Geology Reference
In-Depth Information
dr
de
¼
M
2
ds
ð
6
:
56
Þ
dc
For stricter definition of the factor M, see Kocks (
1970
). The polycrystal factor
M is, in general, a tensor but we consider here only the case in which the stress r is
taken to have only a single nonzero component and in which M can be treated as a
scalar. This case applies to uniaxial tension and compression tests, as well as to the
usual axisymmetric triaxial tests of rock mechanics. In the latter case, if r
1
and r
3
are the greatest and least principal stresses, the stress difference r
1
r
3
is to be
taken as r since the superposition of a hydrostatic pressure has no effect in the
theory.
Under the athermal, perfect plasticity assumption that slip in a given system
obeys Schmid's law (
Sect. 6.1
), that is, occurs at a critical resolved shear stress
regardless of the shearing rate, the Taylor and Sachs models (
Sect. 6.8.1
) provide
upper and lower bounds for M, namely, the Taylor factor M
T
and the Sachs factor
M
S
, respectively. The Sachs factor M
S
is calculated simply as the mean, over all
grain orientations, of the reciprocal of the maximum Schmid factor for the indi-
vidual orientations, since only one slip system is assumed to operate in each grain.
In calculating the Taylor factor M
T
, the minimum set of five slip systems that will
operate to give the prescribed strain in a grain of a given orientation must first be
selected. This is done through an optimization procedure, consisting either of
minimizing the internal work s
k
dc
k
summed over the available slip systems k
(Taylor
1938
#2695) or of maximizing the external work rde (Bishop
1953
,
1954
;
Bishop and Hill
1951
), these extremum conditions being implicit in the formu-
lation of the perfectly plastic model (Chin and Mammel
1969
; Kocks
1970
; Lister
et al.
1978
). Then, from the stress state corresponding to the optimal set of slip
planes, the component parallel to r can be determined and averaged over all grain
orientations to give the macroscopic stress r, from which M
T
can be determined.
Strain hardening can be taken into account by changing the value of s incre-
mentally as the strain is incremented, still under the assumption of athermal
behavior. Certain uniqueness problems can be overcome by introducing a degree
of rate dependence in the single crystal stress-strain relationship (Asaro and
Needleman
1985
).
The calculation of the value of M has been explored most extensively for the
f.c.c.cubic crystal structure, with a single set of 12 crystallographically equivalent
slip systems 11
fg
11
h
;
capable in themselves of satisfying the von Mises cri-
terion. Proceeding as just described, the upper bound or Taylor factor M
T
is found
to be 3.06 for simple extension or compression (principal stresses r, 0, 0) and 1.656
for shear (principal stresses r,
r, 0 and putting r
¼
Ms) (Bishop and Hill
1951
).
The lower bound or Sachs factor M
S
is 2.24 for simple extension or compression
and 1.12 for shear (Cox and Sopwith
1937
; Sachs
1928
). The value M
T
¼
3
:
06 also
applies for the b.c.c. structure with active slip systems 11
fg
11
h ;
decreasing
toward 2.75 as pencil glide comes into effect (Kocks
1970
).
The resolved shear stress needed to operate a given slip system is, in practice,
affected by slip occurring on intersecting slip systems, an effect known as latent
