American Mineralogist, Volume 67, pages 1065-1066,l9E2 Empirical evaluation of fracture toughness:the toughness quartz of Mrcneer M. Wooo Department of Geological Sciences C alifornia S t at e U niv er sity Hayward, California 94542 eNo J. E. We,rolrcn Dep art ment of M at hematic s C alifornia State U niv ersity Hayward, California 94542 Abstract A toughnessrelated parameter,Asv, of monocrystalline, amorphous,and finely poly- crystalline brittle materials is derived from standard sedimentologic size analysis of a crushedsample.Asv is numericallyequal to the areaunder the cumulative-frequency size distribution curve and is related to the fracture toughness, Kc, of the material by the relationship,Asv = 202 + 383 log Kc. A study of quartz shows that toughness generallyincreases with decreasinggrain size and with an increasing degreeof interlockingof the grainsor fibers. Introduction material and standard sedimentologic analysis of Brittle materialsgenerallybreak by catastrophic the resultingproducts. The method is amenableto propagation Griffith flaws. Toughness, parame- rapid analysis, involves no prior knowledge of of a ter often appliedin context to structuralmetals,has physical constantsof the material except density, become a valuable measure of the resistance of and is conceptuallyreadily appreciated. these materials to fracture. The development of Historically the processbenefitsfrom the treat- Griffith-Irwin fracture mechanical principles has ment of Rosin's Law by Geer and Yancy (1938) who givenrise to severalspecificparameters measur- for demonstrate the existenceof a type of probability ing toughness. these, fracture toughness,Kc, a distribution for the size fractions of crushed materi- Of critical value of the stress intensity factor, has als. Protodyakonov (1962) used size distribution gainedwide acceptance. is a material constant data of crushedmaterial to obtain a strengthindex Kc characterizing the inherent difficulty of crack for the material, and Hobbs (1964) and Evans and growth in a material. A discussionof the signifi- Pomeroy (1966) have related similarly derived cance of Kc is given by Tetelman and McEvily strengthindicesto compressive strength.Kwong e/ (1967) and conventional procedures for deriving al. (1949)have related amount of new surface area toughnessparametersare discussedin American produced by crushing brittle materials to fracture Society for Testing Materials (1979). surfaceenergy. Quantitativeaspectsof toughnesshave received Method little attention from geologists. This stems partly from the elaborate procedures for measurgmentof A toughnessrelated parameter,Asv, derives from toughnessparametersand the consequentdearth of a unique analysis of the size distribution of the toughness data for natural materials, and partly crushed product of a rnaterial. from the lack of obvious significance convention- of Initial preparation of a sample consists of hand- al toughness parametersto natural processes. crushingand sieving to obtain approximately 30 g of In the present study an empirical method for startingmaterialin the -2 phi to - 1.5phi sizerange evaluating fracture toughnessis based on a proce- (4 mm to 2.83 mm). An amount of starting material dure of crushing a specific starting size fraction of equivalent weight to a6.666cm3standardvolume in 0003-004)v8210910-1065$02.00 1065 WOOD AND WEIDLICH: FRACTURE TOUGHNESS (numericallyequal in grams to the density multi- undetermined influence on Asy. Values of Asv plied by 6.666)is placed in the 39 mm by 59 mm obtainedwith other crushingand sievingapparatus cylindricalcrushingchamberof a spnx shakertype must be standardized with results from this studv. mixer/mill with three 15.875mm hardened steel balls and crushed for 45 seconds. The crushed Resultsand discussion productis sievedin eight inch U.S. StandardSeries A theoreticalbasis for relatingAsv to traditional sievesof one phi interval between - 1.5phi and 4.5 fracture parameters probably lies in the analysisof phi for 15 minutes on a Combs gyratory sifting cracksproducedby sphericalindenters(Frank and machine. The amount retained on each sieve is Lawn, 1967)in which a relationship between the weighed 0.01g. to productionof aHertzian crack systemand material Weight data may be plotted as a cumulative- toughness is derived from Griffith-Irwin fracture frequency plot with arithmetic ordinate in which mechanics. cumulative percent is plotted against phi for the An empirical relationshipbetweenAsv and frac- sevenone-phiintervalsfrom - 1.5phi to 4.5 phi. A ture toughness, in Kc, is demonstrated Figure I and quantitativetoughnessparameter,Asv, is numeri- Table I for severalstandardmaterials.The regres- cally equal to the area beneath the cumulative- sion line represents statisticallysignificantrela- the frequencycurve. Asv may be comparedvisually on tionship, Asv : 202 + 383 log Kc. Values of Kc the graph or obtaineddirectly from the data by the reportedfor the standards were derived by conven- trapezoidal approximation rule in the form tional testing methods of fracture mechanics as /t.5\ indicated in the referencesin Table 1. Although A s y- o . s ( r 4 _ ,.,+ --l there is a distinct correlation betweenAsv and Kc \ i:_0.5 for the standardmaterialsand experimentalcondi- whereP@;represents cumulativepercentat phi the tions in this study, fracture toughnessis strongly t. influenced by atmospheric moisture (Dunning et Reproducibility of Asy is generally within five al., 1980and Schuyler et al., 1981).Inability to percentof the mean for multiple runs on separates control humidity in the experimentalsituationpre- of a singlehomogenous startingsample.The sieving cludesanalysisof this effect on the value of Asy. process discriminates on the basis of effective The approximately20 g of starting sampleneces- (least)cross-sectional area of the grains and, there- sary to obtain Asy ensuresthat the value obtained fore, variation in grain shape (equidimensional, represents nearly averagevalue of toughness a for tabular, fibrous, etc.) must have an important, but the sample.The use of Asv as an indicator of Kc is particularly valuable when toughness disparity within the sample renders measurements con- by ventionalmethods,which employ a restrictedpor- tion of the sample,susceptibleto large variations. On the other hand,Asv cannotbe usedas a measure of toughness anisotropyand provides only an indi- cation of average toughnessfor anisotropic mono- crystalline materials. For this reason the data for sapphire shown in Figure I are not used in the regression line calculations. Application of the method to polycrystalline material is limited to thosewhosemaximumgrain dimensions consid- are erably less than the minimum diameterof the start- ing size fraction (2.83 mm). Toughness quartz of M#t* To demonstrate applicationof the method,the an Fig. l. Relationship between Asv and fracture toughness, Kc. Regressionline and 95 percent confidencebands for Asv (based toughness parameter,Asv, for severalquartz mate- on "r" distribution) are derived from the arithmetic mean for rials has been determinedand is shown in Table 2. each sampleexcept sapphire. Values of Kc are from references Quartz materials fall into two basic groups by in Table 1. toughness. group with relatively low toughness A of WOOD AND WEIDLICH: FRACTURE TOUGHNESS 1067 Table l Toughnessdata for standard materials Source Character AsvKc^. Source of data (mean) (MPa m-'") Fused quartz G .E . Amorphous 140 0.7 wiederhorn (1969) Spinel Unknown, Monocrystal 257 1.3 Evans and charles (f976) synthetic si3N4 (NC350) Norton Polycrystal 3I4 2. 0 Anstis et 3!. (1981) Sapphire Natural Monocrystal 337 2.r Evans and Charles (f975) c-9606 corning Polycrystal 334 2.5 Anstis et aI. (1981; AI2O3 (AD999) Coors PoLycrystal 408 3.9 Anstis et al. (198I) sic (Nc203) Norton Polycrystal 443 4.0 Anstis et al. (I98I) si3N4 (NCl32) Norton Polycrystal 4s5 4.0 Anstis et aL (1981) Asv : 124to 153comprisesmore coarselycrystal- for non-bandedchalcedony indicates that Asv is line aggregate,monocrystalline, and vitric varieties. apparentlysensitiveto the amount of interlocking A second,tougher, group with Asv : 243 to 296 of fibers with similar morphology. comprises finely polycrystallinevarietieswith high- A study of the toughness many natural materi- of ly suturedgrainsand interlockingfibers. The great- als will provide data for a number of attendant er toughnessof agate, with interlocking fibrous applicationsand investigations.A compilation of texture, compared to flint and chert with granular Asv toughnessdata for lapidary use, the relation- texture bearsan obvious similarity to the toughness shipof toughness crystal structureand to various to relationship between the jade minerals, nephrite natural polycrystalline textures, and the role of and jadeite. Fibrous nephrite is generally tougher toughness(as opposed to hardnessand chemical than the more granularjadeite (Bradt et al., 1973). alteration) in weathering processesare examples The greater value of Asv for agatecompared to Asv that have occurred to us. Table 2. Toughness quartz of Grain morphology Texture Asv (mean) Agate, banded 0.02run by 0.lmm, acicuLar Small interlocking bundles of 296 radiating fibers make-up larger interlocking bundles Flint 0.005mm to 0.01nm, equidimensional Highly sut.ured, granoblastlc 263 Chert 0.0lrun to 0.02mm, equidimensional Highly sutured, granoblastic 253 Chalcedony, 0.02mm by 0.1mm, acicular Equidimensional lcm domains of 243 non-banded radiating fibers. Fibers are Iess interlocking than agate Quartz crystal Monocrystal 153 Fused quartz Amorphous 143 wood opal Amorphous (?) Cellular microstructure of 135 wood preserved Tiger eye 0.08mm by 10mm, aclcular Paralle1 aggregates of highly 135 fractured acicular grains Aventurine 0.1mn to 0.5mn equidimensional Lepidoblastic texture from L24 quartz mica. Little suturing of 0.4mn by 0.4mm by 0.04mm mica quartz 1068 WOOD AND WEIDLICH: FRACTURE TOUGHNESS Acknowledgments Geer,M. R., and Yancy, H. F. (1938)Expression and interpre- Materials for this study have been generouslyprovided by the tation of the size composition of coal. Transactionsof the Advanced Product Division of Corning Glass Works and by AmericanInstitute of Mining and MetallurgicEngineers,130, Norton Company. The manuscript has benefitedfrom comments 250-269. by David B. Marshallof the Materialsand Molecular Research Hobbs, D. W. (1964) Rock compressive strength.Colliery Engi- Division, Lawrence Berkeley Laboratory. neering,41,287-292(not seen,referenced Vutukuri, 1974, in p.79). References Kwong,J. N. S., Adams,J. T., Johnson, F., and Piret,E. L. J. (1949) Energy-newsurfacerelationshipin crushing.Chemical Anstis,G. R., Chantikul, Lawn, B. R., and Marshall, B. P., D. Engineering 45, Progress, 508-516. (1981) A critical evaluation of indentation techniquesfor Protodyakonov, M. (1962)Mechanicalpropertiesand drill- M. measuring fracture toughness: direct crack measurements. I. ability of rocks. Proceedings the 5th Symposiumon Rock of Journal of the American Ceramic Society, 64, 534-53g. Mechanics, Minneapolis, Minn., 103-l18 (not seen, refer- AmericanSocietyfor TestingMaterials(1979) FractureMechan- encedin Vutukuri, 1974, p.73). ics Applied to Brittle Materials.SpecialTechnicalpublication Schuyler, N., Owens,A. D., and Dunning,J. D. (1981) J. The 678. role of surfaceenergy in chemomechanical weakening.EOS, Bradt, R. C., Newnham,R. E., and Biggers, V. (1973) J. The 62, 1040. toughness jade. American Mineralogist,58,727-:732. of Tetelman, S., and McEvily, A. J. (1967) A. Fractureof Structur- Dunning, D., Lewis, W. L., and Dunn, D. E. (1980) J. Chemo- al Materials.John Wiley and Sons, New York. mechanical weakening the presence surfactants. in of Journal Vutukuri, V. S., Lama, R. D., and Saluja,S. S. (1974) Hand- of GeophysicalResearch, 85, 5344-5354. book on MecbanicalPropertiesof Rocks, Volume L Trans Evans, A. G., and Charles, E. A. (1976) Fracture toughness Tech Publications, Ohio. determinationby indentation. Journal of the American Ceram- Wiederhorn,S. M. (1969)Fracture of ceramics.In Mechanical ic Society, 59, 37l-372. and Thermal Properties of Ceramics, National Bureau of Evans, I., and Pomeroy, C. D. (1966)The Strength,Fracture -241. Standards SpecialPublication303, p. 217 and Workability of Coal. PergamonPress, London (nor seen, referenced Vutukuri, 1974,p.78). in Frank, F. C., and Lawn, B. R. (1967) the theory of Hertzian On fracture. Proceedingsof the Royal Society of London, 2994, Manuscript received, November I I, 1981 ; 291-306. acceptedfor publication, May 10, 1982.