Geology Reference
In-Depth Information
variable,
ð Þ
instead of e
¼
er
;
t
;
T
;
y
;
..
ð Þ
in the thermal field. Consequently, experimental work has been
commonly concentrated on determining the steady-state strain rate as a function of
stress and temperature, that is, establishing a ''flow law''. Often this aim has been
pursued, not through the creep test itself, but through experiments at constant
strain rate, taken to strains sufficient for the rate of strain hardening to become
sufficiently small, so that the specimen can be regarded as deforming at a stress
and a strain rate that are both sensibly constant; it is then assumed that this
combination represents a steady state, independent of the testing history and
therefore equivalent to what would eventually be reached in a creep test, in the
absence of a tertiary stage.
enabling
one
to
concentrate
on
e
s
¼
e
s
r
;
T
;
y
;
...
4.4.2 Analysis of ''Steady State'' Deformation
We now consider the relationship commonly sought in triaxial tests for the steady-
state creep rate in the form
e
s
¼
e
s
r
;
T
;
p
;
y
ð
Þ;
where an explicit distinction is made between the stress difference r and the
pressure p
;
as explained in
Sect. 4.2
; y serves as an internal variable covering any
structural factors to be taken into explicit account. In a first approximation, it
greatly facilitates analysis to assume that the dependence of e
s
on each variable can
be considered separately.
The temperature dependence of the steady-state creep rate is normally well
represented
over
limited
ranges
of
temperature
by
an
Arrhenius
form
e
s
/
ð Þ;
where R is the gas constant (8.32 J K
-1
mol
-1
), T is the absolute
temperature, and Q
¼
Ro ln e
s
=
o 1
=ð Þ
is a measure of the degree of temperature
sensitivity, called the apparent or empirical activation energy (see
Sect. 3.1.2
). It
should be recognized that in fitting the experimental results, in this way, all aspects
of the temperature dependence are lumped together and Q may not represent an
actual activation energy for a particular elementary process. For example, a part of
the temperature dependence may arise from variations in elastic modulus with
temperature (which may eventually prove not even to be of exponential form) and
another part from the temperature dependence of a diffusivity (Bird et al.
1969
;
Poirier
1976
, p. 47), in which case the apparent activation energy for steady-state
creep could not, on the above analysis, be identified exactly with the activation
energy for diffusion. It may also be remarked that, at this level of discussion, the
apparent activation energy Q cannot be given a specific thermodynamic desig-
nation, such as enthalpy or Gibbs free energy, until the thermodynamic system has
been adequately defined (Poirier
1976
, pp. 37-38 and 98-106).
The stress dependence of e
s
can generally be represented over limited ranges of
stress either by e
s
/
exp
ð
r
=
r
0
Þ
or by e
s
/
r
n
;
where r
0
and n are constants. The
exp
Q
=
RT
