Geoscience Reference
In-Depth Information
Table 4.1 Constitutive relationship for several deformation mechanisms.
(a) Power-law constitutive relation
This
( σ/μ < 10 3 , μ :
type of
flow law works at
low stresses
shear modulus)
ε =
A μ n
b m exp
RT
H
mechanism
n
m
Diffusion creep
(volume diffusion)
1
2
Diffusion creep
(grain-boundary diffusion)
1
3
Diffusion creep
(reaction controlled)
1
1
Dislocation creep (power-law creep)
3-5 0
22
Grain-boundary sliding + dislocation creep
2 1
23
(b) Exponential constitutive relation
This
( σ/μ > 10 3 , μ :
type of
flow law works at high stresses
shear modulus)
˙
ε
=
B μ 2 exp
σ 0 q s ,0 q 1,
RT 1
H
1 s 2
mechanism
q
s
Discrete obstacle
1
1
Peierls barrier (low stress)
1/2 1
Peierls barrier (high stress)
1
2
limited and its importance in geological processes
is unclear. Various flow laws are summarized in
Table 4.1.
A key point is that the stress-strain distribu-
tion among co-existing phases evolves with strain
and it is frequently observed that the strain par-
titioning changes with strain in such a way that
a weaker phase will accommodate a larger frac-
tion of strain at larger strains (e.g., Bloomfield &
Covey-Crump, 1993). When the weaker phase is
interconnected then a sudden strength drop could
occur leading to shear localization. The lower
mantle of the Earth is a region where this type of
strain localization may occur because it is made
of 20-30% of a weaker phase ((Mg,Fe)O) together
with a stronger phase ((Mg, Fe) SiO 3 perovskite)
(e.g., Yamazaki & Karato, 2001).
4.2.5 Deformation of multi-phase mixtures
(a) Generalities Rocks are in general made of
various minerals. Let us consider deformation of a
two-phase mixture. For simplicity, let us assume
these two phases have isotropic plastic proper-
ties but their strengths are different. How is the
strength of such a mixture related to the strength
of each phase? Obviously, the strength of such an
aggregate is determined by the strengths of each
phase and their volume fraction, but it also de-
pends on the geometry of each phase. Predicting
the average strength of a mixture is difficult be-
cause the strength of a mixture depends strongly
on how stress and strain are distributed among
co-existing phases. Two end-member cases can
be considered, one with homogeneous stress and
another with homogeneous strain. The former
gives the upper limit for the actual strength, and
the latter does the lower imit.
(b) Influence of partial melting Partially molten
material is a typical case of a two-phase mix-
ture. When the volume fraction of a liquid
phase
20-30%),
then such an aggregate behaves like a liquid
with some solid suspensions. If the volume
fraction is less than this limit, an aggregate
deforms like a solid but its resistance to plastic
exceeds
a
certain limit
(
Search WWH ::




Custom Search