M B. Ruggles-Prenn et al International Journal of fatigue 30(2008)502-516 results from prior work [44]are included for comparison. It is seen that all specimens tested in air exhibited increase in d=AK?=A(Yava) tensile strength, irrespective of the fatigue stress level or the where a is the crack size, t is time, A and n are material con- loa ding frequency. However, considerable stiffness loss (18-33%)was observed. The N720/A composite relies on the applied stress intensity factor, Y is a geometric factor and a is the applied stress. If the cyclic and static fatigue g,as evidenced by a decrease in stiffness, and induces by subcritical(slow)crack growth, then the cyclic fatigue the weakening of the fiber-matrix interfaces. By causing lifetime can be predicted from the static fatigue data by additional matrix cracking, prior fatigue in air serves using a linear elastic crack growth model [59]. Evans and maintain the matrix porosity at the minimum level needed Fuller[59] proposed that the ratio of the time to failure un for crack deflection. As a result, prior fatigue in air effec- der cyclic fatigue, taf, to the time to failure under static fa- ively improves the damage tolerance and long-term dura- tigue, Ist, under the same maximum applied stress omax can bility of the composite as demonstrated by an improved be obtained as tensile strength of the pre-fatigued specimens. Conversely, prior fatigue in steam caused reduction in both strength (1) and stifness. Strength loss in steam was limited to 12% and stifness loss, to 20%. Strength and stiffness degrada where the applied stress d(n)=omaxf(n) is a periodic func tion increases with decreasing frequency of prior fatigue. tion of time, t is the period, and 0<fo< I for tensile As the frequency of the 170 MPa tests decreased from o loading. For static loading, the applied stress o(0)=omax to 1.0 Hz, strength loss increased from 4% to 12% and stiff- ness loss, from 7% to 20%. The discrepancy between the For cyclic loading with the ratio R(minimum to maximum retained modulus of a run-out specimen and the decrease stress)and a triangular waveform the periodic function f() hysteresis modulus observed during fatigue testing most is expressed as likely stems from different methods used to determine the R+ R f(t)= or0≤t≤ retained and hysteresis moduli Recent work [44]revealed the degrading effect of steam 1-(1-R)(￥-1)for≤t≤t on fatigue as well as on creep performance of the N720/A In the case of a trapezoidal waveform, where ih is the hold composite at 1200C. Mehrman et al. [45] showed that the time at maximum stress, fL is the loading/unloading time, superposition of a hold time at maximum stress onto a fati- and t=(h IL) is the period, the function f(n) is given by gue cycle drastically reduced cyclic lifetime of N720/A at 1200C in steam. The current study revealed a marked R+(1-R) or0≤t≤ influence of the loading frequency on fatigue life in steam. f()=I for≤t≤h+号(4) These findings suggest that a unique time-dependent failure 1-2(t-4-)forh+是≤t≤ mechanism may be operating under both cyclic and static loading at 1200C in steam The crack growth exponent n is readily obtained from the It is recognized that stress corrosion of the N720 fibers static fatigue data at 1200C in steam [44]. Noting that sta may be the mechanism behind reduced creep resistance of tic fatigue lifetime in steam is related to the applied stress N720/A composite at 1200 C in steam. Earlier studies as: [49-54] suggested that static fatigue (i.e. delayed fracture under a sustained constant load)of silica-based glasses sf=4.5×1026-113 (5) was a chemical process, where subcritical (slow) crack ( the applied stress is in megapascals and time is in seconds) growth resulted from and was controlled by a stress- the crack growth exponent n=11. 3 is determined. Using enhanced chemical reaction between glass and water in R=0.05 and Eqs.(2)-(4), the ratio of cyclic to static life- the environment. Michalske and Bunker [55-57]examined times, r, can be calculated for the triangular waveform as the role of mechanical strain in accelerating chemical reac- well as for the trapezoidal waveforms with the loading/ tions between the Si-o bonds at the crack tip and environ- unloading time tL= l s and hold times fh of 10 and 100 s mental molecules and found that the highly strained Si-o The predicted cyclic fatigue lifetimes are plotted in Fig. 9 onds reacted with water at least 8 orders of magnitude together with the experimental results obtained in steam faster than the unstrained bonds. Michalske and Bunker The results of fatigue tests from prior work [44, 45] are also [57] proposed a quantitative chemical-kinetics-based model shown in Fig. 9 for comparison. It is seen that the predicted to predict the rate of crack growth in silica glass in humid cyclic lifetimes are in excellent agreement with the experi- condition as a function of the applied stress. This model mental results obtained in fatigue tests with the trapezoidal describes a fracture rate law in which the crack growth rate waveform and with the triangular waveform at 0. I Hz, increases exponentially with the applied stress intensity. indicating that there is no apparent cyclic effect on the For many glass and ceramic materials subcritical crack crack growth and that the slow crack growth is growth can be described by a power law [5 the dominant failure mechanism. Conversely, predictionsresults from prior work [44] are included for comparison. It is seen that all specimens tested in air exhibited increase in tensile strength, irrespective of the fatigue stress level or the loading frequency. However, considerable stiffness loss (18–33%) was observed. The N720/A composite relies on the porous matrix for crack deflection and damage toler￾ance. Fatigue cycling promotes progressive matrix crack￾ing, as evidenced by a decrease in stiffness, and induces the weakening of the fiber–matrix interfaces. By causing additional matrix cracking, prior fatigue in air serves to maintain the matrix porosity at the minimum level needed for crack deflection. As a result, prior fatigue in air effec￾tively improves the damage tolerance and long-term dura￾bility of the composite as demonstrated by an improved tensile strength of the pre-fatigued specimens. Conversely, prior fatigue in steam caused reduction in both strength and stiffness. Strength loss in steam was limited to 12% and stiffness loss, to 20%. Strength and stiffness degrada￾tion increases with decreasing frequency of prior fatigue. As the frequency of the 170 MPa tests decreased from 10 to 1.0 Hz, strength loss increased from 4% to 12% and stiff- ness loss, from 7% to 20%. The discrepancy between the retained modulus of a run-out specimen and the decrease in hysteresis modulus observed during fatigue testing most likely stems from different methods used to determine the retained and hysteresis moduli. Recent work [44] revealed the degrading effect of steam on fatigue as well as on creep performance of the N720/A composite at 1200 C. Mehrman et al. [45] showed that the superposition of a hold time at maximum stress onto a fati￾gue cycle drastically reduced cyclic lifetime of N720/A at 1200 C in steam. The current study revealed a marked influence of the loading frequency on fatigue life in steam. These findings suggest that a unique time-dependent failure mechanism may be operating under both cyclic and static loading at 1200 C in steam. It is recognized that stress corrosion of the N720 fibers may be the mechanism behind reduced creep resistance of N720/A composite at 1200 C in steam. Earlier studies [49–54] suggested that static fatigue (i.e. delayed fracture under a sustained constant load) of silica-based glasses was a chemical process, where subcritical (slow) crack growth resulted from and was controlled by a stress￾enhanced chemical reaction between glass and water in the environment. Michalske and Bunker [55–57] examined the role of mechanical strain in accelerating chemical reac￾tions between the Si–O bonds at the crack tip and environ￾mental molecules and found that the highly strained Si–O bonds reacted with water at least 8 orders of magnitude faster than the unstrained bonds. Michalske and Bunker [57] proposed a quantitative chemical-kinetics-based model to predict the rate of crack growth in silica glass in humid condition as a function of the applied stress. This model describes a fracture rate law in which the crack growth rate increases exponentially with the applied stress intensity. For many glass and ceramic materials subcritical crack growth can be described by a power law [58]: da dt ¼ AKn I ¼ A Y r ffiffiffi a  p n ð1Þ where a is the crack size, t is time, A and n are material con￾stants dependent on temperature and environment, KI is the applied stress intensity factor, Y is a geometric factor and r is the applied stress. If the cyclic and static fatigue failure mechanisms are indeed identical and dominated by subcritical (slow) crack growth, then the cyclic fatigue lifetime can be predicted from the static fatigue data by using a linear elastic crack growth model [59]. Evans and Fuller [59] proposed that the ratio of the time to failure un￾der cyclic fatigue, tcf, to the time to failure under static fa￾tigue, tsf, under the same maximum applied stress rmax can be obtained as: r ¼ tcf tsf ¼ s Z s 0 rðtÞ rmax  n dt  1 ð2Þ where the applied stress r(t) = rmaxf(t) is a periodic func￾tion of time, s is the period, and 0 6 f(t) 6 1 for tensile loading. For static loading, the applied stress r(t) = rmax. For cyclic loading with the ratio R (minimum to maximum stress) and a triangular waveform the periodic function f(t) is expressed as: f ðtÞ ¼ R þ ð1 RÞ 2t s for 0 6 t 6 s 2 1 ð1 RÞ 2t s 1  for s 2 6 t 6 s ( ð3Þ In the case of a trapezoidal waveform, where th is the hold time at maximum stress, tL is the loading/unloading time, and s = (th + tL) is the period, the function f(t) is given by: f ðtÞ ¼ R þ ð1 RÞ 2t tL for 0 6 t 6 tL 2 1 for tL 2 6 t 6 th þ tL 2 1 2ð1RÞ tL t th tL 2  for th þ s 2 6 t 6 s 8 >>< >>: ð4Þ The crack growth exponent n is readily obtained from the static fatigue data at 1200 C in steam [44]. Noting that sta￾tic fatigue lifetime in steam is related to the applied stress as: tsf ¼ 4:5  1026r11:3 max ð5Þ (the applied stress is in megapascals and time is in seconds), the crack growth exponent n = 11.3 is determined. Using R = 0.05 and Eqs. (2)–(4), the ratio of cyclic to static life￾times, r, can be calculated for the triangular waveform as well as for the trapezoidal waveforms with the loading/ unloading time tL = 1 s and hold times th of 10 and 100 s. The predicted cyclic fatigue lifetimes are plotted in Fig. 9 together with the experimental results obtained in steam. The results of fatigue tests from prior work [44,45] are also shown in Fig. 9 for comparison. It is seen that the predicted cyclic lifetimes are in excellent agreement with the experi￾mental results obtained in fatigue tests with the trapezoidal waveform and with the triangular waveform at 0.1 Hz, indicating that there is no apparent cyclic effect on the crack growth rate and that the slow crack growth is the dominant failure mechanism. Conversely, predictions M.B. Ruggles-Wrenn et al. / International Journal of Fatigue 30 (2008) 502–516 509