KIM et al: CYCLIC FATIGUE OF BRITTLE CERAMICS Pn generic form [47, 48 K(c)=do(tc)/Fc) where c is the crack dimension measured along the downward crack coordinate s and F(c) is a dimen sionless Greens function, with F< I for any dimin ishing tensile field as(c)< oo at s >0. Suppose now that the indenter is subject to some time-dependent contact load P(n. Assume that a ring crack initiates from a surface flaw of character istic size cr and begins to extend downward accord attain instability by writing u=dc/dr in equation and co th ions(2)and (3) Ro 0 dc/le/FOn Static load SIze c= cF at time I= Ie for cone crack pop-in is determinable from the 0 limiting equilibrium condition K= To in equation (3). Since the extension to depth cF is generally Multi-cycle 10c small compared to the final con we regard this phase of the crack evolution as initiation In the special case of periodic loading to pea load P over n cycles at frequency f, corresponding Fig. 1. Schematic of Hertzian contact of radius r and n cycles at load P. Beyond ith wc to an indentation time I=n/f, equation(4)reduces threshold load. damage consists of ks. surface ring radius Ro plus, under severe loading conditions, sub- (nPN)=[(N) Ro/G(N )uo(2n/RS/To/(1-2v)1 un over n normal contacts at frequency ests, and static loading over prescribed hold times I, are also run where subscript c denotes critical number of cycles Ie to pop-in at fixed P; or, alternatively, critical load p at fixed n. The G and h terms are dimen- sionless integrals [10 D=L0(K/70)(K<To) G(N)= [P(o/P] d(n where To(Ke)is the single-value material toughness defining the upper limit of the velocity range, and N and o are velocity exponent and coefficient, re- H(N)= d(c/Ro)/(c/Ro)Fc/Ro).(6b) spectively. We consider two stages in the kinetic rack evolution in relation to the ensuing strength Equation (5) degradation properties: initiation, determining the ing relative to threshold load at which strength undergoes an initiation in single-cycle contact, i.e. P=Pi at sequent continued strength loss at post-threshold n= 1: 2. 2. Cone crack initiation. The radial tensile stress do acting at the surface coordinate ro of a ne=(P1/P)(fixed P (7b) prospective cone crack from contact with a sphere of radius r at load P(Fig. 1)for a material of pois- The quantity Pi has the functional form sons ratio v is [46 IC,N, Do, Ro, To, v). We will obtain an explicit re- lation for this function in Section 5. Strictly, PI o=1-2)P/RG (2)also depends on cr and ce, but these dependencies are small because of the stabilizing effect of the The corresponding stress-intensity factor has the diminishing stress field in cone initiation, such thatfunction u ˆ u0…K=T0† N …K < T0† …1† where T0 (Kc) is the single-value material toughness de®ning the upper limit of the velocity range, and N and u0 are velocity exponent and coecient, re￾spectively. We consider two stages in the kinetic crack evolution in relation to the ensuing strength degradation properties: initiation, determining the threshold load at which strength undergoes an abrupt strength loss; propagation, determining sub￾sequent continued strength loss at post-threshold loads. 2.2.1. Cone crack initiation. The radial tensile stress s0 acting at the surface coordinate R0 of a prospective cone crack from contact with a sphere of radius r at load P (Fig. 1) for a material of Pois￾son's ratio n is [46] s0 ˆ 1 2 …1 ÿ 2n†P=pR2 0: …2† The corresponding stress±intensity factor has the generic form [47, 48] K…c† ˆ s0…pc† 1=2 F…c† …3† where c is the crack dimension measured along the downward crack coordinate s and F(c) is a dimen￾sionless Greens function, with F < 1 for any dimin￾ishing tensile ®eld ss…c† < s0 at s > 0. Suppose now that the indenter is subject to some time-dependent contact load P0 …t†. Assume that a ring crack initiates from a surface ¯aw of character￾istic size cf and begins to extend downward accord￾ing to equation (1). We may solve for the time to attain instability by writing u ˆ dc=dt in equation (1) and combining with equations (2) and (3): u0‰…1 ÿ 2n†=2p1=2 R2 0T0Š N …tc 0 ‰P 0 …t†ŠN dt ˆ …cF cf dc=‰c1=2 F…c†ŠN …4† where the instability crack size c ˆ cF at time t ˆ tc for cone crack pop-in is determinable from the limiting equilibrium condition K ˆ T0 in equation (3). Since the extension to depth cF is generally small compared to the ®nal cone crack size, we regard this phase of the crack evolution as ``initiation''. In the special case of periodic loading to peak load P over n cycles at frequency f, corresponding to an indentation time t ˆ n=f, equation (4) reduces to …nP N†c ˆ ‰H…N †fR0=G…N †u0Š‰2p1=2 R3=2 0 T0=…1 ÿ 2n†ŠN …5† where subscript c denotes critical number of cycles nc to pop-in at ®xed P; or, alternatively, critical load Pc at ®xed n. The G and H terms are dimen￾sionless integrals [10] G…N † ˆ …1 0 ‰P 0 … ft†=PŠ N d… ft† …6a† H…N † ˆ …cF=R0 cf =R0 d…c=R0†=‰…c=R0† 1=2 F…c=R0†ŠN: …6b† Equation (5) may be further reduced by normaliz￾ing relative to the value of critical load for cone initiation in single-cycle contact, i.e. P ˆ P1 at n ˆ 1: Pc ˆ P1=n1=N …fixed n† …7a† nc ˆ …P1=P† N …fixed P†: …7b† The quantity P1 has the functional form P1… f, N, u0, R0, T0, n†. We will obtain an explicit re￾lation for this function in Section 5. Strictly, P1 also depends on cf and cF, but these dependencies are small because of the stabilizing e€ect of the diminishing stress ®eld in cone initiation, such that Fig. 1. Schematic of Hertzian contact test, with WC sphere of radius r and n cycles at load P. Beyond threshold load, damage consists of cone cracks, surface ring radius R0; plus, under severe loading conditions, sub￾surface quasi-plastic deformation zone. Cyclic fatigue tests are run over n normal contacts at frequency f. Comparative single-cycle tests, and static loading tests over prescribed hold times t, are also run. KIM et al.: CYCLIC FATIGUE OF BRITTLE CERAMICS 4713