Empirical evaluation of fracture toughness: the toughness of


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American Mineralogist, Volume 67, pages 1065-1066,l9E2

Empirical evaluationof fracture toughness:the toughnessof quartz
Mrcneer M. Wooo
Department of Geological Sciences California Stat e U niv ersity Hayward, California 94542
eNo J. E. We,rolrcn
Dep art ment of M athematic s California State University Hayward, California 94542
Abstract
A toughnessrelatedparameter,Asv, of monocrystalline,amorphous,and finely polycrystalline brittle materialsis derived from standardsedimentologicsize analysisof a crushedsample.Asv is numericallyequalto the areaunderthe cumulative-frequencsyize distribution curve and is relatedto the fracture toughness,Kc, of the material by the relationship,Asv = 202 + 383log Kc.
A study of quartz showsthat toughnessgenerallyincreaseswith decreasinggrain size and with an increasingdegreeof interlockingof the grainsor fibers.

Introduction
Brittle materialsgenerallybreak by catastrophic propagationof Griffith flaws. Toughness,a parameter often appliedin contextto structuralmetals,has become a valuable measure of the resistance of these materials to fracture. The developmentof Griffith-Irwin fracture mechanical principles has givenriseto severalspecificparametersfor measuring toughness.Of these,fracture toughness,Kc, a critical value of the stress intensity factor, has gainedwide acceptance.Kc is a materialconstant characterizing the inherent difficulty of crack growth in a material. A discussionof the significance of Kc is given by Tetelman and McEvily (1967) and conventional procedures for deriving toughnessparametersare discussedin American Society for Testing Materials (1979).

material and standard sedimentologic analysis of the resultingproducts.The methodis amenableto rapid analysis, involves no prior knowledge of physical constantsof the material except density, and is conceptuallyreadily appreciated.
Historically the processbenefitsfrom the treatmentof Rosin'sLaw by GeerandYancy (1938)who
demonstratethe existenceof a type of probability distribution for the sizefractions of crushedmaterials. Protodyakonov(1962)used size distribution dataof crushedmaterialto obtain a strengthindex for the material, and Hobbs (1964)and Evans and Pomeroy (1966) have related similarly derived strengthindicesto compressivestrength.Kwong e/ al. (1949)have related amount of new surface area produced by crushing brittle materials to fracture surfaceenergy.

Quantitativeaspectsof toughnesshave received little attention from geologists. This stems partly from the elaborateproceduresfor measurgmentof toughnessparametersand the consequentdearth of toughness data for natural materials, and partly from the lack of obvioussignificanceof conventional toughnessparametersto natural processes.
In the present study an empirical method for evaluatingfracture toughnessis based on a procedure of crushing a specific starting size fraction of

Method
A toughnessrelatedparameter,Asv, derivesfrom a unique analysis of the size distribution of the crushedproduct of a rnaterial.
Initial preparation of a sample consists of handcrushingand sievingto obtain approximately 30g of startingmaterialin the -2 phi to - 1.5phi sizerange (4 mm to 2.83 mm). An amount of starting material equivalentin weightto a6.666cm3standardvolume

0003-004)v8210910-1065$02.00

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WOOD AND WEIDLICH: FRACTURE TOUGHNESS

(numericallyequal in grams to the density multiplied by 6.666)is placed in the 39 mm by 59 mm cylindricalcrushingchamberof a spnx shakertype mixer/mill with three 15.875mm hardened steel balls and crushed for 45 seconds. The crushed productis sievedin eightinch U.S. StandardSeries sievesof one phi interval between- 1.5phi and 4.5 phi for 15 minutes on a Combs gyratory sifting machine. The amount retained on each sieve is weighedto 0.01g.
Weight data may be plotted as a cumulative-
frequencyplot with arithmetic ordinate in which
cumulativepercent is plotted againstphi for the sevenone-phiintervalsfrom - 1.5phi to 4.5 phi. A quantitativetoughnessparameter,Asv, is numeri-
cally equal to the area beneath the cumulative-
frequencycurve.Asv may be comparedvisually on the graphor obtaineddirectly from the databy the trapezoidalapproximation rule in the form

/t.5\

A s y- o . s( r 4 _ ,.,+

--l

\

i:_0.5

[email protected];representsthe cumulativepercentat phi
t.
Reproducibility of Asy is generally within five percentof the meanfor multiple runs on separates of a singlehomogenousstartingsample.The sieving process discriminates on the basis of effective (least)cross-sectionaal reaof the grainsand, therefore, variation in grain shape (equidimensional, tabular,fibrous,etc.) must have an important, but

undeterminedinfluence on Asy. Values of Asv obtainedwith other crushingand sievingapparatus must be standardizedwith resultsfrom this studv.
Resultsand discussion
A theoreticalbasisfor relatingAsv to traditional fractureparametersprobably lies in the analysisof cracksproducedby sphericalindenters(Frank and Lawn, 1967)in which a relationshipbetween the productionof aHertzian crack systemand material toughness is derived from Griffith-Irwin fracture mechanics.
An empiricalrelationshipbetweenAsv and fracture toughnessK, c, is demonstratedin Figure I and Table I for severalstandardmaterials.The regressionline representsthe statisticallysignificantrelationship, Asv : 202 + 383 log Kc. Values of Kc reportedfor the standardswere derivedby conventional testing methods of fracture mechanicsas indicated in the referencesin Table 1. Although thereis a distinct correlationbetweenAsv and Kc for the standardmaterialsand experimentalconditions in this study, fracture toughnessis strongly influencedby atmosphericmoisture (Dunning et al., 1980and Schuyleret al., 1981).Inability to control humidity in the experimentalsituationprecludesanalysisof this effecton the value of Asy.
The approximately20 g of startingsamplenecessary to obtainAsy ensuresthat the value obtained representsa nearly averagevalue of toughnessfor the sample.The use of Asv as an indicatorof Kc is particularly valuable when toughness disparity within the samplerenders measurementsby conventionalmethods,which employ a restrictedportion of the sample,susceptibleto large variations. On the otherhand,Asv cannotbeusedasa measure of toughnessanisotropyand providesonly an indication of averagetoughnessfor anisotropic monocrystallinematerials.For this reasonthe data for sapphireshown in Figure I are not used in the regressionline calculations. Application of the method to polycrystalline material is limited to thosewhosemaximumgraindimensionsareconsiderablylessthan the minimumdiameterof the starting sizefraction (2.83mm).

M#t*
Fig. l. RelationshipbetweenAsv and fracturetoughnessK, c. Regressionline and 95 percent confidencebandsfor Asv (based on "r" distribution) are derived from the arithmetic mean for eachsampleexcept sapphire.Values of Kc are from references in Table 1.

Toughnessof quartz
To demonstratean applicationof the method,the toughnessparameter,Asv, for severalquartz materials hasbeendeterminedand is shownin Table 2. Quartz materials fall into two basic groups by toughnessA. groupwith relativelylow toughnessof

WOOD AND WEIDLICH: FRACTURE TOUGHNESS

Table l Toughnessdata for standardmaterials

Source

Fused quartz Spinel
si3N4 (NC350) Sapphire c-9606 AI2O3 (AD999) sic (Nc203) si3N4 (NCl32)

G .E . Unknown, synthetic Norton Natural corning Coors Norton Norton

Character
Amorphous Monocrystal
Polycrystal Monocrystal Polycrystal PoLycrystal Polycrystal Polycrystal

AsvKc^. (mean) (MPa m-'")

140

0.7

257

1.3

3I4

2.0

337

2.r

334

2.5

408

3.9

443

4.0

4s5

4.0

1067

Source of data

wiederhorn (1969) Evans and charles

(f976)

Anstis et 3!. (1981) Evans and Charles (f975) Anstis et aI. (1981; Anstis et al. (198I) Anstis et al. (I98I) Anstis et aL (1981)

Asv : 124to 153comprisesmore coarselycrystal-
lineaggregatem, onocrystalline,andvitric varieties. A second,tougher, group with Asv : 243 to 296 comprisesfinely polycrystallinevarietieswith highly suturedgrainsand interlockingfibers.The great-
er toughnessof agate, with interlocking fibrous
texture, compared to flint and chert with granular
texturebearsan obvioussimilarityto the toughness relationshipbetween the jade minerals, nephrite andjadeite. Fibrous nephrite is generallytougher than the more granularjadeite (Bradt et al., 1973). The greatervalue of Asv for agatecomparedto Asv

for non-bandedchalcedony indicates that Asv is apparentlysensitiveto the amount of interlocking of fiberswith similar morphology.
A studyof the toughnessof many naturalmaterials will provide data for a number of attendant applicationsand investigations.A compilation of Asv toughnessdata for lapidary use, the relationshipof toughnessto crystalstructureandto various natural polycrystalline textures, and the role of toughness(as opposedto hardnessand chemical alteration)in weathering processesare examples that haveoccurredto us.

Agate, banded
Flint Chert Chalcedony, non-banded
Quartz crystal Fused quartz wood opal
Tiger eye
Aventurine

Table 2. Toughnessof quartz

Grain morphology

Texture

0.02run by 0.lmm, acicuLar

0.005mm to 0.01nm, equidimensional 0.0lrun to 0.02mm, equidimensional 0.02mm by 0.1mm, acicular

Monocrystal Amorphous Amorphous (?)

0.08mm by 10mm, aclcular

0.1mn to quartz
0.4mn by

0.5mn equidimensional 0.4mm by 0.04mm mica

Small interlocking

bundles of

radiating fibers make-up

larger interlocking

bundles

Highly sut.ured, granoblastlc

Highly sutured, granoblastic

Equidimensional lcm domains of

radiating fibers.

Fibers are

Iess interlocking

than agate

Cellular microstructure

of

wood preserved

Paralle1 aggregates of highly fractured acicular grains

Lepidoblastic mica. Little quartz

texture from suturing of

Asv (mean)
296
263 253 243
153 143 135
135
L24

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WOOD AND WEIDLICH: FRACTURE TOUGHNESS

Acknowledgments
Materialsfor this study have beengenerouslyprovided by the Advanced Product Division of Corning Glass Works and by Norton Company.The manuscripthasbenefitedfrom comments by David B. Marshallof the Materialsand MolecularResearch Division, Lawrence Berkeley Laboratory.
References
Anstis,G. R., Chantikul,P., Lawn, B. R., and Marshall,D. B. (1981)A critical evaluation of indentation techniquesfor measuringfracturetoughness:I. direct crack measurements. Journal of the American Ceramic Society, 64, 534-53g.
AmericanSocietyfor TestingMaterials(1979)FractureMechanicsAppliedto Brittle Materials.SpecialTechnicalpublication 678.
Bradt,R. C., Newnham,R. E., and Biggers,J. V. (1973)The toughnessof jade. AmericanMineralogist,58,727-:732.
DunningJ, . D., Lewis, W. L., and Dunn,D. E. (1980C) hemomechanicawl eakeningin the presenceof surfactantsJ. ournal of GeophysicalResearch,85, 5344-5354.
Evans, A. G., and Charles,E. A. (1976)Fracture toughness determinationby indentation. Journalof the American Ceramic Society,59,37l-372.
Evans, I., and Pomeroy,C. D. (1966)The Strength,Fracture and Workability of Coal. PergamonPress,London (nor seen, referencedin Vutukuri, 1974,p.78).
Frank,F. C., and Lawn, B. R. (1967)On the theoryof Hertzian fracture. Proceedingsof the Royal Society of London, 2994, 291-306.

Geer,M. R., and Yancy, H. F. (1938)Expressionand interpretation of the size compositionof coal. Transactionsof the AmericanInstituteof Mining and MetallurgicEngineers,130, 250-269.
Hobbs,D. W. (1964)Rock compressivestrength.Colliery Engineering,41,287-292(not seen,referencedin Vutukuri, 1974, p.79).
Kwong,J. N. S., Adams,J. T., JohnsonJ, . F., andPiret,E. L. (1949)Energy-newsurfacerelationshipin crushing.Chemical EngineeringProgress,45, 508-516.
Protodyakonov,M. M. (1962)Mechanicalpropertiesand drillability of rocks. Proceedingsof the 5th Symposiumon Rock Mechanics,Minneapolis,Minn., 103-l18 (not seen, referencedin Vutukuri,1974p, .73).
SchuylerJ, . N., Owens,A. D., and Dunning,J. D. (1981)The role of surfaceenergyin chemomechanicawl eakening.EOS, 62, 1040.
Tetelman,A. S., and McEvily, A. J. (1967)Fractureof Structural Materials.John Wiley and Sons,New York.
Vutukuri,V. S., Lama, R. D., and Saluja,S. S. (1974)Handbook on MecbanicalPropertiesof Rocks, Volume L Trans Tech PublicationsO, hio.
Wiederhorn,S. M. (1969)Fractureof ceramics.In Mechanical and Thermal Properties of Ceramics, National Bureau of StandardsSpecialPublication303,p. 217-241.
Manuscript received, November I I, 1981; acceptedfor publication, May 10, 1982.

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Empirical evaluation of fracture toughness: the toughness of