Geology Reference
In-Depth Information
hardening, which is not taken into account in the formulation of Schmid's law.
Therefore, Kocks (
1960
,
1970
) has proposed that an appropriate single crystal
curve to choose as the basis for calculating the polycrystal curve on the Taylor
model is that for a symmetrical or ''polyslip'' orientation in which several slip
systems operate simultaneously. Alternatively, a constitutive relation for the single
crystal that explicitly takes into account the latent hardening can be introduced
(Asaro and Needleman
1985
). In contrast, the Sachs model assumes single slip in
the grains and so, logically, a single crystal curve for single slip should be used for
it. If these respective procedures are followed, it is found that the measured
polycrystal stress-strain curve for f.c.c. and b.c.c. materials tends to fall much
nearer to the calculated curve for the Taylor model than that for the Sachs model
(Kocks
1970
). However, the Sachs model is patently defective in not recognizing
that in practice the grains mostly deform by some degree of multiple slip. It has
therefore been proposed that in applying the Sachs model, a single crystal curve
for a polyslip orientation should also be used. Such a ''modified Sachs model''
predicts a polycrystal stress-strain curve much nearer to the measured curve
(Leffers
1979
,
1981
).
In spite of this approximate agreement between observation and the predictions
of the Taylor and modified Sachs models (see Asaro and Needleman
1985
for a
review of recent developments of the Taylor model), the theoretical situation
cannot be accepted as being very satisfactory on at least two grounds:
1. Both models are conceptually defective in that they entail physically unac-
ceptable discontinuities in either force or displacement at the grain scale.
2. There remain severe difficulties in application to lower symmetry materials in
which the von Mises criterion is not met even with the operation of several
crystallographically nonequivalent slip systems, for example, in hexagonal
close-packed materials with basal and prismatic slip feasible but pyramidal slip
too difficult to operate, or in olivine with only three independent slip systems,
all nonequivalent and one rather difficult to operate.
It is desirable, therefore, to develop some way of allowing for heterogeneity of
deformation at the grain scale, while meeting local continuity and equilibrium
requirements. Two approximate methods that aim to meet this need are the self-
consistent and finite-element approaches.
In the self-consistent approach, the individual grain is viewed as an inclusion in a
matrix, the properties of which are assumed to be identical with the macroscopic
properties. The local departures of stress and strain in the individual grain from some
initially assumed macroscopic values are calculated for all grain orientations and
averaged to obtain a new set of macroscopic values, which can then be used to refine the
calculated local departures in the grains, and so on iteratively. The applications to
athermal plasticity by workers such as Hutchinson (
1970
) and Berveiller and Zaoui
(
1978
,
1981
) are developments of earlier applications to elastic-plastic problems
by Kröner (
1961
), Budianski and Wu (
1962
) and Hill (
1965
). An important extension
to creep problems was made by Hutchinson (
1976
,
1977
), from whose papers most of
the following comments are derived.
