Geology Reference
In-Depth Information
=
3
de
1
de
2
h
i
1
=
2
p
2
de
¼
Þ
2
þ
de
2
de
3
Þ
2
þ
de
3
de
1
Þ
2
ð
ð
ð
where e
1
;
e
2
;
e
3
are the principal plastic strain increments. Von Mises' yield cri-
terion for a perfectly plastic material then states that yielding occurs whenever the
effective stress r
reaches a certain value k
;
a constant for the material. This
criterion is observed to be widely applicable for metals when the hydrostatic
component of the stress is not large compared with the effective stress, although
some metals follow more closely the Tresca criterion according to which yielding
occurs when the maximum shear stress r
1
r
ð Þ=
2 reaches a fixed value for a
given material. It seems likely that the von Mises criterion could also be useful for
rocks where nondilatant ductile behavior is involved at low temperature and
moderate pressure, and that the strain-hardening extensions of the theory could
even be useful at higher temperatures where steady state is not reached. However,
there are few observational data for testing this application (see, for example
Robertson
1955
, for marble).
In extending the approach of the theory of plasticity to creep, it is often
assumed, following Odqvist (
1935
), that creep under general stress states can also
be expressed in terms of relationships between the effective stress r
and the
effective plastic strain rate e
;
=
3
e
1
e
2
h
i
1
=
2
p
2
Þ
2
þ
e
2
e
3
Þ
2
þ
e
3
e
1
Þ
2
e
¼
ð
ð
ð
(e
1
;
e
2
;
e
3
are the principal plastic strain rates), that are similar in form to the
relationships found between r and e in uniaxial tests; however, in some cases the
maximum shear stress and maximum shear-strain rate are found to be more
suitable as ''effective'' stress and strain rate, and in other cases more elaborate
forms have been proposed (see brief review by Finnie and Abo el Ata,
1971
). An
example of this approach is that of Nye (
1953
) for the case of high temperature
steady-state creep of ice, which may also be applicable to rocks in particular
circumstances, such as where the range of pressure is not great and solution
transfer effects are not involved. Nye assumed that there is a universal steady-state
relationship e
¼
f r
ðÞ
of the same form as that established in a uniaxial experi-
ment, namely, e
¼
A r
ð
n
;
where A and n are constants (note that the effective
strain rate and stress used by Nye differ by numerical factors from those given
above). This universal relationship is then incorporated in an application of the
theory of plasticity analogous to its extension from perfectly plastic material to one
with isotropic strain hardening for which a universal stress-strain curve is intro-
duced (Malvern
1969
, p. 364); the position- and time-dependent scalar multiplier
dk of the theory of plasticity now becomes dk
¼
3f r
ð =
2r
:
It has also been
suggested that such a procedure can be extended to cover transient creep as well,
and general relationships such as e
¼
C r
ð
n
0
e
ðÞ
p
;
e
¼
C
0
r
ð
n
/
ðÞ;
and e
ij
¼
r
ij
FJ
ð
/
ðÞ
have been proposed, where C
;
C
0
;
n
;
n
0
;
p are constants, /
ðÞ
is a
function of time, equal to unity for steady-state creep, FJ
ðÞ
is a function of
