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RHEOLOGY
I. Introduction
Rheology is the study of flow of matter- in this case, rocks. Our aim is to
describe the general behaviour of rheology by introducing strain rate and what
the creep curve is.
From this we introduce equations that represent the behaviour of different
materials that we can calculate the motion and deformation of a body of rock.
The relationships regarding such behaviour include those such as elastic,
viscous, viso-elastic, elastico-viscous, general linear and non-linear
behaviour.
Mechanisms explain the way in which rheologic behaviour occurs, and some
that have been discussed are fracture, crystal-plastic flow and diffusive mass
transfer.
When carrying out physical experiments on rocks, parameters such as
confining pressure, temperature, strain-rate and pore-fluid pressure need to
be taken into account when using the deformation apparatus, and these will
be discussed with some real-life analogies.
Finally, a case study on the deformation of dunite will look at the parameters
discussed and the outcome of the experiment.
Next- General Behaviour
RHEOLOGY
II. General Behaviour
II.1Strain Rate
Strain rate is the interval of time taken for a certain amount of ( longitudinal)
strain to accumulate. It is defined as elongation per time and is symbolised by
ė:
ė=e/t=∂ l/(lot)
The SI unit for strain rate is [t]-1. The relationship between shear strain rate
and strain rate is:
γ=2 ė
Fig1. Brittle to ductile deformation
II.2 The Concept of Creep
Tests on rock samples show that the behaviour of rocks under stress is far
from simple. The concept of creep, graphically represented as the creep
curve, plots strain as a function of time. Creep describes macroscopically
ductile, slow, continuous flow in which the active deformation mechanisms
can accommodate a constant stress.Three creep regimes are observed1. Primary or Transient creep, where strain decreases with time
after an initial rapid accumulation.
2. Secondary or Steady -state creep,where strain accumulation is
aproximatly linear with time.
3. Tertiary or Accelerated creep, where strain increases with time,
eventually continued loading will lead to failure in the rock.
Fig2. The creep curve
Next- Rheologic Relationships
RHEOLOGY
III. Rheologic Relationships
To describe the various rheologic relationships we first divide the behaviour of
materials into two broad groups- Elastic and Viscous. These are both linear
behaviors and the combination of the two can be used to model some rock
types. For other rock types we have to consider nonlinear behaviour.
III.1 Elastic Behaviour
An important characteristic of elastic behaviour is its reversibility: once the
stress is removed the material returns to its original state. This implies that the
energy used to deform the material remains available for returning the system
to its original state. The ability of rocks to deform elastically lies in nonpermanent distortions of their crystal lattices. Expressing elastic behaviour in
terms of stress and strain we get-
σ =E · e
where e is the rate of elongation and E is a constant of proportionality called
Young’s modulus that describes the slope of the line in the σ – e diagram. The
SI unit of this elastic constant is Pascal (Pa=kgm-1s-2). This equation is also
known as Hooke’s law. A spring is a typical physical model.
Fig1. A spring as a model for elastic
behaviour.
Elastic behaviour can also be described in terms of shear stress-
σs = G·γ
where G is another constant of proportionality called the shear modulus or the
rigidity, and γ is the shear strain.
Elastic behaviour amounts to a very small percent of total strain in naturally
deformed rocks, but elastic behaviour is very important for the propagation of
earthquakes.
III.2 Viscous Behaviour
Viscous behaviour differs from elastic behaviour in that it is irreversible, strain
accumulates as a function of time and when the stress is removed the effects
are permanent. The flow of water in a stream is an everyday example of
viscous behaviour.
Expressing viscous behaviour in terms of stress and strain we get-
σ = η·ė
where η is a constant of proportionality called viscosity (tanθ in the σ-e
diagram) and ė is the strain rate. This is ideal viscous behaviour and is
commonly referred to as Newtonian or linear viscous behaviour (not to be
confused with general linear behaviour discussed later). The SI unit for
viscous behaviour is the unit of stress multiplied by time- Pa · s (kgm-1 s-1)
Fig2. A Dash pot as a pysical example of
viscous behaviour
III.3 Visco-Elastic Behaviour
Visco-elastic behaviour is where the deformation process is reversible but the
accumulation and recovery of the strain is delayed. A physical model would
be a spring and a dash pot placed in parallel. When a stress is applied both
systems move simultaneously, but the dash pot retards the extention of the
spring. When the stress is released the spring will be returned to its original
position but again the dash pot will retard the movement.
The equation for visco-elastic behaviour reflects the addition of viscous and
elastic elements-
σ=E·e+η·ė
Fig3. A physical model of Visco-elastic
behaviour
III.4 Elastico-Viscous Behaviour
Elastico-viscous materials behave elastically at first but with continued stress
they will behave viscously. With the removal of stress we see the recovery of
the elastic component but the viscous component remains. We can model this
behaviour by placing the spring and the dash pot in series. As the stress is
applied the spring will deform after which the stress is transferred to the dash
pot. When the stress is removed the spring returns to its original position but
the dash pot remains where it stoped.
The equation for this behaviour is-
ė = σ/E + σ/η
Where σ is the stress rate. The time taken for stress to reach l/e times its
original value is known as the Maxwell relaxation time (after J.C Maxwell
Scottish physicist 1831-1879). Stress relaxation in this model decays
exponentially.
Maxwell relaxation time = η/G
The earths mantle displays elastico-viscous behaviour in that seismic waves
(an elastic phenomena) are propagated through the mantle, but the mantle
over a long period convects (a viscous behaviour).
Fig4. Physical model of an Elastico-viscous system
III.5 General Linear Behaviour
General linear behaviour is a model that reasonably approximates the
behaviour of actual rocks. It is modeled by placing the elastico-viscous and
the visco-elastic models in series. The first aplication of stress accumulates in
the elastic part of the elastico-viscous model. Continued stress is
accomodated within the ret of the model. With the removal of the stress the
elastic strain is recovered first followed by the visco-elasic component.
However, some strain is permanent (from the elastico-viscous model).
Fig5. Physical model for General Linear behaviour
III.6 Nonlinear Behaviour
All the previous models wored under the assuption that the relationship
between the strain rate and the stress was linear ( ė α σ ), but at elevated
tempuratures experiments have shown that the relationship is not linear. A
physical model for rocks using nonlinear behaviour is a blocg and spring in
parallel. Strain accumulates in the spring until a critical stress is reached
where the block will then move and permanent strain occurs. This sort of
behaviour is also know as Elastic-Plastic Behaviour . Because the slope of the
stress-strain rate curve varies in this model we can no longer talk about
Newtonian viscosity, as the viscosity changes as the slope varies. We can
define efective viscosity though-
ηe=σ /ė
ηe is also known as stress dependent or strain rate dependent viscosity
because it is simply a convenient description of Newtonian viscosity under
known stress or strain conditions.
The equation describing the relationship between strain rate and stress under
nonlinear conditions is-
ė = A· σ n exp(-E*/RT)
where A is an experimentally derived constant, E* is the activation energy
required for creep to occur in crystals, T is the temperature in 0K and R is the
gas constant.
Fig6. Pysical model of Nonlinear behaviour
Next- Mechanisms of Deformation