Geology Reference
In-Depth Information
A model involving the pile-up of dislocations at the grain boundaries is usually
used to rationalize the value m ¼ 2 (Hall 1951 ; Nabarro 1950 ; Petch 1953 ),
although a model involving emission of dislocations from the grain boundary has
also been proposed (Li 1963 ). The value m ¼ 1 has been rationalized using a core
and mantle model (Kocks 1970 ; Mecking 1981b ) and an intermediate case has
been treated by Armstrong et al. ( 1962 ), while Ashby ( 1970 , 1971 ) has given a
treatment in terms of geometrically necessary dislocations. In the pile-up model
for m ¼ 2 ; k is predicted to have the value r c l 2 where r c is the stress required to
initiate flow in a second grain, adjacent to the boundary against which the pile-up
is occurring, and l is the mean dislocation spacing within the grains (Friedel 1964 ,
p. 268; Haasen 1978 , p. 282). It follows that the grain size sensitivity will be most
pronounced at small grain size and low dislocation density. For common metals,
values of k of 0.1-1 MPa m -1/2 are found when m ¼ 2 is used (Armstrong et al.
1962 ). In the case of the core and mantle model for m ¼ 1 ; k is predicted to be 4tDr
where t is the width of the mantle region and Dr is the increase in flow stress in the
mantle above r 0 (Kocks 1970 ; Mecking 1981b ). In Ashby's model, with m ¼ 2 ;
k ¼ aGb ð 2 where a is a constant, G the shear modulus, b the Burgers vector and c
the shear strain in a grain. See further discussion on the value of k by Hansen
( 1983 , 1985 ).
In passing from the athermal deformation regime to the thermal regime at
higher temperatures, the grain size sensitivity tends to be reduced, that is, the value
of m smaller, because of thermal recovery effects in the grain boundary neigh-
borhood (Mecking 1981b ). Eventually, with further increase in temperature, grain
boundary weakening associated with the change from crystal plasticity to atom
transfer and grain boundary sliding mechanisms begins to enter, leading finally to
the strong inverse grain size dependences discussed in Chaps. 5 and 7 .
6.8.4 Relation of Single Crystal to Polycrystal Flow Stresses
The quantitative problem in the theory of the flow of polycrystalline materials by
crystal plasticity is to relate the macroscopic flow stress r of the aggregate to the
flow stress of the single crystal, expressed as the resolved shear stress s on an
active slip system; that is, to evaluate the numerical factor M in the relation
r ¼ Ms
ð 6 : 55 Þ
In an athermal regime, s is the resolved shear stress after a given strain and, in a
steady-state thermal regime, s is the resolved shear stress for flow at a specified
strain rate. In order to obtain the macroscopic stress-strain (r e) curve from the
plot of resolved shear stress s versus resolved shear strain c for the single crystal, a
similar numerical factor, which we take for present purposes to be again M, has to
be used to relate c to e (cf. Eqs. ( 6.1 ) and ( 6.2 )), so that finally the strain-hardening
rate becomes
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