Geology Reference
In-Depth Information
of the properties of the grains, treated as separate individual crystals with their
orientations taken into account, because of two factors:
1. the requirement of compatibility of strain from grain to grain, so that the con-
tinuity of the polycrystalline aggregate is maintained during the deformation
2. the influence of the presence of the grain boundaries on the behavior within the
grains.
These are two separate factors since the strain compatibility requirement
depends primarily only on the relative orientations of the grains and not on their
dimensions, whereas the grain boundary effects introduce a grain size dependence
into the behavior of the polycrystalline aggregate. We shall first consider the strain
compatibility aspect.
Although it is well known that the individual grains of a polycrystalline
aggregate do not deform homogeneously even when the macroscopic deformation
of the aggregate is statistically homogeneous (Barrett
1943
, p. 325; Boas and
Hargreaves
1948
), it is useful initially to view the stress and strain in each grain as
being effectively homogeneous to a first approximation. Taking this view, there are
two classical models for relating single crystal to polycrystal behavior, which
serve as limiting cases or bounds to the actual behavior:
1. The model of Sachs (
1928
), in which it is assumed that the stress in each grain
is equal to the macroscopic stress (as if the grains were loaded in series).
2. The model of Taylor (
1938
), in which it is assumed that the strain in each grain
is equal to the macroscopic strain (as if the grains were deformed in parallel).
Strictly, neither model is physically realistic since the Sachs model leads to
violation of the equations of continuity at the grain scale due to misfit at grain
boundaries, while the Taylor model leads to similar violation of the equations of
equilibrium. A physically valid model has to incorporate some degree of hetero-
geneity in both stress and strain at the grain scale, and we now consider ways in
which this aim has been approached.
The actual departures from a state of homogeneous stress can be represented in
terms of an internal stress, by postulating that the actual local stress results from a
superposition of a uniform applied stress and a locally varying internal stress. The
internal stress may give rise to a residual stress when the macroscopic stress is
removed and so be, in principle, measurable. The internal stress will, of course, be
a function of the strain unless a steady state is established. Its mean value will be a
measure of the amount by which the applied stress has to exceed the flow stress of
an average individual grain in order to achieve macroscopic flow. For use of the
notion of internal stress in the polycrystal context, see Leffers (
1981
) and Ber-
veiller et al. (
1981
).
Heterogeneity of strain within the grains of a polycrystalline aggregate has
often been modeled by postulating a core and mantle structure for the grains. In
this view, the core of the grain is regarded as undergoing a more or less homo-
geneous strain, approximating the macroscopic strain, while the complicated
intergranular adjustments are concentrated in a mantle region in the vicinity of the
