CERAMIC fibrous monoliths (FMs) have emerged as a potential

Transcription

1 journal J. Am. Ceram. Soc., 84 [2] (2001) In-Plane Fracture Resistance of a Crossply Fibrous Monolith John C. McNulty,*, Matthew R. Begley, and Frank W. Zok*, Materials Department, University of California, Santa Barbara, California 93106; and Mechanical Engineering Department, University of Connecticut, Storrs, Connecticut The in-plane fracture resistance of a crossply Si 3 N 4 /BN fibrous monolith in the 0 /90 and 45 orientations is examined through tests on notched flexure specimens. The measurements and observations demonstrate the importance of fiber pullout following fiber fracture. The mechanical response is modeled using a crack-bridging approach. Two complementary approaches to evaluating the bridging law are developed: one based on a micromechanical model of fiber pullout and the other based on the load versus crack mouth opening displacement response of the flexure specimens following fracture of all fibers. Both approaches indicate that the bridging law follows an exponential form, characterized by a bridging strength and an effective pullout length. An assessment of the bridging model is made through comparisons of simulations of the load displacement response with those measured experimentally. I. Introduction CERAMIC fibrous monoliths (FMs) have emerged as a potential low-cost, damage-tolerant material for thermostructural applications. The majority of systems fabricated and tested to date comprise Si 3 N 4 fibers and a BN interphase. 1 7 The damage tolerance is derived in part from the deflection of cracks from the Si 3 N 4 fibers into the surrounding BN. These effects are particularly pronounced under out-of-plane bending of laminated panels wherein the BN interphase provides a contiguous path for delamination, without significant interference from the fibers. In this configuration, the material response closely resembles that of a weakly bonded layered composite in the sense that the load displacement curve exhibits a characteristic saw-toothed shape with load drops occurring as each successive layer cracks and the specimen compliance increases. 6 The present article focuses on the in-plane fracture resistance of a crossply FM. The behavior is expected to differ from that obtained during out-of-plane bending in two ways. (i) Following cracking of the fibers aligned with the loading direction, delamination is resisted by the fibers in the transverse orientation. (ii) The response is anisotropic, varying with the direction of load with respect to the fiber orientation; in contrast, the delamination behavior for out-of-plane bending is relatively insensitive to the fiber architecture, because a contiguous delamination path exists for all orientations. R. Naslain contributing editor Manuscript No Received November 15, 1999; approved September 14, Supported by the U.S. Defense Advanced Research Projects Agency, through a U.S. Department of Energy Interagency Agreement and a subcontract from Argonne National Laboratory (Prime Contract W Eng-98) to UCSB (Contract No ). *Member, American Ceramic Society. University of California. University of Connecticut. The study includes experimental measurements of the fracture resistance of notched flexure specimens in the 0 /90 and 45 orientations as well as simulations based on a crack-bridging fracture model. The parameters characterizing the fracture resistance are the intrinsic toughness, K o, and the bridging-traction law, b (u), where b is the bridging stress and u is the crack-opening displacement. Two complementary approaches to evaluating the bridging law are developed: one based on a micromechanical model of fiber pullout and the other based on the load displacement response of the flexure specimens following fracture of all fibers. Both approaches indicate that the traction law follows an exponential form, characterized by a bridging strength and an effective pullout length. An assessment of the crack-bridging model is made through comparisons of simulations of the load versus crack mouth opening response with those measured experimentally. II. Materials and Experimental Procedures Tests were performed on a Si 3 N 4 /BN FM (Advanced Ceramics Research, Tucson, AZ). The fabrication methods are presented elsewhere. 6 The fibers are in a 0 /90 crossply (Fig. 1). Each fiber has an approximately rectangular cross section, with dimensions t m and t 2 85 m. A contiguous BN interphase, 17 m thick, is present between the fibers. The fiber cross section exhibits variations along the fiber length, resulting from the indentation of the adjacent transverse fibers during pressing in the green state. The indentation also leads to the formation of cusps at many of the triple junctions between longitudinal and transverse fibers. Examples of such cusps are highlighted in Fig. 1. The frequency of the cusps appears to be somewhat irregular, being absent at many of the triple junctions. These geometric features have ramifications in the extent of pullout following fiber fracture, as described below. For comparison, tests also were performed on a monolithic Si 3 N 4 that was fabricated by the same process. The fracture resistance of the FMs was measured using edgenotched three-point flexure tests, oriented to produce in-plane cracking. Typical specimen dimensions were width (W) of 9.9 mm, notch length (a o ) of 5.0 mm, notch width () of 0.4 mm, thickness (B) of 3.4 mm, and span (S) of 40 mm. To assess the degree of anisotropy in fracture resistance, specimens were cut and tested in the 0 /90 and 45 orientations. One side of each specimen was polished prior to testing to facilitate in situ observations. The specimens were instrumented with a gauge to measure crack mouth opening displacement (CMOD) and a gauge to measure load line displacement. Two steel knife edges, 1.4 mm thick, were bonded onto the tensile face adjacent to the notch to facilitate attachment of the CMOD gauge. The tests were performed in a displacement-controlled mode at a rate of 2 m/min. The nominal fracture resistance was calculated from the applied load, the total crack length (taken as the distance from the tensile face to the edge of the last broken fiber), and standard solutions for stress intensity factors in edge-cracked specimens (see, for e.g., Tada et al. 8 ). Furthermore, the work of fracture (WOF), obtained from the area under the load displacement curve, was used as a measure of steady-state toughness. 9,10 In some cases, the specimens were subjected to unload reload excursions at several points in the test for the purpose of evaluating the change in specimen compliance. 367

2 368 Journal of the American Ceramic Society McNulty et al. Vol. 84, No. 2 III. Fracture Behavior Fig. 1. Scanning electron micrograph of the Si 3 N 4 /BN fibrous monolith (backscatter electron imaging mode). Arrow labeled A shows an example of the cusps that are formed at the triple junctions between longitudinal and transverse fibers; arrow labeled B shows a triple point at which a cusp had not formed. Straight (unnotched) specimens of the FMs in the 0 /90 and 45 orientations were tested in four-point flexure to ascertain the Young s moduli and the unnotched strengths. The specimens were instrumented with strain gauges (2 mm 6 mm) on the tensile and compressive faces. The fracture resistance of the monolithic Si 3 N 4 was measured using chevron-notched flexure specimens (shown in the inset of Fig. 2). The fracture toughness was calculated from the load maximum and the solutions for stress-intensity factors developed by He and Evans. 11 Additionally, attempts were made to measure the WOF from the area under the load displacement curve. 9,10 Following testing, the fracture surfaces were examined in a scanning electron microscope (SEM). Moreover, for the 0 /90 notched specimens, the pullout lengths of the fibers were measured from a series of micrographs taken at a tilt angle of 45. Four separate measurements were made on each fiber: one on each of the two broad faces and one on each of the two narrow faces. Essentially all of the fibers on the fracture surface were included, for a total of 1000 measurements per specimen. The measurements were subsequently used in conjunction with a shear lag analysis to obtain the form of the bridging traction law. Similar measurements could not be made on the 45 specimens because of the greater tortuosity of the fracture surface. Additional information on the pullout behavior was obtained by sectioning and viewing fractured specimens along a plane perpendicular to the fracture surface. The results of the unnotched-flexure tests are summarized in Table I. The results indicate that the elastic response is nearly isotropic (within 2%). Furthermore, the strengths in the two orientations are comparable to one another, contrary to the behavior of conventional fiber-reinforced ceramic composites with crossply architectures. Typical results for the variations in the nominal bending stress, nom, with CMOD for the notched FM specimens, are plotted in Fig. 3(a). The curves exhibit an initial rapid rise, accompanied by slight nonlinear deformation; a stress maximum, comparable in value to the unnotched flexural strength; and a subsequent gradual decline. A particularly notable feature is the absence of any precipitous load drops in the postload maximum regime, indicative of stable crack growth. An implication is that the WOF provides a valid estimate of the steady-state toughness, G ss. 9,10 The results of such calculations are G ss J/m 2 in the 0 /90 orientation, and G ss J/m 2 in the 45 orientation (the ranges being from two tests in each orientation). Also shown in Fig. 3(a) are predictions based on linear-elastic fracture mechanics (LEFM) 8 using several assumed values of the fracture toughness, K c. The lack of correlation between the predictions and the measurements demonstrates the shortcomings of LEFM in describing the fracture resistance of the FM. Furthermore, the measured curves diminish less rapidly than the predicted curves, indicating a progressively increasing fracture resistance with increasing crack growth. The corresponding measurements and LEFM predictions of the variation in specimen compliance, C, with CMOD are plotted in Fig. 3(b). Again the correlation between experiment and theory is poor. A notable feature of the experimental measurements is the attainment of a nearly constant compliance at moderate levels of CMOD (0.2 mm), at a value, C s, of about twice the initial value, C o. In contrast, the LEFM predictions yield changes of more than an order of magnitude. The response of the monolithic Si 3 N 4 is plotted in Fig. 2. The fracture toughness obtained from the load maxima in three tests spans the range MPam 1/2. In all cases, the initial stage of crack growth occurs unstably and is accompanied by a load drop of 50% from the maximum. The crack subsequently grows stably, as manifest in a smooth load deflection curve. Estimates of the steady-state fracture resistance ( MPam 1/2 ), derived from the apparent work of fracture, are 50% higher than those based on the load maxima. The discrepancy is due in part to the initial burst of unstable crack growth, a feature that invariably leads to an overestimate of the true WOF. 9 The discrepancy may also be due to a rising fracture resistance curve. A lower-bound estimate of the steady-state toughness is obtained from the work done only during stable crack growth. This calculation is based on the area under the load displacement curve but without the top triangular portion preceding the load drop. The resulting estimates are MPam 1/2, which are slightly higher than the initiation toughness ( MPam 1/2 ). The inference is that the Si 3 N 4 exhibits a slightly rising fracture-resistance curve. The maximum increase in resistance beyond initiation is estimated to be MPam 1/2. This increase is considerably smaller in magnitude than those in the FMs, as demonstrated below. A series of micrographs showing the evolution of cracks in a 0 /90 FM specimen is presented in Fig. 4. The fiber at the notch tip cracks directly ahead of the notch tip (along the plane of maximum tension) at an apparent initiation stress intensity, K i 4.8 MPam 1/2 (Fig. 4(a)). This value is somewhat higher than the Table I. Summary of Unnotched In-Plane Flexural Properties Architecture Unnotched flexural strength, max (MPa) Young s modulus, E (GPa) Fig. 2. Response of chevron-notched monolithic Si 3 N 4 specimens. Notch geometry is shown in the inset, with dimensions in millimeters. 0 /90 186, 230, , 236,

3 February 2001 In-Plane Fracture Resistance of a Crossply Fibrous Monolith 369 Fig. 3. Response of edge-notched flexure specimens. Point A in (a) indicates the CMOD at which all fibers on the surface had broken; the subsequent response is dictated by pullout of the broken fibers. initiation toughness of the monolithic Si 3 N 4 (K o 3 MPam 1/2 ), probably because of the finite notch-root radius. The crack arrests at the interface with the neighboring fiber and causes cracking within the BN parallel to the loading axis. (Similar observations of cracking through the BN, rather than along the Si 3 N 4 /BN interface, have been made previously. 6 ) At a substantially higher stress intensity, K ren 10 MPam 1/2, the next fiber cracks, at a location that is not coplanar with the initial crack. The stress intensity required for renucleation is roughly equal to that associated with the load maximum (11 MPam 1/2 ). The high stress intensity needed for crack renucleation demonstrates the effectiveness of the BN in deflecting the first crack and hence mitigating the stress concentration. Furthermore, the offset between cracks is indicative of the intrinsic variability in the fiber strength. The sequence of fiber cracking, crack arrest, and crack renucleation is repeated until all fibers are broken. The latter event occurs at u o 0.2 mm (Fig. 4(d)), roughly coincident with the point at which the compliance reaches its saturation value, C s C o /2. The lack of significant change in compliance following fracture of all fibers is attributed to the interlocking between the broken fibers. The interlocking also gives rise to a finite load-bearing capacity, at 15% of the peak nominal stress (point A in Fig. 3(a)). Subsequently, the fibers slide past one another, thereby reducing the contact area between adjacent fibers and the applied load for further deformation. The latter portion of the stress CMOD curve provides a direct measure of the pullout response of the broken fibers and can be utilized for determining the relevant bridging law, as detailed below. The corresponding fracture resistance curve for the 0 /90 FM is plotted in Fig. 5. Cracking initiates at a level dictated essentially by the toughness of the Si 3 N 4 itself, because there is initially no mechanism for crack shielding or blunting. The resistance then increases discontinuously to a level of 10 MPam 1/2, a consequence of crack deflection through the BN. It subsequently Fig. 4. A series of optical micrographs taken in situ on a 0 /90 specimen: (a) crack initiation in the fiber at the notch tip at K i 4.8 MPam 1/2 and subsequent crack arrest; (b) renucleation of the crack in the next fiber, at K ren 10 MPam 1/2, followed again by crack arrest; and (c) and (d) continued progression of similar events until all fibers are broken. Fig. 5. Nominal fracture resistance curves for the two 0 /90 FM specimens. increases approximately linearly with crack length to values of K R MPam 1/2 at a crack extension a 3 mm. However, because the bridging zone in the crack wake constitutes a significant fraction of the entire crack length (350% as a/w31), the resistance curve represents a structural response rather than an intrinsic material property. Similar effects associated with large-scale

4 370 Journal of the American Ceramic Society McNulty et al. Vol. 84, No. 2 bridging have been observed in fiber-reinforced ceramic composites 12,13 and metal-toughened ceramics. 14 Notwithstanding this behavior, the results demonstrate that substantial toughening is derived from pullout of the broken fibers in the crack wake. A similar conclusion is obtained from the WOF measurements, which yield estimates of the steady-state fracture resistance of K ss MPam 1/2 for the 0 /90 FM. These values are considerably greater than those that can be rationalized solely on the basis of the toughness of the fibers and crack deflection through the BN. Attempts to make analogous measurements on the 45 FMs were unsuccessful. On the surface, crack growth occurs predominantly through the BN interphase along an irregular zigzag path, without a well-defined crack tip. At the same time, fiber fracture occurs progressively in the bulk of the specimen, especially beyond the load maximum, as evidenced by frequent audible pops during the test. Despite the problems in monitoring crack growth and measuring fracture resistance, it is demonstrated later that the form of the crack-bridging fracture model developed for the 0 /90 FM can be adapted to the 45 FM, through appropriate selection of the parameters characterizing the bridging law and the intrinsic toughness. Low-magnification images of the fracture surfaces are shown in Fig. 6. In the 0 /90 orientation, most of the axial (0 ) fibers are seen to have broken on different planes from their neighbors, consistent with the in situ observations on the side surfaces (Fig. 4). In contrast, failure in the transverse (90 ) laminae occurs through the BN, leaving the adjacent fibers intact on the two respective fracture surfaces. In the 45 orientation, all fibers break at an angle of 45 to the loading direction. The inclination of the fracture planes suggests that the fibers are subject to bending along the fiber axis. Furthermore, the extent of pullout appears greater than that in the 0 /90 orientation. The relative amounts of pullout are consistent with the differences in the WOF: 5600 J/m 2 for the 45 FM and 2700 J/m 2 for the 0 /90 FM. Transverse cross sections through the fracture surfaces reveal several additional features (Fig. 7). In the 0 /90 FM, the pullout lengths of the axial fibers are of the same order as the fiber width (t mm). This correlation can be attributed to the periodic variations in the fiber cross section and the cusps that exist at the triple junctions between longitudinal and transverse fibers. Because of the geometric registry between the cusps and the neighboring transverse fibers, debonding between fibers is limited to distances comparable to the cusp spacing. Moreover, the variations in the fiber thickness in the segments between the cusps are expected to lead to premature disengagement of the broken fibers from their corresponding sockets. That is, complete loss of contact may occur at a crack-opening displacement that is less than the geometric pullout length. Indeed, such effects are confirmed by the subsequent measurements and analyses of the pullout response. In the 45 FM (Fig. 7(b)), fracture follows a more tortuous path. Pullout occurs between individual fibers and along planes composed of multiple fibers. Because the sliding direction is inclined at 45 to the fiber axes, the variation in fiber thickness is not expected to promote fiber disengagement to the extent it does in the 0 /90 FM. IV. Application of Bridging Concepts (1) Preliminaries The preceding results (especially those for the 0 /90 FM) suggest that the fracture resistance can be modeled using established crack-bridging concepts. In the standard bridging model, the Fig. 6. Fracture surfaces of (a) 0 /90 and (b) 45 notched specimens. Fig. 7. Transverse cross sections of fracture surfaces of (a) 0 /90 and (b) 45 notched specimens. Sections were taken approximately halfway between the notch tip and the compressive surface.

5 February 2001 In-Plane Fracture Resistance of a Crossply Fibrous Monolith 371 bridging zone is treated as a planar crack with a continuous distribution of closing tractions. Furthermore, the fracture resistance K R is partitioned into two components: an intrinsic toughness, K o, defined as the crack tip stress intensity needed for continued crack growth, and a bridging contribution, K b, derived from the crack-closing tractions. In the present context, K o is taken to be that associated with the first crack renucleation event, 10 MPam 1/2 for the 0 /90 FM. The fundamental property dictating K b is the bridging law, b (u). K R is calculated using appropriate weight functions in conjunction with the bridging stresses, ensuring that consistency is maintained between the resulting crackopening displacement profile, the bridging-stress distribution, and the crack-tip stress intensity. The key unknown relationship needed for simulating the notched response of the FMs is the bridging law. The following sections describe two complementary approaches for determining this law: one based on a micromechanical model of fiber pullout and the other based on the load CMOD response of FM specimens following fracture of all fibers. Both approaches indicate that the bridging law follows an exponential form, decreasing with increasing crack-opening displacement. This exponential law is then used in simulating the load CMOD response during cracking of the fibers. Fig. 8. Measured pullout length distributions of the 0 /90 FMs. (2) A Micromechanical Model of Fiber Pullout Pullout of the axial fibers in the 0 /90 FMs has close analogies with fiber pullout in fiber-reinforced ceramic composites, with two notable exceptions. First, there is no matrix crack in the FMs from which pullout lengths can be defined. Instead, the axial fibers slide past either neighboring axial fibers within the same lamina or transverse fibers in the adjacent laminae. Second, the pullout lengths on the four faces of the fiber are generally different from one another, each being dictated by the failure location of the adjacent fiber. Accounting for these features, the bridging law can be derived following a standard shear-lag approach. Here, the fibers are assumed to be coupled to one another through a constant sliding stress,, and the fiber cross section taken to be rectangular, with dimensions t 1 and t 2. The resulting law is b t 1 t 2 t 1 t 2 t 1 t 1 t 2 u t 2 gh 1 h 1 u dh 1 gh 2t 1 t 2 h 2 u dh 2 u 2 (1) where h 1 is the pullout length measured along the broad face of the fiber (of width t 1 ), h 2 is the pullout length measured along the narrow face (of width t 2 ), and g(h 1 ) and g(h 2 ) are the probability densities for pullout lengths of h 1 and h 2, respectively. (The factor of 1/2 in front of the second term in brackets arises because the frictional force between adjacent sliding fibers within the same lamina is shared by each of the two fibers, contributing half of the total to each of the two respective fibers.) The distributions g(h 1 ) and g(h 2 ) were obtained from measurements of pullout lengths on two broken specimens of the 0 /90 FM. The measurements are summarized in Fig. 8 in the form of cumulative probability functions Gh o h gh dh These measurements were then combined with Eq. (1) and the average fiber dimensions (t m and t 2 85 m) to construct the bridging laws. The results are plotted in Fig. 9 in terms of a normalized bridging stress, b /. To facilitate further calculations with this bridging law, the collection of terms in brackets in Eq. (1) is assumed to follow an exponential form, given by h* exp(u/h*), where h* is an effective average pullout length. This form is suggested by the Fig. 9. Normalized bridging law constructed from the pullout measurements and Eq. (1). Solid lines represent fits of the exponential approximation (Eq. (2)). present data as well as pullout data for fiber-reinforced ceramic composites. 15,16 With this assumption made, the traction law becomes b 2 h* t exp u (2) h* where t is the geometric mean of the fiber dimensions, defined by 1/t [(1/t 1 ) (1/t 2 )]/2 (t 0.13 mm). Figure 9 shows the fit of Eq. (2) to the curves obtained directly from the measured pullout distributions and Eq. (1). Excellent correlations are obtained using effective pullout lengths, h*, of 130 and 190 m for the two tests. These correlations indicate that the bridging law does indeed follow an exponential form. An estimate of the sliding stress can be obtained by combining Eq. (2) with the fracture energy contribution associated with pullout. For this purpose, the total fracture energy, G ss, is partitioned into two components: G ss G 0 G po where G o is the intrinsic toughness, K 2 o /E, and G po is the energy due to pullout. The latter quantity is related to the bridging law through the expression G po 0 b u du (3) Upon combining Eqs. (2) and (3), the sliding stress is found to be tg po 2h* 2 (4)

6 372 Journal of the American Ceramic Society McNulty et al. Vol. 84, No. 2 Using this result and the relevant values for t, G po, and h*, the inferred sliding stresses are 5 and 7 MPa for the two tests performed on the 0 /90 FM. By comparison, the values obtained from pushout measurements on a similar unidirectional FM are 25 7 MPa for sliding distances 0.7 mm. 17 The discrepancy between these values is mainly due to the premature disengagement of the nonuniform fibers and the resulting overestimation of the contact area obtained from the measured pullout lengths. That is, the pullout length measured on the fracture surfaces is greater than the effective value relevant to Eq. (4), and, hence, the inferred sliding stress is anomalously low. Further evidence of this effect is presented in the subsequent section. (3) Analysis of the Load CMOD Response in the Postcracking Regime An alternative approach to evaluating the bridging law has been developed also. It is based on an analysis of the load CMOD response following complete fiber fracture. In this regime, the response is dictated exclusively by pullout. This approach is applicable to the 0 /90 and 45 FMs and obviates the need for measuring pullout lengths on the fracture surface. Furthermore, it provides a direct measure of the relevant bridging parameters. The analysis is based on beam theory. The specimen is treated as two rigid beams that are linked by a thin plastic hinge. The crack-opening displacement profile within the hinge is assumed to vary linearly across the specimen width, being zero at the compressive face and at a maximum, u u 0, at the location of the knife edges. The profile is thus described by u x u 0x (5) W where x is the distance along the crack from the compressive face and W is the effective beam width (the actual width, W, plus the thickness of the knife edges, 1.4 mm). Across the hinge, the traction distribution is assumed to follow an exponential form, written phenomenologically as b 0 exp u (6) bulk had also fractured). The fits are plotted in Fig. 10(a). In both cases, the form of the predicted curve follows the experimental data extremely well, providing additional support for the exponential softening traction law. The values of the characteristic length inferred from the fits of the two data sets are virtually identical: 70 m. This value is about half of that obtained from the pullout measurements: h* m. The discrepancy is consistent with the notion that the fibers become disengaged prematurely because of their nonuniform cross section. The corresponding strengths from the fits are slightly different from one another, 0 23 and 34 MPa, probably due to material variations. Recognizing the equivalence of Eqs. (2) and (6), the effective sliding stress can be obtained from the characteristic strength and the effective pullout length through 0t (10) 2 The inferred sliding stresses are 23 and 33 MPa: comparable to the ones measured by pushout on the unidirectional material (25 MPa) and considerably higher than the values obtained using the measured pullout lengths and the steady-state fracture energy (Section IV(2)). The former correlation suggests that the present approach to determining the bridging parameters is more reliable than the one based on pullout measurements. Similar fits were performed of the load CMOD response of the 45 FM. Because the CMOD at which fiber fracture is complete could not be determined experimentally, the fits were performed using a variety of CMOD cutoff values, ranging from 0.2 to 0.5 mm. In all cases, the curves fit the experimental data exceedingly well. More importantly, the values of 0 and inferred from the fits are insensitive to the selection of the cutoff CMOD. The relevant values from the two tests are m and MPa. The fact that the inferred lengths are greater in the where 0 is a characteristic strength (related to the sliding stress) and a characteristic length (related to the pullout length). A tacit assumption stemming from the displacement profile and the exponential bridging law is that the neutral axis exists at, or very near, the compressive face of the specimen. The bending moment is then calculated in the usual way, yielding the result Wa o M 0 x exp u ox dx (7) W B0 Upon integration of Eq. (7), the resulting nominal bending stress, nom 6M/(BW a o ) 2, becomes nom 6 0W 2 2 u 2 o W a o 2 exp u ow a o 1 1 u ow a o W W (8) and the corresponding applied load, P 4M/S, is P 4B 0W Su o 1 1 u ow a o W exp u ow a o W (9) Equation (9) was fitted to the load CMOD data for the 0 /90 FM in the regime CMOD 0.3 mm. (The CMOD cutoff was selected to be slightly higher than that associated with fracture of the last fiber on the surface, 0.2 mm, to ensure that all fibers in the Fig. 10. Fitting of the postcracking load CMOD response using Eq. (9): (a) 0 /90 and (b) 45 notched specimens.

7 February 2001 In-Plane Fracture Resistance of a Crossply Fibrous Monolith FM (by a factor of 2) is consistent with the observations of pullout lengths on the fracture surfaces (Fig. 6) and in the transverse cross sections (Fig. 7). However, because of the complex fracture path in the 45 FM, there is no straightforward relationship between the sliding stress and the characteristic bridging strength, analogous to Eq. (10). Notwithstanding this deficiency, the critical strength can be used directly in the bridging law for simulating the response in the cracking regime, as described below. (4) Load CMOD Response of 0 /90 FM during Cracking The load CMOD response of the 0 /90 FM during fiber cracking was simulated using a crack-bridging model. The integral equation governing the bridging traction distribution supplied by the fibers was derived by summing the crack opening, u a, due to the applied load and crack opening, u b, due to the bridging tractions, and equating the result to the total crack opening, u, using the exponential bridging law given in Eq. (6). The crack-tip stress intensity was then calculated for the bridging traction distribution, the applied load, and the appropriate weight functions. This procedure is described in detail in Ref. 18 and outlined briefly in the Appendix. The crack-tip stress intensity was maintained at a constant value, taken to be that for crack renucleation (11 MPam 1/2 ). The values of the bridging parameters, 0 and, were selected to span a range that includes the values inferred in the preceding section. In one case, the characteristic length was taken to be 70 m, and the characteristic strength was varied from MPa. Additionally, for reasons described below, simulations were performed for the same range of 0 and a characteristic length,, which corresponds to a constant bridging stress. In order to account for the 1.4 mm offset in the CMOD gauge relative to the tensile face of the specimen, the predicted CMOD at the real notch mouth was extrapolated linearly from the tensile face to the gauge location. With this adjustment, the predicted CMOD at the gauge location was typically 15% greater than that at the tensile face. Finite-element simulations of the present specimen geometry indicated that the extrapolation procedure yielded acceptable results. The finite-element results were also used to assess the effect of the finite notch-root radius. The result is an 5% increase in the initial specimen compliance. The latter effects, being rather small and only applicable to the initial linear-elastic response, were neglected in the crack-bridging calculations. For very long cracks (approaching the specimen width), convergence problems were encountered in the simulations, a consequence of the rapidly increasing stress intensity. As a result, the simulations were truncated at a crack length of a/w 0.9. Comparisons of the simulated and measured load CMOD responses are presented in Fig. 11(a). The simulations in the regime prior to complete fiber fracture show broad agreement with the measurements, with two exceptions. First, the measured curves exhibit slight nonlinearity prior to the load maximum, likely a result of fracture of the fibers that had been cut in the notching operation and the ensuing debonding through the BN ahead of the fractured fibers. This nonlinearity is not captured by the simulations because of the assumption that crack growth initiates at a crack-tip stress intensity associated with the crack renucleation in the second row of fibers. Second, beyond the load maximum, the measured curves fall off somewhat more gradually than the simulated ones for values of 0 and that are consistent with the pullout-dominated response (20 40 MPa and 70 m). This discrepancy is due to the debonding that occurs ahead of the crack tip prior to fiber fracture, a feature not incorporated explicitly in the bridging model. The simulations for illustrate the sensitivity of the predictions to the characteristic length. In the regime defined by CMOD, the simulations are highly insensitive to. Because the bridging stress decays exponentially with u/, the bridging stress in this regime is essentially constant along the bridged part of the crack, independent of. This insensitivity precludes the use of comparisons at small CMODs for inferring the characteristic Fig. 11. Comparisons of the load CMOD response obtained experimentally (heavy solid lines) and through the crack bridging model: (a) 0 /90 and (b) 45. Dashed lines represent calculations assuming a constant bridging stress. length. Indeed, changing from 70 m to produces rather modest changes in the simulated response. Conversely, when the CMOD is large, the simulations are sensitive to the selection of. For, a limit load is obtained following complete cracking (a/w 1), given by P lim 2B 0W a o 2 (11) S These limits are also plotted on Fig. 11(a) for comparison with the simulations for 70 m. Comparisons were also made of the fracture resistance curves (Fig. 5). Similar correlations and discrepancies were obtained between the simulations and the measurements. That is, the simulated curves are in broad agreement with the measurements, but tend to underestimate slightly the fracture resistance for small crack extensions. This is consistent with the notion that other dissipative mechanisms operate at the crack tip and enhance fracture resistance. (5) Load CMOD Response of 45 FM during Cracking Similar simulations were performed for the 45 FM during cracking. The characteristic length was selected to be 150 m, and the characteristic strength was again varied over the range of MPa, consistent with the values inferred from the postcracking regime. The intrinsic toughness was taken to be 12.7 MPam 1/2, which corresponds to the stress intensity at the load maximum. Additional simulations were performed for. The results are plotted in Fig. 11(b). Somewhat larger discrepancies exist between experiment and theory, especially in the regime immediately following the load maximum. Here the

8 374 Journal of the American Ceramic Society McNulty et al. Vol. 84, No. 2 predicted loads lie substantially below the measured ones, by as much as a factor of 2. The agreement improves at larger values of CMOD ( mm), using 0 30 MPa and 150 m. The discrepancies are likely due to the presence of other toughening mechanisms, such as crack-front debonding, which are not incorporated in the present model. V. Concluding Remarks The fracture resistance of the Si 3 N 4 /BN FM is characterized by a high intrinsic toughness (10 13 MPam 1/2 for the two architectures) and a subsequent increase in fracture resistance with crack growth. The intrinsic toughness is 3 times that of the monolithic Si 3 N 4. The latter difference is due to the deflection of cracks from the fibers into the surrounding BN and the resulting reduction in the crack-tip stress field. Clearly, this behavior is only relevant in the regime in which the crack length is greater than the fiber dimensions. The rising fracture resistance curve is attributable largely to the pullout of broken fibers in the crack wake, although there is some evidence that other dissipative mechanisms operate at the crack tip and further increase fracture resistance. The load CMOD response and the fracture-resistance curve can be simulated using models based on crack-bridging concepts. The relevant bridging law follows an exponential form, characterized by a strength, 0, and a length,, both of which depend on the fiber architecture and orientation. The characteristic length is controlled by the spacing between cusps on the longitudinal fibers. Furthermore, it is influenced by the periodic variations in the fiber cross section, which allow premature disengagement of fibers during pullout. The characteristic strength is dictated by the sliding stress and the pullout length. The preferred method for determining the bridging law involves analysis of the load CMOD response after all fibers have broken. Using this method, the inferred sliding stress is found to be comparable to that obtained by fiber pushout. In the 45 and 0 /90 orientations, the characteristic bridging strength is 1 order of magnitude smaller than the unnotched strength. One implication is that the pullout process (after fiber cracking) does not contribute substantially to the ultimate strength of the FM, except perhaps in the presence of very long cracks or notches. Nevertheless, it contributes substantially to the fracture energy because of the relatively large characteristic length, m. Furthermore, it is expected to impart a high level of damage tolerance under localized loading conditions, such as that associated with impact by foreign objects. Appendix Mechanics of Fiber Bridging In general, the two contributions of crack-opening displacement are given by u a x 4 a a f E a x, â dâ x u b x 8 Ea o a Hx, â, a b x dâ (A 1) (A 2) where f a (x, â) is the weight function for a point force applied at x and H(x, â, a) the Green s function for the given geometry. Upon combining Eq. (A 1) and (A 2) with the bridging law in Eq. (2), the governing equation becomes This result is in a general form in the sense that it is applicable to a variety of crack and loading configurations, ranging from a center crack in tension to edge cracks in bending and/or tension, provided appropriate functions are used to calculate H and f a. Equation (A 3) can be normalized such that the solution for a given load and crack length represents the general solution for all material properties. The result is ln bx a 8 ao Hx, â, a b x dâ 4 ax a f a x, â dâ (A 4) where b b / 0 is the normalized bridging stress, a a / 0 the normalized applied load, x x/w the normalized position along the crack, a a/w the normalized crack length, a o a o /w the normalized notch size, and 0 w/e u o. The parameter completely captures the dependence of the bridging stress profile on the material parameters u o and 0. Equation (A 4) is solved by a numerical technique wherein the bridging-stress distribution is broken into discrete intervals in which the traction is taken to be constant. The result is a nonlinear matrix equation: i ln b A j ij b b i (A 5) where i b is the bridging traction value for the interval centered on x i i (i.e., b (x) b for x i1/2 x x i1/2 ), b i f( a, a o, a) the crack opening at x i, and the matrix A ij f(a o, a) is determined from numerical evaluation of the Green s function on the interval x i1/2 x x i1/2. The summation convention applies. Thus, A ij represents the contribution to the opening at x i due to the bridging stress acting on x j1/2 x x j1/2. Equation (A 5) is solved using a Newton Raphson procedure to minimize the difference F i between the right and left sides. The i bridging stress vector b is found by iteration using the formulas i J ij b F i k (A-6) J ij Fi A ij ij i no sum A 7a b jk b b i k1 b i k b i A 7b i where k is the iteration variable, b is the update to the bridging stress vector, ( i b ) k1 and ( i b ) k are the new and previous bridging stress vectors, respectively, and J ij is the Jacobian matrix. Once Eq. (A 7a) is solved, the bridging-stress vector is updated via Eq. (A 7b), and the procedure continues until the bridging-stress vector does not change, i.e. F j 0. The procedure requires an initial guess for the bridging-traction vector; the choice of ( i b ) 0 exp[(x i ) 2 1] proved effective in ensuring convergence. The quantities of interest, including the bridging-stress distribution, the crack-tip stress intensity factor, and the notch-opening displacement, required discretized points for convergence. Once the bridging-stress distribution has been determined, the crack-tip stress intensity factor, K tip, is calculated in a straightforward manner and is given by K tipw 1/ 2 0 u o w u o 2 aa 1/ 2 Fa 2 w 1/ 2 a 1/ a a o b t f P t,a dt u 0 ln bx 0 8 Ea o a Hx, â, a b x dâ 4 a a f a x, â dâ Ex (A 3) (A 8) where F(a) is the finite-width correction factor for the applied load and f P (t, a) is determined from the point-force stress intensity factor. Equation (A 8) is discretized in a similar fashion to Eq. (A 4), such that the bridging tractions values determined via Eq. (A 7) are used.

9 February 2001 In-Plane Fracture Resistance of a Crossply Fibrous Monolith 375 Finally, it is of interest to determine the crack-opening displacement at the notch mouth, because this is an experimentally measured quantity. Once the bridging stress has been solved, the crack opening in the bridged region of the crack can be calculated directly via the phenomenological bridging law (Eq. (5)). For locations outside of the bridged region, the opening is calculated by subtracting the closure due to the bridging stress from the opening due to the applied load. The crack-opening displacement at the notch mouth, u o, is found via u a o h* 4 f a 0, â dâ 8 a H0, t, a b t dt (A 9) a0 ao where H is again a Green s formula, and varies slightly from H because of a change in the limits of integration. 8,18 Again, this equation is discretized and the bridging traction values acquired via Eq. (A 6) are used. The load CMOD simulations presented in Fig. 11 are obtained in the following way. For a given crack length, an estimate of the applied load needed to achieve K tip K 0 is used to solve Eq. (A 4). The K tip value from this load is then computed using Eq. (A 8); based on this result, the applied load is adjusted until K tip K 0. Once this is accomplished, the notch opening is computed via Eq. (A 9). The procedure is repeated for different crack lengths, thus yielding the desired load CMOD relation. Acknowledgments The authors gratefully acknowledge Advanced Ceramics Research for the provision of the FM materials and the monolithic Si 3 N 4 used in this study, and Drs. W. Coblenz, K. Goretta, and D. Singh for support and helpful comments. References 1 S. Baskaran, S. D. Nunn, D. Popovic, and J. W. Halloran, Fibrous Monolithic Ceramics: I, Fabrication, Microstructure, and Indentation Behavior, J. Am. Ceram. Soc., 76, (1993). 2 S. Baskaran and J. W. Halloran, Fibrous Monolithic Ceramics: II, Flexural Strength and Fracture Behavior of the Silicon Carbide/Graphite System, J. Am. Ceram. Soc., 76, (1993). 3 S. D. Nunn, D. Popovic, S. Baskaran, J. W. Halloran, G. Subramanian, and S. G. Bike, Suspension Dry Spinning and Rheological Behavior of Ceramic-Powder- Loaded Polymer Solutions, J. Am. Ceram. Soc., 76, (1993). 4 G. Hilmas, A. Brady, U. Abdali, G. Zywicki, and J. W. Halloran, Fibrous Monoliths: Non-Brittle Fracture from Powder Processed Ceramics, Mater. Sci. Eng. A, A915, (1995). 5 D. Popovic, G. A. Danko, K. Stuffle, B. H. King, and J. W. Halloran, Relationship Between Architecture, Flexural Strength, and Work of Fracture for Fibrous Monolith Ceramics, Sci. Sintering, 27, (1995). 6 D. Kovar, B. H. King, R. W. Trice, and J. W. Halloran, Fibrous Monolithic Ceramics, J. Am. Ceram. Soc., 80 [10] (1997). 7 R. W. Trice and J. W. Halloran, Elevated-Temperature Mechanical Properties of Silicon Nitride/Boron Nitride Fibrous Monolithic Ceramics, J. Am. Ceram. Soc., 83 [2] (2000). 8 H. Tada, P. C. Paris, and G. Irwin, The Stress Analysis of Cracks Handbook, 2nd ed.; pp Paris Productions, St. Louis, MO, H. G. Tattersall and G. Tappin, The Work of Fracture and its Measurement in Metals, Ceramics, and Other Materials, J. Mater. Sci., 1, (1966). 10 J. I. Bluhm, Slice Synthesis of a Three Dimensional Work of Fracture Specimen, Eng. Frac. Mech., 7, (1975). 11 M. Y. He and A. G. Evans, Three-Dimensional Finite-Element Analysis of Chevron-Notched, Three-Point and Four-Point Bend Specimens ; pp in Fracture Mechanics: Twenty-Second Symposium (Vol. 1), ASTM STP Edited by H. A. Ernst, A. Saxena, and D. L. McDowell, American Society for Testing and Standards, Philadelphia, PA, F. W. Zok and C. L. Hom, Large Scale Bridging in Ceramic Composites, Acta Metall., 38, (1990). 13 F. W. Zok, O. Sbaizero, C. L. Hom, and A. G. Evans, The Mode I Fracture Resistance of a Laminated Fiber Reinforced Ceramic, J. Am. Ceram. Soc., 74, (1991). 14 B. D. Flinn, F. W. Zok, C. Lo, and A. G. Evans, The Fracture Resistance Characteristics of a Metal Toughened Ceramic, J. Am. Ceram. Soc., 76, (1993). 15 P. Brenet, F. Conchin, G. Fantozzi, P. Reynaud, and C. Tallaron, Direct Measurement of Crack Bridging Tractions: A New Approach to the Fracture of Ceramic Composites, Compos. Sci. Technol., 56, B17 B23 (1996). 16 J. C. McNulty and F. W. Zok, Low-Cycle Fatigue of Nicalon-Fiber-Reinforced Ceramic Composites, Compos. Sci. Technol., 59, (1999). 17 D. Singh, Argonne National Laboratory, unpublished research, M. R. Begley and R. M. McMeeking, Fatigue Crack Growth with Fiber Failure in Metal Matrix Composites, Compos. Sci. Technol., 53, (1995).