Physics for Geologists, Second edition

88 Stress and strain
Equations 9.2a above can be written as
The co-ordinate system may not be parallel to the principal stresses and
strains. It can be shown that
Since these are not necessarily principal stresses, there will be three possible
shear stresses:
where y is the shear strain. Adding Equations 9.3a and using the dilatation
term above,
There are other material constants of a similar nature. Consider a wire that
is not coiled supporting a weight mg that is not so large that the proportional
or elastic limit will be exceeded. The ratio of stress to strain is constant, so
mg/A = E/(61/1), where A is the cross-sectional area of the wire. E is called
Young's modulus. It is a special case of Hooke's law and is sometimes called
the modulus of elasticity. The dimensions of Young's modulus are those of
a force on an area (a pressure), ML-~T-~, and the units are pascals (N mP2).
It is essentially uniaxial.
The ratio of shear stress to shear strain, t/y, is called the modulus of
rigidity or shear modulus (G). Strain is dimensionless, being 6111, so the
dimensions are also those of a pressure, and the units, pascals.
The wire also suffers a loss of radius when extended. The modulus that
relates the loss of radius to the increase of length, is called Poisson's ratio:
v = (6r/r)/(61/1). This is dimensionless.
When an elastic body is submerged and a pressure is applied to the fluid
enveloping the body, there is a change of volume. The strain is defined as
a change of volume per unit of volume, so the ratio of the change of pressure
and the strain it causes is 6p/(6V/V). This is the bulk modulus (K), and it
has the dimensions of pressure. Compressibility is the inverse of K.
Copyright 2002 by Richard E. Chapman