Geology Reference
In-Depth Information
The first conclusion from self-consistent theory is that the polycrystal factor M
in (
6.55
) for athermal plastic deformation of f.c.c. materials falls very near to the
upper bound calculated on the Taylor model, thus justifying the use of the Taylor
model where five independent slip systems are available and the flow stress is
insensitive to strain rate. However, the self-consistent results tends to fall signif-
icantly below the athermal upper bound when strain-rate sensitive flow is con-
cerned, and increasingly so as Newtonian flow (power-law stress exponent n
¼
1)
is approached. In the Newtonian case, which is formally analogous to the elastic
case, the self-consistent result is found to fall between the upper and lower bounds
of Hashin and Shtrikman (
1963
), suggesting that the self-consistent theory gives a
fairly accurate estimate of M.
As already pointed out (
Sect. 6.8.2
), the self-consistent calculations show that
deformation in a polycrystal can be achieved with four independent slip systems
when heterogeneity of deformation on the grain scale is allowed, as revealed for
the hexagonal close-packed case for s
c
!1:
However, the result for NaCl-
structure materials with n
¼
1 leads to r
!1;
indicating that the three inde-
pendent 10
fg
11
hi
slip systems alone are insufficient for plasticity even on the
self-consistent model. There is still some reservation in applying this conclusion to
real materials since in the self-consistent model the grains are still treated as
deforming homogeneously within themselves, even though differently from one to
another, and so there may be a little more freedom of accommodation by heter-
ogeneity of deformation in a real material. Peirce et al. (
1982
) have, for example,
made a beginning in treating heterogeneity of deformation within grains but their
approach has not yet been applied to the polycrystal problem.
For examples of the application of finite-element numerical modeling, see Abe
and Nagaki (
1981
) and Peirce et al. (
1982
,
1983
).
So far, we have been considering the relation of the single crystal to poly-
crystalline flow stresses at given strains without taking into account the effects of
the evolution of the structure, except for passing reference to strain hardening. We
next consider the effects of large strains and of the preferred orientations, both
crystallographic and in grain shape that may exist initially or develop during the
deformation.
6.8.5 Preferred Orientations and Large Strains
The local strain compatibility requirements in a polycrystal depend on the grain
shape. The considerations set down in the previous subsections apply primarily to
grain shapes that are more or less equant. When grains of very inequant shape are
involved, fewer independent slip systems are required (four for rod-shaped and
three for disk-shaped grains) because the compatibility requirements at the
boundaries of small area (edge of disk or end of rod) affect only a small fraction of
the volume of the grain and so can be neglected (Honneff and Mecking
1978
;
