Page 592 - Petrophysics

STATIC STRESS-STRAIN RELATION 559

If the magnitudes and directions of the principal stresses can be easily
obtained, it is convenient to use them as reference axes.

STRAIN ANALYSIS

Strain is defined as the compression (positive) or extension (negative)
resulting from the application of external forces, divided by the original
dimension. Two types of strain can be recognized: homogeneous and
heterogeneous. When every part of a body is subjected to a strain of the
same type and magnitude in any direction of the displacement, the strain
is considered homogeneous [7]. The strain is heterogeneous if it is not the
same throughout the body. Strain resulting from extended application of
large stresses at high temperatures is described as finite. If, however,
the strain results from the application of an increment of stress and can
be treated mathematically, then it is defined as infinitesimal strain. The
strain is responsible for inducing body displacement, rotation, and strain.
Shear strain, y, is defined as the angular change in a right angle at a point
in a body and is related to the displacements in the x, y, and z directions.
Assuming that a negative shear strain represents a decrease in the right
angle and a positive shear strain represents an increase in the right angle:

,
where E=, E ~ and are the normal strains.
In matrix operations, it is convenient to use the double suffix notation
and to define yij/2 as Eij. The strain matrix is then [7]:

(9.8)

The shear strains in the three principal planes of strain are zero, and the
normal strains are the principal strains. The greatest and least normal
strains at a point are preferably referred to as the major and minor
principal strains. The principal strains are determined in a similar manner
to principal stresses, i.e. as the roots of Equation 9.3, in which 0 and 7
are replaced by E and y/2, respectively.