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where relative grain sliding is significant, in the so-called superplastic regime; it is
found that e s / d x and the sensitivity to grain size is then specified by the constant
x ¼ o ln e s = o ln d : However, when attention is not specifically directed to struc-
tural, or any other, variables, their influence is subsumed in a ''constant'' factor A
included as a multiplier in the empirical steady-state creep law.
Recapitulating, in the simplest or first analysis, the steady-state creep behavior
is commonly and conveniently represented in the form
e s ¼ Af ðÞ exp Q = RT
ð
Þ
ð 4 : 6 Þ
where f ðÞ¼ exp r = r ð Þ or r n according to which gives the better fit to the
observations, and A is a multiplier to be treated as a constant when only variations
of r and T are concerned or to be expressed as an appropriate function Ap ; ðÞ
when independent variations of pressure or structure are considered. When no
expression of form ( 4.6 ) with a single set of constants can be found to fit the
observations over the whole range of the variables, it is usual to consider this range
in several parts or regimes within each of which the observations can be fitted to an
expression of form ( 4.6 ) with a single set of constants. Such a definition of distinct
steady-state rheological regimes is, of course, an empirical procedure and may
initially appear rather arbitrary but it can take on more physical significance if the
change from one regime to another can be shown also to involve microstructural
changes suggestive of change in the deformation mechanism or in factors con-
trolling the deformation. The microstructural study may, alternatively, lead to the
postulation of models for the deformation mechanism that imply rheological laws
different in form from ( 4.6 ) which it will then be desired to test for fit.
4.5 Instabilities and Localization
So far we have taken it for granted that a specimen will undergo a uniform
deformation when loaded under conditions of uniform stress. However, this is not
always the case and rather restrictive conditions may have to be satisfied before
uniformity of deformation can be expected. When these conditions are not met,
heterogeneity of deformation can develop, giving rise to structural patterns on
various scales, apparently analogous in many cases to the geological features
studied by structural geologists.
In considering instability, it is often convenient to distinguish between shape or
external geometrical instability and material or internal instability (Drucker 1960 );
(Biot 1965 , pp. 192-204) (Paterson and Weiss 1968 ), although the distinction is
not always a clearcut one, however, and in some cases may simply refer to two
different aspects of the same unstable behavior. The first category is typified by the
necking instability in a tensile test specimen or by the Euler buckling of a slender
column in compression, while the second is typified by the development of
Lüders' bands in the deformation of mild steel or of kink bands in crystals
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