Empirical Evaluation of Fracture Toughness: the Toughness of Quartz

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

Empirical evaluation of fracture toughness:the toughnessof quartz

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 California State U niv ersity Hayward, California 94542

Abstract

A toughnessrelated parameter,Asv, of monocrystalline,amorphous, and finely polycrystalline brittle materials is derived from standard sedimentologicsize analysis of a crushedsample. Asv is numericallyequal to the areaunder the cumulative-frequencysize distribution curve and is related to the fracture toughness,Kc, of the material by the relationship,Asv = 202 + 383log Kc. A study of quartz shows that toughnessgenerally increases with decreasinggrain size and with an increasingdegree of interlockingof the grainsor fibers.

Introduction material and standard sedimentologic analysis of products. Brittle materialsgenerally break by catastrophic the resulting The method is amenableto propagationof Griffith flaws. Toughness,a parame- rapid analysis, involves no prior knowledge of physical ter often appliedin context to structuralmetals, has constantsof the material except density, become a valuable measure of the resistance of and is conceptuallyreadily appreciated. process these materials to fracture. The development of Historically the benefitsfrom the treat- (1938) Griffith-Irwin fracture mechanical principles has mentof Rosin'sLaw by Geerand Yancy who givenrise to severalspecific parameters for measur- demonstratethe existenceof a type of probability ing toughness.Of these, fracture toughness,Kc, a distribution for the size fractions of crushed materi- (1962) critical value of the stress intensity factor, has als. Protodyakonov used size distribution gainedwide acceptance.Kc is a material constant data of crushedmaterial to obtain a strengthindex (1964) characterizing the inherent difficulty of crack for the material, and Hobbs and Evans and (1966) growth in a material. A discussionof the signifi- Pomeroy have related similarly derived cance of Kc is given by Tetelman and McEvily strengthindices to compressivestrength. Kwong e/ (1949) (1967) and conventional procedures for deriving al. have related amount of new surface area produced toughnessparameters are discussedin American by crushing brittle materials to fracture Society for Testing Materials (1979). surfaceenergy. aspectsof toughnesshave received Quantitative Method little attention from geologists. This stems partly from the elaborate procedures for measurgmentof A toughnessrelated parameter,Asv, derives from toughnessparameters and 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 obvioussignificance of convention- Initial preparation of a sample consists of hand- al toughnessparameters to natural processes. crushingand sieving to obtain approximately 30 g of In the present study an empirical method for startingmaterial in 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 equivalentin weight to a6.666cm3 standard volume 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 cylindricalcrushing chamber of a spnx shakertype must be standardizedwith resultsfrom 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 fractureparameters probably lies in the analysisof phi for 15 minutes on a Combs gyratory sifting cracksproduced by sphericalindenters (Frank and machine. The amount retained on each sieve is Lawn, 1967)in which a relationship between the weighedto 0.01g. 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 relationshipbetween Asv and frac- sevenone-phi intervals from - 1.5phi to 4.5 phi. A ture toughness,Kc, is demonstratedin Figure I and quantitativetoughness parameter, Asv, is numeri- Table I for severalstandard materials. The regres- cally equal to the area beneath the cumulative- sion line representsthe statisticallysignificant rela- 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 standardswere derivedby conven- trapezoidal approximation rule in the form tional testing methods of fracture mechanics as /t.5\ indicated in the referencesin Table 1. Although - Asy o.s(r4_,., + --l there is a distinct correlation betweenAsv and Kc i:_0.5 \ for the standardmaterials and experimentalcondi- tions in this study, fracture toughnessis whereP@; represents the cumulativepercent at phi strongly t. influenced by atmospheric moisture (Dunning et al., 1980and Schuyleret al., 1981).Inability to Reproducibility of Asy is generally within five control humidity in the experimentalsituation pre- percentof the meanfor multiple runs on separates cludesanalysis of this effect on the value of Asy. of a singlehomogenous starting sample. The sieving The approximately20 g of starting process discriminates on the basis of effective sampleneces- sary to obtain Asy ensuresthat the value (least)cross-sectional area of the grainsand, there- obtained fore, variation in grain shape (equidimensional, representsa nearly averagevalue of toughnessfor the sample.The use of Asv as an indicator of Kc is tabular, fibrous, etc.) must have an important, but particularly valuable when toughness disparity within the sample renders measurementsby con- ventionalmethods, which employ a restrictedpor- tion of the sample,susceptible to large variations. On the other hand,Asv cannotbe usedas a measure of toughnessanisotropy and provides only an indi- cation of average toughnessfor anisotropic monocrystalline 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 thosewhose maximum grain dimensionsare consid- erably lessthan the minimum diameterof the starting sizefraction (2.83 mm). Toughness quartz * of M#t To demonstratean applicationof the method,the Fig. l. Relationshipbetween Asv and fracture toughness,Kc. toughnessparameter, Asv, for quartz Regressionline and 95 percent confidencebands for Asv (based several 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.A group with relativelylow toughnessof 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 153comprises more coarselycrystal- for non-bandedchalcedony indicates that Asv is line aggregate,monocrystalline, and vitric varieties. apparentlysensitive to the amount of interlocking A second,tougher, group with Asv : 243 to 296 of fibers with similar morphology. comprisesfinely polycrystallinevarieties with high- A study of the toughnessof many natural materi- ly suturedgrains and 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- texturebears an obvious similarity to the toughness shipof toughnessto crystal structureand to various 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 occurredto us.

Table 2. Toughnessof quartz

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

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