Geology Reference
In-Depth Information
e
¼
e
0
þ
a ln t
ð
4
:
3
Þ
where the a ln m term has been subsumed into the constant (a ln m
;
in fact, tends to
be negligible compared with the elastic strain). This is the form of logarithmic law
originally proposed by Phillips (
1905
) and widely used (for example, Griggs
1939
). On the other hand, the relation of Lomnitz (
1956
), often quoted by seis-
mologists, is equivalent to (
4.2
). Other relationships for low-temperature creep
have been proposed with more parameters or more terms but need not be elabo-
rated here (Benioff
1951
; Griggs
1939
; Jeffreys
1958
; Michelson
1917
,
1920
).
Time-dependent strain can also occur in elastic vibrations, the theoretical
analysis of which in turn also implies specific forms of creep relation that may be
relevant at very small stresses and low temperatures. Thus, the theory of the
''standard linear solid'' (Zener
1948
, p. 43) leads to a relation of the form
1
e
t
=
t
0
e
¼
e
0
þ
e
t
ð
4
:
4
Þ
where t
0
and e
t
are constants (in this linear theory, e
0
¼
r
=
M
u
and e
t
¼
r
=
M
ð Þ
e
0
where r is the stress and M
u
;
M
R
are the unrelaxed and relaxed elastic moduli,
respectively).
4.4 The Thermal Field
4.4.1 Stress-Strain-Time Relationships
In temperature-sensitive mechanical behavior, the variables elapsed time t and
temperature T assume the same degree of importance as the stress r and strain e
:
The basic relationship to be sought is then of the form
e
¼
er
;
t
;
T
;
y
;
...
ð
Þ
involving the four variables just mentioned as well as independent structural
parameters (or internal variables) y
;
such as initial grain size or concentrations of
impurities. Experimentally, this multivariable situation is normally approached by
observing the relationship only between two of the variables while the others are
held at fixed, known values.
The simplest practical test from the point of view of interpretation is the creep
test in which one measures e
¼
e
ðÞ
at constant r
;
T
;
y
:
The approach, normal at
low temperatures, of measuring the stress-strain curve at constant e
;
T ... is very
useful in exploratory work when it is not known what stress levels are of most
interest but interpretation may be more difficult since the role of time is not fully
represented in the strain rate (for example, static recovery effects may be occur-
ring). Therefore, our discussion here will be in terms of the creep test (note also
that, unless otherwise specified, we shall take r to be the ''true stress'', based on
