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Empirical evaluation of fracture toughness the toughness of quartz

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					                             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).
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