Geoscience Reference
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parameters (temperature, pressure, water content,
stress, grain-size etc.). And the dependence of
these parameters ( A and n ) on physical/chemical
conditions is different among different deforma-
tion mechanisms. In the following, rheological
properties will be described by the strain-rate at a
given stress, or by the effective viscosity ( η eff =
(dislocation density, grain-size) is controlled by
the thermo-mechanical history of the material. In
most cases, however, it is a good approximation
to assume steady-state dislocation density,
and in these cases, dislocation density can be
treated as a parameter that is determined by
the magnitude of applied differential stress (e.g.,
Poirier, 1985),
σ
˙
ε ),
or by the creep strength (i.e., stress needed to
deform a material at a given strain-rate).
Similarly, although there is not much transient
phenomenon in electrical conductivity (except
for the apparent frequency dependence caused by
sample-electrode interaction, see Chapter 5, this
volume), transient deformation is often important
in plastic deformation because the establishment
of steady-state driving force and defect concen-
tration requires finite time or strain. Processes
that control stress distribution and defect con-
centrations (densities) need to be understood to
formulate the flow laws in plastic deformation.
Figure 4.1 illustrates the sensitivity of rheolog-
ical properties to these variables in a schematic
manner. If experimental studies are conducted in
a regime where the flow law is different from the
one appropriate for deformation in the Earth's in-
terior then the experimental results cannot be
extrapolated. Identifying the similarity in the
mechanisms requires extensive and multifaceted
studies, and careful considerations of strategy are
needed to make progress in this area.
b 2 σ
μ
2
ρ
,
(4.2)
where ρ is dislocation density (the total length of
dislocation per unit volume), b is the length of
the Burgers vector (unit displacement associated
with a dislocation), σ is the differential stress and
μ is the shear modulus.
In contrast, steady-state grain-size can be at-
tained only after-long termannealing or after large
strain deformation. Consequently, transient be-
havior plays an important role both in dislocation
and diffusion creep.
4.2.1 Diffusion creep
Theory and experimental observations are
both well established for diffusion creep. Grain-
boundaries are weaker than the bulk of the grains,
and therefore grain-boundary sliding occurs when
deviatoric stress is applied to a polycrystal.
Grain-boundary sliding results in the gradient
in normal stress among grain-boundaries with
different orientations. Gradient in the normal
stress establishes the concentration gradient
in point defects that requires grain-boundary
reactions. This concentration gradient drives
diffusion flux. Consequently, the slower of these
processes controls the rate of deformation. In
most cases, diffusion is the slower process that
controls the rate of deformation (Nabarro, 1948;
Herring, 1950; Raj & Ashby, 1971). In some cases,
reaction at grain-boundaries controls the rate of
deformation. This is a case where diffusion occurs
easily through grain-boundary fluids, a case called
pressure-solution creep (Spiers et al ., 2004).
Diffusion of atoms occurs both inside the
grains and along the grain-boundaries. Because
4.2 Mechanisms of Plastic Deformation
and Flow Laws
Figure 4.2 illustrates two processes of plastic
deformation. In both of them, defects play an
important role. These defects include point de-
fects, dislocations and grain boundaries. Plastic
deformation in minerals at an appreciable rate is
possible only when these defects are present.
Among these defects, point defects are present
in any materials at thermochemical equilibrium
although point defect concentration is locally
modified by the stress. In contrast, both dis-
locations and grain-boundaries are present as
nonequilibrium defects, and their abundance
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