Journal of the European Ceramic Society 19(1999)591-599 o 1999 Published by Elsevier Science Limited Printed in Great Britain. All rights reserved PII:S0955-2219(98)00231-3 0955-2219/99/Ssee front matter Crack Closure stresses in Fiber Reinforced Brittle matrix Composites H. Xu and C. P Ostertag CEE Department, University of California, Berkeley, CA 94720, USA (Received 24 June 1998; accepted 20 September 1998) Abstract able, are the key to achieving such an under- standing. Direct experimental examination of the The fracture toughness of an alumina ceramic and a relative importance of various toughenin continuous SiC fiber reinforced alumina composite mechanisms and the interaction between matrix processed by pressureless sintering was studied in cracks and reinforcing fibers are therefore valuable in a se applied stress intensity factor was obtained as a fied by hot-pressing, chemical vapor, or reaction function of both applied load and crack extension. bonding. Commercialization of any of these pro- Closure stresses across crack surfaces imposed by cesses is hindered by technological and economic grain-bridging and fiber-bridging and hence fracture difficulties. The fiber reinforced composites in this resistance from bridging were studied by both stress study are pressureless sintered, a potential alter intensity factor and J-integral considerations. Theo- native process for fabricating near-net-shape com- retical calculations agree with experimental results. An posites. A composite processed by pressureless average fracture resistance of a40Jm-per fiber and sintering may have different mechanical properties a corresponding toughness of x1-6" per fiber than those processed by other techniques, since the was obtained for fiber elastic bridging before fiber fail- state of residual stresses and the interfacial prop- ure. Fiber-matrix interfacial properties were examined erties may be different. Therefore, it is important to and a technique for evaluating interfacial frictional examine: how pressureless sintering influences shear stress was developed. C 1999 Published by mechanical properties; how the interfacial proper Elsevier Science Limited. All rights reserved ties differ in a pressureless sintered composite from those of a hot-pressed compo Keywords: fracture, AlO3, fibers, SiC, interfaces mechanical properties (i.e. fracture strength) of fibers change in pressureless sintering and how this affects the interactions of matrix cracks with fibers 1 Introduction and if mechanical properties can be improved by pressureless sintering Fracture toughness as high as 40 MPam/2 and In the pressureless sintered fiber reinforced com- 50/2 have been achieved in continuous fiber posite under investigation, both fiber-bridging and reinforced ceramic composites. In these materi- matrix grain-bridging are operative. Grain-localized als, a major portion of the toughness is attributable bridging as a toughening mechanisms in alumina to the work required to elastically elongate the was confirmed directly with in-situ observations of bridging fibers and to frictionally pull the broken crack propagation by optical microscopy, and fibers out of the matrix. A complete understanding scanning electron microscopy. 4. It was observed of reinforcement-toughening mechanisms would that individual bridging grains interlock between include fiber-crack interactions and the relative the crack planes behind the advancing crack tip importance of various toughening mechanisms, sliding frictionally against matrix grains during such as fiber elastic bridging, broken fiber pullout crack opening. It is this frictional sliding that con and matrix grain bridging Micromechanics studies sumes energy and results in a rising resistance of the toughening process, which are not yet avail- curve. The combination of grain-and fiber-bridging enhances the crack growth resistance even further *To whom correspondence should be addressed by exerting additional closure stresses on the crack
Crack Closure Stresses in Fiber Reinforced Brittle Matrix Composites H. Xu and C. P. Ostertag* CEE Department, University of California, Berkeley, CA 94720, USA (Received 24 June 1998; accepted 20 September 1998) Abstract The fracture toughness of an alumina ceramic and a continuous SiC ®ber reinforced alumina composite processed by pressureless sintering was studied in situ in a Scanning Electron Microscope (SEM). The applied stress intensity factor was obtained as a function of both applied load and crack extension. Closure stresses across crack surfaces imposed by grain-bridging and ®ber-bridging, and hence fracture resistance from bridging were studied by both stress intensity factor and J-integral considerations. Theoretical calculations agree with experimental results. An average fracture resistance of &40 J mÿ2 per ®ber and a corresponding toughness of &1.6MPam1/2 per ®ber was obtained for ®ber elastic bridging before ®ber failure. Fiber-matrix interfacial properties were examined and a technique for evaluating interfacial frictional shear stress was developed. # 1999 Published by Elsevier Science Limited. All rights reserved Keywords: fracture, Al2O3, ®bers, SiC, interfaces. 1 Introduction Fracture toughness as high as 40MPam1/2 and 50MPam1/2 have been achieved in continuous ®ber reinforced ceramic composites.1,2 In these materials, a major portion of the toughness is attributable to the work required to elastically elongate the bridging ®bers and to frictionally pull the broken ®bers out of the matrix. A complete understanding of reinforcement-toughening mechanisms would include ®ber-crack interactions and the relative importance of various toughening mechanisms, such as ®ber elastic bridging, broken ®ber pullout and matrix grain bridging. Micromechanics studies of the toughening process, which are not yet available, are the key to achieving such an understanding. Direct experimental examination of the relative importance of various toughening mechanisms and the interaction between matrix cracks and reinforcing ®bers are therefore valuable. Ceramic matrix composites are currently densi- ®ed by hot-pressing, chemical vapor, or reactionbonding. Commercialization of any of these processes is hindered by technological and economic diculties. The ®ber reinforced composites in this study are pressureless sintered, a potential alternative process for fabricating near-net-shape composites. A composite processed by pressureless sintering may have dierent mechanical properties than those processed by other techniques, since the state of residual stresses and the interfacial properties may be dierent. Therefore, it is important to examine: how pressureless sintering in¯uences mechanical properties; how the interfacial properties dier in a pressureless sintered composite from those of a hot-pressed composites; how the mechanical properties (i.e. fracture strength) of ®bers change in pressureless sintering and how this aects the interactions of matrix cracks with ®bers; and if mechanical properties can be improved by pressureless sintering. In the pressureless sintered ®ber reinforced composite under investigation, both ®ber-bridging and matrix grain-bridging are operative. Grain-localized bridging as a toughening mechanisms in alumina was con®rmed directly with in-situ observations of, crack propagation by optical microscopy,3 and scanning electron microscopy.4,5 It was observed that individual bridging grains interlock between the crack planes behind the advancing crack tip, sliding frictionally against matrix grains during crack opening. It is this frictional sliding that consumes energy and results in a rising resistance curve. The combination of grain-and ®ber-bridging enhances the crack growth resistance even further by exerting additional closure stresses on the crack Journal of the European Ceramic Society 19 (1999) 591±599 # 1999 Published by Elsevier Science Limited Printed in Great Britain. All rights reserved PII: S0955-2219(98)00231-3 0955-2219/99/$Ðsee front matter 591 *To whom correspondence should be addressed
92 H. Xu, C. P Ostertag walls. These closure stresses shield the crack tip notch tip going through the indent. After a crack from the applied load. In order for an existing ength of approximately 1300 um was propagated crack to extend, a higher stress needs to be applied the half chevron wedge including the indent was Crack closure stresses associated with fiber- and carefully sawed away, leaving a notch of constant grain-bridging are obtained from crack profiles thickness and a precrack of 300 um from the notch measured at high magnification inside the scanning tip. The crack tip is several hundred microns away electron microscope from the second row of fibers which will serve as bridging ligaments in subsequent in-situ crack pro- pagation studies inside the Sem 2 Experimental Procedure 2.2 SEM measurements of crack propagation 2.1 Specimen preparation The precracked specimens were coated with Au-Pd Both composite samples and monolithic samples to a thickness of 25 mm. A custom designed load were prepared by pressureless sintering. Several ing device was used to apply a mechanical load to sheets of tape-casted Al_O3 green tapes stacked compact tension specimens as crack growth was ogether to form a square specimen with fibers observed in the SEM. After the specimens were embedded between the tapes. The green thickness placed in the device and loaded inside the seM to was approximately 4 mm. The fibers were SCS-2 propagate the crack, events occurring at the brid- SiC fibers(Textron Specialty Materials, MA), with ging sites were monitored and recorded on film and a total diameter of 140 um including a 33 um car. video tapes. Both crack tip propagation and crack bon core and a carbon-rich surface coating of wake grain- and fiber-bridging were observed and 2 um. These fibers were coated with a layer of gold video taped. This was done along the crack paths of 60 mm thickness before they were embedded in for crack opening displacement(COD)measure- the tapes to reduce the chemical reaction between ments. The COD measurements were made only SiC and Al2O3. 6 The square specimens were then well-behaved regions, i.e. at grain facets oriented cold-pressed uniaxially at 13 MPa. After the binder normal to the load axis and located away from any burnout at 500@C for 8h, the specimens were pres- secondary cracking around bridging sites. The sureless sintered at 1600 C for 2 h in air. The grain cracks were rendered highly visible in the second- size of these specimens ranged from 2 to 15 mm, ary electron mode by edge charging. This charging with an average grain size of 8 um. The density was limited the absolute resolution of surface-surface 3.7 gcm-3,93% of the theoretical density. In this separation to about 70 nm, although relative mea- study six fibers were embedded in each specimen, surements could be made to better onm with two fibers in each plane along the specimen Typical COD values were in the range of 50- thickness Monolithic Al2O3 samples without fiber 100 nm which are too small to be measured by reinforcement were processed in the same way to optical techniques. SEM techniques are required s control sp for accurate measurements The sintered square specimens (edge length approx. 18 mm)were ground to a thickness of 1.5- 2mm, and polished with diamond paste to a 6 um 3 Results finish on the surface for crack length measure- ments. Loading holes were drilled and notches 3.1 Observation of bridging sites sawed to form compact tension specimens to In-situ SEM observation revealed that both grain ASTM specifications. The notch was sawed bridging and fiber-bridging were operative along through the first plane of fibers, so that only the the crack paths. Crack propagation occurred in a second plane of fibers served as bridging ligaments discontinuous fashion and the fracture mode was during testing. The first and the third planes of predominantly intergranular. For the monolithic fibers fulfilled the symmetry requirements during specimens, grain-bridging occurred throughout the densification. The notch was cut at an angle of 45 entire crack path. The final crack lengths before to the specimen surface in a half-chevron geometry failure were typically 600 mm with maximum to enhance the stability in the initial crack exten- CODs in the range of 0-4 um. In the fiber rein- sion. Accordingly, the notch tip extended I mm forced samples where both grain-bridging and further on the unpolished surface compared to the fiber-bridging were operative, much longer cracks polished surface. A Vickers indentation, contact with 2 mm length and maximum CODs of 1-4 um load 50N was placed 200 um ahead of the notch were stable under higher applied load, indicating tip on the polished surface, which left a starter the effectiveness of fiber-bridging SEM pictures of crack. The specimen was then precracked in a pre- grain-bridging and fiber-bridging are shown in liminary load cycle to create a crack from the Fig. 1(a) and(b), respectively
walls. These closure stresses shield the crack tip from the applied load. In order for an existing crack to extend, a higher stress needs to be applied. Crack closure stresses associated with ®ber- and grain-bridging are obtained from crack pro®les measured at high magni®cation inside the scanning electron microscope. 2 Experimental Procedure 2.1 Specimen preparation Both composite samples and monolithic samples were prepared by pressureless sintering. Several sheets of tape-casted Al2O3 green tapes stacked together to form a square specimen with ®bers embedded between the tapes. The green thickness was approximately 4 mm. The ®bers were SCS-2 SiC ®bers (Textron Specialty Materials, MA), with a total diameter of 140m including a 33m carbon core and a carbon-rich surface coating of 2m. These ®bers were coated with a layer of gold of 60 mm thickness before they were embedded in the tapes to reduce the chemical reaction between SiC and Al2O3. 6 The square specimens were then cold-pressed uniaxially at 13MPa. After the binder burnout at 500C for 8 h, the specimens were pressureless sintered at 1600C for 2 h in air. The grain size of these specimens ranged from 2 to 15 mm, with an average grain size of 8m. The density was 3.7 g cmÿ3 , 93% of the theoretical density. In this study six ®bers were embedded in each specimen, with two ®bers in each plane along the specimen thickness. Monolithic Al2O3 samples without ®ber reinforcement were processed in the same way to serve as control specimens. The sintered square specimens (edge length approx. 18 mm) were ground to a thickness of 1.5± 2 mm, and polished with diamond paste to a 6m ®nish on the surface for crack length measurements. Loading holes were drilled and notches sawed to form compact tension specimens to ASTM speci®cations. The notch was sawed through the ®rst plane of ®bers, so that only the second plane of ®bers served as bridging ligaments during testing. The ®rst and the third planes of ®bers ful®lled the symmetry requirements during densi®cation. The notch was cut at an angle of 45 to the specimen surface in a half-chevron geometry to enhance the stability in the initial crack extension. Accordingly, the notch tip extended 1 mm further on the unpolished surface compared to the polished surface. A Vickers indentation, contact load 50 N was placed 200m ahead of the notch tip on the polished surface, which left a starter crack. The specimen was then precracked in a preliminary load cycle to create a crack from the notch tip going through the indent. After a crack length of approximately 1300m was propagated, the half chevron wedge including the indent was carefully sawed away, leaving a notch of constant thickness and a precrack of 300m from the notch tip. The crack tip is several hundred microns away from the second row of ®bers which will serve as bridging ligaments in subsequent in-situ crack propagation studies inside the SEM. 2.2 SEM measurements of crack propagation The precracked specimens were coated with Au±Pd to a thickness of 25 mm. A custom designed loading device was used to apply a mechanical load to compact tension specimens as crack growth was observed in the SEM. After the specimens were placed in the device and loaded inside the SEM to propagate the crack, events occurring at the bridging sites were monitored and recorded on ®lm and video tapes. Both crack tip propagation and crack wake grain- and ®ber-bridging were observed and video taped. This was done along the crack paths for crack opening displacement (COD) measurements. The COD measurements were made only in well-behaved regions, i.e. at grain facets oriented normal to the load axis and located away from any secondary cracking around bridging sites. The cracks were rendered highly visible in the secondary electron mode by edge charging. This charging limited the absolute resolution of surface-surface separation to about 70 nm, although relative measurements could be made to better than 30 nm. Typical COD values were in the range of 50± 100 nm which are too small to be measured by optical techniques. SEM techniques are required for accurate measurements. 3 Results 3.1 Observation of bridging sites In-situ SEM observation revealed that both grainbridging and ®ber-bridging were operative along the crack paths. Crack propagation occurred in a discontinuous fashion and the fracture mode was predominantly intergranular. For the monolithic specimens, grain-bridging occurred throughout the entire crack path. The ®nal crack lengths before failure were typically 600 mm with maximum CODs in the range of 0.4m. In the ®ber reinforced samples where both grain-bridging and ®ber-bridging were operative, much longer cracks with 2 mm length and maximum CODs of 1.4m were stable under higher applied load, indicating the eectiveness of ®ber-bridging. SEM pictures of grain-bridging and ®ber-bridging are shown in Fig. 1(a) and (b), respectively. 592 H. Xu, C. P. Ostertag
Crack closure stresses in fiber reinforced brittle matrix composites 593 3.2 In-situ measurement of crack-opening increase from crack closure forces due to grain- displa bride d fiber -bric Crack opening displacements were measured for both monolithic and fiber-reinforced specimens The crack profile when the stress-intensity factor Ka=7。+Tg+Tr Ka, was 4 MPam/ for a monolithic specimen is shown in Fig. 2(a). The crack profile of a specimen with two bridging fibers is shown in Fig. 2(b) for a For monolithic specimens, the third contribution stress intensity factor of 7 MPam/. The positions Tr is zero. The applied stress intensity factor, Ka, is of the two bridging fibers are indicated. The Cod never felt at the crack tip in the composites due to values, together with the fiber-matrix interfacial shielding by grain- and fiber-bridging frictional shear stress, t. enable the evaluation of For the compact tension test in this study, Ka, he bridging force in each individual fiber, as will can be readily calculated by, be shown later 3.3 Applied stress intensity factor Ka=(F/tw/)Y(a/w) The applied stress intensity factor is the driving force for crack propagation, while the toughness of where F is the applied load recorded during crack the material is the resistance to crack propagation. propagation, t and w are the specimen dimensions The resistance to crack propagation results from and a is the crack length measured from the center he intrinsic toughness, To, of the material (related of the loading holes. The crack geometry function to the atomic bond strength), and the toughness Y(a/w)is given by: 7 a5E6王 Distance Behind Crack Tip, x(um) aE: Fig. 1. SEM micrographs of (a) grain bridging at crack inter- Fig. 2. Measured crack opening displacements (COD)at ace behind the crack tip; (b) SiC fiber bridging crack faces crack interfaces:(a) in alumina,(b)in a Sic fiber reinforced
3.2 In-situ measurement of crack-opening displacements Crack opening displacements were measured for both monolithic and ®ber-reinforced specimens. The crack pro®le when the stress-intensity factor, Ka, was 4 MPam1/2 for a monolithic specimen is shown in Fig. 2(a). The crack pro®le of a specimen with two bridging ®bers is shown in Fig. 2(b) for a stress intensity factor of 7MPam1/2. The positions of the two bridging ®bers are indicated. The COD values, together with the ®ber-matrix interfacial frictional shear stress, , enable the evaluation of the bridging force in each individual ®ber, as will be shown later. 3.3 Applied stress intensity factor The applied stress intensity factor is the driving force for crack propagation, while the toughness of the material is the resistance to crack propagation. The resistance to crack propagation results from the intrinsic toughness, To, of the material (related to the atomic bond strength), and the toughness increase from crack closure forces due to grainbridging, Tg. and ®ber-bridging. Tf. At equilibrium Ka To Tg Tf 1 For monolithic specimens, the third contribution Tf is zero. The applied stress intensity factor, Ka, is never felt at the crack tip in the composites due to shielding by grain- and ®ber-bridging. For the compact tension test in this study, Ka, can be readily calculated by, Ka F=tw1=2 Y a=w 2 where F is the applied load recorded during crack propagation, t and w are the specimen dimensions and a is the crack length measured from the center of the loading holes. The crack geometry function Y a=w is given by:7 Fig. 1. SEM micrographs of (a) grain bridging at crack interface behind the crack tip; (b) SiC ®ber bridging crack faces. Fig. 2. Measured crack opening displacements (COD) at crack interfaces: (a) in alumina, (b) in a SiC ®ber reinforced alumina compact tension specimen. Crack closure stresses in ®ber reinforced brittle matrix composites 593
H. Xu, C. P Ostertag Y(a/)=(2+a/)/(1-a/m32 086+464(a/)-1332(a/n)2 +1472(a/)3-56a/m)4 For both the monolithic specimens and fiber rein forced specimens, Ka was calculated as a function of the applied load and the crack extension. For he monolithic specimens, Ka, reached a maximum value (after which the specimen broke) of 43 MPam/2. With two bridging fibers, a m Ka of 7.4 MPam/was obtained. The increase in toughness with crack extension for one of the fiber reinforced composites is shown in Fig 3 4 Analysis and Discussion Crack Extension△c[um Fig. 3. Measured toughness as a function of crack extension Consider a crack that moves in the matrix towards for a fiber-reinforced alumina specimen with two fibers in the a fiber. Fiber debonding can occur either ahead of bridging zone the crack front before the crack reaches the fiber or after the crack passes the fiber, depending on the elastic modulus difference between the matrix and the fiber. If the difference is large enough, fiber- than that of elastic bridging. For composites rein- matrix debonding may occur before the crack forced uniformly with a substantial volume percent reaches the fiber due to strain mismatch. On the (i.e. 30%)of fibers, elastic bridging and frictional other hand, if the elastic moduli are similar, as in pullout occur simultaneously during crack propaga- this study (Em=400 GPa, Er=410 GPa), strain tion. Therefore, their contributions to crack resistance mismatch is small and fiber-matrix debonding can not be measured separately For specimens a occurs only after the crack meets the fiber and is deflected along the interface or around the fibers. during stable crack propagation. When one fiber Subsequent increases in applied load drive the failed, the other fiber and the specimen broke crack past the fiber, with an increase in debonding immediately due to the sudden increase in stress length Ld, and an increase in Cod at the fiber Therefore, measurement of contribution from fiber position. Accordingly, the tensile strain in the fiber frictional pullout after fiber failure was not taken s enhanced which increases the bridging str ress This allowed the measurement of toughness increase across the crack walls yielding the R-curve beha- from elastic bridging alone, the contribution from vior. The debonded part of the fiber is partially pullout after fiber failure was not a factor pulled out of the matrix. Fiber failure usually The debonded part of the fiber that provides the occurs away from the crack plane due to a dis- closure stress is elongated elastically and therefore tribution of strength in fibers. The broken fiber is in tension. At the crack plane, the tensile stress in then pulled out of the matrix upon further opening the fiber as a function of half crack opening of the crack. The term 'pullout' has been used in u, oru), acts to pull the fiber out of the matrix literature to describe only the toughening process This is opposed by the frictional stress at the fiber after fiber failure when the broken fiber is being matrix interface pulled out of the matrix. The term'bridging has been used to describe the toughening process or(ur= t2 RLd before fiber failure. Strictly speaking, both bridging and pullout occur throughout the entire process of where R is the fiber radius, and r is the frictional toughening, both before and after fiber failure To shear stress between the debonded fiber of length avoid confusion the term 'elastic bridging will be Ld and the matrix. The tensile stress in the fibers ed in this paper to describe toughening before fiber may also be expressed as a function of the crack failure, and frictional pullout to describe toughening opening displacement, u,(see Appendix a after fiber failure. Usually the fiber pullout length, Lp, is much larger than the maximum COD before o/()=(4Eu/R)2 (5) fiber failure. and the crack resistance as a result of frictional pullout may be comparable to or larger where Er is the Youngs modulus of the fiber
Y a=w 2 a=w= 1 ÿ a=w 3=2 0886 464 a=w ÿ 1332 a=w 2 1472 a=w 3 ÿ 56 a=w 4 3 For both the monolithic specimens and ®ber reinforced specimens, Ka was calculated as a function of the applied load and the crack extension. For the monolithic specimens, Ka, reached a maximum value (after which the specimen broke) of 4.3MPam1/2. With two bridging ®bers, a maximum Ka of 7.4 MPam1/2 was obtained. The increase in toughness with crack extension for one of the ®ber reinforced composites is shown in Fig. 3. 4 Analysis and Discussion Consider a crack that moves in the matrix towards a ®ber. Fiber debonding can occur either ahead of the crack front before the crack reaches the ®ber or after the crack passes the ®ber, depending on the elastic modulus dierence between the matrix and the ®ber. If the dierence is large enough, ®bermatrix debonding may occur before the crack reaches the ®ber due to strain mismatch. On the other hand, if the elastic moduli are similar, as in this study (Em=400 GPa, Ef=410 GPa), strain mismatch is small and ®ber-matrix debonding occurs only after the crack meets the ®ber and is de¯ected along the interface or around the ®bers. Subsequent increases in applied load drive the crack past the ®ber, with an increase in debonding length Ld, and an increase in COD at the ®ber position. Accordingly, the tensile strain in the ®ber is enhanced which increases the bridging stress across the crack walls yielding the R-curve behavior. The debonded part of the ®ber is partially pulled out of the matrix. Fiber failure usually occurs away from the crack plane due to a distribution of strength in ®bers. The broken ®ber is then pulled out of the matrix upon further opening of the crack. The term `pullout' has been used in literature to describe only the toughening process after ®ber failure when the broken ®ber is being pulled out of the matrix. The term `bridging' has been used to describe the toughening process before ®ber failure. Strictly speaking, both bridging and pullout occur throughout the entire process of toughening, both before and after ®ber failure. To avoid confusion the term `elastic bridging' will be used in this paper to describe toughening before ®ber failure, and frictional pullout to describe toughening after ®ber failure. Usually the ®ber pullout length, Lp, is much larger than the maximum COD before ®ber failure, and the crack resistance as a result of frictional pullout may be comparable to or larger than that of elastic bridging. For composites reinforced uniformly with a substantial volume percent (i.e. 30%) of ®bers, elastic bridging and frictional pullout occur simultaneously during crack propagation. Therefore, their contributions to crack resistance can not be measured separately. For specimens in this study only two ®bers are bridging the crack during stable crack propagation. When one ®ber failed, the other ®ber and the specimen broke immediately due to the sudden increase in stress. Therefore, measurement of contribution from ®ber frictional pullout after ®ber failure was not taken. This allowed the measurement of toughness increase from elastic bridging alone, the contribution from pullout after ®ber failure was not a factor. The debonded part of the ®ber that provides the closure stress is elongated elastically and therefore in tension. At the crack plane, the tensile stress in the ®ber as a function of half crack opening u; f u, acts to pull the ®ber out of the matrix. This is opposed by the frictional stress at the ®bermatrix interface: f uR2 2RLd 4 where R is the ®ber radius, and r is the frictional shear stress between the debonded ®ber of length Ld and the matrix. The tensile stress in the ®bers may also be expressed as a function of the crack opening displacement, u, (see Appendix A): f u 4Ef u=R 1=2 5 where Ef is the Young's modulus of the ®ber. Fig. 3. Measured toughness as a function of crack extension for a ®ber-reinforced alumina specimen with two ®bers in the bridging zone. 594 H. Xu, C. P. Ostertag
Crack closure stresses in fiber reinforced brittle matrix composite 595 4.1 Fracture strength of fibers in matrix radial compression when the specimen is cooled The fracture strength of the as-received SCS-2 Sic from the sintering temperature. In this case t can fibers was measured to be 4 GPa at a gauge length be expressed by of 20 mm. When they are embedded in the A12O3 matrix and sintered in air. the fracture strength of the fibers degrades due to fiber grain growth and where oR is the residual stress at the interface in the chemical reactions. Fiber degradation due to che- fiber radial direction, and u is the frictional coeffi mical reaction was reduced in this study by a pro- cient at the debonded interface. There are three tective gold coating applied to the fibers before possible reasons for the high value of t in the spe- embedding them in the matrix cimens of this study. First, the large difference in If we consider the fiber fracture strength to be thermal expansion coefficients between the SiC fiber single-valued(Weibull modulus m=oo), fiber fail- and the alumina matrix (am-ap> 4 x 10-6/C) ure will occur between the crack faces where the results in a high value of oR when the specimen tensile stress in the fiber is a maximum. However, cooled from the sintering temperature of 1600C to fibers exhibit a strength distribution due to the room temperature. Assuming the residual stress presence of flaws and fiber failure usually occurs above 1200C relaxes due to creep, an estimate of away from the crack plane, resulting in fiber pull- the residual stress at room-temperature results in out. The fiber pullout length,Lp, corresponds to OR=E△a△T≈(400GPa)4×10-6C)=19GP the distance from the crack plane, Zf, at which The composite materials reported in literature, fiber failure occurs. The tensile stress in the fiber at with glass matrices in most cases, have much failure at a distance Zf, is the fracture strength of smaller thermal mismatches between the matrices the in-situ fiber, S. The fracture strength of these and fibers. -14 The second reason for the high fric- embedded fibers can be evaluated for each indiv tional stress may by associated with the pressure- dual fiber by measuring the mirror radius, am, on less sintering process. Most fiber reinforced the fiber fracture surface, as demonstrated by composites are processed by hot-pressing, while the Thouless et aL. 8 specimens in this study are processed by pressure- less sintering. In hot-pressing the residual stresses S=3.5K/d2 (6) result mainly from thermal expansion mismatchIn sintering the matrix shrinks 30 to 40 vol% while where Ko is the fiber fracture toughness which is the fibers, being fully dense do not shrink at all approximately 2 MPam2. The fracture strength This gives rise to an additional compressive stress for each individual fiber can thus be calculated and at the interface perpendicular to the fibers which an average value can be obtained. In this study do not relax completely by creep. Thirdly, a thin S=2 GPa was obtained for the in-situ fiber reaction layer between the fibers and the matrix strength after the sintering process was found after sintering. This may increase the roughness of the interface after debonding 4.2 Interfacial frictional shear stress increasing the frictional coefficient, u, between the The interfacial shear stress t can be obtained as sliding fiber and the matrix. Therefore, the higher shown in Appendix B value of t may be associated with higher values of both oR and u. Higher values of t are of practical importance in achieving high short-crack toughness. 6 A number of techniques have been proposed to T= REruo/42[1-(1-2SLp/Eruo) 1(7) measure the interfacial shear stress, t. These tech- niques may not be well suited for the fiber reinforced composites in this study with high interfacial here uo is the half Cod at the fiber position when tional shear stress. Fiber cracking can occur in fiber failure occurs. In this equation S can be eval- push-in tests when the load is high. The slice- uated as described in Section 4.1, uo and Lp can be compression technique, 4 is not effective when the measured experimentally for each fiber, and Er and Youngs modulus of fibers is close to that of the R are known constants. Using this equation an matrix, as in this study. Furthermore, various average value of [=1-3 GPa was obtained for spe- techniques often reveal very different results (i.e. the simens in this stud value of t by the slice-compression technique was An interfacial frictional shear stress of 1. 3 GPa is an order of magnitude smaller compared with the large compared to values of generally <100 MPa fiber pull-out technique for the same composite reported in the literature.9-14 When the thermal The technique for measuring t described in this expansion coefficient of the matrix is larger than section is believed to be fairly dependable compared hat of the fibers as in this study, the fibers are in to other techniques, especially when t is large
4.1 Fracture strength of ®bers in matrix The fracture strength of the as-received SCS-2 SiC ®bers was measured to be 4 GPa at a gauge length of 20 mm. When they are embedded in the A12O3 matrix and sintered in air, the fracture strength of the ®bers degrades due to ®ber grain growth and chemical reactions. Fiber degradation due to chemical reaction was reduced in this study by a protective gold coating applied to the ®bers before embedding them in the matrix. If we consider the ®ber fracture strength to be single-valued (Weibull modulus m=1), ®ber failure will occur between the crack faces where the tensile stress in the ®ber is a maximum. However, ®bers exhibit a strength distribution due to the presence of ¯aws and ®ber failure usually occurs away from the crack plane, resulting in ®ber pullout. The ®ber pullout length, Lp, corresponds to the distance from the crack plane, Zf, at which ®ber failure occurs. The tensile stress in the ®ber at failure at a distance Zf, is the fracture strength of the in-situ ®ber, S. The fracture strength of these embedded ®bers can be evaluated for each individual ®ber by measuring the mirror radius, am, on the ®ber fracture surface, as demonstrated by Thouless et al. 8 S 35Kf=a1=2 m 6 where Kf is the ®ber fracture toughness which is approximately 2 MPam1/2. 8 The fracture strength for each individual ®ber can thus be calculated and an average value can be obtained. In this study S=2 GPa was obtained for the in-situ ®ber strength after the sintering process. 4.2 Interfacial frictional shear stress The interfacial shear stress can be obtained as shown in Appendix B: REfuo=4L2 p1 ÿ 1 ÿ 2SLp=Efuo 1=2 2 7 where uo is the half COD at the ®ber position when ®ber failure occurs. In this equation S can be evaluated as described in Section 4.1, uo and Lp can be measured experimentally for each ®ber, and Ef and R are known constants. Using this equation an average value of =1.3 GPa was obtained for specimens in this study. An interfacial frictional shear stress of 1.3 GPa is large compared to values of generally 4 10ÿ6= C results in a high value of R when the specimen is cooled from the sintering temperature of 1600C to room temperature. Assuming the residual stress above 1200C relaxes due to creep, an estimate of the residual stress at room-temperature results in R E T (400 GPa)(410ÿ6 / C)=1.9 GPa. The composite materials reported in literature, with glass matrices in most cases, have much smaller thermal mismatches between the matrices and ®bers.9±14 The second reason for the high frictional stress may by associated with the pressureless sintering process. Most ®ber reinforced composites are processed by hot-pressing, while the specimens in this study are processed by pressureless sintering. In hot-pressing the residual stresses result mainly from thermal expansion mismatch. In sintering the matrix shrinks 30 to 40 vol% while the ®bers, being fully dense, do not shrink at all. This gives rise to an additional compressive stress at the interface perpendicular to the ®bers which do not relax completely by creep.15 Thirdly, a thin reaction layer between the ®bers and the matrix was found after sintering. This may increase the roughness of the interface after debonding, increasing the frictional coecient, , between the sliding ®ber and the matrix. Therefore, the higher value of may be associated with higher values of both R and . Higher values of t are of practical importance in achieving high short-crack toughness.16 A number of techniques have been proposed to measure the interfacial shear stress, . These techniques may not be well suited for the ®ber reinforced composites in this study with high interfacial frictional shear stress. Fiber cracking can occur in push-in tests when the load is high.17 The slicecompression technique11,14 is not eective when the Young's modulus of ®bers is close to that of the matrix, as in this study. Furthermore, various techniques often reveal very dierent results (i.e. the value of by the slice-compression technique was an order of magnitude smaller compared with the ®ber pull-out technique for the same composite11). The technique for measuring described in this section is believed to be fairly dependable compared to other techniques, especially when is large. Crack closure stresses in ®ber reinforced brittle matrix composites 595
596 H. Xu, C. P Ostertag 4.3 Closure stresses associated with grain-and fiber- 4.4 Grain-bridging plus fiber-bridging bridging For a material with both grain- and fiber-bridging, the toughness and the crack resistance have three 4.3.1 Grain-bridging major contributions: the intrinsic fracture tough Grain localized bridging at the crack interface ness of the matrix(To); grain bridging and pullout behind the advancing crack tip increases the crack (Tg); and fiber bridging and pullout (Tr). Thus growth resistance of the material. When the crack opens, grain pullout occurs and the fraction at the T=70+7g+Tr (1la) sliding matrix-grain interface contributes to the or resistance. Grain-bridging, exerts a closure stress R=R。+Ra+Rf (11b) across the crack faces. The closure stress o(u) as a function of u can be expressed as 8 Where r。=T。2/ELR=T/E, and Re=R(elas o)=om(1-u/u) (9) tic bridging)+RP(broken-fiber pullout), E E/( The specimens in this study with two bridging fibers only the resistance due to elastic where om is the maximum closure stress which fiber-bridging was measured and calculated, Rp occurs at the crack tip(u=O), and u is the half was not included This allowed direct evaluation of Cod at the end of a saturated bridging zone he effects of elastic fiber-bridging When steady state is reached, the bridging zone For a composite with a substantial volume frac is saturated. When the increase in Ka with crack tion of reinforcing fibers uniformly aligned along length is faster than the increase in toughness, T, the crack plane, the bridging effect can be repre- the specimen fails when the crack length reaches a sented by a continuous closure stress, P, over the critical value. In this case the bridging zone is not fiber bridging zone of length xr. The crack growth saturated, and the maximum crack opening before resistance can be calculated by the J-integral failure, 2Ub, is less than the steady state opening 2u. The crack resistance can be calculated using qn(9)and the J-integral Rr=2 P(u) (12a) R2=2|n(1-u/)dn where ur is the crack opening at the end of the fiber bridging zone. When u=u(x)is known as a func (10a) tion of x(x is the coordinate along the crack lengt =20mb(1-b/2u) with the origin at the crack tip), eqn(12a) can be written as: When steady, state is reached, ub=u*, therefore R Rr=2 a(x)(du/dx)dx (12b) For the Al2O3 compact tension specimens in this study, experimental measurements gave where o(x) is the closure stress, o=p. Here p is ub 0.22 um. The steady state half COD, u, is a expressed as a function of u, while o is expressed as fraction of the grain size, L In this study, L was a function of x, i.e. o(x=plu(x)). The resistance 8um. Taking literature values for om=79 MPa+ can hence be calculated when the closure stress is and'bridging rupture' strain E1, equal to 0.18 we known as a function of Cod or as a function of obtain u=0.4um (u=E1 L/2). Inserting these crack length. Usually, the closure stress is known parameters into eqn(10a)results in Rg=22 Jm as a function of x. since u is too small to be mea- For, Al2O3 with its intrinsic resistance Ro of 17 sured by conventional techniques. But in this Jm-,9the total resistance=Ro+Rg is 39Jm-, study, due to the advantage of the SEM in-situ corresponding to a toughness value T=(ER) of measurements, the closure stress can be obtained 4.2 MPam /2. The calculated toughness value agrees both as a function of x and u well with the experimental measurement of Alternatively, the toughness can be calculated Ka =43 MPam 2. Therefore, the grain bridging using the Barenblatt relation. 0 In the small brid model gives a toughness value consistent with that ging zone limit, where xf <c, the Barenblatt obtained from the stress-intensity approach equation evaluates the toughness increase from the When steady state is reached, inserting the para- closure stress as meters above into eqn (10b) results in R:-=28 Jm-, yielding a total crack resistance rof 45 Jm, and a corresponding maximum toughness T=(2/n)2|a(x)/x/d T of 4.4 MPam /2
4.3 Closure stresses associated with grain-and ®berbridging 4.3.1 Grain-bridging Grain localized bridging at the crack interface behind the advancing crack tip increases the crack growth resistance of the material. When the crack opens, grain pullout occurs and the fraction at the sliding matrix-grain interface contributes to the resistance. Grain-bridging, exerts a closure stress across the crack faces. The closure stress u as a function of u can be expressed as18 u m 1 ÿ u=u 9 where m is the maximum closure stress which occurs at the crack tip u O, and u is the half COD at the end of a saturated bridging zone. When steady state is reached, the bridging zone is saturated. When the increase in Ka with crack length is faster than the increase in toughness, T, the specimen fails when the crack length reaches a critical value. In this case the bridging zone is not saturated, and the maximum crack opening before failure, 2Ub, is less than the steady state opening 2u. The crack resistance can be calculated using eqn (9) and the J-integral: Rg 2 u b 0 m 1 ÿ u=u du 2mub 1 ÿ ub=2u 10a When steady, state is reached, ub u, therefore R g mu 10b For the Al2O3 compact tension specimens in this study, experimental measurements gave ub 022m. The steady state half COD, u, is a fraction of the grain size, L. In this study, L was 8m. Taking literature values for m=79 MPa4 and `bridging rupture' strain "1, equal to 0.118 we obtain u=0.4m u "1L=2. Inserting these parameters into eqn (10a) results in Rg=22 J mÿ2 . For, Al2O3 with its intrinsic resistance Ro of 17 J mÿ2 , 19 the total resistance R RoRg is 39 J mÿ2 , corresponding to a toughness value T E0 R 1=2 of 4.2MPam1/2. The calculated toughness value agrees well with the experimental measurement of Ka=4.3 MPaml/2. Therefore, the grain bridging model gives a toughness value consistent with that obtained from the stress-intensity approach. When steady state is reached, inserting the parameters above into eqn (10b) results in R g=28 J mÿ2 , yielding a total crack resistance R of 45 J mÿ2 , and a corresponding maximum toughness T of 4.4MPam1/2. 4.4 Grain-bridging plus ®ber-bridging For a material with both grain- and ®ber-bridging, the toughness and the crack resistance have three major contributions: the intrinsic fracture toughness of the matrix To; grain bridging and pullout Tg; and ®ber bridging and pullout Tf. Thus: T T0 Tg Tf 11a or R Ro Rg Rf 11b Where Ro To 2=E0 LR T2=E0 , and Rf Re r (elastic bridging)Rp f (broken-®ber pullout), E0 E= 1 ÿ 2. The specimens in this study with two bridging ®bers only the resistance due to elastic ®ber-bridging was measured and calculated, Rp f was not included. This allowed direct evaluation of the eects of elastic ®ber-bridging. For a composite with a substantial volume fraction of reinforcing ®bers uniformly aligned along the crack plane, the bridging eect can be represented by a continuous closure stress, p, over the ®ber bridging zone of length xf. The crack growth resistance can be calculated by the J-integral: Rf 2 uf 0 P udu 12a where uf is the crack opening at the end of the ®ber bridging zone. When u u x is known as a function of x x is the coordinate along the crack length, with the origin at the crack tip), eqn (12a) can be written as: Rf 2 xf 0 x du=dxdx 12b where x is the closure stress, p. Here p is expressed as a function of u, while is expressed as a function of x, i.e. x p u x. The resistance can hence be calculated when the closure stress is known as a function of COD or as a function of crack length. Usually, the closure stress is known as a function of x, since u is too small to be measured by conventional techniques. But in this study, due to the advantage of the SEM in-situ measurements, the closure stress can be obtained both as a function of x and u. Alternatively, the toughness can be calculated using the Barenblatt relation.20 In the small bridging zone limit, where xf c, the Barenblatt equation evaluates the toughness increase from the closure stress as: Tf 2= 1=2 xf 0 x=x1=2 dx 13 596 H. Xu, C. P. Ostertag
Crack closure stresses in fiber reinforced brittle matrix composites 597 The results of eqns(12)and(13)should agree with Alternatively, with the measured values of xi and each other to be consistent x?and the calculated closure stress of o=197 MPa When there are only a few fibers bridging a nar- eqn (15) results in Tr=3.1 MPam /2and row region of the crack, a continuous closure stress T=To +T+ Tr or 7.5 MPam/2. Both results are can be approximated over the narrow bridging consistent with the experimental value of the region, x1 <x<x2. If the fibers are far apart from applied stress intensity factor, Ka=7. 4 MPam/2 each other, it is more appropriate to use point for An increase in crack growth resistance of 76 ces instead of a continuous stress. In this study, the Jm- and a corresponding fracture toughness two bridging fibers were very close to each other in increase of 3.2 MPam/2 from the elastic bridging a narrow bridging zone of 500 um. Accordingly, of two fibers is remarkable, compared to an the use of a continuous closure stress within the increase in fracture toughness of only 0.2 MPam/2 narrow bridging zone, xI <x<x2, is appropriate by one bridging fiber in a glass matrix 12-14This and the crack resistance due to fiber-bridging is increase in fracture toughness in our study attributed to the large interfacial fictional stress .t compared with those in glass specimens, where t is R=2 p(u)du=2 a(x)(du/dx)dx (14) so low that the embedded fiber can be pulled out of the matrix completely without breaking. The accompanying crack resistance is practically unu- where 2ul, and 2u2 are the CODs at xI and x2, sable in industrial applications since the crack respectively. Both u and x are measured experi- opening is too large (in the order of the length of mentally the embedded fiber). Therefore optimum values of Alternatively, inserting this narrow bridging T, not minimum values of T, need to be investigated zone into the Barenblatt equation gives to achieve maximum crack resistance within th maximum practical COD. To achieve a steep rising R-curve, this work based on elastic bridging sug r=(2/x)2|o(x)/x/d (15) gests the need to increase t in continuous fiber reinforced composites with low t values The final debond length, Lfa, of fibers bef ore For specimens with fiber bridging, a stable crack of failure can be calculated from Appendix B c2 mm length was obtained with a maximum u of 0.7 um. Since this COd exceeds u*=0.4 um L=2Lp/(1-(1-20Lp/EFn)/](16) steady-state grain-bridging was achieved. As Numerical calculations using the measured values shown in Fig. 3(a)and(b), the crack profile differs of Lp, or and u result in Lfd=90 um on average for for fiber reinforced specimens compared to the the specimens studied. This debond length is small monolithic samples due to the closure stress compared to the fiber diameter due to the large from fiber-bridging. Since the deviation of the value of t crack profile from linear is slight, the result for monolithic Al2O3 obtained from eqn(10b)was used to represent the grain-bridging contribution here. 5 Summary Since steady-state was reached, Rg=R:-28Jm The maximum tensile stress in each bridging Alumina specimens reinforced with con- fiber before failure was calculated from eqn(5) tinuous SiC fibers were processed by pres- using the measured value of cod and the calcu cureless sintering and exhibit both grain- and lated value of t. The average value of or was fiber-bridging 3.2 GPa. This value exceeds the fiber fracture Crack propagation and crack profile mea- strength S because or corresponds to the fiber stress surements were performed in situ inside the at the crack plane with maximum tensile strain SEM. The crack profile measurements provide hile the fiber failure occurred at a weak point of information on crack tip shielding due to the fiber away from the crack plane. The tensile grain-and fiber-bridging force in the fibers at the crack plane divided by the Crack resistance and fracture toughness as a bridging zone area results in the closure stress function of crack extension were computed by within the bridging zone. The closure stress thus both stress intensity considerations and the J- obtained is p=o=197 MPa. Using this value and integral, together with the applied stress the measured COd values, eqn (14)results in intensity factor. The results were shown to be Re=76 Jm-2 and R=Ro+R*+R=121 J m-2 self-consistent Correspondingly, a total toughness T=(Re) of The fracture strength of fibers in the matrix 7.3 MPam/2 was obtained were evaluated. A technique for measuring the
The results of eqns (12) and (13) should agree with each other to be consistent. When there are only a few ®bers bridging a narrow region of the crack, a continuous closure stress can be approximated over the narrow bridging region, x1 x x2. If the ®bers are far apart from each other, it is more appropriate to use point forces instead of a continuous stress. In this study, the two bridging ®bers were very close to each other in a narrow bridging zone of 500m. Accordingly, the use of a continuous closure stress within the narrow bridging zone, x1 x x2, is appropriate and the crack resistance due to ®ber-bridging is: Rf 2 u 2 u1 p udu 2 x 2 x1 x du=dxdx 14 where 2u1, and 2u2 are the CODs at x1 and x2, respectively. Both u and x are measured experimentally. Alternatively, inserting this narrow bridging zone into the Barenblatt equation gives: Tf 2= 1=2 x 2 x1 x=x1=2 dx 15 For specimens with ®ber bridging, a stable crack of &2 mm length was obtained with a maximum u of 0.7m. Since this COD exceeds u 04m, steady-state grain-bridging was achieved. As shown in Fig. 3(a) and (b), the crack pro®le diers for ®ber reinforced specimens compared to the monolithic samples due to the closure stress from ®ber-bridging. Since the deviation of the crack pro®le from linear is slight, the result for monolithic Al2O3 obtained from eqn (10b) was used to represent the grain-bridging contribution here. Since steady-state was reached, Rg R g=28 J mÿ2 . The maximum tensile stress in each bridging ®ber before failure was calculated from eqn (5) using the measured value of COD and the calculated value of . The average value of f was 3.2 GPa. This value exceeds the ®ber fracture strength S because f corresponds to the ®ber stress at the crack plane with maximum tensile strain, while the ®ber failure occurred at a weak point of the ®ber away from the crack plane. The tensile force in the ®bers at the crack plane divided by the bridging zone area results in the closure stress within the bridging zone. The closure stress thus obtained is p =197MPa. Using this value and the measured COD values, eqn (14) results in Rf=76 J mÿ2 and R Ro R g Rf=121 J mÿ2 . Correspondingly, a total toughness T RE0 1=2 of 7.3MPam1/2 was obtained. Alternatively, with the measured values of x1 and x2 and the calculated closure stress of =197MPa, eqn (15) results in Tf=3.1MPam1/2 and T To T g Tf or 7.5 MPam1/2. Both results are consistent with the experimental value of the applied stress intensity factor, Ka=7.4MPam1/2. An increase in crack growth resistance of 76 J mÿ2 and a corresponding fracture toughness increase of 3.2 MPam1/2 from the elastic bridging of two ®bers is remarkable, compared to an increase in fracture toughness of only 0.2 MPam1/2 by one bridging ®ber in a glass matrix.12±14 This increase in fracture toughness in our study is attributed to the large interfacial ®ctional stress, compared with those in glass specimens, where is so low that the embedded ®ber can be pulled out of the matrix completely without breaking. The accompanying crack resistance is practically unusable in industrial applications since the crack opening is too large (in the order of the length of the embedded ®ber). Therefore. optimum values of , not minimum values of , need to be investigated to achieve maximum crack resistance within the maximum practical COD. To achieve a steep rising R-curve, this work based on elastic bridging suggests the need to increase in continuous ®ber reinforced composites with low values. The ®nal debond length, Lfd, of ®bers before failure can be calculated from Appendix B Lfd 2Lp=1 ÿ 1 ÿ 2fLp=Efu 1=2 16 Numerical calculations using the measured values of Lp; f and u result in Lfd=90m on average for the specimens studied. This debond length is small compared to the ®ber diameter due to the large value of . 5 Summary . Alumina specimens reinforced with continuous SiC ®bers were processed by pressureless sintering and exhibit both grain- and ®ber-bridging. . Crack propagation and crack pro®le measurements were performed in situ inside the SEM. The crack pro®le measurements provide information on crack tip shielding due to grain- and ®ber-bridging. . Crack resistance and fracture toughness as a function of crack extension were computed by both stress intensity considerations and the Jintegral, together with the applied stress intensity factor. The results were shown to be self-consistent. . The fracture strength of ®bers in the matrix were evaluated. A technique for measuring the Crack closure stresses in ®ber reinforced brittle matrix composites 597
H. Xu, C. P Ostertag fiber-matrix interfacial frictional stress t was 17. Brun, M. K and Singh, R N, Effect of thermal expan- developed. The importance of the interfacial sion mismatch and fiber coating on the fiber /matrix inter frictional stress was discussed facial shear stress in cera trix composites. Ad Ceran.Mat,1988,3(5),506-509 A crack resistance of x 40 m- per fiber due 18. Bennison, S J and Lawn, B.R., Role of interfacial grain to elastic fiber-bridging was obtained for the bridging sliding friction in the crack-resistance and fiber reinforced composites strength properties of nontransforming ceramics. Acta Metall.,1989,37(10),26592671 19. Chantikul, P, Bennison, S. J and Lawn, B. R, Role of grain size in the strength and r-curve properties of alu- mina.J.Am. Ceran.Soc.,1990,73(8),2419-2427 References 20. Barenblatt, G. I, The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mechan, 1962, 7,55- 1. Zok. F. Sbaizero. O. Horn. C. L. and Evans. A. G. Mode i fracture resistance of a laminated fiber-reinforced ceramic.J. Am. Ceram Soc., 1991, 74(1), 187-193 2. Nair. S. V. Gwo. TJ. Narbut, N. M. and Kohl. L G Mechanical behavior of a continuous sic fiber reinforced RBSN-matrix composite. J. Am. Ceram. Soc., 1991 Appendix A 7410),2551-2558 and Hockey, B J, Crack-interface grain bridging as a From eqn(4) in the text, it can be obtained that fracture resistance mechanism in ceramics: I. Experi- 9 279-289 dy on alumina. J. Am. Ceram. Soc.. 1987, 70(4). or(u)= lat/R (A1) Rodel, J, Kelly, J. F and Lawn, B.R., In situ measure mets of bridging crack intesacesigothe scann ing electio Taking z as the coordinate axis in the direction 5. Frei, H and Grathwohl, G, New test method for engi- parallel to the fiber with the origin at the crack ring ceramics- in, situ microscopy investigation. plane, the tensile stress, o(Z, in the fiber at an 6. Ostertag, C. P, Chemical reaction between Al,Oa and sic arbitrary position, Z, within the debonded region at the fiber/matrix interface. Unpublished work can be obtained. since 7. ASTM designation E 399, standard test method for plane strain fracture toughness of metallic materials. In ASTM Standards, Part 10. American Society for Testing and o(z)R= 2RI(Ld-Z Materials, Philadelphia, PA, 1981 8. Thouless. M. D. Sbaizero O. Sig. L.S. and Evans. A. G therefore Effect of interface mechanical properties on pullout of SiC fiber reinforced lithium aluminum silicate glass cera- g(z=2t(Ld-Z/R (A3) mic.J.Am. Ceran.Soe.1989,72(4),525-532 9. Weihs, T. P. and Ni, W. D, Experimental examination of the push-down technique for measurina the sliding Then the tensile strain in the fiber at position Z is resistance of silicon carbide fibers in a ceramic matrix. J Am. Ceran.Soc.,1991,74(3),524-534 10. Kerans, R.K., Hay, R. S and Pagano, N.J., The role of E2=o(z)/Er= 2t(Ld-Z)/REf (A4) he fiber-matrix interface in ceramic composites Ceramic Bulletin,l989,68(2),548-553 The half crack opening displacement, u, can thus friction and debond strength of aligned ceramic matrix be calculated using eqn(A4) composites. In Euro-Ceramics, Vol 13, Engineering Cera- G de With,R A Terpstra and M. Metselear, u=ezdz= 2t(Ld-Z)/Er Rdz=tLd2/Er R) Applied Science, London, 1989, 453-464 12.C W. Fuller Jr. E. F. and Swanson. P. Fr echanics characterization of crack/fiber interacti 1987,8(7-8),258-264 13. Coyle,T.W,Palamides, T.R.Freimaxs' 1987 y in The debonding length can then be obtained from W. Fuller. Jr E.R. and Deshmukh. U. V. Crack-fiber ceramic matrix composites. Proceedings of the 1987 Yor- eqn(A5)as theast Regional meeting of TMS, High Temperature Structural Composites: Synthesis, Characterization, and Properties. HoBoken, NJ, 1987, p. 1458-1464 Ld=(Er Ru/r) Butler. L, Ph D. thesis, Mechanical properties of fibre reinforced glass matrix composites. Lehigh University, By inserting eqn(A6) into(Al), the tensile stress in 15. Ostertag. C. P, Drescher-Krasicka, E, Novel residual the fiber between the crack faces can be expressed stress measurement techniques in fiber reinforced compo- as a function of u as sites.J Mat. Sci., in press for publication 16. Xu, H, Ostertag. C. P, Braun, L M. and lloyd, I, Short crack mechanical properties and failure mechanisms of o1(u)=2L/R=(4Eu/R)1/2(A7) Si3N4 matrix SiC fiber composites. J. Am. Ceram. Soc., 1994,77(7,1889189 which is eqn(5)in the text
®ber-matrix interfacial frictional stress was developed. The importance of the interfacial frictional stress was discussed. . A crack resistance of &40 J mÿ2 per ®ber due to elastic ®ber-bridging was obtained for the ®ber reinforced composites. References 1. Zok, F., Sbaizero, O., Horn, C. L. and Evans, A. G., Mode I fracture resistance of a laminated ®ber-reinforced ceramic. J. Am. Ceram. Soc., 1991, 74(1), 187±193. 2. Nair, S. V., Gwo, T. J., Narbut, N. M. and Kohl, L.G., Mechanical behavior of a continuous SiC ®ber reinforced RBSN-matrix composite. J. Am. Ceram. Soc., 1991, 74(10), 2551±2558. 3. Swanson, P. J., Fairbanks, C. J., Lawn, B. R., Mai, Y. W. and Hockey., B. J., Crack-interface grain bridging as a fracture resistance mechanism in ceramics: I. Experimental study on alumina. J. Am. Ceram. Soc., 1987, 70(4), 279±289. 4. Rodel, J., Kelly, J. F. and Lawn, B. R., In situ measurements of bridging crack interfaces in the scanning electron microscope. J. Am. Ceram. Soc., 1990, 73(11), 3313±3318. 5. Frei, H. and Grathwohl, G., New test method for engineering ceramicsÐin situ microscopy investigation. Ceram. Forum Int., 1991, 68, 27±33. 6. Ostertag, C. P., Chemical reaction between Al2O3 and SiC at the ®ber/matrix interface. Unpublished work. 7. ASTM designation E 399, standard test method for planestrain fracture toughness of metallic materials. In ASTM Standards, Part 10. American Society for Testing and Materials, Philadelphia, PA, 1981. 8. Thouless, M. D., Sbaizero, O., Sig, L. S. and Evans, A. G., Eect of interface mechanical properties on pullout of a SiC ®ber reinforced lithium aluminum silicate glass ceramic. J. Am. Ceram. Soc., 1989, 72(4), 525±532. 9. Weihs, T. P. and Nix, W. D., Experimental examination of the push-down technique for measurina the sliding resistance of silicon carbide ®bers in a ceramic matrix. J. Am. Ceram. Soc., 1991, 74(3), 524±534. 10. Kerans, R. K., Hay, R. S. and Pagano, N. J., The role of the ®ber-matrix interface in ceramic composites. Ceramic Bulletin, 1989, 68(2), 548±553. 11. Shafry, N. Brandon, D. G. and Terasaki, M., Interfacial friction and debond strength of aligned ceramic matrix composites. In Euro-Ceramics, Vol. 13, Engineering Ceramics, eds G. de With, R. A. Terpstra and M. Metselear, Elsevier Applied Science, London, 1989, 453±464. 12. Coyle, T. W., Fuller Jr, E. F. and Swanson, P., Fracture mechanics characterization of crack/®ber interactions in ceramic matrix composites. Ceramic. Engr. and Sci. Proc., 1987, 8(7±8), 258±264. 13. Coyle, T. W., Palamides, T. R., Freiman, S. W., Fuller, Jr, E. R. and Deshmukh, U. V., Crack-®ber interactions in ceramic matrix composites. Proceedings of the 1987 Yortheast Regional meeting of TMS, High Temperature Structural Composites: Synthesis, Characterization, and Properties. HoBoken, NJ, 1987, p. 1458±1464. 14. Butler. L., Ph.D. thesis, Mechanical properties of ®bre reinforced glass matrix composites. Lehigh University, Bethlehem, PA. 1993. 15. Ostertag, C. P., Drescher-Krasicka, E., Novel residual stress measurement techniques in ®ber reinforced composites. J Mat. Sci., in press for publication. 16. Xu, H., Ostertag, C. P., Braun, L. M. and Lloyd, I., Short crack mechanical properties and failure mechanisms of Si3N4 matrix SiC ®ber composites. J. Am. Ceram. Soc., 1994, 77(7), 1889±1896. 17. Brun, M. K. and Singh, R. N., Eect of thermal expansion mismatch and ®ber coating on the ®ber/matrix interfacial shear stress in ceramic matrix composites. Ad. Ceram. Mat., 1988, 3(5), 506±509. 18. Bennison, S. J. and Lawn, B. R., Role of interfacial grainbridging sliding friction in the crack-resistance and strength properties of nontransforming ceramics. Acta Metall., 1989, 37(10), 2659±2671. 19. Chantikul, P., Bennison, S. J. and Lawn, B. R., Role of grain size in the strength and R-curve properties of alumina. J. Am. Ceram. Soc., 1990, 73(8), 2419±2427. 20. Barenblatt, G. I., The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mechan., 1962, 7, 55± 129. Appendix A From eqn (4) in the text, it can be obtained that: f u Ld =R A1 Taking Z as the coordinate axis in the direction parallel to the ®ber with the origin at the crack plane, the tensile stress, Z, in the ®ber at an arbitrary position, Z, within the debonded region can be obtained. Since zR2 2R Ld ÿ Z A2 therefore Z 2 Ld ÿ Z=R A3 Then the tensile strain in the ®ber at position Z is "z z=Ef 2 Ld ÿ z=REf A4 The half crack opening displacement, u, can thus be calculated using eqn (A4) as: u Ld o "zdZ Ld o 2 Ld ÿ Z=EfRdZ Ld2= EfR A5 The debonding length can then be obtained from eqn (A5) as: Ld EfRu= 1=2 A6 By inserting eqn (A6) into (A1), the tensile stress in the ®ber between the crack faces can be expressed as a function of u as: 1 u 2Ld =R 4Ef u=R 1=2 A7 which is eqn (5) in the text. 598 H. Xu, C. P. Ostertag
Crack closure stresses in fiber reinforced brittle matrix composite 599 Appendix b or()=(4EFu/R)1/2 If Lfd is the maximum debonding length before Combining eqns(bl)and(B2)gives fiber failure, and 2uo is the maximum COd before fiber failure, then at the crack plane where Z= 0: S=a(O)(Lrd-Lp)Lrd or(0)=2tLdf/R Similarly, the stress in the fiber at the fiber position From eqns(bl)B4), it can be obtained that just before fiber failure is: 4(Lp13/R2+4(LpS-l0EF)R+S2=0(B5) S=2I(Lfd-Lp/R (B2) By solving eqn(B5) for t, eqn(7)in the text can be with Lp the pull-out length of the fibers. Right obtained By combining eqns(B2)and(7,(16)in before fiber failure, eqn(A7)becomes: the text can be obtained
Appendix B If Lfd is the maximum debonding length before ®ber failure, and 2u0 is the maximum COD before ®ber failure, then at the crack plane where Z O: f 0 2 Ldf=R B1 Similarly, the stress in the ®ber at the ®ber position just before ®ber failure is: S 2 Lfd ÿ Lp=R B2 with Lp the pull-out length of the ®bers. Right before ®ber failure, eqn (A7) becomes: f uo 4Ef uo=R 1=2 B3 Combining eqns (B1) and (B2) gives: S 0 Lfd ÿ LpLfd B4 From eqns (B1)±(B4), it can be obtained that: 4 Lp 2 =R2 4 LpS ÿ u0Ef=R S2 0 B5 By solving eqn (B5) for , eqn (7) in the text can be obtained. By combining eqns (B2) and (7), (16) in the text can be obtained. Crack closure stresses in ®ber reinforced brittle matrix composites 599