KIM et al: CYCLIC FATIGUE OF BRITTLE CERAMICS the crack spends most of its lifetime in the inter- crack [equation(8)it is now the final stage of mediate stages of growth [38, 47 growth that is rate limiting. 2.2. 2. Cone crack propagation Equations (10)and(Il) indicate that relative to of fatigue for specimens containing well-developed cyclic contact tests, static contact tests run at the cone cracks has been given in an earlier study [20]. same maximum load P yield the same strengths a We reproduce that analysis here, with minor modifi- at anequivalent"contact time cations The stress-intensity factor for (13) cone cI may be approximated by the traditional roesler re- noting that GcsI [0sP'(OsP and Gs=l lation [28, 49] (P= P=constant) in equation(6a)(Appendix) where z is a dimensionless geometry coefficie 3. CONTACT FATIGUE EXPERIMENTS This relation is a far-field approximation, vali 3. Materials selection the limit c> Ro--note that it is independent of Ro Soda-lime glass laths 90 mmx 19.0 mmx5.65 mm and sphere radius r. For a time-dependent load P(n we insert equation(8) into equation (1) were cut with a diamond scribe from as-receive obtain the cone crack size c after time I commercial window plate. The laths were beveled at their top edges(the edges opposite the scribed 3N/ dc= Do(z/To) [P(]dr surfaces) to minimize edge failures in the sub- sequent strength tests, and annealed to re spurious residual stresses [28 with co the equilibrium crack size at K=To in Bars 3 mmx 4 mm x 25 mm of a feldspathic por equation(8)at critical time l= ne/f to pop-in. celain in use for dental restorations (Vita Mark Il For periodic loading over n cycles at frequency f, Vita Zahnfabrik, Bad Sackingen, Germany)were we integrate to obtain ground from large blocks. Top surfaces of the bars were finished with a I um diamond paste polish 3N/2+=[(3N/2+1)G(N )uonf(zPTo)"(10) The microstructure consists of a glass matrix with in the limit n>ne and N> I, with G(N) defined in reinforcing sanidine crystals and residual frit par- equation(6a) ticles of dimension 1-7 um [25, 26 The"inert"strength of a specimen containing a Bars of the same dimension and surface finish as well-formed cone crack is given by the familia lin were prepared from blocks of a bear- Griffith relation ing-grade silicon nitride from an earlier study(there eferred to as M-Si3N4)[35]. In this material the (11) microstructure consists of a bimodal distribution of elongated B grains of length 4.0 um and diameter where y is a crack geometry parameter. Below the 0.5 um(70 vol %)and equiaxed a grains of mean threshold for cone crack initiation [equations(7a) size 0.5 um (20 vol %) with oxynitride glassy and(7b)). the strength is considered to be deter- bonding phase(e10 vol % mined by the surface flaw size, c= cr. Above the threshold, the functional dependence an, P) is 3.2. Contact lesting determined by inserting equation (10)into equation The center regions of the top surfaces were sub- (11). Normalizing relative to the value of the jected to contact testing. For the glass specimens, a strength a=0I for contacts that produce cone light pre-abrasion was first made at the prospective cracks at critical load P= Pi at n= l, we obtain, contact sites with 400-mesh Sic grit to provide an in the limit N> 2/3 adequate density of flaws for ring-crack nucleation 28, 30, 31]. In the other materials, the intrinsic flaw a=01(P1/n/P)(fixed m) (12a) population associated with the microstructure obviated the need for any such pre-abrasion treat- n=I(P,/PGo](fixed a). (12b) ment 25,35 The contacts were made using tungsten carbide Note that even though n may vary over many dec- (wC) spheres of radius r=3.18 or 1.98 mm on a ades, a is not expected to be sensitive to n for typi- servo-hydraulic universal testing machine (Model cally large values of M 8502, Instron Corp, Canton, MA). Cyclic tests As in the preceding subsection, the quantity an were run in repeat loading in haversinusoidal an be expressed in functional form, form up to n=10 at frequency f=10 Hz, between aI(, N, Uo. Ro, To v). Again, an explicit relation for specified maximum and minimum loads(the latter, this quantity will be derived in Section 5. Strictly, typically 20 N, primarily to prevent the contact I also depends on co, but this dependency is small from wandering). Comparative static tests because of the stability of the well-developed cone also made over hold times t=n/f(Fig. 1), to iso-the crack spends most of its lifetime in the inter￾mediate stages of growth [38, 47]. 2.2.2. Cone crack propagation. A simple analysis of fatigue for specimens containing well-developed cone cracks has been given in an earlier study [20]. We reproduce that analysis here, with minor modi®- cations. The stress±intensity factor for the cone crack may be approximated by the traditional Roesler re￾lation [28, 49] K ˆ wP=c3=2 …8† where w is a dimensionless geometry coecient [50]. This relation is a far-®eld approximation, valid in the limit cwR0Ðnote that it is independent of R0 and sphere radius r. For a time-dependent load P0 …t† we insert equation (8) into equation (1) to obtain the cone crack size c after time t …c c0 c3N=2 dc ˆ u0…w=T0† N …t tc ‰P 0 …t†ŠN dt …9† with c0 the equilibrium crack size at K ˆ T0 in equation (8) at critical time tc ˆ nc=f to pop-in. For periodic loading over n cycles at frequency f, we integrate to obtain c3N=2‡1 ˆ ‰…3N=2 ‡ 1†G…N †u0n=f Š…wP=T0† N …10† in the limit nwnc and Nw1, with G(N) de®ned in equation (6a). The ``inert'' strength of a specimen containing a well-formed cone crack is given by the familiar Grith relation s ˆ T0=cc1=2 …11† where c is a crack geometry parameter. Below the threshold for cone crack initiation [equations (7a) and (7b)], the strength is considered to be deter￾mined by the surface ¯aw size, c ˆ cf. Above the threshold, the functional dependence s(n, P) is determined by inserting equation (10) into equation (11). Normalizing relative to the value of the strength s ˆ s1 for contacts that produce cone cracks at critical load P ˆ P1 at n ˆ 1, we obtain, in the limit Nw2=3 s ˆ s1…P1=n1=NP† 1=3 …fixed n† …12a† n ˆ ‰…P1=P†…s1=s† 3 Š N …fixed s†: …12b† Note that even though n may vary over many dec￾ades, s is not expected to be sensitive to n for typi￾cally large values of N. As in the preceding subsection, the quantity s1 can be expressed in functional form, s1… f, N, u0, R0, T0, n†. Again, an explicit relation for this quantity will be derived in Section 5. Strictly, s1 also depends on c0, but this dependency is small because of the stability of the well-developed cone crack [equation (8)]Ðit is now the ®nal stage of growth that is rate limiting. Equations (10) and (11) indicate that relative to cyclic contact tests, static contact tests run at the same maximum load P yield the same strengths s at an ``equivalent'' contact time tS ˆ GCn=f …13† noting that GCR1 [0RP0 …t†RP] and GS ˆ 1 (P0 ˆ P ˆ constant) in equation (6a) (Appendix). 3. CONTACT FATIGUE EXPERIMENTS 3.1. Materials selection Soda-lime glass laths 90 mm19.0 mm5.65 mm were cut with a diamond scribe from as-received commercial window plate. The laths were beveled at their top edges (the edges opposite the scribed surfaces) to minimize edge failures in the sub￾sequent strength tests, and annealed to remove any spurious residual stresses [28]. Bars 3 mm4 mm25 mm of a feldspathic por￾celain in use for dental restorations (Vita Mark II1 , Vita Zahnfabrik, Bad Sackingen, Germany) were ground from large blocks. Top surfaces of the bars were ®nished with a 1 mm diamond paste polish. The microstructure consists of a glass matrix with reinforcing sanidine crystals and residual frit par￾ticles of dimension 1±7 mm [25, 26]. Bars of the same dimension and surface ®nish as the porcelain were prepared from blocks of a bear￾ing-grade silicon nitride from an earlier study (there referred to as M-Si3N4) [35]. In this material the microstructure consists of a bimodal distribution of elongated b grains of length 4.0 mm and diameter 0.5 mm (170 vol.%) and equiaxed a grains of mean size 10.5 mm (120 vol.%), with oxynitride glassy bonding phase (110 vol.%). 3.2. Contact testing The center regions of the top surfaces were sub￾jected to contact testing. For the glass specimens, a light pre-abrasion was ®rst made at the prospective contact sites with 400-mesh SiC grit to provide an adequate density of ¯aws for ring-crack nucleation [28, 30, 31]. In the other materials, the intrinsic ¯aw population associated with the microstructure obviated the need for any such pre-abrasion treat￾ment [25, 35]. The contacts were made using tungsten carbide (WC) spheres of radius r ˆ 3:18 or 1.98 mm on a servo-hydraulic universal testing machine (Model 8502, Instron Corp., Canton, MA). Cyclic tests were run in repeat loading in haversinusoidal wave￾form up to n ˆ 107 at frequency f=10 Hz, between speci®ed maximum and minimum loads (the latter, typically 20 N, primarily to prevent the contact from ``wandering''). Comparative static tests were also made over hold times t ˆ n=f (Fig. 1), to iso- 4714 KIM et al.: CYCLIC FATIGUE OF BRITTLE CERAMICS