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Continuum mechanism: Rheology
A
Idealizations of Mechanical Behavior
There are various idealized types of material behavior, which approximate the behavior of actual
materials under certain special circumstances.
Elastic Behavior (Hookean Solid)
This is a behavior in which stress σ and strain e are linearly related,
σ(t) = µe(t).
The proportionality constant µ has dimensions of stress (force/unit area) and is called an elastic
modulus.
(a)
(c)
Stress, σ
Slope µ
Strain, ε
Strain, ε
Stress, σ
(b)
Time, t
Figure 1: (a) Symbolic representation of an elastic element, (b) elastic relation between stress and
strain, and (c) elastic response to a time dependent stress.
This behavior in one dimension is like an ideal spring and is symbolized by a spring as in
Fig. 1a. Fig. 1b shows the relationship between σ and e and (Fig. 1c) shows the resulting strain in
an experiment in which a load is applied at t = 0 and removed at t = t1 . For any rod there is a
unique state length (lo) that is always recovered when the force is removed.
All real materials display some strain in response to changes in stress that is delayed (noninstantaneous), but nevertheless completely recoverable. This is commonly termed ”delayed elasticity”. It is discussed extensively later in this chapter.
One can also imagine a non-linear generalization of elastic behavior, in which the initial length
is always recovered on removal of the load and there is a unique relationship between σ and e, but
it is non-linear. One can visualize this behavior as µ not being a specified constant, but depending
on e (or σ ).
Some information on the physics of elastic behavior is presented in Turcotte and Schubert
(2002), Chapter 7-2. It is usually restricted to small strains less than about 10−2 to 10−3 . It is
a behavior in which the arrangement of atoms and/or molecules in the structure is not changed,
but the deformation occurs by stretching of the inter-atomic bonds without breaking them and
disrupting the crystal structure.
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Continuum mechanism: Rheology
Viscous Behavior (Newtonian Fluid)
This is a behavior in which stress σ and the rate of strain e˙ = de/dt are linearly related
σ(t) = η e(t).
˙
The constant η has dimensions of stress-time and is called viscosity.
(a)
(c)
Stress, σ
Slope η
Strain, ε
Stress, σ
(b)
Strain rate, dε/dt
Time, t
Figure 2: (a) Symbolic representation of an viscous element, (b) viscous relation between stress and
strain rate, and (c) viscous response to a time dependent stress.
In one dimension, this behavior is often symbolized as a “dash pot” (Fig. 2a). Fig. 2b shows
the relationship between σ and e˙ and Fig. 2c shows the resulting strain in an experiment in which
load is applied at t = 0 and removed at t = t1 . For this material, the straining stops when the
load is removed, but the rod has experienced an accumulated strain, its length being different than
before the experiment. The deformation is non recoverable.
As in the case of elasticity, there is a non-linear generalization in which η depends on e˙ (or σ).
The fluid behavior of solids is commonly called creep and is usually non linear. Highly non-linear
creep is sometimes referred to as viscoplastic or just plastic.
An important characteristic of viscous behavior, whether linear or non-linear is the possibility
of arbitrarily large deformations. Deformation continues as long as stress is non zero. A behavior
that allows unlimited deformation is called fluid.
Turcotte and Schubert (2002) discuss some of the physics behind fluid behavior of solids in
Chapter 7 (7-3 and 7-4). The non-recoverable fluid-like deformation of a crystalline solid can
proceed without disrupting the structure by the motion of defects that break and reform bonds.
Failure
Failure is a general term referring to permanent non-elastic deformation of a solid that comes about
from either plastic flow or fracture.
Perfect Plasticity is an idealization of plastic behavior for which there exists a critical stress
value called the yield stress σ0 such that if the stress σ < σo , there is no deformation, but for
σ = σo , the material can strain at any rate.
Schematically this can be visualized as a block on a tabletop held in place by static friction
(Fig. 3a). There is no unique relationship between stress and strain rate (Fig. 3b) or strain (Fig. 3c).
Throstur Thorsteinsson (th@turdus.net)
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Continuum mechanism: Rheology
(b)
(c)
Stress, σ
(a)
Strain, ε
Stress, σ
s
s
Strain rate, dε/dt
Time, t
Figure 3: (a) Symbolic representation of a perfectly plastic element, (b) plastic relation between stress
and strain rate, and (c) plastic response to a time dependent stress.
Stress, σ
The actual strain which accumulates during the time σ = σo depends on the way the force F is
applied (how fast you pull). A variation on this type of behavior is called work hardening and it
results in a one-one relationship between stress and strain, when deformation is taking place. As
s
Strain, ε
Figure 4: Relation between stress and strain in a plastic body that ”work hardens”.
before, there is no strain until the stress reaches the yield stress. However, for straining to continue
σ must be progressively raised above σo as if the apparent yield stress were increased by the prior
accumulated strain.
Fracture refers to the destructive failure by the formation of one or more surfaces traversing
the material and an associated drop in stress to low values with increasing strain. The failure
commonly sets on at some critical stress value that is referred to as the strength. It is common to
speak of ductile or brittle fracture.
Ductile refers to a behavior where there is substantial non-recoverable “plastic” strain before
any loss of strength occurs. Brittle refers to behavior where no plastic strain happens before loss
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Continuum mechanism: Rheology
of strength.
B
Mechanical Behavior of Real Materials
We can find real materials in our everyday experience that can be easily classify as behaving
according to a single one of the above idealizations. Spring steel is a good example of elastic
behavior. When stress is applied it deforms a certain amount and it springs back when the force is
removed. Every day liquids (water, honey) are good examples of viscous behavior. Lead provides
an example of plastic behavior and ductility. Glass is brittle. On the other hand we find some
material in some conditions that defy simple classification. Silly putty at room temperature and
pressure is a notable example that is easily available for experimentation. A sphere of it when
dropped on the floor will bounce, thus showing elastic (springy) behavior. However, if it is thrown
very hard, it loses its shape by a non-recoverable plastic deformation and does not bounce nearly
as high as would be expected with perfectly elastic behavior. If a sphere of it is set on the table, in
a short time (minutes) it will flatten out, much like a highly viscous fluid. This illustrates how the
size of the stress and the time scale of interest affect the dominant type of behavior. In addition, the
type of behavior is affected by temperature and pressure. Silly putty is unusual in that transitions
between the basic types of idealized behavior occurs over a fairly small range of stress magnitude
and time scale. However, most real materials have the potential to exhibit any of these behaviors
depending on temperature, pressure, stress, and time scale. The hope is that in certain applications
specific aspects of the behavior can be singled out as most important, although there is usually
some degree of combined behavior.
To represent a combined behavior, the mechanical idealizations discussed above can be combined
in a single multi-process model schematically illustrated in Fig. 5. In this picture the arrangements
Figure 5: Combined model incorporating elastic, delayed elastic, viscous and failure behaviors.
of the elements have the following interpretations. Those elements in series support the same
force (stress), but their contributions to the elongation (strain) add. Those elements in parallel
experience the same elongation (strain) and the total force is shared between them. The simple
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Continuum mechanism: Rheology
combination of elastic and viscous elements in parallel results in a recoverable elastic strain that is
delayed by a viscous process.
In this model elasticity, delayed elasticity, viscous deformation and failure all potentially contribute to the strain. The different components are more or less important depending on time-scale,
temperature, pressure and stress level. Generally viscosity is strongly dependent on temperature;
however, elasticity is only weakly dependent on temperature. Thus, if the temperature is high
or the time scale long, the strain contribution from the viscous element becomes more significant
than arising from the instantaneous or delayed elastic elements. If the stress is low enough, the
failure process is inactive and the combination of viscous and elastic elements control the straining.
As discussed below the failure may be sensitive to pressure (and pore pressure), which can then
influence the dominant mode of behavior.
The model represented by Fig. 5 contains all of the qualitative behavior we have discussed. One
can imagine more complex combinations of mechanical elements to represent multiple processes of
delayed elasticity or failure, which could occur in real materials. However, such complex rheological
models are usually not especially helpful in increasing understanding.
C
Factors Affecting Mechanical Behavior
We now discuss qualitatively some of the effects from stress, pressure, temperature, and time scale
on the type of behavior.
Effects of Stress Level
Increasing stress level tends to lead to transition from elastic or viscous behavior to plastic flow
or fracture. For example, in most unfractured rock at room temperature and pressure a basically
elastic behavior exists as long as the strain does not exceed about 10−3 to 10−2 . Typical elastic
moduli for rocks are about 1010 to 1011 Pa (105 to 106 bar), so these strains correspond to stress
in the range 107 to 109 Pa (102 to 104 bars). At larger strains (or stress) some non-recoverable
plastic strain becomes apparent or fracture occurs. This introduces the concept of an elastic limit.
Above the elastic limit failure occurs.
Effects of stress geometry
Although the present discussion focuses on a one-dimensional view of stress and strain it is important to recognize that the three-dimensional configuration does have some effects on the qualitative
behavior of materials. Materials may behave differently in extension and shear. For example, simple real liquids show viscous behavior in shear and elastic behavior in expansion and contraction.
Similarly in solids, proportionality constants relating elastic deformation and stress are numerically
different for different kinds of loading. Also there are big differences in failure behavior depending
on whether there is compression, tension, or shear.
Effect of Confining Pressure
In the uniaxial experiment we can apply a pressure to the walls of the cylinder, which is referred
to as confining pressure. The effective axial stress is the difference between σ and σp .
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Continuum mechanism: Rheology
Figure 6: Loading in shear and isotropic stress (pressure).
Figure 7: Schematic of one-dimensional loading with confining pressure.
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Continuum mechanism: Rheology
The effect of confining pressure is most important for the high stress range. For most rocks at
low confining pressure, fracture occurs before any significant amount of plastic strain has occurred,
that is, rocks generally show brittle behavior when the elastic limit is exceeded. However, at high
confining pressure, fracture is suppressed and substantial non-elastic strain can occur. The sketch
below is taken from Jaeger “Elasticity, Fracture and Flow,” pg. 52, and shows σ vs. e for limestone
under various confining pressures p. It shows a transition from a brittle behavior, where at a
Figure 8: Deformation depending on confining pressure.
strain of several x10−2 the material loses strength because of fracture, to ductile behavior (with
a work hardening like property) caused by a change in confining pressure. This effect of pressure
is sometimes explained as a result of high friction on existing or potential crack surfaces caused
by the high pressure, and this suppresses loss of strength by cracking. In this particular case, the
non-recoverable plastic-like strain is a ductile like behavior from a cataclastic type deformation,
which physically is quite different from plastic deformation of a metal. In general, the brittleductile transition is quite complex. A short review is given by Nur, “Tectonophysics,” Sec. 3.1, pg.
291-295. The transition is highly material and temperature dependent. For example, some metals
are ductile to low confining pressure and temperature, but quartz remains brittle even at confining
pressures of 10 to 20 kbar when the temperature is low. Most rocks are intermediate with some
degree of ductility at pressures above a few kbar.
The effect of confining pressure is influenced by fluid pressure in the pores of the material
(called ”pore pressure”). Pore pressure tends to balance confining pressure. The effect of confining
pressure can often be taken into account through an ”effective pressure”, which is discussed later.
Effect of Temperature
Increasing temperature tends to promote a transition from elastic or plastic behavior to a fluid
viscous-like behavior. Qualitatively this occurs because elastic behavior is not very sensitive to
temperature, but viscous behavior is highly sensitive. This is discussed somewhat in T & S, Chapter
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Continuum mechanism: Rheology
7. Elastic moduli usually decrease slightly with increasing temperature. On the other hand the
apparent viscosity (defined as σ/e˙ once steady deformation rate has been achieved) usually shows
an exponential dependence on temperature such that an increase in temperature decreases the
viscosity strongly. In typical solids, if the temperature is larger than 1/2 the melting temperature,
then the contribution from a viscous type deformation exceeds the elastic contribution even for
fairly low stresses and modestly short time scale. In solids, this type of behavior is called hightemperature creep. Commonly, creep shows a non-linear relationship between e˙ and σ.
Effect of Time Scale
If one examines deformation on a long time scale, there is a greater chance that the continuing
strain associated with fluid-like behavior will add up to be the major contribution to the total
strain. Increasing the time scale tends to have similar consequence as increasing the temperature.
The time scale also arises in consideration of the rate of application of stress and/or the strain rate
in conditions that load to fracture. If the load is applied rapidly and/or strain rate is high, failure
by fracture occurs at relatively high stress (promoting a high strength) and at relatively low plastic
strain (promoting a brittle behavior).
D
Rheological Description of the Earth
The following schematic shows some aspects of the mechanical behavior of the earth, which illustrates some of the consequences of increasing temperature and pressure with depth and the wide
range of time scale. On the short time scale as represented for example, by seismic wave propagation, we find a predominantly elastic behavior, but attenuation shows the existence of some viscous
process even on this short time scale. On a much longer time scale such as in the range of 103 to
104 years associated with isostatic rebound of the ground surface after melting of ice sheets or 106
to 108 years associated with large scale plate displacements, it is certain that parts of the mantle
can undergo large deformations characteristic of fluid behavior. At the same time the existence of
deep focus earthquakes in the upper mantle suggests that stored elastic strain energy can persist
in the presence of stress, which would not be possible in an ideal viscous fluid. Therefore, on both
the short and long scale, we have evidence of a combined behavior, which is termed viscoelastic.
It is a combination of stored elastic strain energy and failure by fracture that makes earthquakes
possible. With reference to Fig. 5, as strain accumulates, stress and elastic strain rise until the
stress is so high that failure occurs. Then the elastic energy may be released. Although the four
basic idealizations of material behavior find applicability to the earth in certain situations depending on location (stress, pressure, temperature) and time scale, a more complex combined behavior
underlies many important geophysical phenomena.
E
Experimental Description of Viscoelastic Behavior
A behavior consistent at least qualitatively with the range of viscoelastic behavior seen in the earth
and also in materials studied in the lab can be described by the following experiment. A stress σ
is applied to the rod at t = 0 and held constant thereafter. This is called a creep experiment and
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Continuum mechanism: Rheology
the resulting strain e versus time is called a creep curve. The behavior that is commonly found is
sketched in Fig. 9. Upon application of the stress there is an instantaneous strain, as for example
Figure 9: Strain response to application of a step in stress.
for an ideal elastic material. However, there is a continued increase in strain rate that decays
asymptotically over a time scale represented by t1 to a residual non-zero rate of strain as would be
expected of a viscous fluid.
In reality the results of such an experiment often look like Fig. 10. and may be terminated
by fracture. The tertiary creep can arise from a number of sources of geometrical (e.g. necking)
or structural (formation of cracks, recrystallization) origin. We will neglect the complication of
tertiary creep.
In our mind we can produce the above behavior with rheological model of Fig. 5 at low stress
with out a failure process. The instantaneous strain would be associated with the spring; the
residual strain rate would be associated with the dash pot; and the transition could be associated
with a spring whose response is retarded (delayed) by the action of a dash pot. The reduced model
is shown in Fig. 11 with labels µ1 , µ2 , η1 and η2 for the moduli of the springs the viscosities of the
dash pots.
We could pick the four parameters µ1, η1, µ2, η2 so the rheological model would give the creep
curve: µ2 being determined by the instantaneous strain at t = o, µ1 by the delayed elastic strain,
η2 by the strain rate at t → ∞, and η1 by the time scale for completion of the delayed elastic
response.
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Continuum mechanism: Rheology
Figure 10: Schematic of stages of creep in a creep experiment.
Figure 11: Viscoelastic model including elastic, delayed elastic, viscous (fluid) behaviors.
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Continuum mechanism: Rheology
Behavior of the Viscoelastic Model
We can work out the behavior for a particular choice of the four parameters as follows. Consider
the contribution to the total strain from each of the elements. Starting from the right, the dash
pot η2 contributes strain σt/η2 which is obtained by integration of the rate of strain e˙ = σ/η2
appropriate to the viscous behavior. A second contribution comes from the spring and is σ/µ2.
A third contribution comes from the dash pot η1 and spring µ1 in parallel. In this combination
we view the force as being shared between η1 and µ1 but the displacements are equal. Let the
contribution to the strain be ev , then we have in general
σ(t) = µ1 ev (t) + η1e˙v (t).
We want to solve this equation for ev for the case σ(t) = 0 for t < 0 and σ = constant for t ≥ 0.
This differential equation can be solved in a variety of ways. The simplest method is to use our
intuition about what should happen and how the solution should look for the special loading we are
talking about. It is fairly obvious that after a long time the spring µ1 will eventually be stretched
to the extent that it supports all of the applied load, and the deformation will stop. In this case
we should get
σ
ev =
f or t → ∞.
µ1
Indeed, we see that σ/µ1 is a particular solution to the differential equation for constant σ. The
homogeneous equation shows that ev should have an exponential time dependence; a solution to
the homogeneous equation is
µ1
exp −
t .
η1
The solution then can be expressed as
ev (t) =
σ
+ A exp
µ1
µ1
−
t .
η1
The requirement that ev (0) = 0 determines A and we have the result
µ1
σ
ev (t) =
1 − exp −
t .
µ1
η1
The behavior of the whole system is now obtained by adding up all of the contributions to the
strain.
σ
µ1
σ
σ
e(t) = t +
1 − exp − t .
+
(1)
η2
µ2 µ1
η1
The viscoelastic model that we have just discussed contains within it a quite general viscoelastic
behavior. It also contains within it several simpler models that are often considered in textbooks
and that we should note (for historical interest). The combination of the pair elements µ2 , η2 is
called Maxwell fluid, which is given by our somewhat more general model when µ1 → ∞. The
combination of the pair of elements µ1 , η1 is called a Voight solid, and its behavior is given by our
result when µ2 and η2 → ∞. A third model called a standard linear solid is represented by the
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Continuum mechanism: Rheology
combination η1, µ1 , µ2 and is described by our result when η2 → ∞. Also we can at this point
express the distinction between solid-like and fluid-like behavior. Solid-like means deformation is
bound and stops under application of a constant stress. Fluid-like means deformation continues
indefinitely under a constant stress.
F
Linear Viscoelastic Creep and Relaxation Functions
The creep experiment describes the response of a material to a step loading. How can one determine
the response to a general history of loading started at t = 0 (σ(t) = 0 t < 0, σ(t) 6= 0 t ≥ 0)?
We make the following reasonable assumptions:
(i) a change in stress at a time t0 causes a change in strain at later time t which is proportional to
the change in stress, that is
de(t) = Cdσ(t0 )
t > t0
(ii) the proportionality constant depends only on the elapsed time interval, that is
C = C(t − t0 )
The basic idea is illustrated in Fig. 12. Represent the increment in stress at t0 as dσ(t0) = σ 0 (t0)dt0
Figure 12: Change in strain versus time caused by an increment in stress at a specified time.
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Continuum mechanism: Rheology
and integrate from t0 = 0 up to t to get
e(t) =
Z
t
C(t − t0 )σ 0(t0 )dt0
0
where we suppose e(0) = 0 because there has been no loading prior to t = 0.
C(t − t0 ) is called the creep function of the material; it is a measure of how well a material
remembers a change in loading condition. Physically it is the strain versus time produced by the
special loading we have already considered, that is the sudden application of stress (σ = 1) at t = 0
and holding it constant thereafter. Formally we can show this since for
(
0t<0
σ(t) =
1t≥0
then we can use σ 0 (t) = δ(t), and
e(t) =
Z
t
C(t − t0 )δ(t0)dt0 = C(t).
0
Therefore if assumptions i) and ii) hold true, then the creep curve from a creep experiment completely characterizes the viscoelastic behavior of the material.
A similar type of functional description characterizing a viscoelastic behavior is given by the
relaxation function k(t), which is the stress σ(t) needed to produce a unit strain at t = 0 and
hold it constant thereafter. Imposing a step strain and measuring the stress is called a relaxation
experiment. The stress history resulting from an arbitrary history of imposed strain for t ≥ 0 is
given by
Z t
de(t0) 0
σ(t) =
k(t − t0 )
dt
dt
0
which is derived based on assumptions analogous to i) and ii) used for the creep function. This
approach is sometimes useful from an experimental point of view, because some experimental
techniques impose strain rather than stress. A common experiment is the so-called constant strain
rate experiment (de/dt = constant). The stress needed to produce a step strain of magnitude e in
a standard linear solid (η2 = ∞) is easily shown to be
µ 1 µ2
µ2
µ 1 + µ2
σ(t) = e
1+
exp −
t
,
(2)
µ1 + µ 2
µ1
η1
which gives the relaxation function for this model (Figure 13). The form of this function is illustrated below. At first all of the strain is contributed by the spring µ2 so initially the stress is µ2 e.
This stress acting on the combination η1µ1 causes it to deform at a rate initially of µ2 e/η1. As it
deforms the strain builds up in µ1 and is reduced in µ2 . Eventually as µ1 supports more and more
load, the load on η1 drops asymptotically to zero. In this final condition the same stress σ∞ acts
on both µ1 and µ2 and their individual strains σ∞ /µ1 and σ∞ /µ2 add to give the total strain e, so
σ ∞ σ∞
σ∞
e=
+
→
=
µ1
µ2
e
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1
1
+
µ1
µ2
−1
=
µ 1 µ2
.
µ1 + µ 2
13
Continuum mechanism: Rheology
Figure 13: Stress caused by imposing a step in stain.
The description of a viscoelastic material in terms of its creep function (or relaxation function)
is more direct from an experimental point of view and also more general than a description in terms
of a rheological model built from a combination of a few springs and dash pots. For example, a
creep curve like that shown in Fig. 9 might have a transient response (delayed elasticity) that does
not approach its final value as an exponential decay with a single item constant. It is common for
real materials to have several mechanisms of delayed elasticity with different time scales. As long as
the linearity assumptions hold, the creep function provides a description of the behavior. Normally
the linearity assumptions are very good for low stress and short time scales. In solids, the fluid-like
viscous (creep) behavior is generally non-linear. Similarly, a threshold process like failure is highly
non-linear. When these non-linear processes contribute to the deformation, the generalization of a
creep curve to an arbitrary history of loading is not possible without other information.
Energy Change
The rate of work or power input into a deforming rod is F u,
˙ where as before F is the force on the
end and u˙ is the rate of elongation. The power/unit volume is
F u˙
·
= σ e˙
A l
Therefore the total energy input/unit volume between t = 0 and t is
Z t
∆E(t) =
σ(t0)e(t
˙ 0 )dt0.
(3)
0
For example consider simple elastic and viscous behaviors with the assumption that the strain
is initially zero.
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Continuum mechanism: Rheology
Elastic behavior described by
σ(t) = µe(t)
predicts
∆E(t) =
Z
t
µe(t0 )e(t
˙ 0 ) dt0 =
0
1 2
σ 2(t)
µe (t) =
2
2µ
σ(t) is determined by e(t) and the net energy change is zero, when strain or stress is zero. ∆E(t)
represents recoverable elastic strain energy.
For viscous behavior
σ(t) = η e(t)
˙
Z t
2
∆E(t) =
η e(t
˙ 0 ) dt0
0
∆E(t) always increases; for any non-zero stress, work is done on the body and energy is dissipated.
Harmonic Loading of a Viscoelastic Material
In harmonic loading
σ = σo eiwt
(σo real)
and we may assume
e = eo eiwt
where eo may be complex, in order to allow for a possible difference in phase between σ and e. We
may define a complex modulus that is analogous to elastic modulus
m ≡
σo
= |m| eiδ
eo
where
tan δ = Im(m)/Re(m).
δ may be interpreted as a lag in phase of strain behind stress.
Let us calculate the energy input as a function of time. Taking real parts
σ(t) = σo cos wt
Then using eo =
σo −iδ
e
|m|
e(t) = (σo /|m|) cos (wt − δ)
e(t)
˙
= − (wσo /|m|) sin (wt − δ)
Thus, from Eq. 3
wσ 2
E(t) = Eo o
|m|
Z
t
cos wt0 sin (wt0 − δ)dt0
0
Now
sin (wt − δ) = sin wt cos δ − sin δ cos wt
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Continuum mechanism: Rheology
and
sin wt cos wt =
1
sin 2wt
2
so we writw
1
0
2
0
t
sin 2wt cos δ − cos wt sin δ dwt0
2
0
Z
σ02
σo2 w
=
t cos2 wt0 dwt0 sin δ + Eo
(cos 2 wt − 1) cos δ +
4|m|
|m| 0
σ2
E(t) = Eo − o
|m|
Z
w
this part oscillatory
this part monotonic
(storage)
(dissipation)
Figure 14 illustrates the behavior schematically. The full minimum to maximum amplitude of the
Figure 14: Energy change with time caused by a sinusoidally varying stress.
first term gives the maximum recoverable (stored) energy and the second term integrated over
one period (2π/w), which is easily done noting that the average value of cos2 wt = 12 , gives the
non-recoverable energy input per cycle. Now we compute Q−1
Q−1 =
∆E
= tan δ.
2πE
(4)
tan δ is often referred to as the internal friction, and the process of dissipation during cyclic loading
is called mechanical relaxation.
We see that energy loss is associated with the strain being out of phase (lagging) the stress.
Any delayed strain (e.g. Fig. 9 leads to dissipation of energy and non-zero Q−1 .
For a rheological model like Fig. 11 and creep curve as in Eq. 1 such that temperature is low
enough so that the long time scale strain rate is zero (η2 = ∞) (i.e. a standard linear solid),
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Continuum mechanism: Rheology
the lag of the strain behind the stress is caused by a delayed elasticity. On a very short time scale
the apparent modulus of the material is just µ2 but on a longer time scale the apparent modulus
is µ1 µ2 /(µ1 + µ2 ) as can be seen from the creep function (Eq. 1). At intermediate time scaled,
one expects the apparent modulus to depend on frequency w, that is, there is dispersion. It is in
this range of dispersion that the strain cannot keep pace with the stress and dissipation occurs.
Therefore, we may anticipate the following behavior. Here τ represents a characteristic period for
Figure 15: Dependence of elastic modulus and energy dissipation on frequency of loading for a single
process of delayed elasticity.
which dissipation is a maximum.
The physical basis for the form of the behavior can be understood in terms of the rheological
model of Fig. 11. At very high frequency (w large), the response is determined by µ2 alone because
the combination µ1 and η1 does not respond fast enough. The dash pot η1 does not deform and
therefore does not dissipate any energy. At very low frequency the strain response is given by the
sum for both springs
1
1
µ 1 µ2
σ(
+ ) = σ/(
).
µ1
µ2
µ1 + µ 2
In this case the dash pot η1 deforms, but the rate of deformation is so slow that negligible stress
is required, and the entire stress is supported by the spring µ1 . Consequently, the energy input to
the dash pot is negligible. It is only on an intermediate time scale that the deformation rate and
the stress supported by the dash pot η1 are both large enough that there is work done on it.
Independent of the specific model we can understand how dispersion and attenuation go together. Variation of elastic modulus with frequency implies a non-instantaneous strain process that
in turn implies a lag between strain and stress.
The complex modulus m and the corresponding quantities |m| and tan δ can be worked out
Throstur Thorsteinsson (th@turdus.net)
17
Continuum mechanism: Rheology
explicitly in terms of µ1 , η1 and µ2 for the standard linear solid. One finds
tan δ =
τ c − τr
wτ
τ
1 + w2τ 2
(5)
where τc = η1/µ1 is the characteristic time in a creep experiment (Eq. 1), τr = η1 /(µ1 + µ2 ) is
the characteristic time in a relaxation experiment (Eq. 2), and
√
τ = τr τc
which is the geometric mean of the characteristic times for the creep and relaxation experiments.
Also
n
o1
2
2
(1 + w2τ 2 ) + w2 (τc − τr )2
µ1 µ2
|m| =
(6)
2
2
|1 + w τr |
µ1 + µ 2
This shows
µ1 µ2
|m| →
as w → 0
µ1 + µ 2
|m| →
τ 2 µ1 µ2
µ1 µ2
τc
=
= µ2 as w → ∞.
2
τ r µ1 + µ 2
τr (µ1 + µ2 )
When tan δ and |m| are plotted against w, they look very much as in the Fig. 15: tan δ is a
bell shaped curve with maximum at w = 1/τ and |m| shows a maximum rate of change at this
frequency. The particular relationship for the standard linear solid is called Debye dispersion and
arises in other relaxation processes involving delayed effects such as in dielectric relaxation, electric
circuits, absorption of light, etc.
Mechanical relaxation in some substances can show Debye dispersion with a single peak, as for
a standard linear solid. An example is a single crystal of ice which as maximum dissipation in the
kilohertz range, the actual peak frequency depending strongly on temperature. Most substances
are more complex and show a spectrum of relaxation frequency because of a number of different
physical mechanisms that can cause energy dissipation at different time scales. The internal friction
spectrum provides another option in addition to the creep and relaxation functions for describing
the viscoelasticity of a material.
The physics of internal friction is interesting and a good reference discussing thisfrom a geophysics point of view is Jackson and Anderson, 1970, “Physical Mechanisms of Seismic-Wave Attenuation” Reviews of Geophysics and Space Physics, Vol. 8, no. 1, pgs. 1-63. Some examples
of mechanisms are thermoelastic relaxation (delayed thermal strain from strain heating and heat
flow), atomic scale diffusion processes (stress induced ordering, dislocation damping), grain boundary relaxation (time dependent slip on grain boundaries), and partial melting (delayed strain in
fluid components).
The internal friction (Q−1 ) in the earth has commonly been assumed to be relatively frequency
independent although this is not established very well. Distribution of Q in earth is discussed by
Anderson and Hart (1978 JGR, V. 83, No. B12, pg. 5869-5882) and Proceedings of the Stanford
Conference (1980, JGR, V. 85, No. B10). However, the lack of observational definition of a
frequency dependence is probably associated with the limited resolution and bandwidth of seismic
data. A more recent discussion of Q is given in Chapter 14 of ”Theory of the Earth” by Don L.
Anderson (1989).
Throstur Thorsteinsson (th@turdus.net)
18
Continuum mechanism: Rheology
READING ASSIGNMENT
Rheology
Turcotte and Schubert: Sections 7-1, 7-6, 7-9, & 7-10
PROBLEMS
1 Write down the stress-strain relation for Maxwell and Kelvin models
2 Write down the stress-strain relation for two dashpots (η1 and η2) in a) series, and b) parallel.
3 Describe the properties of a Voight solid (µ2 and η2 → ∞ in the visco-elastic equation), that
is make a strain vs. time plot for a step in stress (constant stress applied at t = 0).
4 Describe the properties of a standard linear solid (η2 → ∞ in the visco-elastic equation).
PROBLEMS
Energy change
• Determine the complex modulus m = |m|eiδ for harmonic loading of a standard linear solid
and show that tan δ and |m| are given by Eqs. 5 and 6
• Explain the effects water and melt have on the rheology of mantle materials.
• Explain Q in Earth.
• Explain brittle and ductile deformation.
Turcotte, D. L. and G. Schubert. 2002. Geodynamics: Application of continuum physics to geological
problems. Cambridge University Press, 2nd edition.
Throstur Thorsteinsson (th@turdus.net)
19