Nonequilibrium and Nonlinear Dynamics in Geomaterials I : The Low Strain Regime

0.57 MB·English

by  Donatella Pasqualini

LA-UR-05-3335 JOURNAL OF GEOPHYSICALRESEARCH,VOL. ???, XXXX, DOI:10.1029/,
Nonequilibrium and Nonlinear Dynamics in Geomaterials I :
The Low Strain Regime
1 2 3 4
Donatella Pasqualini, Katrin Heitmann, James A. TenCate, Salman Habib,
6
5 3
0 David Higdon, and Paul A. Johnson
0
2 Abstract. Members of a wide class of geomaterials are known to display complex and
n fascinating nonlinear and nonequilibrium dynamical behaviors over a wide range of bulk
−7
a strains, down to surprisingly low values, e.g., 10 . In this paper we investigate two sand-
J stones, Berea and Fontainebleau, and characterize their behavior under the influence of

6 very small external forces via carefully controlled resonant bar experiments. By reduc-
ing environmental effects due to temperature and humidity variations, we are able to sys-
] tematically and reproducibly study dynamical behavior at strains as low as 10−9. Our
ci study establishes the existence of two strain thresholds, the first, ǫL, below which the
s material is essentially linear, and the second, ǫM, below which the material is nonlin-
-
l ear but where quasiequilibrium thermodynamics still applies as evidenced by the suc-
r
cess of Landau theory and a simple macroscopic description based on the Duffing os-
t
m cillator. At strains above ǫM the behavior becomes truly nonequilibrium – as demon-
. strated by the existence of material conditioning – and Landau theory no longer applies.
t
a The main focus of this paper is the study of the region below the second threshold, but
m we also comment on how our work clarifies and resolves previous experimental conflicts,
as well as suggest new directions of research.
-
d
n
o 1. Introduction 1998];wewillrefertothisastheclassicaltheoryofnonlinear
c elasticity or simply as Landau theory.
[ Geomaterials display very interesting nonlinear features, Many geomaterials, such as sandstones, belong to the

diverse aspects of which have been investigated over a long general class of nonlinear materials. The second and third
1 periodoftimeforarecentoverviewsee,e.g., Ostrovksy and panelinFigure1displayresonantbarresultsfortworepre-
v Johnson,2001andreferencestherein]. Astandardtechnique sentative samples, Berea and Fontainebleau. In both cases
0 usedtostudythesenonlinearfeaturesistheresonantbarex-
the shift in the resonance frequency is very large and the
2 periment [Clark, 1966; Jaeger and Cook, 1979; Carmichael,
resonance frequency decreases with drive amplitude. The
1 1984; Bourbie et al.,1987]. Intheseexperimentsalongrod
strengthofthenonlinearresponseinthesematerialsisvery
1 of the material under test is driven longitudinally and its
large, orders of magnitude more than for metals. Conse-
0 amplitude and frequency response monitored. For a linear
quently, it is important to check whether Landau theory
6 materialtheresonancefrequencyoftherodisinvariantover
still applies to these materials, and, if so, over what range
0 a very wide range of dynamical strain. An example of this
/ behaviorisshownintheresultsfromoneofourexperiments of strains.
t onAcrylicinthetoppanelofFigure1: increasingthestrain It is widely believed that geomaterials behavedifferently
a upto210−6leavestheresonancefrequencyunchanged(note thanweaklynonlinearmaterialsbecauseoftheircomplexin-
m thatth·ex-axisshowsthechangeintheresonancefrequency, ternalstructure. Theyareformedbyanassemblyofmoreor
- ∆f,andnottheresonance frequencyitself.) Theresonance less rigid “grains” connected via a muchsofter “bond” net-
d frequency of a rod made from a nonlinear material such as work of varying porosity. The grains make up a large frac-
n Berea sandstone behaves quite differently: When a driving tion of the volume, between 80 and 99%. Individual grains
o forceisapplied totherod,thefrequencyeitherincreases or canbeverypure(asinthecaseofFontainebleau, 99+%
c decreases (the modulus either hardens or softens) depend- quartz)ormadeupfromseveraldifferentcompone∼nts(asin
v: ing on the precise properties of the material. This phe- thecaseofBerea: 85%quartz,8%feldspar,plussmallquan-
i nomenon is well-known and a theoretical description based tities of other minerals). Most of these materials are quite
X on quasiequilibrium thermodynamics and nonlinear elastic- porous and their behavior changes dramatically under the
ityhasexistedforalongtime[seee.g.,LandauandLifshitz,
r influence of environmental effects, such as temperature [see
a e.g. Sheriff, 1978] or humidity [see e.g., Gordon and Davis,
1968; O’Hara, 1985; Zinszner et al., 1997; Van den Abeele
1EES-9,UniversityofCalifornia,LosAlamosNational etal.,2002]. Thissensitivitytotheenvironmentmakescon-
Laboratory,LosAlamos,NewMexico87545 trolled studies difficult, as the experiments must be carried
2ISR-1,UniversityofCalifornia,LosAlamosNational outinsuchawaythattheseeffectsaredemonstrablyunder
Laboratory,LosAlamos,NewMexico87545 control.
3EES-11,UniversityofCalifornia,LosAlamosNational Anotherdifficultyinmeasuringthefrequencyresponseof
Laboratory,LosAlamos,NewMexico87545
sandstones arises from the brittleness of rocks. If the sam-
4T-8,UniversityofCalifornia,LosAlamosNational
plesaredriventoohard,microcrackscanbeinducedandthe
Laboratory,LosAlamos,NewMexico87545
5D-1,UniversityofCalifornia,LosAlamosNational resulting behavior of the material can change dramatically.
In addition, driving can also induce long-lived nonequilib-
Laboratory,LosAlamos,NewMexico87545
riummacrostates thatrelaxbackoveralongperiodoftime
( hours). Thus,it is important toensure–byrepeating a
∼
Copyright2008bytheAmericanGeophysical Union. given drive protocol on thesame sample and verifying that
0148-0227/08/$9.00 thematerialresponsedoesnotchangefrom oneexperiment
1
X - 2 PASQUALINIET AL.: NONEQUILIBRIUMANDNONLINEARDYNAMICS INGEOMATERIALS
to the next – that the samples have not been altered from ite,andOlivinebasalt,atstrainsbetween10−9 <ǫ<10−3.
their original condition and the environment is unchanged Their main objective was to measure the loss factor Q−1
over the set of observations. The experiments described in (or the internal friction φ in their terminology) as a func-
this paper were carried out in this way. Furthermore, the tion of strain and the ratio of stress and strain. In order
very low strain values ensured that sample damage rarely to cover the large strain range they divided their experi-
occurred. ments in two components: for 10−9 < ǫ < 10−5 they used
One goal of this work is to clarify, using new and exist- thedrivenfrequencymethod,drivingtherocksatveryhigh
ing data, conflicting observations in the literature, and to frequencies,andfor10−5 <ǫ<10−3 theymadedirectmea-
present a description of the “state of the art” at low strain surementsofthestress-straincurve. Theirmainfindingsare
amplitudes. Here we restrict ourselves mainly to the ques-
the following. (i) The loss factor is quite insensitive to the
tionofdynamicnonlinearityanddonottakeuptheequally
strain amplitude, diverging from a constant value only at
important question of the nature of loss mechanisms and
high strains. At these high strains they conclude that this
their connection and interaction with the nonlinear (com- increase in Q−1 istheresult of internaldamage. (ii) Q−1 is
pliant) behavior underlyingthefrequency shift.
highly structure sensitive, i.e., it is sensitive to the details
Inthepast,severaldifferentgroupshavecarriedoutreso- of themicrostructure of therock. (iii) Q−1 increases as the
nantbarexperiments. GordonandDavis[1968]investigated
temperature increases. They conclude that this increase is
alargesuiteofcrystallinerocks,includingQuartzite,Gran-
duetograin-interfacedisplacement,andthereforealteration
of the internal structure of the rock. (iv) At large strains
they find static hysteresis with end-point memory.
Following up on Gordon and Davis [1968], McKavanagh
6
−0 Acrylic andStacey[1974]andBrennan andStacey[1977]performed
e
2.0 another set of stress-strain loop measurements on granite,
basalt, sandstone, and concrete. Their main objective was
the measurement of stress-strain loops below strain ampli-
tudes of ǫ = 10−5, since Gordon and Davis [1968] had re-
e e−06 portedthatQ−1abovethislimitwasnolongeralinearfunc-
1.0 tion of the applied strain. McKavanagh and Stacey [1974]
were able to go down to strains of 10−6. (Note that this
level is still above the strain at which we found nonequilib-
rium effects to be important, TenCate et al. [2004].) At
0
+0 these strains they found that the hysteresis loops for sand-
e
0.0 −100 −50 0 50 100 stone were always cusped at the ends. Another interesting
result was that below a certain strain amplitude the shape
D f [Hz]
oftheloopbecameindependentoftheappliedstrainampli-
tude. Fromthistheyconcludedthatevenattheverysmall-
est strain amplitudes, cusps should continue to be present
6
e−0 Berea instress-strain loops. (However,Brennan and Stacey [1977]
2.0 noted that for granite and basalt, the stress-strain loops
do become elliptical for strains lower than 10−6.) In view
of our recent results [TenCate et al., 2004] this conclusion
might have been drawn without having enough evidence at
6
e e−0 lowenoughstrain amplitudes. Wereturntothispointlater
1.0 in Section 7.
Winkler et al. [1979] conducted experiments with Mas-
silon and Berea sandstone at strain amplitudes between
10−8 and10−6. Themaingoalwastodeterminethestrains
0
+0 atwhichseismicenergylossescausedbygrainboundaryfric-
e
0.0 −40 −20 0 20 40 tion become important but softening of the resonance fre-
quency with strain amplitude was also investigated. They
D f [Hz]
concludedthatthelossesareonlyimportantatstrainslarger
than were investigated. Additionally, they found that the
twosandstonesinvestigateddisplayednonlinearfeaturesde-
6
e−0 Fontainebleau pendent on several external parameters, such as water con-
2.0 tent or confining pressure. They find that the loss factor is
independentofstrain belowstrainsof5 10−7 whileatrela-
tivelargestrain(>10−6)thereisaclea·rincrease,inagree-
ment with Gordon and Davis [1968]. The main drawback
6
e e−0 of the experiments by Winkler et al. [1979] is the relative
1.0 lackofdatapoints,especially intheverylowstrainregime;
the quality of the repeatability of their measurements on
the same sample is also not shown. In this respect, our
work significantly improveson previousresults; we increase
0
+0 thenumberof measurement pointsin thelow strain regime
e
0.0 −30 −20 −10 0 10 20 30 by a factor of five in comparison to Winkler et al. [1979],
allowing a more robust analysis of thedata.
D f [Hz]
More recently, Guyer et al. [1999] and Smith and Ten-
Cate[2000]analyzedasetofresonantbarexperimentswith
Figure 1. Resonance curves for Acrylic, Berea, and
Berea sandstone samples also at low strains. The conclu-
Fontainebleau at different drives. Acrylicis a linear ma-
sions they reached, however, were in strong disagreement
terial used as a control in the experiments. Nonlinearity
with the older results of, e.g., Winkler et al. [1979]. In-
is evidenced in Berea and Fontainebleau samples by the
stead of the expected quadratic behavior of the frequency
shift in thepeak of theresonance curves.
PASQUALINIET AL.: NONEQUILIBRIUMANDNONLINEARDYNAMICS INGEOMATERIALS X - 3
shift with driveatverylow strains–anessential prediction 35 cm long. As established by X-ray diffraction measure-
of Landau theory – they reported an ostensibly linear de- ments, the Fontainebleau sandstone is almost pure quartz
pendence,claimed toholddowntothesmallest strains. We (>99% withtraceamountsofothermaterials); Berea sand-
notethatsuchalinearsofteninginseveralmaterialsamples stone is less pure having only 85 8% quartz with 8 1%
was also reported in Johnson and Rasolofosaon [1996] (see feldspar and 5 1% kaolinite and±approximately 2% o±ther
alsoreferencestherein),albeitatsignificantlyhigherstrains. constituents. ±Fontainebleau sandstone has grain sizes of
This surprising behavior was claimed to be consistent around 150 µ and a porosity of 24%. Berea sand-
withpredictionsofaphenomenological modeloriginally de- stonesamples havegrain sizeswhich≈aresomewhat smaller,
velopedtoexplain(static)hystereticbehavioringeomateri- 100 µ, with a porosity of about 20%.
als at veryhigh strains [thePreisach-Mayergoyz space (PM ≈
A small Bruel&Kjær 4374 accelerometer is carefully
space) model]. In this model a rock sample is described in
bondedtooneendofeachcoresamplewithacyanoacrylate
terms of an ensemble of mesoscale hysteretic units [McCall
glue (SuperGlue gel, Duro). The accelerometers are an in-
and Guyer, 1994; Guyer et al., 1997]. By applying the PM
dustry standard, and are well characterized. With perfect
space model to low-strain regimes, a linear dependence of
bonding between accelerometer and rock, the accelerome-
the frequency shift with drive can be obtained. By its very
ter – and the associated B&K 2635 Charge Amp – has a
nature, the model also predicts the existence of cusps in
flat frequency and phase response to 25 kHz. With poor
low-amplitudestress-strainloops. AsdiscussedinSection7,
bonds,theupperfrequencylimit of theflat responsedrops.
however,wedonotdetectcuspsinstress-strainloopsatlow
Thus, great care is taken to establish a good bond between
strains.
Motivated partly bythesevery different findingson sim- accelerometer and sample. Each accelerometer is first qual-
ilar sandstones and with similar experimental set-ups, we itatively tested (i.e., finger pressure) to be sure of a strong
embarkedonasetofwell-characterized resonant-barexper- bond. Furthermore, before the samples are placed in the
iments using Fontainebleau and Berea sandstone samples environmentalisolationchamber(discussedbelow)formea-
TenCate et al. [2004]. Broadly speaking, our findings for surements,acomparisonoftheaccelerometerresponsewith
theresonance frequency shift confirm theoriginal resultsof alaservibrometer(Polytec)ismadeandaccelerometersare
Winkler et al. [1979]; below a certain strain threshold ǫM rebonded if the frequency responses differed noticeably. In
bothsandstonesdisplayedtheexpectedquadraticbehavior. any case, it is important to point out that for the samples
In addition, we were able to show that previous claims of usedinthisstudy,alloftheresonancefrequenciesarebelow
a linear shift at high strains are actually an artifact due to 3 kHz, nearly an order of magnitude below the upper fre-
thematerialconditioningmentionedaboveatstrainshigher quencyflatresponselimitfortheaccelerometer/chargeamp
thanǫM,andthatasimplemacroscopicDuffingmodelpro- combination.
vides an excellent mathematical description of the experi- Thesourceexcitationisprovidedbya0.75cmthickpiezo-
mental data without going beyond Landau theory (as PM- electric disk epoxied (Stycast 1266) to the other end of the
space models explicitly do, by adding nonanalytic terms to sample core and backed with an epoxied high impedance
the internal energy expansion). Thus, we established that, backload (brass) toensurethat most of theacoustic energy
to the extent macro-reversibility holds, the predictions of couplesintotherocksampleinsteadofthesurroundingenvi-
classical theory are in fact correct.
ronment. Resonances in thebackload (>50 kHz) are much
In this paper we extend our previous analysis by adding
higher than the frequencies and resonances of the sample
an investigation of energy loss (via the resonator quality
and thusare not excited in our experiments.
factorQ),dynamicalstress-strain loops, andharmonicgen-
For all the experiments described here, the lowest order
eration. We carry out the same experiment several times
longitudinal mode (the first Pochhammer mode) is excited.
withthesamesampletodemonstrateenvironmentalcontrol
(Wenotethatthemassofthebrassbackloadlowersthecen-
andrepeatability. ThedataanalysisisbasedonaGaussian
ter frequencies of the Pochhammer mode resonances some-
processmodeltoavoidbiasingfrom nonoptimalfittingpro-
what but does not affect the shape of a resonance curve.)
cedures applied to experimental data. The Duffing model
Resonancecurvesareeasytomeasureandanalyzeandfairly
introduced in our previous work is shown to be nicely con-
sistentwiththenewerresults. Thepredictionsofthismodel highstrainscanbeattainedwithoutrequiringahigh-power
for the quality factor, the frequency shift, and hysteresis amplifier (with its frequently accompanying nonlinearities).
cusps (null prediction) all hold within experimental error For the Fontainebleau sandstone the lowest resonance fre-
at strains below ǫM. At higher strains, this simple model quencyis around 1.1 kHz;for theBerea sandstone thelow-
breaks down – as it must – due to the (deliberate) exclu- estresonancefrequencyisaround2.8kHz. Measuredvalues
sion of nonequilibrium effects. Finally, we have reanalyzed
a subset of the data which were taken in 1999 [Smith and
TenCate,2000] andhadledtoverydifferentconclusionsfor
-10
Bereasamples. Weshowthattheinterpretationofthedata 1.6x10
intheearlierpaperswasincorrectanddemonstratethatthe
experimental data are actually in good agreement with our 1.4
present findings[this paper and TenCate et al., 2004].
The paper is organized as follows. First, in Section 1
1.2
we describe the experimental set-up in some detail. Next,
in Section 3weexplain how weanalyze thedata, especially
howwedeterminethepeaksoftheresonancecurvesandhow eain 1.0
ourprocedureallowsustodeterminerealisticerrorbars. In Str
Section 4 we discuss the results from the experiments. A 0.8
simpletheoreticalmodelthatdescribestheexperimentalre-
sultsispresentedinSection5. Weconfrontpreviousfindings
0.6
inverysimilarexperimentswithournewresultsinSection6
and conclude in Section 7.
1140 1145 1150 1155 1160 1165
2. Experiments Frequency [Hz]
Figure 2. Low-amplitude drive resonance curve for
The samples used in the experiments are thin cores of
FontainebleauandBereasandstone1,2.5cmindiameterand Fontainebleausandstone. ThesolidcurveisaLorentzian
fit to thedata points.
X - 4 PASQUALINIET AL.: NONEQUILIBRIUMANDNONLINEARDYNAMICS INGEOMATERIALS
for the quality factor Q of these resonances are about 130 theresonancefrequencyis onlyabout 1.6 10−10,theshape
for the Fontainebleau sandstone sample and about 65 for of the resonance curve is clear with only m·inimal noise ob-
theBereasandstonesample. ThelowestorderPochhammer scuration: a Lorentzian curve is an extremely good fit to
modehasbothcompressionalandshearcomponentsbutthe the data as shown by the solid line. (Error bars are not
motionisneverthelessquasi-one-dimensionalandthebulkof shown for clarity.) With computer control and long-term
thesample participates in the wavemotion associated with temperature stability duetotheisolation chamber,this ex-
perimental setup permits long enough times to take data
the resonance. As higher-order Pochhammer modes begin
over a large – and an order of magnitude lower – range of
to resemble surface waves, only the very lowest frequency
strains not studied previously.
modes are examined here.
The samples are suspended at two points with loops of
synthetic fiber (dental floss) or thin O-rings. Different sus- 3. Data Analysis
pension points slightly alter the lowest Pochhammer mode
resonancefrequenciesbutthesedifferencesaremuchsmaller Thebasicquantitiesmeasuredinaresonanceexperiment
than differences caused by even slight changes of temper- arethefrequencyf andtheaccelerometervoltageV,which
ature; moreover, and perhaps more importantly, once the is automatically converted into acceleration u¨. It is con-
bar is mounted, the resonance frequencies do not change venient to translate the acceleration to a strain variable in
order tomakethecomparison of different samples with dif-
with increasing drive levels when tested with a standard
ferent lengths easier. As stated earlier, we employ the con-
(an acrylic bar). Suspended in this way (stress-free ends) vention ǫ = u¨/(4πLf2), where L is the length of the bar.
the sample’s lowest Pochhammer resonance frequency cor-
Thesemeasurementsleadtoresonancecurvesasshowne.g.,
responds to roughly a half-wavelength in the sample.
in Figure 1. The task now is to determine the peaks of
Since most rocks are extremely sensitive to temperature
theresonancecurves,trackingtheshiftoftheresonancefre-
andtemperaturechanges[Ide,1937]–withrelaxationtimes
quency as a function of thestrain as displayed in Figure 3.
ofseveralhours–wehavebuiltasample chamberforeffec- In the past, different methods have been suggested to
tive environmental isolation. An inner 3/4-inch-wall plexi- analyzedatafromlow-strainresonantbarexperiments[ear-
glass box withcaulked seamsholdsboth thesamples which lier attempts include Guyer et al., 1999; Smith and Ten-
are suspended from the top of the box. Air-tight electrical Cate,2000]. Inthispaperweuseastatisticalanalysisbased
feedthroughsareavailablefordriverandaccelerometercon- on a nonparametric Gaussian process to model the strain
nections. The entire chamber is flushed with N2 gas and ǫ as a function of the driving frequency f. The flexibility
then placed inside another (larger) plexiglass box and sur- of the Gaussian process model for strain allows for estima-
rounded with fiberglass insulation and sealed. The inner tion oftheresonance frequencyandresulting strain (f∗,ǫ∗)
sample chamber also sits on top of gel pads for vibration without assuming a parametric form for the dependence of
isolation. The complete isolation chamber is placed in a strain on driving frequency. Drawbacks of using a para-
metric model can include understated uncertainties regard-
room whose temperature is controlled with a thermostat
ingresonancequantities(f∗,ǫ∗)andexcessivesensitivityto
andtypicallyvariesbynomorethan3degreesC.Measured
measurementsfarawayfromtheactualresonancefrequency.
resonancefrequenciesofsamplesinthisboxhavebeenstable
Thenonparametricmodelingapproachavoidsbothofthese
to within 0.1 Hz.
possible pitfalls.
To get the most precise measurements possible, we use
an HP 3325B Frequencysynthesizerwith acrystal oven for Foragivenexperiment,observations(fi,ǫi), i=1,...,n
aretaken. Theobservedstrainismodeledasasmoothfunc-
frequencystabilityasthesignalsource. Thesignalfromthe tion of frequency pluswhite noise δ:
HP3325BisfedintothereferenceinputofanEG&G5301A
Lock-Inamplifierwhichcomparesthatreferencesignalwith ǫi =z(fi)+δi, i=1,...,n, (1)
themeasuredsignalfromtheaccelerometerviaaB&K2635
charge amplifier. The whole experiment,including data ac- where the smooth function z(f) is modeled as a Gaussian
quisition,iscomputercontrolledviaLabVIEWandaGPIB process and each δi is modeled as an independent N(0,σ2)
bus. To drive thesource, thesignal from theHP frequency deviate. TheGaussianprocessmodelforz(f)isassumedto
synthesizerisfed intoaCrown StudioReferenceIamplifier haveanunknownconstantmeanµandacovariancefunction
and matched to the (purely capacitive) piezoelectric trans- of theform
ducerviaacarefullyconstructedandtestedlinearmatching
transformer. C[z(fi),z(fj)]=σz2ρ−|fi−fj|2. (2)
To test all the electronics for linearity, we have con-
structed several known linear sample standards of nearly
identical geometry to the rock samples. The density,sound
speed, and Q’sof thesamples are chosen such that theme- 0
chanical impedances ρ c are similar to those of the rock
·
samples. These “standard” samples are driven with iden-
tical source/backloads and at levels similar to those expe- 5
−
rienced by the rock samples. No nonlinearities have been Acrylic
seeWn;irtehsuthltespforersaenntaicsroylalitciornodsyasrteemsh,owwenhinavFeivgeurriefie1d. long- f [Hz] −10 BFoenretaainebleau
term frequency stability of the samples to 0.1 Hz (corre- D
±
sponding to a long-term thermal stability inside the cham-
ber of 10 mK), which is close to how well the peak of the 15 (a)
−
frequency response curve can be determined at the lowest
levels of strain shown in this paper. To test the sensitivity
of the Lock-In amplifier and assembled apparatus, we have 20
−
measuredaresonancecurveontheFontainebleausampleat
5e−08 1e−07 2e−07 5e−07 1e−06 2e−06
an extremely low drive level. The result is shown in Fig-
e
ure 2. The acceleration measured by the accelerometer has
beenconvertedtostrain(theopencircles)usingthedriving
frequency f via ǫ= u¨/(4πLf2) following the convention in Figure 3. Resonance frequency shift ∆f as a function
of the effective strain ǫ for the three samples shown in
TenCate et al. [2004]. Even though the peak strain near
Figure 1.
PASQUALINIET AL.: NONEQUILIBRIUMANDNONLINEARDYNAMICS INGEOMATERIALS X - 5
9 proachtosamplerealizationsfromtheposteriordistribution
0
e− ofz(f)overadensegridofpointsintheneighborhoodofthe
1.95 resonance frequency f∗ [Banerjee et al., 2004]. From each
(a) oftheseMCMCrealizationsofz(f)theresonancefrequency
9 f∗ and the corresponding maximum strain ǫ∗ = z(f∗) are
−0 recorded. This creates a posterior sample of pairs (f∗,ǫ∗)
e
85 which are given by the dots in Figure 4(a). Figures 4(b)
1. and 4(c) show the posterior uncertainty for f∗ and ǫ∗ sep-
e arately with histograms of these posterior samples. We use
9
−0 the posterior mean as point estimates for f∗ and ǫ∗. Later
e
75 in the paper we use error bars that connect the 5th and
1. the95thpercentilesoftheposteriorsamplestoquantifythe
uncertainty in our estimates.
9
0
−
e
5
6 4. Experimental Results
1. 1153 1154 1155 1156 1157 1158 1159
Frequency [Hz] 4.1. Memory Effects and Conditioning
We have recently established the existence of two strain
0 regimes [TenCate et al., 2004]. As mentioned earlier, in
5
the first regime (strains below ǫM) the material displays a
reversible softening of the resonance frequency with strain,
40 (b) while in the second regime, (nonequilibrium) memory and
conditioning effects become apparent. The second regime
30 is entered at the strain threshold ǫM which dependson the
nts materialandtheenvironment(e.g.,temperature,saturation
ou etc.). To determine ǫM for these samples, the following ex-
c 20 periments are performed.
A reference resonance curve is obtained at the lowest
0 strain possible. The resonance frequencyis determined and
1
usedasareferencefrequencyf0 forthefollowingprocedure.
The source excitation level is increased, a new resonance
0 curve is obtained, and then followed immediately by drop-
1155.8 1155.9 1156.0 1156.1 1156.2 ping the excitation level back in an attempt to repeat the
Frequency [Hz] reference resonance curve. If there are no memory effects,
the repeated curve’s resonance frequency should match the
initial reference frequency. If memory effects are at play,
they will persist and the repeated curve’s peak resonance
frequency will be lower than the original. An example of
0
5 thisisshowninFigure5. Thisprocedureisrepeatedforin-
crementallyincreasingexcitationlevelsuntilmemoryeffects
40 (c) becomemeasurable. Theexcitationlevel(andstrain)where
memory effects first become noticeable defines ǫM for that
s 0 sample.
ount 3 TheexistenceofthetworegimesdelineatedbyǫM iscru-
c cialtounderstandingandinterpretingthedynamicalbehav-
0
2 ior of geomaterials. Although it is possible to describe the
0
1
0 0
1.900e−09 1.910e−09 1.920e−09 1.930e−09
e
5
Figure 4. (a) Resonance curve for Fontainebleau at a
sptorsatienri∼or2·s1a0m−p9l.eTohfpeaciernst(rfa∗l,cǫl∗u)sttheratodfedfiontseitshtehreesMonCaMncCe f [Hz] 10
peak. Frequency peak distribution (b) and frequency D
peakstraindistribution(c)fromtheMCMCanalysisfor
5
thesame resonance curveshown in (a). 1
0
Themodelspecificationiscompletedbyspecifyingpriordis- 2
tributionsfortheunknownparametersσ2,µ,σ2,andρ. Af- 5e-08 1e-07 2e-07 5e-07 1e-06 2e-06
z
tershiftingandscalingthedatasothatthefi’sarebetween e
0 and 1, and the ǫi’s have mean 0 and variance 1, we fix µ
tobe 0and assign uniform priors over thepositive real line Figure 5. Example of resonance frequency shift show-
to σ−2 and σz−2, and a uniform prior over [0,1] to ρ. ing the conditioning effect. The drive is increased up to
The resulting analysis gives a posterior distribution for astrainof210−6 andafterwardstherockisdrivenagain
the unknown function z(f) which we take to be the reso- ·
at the lowest strain. The black dot shows the value of
nance curve. This posterior distribution quantifies the up-
the resonance frequency peak after the last drive appli-
dateduncertaintyaboutz(f)giventheexperimentalobser-
cation. The difference between the two values for ∆f at
vations. Weuse aMarkov chain MonteCarlo (MCMC) ap-
theloweststraindemonstratestheeffectofconditioning.
X - 6 PASQUALINIET AL.: NONEQUILIBRIUMANDNONLINEARDYNAMICS INGEOMATERIALS
nonlinearityofthematerialatstrainsbelowǫM withclassi-
caltheory[Landau and Lifshitz,1998], aboveǫM theexper- 2
imental results are complicated by conditioning effects due 0.
to the nonequilibrium dynamics of the rock. Disentangling
theintrinsicnonlinearityofthematerialandthesenonequi-
librium effects is very difficult and the frequency shifts in 0.0
dynamical experiments at strains above ǫM do not have z]
a simple interpretation. In particular, classical elasticity H
tchaneonroytabsesuamppeslietdheirnmtohdisyneassmenictiraellvyernsiobnieliqtyuilaibnrdiutmherseitfuorae- Df [ −0.2 Fontainebleau
tion. By the same token, classical theory cannot be tested
by experiments carried out in this regime. As discussed in 4
theIntroduction,previousexperimentaldatawereinterpre- −0.
tated without properly taking the existence of these differ-
ent regimes into account [e.g., Guyer and Johnson, 1999].
This,alongwithincorrectanalysisoftheexperimentaldata 2e−09 5e−09 1e−08 2e−08 5e−08 1e−07 2e−07
(seethediscussionbelow),ledtoclaimingevidencefornon-
classical behavior where in fact none existed. Nevertheless,
it is clear that a new theoretical framework for the second
regime,onethatcombinesnonlinearitywithnonequilibrium
dynamics, is definitely needed. 4
0.
2
0.
0
0 0.
(cid:3)-5 (a) Df [Hz] 0.4−0.2
− Berea
f [Hz] 10 −0.6
D(cid:3)- 8
0.
Nonlinearity −
5 Nonlinearity +
1 Nonequilibrium 1e−09 5e−09 2e−08 5e−08 2e−07 5e−07
(cid:3)-
e
0
(cid:3)-2 Figure 7. Resonance frequency shift ∆f as a function
5e-8 1e-7 2e-7 5e-7 1e-6 2e-6 oftheeffectivestrainǫforFontainebleauandBereasam-
e ples for ǫ < ǫM. The solid lines represent predictions of
atheoretical modelincorporatingaDuffingnonlinearity,
Eqn. (23). Two different sets of data points obtained
0 fromthesamesamplesareshowntodemonstratethero-
bustnessofthemeasurements. Notethelogarithmicscale
on thex-axis.
5 (b)
(cid:3)-
Hz] (cid:13) Figures 6 (a) and (b) show our data for the resonance
f [ 10 frequency shifts versus strain for Berea and Fontainebleau
D(cid:3)- samples respectively. The first regime, where the material
Nonlinearity displays only the intrinsic reversible nonlinearity is shown
Nonlinearity +
5 Nonequilibrium in the unshaded area, whereas the regime which combines
1
(cid:3)- nonlinearandnonequilibriumdynamicaleffectsisshadedin
gray. The strain threshold for Berea is ǫM 510−7 and
210−7 for Fontainebleau under the present≃exp·erimental
(cid:3)-20 co·nditions. The data points in the gray region are history-
5e-8 1e-7 2e-7 5e-7 1e-6 2e-6 dependent, and change depending on the way the experi-
e mentalprotocol isimplemented,whereas thedatapointsin
the unshaded region are insensitive to such changes, pro-
Figure 6. Resonancefrequencyshift versusstrain. The vided one begins with the rock in an unconditioned state.
firstregime wherethematerial displaysonly anintrinsic For the remaining part of the paper we will focus only on
reversiblenonlinearityisshownunshaded,andthesecond the intrinsic nonlinear regime which is uncontaminated by
regime whichcombinesnonlinearandnonequilibrium ef- conditioningeffectsandallowsforasimpleinterpretationof
fects is shaded in gray. The threshold strain for Berea is the experimental data.
ǫM 510−7 (a) and for Fontainebleau is ǫM 210−7
(b).≃Sin·ce ǫM is not only a material specific≃con·stant 4.2. Intrinsic Nonlinearity
but can also depend on environmental variables, such as
Inthissectionwedescribeexperimentalresultsforstrains
temperatureand humidity,we show theregime in which
nonlinearity and nonequilibrium are mixed, not as one below ǫM. In this regime, the data are free from memory
andconditioningeffectsandthesamplesdisplayareversible
solid block, but rather as a region in different shades of
softening of the resonance frequency with strain. For this
gray. It is important to note that the data points in the
reason it is possible to speak of – and analyze – the in-
shaded regions depend on the (temporal) experimental trinsic nonlinearity of the material. As discussed in some
protocolwhereasthedatapointsintheunshadedregions detailintheIntroduction,theprevioushistoryofresonance
characterize an invariant behavior.
PASQUALINIET AL.: NONEQUILIBRIUMANDNONLINEARDYNAMICS INGEOMATERIALS X - 7
measurements and the analysis of the associated results is ǫM,beyondwhichvalueconditioningeffectsalsoenter. This
somewhat confusing. On the one hand, there are claims behavior can be fully described by classical nonlinear the-
that geomaterials display essentially nonclassical nonlinear ory. At very low strains, ǫL 10−8 10−7 (lower end for
elasticbehaviordowntoverylowstrains(10−8)[Guyer and Fontainebleau, upper end for∼Berea)−the samples are effec-
Johnson, 1999] with no evidence for a crossover to elastic tively in a linear elastic regime. At these low strains there
behavior. On the other hand, earlier findings [Winkler et is no discernible dependence of the resonance frequency on
al., 1979], albeit with generous error bars, are inconsistent the strain – the materials behave linearly to better than 1
with theseclaims. part in 104. Our results are in qualitative agreement with
Inordertoinvestigatethisissueinasystematicandcon- previousworkbyWinkler[Winkleret al.,1979],butincon-
trolled fashion, we carried out repeatable resonance barex- tradiction with other results, Guyer et al. [1999]; Guyer
perimentsatstrainsaslowas10−9 followingtheexperimen- and Johnson [1999], and Smith and TenCate [2000]. We
tal protocols discussed above; these strains are an order of will study thiscontradiction in detail in Section 6.
magnitude lower than those previously investigated.
The results for the resonance frequency shift ∆f, ∆f = 4.3. Quality Factor
f0 Ω/2πwhereΩisthe(linear)resonanceradianfrequency,
− Energy loss in solids is mostly characterized by a
as a function of the effective strain ǫ for Fontainebleau
frequency-independent loss factor (“solid friction”) in con-
and Berea sandstone samples are shown in Figure 7. The
measured strain for Fontainebleau ranges from 2 10−9 to trast to liquid friction. Nevertheless, rocks are known to
ǫM ≃ 2 · 10−7 and from 2 · 10−9 to ǫM ≃ 5 ··10−7 for dfliusipdlalyoacdhianrgac[tee.rgi.s,tiBcsoronf,liq1u94id1]frwicittihonaansaassfuoncicattieodnodfeppeonre-
Berea. We observe a resonance frequency shift of 0.45 Hz
dence of the loss factor 1/Q (Q is also termed the quality
for Fontainebleau and 0.5 Hzfor Berea in theregime below
factor) on the frequency. It appears that the unusual na-
ǫM. The error bars shown in Figure 7 are calculated using ture of wave attenuation in geosolids remains to be fully
the MCMC analysis as described in Section 3. The strain
studiedandunderstood[Cf. Knopoff and McDonald,1958].
error bars are smaller than thesymbols used in the figures.
AspointedoutbyKnopoffandMcDonald,afrequencyinde-
Theerrorbarsfor∆f forBereaarelargerthantheonesfor
pendentQcannotbeexplainedbyalineartheoryofattenu-
Fontainebleau because of the smaller Q for the Berea sam-
ation,however,itisunlikelythatthenonlinearityshouldbe
ple: theBerearesonancecurvesaremuchwider,makingthe
peak determination more uncertain. The solid lines in Fig-
ure7 represent theprediction of a theoretical model with a
Duffingnonlinearity described in detail in Section 5.
We find that the resonance frequency softens quadrati-
callywithincreasingdriveamplitudeuntilthestrainreaches (a) Berea
3
2
s]
Volt 1
2 n [
0.0 atio 0
er
el 1
c −
c
0 A
o 0.0 −2
G/
DG 3
2 −
0
0.
− −6 −4 −2 0 2 4 6
(a)
Drive [Volts]
4
0
0.
−
0.0e+00 5.0e−08 1.0e−07 1.5e−07 2.0e−07 2.5e−07
e (b) Fontainebleau
5
0.
s]
06 Volt
0. n [
o 0
0.04 erati 0.
el
0.02 Acc
Qo 0.5
Q −
D 2
0
0.
−
−1.0 −0.5 0.0 0.5 1.0
(b)
Drive [Volts]
6
0
−0. Figure 9. Acceleration versus drive amplitude for the
0.0e+00 5.0e−08 1.0e−07 1.5e−07 2.0e−07 2.5e−07 (a) Berea and (b) Fontainebleau samples. The accel-
e eration and the drive voltage are proportional to the
strain and the stress respectively. Berea: strain ampli-
Figure 8. Fontainebleau: (a) Variation of the width Γ tude2.5 10−7 atafrequencyof2754.5Hz;Fontainebleau:
oftheresonancecurvepeak. (b)Variationofthequality strain am· plitude 10−7 at a frequency of 1154 Hz. Note
factor Q with strain. theabsence of cusps.
X - 8 PASQUALINIET AL.: NONEQUILIBRIUMANDNONLINEARDYNAMICS INGEOMATERIALS
associated with amplitudesince evenfor verysmall strains, in geomaterials via PM space models which are based on
Q remains finite. static-hysteretic building blocks. In previous work, it has
Inthepresentworkwedonotfocusonthedependenceof been argued that these models provide a correct descrip-
Q on frequency at small strains, but investigate the depen- tion of the dynamics of rock even at small strains [McCall
denceonstrainamplitudeasanalternativeprobeofdynam- and Guyer, 1994]) and at high frequencies [Cf. Guyer et
ical nonlinearity for effective strains ǫ < ǫM. We measure al., 1999]. Our dynamical experiments allow us to analyze
theQ from theamplitude resonance curvesdirectly,using stress-strain loops at very low strains in the kHz frequency
ω0 2
Q= [1+ (1/Q )] (3)
Γ O
where ω0 = 2πf0 and Γ is the width of the response curve
6
measured at the points a0/√2 where a0 is the peak ampli- on +0 Plexiglass
tude. ThisdefinitionofQisstrictlyvalidonlyforlinearsys- ati 1e
er
tems but, as will be discussed further below, at low strains el
c
the amplitude response curves are effectively those of a lin- Ac 4
ear system, albeit with a peak frequency shift. At leading m of 1e+0
order, the Q as defined in (3) is independent of the nature or
of theloss mechanism (solid or liquid friction). nsf
pspliotTnuhsdeeeclrouesrssvpeof.ancWsteoerpceteahrktuasfirndelqeyupeeexnnpcdeysctaonnitdttwtohocehvwaanirdgiatehbalesΓsa,offtuhtnehcetairmoen-- ourier Tra 1e+02
F
of the strain simply because ω0 is a function of the strain 0
0
amplitude. This is, however, a very small change, fraction- +
allyoforder10−4. Asidefromthisexpectedvariation,what 1e 0 1000 2000 3000 4000 5000 6000
is of more interest is whether Γ is also a function of the Frequency [Hz]
strain.
In Figure 8(a) we show measurements of the variation
in the relative width ∆Γ/Γ0 for the Fountainebleau sam-
ple. Asmentionedearlier,werestrictourselvestothestrain 8
n 0
nreognimeqeuibliebloriwumǫMeffteocptsr.evTenhtecwoindttahmΓincaatinononolfythbeermeseualstusrbedy eratio 1e+ Berea
toan accuracy of 1%, theerrorbarsbeingobtained from cel
MthCeMreCsulatnsaolfyFsiisguo∼freth8e(ar)esdoenmaonncsetrcautrevetsh.atT∆oΓth/iΓs0aicscuesrsaecny-, m of Ac 1e+06
tiallyconstant(exceptforthesinglehigheststrainpoint)as or
sf
is the case for linear systems. This result is also consistent n
winitShectthioenp5re.dictions of the Duffing model discussed below urier Tra 1e+04
Themeasurementoftherelativechangeinqualityfactor o
F
is shown in Figure 8(b) and, given the smallness of the fre- 2
0
quency peak shift, simply reflects the behavior of ∆Γ/Γ0. e+
1
Wenotethatexceptforthehigheststrainpoint,ourresults 0 2000 4000 6000 8000
are in agreement with a strain-independent quality factor Frequency [Hz]
withinthedisplayederrors. Ourresultsthereforecontradict
Guyer et al. [1999] who found a linear dependence of Q on 8
0
strain amplitude (over a similar strain range as measured +
e
here). To summarize, to the extent that we have investi- 1
n
uganteexdpetchteedstrbaeihnadvieoprenhdasenbceeenoffoauconuds.tic losses (ǫ<ǫM), no eratio +06 Fontainebleau
cel 1e
c
4.4. Stress-Strain Loops and Harmonic Generation of A 4
m +0
oneAwtovuerldyeloxwpescttratihnesraensodnaatntthbeafrresqyusteenmcietsoobfeinetsesreensttiahlleyrea, nsfor 1e
a
dtoambpeedan,derlilviepnseh.arTmhoinsicisosicnillcaotnotrraanstdttohethhyestseitruesaitsiocnurvine urier Tr 1e+02
(quasi)static hysteresis where “pointed” or “cusped” loops o
F
are observed duetosources of inelasticity that donot fit in 0
0
to the simple viscoelastic model. Whether low strain loops e+
1
at some point become elliptical was investigated by MacK- 0 500 1000 1500 2000 2500 3000 3500
avanagh and Stacey [1974] whocametotheconclusion that Frequency [Hz]
thiswasnotthecaseatstrains 10−5 forsandstoneandin-
∼
deedthat,“–cuspedloopsextendtoindefinitelysmallstrain Figure 10. Fourier transform of the acceleration
amplitudes”. Ontheotherhand,BrennanandStacey[1977] taken at the resonance frequency for Acrylic, Berea and
foundthatforgraniteandbasalt,loopsbecameellipticalat
Fontainebleau (semilog plot). Acrylic: nominal strain
strainvalueslowerthan10−6. Thesestatementsweremade of 2.6 10−6 at frequency 2120Hz; Berea, 2.5 10−7 at
with data taken at low frequencies, less than 0.1 Hz, thus 2754.5·Hz; Fontainebleau, 10−7 at frequency ·1154 Hz.
do not directly apply to our experiment unless the under-
The dashed lines show the positions of the first, second
lying sources of inelasticity continue to be relevant at high
and third harmonics. Harmonic generation is not de-
frequencies.
tected. The two spikes which occur in Plexiglass and
Experimental evidence for cusped stress-strain loops led
Berea are due to the residual nonlinearity of the experi-
to the theoretical description of nonequilibrium dynamics
mental apparatus.
PASQUALINIET AL.: NONEQUILIBRIUMANDNONLINEARDYNAMICS INGEOMATERIALS X - 9
rangeandtodetecttheexistenceofpointedorcuspedloops. 5.1. Multiscale Analysis
AsevidentinFigure9below,theloopsareellipticalwithno
Since the displacement u is small we can solve the equa-
evidence for cuspy behavior. Thus, we find no evidence to
supporttheexistenceof“nonlinear”dissipationmechanisms tion of motion (4) analytically and predict the softening of
– as invoked in PM space models – at kHz frequencies. In the frequency with the drive amplitude. We employ mul-
contrast,predictionsofthesimpleDuffingmodelintroduced tiscale perturbation theory to obtain a useful closed-form
inTenCateetal. [2004]anddescribedindetailinSection5, solution to Eqn. (4). In the following we describe how this
are completely consistent with thedata. approach works and howto extract modelparameters from
Ourexperimentalresultsareshown inFigure 9. Weplot experimental data. [For a complete derivation of multiscale
theaccelerationversustheamplitudeofthedriveappliedto perturbation theory see Nayfeh, 1981.] While one can of
the bar for both the (a) Berea and (b) Fontainebleau sam- course solve Eqn. (4) numerically, the analytic approach
ples. InthecaseofFontainebleau,thestrain is1 10−7 ata yields simple formulae which provide much better physical
frequencyof1154 Hzwhile forBerea thestrain is·2.5 10−7
atafrequencyof2754.5Hz.2 Theaccelerationandthe·drive intuition.
AnaiveapproachtosolvingEqn. (4)wouldbeastraight-
amplitude are proportional to the strain and the stress re-
forward expansion of thedisplacement in theform
spectively. The acceleration and the drivevoltage are mea-
suredasfunctionsoftimeandthetimeseriesisstoredonce
steady state was attained. In Figure 9, a piece of the time u(t,α)=u0(t)+αu1(t)+···. (5)
series is displayed and the acceleration shifted to obtain it
180◦ outofphasewiththedrivevoltage. Forbothsamples, This ansatz is justified for small displacements. Inserting
thereis no evidence for cusps in the stress-strain loops. the expansion of u in the equation of motion and keeping
Another important question is whether the nonlinearity onlytermsof (α)leadstotwodifferentialequationsforu0
O
evidenced by the peak frequency shift can also be detected and u1:
by searching for harmonic generation in resonant bar and
wavepropagationexperiments. Theinterpretationofresults u¨0+Ω2u0 =Fsin(ωt), (6)
fromwavepropagationexperimentsissomewhatambiguous 2 3
[Meeganetal.,1993,TenCateetal.,1996]duetoexperimen- u¨1+Ω u1 =−2µu˙0−γu0, (7)
talcomplications(e.g.,reflectivelosses). However,harmonic
which are simply harmonic oscillators with an inhomogene-
detection in (potentially much cleaner) resonant bar exper-
iments has been previously reported (Cf. Johnson et al. ity on the right hand side. The equation for u0 (6) can be
[1996]). These authors found substantial harmonic genera- solved immediately and the solution inserted into the right
tion in rock samples – includingBerea and Fontainebleau – handsideoftheequationofmotionforu1 (7)specifyingthe
at strains as low as 10−7. inhomogeneity for u1 completely. The solution for u1 can
In this paper, we present our results in a search for har- now be determined and a perturbative solution for u itself
monics at strains ǫ < ǫM. Figure 10 shows spectral mea- can be obtained by inserting u0 and u1 into Eqn. (5). A
surements for a linear material (acrylic) and the two rock detailedanalysisofthissolutionforu(t)leadstothefollow-
samples. The dashed lines indicate where the first, second, ingresult: forspecificvaluesofωresonancesoccur,thecase
andthirdharmonicsofthefundamentalareexpectedtoap- ω Ω leading to a primary resonance causing the solution
pear (these are not the higher Pochhammer modes). In all for∼u to diverge. To determine a solution for Eqn. (4) free
threecasesweobservenoevidencefortheexistenceofhigher
fromthisproblem,themethodofmultiplescalescanbeused
order harmonics. The two small spikes which occur in the
[Nayfeh, 1981]. The idea is the following: besides assuming
dataforPlexiglass (acrylic) andBerea are duetotheresid-
thatthedisplacementissmall,wealsoassumethatthenon-
ual nonlinearity of theexperimental apparatus.
linearityissmall. Inadditionweassumethattheexcitation,
thedamping, andthenonlinearity areall of thesameorder
5. The Model in α. This leads to a modified equation of motion for u:
In this section we introduce a simple phenomenological 2 3
u¨+Ω u+2αµu˙ +αγu =αFsin(ωt). (8)
model which describes the nonlinear behavior of the rock
samples under consideration. This model does not include
Further we introduce two time scales, a slow scale T1 =αt
a treatment of memory and nonequilibrium effects and is
therefore not meant to apply in the regime where these ef- andafasttimescaleT0=twhichleadstoatransformation
of the derivativesof theform
fects become important, i.e. for strains greater than ǫM. A
morecomplexmodelwhichappliesalso tothehigherstrain
d
regimes will be described elsewhere. As shown by us pre- = D0+αD1, (9)
dt
viously (TenCate et al., [2004]), a quartic (Duffing) poten-
tyiiaelldnsonrelisnueltasrittyhaatuagcmcuenratitneglyaddeascmripbeedthhaermdaotnaicinostchilelaltoowr ddt22 = D02+2αD0D1+···, (10)
strain regime. This model predicts a quadratic softening of
theresonancefrequencyasafunctionofdriveamplitude,as with Di =∂/∂Ti. Expandingu in the form
expected from the theory of classical nonlinear elasticity.
The equation of motion for the displacement is taken to u=u0(T0,T1)+αu1(T0,T1) (11)
be:
andkeepingagain onlytermsoforderαleadstothefollow-
u¨+Ω2u+2µu˙ +γu3 =Fsin(ωt), (4) ing set of differential equation for u0 and u1:
2 2
where γ < 0 leads to a softening nonlinearity as observed D0u0+Ω u0 = 0, (12)
in theexperiment (e.g., Figure 1). The drivingforce on the 2 2
D0u1+Ω u1 = 2D0D1u0 (13)
righthandsiderepresentsthedriveappliedtotherodsinthe −
3
experiment. ThefrequencyΩisthe(unshifted)harmonicos- 2µD0u0 γu0+Fsin(ωT0).
− −
cillatorfrequency(forγ,µ=0)andµisthelineardamping
coefficient. In the following we briefly discuss a convenient The difference with the previous naive expansion becomes
analytic approximation for thesolution of Eqn. (4). clear immediately: While earlier the driving force was part
X - 10 PASQUALINIET AL.: NONEQUILIBRIUMANDNONLINEARDYNAMICS INGEOMATERIALS
of the differential equation for u0, it is now part of the in- monic oscillator response, with a frequency shift and peak
homogeneity of u1. A general solution for u0 is given by height dependenton thedriveamplitude.
u0 =A(T1)eiT0 +A¯(T1)e−iT0. (14) 5.2. Constraints on the Model Parameters from the
Experimental Data
InsertingEqn. (14)intothedifferentialequationforu1 (13)
TheDuffingmodelpredictsan invariant resonancecurve
yields
width Γ, therefore we first measure this quantity from the
D02u1+Ω2u1 = (2iA′Ω+2iµAΩ+3A2A¯γ)eiΩT0 experimental resonance curves. Consistent with the above
−
A3γe3iΩT0 + 1FeiωT0+c.c. (15)
− 2
7
0
−
Since we are only interested in the case ω Ω, i.e., driv- e
ingneartotheresonancefrequencyweintrod∼uceadetuning 2.5
parameter (a)
ω=Ω+ασ ωT0=ΩT0+σT1. (16) 07
⇒ e−
5
e 1.
Inserting this expression into the differential equation (15),
expressing A in the polar form A = 1/2aexpiβ, defining a
new parameter φ=σT1 Ωβ and φ′ =σ Ωβ′, and elim-
− − 8
inating the secular terms from the resulting equation, we 0
−
e
arrive at thefollowing solution for u(t): 0
5.
u = acos(ωt φ)+ (α), (17) −10 −5 0 5 10
− O D f [Hz]
1F
a′ = aµ+ sinφ, (18)
− 2Ω
3γa3 1F
aφ′ = aσ + cos(φ). (19)
− 8 Ω 2Ω
After a sufficiently long time, a and φ will reach a steady- 7
−0 (b)
state hence their derivatives will vanish and the left hand 4e
sides of Eqns. (18) and (19) will be zero. Squaring the
equationsandaddingthemleadstotheso-calledfrequency- 07
response equation e 3e−
Ω2µ2a2+a2(cid:16)σΩ− 83a2γ(cid:17)2 = 41F2. (20) 2e−07
This equation can besolved with respect to σ
σ= 3a2γ 1 F2 4µ2a2Ω2. (21) −10 −5 D f [0Hz] 5 10
8 Ω ± 2aΩp −
As σ has to be real, the maximum value for a (which we
label a0) and therefore the peak of the response curve can
be immediately determined:
(c)
2 2 2 2 F
F =4µ a0Ω0 a0 = , (22)
⇒ 2µΩ 7
0
−
e
5
and therefore 7
e 4.
3F2γ
σ0 = 32µ2Ω3. (23)
7
0
Thus, the model predicts a quadratic softening of the fre- e−
5
quencywiththedriveamplitudeF. Themodelalsopredicts 4.6
theinvarianceoftheresonancecurvewidthΓforanystrain. −6 −4 −2 0 2 4 6
Solving Eqn. (20) for σ and substitutinga=a0/√2we ob- D f [Hz]
tain
Figure 11. Average strain amplitude ǫ as a function
Γ=2µ. (24) of drive frequency for Fontainebleau (a) and Berea (b)
and(c). Thereferencecenterfrequencyis1155.98Hzfor
Notethattheapproximationignorescorrectionsof (1/Q2). Fontainebleau and 2765.179 Hzfor Berea. The open cir-
ThesearenumericallysmallonthescaleoftheexpOerimental clesaretheexperimentaldata;thefilledcirclesmarkthe
peakpositions. Thesolidlinesaretheoreticalpredictions
errors. At this leading order of the approximation, the ef-
from Eqn. (20). Figure(c)showsindetailtheresonance
fectofthenonlinearityissimplytoproduceaneffectivehar-
curveat thehighest strain for Berea.