Journal of the American Ceramic Society--Parthasarathy and Kerans Vol. 80. No 8 L.5 .0I0U20.D30 Rudius fum) Roughness Anplitude (um) Fig 9. In the Nicalon/MAS-L system, the effect of fiber radius or Fig. 7. Fiber strength degradation during processing ca the pullout length has been measured experimentally. Dependence or debond lengths significantly, as shown in this plot for the fiber radius is shown, compared with the prediction of the model ystem. Effect of the maximum fiber stress(strength) is obs (debond length), for two situations: one where the roughness param- ge at lower roughness amplitudes(as-received Nicalon eters are held constant, and another where they scale with the fiber nificant even at a roughness amplitude of 0.03 um(the radius. Predictions using the constant roughness parameters match the reported roughness amplitude measured indirectly from composite be. trend of experimental data better. See text for possible implications. havior). model is obtained by using a piecewise linear fit, as indicated ralue thus extracted will have a tendency to be rather high. in Fig. 11(a). However, this will involve multiple parameters From Fig. 10(c), even a very high matrix cracking strain would (rather than one)to define the debond length-fiber stress rela- yield a T value of -60 MPa at a fiber stress of 500 MPa tionship (composite stress of 200 MPa); this is significantly higher than what would be obtained using fiber pushout tests in the sliding Ld=a。+m1or(foro。<0<o1) regime(Fig. 10(b). It is evident that tests that sample different portions of the curve will give different values for T in any Ld=m2(r-01)+m11+oa(foro1<σ<02) constant-T approximation Ld=m3(0-02)+m2(02-01)+m11+。 It is clear from Figs. 10(a(c) that it is almost impossible to (fora>2)(14) define a unique value of T, as determined from experimental tests; this is consistent with the data of Eldridge et al, 38 which Although not especially elegant, it allows a reasonable com- are shown in Fig. 5(b). However, Fig. 10(c)also shows that it romise between accuracy and computational tractability F toler be possible to choose a constant T value that will yield a ure I(b)shows how a piecewise linear fit also may be used to describe frictional sliding of completely debonded fiber/matrix a range of fiber stress of interest, say 0-2 GPa. If such an interfaces. In this plot, the slip length over which the fiber approach were to be used, a protocol for choosing an effective stress is transferred to the matrix is plotted against fiber stress T value would be required. The best choice for T would be one that predicts the composite behavior well. The model of Cur- is equal to zero tin, for composite stress-strain behavior was used to evalu The above-mentioned approximation was examined for its ate the choice of an effective T valu utility in predicting the composite behavior. The relationship between slip length and fiber stress can be used to estimate the (5) Composite Stress-Strain Behavior tensile stress-strain behavior of a composite using the analysis To use the analysis of Curtin, one needs a relationship be- of Curtin. ,32 This analysis focuses on the composite behavior tween the fiber stress and the debond length or slip length. We after matrix cracking with the fiber/matrix interface completely suggest that a good approximation to the progressive roughness debonded. The matrix contribution is neglected, so the curve is roughness 1.2 100- 08 0+R2 00010020030.04 00001000200030.004 Ruugluness Amplitude (un) Roughness Amplitude/Fiber radius Fig 8. Fiber radius has a large effect on the debond length(Fig. 8(a)); however, if the period of the roughness is scaled with changes in roughness, and the debond length and roughness amplitude are normalized with respect to the fiber radius, there is a negligible effect of fiber radius(Fig. 8(b)) Thus, if roughness parameters scale with the fiber radius, then there is no effect, which is consistent with intuition.value thus extracted will have a tendency to be rather high. From Fig. 10(c), even a very high matrix cracking strain would yield a
value of ∼60 MPa at a fiber stress of 500 MPa (composite stress of 200 MPa); this is significantly higher than what would be obtained using fiber pushout tests in the sliding regime (Fig. 10(b)). It is evident that tests that sample different portions of the curve will give different values for
in any constant-
approximation. It is clear from Figs. 10(a)–(c) that it is almost impossible to define a unique value of
, as determined from experimental tests; this is consistent with the data of Eldridge et al.,38 which are shown in Fig. 5(b). However, Fig. 10(c) also shows that it may be possible to choose a constant
value that will yield a tolerable fit to the debond length–fiber stress relationship over a range of fiber stress of interest, say 0–2 GPa. If such an approach were to be used, a protocol for choosing an effective
value would be required. The best choice for
would be one that predicts the composite behavior well. The model of Cur￾tin1,32 for composite stress–strain behavior was used to evalu￾ate the choice of an effective
value. (5) Composite Stress–Strain Behavior To use the analysis of Curtin, one needs a relationship be￾tween the fiber stress and the debond length or slip length. We suggest that a good approximation to the progressive roughness model is obtained by using a piecewise linear fit, as indicated in Fig. 11(a). However, this will involve multiple parameters (rather than one) to define the debond length–fiber stress rela￾tionship: Ld = o + m1f (for o <  < 1) Ld = m2
f − 1) + m11 + o
for 1 <  < 2) Ld = m3
f − 2 + m2
2 − 1 + m11 + o (for  > 2) (14) Although not especially elegant, it allows a reasonable com￾promise between accuracy and computational tractability. Fig￾ure 11(b) shows how a piecewise linear fit also may be used to describe frictional sliding of completely debonded fiber/matrix interfaces. In this plot, the slip length over which the fiber stress is transferred to the matrix is plotted against fiber stress. For this case, the slip length in Eq. (14), Ls, replaces Ld and ° is equal to zero. The above-mentioned approximation was examined for its utility in predicting the composite behavior. The relationship between slip length and fiber stress can be used to estimate the tensile stress–strain behavior of a composite using the analysis of Curtin.1,32 This analysis focuses on the composite behavior after matrix cracking with the fiber/matrix interface completely debonded. The matrix contribution is neglected, so the curve is Fig. 8. Fiber radius has a large effect on the debond length (Fig. 8(a)); however, if the period of the roughness is scaled with changes in roughness, and the debond length and roughness amplitude are normalized with respect to the fiber radius, there is a negligible effect of fiber radius (Fig. 8(b)). Thus, if roughness parameters scale with the fiber radius, then there is no effect, which is consistent with intuition. Fig. 9. In the Nicalon/MAS-L system, the effect of fiber radius on the pullout length has been measured experimentally. Dependence on fiber radius is shown, compared with the prediction of the model (debond length), for two situations: one where the roughness param￾eters are held constant, and another where they scale with the fiber radius. Predictions using the constant roughness parameters match the trend of experimental data better. (See text for possible implications.) Fig. 7. Fiber strength degradation during processing can affect the debond lengths significantly, as shown in this plot for the Nicalon/SiC system. Effect of the maximum fiber stress (strength) is observed to be large at lower roughness amplitudes (as-received Nicalon) and is sig￾nificant even at a roughness amplitude of 0.03 m (the maximum reported roughness amplitude measured indirectly from composite be￾havior). 2050 Journal of the American Ceramic Society—Parthasarathy and Kerans Vol. 80, No. 8