JOURNAL O£ MAIERIALS SCIENCE LETTERS 6 (1987) 462-464 Lifetime prediction of ceramic materials subjected to static loads XIAO-ZHI HU, YIU-WlNG MAI, BRIAN COTTERELL Department of Mechanical Engineering, University of Sydney, Sydney, New South Wales 2006, Australia Structural components made from many engineering ceramics show slow crack growth in pre-existing cracks under static loads prior to eventual fracture. The conventional approach to predict lifetimes (tf) of ceramics due to applied static stresses (o-~) is to con- sider the growth of a predominant single crack from some initial flaw size (a~) to a critical size (at) accord- ing to the fundamental crack-growth law given by [1] da/dt = AK ~ = A(~raYal/2)" (1) Here, da/dt is the crack velocity, Kis the applied crack tip stress intensity factor, Y is a geometry correction factor depending on crack size and specimen dimen- sions and (A, n) are numerical constants for a given material-environment system. Equation 1 can be obtained from independent experiments using double- cantilever-beam geometries for which crack velocities are easily measured. The critical flaw size, af, is related to the material's toughness, K~c, by K(a = aO = Kk. Thus, integration of Equation 1 for constant 0-, gives = 2a~-'/2/[(n - 2) Y"A] (2) provided (ai/af) "/2-1 ~ 1. Although (A, n) can be evaluated unambiguously from Equation 1 the initial flaw size a~varies according to the Weibull distribution of the inert strength o-0. The Weibull failure probability (F) at a~ is given by: F(ao) = 1 - exp [-(%/a.) m] (3) where o-. is a normalizing parameter, m is the Weibull modulus, and ao = KIo/Ya~/2 (4) Equation 2 for lifetime predictions therefore becomes tr0-~ = 2s = 20", 2 / 1 \(, 2)/m/ × lnLl_~ ) /[A(n- 2)Y2Kr~ -2] (5) This expression allows the applied stress-probability of failure-lifetime diagrams to be constructed. Equations 2 and 5 are valid for both static tensile and static bending loads. In the latter case (i.e. bending) the largest single flaw is considered to be situated at the tensile surface where the applied stress is aa. There is, however, a major problem with this con- ventional approach. Specimens usually contain natural flaws whose sizes are not constant and in the case of static bending loads the flaws do not experience the same stresses. A statistical theory of fracture is 462 required to predict lifetimes to failure. Of course, if the flaws are subjected to the same uniform stress as in the case of static tensile loads, the largest flaw still con- trols the time to failure and there is no difference between the single crack and the statistical multiple cracks approaches. But because of the ease of loading arrangements, static bending tests are usually preferred to static tensile tests in the laboratory for evaluation of failure lifetimes of brittle ceramic materials. It is the purpose of this communication to point out the poss- ible inaccuracies involved in lifetime predictions if the single crack theory using Equations 2 and 5 is used in non-uniform bending stress situations such as subjected to static three-point and four-point bending loads. By assuming the flaws to be randomly distributed within the volume of stressed material, V, and the flaw size distribution q(a) to be described by a Pareto function, q(a) = Or~-~a ° a o ~ a (6) where Or is the flaw density and ao is the smallest flaw, we can show that the failure probability at stress, o- a is [2] f(Ga) -~- 1 -- exp [-- fv f£(oa)q(a)dadVl (7) Using Equation 6 in Equation 7 and integrating over V, we can recover Equation 3 for which ~c = aa and = ~r0[2(m + 1)/Vof] 1/m (four-point pure bend) 0-, = ~012(m + 1)2/Vor] L/m (three-point bend) (8) where ~0 is given by Equation 4 when a = a0. In the statistical treatment it is realized that the flaws will grow with time (t) under a given applied stress (ea). This means that the flaw-size distribution function will also vary with time, changing from q(a) to q(a, t). It can be shown [3] that (~a) q(a, Oda - r(o,,t) q ( a ) d a (9) if there is no flaw nucleation and that ar(aa, t) = [1 + (n/2 -- l)Ay2K~22~t]2/(2-')a(a,) (10) The failure probabilities can now be derived from Equation 7 by replacing q(a) with q(a, t). Equation 9 is then used to facilitate integration because q(a) is readily determined from Equation 6. Thus, in a beam of rectangular cross-section we obtain for the pure bending case, 0261-8028/87 $03.00 + .12 © 1987 Chapman and Hall Ltd.