June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1355 10000000 ◆950°C 1000000 Fiber data Fit to Composite Data 100000 10E08100E07100E-061.00E-051.00E041.00E03 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 various temperatures((◆)950°cand(■)1050°C load is carried by the fiber, thus, the fiber stress is given as capp/, When the fiber breaks, slip along the fiber/matrix interface occurs wherefis the fiber volume fraction. At a distance from the matrix over a fiber slip length, Is, ack, stress is transferred from the fiber to the matrix through the nterface frictional stress as fr(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 fiber stress, as a function of distance from a matrix crack, is given where 8 is the distance over which interfacial slip occurs and is defined ()=7(1-2.)(r0<H<b (12a) r(=) (for 8<E=D (12b) and Es Em, and Ec are the Youngs moduli of the fiber, matrix, and omposite, respectively (with Ec =Er+(1-nEm), and r is the where T=Uanp/. This stress profile and the above-described fiber radius. For 28, the fiber stresses and strains are equal and notations are illustrated in Fig. 5. Under the stress state given in the stress carried 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 app follows from Eq(6) (10) q(,4)=1-ex-(m+1)rm(1+m-)(3) Fiber stress profile Matrix Crack matic of time-dependent flaw growth under the spatially varying stress on a fiber around a matrix crack, the crack-growth rate is dependent itial crack size and the stress acting on it
load is carried by the fiber; thus, the fiber stress is given as app/f, where f is the fiber volume fraction. At a distance z from the matrix crack, stress is transferred from the fiber to the matrix through the interface frictional stress as z app f 2z r for z (8) where is the distance over which interfacial slip occurs and is defined as r app 2f 1 f Em Ec (9) and Ef , Em, and Ec are the Young’s moduli of the fiber, matrix, and composite, respectively (with Ec fEf (1 f)Em), and r is the fiber radius. For z , the fiber stresses and strains are equal and the stress carried by the fiber regains the constant far-field value: ff app Ef Ec (10) When the fiber breaks, slip along the fiber/matrix interface occurs over a fiber slip length, ls, ls r app 2f (11) 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 fiber stress, as a function of distance z from a matrix crack, is given as z T 1 z ls for 0 z (12a) z app Ef Ec for z (12b) where T app/f. This stress profile and the above-described notations are illustrated in Fig. 5. Under the stress state given in Eqs. (12), the probability of fiber failure q over a length 2ls at the applied stress app follows from Eq. (6) as qT˜, lsT˜ 1 exp 1 m 1T˜ m 1 1 mm 1 (13) Fig. 4. Monkman–Grant plot (log (lifetime) versus log (strain rate)) for fibers (indicated by dashed lines; data from DiCarlo and Yun8 ) and the composite at various temperatures (() 950°C and () 1050°C). 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 dependent on both the initial crack size and the stress acting on it. June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1355
1356 urnal of the American Ceramic Society-Halverson and Curtin Vol. 85. No. 6 Here, we have introduced the dimensionless parameters Tand 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 (14a) dIaS (14b) where a is the crack size and k is the stress intensity factor; B is the slow-crack-growth exponent and A is a rate constant, each of dooT+D) which generally is dependent on temperat which is a (14c) function of the current stress T, crack size a, and a geometric factor A length 21s is chosen because only fiber breaks within +l, of a K= YTa (19) matrix crack will influence tensile failure of the composite at this matrix crack(see below) lure is determined using the expression K=Klc. Thus, the Depending on the Weibull modulus m and the stiffness of the nsile strength(o, at initial crack size ai) can be related to matrix, as captured by the parameter c, there is a size a(n)and flaw strength o(n)at time r by combining eqs between fiber failure within the linearly decreasing portion of the (19)and integrating, to yield ess field and the It far-field region. For low values of mo failure in eld stress region is negligible and the probability of fiber accurately represented by (=2-c|°dr q(4)=1-exp-(m+门)产 (15) C=I5-1JAYK1-2 (20b) which is the single matrix crack"result that was derived by Thouless and Evans, as well as other researchers. For high values where r is assumed to be constant over time. Inverting Eq (20a) of mo"t, the matrix modulus is small, relative to the fibers (i.e yields the initial strength required to provide a current strength of EC FeD; thus, the length 8 is small and the far-field stress region g(n after the given stress history 1(o)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=1 (corresponding to a matrix modulus of zero ), the probability of GG)=a(0-2+元pd fiber failure is given as q(,(⑦)=1 Here we have introduced a nondimensional time t and a normal ized strength o, given by which is the result obtained by Curtin for the case of saturated closely spaced matrix cracking. Thus, the value of mo"+ in Eq (22a) (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 G appropriate limits (22b) In real materials, multiple matrix cracks typically exist with some variable spacing. If two adjacent cracks are spaced by a where T is as defined previously in Eq.( 14a) distance i that is less than a fiber slip length l, then eqs If some stress history T(o)is applied to a fiber, the probability apply over a length x/2. In this case, etermine the that the fiber will fail is the probability that the initial strength of fiber failure around each matrix crack only within of the fiber is less than the initial strength given by Eq. (21) i/2, which corresponds to setting the limits of the position integral That is, for a fiber element of length 8= under some stress in Eq (6)to /=+/2. Then, the cumulative probability of failure history TO), the probability of failure over the applied stress Is increment from o(n) to o()+ So(n) is obtained by substituting Eq(21)into Eq (5) T p(t),6o(1),8-)= 86(l) ×{1+a|(m+1)-1-m(1-a) Following arguments identical to those used in the quasi-static case, the probability of failure of a fiber, with respect to time, under the stress profile of Eqs. (12)is then given by (17a) q(1,120 , xP(r)°dr for5<8(m)(17b) m(B-2) These results will be used below to determine the quasi-static +a+28-2+at(r)d composite damage evolution and quasi-static stress-strain curve up to failure. Equation (17) is the first main result of our analysis. B) Time-Dependent Behavior: We now assume that the where a change of variables from to x =1-Ehls has been fiber strength degrades with time, because of the slow crack performed. Figure 6 shows the ratio of mean fiber lifetimes for
Here, we have introduced the dimensionless parameters T˜ and , as well as the characteristic stress c: 3 T˜ T c (14a) ff z 0 f Ef Ec (14b) c 0 ml0 r 1/m 1 (14c) A length 2ls is chosen because only fiber breaks within ls of a matrix crack will influence tensile failure of the composite at this matrix crack (see below). Depending on the Weibull modulus m and the stiffness of the matrix, as captured by the parameter , there is a competition between fiber failure within the linearly decreasing portion of the stress field and the constant far-field region. For low values of mm 1 , failure in the far-field stress region is negligible and the probability of fiber failure is accurately represented by qT˜, lsT˜ 1 exp 1 m 1T˜ m 1 (15) which is the “single matrix crack” result that was derived by Thouless and Evans,38 as well as other researchers. For high values of mm 1 , the matrix modulus is small, relative to the fibers (i.e., Ec fEf ); thus, the length is small and the far-field stress region is realized very close to the matrix. Then, fiber failure will occur almost equally over the entire region and, in the limit of 1 (corresponding to a matrix modulus of zero), the probability of fiber failure is given as qT˜, lsT˜ 1 exp T˜ m 1 (16) which is the result obtained by Curtin3 for the case of saturated, closely spaced matrix cracking. Thus, the value of mm 1 in Eq. (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 appropriate limits. In real materials, multiple matrix cracks typically exist with some variable spacing. If two adjacent cracks are spaced by a distance x that is less than a fiber slip length ls, then Eqs. (12) only apply over a length x/2. In this case, we determine the probability of fiber failure around each matrix crack only within the region x/2, which corresponds to setting the limits of the position integral in Eq. (6) to l x/2. Then, the cumulative probability of failure is qT˜, x 2 1 exp 1 m 1T˜ m 1 1 m m 1 x 2ls 1 m1 for app x 2 ls (17a) qT˜, x 2 1 exp 1 m 1T˜ m 1 1 1 x 2ls m 1 for x 2 app (17b) These results will be used below to determine the quasi-static composite damage evolution and quasi-static stress–strain curve up to failure. Equation (17) is the first main result of our analysis. (B) Time-Dependent Behavior: We now assume that the fiber strength degrades with time, because of the slow crack growth of pre-existing flaws in the fiber. The slow-crack-growth rate is represented by a Paris law: da dt AK (18) where a is the crack size and K is the stress intensity factor; is the slow-crack-growth exponent and A is a rate constant, each of which generally is dependent on temperature. K, which is a function of the current stress T, crack size a, and a geometric factor Y, is given by K YTa1/ 2 (19) Flaw failure is determined using the expression K KIc. Thus, the initial tensile strength ( i at initial crack size ai ) can be related to the flaw size a(t) and flaw strength (t) at time t by combining Eqs. (18) and (19) and integrating, to yield t i 2 C 0 t Tt dt 1/ 2 (20a) C 2 1AY2 KIc 2 (20b) where Y is assumed to be constant over time. Inverting Eq. (20a) yields the initial strength required to provide a current strength of (t) after the given stress history T(t) on the flaw: ˜ i˜t ˜t 2 0 ˜t T˜t ˜ dt ˜ 1/ 2 (21) Here, we have introduced a nondimensional time t ˜ and a normalized strength ˜, given by t ˜ tC c 2 (22a) and ˜ c (22b) where T˜ is as defined previously in Eq. (14a). If some stress history T(t) is applied to a fiber, the probability that the fiber will fail is the probability that the initial strength of the fiber is less than the initial strength given by Eq. (21). That is, for a fiber element of length z under some stress history T(t), the probability of failure over the applied stress increment from (t) to (t) (t) is obtained by substituting Eq. (21) into Eq. (5): pf t, t, z m i t m 1 l0 0 m z i t (23) Following arguments identical to those used in the quasi-static case, the probability of failure of a fiber, with respect to time, under the stress profile of Eqs. (12) is then given by qT˜, ls, t ˜ 1 exp T˜ 1 x 2 T˜ 2 0 t xT˜t dt m/ 2 dx m 1 T˜ 2 0 t 2 T˜t dt m/ 2 (24) where a change of variables from z to x 1 z/ls has been performed. Figure 6 shows the ratio of mean fiber lifetimes for 1356 Journal of the American Ceramic Society—Halverson and Curtin Vol. 85, No. 6