HENAGER et al. SUBCRITICAL CRACK GROWTH: PART I 3735 not uniform throughout the thickness of the com- The permanent strain at zero load, ET(o), includes con- posite. We were unable to quantify this finding but tributions from misfit relief and sliding, E6, and per- this would seem to be a consequence of the 2D woven manent plastic strain, eB(O), such that architecture, which results in overlapping 0/90 fiber tows and relatively large volumes of CVI SiC matrix HO=E+ EB() material containing crossover voids 144. We often observe cracking associated with these larger micro- Given that e is recovered on unloading, that eo is structural features. Still, every crack path through approximately equal to es, and that e<ep(o), then these composite materials must be bridged by fibers eP() is essentially equal to EB(), as observed. This in several layers permanent strain, EB() is equated with the damage 43. Accumulated plastic strain zone strain, Epz, calculated in equation (7)and is assumed to be due to fiber creep in the damage zone Our hypothesis also implies that there is a relation- ship between crack growth and the measured perma- and permanent plastic strain is explored in Fig. 7a, nent plastic strain if the crack growth is controlled by where the time-dependent permanent strain(Table 2) fiber creep. We observed that crack propagation is is shown as a function of time-dependent crack length accompanied by permanent specimen deflection, or (Table 3). The relationship is nearly a linear one sug- strain,in proportion to test duration, or crack length, gesting that crack propagation and permanent strai at constant load and temperature Table 2). The per- are related by a common mechanism manent specimen strains are not recovered on Assuming both permanent strain and crack exten- unloading(Fig. 5 and Table 2). We estimate these sion are caused by fiber creep, then these two quan- strains by calculating the strain in the damage zone, tities should reflect similar kinetics to each other and Epz, due to the multiple cracking as follows: to fiber ci ship is explored(Fig. 7b) by plotting crack length and strain as functions of 1152 time and applying the Sherby-Dorn function in a non- where the first bracketed term is the bending formula 0.025 for strain as a function of midpoint displacement, 8n and s is the outer span length. The term(w-ao)cor-002 rects for the notch depth, ao is the initial notch length, a and Epz is then the strain at the notch root. The second 0.015 term corrects for the fact that the strain is assumed tg occur only in the damage zone of width nd, where 3 0.01 Linear Fit the number of cracks (Table 3), and s/2 is the inner span. There is an initial stress concentration due to 0.005 the notch that falls off as the cracks propagate into 0.002 Once the cracks are a distance ao from the blunt notch Crack Length (m) the stress concentration of the notch has decreased from 4.0 to 1. 5 [57] However, this stress concen tration is accounted for by using the experimental dis placement data in Table 2, which reflect crack exten- of the notch. these strains are tabulated in table 2 E using values for nd equal to 1. 24x10-3 m and oo 0.02 Two of the Hi-c tests. where more detailed unloading-reloading tests were performed, show that measured strains at load and after unloading are the 0.001 same. The total specimen strain at load and as a func- 10610 tion of time, E,(O), can be written as the sum of elastic Time(s strain,e, inelastic loading strain due to matrix crack ing and misfit strain relief, es, and time-dependent Fig. 7,(a)Relationship between calculated g strain and plastic) strain, ep(o),as crack length for sectioned CG-C specimens after testing at 73 K(b) Crack length time showing similarities in kinetics and Sherby -Dorn fit (8) (dashed and dotted curves) for CG-C specimens at 1373HENAGER et al.: SUBCRITICAL CRACK GROWTH: PART I 3735 not uniform throughout the thickness of the com￾posite. We were unable to quantify this finding but this would seem to be a consequence of the 2D woven architecture, which results in overlapping 0/90 fiber tows and relatively large volumes of CVI SiC matrix material containing crossover voids [44]. We often observe cracking associated with these larger micro￾structural features. Still, every crack path through these composite materials must be bridged by fibers in several layers. 4.3. Accumulated plastic strain Our hypothesis also implies that there is a relation￾ship between crack growth and the measured perma￾nent plastic strain if the crack growth is controlled by fiber creep. We observed that crack propagation is accompanied by permanent specimen deflection, or strain, in proportion to test duration, or crack length, at constant load and temperature (Table 2). The per￾manent specimen strains are not recovered on unloading (Fig. 5 and Table 2). We estimate these strains by calculating the strain in the damage zone, eDZ, due to the multiple cracking as follows: eDZ  48(w
a0)dmp 11s2 
s 2nd¯ (7) where the first bracketed term is the bending formula for strain as a function of midpoint displacement, dmp, and s is the outer span length. The term (w
a0) cor￾rects for the notch depth, a0 is the initial notch length, and eDZ is then the strain at the notch root. The second term corrects for the fact that the strain is assumed to occur only in the damage zone of width nd¯, where d¯ is the mean crack spacing in the damage zone, n is the number of cracks (Table 3), and s/2 is the inner span. There is an initial stress concentration due to the notch that falls off as the cracks propagate into the uniform bending field of the SENB specimen. Once the cracks are a distance a0 from the blunt notch the stress concentration of the notch has decreased from 4.0 to 1.5 [57]. However, this stress concen￾tration is accounted for by using the experimental dis￾placement data in Table 2, which reflect crack exten￾sion and subsequent SENB deflection in the presence of the notch. These strains are tabulated in Table 2 using values for nd¯ equal to 1.24×10
3 m and 1.45×10
3 m for CG-C and Hi-C materials, respect￾ively. Two of the Hi-C tests, where more detailed unloading–reloading tests were performed, show that measured strains at load and after unloading are the same. The total specimen strain at load and as a func￾tion of time, eT(t), can be written as the sum of elastic strain, ee , inelastic loading strain due to matrix crack￾ing and misfit strain relief, es , and time-dependent (plastic) strain, ep (t), as eT(t) ee  es  ep (t) (8) The permanent strain at zero load, e0 T(t), includes con￾tributions from misfit relief and sliding, es 0, and per￾manent plastic strain, ep 0(t), such that e0 T(t) es 0  ep 0(t) (9) Given that ee is recovered on unloading, that es 0 is approximately equal to es , and that es
ep (t), then ep (t) is essentially equal to ep 0(t), as observed. This permanent strain, ep 0(t) is equated with the damage zone strain, eDZ, calculated in equation (7) and is assumed to be due to fiber creep in the damage zone only. The relationship between measured crack length and permanent plastic strain is explored in Fig. 7a, where the time-dependent permanent strain (Table 2) is shown as a function of time-dependent crack length (Table 3). The relationship is nearly a linear one sug￾gesting that crack propagation and permanent strain are related by a common mechanism. Assuming both permanent strain and crack exten￾sion are caused by fiber creep, then these two quan￾tities should reflect similar kinetics to each other and to fiber creep. This relationship is explored (Fig. 7b) by plotting crack length and strain as functions of time and applying the Sherby–Dorn function in a non￾Fig. 7. (a) Relationship between calculated bending strain and crack length for sectioned CG-C specimens after testing at 1373 K. (b) Crack length and bending strain as functions of time showing similarities in kinetics and Sherby–Dorn fits (dashed and dotted curves) for CG-C specimens at 1373 K.