Studying the material properties
and conditions in which a body of
material is subjected and the
subsequent behavior of the
material is called the dynamics or
mechanics of rock deformation. It
requires that we know the
mathematical equations that
describe the relationships among
factors such as stress, strain,
strain rate, temperature, and

pressure. These equations are
known as constitutive equations.
They describe the bulk, or
macroscopic, behavior of the rock.
16.1 Continuum Models of
Material Behavior
i. Elastic materials: such materials
deform by an amount proportional to

the applied stress, but when the
stress is released the material returns
to its original undeformed state. This
is called recoverable deformation.
There is a linear relationship
between stress and strain.
(16.1) n  Een s  2es

ii. Viscous materials: Such materials
exhibit nonrecoverable, linear
deformation.
 A mechanical analogue
for viscous behavior is a device called
a dash pot, which consists of a fluidfilled cylinder containing a porous
piston. When a force is applied
across the system the motion of the
piston is governed by the rate at
which fluid can flow through the pores
int he piston.
(16.2)

Ýs
Ýn s  2
n(Dev)  2
iii. Plastic materials: such materials
commonly undergo no permanent
deformation
if the applied stress is
smaller than a characteristic yield
stress, but flow readily at or slightly
above the yield stress.
Mathematically, plastic behavior is
idealized by assuming that there is no
deformation at all (the material is rigid)
below the yield stress and that during
the deformation the stress cannot rise
above the yield stress except during
acceleration of the deformation. This
model describes perfectly plastic
material or rigid-plastic material.
(16.3 The von Mises yield criterion)
s  K


iv. Power-Law Materials

Rigid-plastic behavior is an
idealization of material behaviorthat is
convenient but not strictly observed in
real materials. Many materials show
strain rates proportional to the
differential stress raised to some
power n, usually between 3 and 5.
This type of behavior is called powerlaw rheology.
Ýn  A((Dif ) ) n
(16.4) 
v. Other continuum models for
material behavior
Visco-elastic materials are
characterized by displaying both
viscous and elastic properties. A
Maxwell material behaves like an
elastic spring and a viscous dash pot
connected in series. firmo-viscous,
Kelvin, or Voigt materials are elastic to
the extent that the equilibrium strain is
a linear function of the applied stress
and is recoverable, but the strain rate
is governed by a viscous response.
Elastic-plastic (Prandtl)materials also
show a combination of recoverable
elastic strain and permanent
deformation. Visco-plastic (or
Bingham) materials display linear
viscous behavior only above a yield
stress such as characterizes plastic
materials.
16.2 Experiments on friction and
cataclastic flow: implications for
faulting
i. Static Rock Friction
coefficient of static friction
(16.5)
*s
 
n
Byerlee’s law describes the friction
for most rocks at higher normal

1
stresses
(16.6 Byerlee’s law)
s  0.85n [MPa] for
5MPa  n  200MPa
n  50  0.6n [MPa] for
n  200MPa
ii. Dynamic Rock Friction and
Cataclastic Flow
Displacement along a fault generally
involves the frictional behavior of a
layer of crushed rock, called gouge or
cataclasite, that accumulates
between the solid sliding surfaces,
and so the sliding behavior of such a
system is dominated by the
cataclastic flow of this material.
Displacement versus time curves for
creep experiments exhibit two
patterns of velocity with time: either
there is a continual decrease of
velocity with time, or there is a
continual increase leading to
catastrophic unstable slip. The
boundary between stable sliding and
unstable sliding is a function of both
temperature and pressure.
iii. Relationship to Fault Movement
Previous experimental results
resemble the behavior of natural
faults. The potential for stick-slip
behavior, and presumably for
earthquakes, depends not only on the
characteristics of the fault surface,
represented by the frictional
properties of the surface, but also on
the stiffness of the rock bodies on
either side of the fault. A detailed
understanding of the mechanics of
cataclastic flow and friction may hold
the key to understanding the
mechanical behavior of faults and
earthquakes.
16.3 Experimental Investigation of
Ductile Flow
Experiments are preformed at either
constant stress (creep experiments)
or constant strain rate. For
investigating the ductile flow of rocks,
experiments at constant stress are
preferable to those at constant strain.
EPS116 Chapter Summary 2011
Jasmine Mason
The homologous temperature is
defined by the ratio of the temperature
of the material to its melting
temperature. Cold working is the
deformation at homologous
temperatures below about .5.
Deformation at homologous
temperatures above .5 is called hot
working. Decreasing creep rate at
constant stress is called work
hardening or strain hardening. In
this phase of the deformation, the
material becomes less ductile with
increasing strain.
16.4 Steady-State Creep
Steady state or secondary creep
occurs when the creep rate stabilizes
the
at a constant value. It represents
long-term deformation processes that
occur within the Earth. These
different creep laws reflect the
different processes that dominate at
different levels of stress.
i. The Moderate-Stress Regime:
Power-Law Creep
The constitutive equation that
accounts for most moderate-stress
steady-state deformation observed in
the lab is the power-law equation.
(16.7)


(Dif )
Ýn(ss)  A1 ((Dif ) ) n exp[

Ý(ss)
 K1 
n
1/ n
*
E
]
RT
E*
exp[
]
nRT
ii. The High-Stress Regime: The
Exponential Creep Law
At high stresses, the strain rate
becomes increasingly sensitive to
differential stress as the stress
increases.
(16.9)
Ýn(ss)  A2 exp[ (Dif ) ]exp[

E *
]
RT 
iii. The Low-Stress Regime: PowerLaw Creep with Low n
At differential stresses roughly below
20MPa, the constitutive equation is
similar to (16.7) but in this case the
stress exponent n commonly has a
value between 1 and 2.
16.5 The Effects of Pressure, Grain
Size, Chemical Environment, and
Partial Melt on Steady-State creep
i. The Effect of Pressure
Pressures at crustal depths can be
ignored. However, within the mantle,
pressure effects become more
important. Adding a pressure term
into (16.7) includes the effects of
pressure on steady-state creep.
(16.11)
Ýn(ss)  A4 ((Dif ) ) n exp[

(E *  pV * )
]
RT
ii. The Effect of Grain Size
When coarse grained solids are
deformed at low homologous
temperatures in the moderate to high
stress regime, the yield stress tends
to decrease slightly with increasing
grain size. The opposite effect is
observed for fine-grained materials in
the low-stress regime.
(16.15)
Ýn(ss)  A6 d b ((Dif ) ) n exp[

(E *  pV * )
]
RT
iii. The Effects of Chemical
Environment
The chemical environment of
deformation has a huge effect on the
rheology of rocks, in particular the
fugacities of water and oxygen.
Water weakening or hydrolytic
weakening is the solution of water in
the lattices of silicates at elevated
temperatures and pressures that
reduces the activation energy for
creep.
(16.18)
(E *  pV * )
Ýn(ss)  A7 d b f Hr 2O ((Dif ) ) n exp[

]
RT
iv. The Effect of Partial Melt
The effect of a partial melt on a body
of rock is most pronounced if the melt
wets the boundaries of the mineral
grains in the rock and least
pronounced if the melt collects in
pores at grain corners.
i. Interpretation of Structures of
Ductile Deformation with Relation to
Finite Strain, Instantaneous Strain
Rate, and Stress
It is preferable to discuss the origins
of deformational structures in terms of
the orientation of the principal finite
strains. This is because they provide
information about the finite strain,
which is the sum of all the
deformations that a given body of rock
has experienced, and in general, they
do not provide information about the
strain rate or instantaneous strain.
Experiments have mostly been done
on monomineralic rocks, and
experiments on polymineralic rocks
have not been entirely successful
because a small percentage of a
significantly weaker mineral can
strongly affect the ductile behavior of
a rock. The difficulty in predicting this
fundamentally restricts our attempts to
model accurate the behavior of Earth
Materials.
ii. Experimental versus Geologic
Strain Rates
Geologic strain rates are generally 4
to 10 orders of magnitude lower than
the experimental rates. How, then,
can we tell whether the constitutive
equations we determine in the lab are
pertinent to geologic deformation?
First, you can look at microscopic and
submicroscopic structures in the
crystal lattices that characterize the
operation of deformation mechanisms.
If similar microstructures are found in
nature, then we can conclude that the
same mechanisms operated under
the two sets of conditions. Secondly,
one can compare observed natural
deformation rates with the predictions
based on lab determined constitutive
equations.
References & Resources
Robert J. Twiss, Eldridge M. Moores,
Structural Geology 2nd edition, (W. H.
Freeman), p. 297-317, 2006
16.6 Application of Experimental
Rheology to Natural Deformation
EPS116 Chapter Summary 2011
Jasmine Mason
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