Evaluating Hysteresis in Earth Materials Under Dynamic Resonance

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. Bll, PAGES 25,139-25,147, NOVEMBER 10, 1996

Evaluating hysteresis in earth materials under dynamic resonance

Abraham Kadish and Paul A. Johnson 1 Earth and EnvironmentalSciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico

Bernard Zinszner Institut Frangais du P•trole, Rueil Malmaison, France

Abstract. A lumped parametermodel is derivedfor studyinghysteretic effects in resonantbar experimentson rock. The model usesequations of state obtainedby approximatingclosed hysteresis loops in the stress-strainplane by parallelograms.The associatedapproximate nonlinear state relations have a soundspeed (modulus) that takes two values.Assuming hysteresis and discretememory to be the primary nonlinear mechanisms,periodic solutions corresponding to theseequations of state are obtained analyticallyfor single-frequencycontinuous wave drivers,and their frequencyspectral densitiesare analyzed.In this simple approximation,if hystereticcontributions to the signalspeed are a correctionto the linear elasticsignal speed (i.e., the parallelogramis narrow), the model predictsthat the spectraldensity at even multiplesof the source frequencyis zero, and an approximate"pairing" of amplitudesis predictedfor odd harmonicmultiples. Comparison of the model spectrumwith experimentaldata showsthe model to be qualitativelycorrect. We concludethat hysteresisis an important mechanism in rocks.We considerthe model to be a prototype.

Introduction characteristics,together with the poor predictionsobtained from classicalnonlinear wave theory [Guyeret al., 1995a, b] Nonlinear elasticwave propagationexperiments have been suggestthat hysteresisand discretememory may play an im- conductedat Los Alamos National Laboratoryand the Institut portant role in propagatingand standingwaves. There is con- Frangaisdu P•trole as part of an effort to determine the non- siderableexperimental evidence suggesting that hysteresisand linear stateparameters in earth materials[e.g., Meegan et al., discretememory and/or other nonlinear phenomenastrongly 1993; Johnson et al., 1996; B. Zinszner et al., Influence of effectpropagating and standingwaves at strainlevels as low as changein physicalstate on elasticnonlinear responsein rock: 10-7 underambient conditions [Ostrovsky, 1991; Guyer et al., Effectsof effectivepressure and water saturation,submitted to 1995a;Johnson et al., 1996]. Journalof GeophysicalResearch, 1996]. Three typesof experi- We introducea simplelumped parametermodel to evaluate mentsare being conductedwith this goal in mind: staticstress- the effectsof hysteresisin resonantbar experiments.It will be strain, dynamicpulse propagation, and dynamicresonance ex- demonstratedthat the lumped parameter model (also "zero periments [see, e.g., Johnsonand Rasolofosaon,1996]. The dimensional")predicts resonant periodic motions whose spec- dynamic experimental evidence suggeststhat more familiar tral amplitudesexhibit qualitativeproperties similar to those approachesto modeling a nonlinear resonantsystem, such as observedin experiments.The term hysteresisis usually em- the Dutting equation [see Stoker, 1950], do not predict the ployed in two contexts:history-dependent multivalued state empirical behavior of the harmonic spectrum and resonant relationsin nonlinearmaterials (e.g., betweenstress and strain frequencyalteration for earth materials[Guyer et al., 1995a,b; in rocks and between the magneticfield and magneticinduc- Johnsonet al., 1996]. That is, classicalpredictions of harmonic tion in magneticmaterials) and the discontinuousrelations distributionand resonantfrequency shift as a functionof drive observedbetween input and output experimentalparameters amplitudein resonanceexperiments on earth materialsdo not match observation. This same conclusion has also been ob- (e.g., frequencyversus amplitude) in drivennonlinear systems. Both of these hystereticphenomena are observedin rocks. tained from pulse mode wave experiments[Ten Cate et al., Because the latter form is common to almost all nonlinear 1996; Van Den Abeele and Johnson, 1996; Kadish, 1995; Kadish et al., 1996]. systems(e.g., Dutting'sequation), its presencecannot be used to deduce the existence of a multivalued state relation. The It is clear from statictests on earth materialsthat hysteresis and discretememory are characteristic[Boitnott, 1993; Hol- lumped parameter model describedin this paper usesa multivalued stress-strain relation that is amenable to mathematical comb, 1978, 1981; Gardneret al., 1965;Birch, 1966]. These two analysisof oscillationsdriven in resonant bar experiments. Although the modelingprocedure accommodates other non- 1Alsoat Departmentd'Acoustique Physique, Universitfi Pierre et linearities, the multivalued nature of the state relation is the Marie Curie (Paris 6), Paris. only nonlinearityconsidered here in order to facilitate analysis. Copyright 1996 by the American GeophysicalUnion. (It is also the only sourceof dissipationconsidered.) Conse- Paper number 96JB02480. quently,one doesnot expectcomplete quantitative agreement 0148-0227/96/96JB-02480509.00 with data from nonlinearresonant bar experiments.However,

25,139 25,140 KADISH ET AL.: EVALUATING HYSTERESIS IN EARTH MATERIALS

cracks,grain-to-grain contacts, etc. The discontinuitiesin signal speedproduce shocksnot seen in the limit of ideal elas- ticity.The shocksare space-timepaths across which derivatives of stressand particlevelocity are not continuous.Evidently, it is necessaryto accountfor local historiesof stressfor accurate numerical simulations of pulse propagation in hysteretic -(•max materials. +(•max The simplestmathematical models of signalspeed in a hys- tereticmaterial are obtainedby approximatingcurvilinear hys- teresisloops by parallelograms,an exampleof which is illus- trated in Figure 1. In this paper approximationsof the form

p*de =c 1+ sgn at h_+)(2) Figure 1. The model hysteresisloop usedin this paper, for the specialcase where o-o = 0 (parallelogram),and a more areused. Here, c• isthe squareof theconstant "elastic" signal physicallyrealistic version (bold line) in stress-strain(o--e) speedof the medium,and the h are hysteresisconstants (sub- space.For the approximatingparallelogram, there are pre- scriptsrefer to the signof the sgnfunction). ciselytwo signalspeeds (equivalently elastic moduli). The ar- Effectively,the medium behaveslike two different elastic rowheadsindicate direction of increasingtime. materials.The sgnfunction and the h _+provide a mathematical prescriptionfor switchingbetween one material and the other. The signsof the hysteresisconstants determine dissipative the model does allow a qualitativedetermination of the rela- propertiesof hysteresis.If they are negative,as will be assumed tive strengthsof hysteresisvis-h-vis other nonlineareffects. In in this paper, hysteresisis a dissipativemechanism. It is the particular,the hystereticspectral signature exhibits unique and only dissipativemechanism and nonlinear effect includedin easilyidentified properties for weak nonlinearity. this treatment. The sgn function of the time derivative of In the following sectionan overviewof the theory is pro- (o-- 0-0)2 determinesthe timesat whichthe changein elastic vided, but the details have been placed in the appendices. behavior occurs.The values of the hysteresisconstants h+_ Spectraldata from recent resonantbar experimentsare pre- togetherwith the elasticsignal speed determine the slopes(i.e., sentedin the subsequentsection. A discussionof the lumped moduli) of the stressstrain relation on the parallelogramap- parametermodel and the pertinentspectral properties it pre- proximationto the hysteresisloop as shownin Figure 1 for the dictsfor resonantbar experimentsfollows the data section. specialcase o- o -- 0. During propagation,the speedsare piece- wise constantin time, greatly simplifyinganalysis. Wave Equations With Hysteresis Followingthe arrowheadson the curvilinearloop indicating One-dimensionalnonlinear pulse propagationin ideal and the directionof time in Figure 1, abruptchanges in slopeoccur nonidealelasticity is governedby coupledwave equationsfor when(o- - 0-0)2 is maximum,Cr2max . These abrupt changes are the stresso-and particle velocity v. The local signal speeds, physical.(When the maximumis achieved,the time derivative csig,al, are given by of (o- - 0-0)2 changessign and the materialswitches elastic states.)Figure 1 alsoshows that the slopesat commonvalues of o-depend on whether (o- - 0-0)2 is increasingor decreasing Csignal • • lO, (1) with time. With the parallelogramapproximation to the closedloop, where e is the local strain and p* is the massdensity of the discontinuitiesin signalspeed also occur when o- - o-o changes unstressedsample. In nonideal materialsthe stress-strainre- sign and require specialtreatment. These discontinuitiesare lation is not one-to-one.Hysteresis and other plasticityphe- not physical.Their presenceis a price one paysfor the math- nomenaresult in multivaluedstate relations (in the remainder ematical simplificationgained by introducing the parallelo- of this paper, use of the term hysteresiswill be restrictedto gram approximationto the true hysteresisloop. Equation(2) mean phenomenaassociated with multivaluedstate relations as opposedto hysteresisof the resonancefrequency-amplitude could be made more physicallyrealistic by smoothingto pro- response).An exampleof sucha relation is illustratedin Fig- vide continuoussignal speeds at o- = o-o, as illustratedby the ure 1. The curvilinearloop in the figure representsan empir- curvilinear path in Figure 1; however, the analysisis much ical path. Arrows along the path indicate direction with in- more difficult and, we believe, would not add significantlyto creasingtime. The counterclockwisedirection corresponds to the physicscontained in the model. Equation (2) may be re- hysteresisbeing dissipative.The parallelogramin the figure is gardedas a limitingcase of expressionsnot havingthis discon- a convenientapproximation, which is discussedin detail below. tinuity. In order to avoid nonphysicalshocks when using(2), Sufficientlysimple periodic evolution of stressand strain re- continuityof first derivativesof stressand particlevelocity will sultsin repetitionof closedloops of the type shownin Figure be imposedwhen o- - o-o changessign. 1. As illustratedby the curvilinearloop, hystereticstate rela- In Appendix A, a lumped parameter model for temporal tions usually exhibit discontinuousvariations in local signal oscillationsdriven in resonantbar experimentsis derivedfrom speed(or modulus)when the time derivativeof stresschanges propagationequations with signalspeeds modeled using (2). sign. The material respondsdifferently to increasesin stress The nonlinear oscillator equations for the evolution of the than it doesto decreases.The hystereticresponse is generally dimensionlessscaled lumped stresss(t) and particlevelocity attributed to the compliantfeatures in the material such as w(t) are (seeAppendix A) KADISH ET AL.: EVALUATING HYSTERESIS IN EARTH MATERIALS 25,141

unboundedstresses and particlevelocities. For 0 Frequency expansionin small/5, (1) terms correspondingto even multi- tudesof l• are absent,i.e., have zero amplitude(see equation (C3)), and (2) there is an approximatepairing of the acceleration amplitudesof termscorresponding to higherodd multiplesof l• (i.e., ratiosof the amplitudesof the 51• and 71• terms are equalto 1 within 2%, while thoseof the 91• and 1ll• terms, etc.,are equalto 1 within <1% (seeequations (C5) and (C6)). These resultsare illustratedqualitatively in Figure 2. The top portion of the figure showsa the spectrumof the driver. I I i i The bottomportion of the figureshows a detectedspectrum of I 3 5 7 9 11 the type predictedby the model. Frequency Figure 2. (top) Single-frequencysource spectrum, and (bot- tom) detectedspectrum of the type predicted. Comparison With Experimental Data A schematicof a resonantbar experimentis shownin Figure 3a along with the typical characterof nonlinear responsein resonancein rock (Figure3b). The bar, occupying0 < z < L, is drivenat z = 0 with a periodicvelocity v(0, t) while the end (3) at z = L is stressfree, o-(L, t) = 0. The resonancepeak shifts downwardwith increasingdrive level as a result of the nonlinear responseof the material.The plot alsoillustrates the hys- where v(0, t) is the particlevelocity applied at the end of the teretic nature of the frequency-amplituderesponse of a reso- bar at z = 0. Here we note that both /5 and 11 are positive nant bar drivenat nonlinearlevels; the upgoingand downgoing constantsthat are propertiesof the material. 11 is an angular frequencycurves are very different from each other when the frequency,and/5 is a hystereticstrength parameter. The plus bar is driven at nonlinear levels. and minussigns appearing in front of/5 and as superscriptsof Figure 4 illustratesexperimental harmonic measurements s(t) and w(t) are thoseof the sgnfunction in (2). from three rock types:Fontainebleau sandstone, Lavoux lime- In the absenceof hysteresis,/5 = 0, and applicationof a stone, and Meule sandstone[see, e.g., Lucet and Zinszner, sourcefunction v(0, t) havingangular frequency 11 produces 1992]. The detectedacceleration is plotted on the horizontal

z=O z=L Frequency Time Swept • -- Averaged CW Drive Acceleration v (O,t) o (L,t)= 0

145 B 125

105

85 A

1290 1310 1330 1350 1370 1390 1410 Frequency Figure 3. Illustrationsof (a) resonantbar experimentalconfiguration and (b) typicalresonant frequency response.In the bottomplot, dotted (solid) curvesshow the responseas the sourcefrequency is increased (decreased)while the amplitudeis held constant.The curvelabeled "A" is an amplificationof the response of a verylow amplitudesource in order to illustrateits behavior.The peak at B illustratesthe responseto the largestdrive level. 25,142 KADISH ET AL.: EVALUATING HYSTERESIS IN EARTH MATERIALS

Meule Sandstone characteristicsof rock.As shownin Figure 4, for instance,the

-20 ' I ' ' I ' ' I ' third harmonicamplitude is generallythe largest.This is a very m3/ml common observationin rocks in general. In all of the rock -30 samplesshown in Figure 4 there is a dominancein amplitude [] o o [] o o m2/ml of the odd harmonicsover the neighboringeven harmonics; -40 o [] o i.e., •o3is almostalways larger than •o2and •o4,•o5 is always o -50 x larger than •o4and o•6,etc. m5/mlx If hysteresisis important,the theoryof the precedingsection -60 shouldbe mostaccurate at low amplitudewhere other sources of nonlinearityare lessimportant. There were no data avail- -70 able at very low amplitude.Consequently, linear fits of the

-80 I , I • , I • , I • availabledata were extrapolatedto low amplitude.Figure 5 1.5 10'6 3 10'6 4.5 10'6 6 10'6 illustratesratios of •o7/•o5for the three rock typesobtained Strain from the data illustratedin Figure 4, together with a least squareslinear fit. Extrapolationof the linear fits to zero accel-

Lavoux Limestone erationyields ratios of about 1.77 for Fontainebleau,0.082 for Lavoux, and 0.27 for Meule. Note that as the amplitudeof the -10 ' ' I ' ' ' I * ' ' I ' ' ' I ' ' ' [ drive decreases, the ratios increase in all cases. While these [] [] [] [] 0 000C[:] 0 m -20 [] extrapolationsdo not exhibitthe predictedpairing of harmon- [] ics(ratio = 1), the dataand slope of the fit are compatiblewith • -30 xXX'x hysteresisdominating other nonlineareffects at low amplitude. X xX - n- -40 x m5/•l If only classicalnonlinearities were present,these ratios would X decreaseto zero with decreasingamplitude. -50 x Although the theory presentedhere does predict general m2/m•, o ' -60 characteristicssuch as the dominancein amplitudeof neigh- •" m4/ml - boring odd harmonics,modeling of the state relation must be + + ++a extendedto includeother nonlinearphenomena as well. .80 , , , m , , , m , , , I , , , m , , , I- 2.9 10'65.8 10'68.7 10'61.1610'•.45 10's Strain Discussion and Conclusions Lumped parameter models of resonant bar experiments Fontainebleau Sandstone (equations(A1)-(A4)) were obtainedfrom one-dimensional equationsof conservationof massand momentum for nonideal - 4 0 m3/ml [] elastic materials whose state relation is hysteretic.Periodic [] solutionswere obtainedanalytically, and their spectrawere -45 [] O0 [] [] ' [] comparedwith the experimentaldata. - 5 0 [] m2/ml The lumped parameter modelswere applied to materials -55 [] o o o oo o o o drivenin resonancefrom one boundaryand free at the other. :0 0 0 This configurationis the one used in nonlinearresonant bar - 6 0 - m5/ml x x x x x x experiments.Periodic solutionsdriven by a single-frequency sourcewere obtainedfor approximatingstress-strain loops. - d,• I[[ m For analyticalpurposes, the simplestclosed hysteresis loop in -75-6570I ,,,,,,,, e• •o, , •-•mT/ml . . • , • , i rolmi mßi stress-strain coordinates is a parallelogrambecause it corre- 1.4 10'6 2.8 10'64.2 10'6 5.6 10'6 7 10'6 spondsto a stress-strainrelation with only two signalspeeds. Strain For weak hysteresis,the theorypredicts the amplitudeof resonant motionsto scaleinversely with the strengthof the hys- Figure 4. Illustration of Young's mode resonantbar experi- teresis.(This is becausehysteresis was the only dampingmech- mental spectralratios for three different rock samples:(a) Meule sandstone,(b) Lavouxlimestone, and (c) Fontainebleau anism included in this treatment; see equation (C3)). In sandstone.Note the dominancein amplitudeof the third har- addition,at odd multiplesof the sourcefrequency, the spectra monic over the secondin each case. In the limestone, both the of thesesolutions exhibit pairing: the ratiosof the amplitudes third and fifth harmonicsare larger than the secondharmonic. of the seventhto fifth, eleventhto ninth, etc., are very closeto The other symbolsrepresent higher harmonics. unity.Because the only nonlinearitypresent is hysteresis,even harmonicshave zero amplitudein this approximation.If other nonlinearitieswere includedin the model, e.g., nonconstant axis, and the harmonic ratios, •oi/•ol, of the ith frequency elasticsignal speed, even harmonics would be presentand the multiple of the sourcefrequency are plotted on the vertical pairingof odd harmonicamplitudes would be weakened. axis.The harmonicamplitudes were obtainedfrom a Fourier The pairing of odd harmonicamplitudes predicted by the transformof the time signalmeasured at peak resonancefor theory was qualitativelyconfirmed by experiment;i.e., linear eachsuccessive resonance peak with increasingdrive level. The extrapolationof the •o7to •o5ratio to low-amplitudedrive was data are reliable to -72.25 dB. The rocks were under ambient consistentwith theory.(Note that higherfrequencies could not conditions for the measurements. be usedbecause these data were beneaththe noisefloor.) The The three rock typeshave very differentmineralogy, crack, linear extrapolationto low amplitudeis not conclusive;how- and grain contactstructure; however, they all illustratetypical ever, as is shownin Figure 5, the extrapolationsindicate an KADISH ET AL.: EVALUATING HYSTERESIS IN EARTH MATERIALS 25,143

Meule Sandstone

0.15

0.13 ••ercept=0.27 .

0.12 0 '

0.11

0.1 -.

0.09

0.08

0.07 .... i .... i .... I .... i .... i .... i .... i .... 80 85 90 95 I O0 I 05 I I 0 I I 5 I 20

Acceleration, m/s•

Lavoux Limestone

0.08 .... i .... i .... i .... i .... i .... o

intercept = 0.082 0.07

0.06 o

0.05

0.04

0.03 , , , , I , ß • , I ß • , ß I . . ß ß I ß I , , I, ß I , 100 200 300 400 500 600

Acceleration, m/s2 Figure 5. Ratio of measuredharmonic amplitudes o•7/o•5 versus detected acceleration for (a) Meule sandstone,(b) Lavouxlimestone, and (c) Fontainebleausandstone. Linear fit shownand interceptof fit notedon plots. 25,144 KADISH ET AL.: EVALUATING HYSTERESIS IN EARTH MATERIALS

Fontainebleau Sandstone

1.2 ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' '

intercept -- 1.77

0.8

0.6

0.4

, , I , , , 60 80 100 I 20 140 160

Acceleration, m/s Figure 5. (continued)

increasingratio as amplitude decreases.In the absenceof where e(z, t) is the strain; a state, or stress-strain,relation hysteresis,ratios of spectralamplitudes would be expectedto between•r and e hasbeen assumed;and p* is the massdensity approachzero with decreasingamplitude on the basisof clas- in the absenceof stress,where the laboratorycoordinate x of sicalperturbation analysis [Stoker, 1950]. Even with hysteresis, an elementis a functionof its initial position,or Lagrangian at low drive levelsthe absoluteharmonic amplitudes approach coordinatez, so that x = x(z, t) (see Figure 3). The particle zero, but their ratio, accordingto this model, does not. Con- velocityis the partial derivativeof x with respectto t. sequently,the extrapolationsof the observationsare consistent In thisappendix and the onesthat follow,the expressionfor with hysteresisbeing a dominantmechanism and suggestthat d(r/de givenby equation(2) of the main text is used.In this care should be taken when applyingclassical perturbation appendixa lumpedparameter model for resonantbar experi- methodsto hystereticmaterials. mentsis derivedusing (A1). The method,described below, is The lumpedparameter model derived here can be usedwith equivalentto a spatialaveraging of (A1). more completestate relations.The only dissipativeand non- The lumpedparameters for the particlevelocity and stress linear mechanismconsidered was that due to hysteresis.The for a bar as illustratedin Figure3a will be denotedby v(t) and modelcan accommodateother dampingmechanisms and non- •r(t), respectively.It will be assumedthat a drivingparticle linearities.Further development of the modelwill seeka more velocity,v(0, t) is appliedat one end of the bar, z = 0, and completeaccount of observedspectral properties by expanding that the other end of the bar, z = L > 0, is stress-free. We the prototypeto includeadditional phenomena. reducethe dimensionalityof the systemgiven in (A1) by re- placingspatial derivativesby differencequotients using the gradientscale lengths Z,, and Z,, definedby AppendixA: Lumped Parameter Model 0 vñ(t) - v(O, t) The simplestone-dimensional model for studyingcompres- v(z t)• sionalwave propagation in elasticand elastic-plasticmedia is Oz ' z•, (A2) the first-order2 x 2 systemconsisting of the equationsof continuityand force balancefor the stress,•r(x, t), and par- 0 •r-*(L, t) - •r(t) •rñ(t) ticle velocity,v(x, t) (the laboratoryposition coordinate is x, Oz(r(z, t) = Z•, - Z•, ' and time is t). In Lagrangiancoordinates (i.e., coordinates fixed in the material), respectively.The superscriptsrefer to the signof the sgnfunction in (2). The time derivativesof (A1) are replacedby time I O•r Ov Ov derivativesat valuesof stressand particle velocityat values =0 p* .... 0 (A1) d•r/de ot Oz Ot Oz betweenthe lumpedparameter and knownboundary values KADISH ET AL.: EVALUATING HYSTERESIS IN EARTH MATERIALS 25,145

0 d appearingin (3) are obtainedfrom the definitions Otv(z, t) • • [Oñv(0,t) + (1- O•,)v-+(t)] (A3) 0 d d p 0,,(1- 0,)Z, s+-(t) (A9) --o.(z,Ot t) • • [O•,o.(t)+ (1 - O•,)o.-+(L,t)] = •-/[O•,o.(t)]. 1 The O are intermediatevalue parameters(i.e., if the O are (v+-(t)- v(O, t)) = (1- 0,,)w+-(t) assigneddifferent values between zero and one, the quantities Assuming•-•2 is positive,we musthave 0 < 8 for hysteresisto beingdifferentiated take valuesbetween the boundarydata be dissipative.Therefore the gradientscale length associated and the lumpedparameters). Substituting (A2) and (A3) in with an increasingmagnitude of stressis lessthan that associ- (A1) yieldsa coupledsystem of first-orderordinary differential ated with a decreasingmagnitude of stress,and the principle equationsfor the lumpedparticle velocity parameter v(t) and dynamiceffect of hysteresisin the modelis seento be associ- stressparameter o.(t). atedwith an abruptchange in effectivegradient scale lengths d 1 do- for stress.Hysteresis will be saidto be weakif the differencein dt[O•,o.(t) ] v-+(t)- v(0,t)] :0 thesegradient scale lengths is small(i.e., if 8 is small).

(A4) d o.+-(t) AppendixB: Determinationof the Stress dt[O•v(0, t) + (1- O?•)v-+(t)]+ p'Z5 = 0 In thisappendix we obtainan asymptoticexpansion for the stresswhen 8 is smalland the displacementsource term for the Constraintsexist that limit the numberof independentgra- resonantbar is v(0, t) = -l/ sin lit. The periodicstress dient scalelengths and intermediatevalue parameters in the generatesa simpleclosed parallelogram in the stress-strain system.Because the termsthat are not time derivativesare plane(see Figure 1). To simplifythe presentation,we intro- integrablewhen the derivativeof the stresschanges sign, the ducedimensionless parameters r, S(r), and W(r) in placeof termsbeing differentiatedare continuous.This impliesthat time, stress,and particlevelocity, given by thereare onlytwo independentintermediate value parameters r--lit

O•+, = O,?= O•, O,+•= O• -= O• (AS) Z•,fia-+( r) S-+(r)= O,•(1 - O,,) Vp,C 2 (B1) Moreover,because the time derivativeof o.(t) is continuous when o.(t) changessign, the first relationin (A4) yieldsthe constraint -- ( - 03 v For clarity,the samephysical lumped parameter symbols have p*Z•,3•1 +: p*Z - (A6) been used even thoughthe independentvariable t has been replacedby r. With thesevariables, the equationfor the di- Equation(A6), togetherwith the secondrelation in (A5) and mensionlessstress (see (A7)) is continuityof v(t), impliesthat the firstderivative of the o.(t) d 2 is a continuous function of time. It must therefore vanish at d•.2S--+(• -)+ (1 -+- 8)2S-+(• ') = COS •' (B2) maximaand minima of o.(t). At stationaryvalues of o.(t), the v(t) - v(O, t) : O. W(r) maybe foundfrom S(r) using(A4). Differentiationof (A4) using(A5) and (A6) yieldsoscillator Our procedurefor determiningS(r) with period2,r is as equationsfor the stressand particlevelocity parameters: follows:We use a clock for times T given by r = T + (I) - ,r/2 (B3) •-td2-2 o.+-p t) + 112-+ p* 0,,(1-0•,)Z 7,dt v(O, t) which is set so that dS(T)/dT = 0 at T = +w/2 with dS(T)/dT > 0 on the half-periodinterval -w/2 < T < d 2 + ,r/2, andwe requireS ( - ,r/2) = - S( + ,r/2). The phase (A7) dt2 (v-+(t)- v(O, t)) + ll2_+(vñ(t)-v(O,t)) dependsupon 8. Let To be the time(to be determined)in the interval - ,r/2 < T < + ,r/2 at whichS (To) = 0. Requiring 1 d 2 periodicity,the stresson -,r/2 < T < +,r/2 and (B2) (1- O,,)dt 2 v(O, t) impliesthat the continuationonto + ,r/2 < T < +3,r/2 is givenby S(T) = -S(T - ,r) (seeFigure B1), with obvious where periodiccontinuation of thesolution for all times.Again using the same symbolfor the stressparameter,

d 2 dr2 St(T) + (1+_ 8)2S+-(T) = -sin (r + cI)) (B4) 1 C 2 (1 _+8) 2 = 112(1_+ 8) 2 (A8) O,•(l - O•,) Z•, Z,• The solutionsto (B4) for -,r/2 < T < + ,r/2 are givenby [(1 _+8) 2- 1]S+-(T)=A t cos[(1 _+a)(r- r0)] definesthe frequencyparameters appearing in (A7) and in equation(3) of the maintext. The scaledlumped parameters + B -+sin [(1 _+8)(r- r0)] + sin (r + cI)) (B5) 25,146 KADISH ET AL.' EVALUATING HYSTERESIS IN EARTH MATERIALS

ds/d{ In terms of the dimensionlessparameters of Appendix B, (A4) becomes

d d •o w+(r)-- • s+(r) (1-i- (•)2S+-(r) = • w+(r) (C1)

Sincethe dimensionlessstress and strain have period 2,r in T, it is natural to representthem as a Fourier seriesin sine and cosine functions periodic in T on -,r < T < + ,r. Odd- numbered harmonicswill not appear in the seriesbecause s(r + = -s(r). Up to and includingterms of order one, the first equationof (C1) is

dT V = (1 - O•,)cos (T + •) ,•o+• d[(1-Ov)v-+(T) 1 + (l q- a)2S-+(T)• (l - Ov) cos T

Figure B1. Phasespace representation of the closedhyster- esisloop from Figure 1 for the evolutionof the dimensionless sinT+• TsinT+ 1- cost stress. Arrowheads on the curve indicate the direction of vari- ation. The plusand minussigns in quadrantscorrespond to the signof the productof stressand its derivativein the quadrant. + • [ _+(T cosT + sinT) ] (C2) The period is 2,r. Two pointson the solutionpath lyingon the samestraight line thoughthe origin are equallydistant from Since the only odd function in T on -,r/2 < T < + ,r/2 in the origin and separatedin time by (C2) is the term that variesinversely with/5, the onlyterms of order 1 in the Fourier series for the acceleration are odd harmonicsof cosinefunctions. Writing Equation(B5) containssix 8-dependentconstants, ,4---, B---, To and •, that must satisfythe six symmetryand continuity constraints on the interval' dT V • -•- sinT + • (X2n+lCOS(2n + 1)T d[(1-O•,)v(T)] rr/4 n=0 • d d (c3) S-(-'rr/2) = -S+(+,r/2), dT S+ (To) = •-f S-(To), for T on -,r < T < +,r, one finds for n > 0 (B6) d d 1 (2n + 1)2 St(To)= O, d• S+(+*r/2) = • S-(-,r/2)= 0 a2,+,= • [n(n+ 1)]2 + n n even (C4) 1 (2n + 1)2 For/5 << 1, (B6)yields a2n+,= • [n(n+ 1)]2- (n+ 1) n odd T0= •-- 8+O(82) •=5Tø+O(82) Settingn = 2k for k = 1, 2, 3,---, one finds

1 1 1 1 S+(T) = •,r/4 sinT + •1{(•) T- sinT I[cr2(2•'+')+ll-Icr2{2k)+•[I- 5 (2k)(2k + 1)2(2k + 2) 32 k 4 (C5) + 1-• •- r cost +O(8) (B7) where the asymptoticexpression is intended for k >> 1. Settingn = 2k + 1 for k = 1, 2, 3, ..., one finds that for S-(T)=•sinT+•w/4 1{(•)T+ sinT large k 1 1 + 1-• •+T cost +O(a) 1l"2,2•+,,+,[-I-=,,_•+=)+,11 • 32k = (C6) The linear time dependencein the asymptoticexpansions for consequently,a "pairing" of amplitudesis obtained(see Figure s(T) in 8 is due to expansionsof trigonometricfunctions hav- 2). The pairing occursfairly early in the seriesand becomes ing 8-dependentarguments. morepronounced. For example,for n = 2, 3 (k = 1 in (C4)) andn = 4, 5 (k = 2 in(C4))

Appendix C: Acceleration Frequency Spectrum 0.98, Icr,,/cr01• 0.996 (C7) The time seriesand spectrafrom resonantbar experiments while Ic•7/c•9l• 1.68. are usually obtained from accelerometermeasurements. In For n = 0, one obtains this appendixwe use (A4) and the stressgiven by (B7) to obtain the lead terms up to order 1 in the expansionof the (C8) accelerationspectrum for weak hysteresis. KADISH ET AL.: EVALUATING HYSTERESIS IN EARTH MATERIALS 25,147

Acknowledgments. We thank Koen Van Den Abeele, Robert strain intervals from laboratory studies,Nonlinear ProcessesGeo- Guyer, Katherine McCall, Thomas Shankland,Patrick N.J. Rasolo- phys.,3, 77-88, 1996. fosaon,and JamesTen Cate for helpful discussions,and Michel Mas- Johnson, P. A., B. Zinszner, and P. N.J. Rasolofosaon, Resonance and son for experimentalassistance. This work was performedunder the nonlinearelastic phenomena in rock,J. Geophys.Res., 101, 11,553- auspicesof the Officesof BasicEnergy Research: Scientific Computing 11,564, 1996. and Engineeringand Geoscience,U.S. Department of Energy,with Kadish,A., Informationpaths and the determinationof staterelations the Universityof California.Support was also provided by the Institut from displacementmeasurements of elastic rods,J. Acoust.Soc. Franqaisdu P6trole. We thank K. Van Den Abeele for critical review Am., 97, 1489, 1995. of the manuscript.We also are grateful to the two reviewers,who Kadish,A., J. Ten Cate, and P. A. Johnson,Frequency spectra of dramaticallyimproved the manuscript. nonlinearpulse-mode waves, J. Acoust.Soc. of Am., in press,1996. Lucet, N., and B. Zinszner,Effects of heterogeneitiesand anisotropy on sonicand ultrasonicattenuation in rocks,Geophysics, 57, 1018- References 1026, 1992. Meegan, G. D., P. A. Johnson,R. G. Guyer, and K. R. McCall, Birch, F., Compressibility;Elastic constants, in Handbookof Physical Observations of nonlinear elastic wave behavior in sandstone, J. Constants,edited by S. P. Clark, Jr., pp. 97-174, Geol. Soc.of Am., Acoust. Soc. Am., 94, 3387-3391, 1993. Boulder, Colo., 1966. Boitnott, G. N., The role of rock joints in nonlinearattenuation, in Ostrovsky,L. A., Wave processesin media with strongacoustic non- Proceedingsof the NumericalModeling for UndergroundNuclear Test linearity,J. Acoust.Soc. Am., 90, 3332-3337, 1991. Stoker,J. J., NonlinearVibrations in Mechanicaland ElectricalSystems, MonitoringSymposium, edited by S. R. Taylor and J. R. Kamm, pp. vol. II, Interscience, New York, 1950. 121-134, Los Alamos Natl. Lab., Rep. LA-UR-93-3839, 1993. Ten Cate, J., K. Van Den Abeele, T. J. Shankland, and P. A. Johnson, Gardner, G., M. R. Wyllie, and D. M. Droschak,Hysteresis in the velocity-pressurecharacteristics of rocks,Geophysics, 30, 111-116, Laboratorystudy of linearand nonlinearelastic pulse propagation in 1965. sandstone,J. Acoust.Soc. Am., in press,1996. Guyer, R. A., K. R. McCall, P. A. Johnson,P. N.J. Rasolofosaon,and Van Den Abeele, K. E.-A., and P. A. Johnson,Elastic pulsed wave B. Zinszner,Equation of statehysteresis and resonantbar measure- propagationin media with secondor higher order nonlinearity,II, mentson rock, in InternationalSymposium on Rock Mechanics,ed- Simulation of experimental measurementson Berea sandstone, Jour. Acoust. Soc. Am., 99, 3346-3352, 1996. ited by J. J. K. Daemen and R. A. Schultz,pp. 177-181, A. A. Balkema, Rotterdam, Netherlands, 1995a. Guyer, R. A., K. R. McCall, and G. N. Boitnott, Hysteresis,discrete P. A. Johnson and A. Kadish, Earth and Environmental Sciences memoryand nonlinearwave propagationin rock, Phys.Rev. Lett., Division, Mail Stop D443, Los Alamos National Laboratory, Los 74, 3491-3494, 1995b. Alamos, NM 87545. (e-mail: [email protected]; Holcomb, D. J., A quantitativemodel of dilitancy in dry rock, J. [email protected];URL: http://www.ees4.1anl.gov/nonlinear) B. Zinszner, Institut Franqais du P6trole, B.P. 311-92506, Rueil Geophys.Res., 83, 4941-4950, 1978. Holcomb, D. J., Memory, relaxation,and microfracturingin dilatant MalmaisonCedex, France. (e-mail: [email protected]) rock, J. Geophys.Res., 86, 6235-6248, 1981. Johnson, P. A., and P. N.J. Rasolofosaon, Manifestation of nonlinear (ReceivedAugust 7, 1995;revised July 22, 1996; elasticityin rock: Convincingevidence over large frequencyand acceptedAugust 7, 1996.)