Geology Reference
In-Depth Information
exponential form tends to be found at relatively high stresses (where it can
alternatively be represented as e
s
/
sinh r
=
r
ð Þ
), while the power law tends to give
better fits at lower stresses. A two parameter representation, e
s
/
sinh r
=
r
ð ½
n
;
has been proposed to give a wider range of fit with one expression, covering both
exponential and power law behavior (Garofalo
1965
), but this in turn can be
inadequate over the full range of interest, and so it would seem better for empirical
representation to use several single parameter expressions that give best fits in
limited ranges of stress, especially if these regimes can be correlated on micro-
structural evidence with changes in mechanism. Sometimes an exponential stress
dependence is represented in the form e
s
/
exp
rV
0
=
R
ð Þ
with a view to rep-
resenting the role of stress in assisting thermally activated processes as contrib-
uting an energy rV
0
;
where the constant V
0
has the dimensions of volume.
For the pressure dependence of e
s
;
the form e
s
/
exp
pV
=
R
ð Þ
is used, with
an implication that its role is through influencing thermally activated processes, as
in diffusion. The degree of pressure dependence is thus measured by the constant
V
;
called the activation volume and defined as V
¼
RTo ln e
s
=
op
:
There is
potentially some confusion in terminology here since the quantity V
0
representing
the stress dependence of e
s
;
introduced in the previous paragraph, is also com-
monly called an activation volume, especially in the metallurgical literature.
Sometimes this ambiguity is resolved by writing V
0
¼
Ab (b being the Burgers
vector for given dislocations in the crystals) and calling A the activation area.
However, it is simpler to distinguish V
0
and V
as activation volumes for stress and
pressure dependence, respectively.
The concept of structure dependence, represented by a term in y
;
calls for some
preliminary comment. Rheological behavior clearly involves microstructural
aspects such as mobility of crystal defects, processes at grain boundaries and the
presence of impurities, and creep rates may be influenced by the changes in
concentration of these entities, which are covered by the generic term ''structure''.
However, in discussing the role of structural factors one must distinguish, at least
in principle, between those which play the role of independent variables, under the
control of the experimenter and those which are dependent or uncontrolled.
Quantities specifying structural features that persist unmodified through the range
of strain under consideration can be treated as independent variables; examples
are: the initial grain size in cases where recrystallization is absent or minor, a
concentration of crystal defects that is continuously equilibrated with respect to an
externally controlled environment (for example, a fixed oxygen fugacity), the
concentration of an impurity in solid solution or forming a separate phase, and the
relative amounts of the phases in a multiphase material. In contrast, variables such
as mobile dislocation density, subgrain size, and recrystallized grain size in situ-
ations where the dislocations, subgrains, and grains have been generated during the
deformation are, in general, dependent variables. Only those quantities that can be
considered as independent variables in a given test are covered by the symbol y in
this section. An example of a structure dependence is given by the dependence of
the steady-state creep rate on grain size d which is observed at high temperatures
