Geology Reference
In-Depth Information
grain boundary. Such a model is sometimes treated as a two-phase view of the
polycrystalline aggregate (Mecking
1981b
). In a careful analysis in core and
mantle terms, the strain in the core region must be acknowledged as itself
departing from the macroscopic strain by a small amount that is related to the
internal stress discussed above (see, further,
Sect. 6.8.4
). The mantle region is
distinguished from the core by a more pronounced development of multiple slip
and lattice rotation and, in metals, is reported to be of the order of tens of
micrometers in thickness (Leffers
1981
). The core and mantle distinction has been
especially emphasized by some writers in relation to high temperature creep (for
example, Gifkins
1974
,
1978
) but it is not clear how widely useful the concept is.
Another approach to the description of the heterogeneity of deformation and its
accommodation is in terms of geometrically necessary dislocations (Ashby
1970
,
1971
; Mecking
1981b
).
6.8.2 Multiplicity of Micromechanisms: The von
Mises Criterion
If two grains, initially in contact along a common boundary, are individually
subjected to simple shearing on planes that are not parallel between one grain and
the other, then after the deformation the grains will no longer fit together at a
common boundary without further adjustments in shape or rotation. In order to
maintain continuity at the grain boundary it is necessary, in general, that adjoining
grains of different orientation undergo deformation by a combination of shears that
gives the equivalent of more than one simple shear in each grain.
In attempting to answer the question of how many microshear mechanisms are
required to operate within a grain in order to achieve strain compatibility in an
aggregate, it is usual to start by enunciating the criterion of von Mises (
1928
).
Viewing the micro mechanisms as interpenetrable or superposable simple shears,
von Mises showed that, in order to achieve an arbitrary homogeneous strain, five
independent shears are required (independent in the sense of making possible
certain deformations that cannot be achieved with any combination of the other
available shear mechanisms). Applying this criterion to deformation by multiple
slip in an aggregate of randomly oriented grains in which the strain in each grain is
the same as the macroscopic strain (homogeneous strain or Taylor model), it
requires that, in general, five independent slip systems must operate in each grain.
Methods for determining the number of independent systems in a given set of
crystallographic slip systems, and some results, are set out by Groves and Kelly
(
1963
), Kocks (
1964
) and Paterson (
1969
).
It is to be noted that, for symmetry reasons, the number of independent slip
systems within a given set of crystallographically equivalent slip systems may be
substantially less than the multiplicity of the set. For example, there are 12
