Geology Reference
In-Depth Information
taking no account of what might be happening on the microscopic scale. The
microscopic aspect of deformation will be touched only in the final section, forming a
4.2 Phenomenological Approach
In the phenomenological approach to plastic deformation, the primary mechanical
variables are the stress r and the strain e
:
These are tensorial quantities, normally
fully representable by six independent components (see texts such as Jaeger
1962
;
Means
1976
for simple introductions to stress and strain; also see texts such as Nye
1957
; Reid
1973
).
In general, stress is a measure of the intensity of force acting on a 3-dimensional
(3-D) element of the body. Its complete description requires nine components, six
of which can be shown to be independent. The full specification of stress is
therefore in the form of a symmetrical second-rank tensor r
¼
r
ij
;
i
;
j
¼
1
;
2
;
3
:
The stress r can often usefully be viewed as consisting of two parts, one
representing the hydrostatic aspect of behavior and relevant to changes in volume,
and the other representing the non-hydrostatic aspect and relevant to changes in
shape. Formally, the stress tensor r
ij
can always be written as the sum of a
hydrostatic component
3
r
ii
and a deviatoric component r
ij
3
r
ii
:
However, in
experimental studies and in geology, it is often convenient to use a slightly dif-
ferent resolution of the stress tensor, which we shall exemplify here in the case of
an axisymmetric stress state, the one most commonly involved in experimental
work. The principal components of the axisymmetric total stress tensor, r
1
;
r
2
;
r
3
are then written as
r
1
¼
p
þ
r
r
2
¼
p
r
3
¼
p
ð
4
:
1
Þ
where p is the ''confining pressure'' and r is the axial stress difference or ''differential
stress''. The hydrostatic component of the stress is now
and
p
þ
3
;
p
þ
3
;
p
þ
3
:
In experimental work with axisym-
metric stress, the components r
2
and r
3
are normally generated by applying a
hydrostatic pressure p to the faces of the specimen parallel to the 1 axis through the
agency of a fluid or weak solid. This pressure is properly called the confining pressure
and is to be distinguished from the hydrostatic component of the stress. It is con-
ventional in rock mechanics to designate compressive stresses as positive, in contrast
to engineering convention where tensile stress is positive.
In general, strain consists of the displacement of points in a 3-D space relative
to a 3-D reference frame. Its complete description therefore requires nine com-
ponents, six of which can be shown to be independent. Therefore, the complete
specification of strain is in the form of a symmetric second-rank tensor e
¼
2r
3
;
3
;
3
the deviatoric component is
