Physics for Geologists, Second edition

9 Stress and strain
Pressure, p, is a surface force divided by the area on which the force is
exerted [MLT-~/L~ = ML-~T-~]. Stress, S or a, is also a force divided
by the area it acts on, but it is three-dimensional. More properly in some
contexts, pressure and stress are the limiting value of the ratio as the area
becomes very small,
S = lim (@/&A).
GA-too
Force, as we saw on page 12, is a vector and we can resolve a force into
its x, y, z components by vector arithmetic. A surface force can be resolved
into a normal component on the surface, and a shear component in the
surface. Can we resolve a stress? Yes, but not by vector arithmetic. In fact,
six components are needed to define a stress completely, and stresses are
tensors. Tensors cannot be resolved by vector arithmetic.
Take a cube and apply a pressure between two opposing faces. Within
the cube there will be a state of stress in three dimensions. More generally,
the cube may have any orientation with respect to the force applied. On
each face there will be a normal and two possible shear directions in the
co-ordinate system, giving eighteen in all. The cube is in equilibrium, so we
need only consider the forces acting on three sides, giving nine components
as in the following matrix:
a, tXy t,, acting along the x axis
ty, ay tyZ acting along the y axis
t,, tZy aZ acting along the z axis.
These further reduce to six quantities because tXy = ty,, txz = t,, and
ty, = tZy when the object is not subjected to a net turning couple.
The stress ellipsoid is a convenient way of describing the state of stress in
a solid. The stress ellipsoid is defined by three mutually perpendicular axes
and the stresses along them. The axes are called the principal directions,
and the magnitudes of the stresses in these directions are called the principal
stresses, designated a,, a, and a, (z being by convention vertical). These
Copyright 2002 by Richard E. Chapman