Fracture Stress as Related to Flaw and Fracture Mirror Sizes | EMA 6715, Exams of Materials science

Download Fracture Stress as Related to Flaw and Fracture Mirror Sizes | EMA 6715 and more Exams Materials science in PDF only on Docsity! 304 Journal of The American Ceramic Society-Bansal and Duckworth Vol. 60, No. 7-8 homogeneous glass. The slight decrease in etch rate observed after the prolonged heat-treatment time was probably caused by the spheroidization which is known to occur. On the other hand, glasses be- longing to the second group (glass 111) show practically no change in etch rate with increasing phase separation because they produce a dispersed alkali-borate phase em- bedded in a silica matrix. This type of structure is shown in Fig. 4(B) for glass HI. With this struc- ture, the continuous high-silica matrix controls the overall etch rate, therefore, a drastic change in the etch rate is not expected to occur via phase separation. A similar explanation for the chemi- cal durability of phase-separated opal glasses in a commercial de- tergent solution recently has been given by Flannery et al. lo (2) The Effect of Phase Com- position on Etch Rate One of the prime objectives of the present research is to identify-among various pos- sibilities such as size, volume fraction, and composition of sep- arated phases-the dominant fac- tor which controls the etch rate of the phase-separated glasses. The foregoing experimental observa- tions indicate that the composi- ‘2’3 Fig. 6. Metastable immisci- bility boundary of SiOz- BZ0,-Na20 system, with tie line (after Refs. 11 and 12). tion of the chemically less durable separated phase is the dominant factor affecting the etch rate of the glasses with interconnected structure, i.e. (A) An etch rate change is accompanied by a change of the integrated intensity of X-ray small-angle scattering, as shown in Fig. 3 ( A ) . (B) The constant etch rate obtained after a long heat-treatment time is characteristic of the equilibrium composition. This interpretation was further confirmed by the fact that the same constant etch rate can be approached both from higher and lower values. (C) A constant etch was observed for 8 to 18 mol% Na,O- SiO, glasses, as shown in Fig. 5. In this glass system at 600°C, the volume fraction changes continuously from 8 to 18 mol% Na,O. The quantitative test of this conclusion can be made by studying the temperature dependence of the etch rate of phase-separated glass VI (Table 11). In Fig. 6, the immiscibility boundary of the SiOp-B203-Na20 system determined by Haller er al. ‘l and the tie line through the composition of glass VI determined by Mazurin and Streltsina” are reproduced. According to Fig. 6, the silica con- tents in the alkali-borate-rich phase of glass VI are 261, 56, 49, and 45 wt% at 700°, 650°, 600°, and 550°C, respectively. Etch rates of glasses with equivalent amounts of silica content can be esti- mated from Fig. 1 and are shown in Table 11. These values show the same trend as the experimentally determined etch rate of this glass. The difference between the observed and predicted etch rates is probably caused by the uncertainty in the immiscibility bound- ary’* and tie line^.'^*'^ These observations indicate that the HF etch rate of glasses with the interconnected microstructure is pri- marily controlled by the etch rate of the chemically less durable alkali-borate-rich phase. This conclusion is reasonable if it is con- sidered that the less durable phase is etched rapidly and the silica- rich, chemically durable phase is attacked from the newly created interface. Acknowledgments: Landermann for experimental assistance. We thankc. F. Atioaaforelectmn microscopy and J. B. References N. P. Danilova, 0. V. Mazurin, and T. S. Tsekhomskaja; pp. 825-41 in LXth International Congress on Glass, Vol. 1. Institut du Verre, Paris, 1971. * 8. F. Howell, J. H. Simmons, and W. Haller, “Loss of Chemical Resistance to Aqueous Attack in a Borosilicate Glass Due to Phase Separation,” Am. Ceram. Soc. Bull, 54 [8] 707-709 (1975). V. A. Borgman, V. K. Leko, and V. K. Markaryan; pp. 66-68 in Stplcture,of Glass, Vol. 8. Phase-Separation Phenomenain Glasses. Edited by E. A. Porai-Koshits. Translated by E. Boris Uvarov. Consultants Bureau, New York, 1973. * (a) R. 1. Charles, ‘’Dissolution Behavior of Micropomus Glass,” J. Am. Ceram. (b) R. J. Charles, “Phase Separation in Borosilicate Glasses,’’ ibid., [I 11559-63. E. B. Shand, Glass Engideering Handbook, 2d ed. McGraw-Hill, New York. SOC., 47 [3] 154-55 (1964). I9TR Werner Espe. Materials of High Vacuum Technology, Vol. 2; pp. 1-66. Perga- ‘ M. Tomozawa, “Liquid Phase Separation and Crystal Nucleation in LizO-SiOz mon, Oxford, 1%8. Glasses.’’ Phvs. Chem. Glasses. 13 161 161-66 (1972). * M. Tomohiwa and T. T k o r i , ‘*&face Layer ofphase-Separated Borosilicate Glasses”; unpublished work. J. H. Simmons, S. A. Mills, and A. Napolitano, “Viscous Flow in Glass During Phase Separation,” J. Am. Cerom. Soc., 57 [3] 109-17 (1974). lo J. E. Flannery, W. H. Dumbaugh, and G. B. Carrier, “Improving the Opacity of Phase-Separated al Glasses by Alterations of the Interfacial Tension,” Am. C e r m . Soc. Bull., 54 [121066-68, 1071 (1975). l1 W. Haller, D. H. Blackburn, E. F. Wagstaff, and R. J. Charles, “Metastable Immiscibility Surface in the System N&0-BZ03-Si02.” J . Am. C e r m . Soc., 53 [I] 34-39 (1970). “0. V. MazurinandM. V. Streltsina, “DeterminationofTie-LineDirections inthe Metastable Phase-Separation Regions of Ternary Systems,” J . Non-Crysf. Solids, 11 [3] 199-218 (1972). I s G. R. Srinivasan, I. Tweer, P. B. Macedo, and A. Sarkar, “Phase Separation in SiO2-BzOS-Naz0 Systems,” ibid., 6 [3] 221-39 (1971). S. Scholes, “Tie Lines in the Metastable Immiscibility Region of the N%O-B203-Si02 System,” ibid.. 12 [2] 266-70 (1973). Fracture Stress as Related to Flaw and Fracture Mirror Sizes GIRRAJ K. BANSAL* and WINSTON H. DUCKWORTH* Battelle Memorial Institute, Columbus Laboratories, Columbus, Ohio 43201 Fractographically observed critical-flaw boundaries in strength-tested specimens of 2 polycrystalline ceramics were used in calculating critical stress-intensity factors (KIc). Each ceramic exhibited a Klc which had little or no dependence on critical-flaw characteristics and which agreed with the value obtained from independent determinations on artificially pre- cracked specimens. Analyses both of fracture-mirror sizes and ofwaterenhanced subcritical crack growth data supported the evidence that K,c is a material constant. The fracture-mirror analysis further indicated that the parameter AIKlc, where A is the fracture-mirror constant, is a dimensionless, material- independent constant. Received May 14, 1976; revised copy received December 20, 1976. ‘Member, the American Ceramic Society. & y d by the Office of Naval Research under Contract No. N00014-73-C- Ju1.-Aug. 1977 Fracture Stress as Related to Flaw and Fracture Mirror Sizes 305 I. Introduction R materials that exhibit brittle behavior, strengths of indi- F" vidual specimens vary because of differences in the critical flaws governing fracture. Another possible source of specimen-to- specimen variations in strength is that the critical stress-intensity factor, Klc, depends on local conditions at the fracture-initiating site. Thus, Baratta er al. found that values of Klc obtained from strength and flaw size measurements on individual specimens of an Si3N, ceramic were 20 to 50% lower than the independently mea- sured Klc value. Similarly, Rice' has reported that the effective fracture surface energy, yi,* varies with microstructural conditions at the fracture-initiating site in polycrystalline ceramic specimens and suggested that yi and, hence, Klc should not be viewed universally as an independently measurable material constant. In the present research, K l c governing room-temperature fracture in individual specimens of 2 dense, fine-grained polycrystalline ceramics was assessed from strength and critical-flaw mea- surements. Fracture conditions were varied to provide a range of strengths and critical-flaw characteristics for each ceramic. TheK,, values obtained from the strength tests were compared with those from independent measurements on artificially precracked speci- mens. The Griffith-Irwin relation used in the determinations of Klc from strength tests is3: u f = ( Z / Y ) ( K l c l G) (1) where uf is the applied stress at the origin when fracture occurs, a the flaw depth in the case of a surface flaw and the half-depth in the case of a subsurface flaw, and Y and Z are dimensionless parame- ters. In the case of flaws much smaller than the specimen dimen- sions, Y -2.0 for a surface flaw and 1.77 for a subsurface flawt; Z , the flaw-shape parameter, is the product of 2 independent dimen- sionless parameters, Z , and Zd.3*6,7 The parameter Z , varies with the flaw shape in the fracture plane, as shown in Fig. 2 of Ref. 3. For failures from 2-dimensional flaws with sharp tips, & = I . In the case of 3-dimensional flaws (e.g. pores and inclusions) of radius R, Zd varies with the ratio L/2R, where L is the length of the sharp- tipped radial crack which extends from the flaw periphery in the fracture plane. It can be seen from Fig. 7 of Ref. 6 that z d , hence Z , becomes large when L/2R is small, indicating that an inclusion or pore, per se, is not a severe flaw; it becomes severe only when a radial crack extends from it. In defining the fracture-initiating site, one sometimes needs to know how the flaw shape is affected by the differential K I along the flaw periphery in the fracture plane before the flaw becomes critical, i.e. before K l c is reached. Only in the case of a circular (penny- shaped) flaw is the K I nearly invariant at all points on the flaw periphery.$ For an elliptical flaw, KI is greatest at the end of the minor axis, hence the flaw will tend to become circular as it approaches critical size.3 Fracture-mirror measurements were also used for investigating possible variability of Klc. Each critical flaw is surrounded by a mirror whose radius, r , is empirically related to the fracture stress, uf, as follows: uf= A / I/; (2) where A is the mirror constant for the particular material.9*'0 Com- bination of Eqs. (1) and (2) gives: ( Z / Y ) V%= AIKlc (3) As can be seen from Eq. (3), the parameter A/Klc becomes a dimensionless constant when K l c is a material constant, and the parameter can be evaluated solely from fractographic observations. *yt=Klc'/2E, where E is Young's modulus. tThese values of Y are for thmugh-the-thickness, slit cracks (Ref. 4). The analysis (Ref. 5 ) of a semicircular part-through surface crack, however, indicates that for Z=r/ZinEq. ( I ) , Y=1.85at 8=r/2and2.14at 8=0, withanaveragevalueof2.0 occumng at 8=0.45n. The angle 8 is as shown in Fig. 1 of Ref. 3. $The assumption that K , is invariant at all points on the periphery of a circular flaw is notstrictlycorrect. Asamatteroffact.K,atanysnessis -15%higherat8=0thanthat at O=?r/2. The fracture will, therefore, initiate at O=O foran ideal penny-shaped flaw (Refs. 5 and 8). Table I. Room-Temperature Properties of Polycrystalline Ceramics Studied - ~ ~ . . ~ ~ ~ ~ _ _ Material E (GNm-2) Klc (MNrn-,'? y,=KIC2/2E (J/mz) Glass-ceramic 1142 1 2.3820.08 24.8 Sintered alumina 3 1 8 5 3 3.84r0.05 23.2 Also, if the left side of Eq. (3) is material independent, A / K I c will also be independent of the material. 11. Materials and Procedures The dense, commercial polycrystalline ceramics investigated were a glass-~eramic~ (average grain size ==2 pm) and a conven- tionally sintered aluminan (average grain size =5 pm). The mate- rials were characterized at room temperature (Table I). The values of E were obtained by measuring average strain as a function of stress in uniaxial compression. The double-torsion method" was used to determine Klc, using artificially precracked specimens tested in dry Nz at acrosshead speed of 0.5 cm min-'. The measured Klc value for each ceramic agrees well with that obtained by other investigators. l2 Room-temperature bend strengths were determined using rectan- gularspecimensO.1 by0.2by 1.5in. and0.5 by l.Oby7.5in.and the bend fixture described by Hoagland er al. l3 All specimens were finish-ground parallel to the span direction, using a 320-grit diamond wheel. Specimens were tested in dry Nz and at a stress rate of 100 M N m P s-l to restrict subcritical crack growth and in dis- tilled water at a stress rate of 4 MNm-'s-I to enhance such growth. Fracture stress, uf, was calculated using the expression at= MCII, where M is the applied moment, C is the distance of the fracture origin from the neutral axis, and I is the cross-sectional moment of inertia. Fracture surfaces of specimens broken in strength testing were examined microscopically. Fracture origins and fra~ture-mirror~.'~ boundaries (beginning of the hackle region) were identified using an optical microscope. The critical-flaw boundaries were defined using stereo scanning electron microscopy (SEM). Critical flaw boundaries in specimens tested under water could not be identified clearly even with the use of stereo micrographs. For these cases, the flaw boundaries were assumed to be ~emicircular,~ and anticipated flaw sizes were calculated using r /a =9.1° When these anticipated flaw sizes were used for guidance on where to search, critical flaw boundaries were identified from abrupt changes in the surface topography seen in SEM examinations. 111. Results About 50 specimens of each ceramic were strength tested and =20 of each were subjected to fractographic examinations. Data and observations are reported here for 9 of these specimens, selected on the basis of representing the various kinds of flaw origins ob- served. Unreported results obtained from the other specimens only serve to support the findings presented herein. Table I1 gives the stress at the fracture site ( uf) along with values for the parameters (u, Z , and Y ) which describe the observed critical flaws in each specimen. The Klc values reported in Table I1 were calculated using Eq. (1). Measured mirror radii, r , are plotted as a function of uf in Fig. 1 . The plots indicate that Eq. (2) is obeyed by each ceramic, with A equal to 5.7 and 9.0 MN~I-~' ' for the glass- ceramic and sintered A1,03, respectively.** Each specimen in Table I1 is identified by one or more (stereo) micrographstt showing the fracture origin (Figs. 2 through 10). SPyroceram 9606, Corning Glass Works, Corning, N.Y. llAlsimag 614, Technical Ceramic Rodwts Div., 3M Co., Chattanooga, Tenn. **As noted in Fig. 1 , the data for the @ass-ceramic fall into 3 separate groups (1,2, and 3) which are the strengths for specimens failing from surface origins in dry Nt. subsurface origins in dry N,, and surface origins in water, respectively. There was little variation in strengths for each test condition (Ref. 14). ttOptical micrographs of the glass-ceramic and sintered AbO, specimens are pre- sented in Refs. 14 and IS. respectively. 308 Journal of The American Ceramic Society-Bansal and Duckworth Vol. 60, NO. 7-8 Fig. 6. Stereo scanning electron micrographs of surface fracture origin in glassceramic (billet C) specimen tested in water. Fig. 7. Fracture initiation from interlinking of several “surface” flaws in sintered A1,03 specimen tested in dry N,. ( A ) Scanningelectron micrograph showing semicircular machine flaws =20 pm deep at M and subsurface pore 50 pm long which linked prior to catastrophic fracture to give critical- flaw boundary sketched in (B). Fig. 8. Fracture initiation from elongated surface pore in sintered A1,Oz specimen tested in dry N,. Fracture initiated at A , i.e. along minor axis. ( F ) Figure 7: (Sintered A1,03 tested in dry N 3 Two machine flaws, each ==20 pmdeep in the tensile surface, and two subsurface pores linked to give the critical-flaw boundary which is ~ 6 0 p m deep and 135 p m wide, as shown in the sketch. For this flaw configuration, Z , = 1.49. A small contribution of Zd on Z attributed to the pore involvement is ignored and the flaw treated as being 2-dimensional. (Sintered A1& tested in dry N,) An elongated pore is seen just below the tensile surface at the fracture origin. Ligaments between the surface and the pore probably fractured first, (G) Figure 8: JuL-Aug. 1977 Fracture Stress as Related to Flaw and Fracture Mirror Sizes 309 Fig. 9. Stereo scanning electron micrographs of surface-grain fracture origin in sintered Alz03 specimen tested in dry No. Note misorientation between fracture surface and flaw plane; fracture markings at A indicate that fracture initiated from machine flaw at M. Fig. 10. Stereo scanning electron micrographs of surface fracture origin in sintered A1203 specimen tested in 3-point bending and in water. Note semicircular machine flaw -15 Fm deep at M; tensile surface is on left of each micrograph. causing a radial crack to form at A in the sketch. Because no extensive radial crack is observed at A, the length of the radial crack is assumed to be =5 pm, i.e. the average grain size in the material. This results in a flaw width of =I20 pm; Zd==18.* T h e a h ratio for this flaw is 3.4, for which the corresponding Z, value cannot be obtained directly from Fig. 2 in Ref. 3. However, the Z , value corresponding to c/a=0.3 can be used in Eq. (l), provided the smaller dimension of the flaw is used for a; i.e. its half-width (60 *'Ibis is only an approximate value because an elongated pore is involved here. pm) rather than its depth (200 pm) is used.3 For r/a=0.3, Z,= 1.10, hence Z z 1 . 1 0 x 1 . 1 8 ~ 1.30. (H) Figure 9: (Sintered A120, tested in dry N2) A small ma- chine flaw at M apparently initiated fracture in the existing large grain, and the crack, formed by the cleavage of the grain, was widened by subcritical microcracking caused by the differential K , along its periphery until the critical-flaw boundary, as observed by stereo SEM, was reached. The final flaw is a 2-dimensional crack =60 pm deep and -130 p m wide, for which Z=Z,-1.50. (I) Figure 10: (Sintered A1203 tested in water) The critical- 310 Journal of The American Ceramic Table III. Critical Flaw Depths and Strengths in Water Critical flaw depth (rm) Strength (MNm-z) Material Calculated Observed Calculated Observed Glass-ceramic C 91* 80-85 210* 204r3 ~ Sintered AIZO3 (3-point bend) l l O + 110-I20 288’ 295 * 9 *Calculated using al=40 p n and ulc=317 MNm-2 (Ref. 14). ?Calculated using a,=55 p m and ulc=408 MNm-2 (Ref. 15). flaw boundary as revealed by the calculated flaw depth using rla -9 is indicated by the arrows. Apparently, a surface flaw of about the same size as in Figs. 7 and 9 extended to a depth of - 1 10 Pm.* IV. Discussion As noted before, fractures in 20 specimens were examined in detail. Equation ( I ) was used to calculate Klc values for each specimen from strength and critical-flaw measurements. Also, the fractographically assessed parameter, (Z/Y) a, was used to in- vestigate whether Klc was a material constant and whether the parameter A /Klc was a material-independent constant, in accor- dance with Eq. (3). Data reported in Table I1 on 9 specimens reflect the results of these detailed examinations. This approach involved carefully identifying and interpreting features on each fracture surface where critical-flaw boundaries sometimes could not be distinguished clearly on initial examina- tions of stereo micrographs. Considerations of moisture-assisted slow crack growth and consequent strength degradation offered an independent check on whether the fractographically assessed critical-flaw sizes were correct without requiring a detailed evalua- tion of critical-flaw boundaries in each individual specimen. Data for this independent check were provided by strength tests in dry Nz and water for specimens of both the glass-ceramic and sintered AlzO3 obtained in 2 separate studie~.’~.’~ Both the glass- ceramic14 and the sintered alumina15 specimens which failed from surface flaws exhibited considerably lower strengths when tested in water than in dry Nz because of water-enhanced slow crack growth. When one assumes that the initial flaw depth, ai, for water- enhanced crack growth is the same as that of the critical flaw which governed strengths of specimens tested in dry N,, the critical-flaw depth, a,, after water-enhanced slow crack growth can be pre- dicted.17 Similarly, average strengths in water (af) can be related to the dry Nz strengths ( u ~ ~ ) . ’ ~ The predictions require independent measurement of crack velocity ( V ) as a function of K, in the environment that produced slow crack growth. The double-torsion technique” was used to obtain the V-KI curves in distilled water at room temperature. Two specimens each of the glass-ceramic and the sintered alumina were used. The following expected relation described the observed growth rate in water reasonably V=AKln (4) with A and n having values of and 56, respectively, for the glass-ceramic, and Relations between the initial (aJ and final flaw (a,) depths, and between fracture stress after subcritical crack growth (ur) and frac- ture stress in the absence of such growth (alc) are as follows: and 42 for the sintered alumina.I5 where u is the stress rate (-4 MNmP/s) for tests in water, and Y, Z, and Klc are as described earlier, i.e. Y-2.0 and Z = d 2 *A small contribution of the pore (at P, associated with the large grain) to the critical-flaw depth is ignored in the interest of maintaining a nearly semicircular critical-flaw boundary (Ref. 3). If, however, the pore contribution is considered in the analysis,theuseofflaw area(Ref. 3) insteadofflawdepthismoreapproDriate(Ref. 3); this usage suggests an error of -5% in the calculated Klc value. Society-Bansal and Duckworth Vol. 60, No. 7-8 for a semicircular surface flaw. These relations assume that K,c is a material constant. Critical flaw depths and fracture stresses calcu- lated using Eqs. (5) and (6), respectively, are given in Table 111, along with those observed. V. Summary and Conclusions For each ceramic, Klc calculated from strengths and flaw size measurements on individual specimens were reasonably constant and agreed with Klc values independently determined using speci- mens having artificial cracks. This finding suggests that Klc for these polycrystalline ceramics is a material constant, independent of local conditions at the fracture origin, and that the only factor which would cause strength variations among specimens is a difference in the sizes of critical flaws causing fracture. Equation (1) ade- quately accounted for how strength was affected by observed critical-flaw characteristics, including the fact that large grains or pores which initiated fracture can be significantly smaller than associated critical flaws. The fracture-mirror measurements provide further evidence that Klc is a material constant and the fractographic interpretations of critical-flaw boundaries are reasonable. Values of the fractograph- ically assessed parameter (Z/Y)(r/a)1’2 ranged from 2.17 to 2.37 with no consistent variation among the specimens, indicating that the parameter is a material-independent constant. Furthermore, these values agree with A/Klc values of 2.39 and 2.34 which were calculated for the glass-ceramic and the sintered alumina, respec- tively, using independent measurements of Klc. The good agreement of observed strengths and critical-flaw sizes in water with those calculated using a fracture-mechanics approach lends confidence to the fractographic identifications and mea- surements of critical flaws, and also provides further evidence that Klc is a material constant in the fracture of each ceramic. Acknowled ments: The authors are particularly thankful to A. H. Heuer and A. R. Rosenfieh for their independent suggestions covering the concept of the critical stress-intensity factor as a material property. The assistance of Battelle staff members R. D. Kreachbaum, I. F. Schofield, and A. I. Skidmore, respectively. in machining, testing, and fractography of specimens is much appreciated. Helpful discussions with D. E. Niesz and L. G. McCoy are gratefully acknowledged. Ten Floyd helped in preparation of the manuscript. The research was monitored for the Office of Naval Research by A. M. Diness, whose guidance and support are gratefully acknowledged. References F. I. Baratta, G. W. Driscoll, and R. N. Katz; pp. 445-76 in Ceramics for High Performance Ap lications. Edited by I. I. Burke, A. E. Gomm, and R. N. Katz. Brook Hill, Chestnut &I, Mass., 1974. R. W. Rice; pp. 323-45 in Fracture Mechanics of Ceramics, Vol. 1. Edited by R. C. Bradt. D. P. H. Hasselman, and F. F. Lange. Plenum, New York, 1974. G. K. Bansal, “Effect of Flaw Shape on Strength of Ceramics,’’ J . Am. Ceram. Soc., 59 [I-21 87-88 (1976). ‘ W. F. Brown, Jr. and I. E. Srawley, Plane Strain Crack Toughness Testing of High-Strength Metallic Materials; pp. 1-15. Am. SOC. Tesf. Mater., Spec. Tech. Plrbl., No. 410, 1967. F. W. Smith, A. F. Emery, and A. S. Koba ashi, “Stress Intensity Factors for Semicircular Cracks: II,” J . Appl. Mech., 34 [4i953-60 (1967). A. G . Evans and G. Tappin, “Effects of Microsmcture on the Stress to Propagate Inherent Flaws,” Prac. Br. Ceram. Soc., 1972, No. 20, pp. 275-97. ’ 0. L. Bowie, “Analysis of an Infinite Plate Containing Radial Cracks Originating at the Boundary of an Internal Circular Hole,” J . Math. Phys., 35 [l I] 60-71 (1956). 1. I. Petmvic and M. G. Mendiratta, “Mixed-Mode Fracture from Controlled Surface Flaws in Hot-Pressed Si,N,,” J. Am. Ceram. Soc., 59 [3-41 163-67 (1976);, H. P. Kirchner and R. M. Gruver, “Fracture Mirmrs in Alumina Ceramics, Philos. Mag., 27 [6] 1433-46 (1973). lo G. K. Bansal, “On Fracture-Mirmr Formation in Glass and Polycrystalline Ceramics,” Philos. Mag.. 35 [4] 935-44 (1977). l1 D. P. Williams and A. G. Evans, “A Simple Method for Studying Slow Crack Growth,” J . Tes t Eval., 1 [4] 264-70 (1973). l2 J. I. Mecholsky, Jr., S. W. Freiman, and R. W. Rice, “Fracture Surface Analysis of Ceramics,” J . Marer. Sci., 11 81 1310-19 (1976). l 3 R. G. Hoagland, C. W. Marschall, and W. H. Duckworth, “Reduction of Errors in Ceramic Bend Tests,” J . Am. Ceram. Soc., 59 [5-61 189-92 (1976). I‘ G. K. Bansal, W. H. Duckworth, and D. E. Niesz, “Strength-Size Relationships in Ceramic Materials: Investigation of a Commercial Glass-Ceramic,’’ Am. Ceram. SOC. Bull., 55 [3] 289-92. 307 (1976). G. K. Bansal, W. H. Duckworth, and D. E. Niesz, “Strength-Size Relations in Ceramic Materials: Investieation of an Alumina Ceramic.” J . Am. Ceram. Soc.. 59 - [Il-121472-78 (1976). tions. Am. SOC. Test. Mater.. Soec. Tech. Publ.. No. 381. 1965. P. C. Paris and G. C. Sih; p. 67 in Fracture Toughness Testing and Its Applica- A. G. Evans and H. John&, “Fracture Stress and its &pendence on Slow Crack I s A. G. p a n s , “Slow Crack Growth in Brittle Materials Under Dynamic Loading Growth,” J . Marer. Sci., 10 [2] 214-22 (1975). Conditions, I n f . J . Fracf., 10 [2] 251-59 (1974).