June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 355 10000000 1050°C 1000000 Fiber Data - Fit to Composite Data 100000 10000 1.00E081.00E-071.00E061.00E05100E041.00E03 Steady-State Strain Rate(1/s) Fig. 4. Monkman-Grant plot (log(lifetime)versus log(strain rate)) for fibers(indicated by dashed lines; data from DiCarlo and Yun")and the composite at vanous temperatures(◆)950Cand(1050°C load is carried by the fiber; thus, the fiber stress is given as cann f. When the fiber breaks, slip along the fiber/matrix interface occurs where f is the fiber volume fraction. At a distance the matrix over a fiber slip length. I rack, stress is transferred from the fiber to the through the interface frictional stress as . 2Tf (11) 27z forl≤8) (8) that is equal to the distance at which the stress in Eq.(8)would reach a value of zero if not cut off by the far-field stress. Thus, the where 8 is the distance over which interfacial slip occurs and is as er stress. as a function of distance z from a matrix crack is given defined as o(3)=71-1)(ok≤6 rooo[(I-DE Tf E (2)=0mE < (12b) and E Em, and Ec are the Youngs moduli of the fiber, matrix, and ively (with EC=Er+(1-DEm, and r is the where T- App/ f. This stress profile and the above-described fiber radiu 28, the fiber stresses and strains are equal and notations are illustrated in Fig. 5. Under the stress state given in the stress by the fiber regains the constant far-field value Eqs. (12), the probability of fiber failure q over a length 2. at the applied stress oapp follows from Eq(6)as E (10) -(1+ma"+)(13) Fiber stress profile 2T Er E Matrix Crack Fig. 5. Schematic of time-dependent flaw growth under the spatially varying stress on a fiber around a matrix crack; the crack-growth rate is de on both the initial crack size and the stress acting on it
Journal of the American Ceramic Sociery -Halverson and Curtin Vol. 85. No 6 1356 Here, we have introduced the dimensionless parameters T and a, growth of pre-existing flaws in the fiber. The slow-crack-growth as well as the characteristic stress o. rate is represented by a Paris law T da (14b) crack size and K is the stress intensity factor; p is G(z=0) growth exponent and a is a rate constant, each of ly is dependent on temperature. K, which is a (14c) of the current stress T, crack size a, and a geometric factor Y, is given by A length 2I, is chosen because only fiber breaks within t/, of a K=YTa matrix crack will influence tensile failure of the composite at this Flaw failure is determined using the expression K=Kle. Thus, the Depending on the Weibull modulus m and the stiffness of the initial tensile strength(o, at initial crack size a, )can be related to matrix crack(see below) matrix,as captured by the parameter a, there is a competition the flaw size afn) and flaw strength o() at time r by combining Egs to vield between fiber failure within the linearly decreasing portion of the stress field and the constant far-field region. For low values of 1B-2 failure in the far-field stress region is negligil probability of fiber failure is accurately represented by o(n=o8-2-c T(r)adr q(7,()=1-exp-(m+1 C=(5-1)AFK2 (20b) which is thesingle matrix crack" result that was derived by Thouless and Evans, as well as other researchers, For high values here y ed to be constant over time. Inverting Eq.(20a) of ma"*, the matrix modulus is small, relative to the fibers (i. e. yields the trength required to provide a current strength of Ee RE, thus, the length 8 is small and the far-field stress region a(t)after n stress history T(n on the flaw: is realized very close to the matrix. Then, fiber failure will occur almost equally over the entire region and, in the limit of a= d =o(-+1(?)di (21) ( corresponding to a matrix modulus of zero), the probability of fiber failure is given as q(7,(7)=1-exp(-7) (16 Here, we have introduced a nondimensional time i and a normal- th given by which is the result obtained by Curtin for the case of saturated, (22a) closely spaced matrix cracking. Thus, the value of mo" in E (13)ranges from zero to m and represents the influence of the matrix modulus on the probability of fiber failure, reducing to the single matrix crack "and"multiple matrix crack"cases in the (22b) In real materials, multiple matrix cracks typically exist with If two adjacent crack paced by a where T is as defined previously in Eq (14a) distance i that is less than a fiber slip length 1,, then Eqs. (12)only apply over a length 2. In this case, we determine the probability that the fiber will fail is the probability that the initia of fiber failure around each matrix crack only within the region of the fiber is less than the initial strength given by Eq.(21) i2, which corresponds to setting the limits of the position integral That is, for a fiber element of length 8z under some stress in Eq. (6)to /=t/2. Then, the cumulative probability of failure history /(n), the probability of failure over the applied stress increment from o(n) to o(n)+ So(n) is obtained by substituting (t) p(or(,.bo(),6x)= 3 {1+a(m+1) Following arguments identical to those used in the quasi-static he probability of failure of a fiber, with respect to time. r the stress profile of Eqs.(12)is then given by fonm)<5<4)(1a) x-79-2+x"T(r)dr results will be used below to determine the +a"2+a7 te damage evolution and quasi-static stress- failure. Equation(17) is the first main result of o (B) Time-Dependent Behav or: We now assun where a change of variables from z to x= I-al. has been er strength degrades with time, because of the performed. Figure 6 shows the ratio of mean fiber lifetimes for