize that although the examples given here
will be in three dimensions, the formalism developed will be applicable to any number of dimensions.
12.2 Components
Up to this point, all that I’ve done is to make some rather general statements about tensors and I’ve
given no techniques for computing with them. That’s the next step. I’ll eventually develop the complete
apparatus for computation in an arbitrary basis, but for the moment it’s a little simpler to start out
with the more common orthonormal basis vectors, and even there I’ll stay with rectangular coordinates
for a while. (Recall that an orthonormal basis is an independent set of orthogonal unit vectors, such as
ˆx,yˆ,zˆ.) Some of this material was developed in chapter seven, but I’ll duplicate some of it. Start off
by examining a second rank tensor, viewed as a vector valued function