Geology Reference
In-Depth Information
Compatibility conditions will normally contain a presumption of constancy of
volume apart from elastic effects. Failure to meet the compatibility conditions will
tend to lead to the formation or elimination of voids, that is, to volume changes and
hence to a much stronger pressure dependence in the flow law than might other-
wise appear (the diffusion coefficient would normally introduce only slight pres-
sure dependence, although appreciable pressure dependence may arise from the
reaction rate in case of reaction-controlled flow where, for example, solubility in a
fluid is a parameter in the reaction kinetics).
Unless the geometry of the source/sink system is suitably specialized, compati-
bility conditions will require local variability in the rate at which material is trans-
ferred; for example, where an asperity or change in orientation tends to obstruct
geometrically necessary sliding on a grain boundary, the rate of transfer will need to
be accelerated locally in order to accommodate the sliding. Such variability is
unlikely to be consistent with a homogeneous stress distribution, in which the only
variation in normal stress component across given sources and sinks arises from
variations in orientation, and therefore the stress distribution will necessarily be
heterogeneous on the domain or grain scale and on finer scales. If the local strain rate
is linear in the stress difference, a theoretical treatment in terms of a homogenous
stress, such as given in this and subsequent sections may represent a suitable aver-
aging procedure for many purposes but it could contain more serious error in non-
linear cases or in describing specific local aspects of linear cases.
It is also to be emphasized that the maintenance of fit between domains through
the compatibility conditions imposes such an intimate interdependence between
the shape changes of the individual domains and the relative displacements of the
domains that these two effects must be regarded as two aspects of the same basic
process. The sliding at interfaces is thus an integral part of the atomic transfer
deformation process, which can be viewed equally as based on the material
transfer or the relative domain displacement aspects. This equivalence has been
stated particularly clearly by Lifshitz (
1963
) and by Raj and Ashby (
1971
).
5.3 Nabarro-Herring Creep
The model for atomic transfer or diffusion creep first put forward was that based on
grain boundaries as the sources and sinks and on volume diffusion through the
grains as the transfer mechanism, with diffusion control, propounded by Nabarro
(
1948
) and Herring (
1950
). The Nabarro-Herring creep law can be obtained from
(
5.7
) by taking the volume of the grains, V
;
to be or order d
3
and the cross-
sectional area and length of the diffusion path, A
t
and l
;
to be of order d
2
and d
;
respectively, so that lV
=
A
t
d
2
:
Then, with c
¼
1
=
V
m
;
(
5.7
) becomes
e
¼
C
NH
V
m
D
v
RT
r
1
r
3
d
2
ð
5
:
9
Þ
