Geology Reference
In-Depth Information
e
ij
;
i
;
j
¼
1
;
2
;
3
:
''Engineering'' components of strain are commonly used in
reporting experimental work. Thus, normal strains e are given as relative elon-
gation or shortening based on the original length l
0
(e
¼
Dl
l
0
;
using the rock
mechanics convention that shortening strain is positive). Shear strains are given by
c
¼
tan h
¼
l
0
where s is the shearing displacement and l
0
is the reference length
normal to the shearing displacement. However, when specifying finite deforma-
tions, the engineering definition of normal strain has the disadvantage that a given
increment of strain has different physical significance as the actual length changes.
Therefore, for theoretical discussion, it is better to define the strain by integration
of strain increments each of which is the relative elongation based on the current
length. This leads to the definition of the so-called natural strain e
¼
ln
l
l
0
(in the
convention of shortening strain being positive). Similarly, in finite deformations,
the reference area for normal stresses changes with the deformation and it is
desirable to specify the normal stress components as ''true stress'' calculated on
current rather than original cross-sectional area.
The primary environmental variable is normally the temperature T
:
However, it
is also often convenient, when relating the stress difference r to the deformation in
axisymmetric deformation, to treat the ''pressure'' as a distinct environmental
variable, specifying it as the confining pressure, p
;
or the hydrostatic component of
the total stress, p
þ
3
;
as appropriate. Other environmental variables that may be
relevant at times are the pore pressure and the activities of chemical components in
reservoirs available to the specimen.
The other quantities that enter into any relationship describing the mechanical
behavior are the material parameters representing the characteristic properties of
the particular material. In practice, they are defined in a largely empirical way,
dictated by the forms of relationship between the mechanical variables that are
found best to describe the observed behavior. Also, particular parameters may be
needed to describe different structural states of a given type of material (for
example, grain size).
We now consider some general relationships between the mechanical variables
that apply in the intermediate athermal and high-temperature thermal fields depicted
in Fig.
4.1
. In practice, the boundary between these fields may be rather diffused but
for highly ductile materials such as metals it can commonly be taken as being roughly
one-half of the absolute melting point for materials that melt simply.
4.3 The Athermal Field
4.3.1 The Stress-Strain Curve
For simplicity, we initially only consider behavior in a simple compression or
tensile test, with or without superposed confining pressure p
:
The state of stress can
