August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2051 ssentially that of an already microcracked composite--that is, 0.5L the stress-strain behavior during a test run after an initial load (for o<ou ing to the strain that is required for complete matrix micro- cracking. In this state, the composite is loaded with no notable 0 features in the stress-strain curve until it begins to bend over +1(foro1<<o2) s a result of fiber failures, until the ultimate strength is at tained. Thereafter, the remaining intact fibers continue to fail as the composite stress decreases. As per Curtin's analysis, the 0.5L-m101-m2{(a2-01) remote applied stress, oapp, is related to the stress on unbroken fibers, U, and the stress on broken(and sliding-out) fibers, o, (fora>2)(17 rough the volume fraction, f, of fibers and the fraction of roken fibers, q A fiber strength g. of 2 GPa with a Weibull modulus m of 4, over a reference gauge length Lo of 25.4 mm, was used for (-q) go (15) the calculations. In Fig. 12(a), the stress-strain curves that were obtained using the piecewise linear fit are shown for a smooth nd g is given by fiber and one with a roughness amplitude h of 20 nm. The plots include debond lengths at the ultimate stress. in units of fiber diameters and millimeters. The fiber roughness enhances the ultimate strength of the composite, provided that the roughne does not affect the fiber strength(e. g, through stress raisers where o is the fiber strength that is measured over a reference This might rationalize the observations of Naslain et al. 19 who gauge length Lo and m is the Weibull modulus. In Eq (16), Ls found that the composite strength increased as the shear is the distance over which slip occurs at a fiber stress of o, as strength of the interface increased, which, in turn, could be shown in Fig. 11(b)and described by Eq (14)(with o =0 attributed to microcracking of the interphase which causes According to Curtin, 32 the average stress ob(supported by the increased interfacial roughness. Thus, roughness may have a broken fibers)is the fiber stress for a slip length of 0.5Ls. Thus, beneficial effect if it does not affect the fiber properties. .4 NicxlunSsiC-Siuule Fiber Puslluut 0.8 AicaloniSIC-Single Fiter sliding 0.3 06 (Peak load) shang with 0.2 0.4+2m Progressiv E0.1 (b=20;m 0.2 G +(Sliding Load)! o2 0 43 01000200030004000 1500300045006000 Fiber Stress(MPa) fiber stress(Mfa) 0.4 Nicalon'SK--Multinhtr tension 0.3 Progressive τ(MPa) 0.2 (Matrix 0. 4 0 1000200030XX Fiber stress(MPa) ig. 10. Constant shear stress(T) approximation is evaluated for its validity of use as a unique parameter, the value obtained for T using a fiber shout test is dependent on whether the peak stress(shown in Fig. 10(a)or plateau stress(frictional sliding, shown in Fig. 10(b)) is used. From Figs. 10(a)and (b), it also is observed that the T value thus extracted is dependent on the specimen thickness (equal to the debond/sliding length) From Fig. 10(c), it is observed that the T value obtained using the matrix crack spacing during a composite tensile test also is a function of the stress at which the crack spacing (debond length x 2)is measured. Because the fiber stress during matrix cracking is typically low, the extracted T value will have a tendency to be high(T= 60 MPa at 500 MPa, a composite stress of 200 MPa)essentially that of an already microcracked composite—that is, the stress–strain behavior during a test run after an initial load￾ing to the strain that is required for complete matrix micro￾cracking. In this state, the composite is loaded with no notable features in the stress–strain curve until it begins to bend over, as a result of fiber failures, until the ultimate strength is at￾tained. Thereafter, the remaining intact fibers continue to fail as the composite stress decreases. As per Curtin’s analysis, the remote applied stress, app, is related to the stress on unbroken fibers, f , and the stress on broken (and sliding-out) fibers, b, through the volume fraction, f, of fibers and the fraction of broken fibers, q: app = f
1 − qf + qb] (15) and q is given by q = 2Ls Lo
f o m (16) where o is the fiber strength that is measured over a reference gauge length Lo and m is the Weibull modulus. In Eq. (16), Ls is the distance over which slip occurs at a fiber stress of f , as shown in Fig. 11(b) and described by Eq. (14) (with o 0). According to Curtin,1,32 the average stress b (supported by the broken fibers) is the fiber stress for a slip length of 0.5Ls. Thus, b = 0.5Ls m1
for  < 1) b =
0.5Ls − m11 m2 1
for 1 <  < 2) b = 0.5Ls − m11 − m2
2 − 1 m3  + 2 (for  > 2) (17) A fiber strength o of 2 GPa with a Weibull modulus m of 4, over a reference gauge length Lo of 25.4 mm, was used for the calculations. In Fig. 12(a), the stress–strain curves that were obtained using the piecewise linear fit are shown for a smooth fiber and one with a roughness amplitude h of 20 nm. The plots include debond lengths at the ultimate stress, in units of fiber diameters and millimeters. The fiber roughness enhances the ultimate strength of the composite, provided that the roughness does not affect the fiber strength (e.g., through stress raisers). This might rationalize the observations of Naslain et al.,19 who found that the composite strength increased as the shear strength of the interface increased, which, in turn, could be attributed to microcracking of the interphase, which causes increased interfacial roughness. Thus, roughness may have a beneficial effect if it does not affect the fiber properties. Fig. 10. Constant shear stress (
) approximation is evaluated for its validity of use as a unique parameter; the value obtained for
using a fiber pushout test is dependent on whether the peak stress (shown in Fig. 10(a)) or plateau stress (frictional sliding, shown in Fig. 10(b)) is used. From Figs. 10(a) and (b), it also is observed that the
value thus extracted is dependent on the specimen thickness (equal to the debond/sliding length). From Fig. 10(c), it is observed that the
value obtained using the matrix crack spacing during a composite tensile test also is a function of the stress at which the crack spacing (∼debond length × 2) is measured. Because the fiber stress during matrix cracking is typically low, the extracted
value will have a tendency to be high (
≈ 60 MPa at 500 MPa, a composite stress of 200 MPa). August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2051