《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_CRITERIA FOR CRACK DEFLECTION

1998 Elsevier Science Ltd. All rights reserved Printed in Great Brita PII:S0266-3538(98)00043-8 0266-3538/98/S-see front matter ELSEVIER CRITERIA FOR CRACK DEFLECTION/PENETRATION CRITERIA FOR FIBER-REINFORCED CERAMIC MATRIX COMPOSITES BK.Ahn, aw. A Curtin, a*T A Parthasarathy &RE.Dutton "Department of Engineering Science and mechanics, Virginia Polytechnic Institute and state University, Blacksburg, VA 24061, USA mAterials Directorate, Wright Laboratory, WL/MLLM Wright-Patterson AFB, OH 45433, USA (Received 15 July 1997; revised 12 December 1997; accepted 10 February 1998) Abstract 1 INTRODUCTION Deflection of a matrix crack at the fiber/matrix interface is the initial mechanism required for obtaining enhanced eramic materials are attractive for use in high-tem toughness in ceramic-matrix composites(CMCs). Here, perature applications because of their high strength and energy release rates are calculated for matrix cracks that low density. However, their low fracture toughness, or either deflect or penetrate at the interface of an axisym- poor resistance against crack propagation, restricts the metric composite as a function of elastic mismatch, fiber use of monolithic materials to a large extent. The rein volume fraction, and length of the deflected or penetrated forcement of ceramics with ceramic fibers has been rack. The energy release rates for the competing fracture shown to be a very effective way of improving tough modes are calculated numerically by means of the axi- ness, and the behavior of cracks at the fiber/matrix symmetric damage model developed by Pagano, which interface is known to be the key factor for obtaining the utilizes Reissner's variational principle and an assumed enhanced toughness. The first fracture mode in a Cm stress field to solve the appropriate boundary value pro- is matrix cracking. If the interface is weak enough for blems. Crack deflection versus penetration is predicted by the matrix crack to be deflected along the interface, the using an energy criterion analogous to that developed by fibers remain intact and the composite can be tough. If He and Hutchinson. Results show that, for equal crack he interface is too strong, the matrix crack penetrates extensions in deflection and penetration, crack deflection into the fibers and the composite is brittle like a mono- is more difficult for finite crack extension and finite fiber lithic ceramic. Therefore, the crack propagation beha- volume fraction than in the He and Hutchinson limit of vior at the interface is critical to toughening in CMCs zero volume fraction and/or infinitesimal crack extension There are at least three possible crack paths for a Allowing for different crack extensions for the deflected matrix crack at the fiber /matrix interface in an axisym and penetrating cracks is shown to have a small effect at metric composite. Figure I shows the simplest possible larger volume fractions. Fracture- mode data on model failure paths: crack deflection on one side of the inter- omposites with well-established constitutive properties face(singly deflected crack); crack deflection on both show penetration into the fibers (brittle behavior), as sides (doubly deflected crack); and crack penetration predicted by the present criteria for crack extensions lar- across the interface. Which of these three paths the ger than 0-2% of the fiber radius and in contrast to the He crack selects is the central issue of the present work, and and Hutchinson criterion, which predicts crack deflection. of much previous work. Stress and energy criteria are This result suggests that the latter criterion may over- typically used to determine the crack path: the former is estimate the prospects for crack deflection in composites governed by the local asymptotic stress field at the with realistic fiber volume fractions and high ratios of interface, while the latter is based on the differences of fiber to matrix elastic modulus. C 1998 Elsevier Science work of fracture along possible alternative crack Ltd. All rights reserved paths.- The recent development of effective techniques to measure the interface toughness has made it possible Keywords: A. ceramic matrix composites, B. fracture to use the energy criterion more easily while measuring toughness, fiber/matrix interface, crack deflection/pene- the interface strength still remains difficult to perform. 3.4 tration, energy criterion Here we adopt the energy approach, as described below From the energy perspective, a crack will grow when *To whom correspondence should be addressed at: Division the energy available in the stress field around it, which is of Engineering, Brown University, Providence, RI 02912, relieved as the crack grows, is sufficient to make up for USA the loss in energy upon creation of the new crack surface

CRITERIA FOR CRACK DEFLECTION/PENETRATION CRITERIA FOR FIBER-REINFORCED CERAMIC MATRIX COMPOSITES B. K. Ahn,a W. A. Curtin,a * T. A. Parthasarathyy & R. E. Dutton a Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA b Materials Directorate, Wright Laboratory, WL/MLLM Wright-Patterson AFB, OH 45433, USA (Received 15 July 1997; revised 12 December 1997; accepted 10 February 1998) Abstract De¯ection of a matrix crack at the ®ber/matrix interface is the initial mechanism required for obtaining enhanced toughness in ceramic-matrix composites (CMCs). Here, energy release rates are calculated for matrix cracks that either de¯ect or penetrate at the interface of an axisym￾metric composite as a function of elastic mismatch, ®ber volume fraction, and length of the de¯ected or penetrated crack. The energy release rates for the competing fracture modes are calculated numerically by means of the axi￾symmetric damage model developed by Pagano, which utilizes Reissner's variational principle and an assumed stress ®eld to solve the appropriate boundary value pro￾blems. Crack de¯ection versus penetration is predicted by using an energy criterion analogous to that developed by He and Hutchinson. Results show that, for equal crack extensions in de¯ection and penetration, crack de¯ection is more dicult for ®nite crack extension and ®nite ®ber volume fraction than in the He and Hutchinson limit of zero volume fraction and/or in®nitesimal crack extension. Allowing for di€erent crack extensions for the de¯ected and penetrating cracks is shown to have a small e€ect at larger volume fractions. Fracture-mode data on model composites with well-established constitutive properties show penetration into the ®bers (brittle behavior), as predicted by the present criteria for crack extensions lar￾ger than 0.2% of the ®ber radius and in contrast to the He and Hutchinson criterion, which predicts crack de¯ection. This result suggests that the latter criterion may over￾estimate the prospects for crack de¯ection in composites with realistic ®ber volume fractions and high ratios of ®ber to matrix elastic modulus. # 1998 Elsevier Science Ltd. All rights reserved Keywords: A. ceramic matrix composites, B. fracture toughness, ®ber/matrix interface, crack de¯ection/pene￾tration, energy criterion 1 INTRODUCTION Ceramic materials are attractive for use in high-tem￾perature applications because of their high strength and low density. However, their low fracture toughness, or poor resistance against crack propagation, restricts the use of monolithic materials to a large extent. The rein￾forcement of ceramics with ceramic ®bers has been shown to be a very e€ective way of improving tough￾ness, and the behavior of cracks at the ®ber/matrix interface is known to be the key factor for obtaining the enhanced toughness.1 The ®rst fracture mode in a CMC is matrix cracking. If the interface is weak enough for the matrix crack to be de¯ected along the interface, the ®bers remain intact and the composite can be tough. If the interface is too strong, the matrix crack penetrates into the ®bers and the composite is brittle like a mono￾lithic ceramic. Therefore, the crack propagation beha￾vior at the interface is critical to toughening in CMCs. There are at least three possible crack paths for a matrix crack at the ®ber/matrix interface in an axisym￾metric composite. Figure 1 shows the simplest possible failure paths: crack de¯ection on one side of the inter￾face (singly de¯ected crack); crack de¯ection on both sides (doubly de¯ected crack); and crack penetration across the interface. Which of these three paths the crack selects is the central issue of the present work, and of much previous work. Stress and energy criteria are typically used to determine the crack path: the former is governed by the local asymptotic stress ®eld at the interface, while the latter is based on the di€erences of work of fracture along possible alternative crack paths.2±5 The recent development of e€ective techniques to measure the interface toughness has made it possible to use the energy criterion more easily while measuring the interface strength still remains dicult to perform.3,4 Here we adopt the energy approach, as described below. From the energy perspective, a crack will grow when the energy available in the stress ®eld around it, which is relieved as the crack grows, is sucient to make up for the loss in energy upon creation of the new crack surface. Composites Science and Technology 58 (1998) 1775±1784 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0266-3538(98)00043-8 0266-3538/98/$Ðsee front matter 1775 *To whom correspondence should be addressed at: Division of Engineering, Brown University, Providence, RI 02912, USA

1776 B. K Ahn et al crack, and hence hold in the limit of ad, ap-0. For identical elastic properties, i=1/2 but for fibers stiffer than the matrix 2< 1/2 while for matrix stiffer than the fibers 1>1/2. Therefore, taking the limit of zero crack extension in either case leads to either (i) zero energy release rate (< 2)and hence no possible cracking or (ii) infinite energy release rate (i>1/2) ence cracking at any finite stress level To overcome the basic difficulties evident from the above results, He and Hutchinson(HH) proposed a nice a to consider the ratio Ga/Gp with ad=ap, in which ae e concept that led to an analytic and finite result for assessing deflection versus penetration. HH propose the crack extension length drops out of the problem Singly deflected crack Doubly deflected crack Penetrated crack Furthermore, HH proposed that crack deflection would Fig 1. Three potential fracture modes at a fiber/matrix inter- occur if Gd/Gp> Td/Tp. For a penetrating crack at the fiber surface, Tp= r where T, is the critical energy release rate or surface energy of the fiber, and for a To predict crack growth thus requires an ability to cal- deflecting crack at the interface, Id= ri where Ti is the culate the energy release rate, G, or elastic energy surface energy of the interface normally lower relieved per unit area of crack advance, and a knowl- than Im of matrix. Hence, the deflection criterion at the edge of the underlying surface fracture energy, T, cre- fiber/matrix interface is ated as the crack grows. Here, we denote by Ga and rd the energy release rate and surface energy for the case of Ga/Gp <Ti/rf deflection, and by Gp and Tp the corresponding quan- tities for penetration. If Gd> rd the crack can deflect while if Gp>Tp the crack can penetrate the fiber. It is This criterion was then studied in considerable detail by not clear which path is selected if both conditions are He and Hutchinson under certain conditions. They stu satisfied, and in fact other fundamental problems arise died a planar interface under plane strain and traction for elastically-mismatched materials, as discussed briefly boundary conditions, with isotropic'matrix'and'fiber below Their analysis implicitly assumed that the crack size in For a bi-material interface under plane strain condi- the 'matrix'is semi-infinite and, as noted above, the tions, the stresses in the system depend on two basic crack extensions are considered infinitesimal. The spe material combinations the dundurs parameters' cial case of B=0 was studied although limited result suggested that the deflection criterion was only weakly affected by the value of B relative to its dependence on E(1-m)-Em(1-v a. Singly and doubly deflected interface cracks were EA1-12+Em( considered within the limitations of plane strain. HH E/(l,m(1-2vm)-Em(1+v(1-2y () also considered cracks approaching the interface at E(1-m)+Em(1-v) oblique angles. The result of hH for the perpendicular, doubly deflected crack is shown as a solid line in Fig. 5. (The result shown here is adopted from Ref. 5 which provides the corrected result of the original work by HH For a crack perpendicular to the interface and under in Ref. 2.) applied load parallel to the interface, the energy release In the present paper, we adopt the hh deflection cri- rates as a function of crack extension ad along the terion based on energy and a ratio of energy release interface and a, into the fiber are well-known to be of rates. We then investigate, using a numerical technique the forms2 developed by Pagano that employs Reissners varia- tional principle, the dependence of the deflection criter- ion on crack extension lengths ad, ap and on fiber (2) volume fraction V for an axisymmetric fiber/matrix interface geometry. We restrict the problem to a per- pendicular matrix crack impinging onto the interface In eqn(2), kr is a Mode I stress-intensity-like factor, d and to the doubly-deflected crack case shown previously and c are complex functions of the Dundurs' para- to be the dominant fracture mode. In the limits of small neters, and the exponent i is also a function of the ad, ap and small V accessible numerically, we reproduce elastic mismatch between fiber and matrix. The above the corrected hh results which also validates the use of forms arise from the asymptotic near-tip field of the the relatively new numerical technique. We also

To predict crack growth thus requires an ability to cal￾culate the energy release rate, G, or elastic energy relieved per unit area of crack advance, and a knowl￾edge of the underlying surface fracture energy, ÿ, cre￾ated as the crack grows. Here, we denote by Gd and ÿd the energy release rate and surface energy for the case of de¯ection, and by Gp and ÿp the corresponding quan￾tities for penetration. If Gd  ÿd the crack can de¯ect while if Gp  ÿp the crack can penetrate the ®ber. It is not clear which path is selected if both conditions are satis®ed, and in fact other fundamental problems arise for elastically-mismatched materials, as discussed brie¯y below. For a bi-material interface under plane strain condi￾tions, the stresses in the system depend on two basic material combinations, the Dundurs parameters6  ˆ Ef…1 ÿ v2 m† ÿ Em…1 ÿ v2 f † Ef…1 ÿ v2 m ‡ Em…1 ÿ v2 f † ; 2 ˆ Ef…1vm†…1 ÿ 2vm† ÿ Em…1 ‡ vf†…1 ÿ 2vf† Ef…1 ÿ v2 m† ‡ Em…1 ÿ v2 f † …1† For a crack perpendicular to the interface and under applied load parallel to the interface, the energy release rates as a function of crack extension ad along the interface and ap into the ®ber are well-known to be of the forms2 Gd ˆ d…; †k2 Ia1ÿ2l d ; Gp ˆ c…; †kI 1a1ÿ2l p …2† In eqn (2), kI is a Mode I stress-intensity-like factor, d and c are complex functions of the Dundurs' para￾meters, and the exponent l is also a function of the elastic mismatch between ®ber and matrix. The above forms arise from the asymptotic near-tip ®eld of the crack, and hence hold in the limit of ad; ap ! 0. For identical elastic properties, l ˆ 1=2 but for ®bers sti€er than the matrix l < 1=2 while for matrix sti€er than the ®bers l > 1=2. Therefore, taking the limit of zero crack extension in either case leads to either (i) zero energy release rate …l < 1 =2† and hence no possible cracking or (ii) in®nite energy release rate …l > 1=2† and hence cracking at any ®nite stress level. To overcome the basic diculties evident from the above results, He and Hutchinson (HH) proposed a nice concept that led to an analytic and ®nite result for assessing de¯ection versus penetration.2 HH proposed to consider the ratio Gd=Gp with ad ˆ ap, in which case the crack extension length drops out of the problem. Furthermore, HH proposed that crack de¯ection would occur if Gd=Gp > ÿd=ÿp. For a penetrating crack at the ®ber surface, ÿp ˆ ÿf where ÿf is the critical energy release rate or surface energy of the ®ber, and for a de¯ecting crack at the interface, ÿd ˆ ÿi where ÿi is the surface energy of the interface which is normally lower than ÿm of matrix. Hence, the de¯ection criterion at the ®ber/matrix interface is Gd=Gp < ÿi=ÿf …3† This criterion was then studied in considerable detail by He and Hutchinson under certain conditions. They stu￾died a planar interface under plane strain and traction boundary conditions, with isotropic `matrix' and `®ber'. Their analysis implicitly assumed that the crack size in the `matrix' is semi-in®nite and, as noted above, the crack extensions are considered in®nitesimal. The spe￾cial case of  ˆ 0 was studied, although limited results suggested that the de¯ection criterion was only weakly a€ected by the value of  relative to its dependence on . Singly and doubly de¯ected interface cracks were considered within the limitations of plane strain. HH also considered cracks approaching the interface at oblique angles. The result of HH for the perpendicular, doubly de¯ected crack is shown as a solid line in Fig. 5. (The result shown here is adopted from Ref. 5 which provides the corrected result of the original work by HH in Ref. 2.) In the present paper, we adopt the HH de¯ection cri￾terion based on energy and a ratio of energy release rates. We then investigate, using a numerical technique developed by Pagano that employs Reissner's varia￾tional principle, the dependence of the de¯ection criter￾ion on crack extension lengths ad, ap and on ®ber volume fraction Vf for an axisymmetric ®ber/matrix interface geometry. We restrict the problem to a per￾pendicular matrix crack impinging onto the interface, and to the doubly-de¯ected crack case shown previously to be the dominant fracture mode. In the limits of small ad, ap and small Vf accessible numerically, we reproduce the corrected HH results, which also validates the use of the relatively new numerical technique. We also Fig. 1. Three potential fracture modes at a ®ber/matrix inter￾face. 1776 B. K. Ahn et al

Crack deflection/penetration criteria for fiber-reinforced ceramic matrix composites 1777 demonstrate the insensitivity of the deflection criterion substrate cracking. Their result showed that thermal on B. At finite ad, ap and Vf, we find that the ratio of Gal expansion misfit can be significant in systems with pla G, decreases well below the HH limit for a>0, imply nar interfaces such as layered materials and thin film ing that deflection is more difficult than previously structures, but in fiber-reinforced composites the effect anticipated in this regime. In fact, we find Ga/Gp-0 as of misfit is expected minimal because of the coupling a-1 for fixed fiber and matrix Poisson,s ratios. between axial and radial residual stresses. Of some Relaxing the assumption of ad=ap we find that, at importance and relevance to the present work, HEH V=1%, the ratio Ga/Gp depends weakly on the ratio demonstrated that when residual stresses are present of aa ap with a scaling close to the asymptotic behavior the ratio of Ga/Gp is always dependent on ad and ap; the (ad/ap) predicted from eqn(2), while at a more rea- convenient cancellation obtained in the absence of resi- listic value of V=40% the dependence is even weaker. dual stresses does not occur. Thus, the deflection criter- Finally, we compare various predictions of deflection ion is an explicit function of both ad and ap. In the and penetration to experimental results on model com- present work we show that, for realistic volume frac- posites with well-established constitutive properties For tions and small to moderate crack extensions, the ratio these systems, the hh result predicts crack deflection Ga Gp depends on the crack extension even in the whereas the present results predict penetration over a absence of residual stresses. Furthermore, these finite wide range of crack extension lengths. In the experi length effects are probably larger than the effects ments, the real materials exhibit clear penetration, indi- obtained from realistic residual stress levels obtainable cating that the hh criterion overestimates the prospects in ceramic composites, particularly for the axisymmetric for crack deflection. The present results are consistent geometr with the data, indicating that finite volume fraction and The remainder of this paper is organized as follows finite ad and ap may play a key role in determining In Section 2, we define the specific problem to be solved deflection versus penetration. Ambiguity in the correct and briefly introduce the basic concepts of Pagano's choice for ad=ap, however, precludes any ability to Axisymmetric Damage Model and Reissner's varia- draw more definite conclusions on this issue until fur- tional principal. Section 3 contains detailed results on ther work is done and possibly new ideas are developed. deflection versus penetration. Comparison of the var Before proceeding with the present work, we close the ious deflection criteria with new experimental data is introduction with a brief presentation of recent impor- presented in Section 4. In Section 5, we summarize and tant work on this problem. Gupta et al. extended discuss directions for future work and Hutchinson,s work to the area of anisotropic materials for the case of a crack approaching perpend cular to the interface. They also derived a strength cri 2 MODELING OF FIBER/MATRIX INTERFACE terion for crack deflection, and confirmed their analysis CrACKs by using laser spallation experiment. Gupta et al.con- cluded that it is impossible to provide generalized dela- 2.1 Problem description mination charts as a function of a alone. Instead, they In order to predict the crack path at the fiber/matrix have tabulated the required values of the interface nterface, an axisymmetric microcomposite consisting of strength and fracture toughness for delamination in a isotropic, elastic fiber and matrix is considered. Fig. 2 number of composite materials. A later work by Marti- shows the fiber /matrix interface model with the appro nez and Gupta(MG) showed in detail that the energy priate boundary conditions. The main matrix crack lies criterion for deflection is sensitive to the material ani- in a plane perpendicular to the fiber and extending to sotropy and has corrected the previous work by HH. In the interface, where it may subsequently deflect along contrast to He and Hutchinson,s results, their calcula- the interface or penetrate into the fiber. Energy release tions show that the GaGp for the doubly deflected crack rates for deflection(Ga)and penetration(Gp) are calcu- higher than Ga Gp for singly deflected crack when lated and the ratio of Ga/Gp as a function of Dundurs Dundurs' parameter a is larger than zero. That is, the parameters a and B is determined. Detailed description doubly deflected crack is the dominating crack mode. on the boundary conditions in Fig. 2 is presented later Martinez and Gupta also examined the effect of aniso- in this section. tropy on the crack deflection by manipulating the ani To obtain energy release rates requires three calcula sotropy-related parameters including the other tions. First, with only the initial matrix crack extending Dundurs' parameter, B. They showed that B fiber /matrix interface, we calculate the elastic assumption may overestimate the Ga G, versus a beha- potential energy in the system at a fixed remote applied vior by 20-25% over the range B=-0-202 load. This gives the initial reference energy Wr. Second Following the work of Martinez and Gupta, He we advance the crack into the fiber by an amount ap and al.(HEH)provided a corrected result for the ratio G calculate the potential energy for this longer, penetrated Gp for the doubly deflected crack. More importantly, crack, Wp. Third, we advance the initial crack at a righ HEH investigated the influence of the residual stre angle along the interface by an equal amount ad to form n the competition between interface cracking the incipient deflected crack, and calculate the potential

demonstrate the insensitivity of the de¯ection criterion on . At ®nite ad, ap and Vf, we ®nd that the ratio of Gd/ Gp decreases well below the HH limit for  > 0, imply￾ing that de¯ection is more dicult than previously anticipated in this regime. In fact, we ®nd Gd=Gp ! 0 as  ! 1 for ®xed ®ber and matrix Poisson's ratios. Relaxing the assumption of ad ˆ ap we ®nd that, at Vf ˆ 1%, the ratio Gd/Gp depends weakly on the ratio of ad/ap with a scaling close to the asymptotic behavior …ad=ap† 1ÿ2l predicted from eqn (2), while at a more rea￾listic value of Vf ˆ 40% the dependence is even weaker. Finally, we compare various predictions of de¯ection and penetration to experimental results on model com￾posites with well-established constitutive properties. For these systems, the HH result predicts crack de¯ection whereas the present results predict penetration over a wide range of crack extension lengths. In the experi￾ments, the real materials exhibit clear penetration, indi￾cating that the HH criterion overestimates the prospects for crack de¯ection. The present results are consistent with the data, indicating that ®nite volume fraction and ®nite ad and ap may play a key role in determining de¯ection versus penetration. Ambiguity in the `correct' choice for ad ˆ ap, however, precludes any ability to draw more de®nite conclusions on this issue until fur￾ther work is done and possibly new ideas are developed. Before proceeding with the present work, we close the introduction with a brief presentation of recent impor￾tant work on this problem. Gupta et al.3 extended He and Hutchinson's work2 to the area of anisotropic materials for the case of a crack approaching perpendi￾cular to the interface. They also derived a strength cri￾terion for crack de¯ection, and con®rmed their analysis by using laser spallation experiment. Gupta et al. con￾cluded that it is impossible to provide generalized dela￾mination charts as a function of  alone. Instead, they have tabulated the required values of the interface strength and fracture toughness for delamination in a number of composite materials. A later work by Marti￾nez and Gupta (MG)4 showed in detail that the energy criterion for de¯ection is sensitive to the material ani￾sotropy and has corrected the previous work by HH. In contrast to He and Hutchinson's results, their calcula￾tions show that the Gd/Gp for the doubly de¯ected crack is higher than Gd/Gp for singly de¯ected crack when Dundurs' parameter  is larger than zero. That is, the doubly de¯ected crack is the dominating crack mode. Martinez and Gupta also examined the e€ect of aniso￾tropy on the crack de¯ection by manipulating the ani￾sotropy-related parameters including the other Dundurs' parameter, . They showed that  ˆ 0 assumption may overestimate the Gd/Gp versus  beha￾vior by 20ÿ25% over the range =ÿ0.20.2. Following the work of Martinez and Gupta,4 He et al.5 (HEH) provided a corrected result for the ratio Gd/ Gp for the doubly de¯ected crack. More importantly, HEH investigated the in¯uence of the residual stresses on the competition between interface cracking and substrate cracking. Their result showed that thermal expansion mis®t can be signi®cant in systems with pla￾nar interfaces such as layered materials and thin ®lm structures, but in ®ber-reinforced composites the e€ect of mis®t is expected minimal because of the coupling between axial and radial residual stresses. Of some importance and relevance to the present work, HEH demonstrated that when residual stresses are present, the ratio of Gd/Gp is always dependent on ad and ap; the convenient cancellation obtained in the absence of resi￾dual stresses does not occur. Thus, the de¯ection criter￾ion is an explicit function of both ad and ap. In the present work we show that, for realistic volume frac￾tions and small to moderate crack extensions, the ratio Gd/Gp depends on the crack extension even in the absence of residual stresses. Furthermore, these ®nite length e€ects are probably larger than the e€ects obtained from realistic residual stress levels obtainable in ceramic composites, particularly for the axisymmetric geometry. The remainder of this paper is organized as follows. In Section 2, we de®ne the speci®c problem to be solved and brie¯y introduce the basic concepts of Pagano's Axisymmetric Damage Model and Reissner's varia￾tional principal. Section 3 contains detailed results on de¯ection versus penetration. Comparison of the var￾ious de¯ection criteria with new experimental data is presented in Section 4. In Section 5, we summarize and discuss directions for future work. 2 MODELING OF FIBER/MATRIX INTERFACE CRACKS 2.1 Problem description In order to predict the crack path at the ®ber/matrix interface, an axisymmetric microcomposite consisting of isotropic, elastic ®ber and matrix is considered. Fig. 2 shows the ®ber/matrix interface model with the appro￾priate boundary conditions. The main matrix crack lies in a plane perpendicular to the ®ber and extending to the interface, where it may subsequently de¯ect along the interface or penetrate into the ®ber. Energy release rates for de¯ection (Gd) and penetration (Gp) are calcu￾lated and the ratio of Gd/Gp as a function of Dundurs parameters  and  is determined. Detailed description on the boundary conditions in Fig. 2 is presented later in this section. To obtain energy release rates requires three calcula￾tions. First, with only the initial matrix crack extending to the ®ber/matrix interface, we calculate the elastic potential energy in the system at a ®xed remote applied load. This gives the initial reference energy Wr. Second, we advance the crack into the ®ber by an amount ap and calculate the potential energy for this longer, penetrated crack, Wp. Third, we advance the initial crack at a right angle along the interface by an equal amount ad to form the incipient de¯ected crack, and calculate the potential Crack de¯ection/penetration criteria for ®ber-reinforced ceramic matrix composites 1777

1778 B. K Ahn et al Matrix Fiber Crack extending to z Debond Penetrati Matrix T lri=o Fiber tafO,r0 Crack Or}=t:,r)=0 Fig. 2. Debond and Penetration at fiber/matrix interface in energy for the deflected crack, Wa. The energy release rates are then calculated from the energy differences and the area of crack growth as G=("=H)1n4)Gn=(H4-w)1(r Fig 3. Schematic of ADM concentric cylinders. where ry is the fiber radius. The dependence of Ga/Gp cracks then simply define a particular boundary condi- versus a and B is then used as the deflection criterion tion along the surface of any region. The elasticity pro- th The crack extension is taken to be 0002 rf, blem within each region subject to appropriate 0-01 rr and 0-025 rf. To investigate the effect of different boundary conditions is solved using the Re eisner varia crack extensions in two types of cracks, we also fix ap at tional equation, as discussed briefly below 0-01 rand adopt various ad of 0-025 r, 0-01 rand 0-002 Reissner has shown that minimizing the functional rr. We use a 1% fiber volume fraction to approach the semi-infinite matrix crack limit. and a 40% volume fraction to model a realistic composite J=Fd-|rds:F=与+与)-W(5) 2.2 Numerical technique: axisymmetric damage model The axisymmetric damage model(ADM) was developed by Pagano to predict the stress and displacement dis- with respect to both stresses and displacements leads to tributions in composite constituents as well as the strain both the field equations and boundary conditions of lin- energy and energy release rates for cracked composites ear elasticity theory. In eqn(5), w is the complementary in which damage modes include fiber breaks, annular energy density, T; is the prescribed traction, and o; and 5 cracks in coatings or matrix, and debond cracks at are the stress and displacement components, respec- interfaces. Concentric cylindrical/annular elements are tively, in Cartesian coordinates. A comma after a sub used to model the fiber and matrix materials. Additional script represents a derivative with respect to the concentric annuli (radii r1, r2,.)can be introduced, as indicated coordinate, and Einsteins summation conven well as longitudinal sections(=1, 22,...), if necessary for tion is understood. V is an arbitrary volume enclosed by computational purpose, as indicated schematically in the entire surface S, while S is the portion of the Fig 3. This discretization forms regions bounded by ri boundary on which one or more traction components and ri+I and =; and E+I. The annuli, or layers, and are prescribed. The body forces have been neglected in sections are chosen so that cracks introduced into thethe formulation In the adm the stress field in each metry always lie along the boundaries of sections annular region is assumed to be one where oe and o: are (for transverse cracks)or layers(for longitudinal cracks) linear in the radial coordinate r, while the forms of and span one or more complete sections/ layers. The or and Tr are chosen to satisfy the axisymmetric

energy for the de¯ected crack, Wd. The energy release rates are then calculated from the energy di€erences and the area of crack growth as Gp ˆ…WpÿWr† =…r2 f ÿ…rfÿap† 2 † ; Gd ˆ…WdÿWr† =…2rfad† …4† where rf is the ®ber radius. The dependence of Gd/Gp versus  and  is then used as the de¯ection criterion with ad ˆ ap. The crack extension is taken to be 0.002 rf, 0.01 rf and 0.025 rf. To investigate the e€ect of di€erent crack extensions in two types of cracks, we also ®x ap at 0.01 rf and adopt various ad of 0.025 rf, 0.01 rf and 0.002 rf. We use a 1% ®ber volume fraction to approach the semi-in®nite matrix crack limit, and a 40% volume fraction to model a realistic composite. 2.2 Numerical technique: axisymmetric damage model The axisymmetric damage model (ADM) was developed by Pagano to predict the stress and displacement dis￾tributions in composite constituents as well as the strain energy and energy release rates for cracked composites in which damage modes include ®ber breaks, annular cracks in coatings or matrix, and debond cracks at interfaces. Concentric cylindrical/annular elements are used to model the ®ber and matrix materials. Additional concentric annuli (radii r1, r2, ...) can be introduced, as well as longitudinal sections (z1, z2, ...), if necessary for computational purpose, as indicated schematically in Fig. 3. This discretization forms regions bounded by ri and ri+1 and zj and zj+1. The annuli, or layers, and sections are chosen so that cracks introduced into the geometry always lie along the boundaries of sections (for transverse cracks) or layers (for longitudinal cracks) and span one or more complete sections/layers. The cracks then simply de®ne a particular boundary condi￾tion along the surface of any region. The elasticity pro￾blem within each region subject to appropriate boundary conditions is solved using the Reissner varia￾tional equation, as discussed brie¯y below. Reissner7 has shown that minimizing the functional J ˆ … V FdV ÿ … S0 TiidS; F ˆ 1 2 ij…i;j ‡ j;i† ÿ W …5† with respect to both stresses and displacements leads to both the ®eld equations and boundary conditions of lin￾ear elasticity theory. In eqn (5), W is the complementary energy density, Ti is the prescribed traction, and i and i are the stress and displacement components, respec￾tively, in Cartesian coordinates. A comma after a sub￾script represents a derivative with respect to the indicated coordinate, and Einstein's summation conven￾tion is understood. V is an arbitrary volume enclosed by the entire surface S, while S0 is the portion of the boundary on which one or more traction components are prescribed. The body forces have been neglected in the formulation. In the ADM, the stress ®eld in each annular region is assumed to be one where  and z are linear in the radial coordinate r, while the forms of r and rz are chosen to satisfy the axisymmetric Fig. 2. Debond and Penetration at ®ber/matrix interface in axisymmetric geometry. Fig. 3. Schematic of ADM concentric cylinders. 1778 B. K. Ahn et al

Crack deflection/penetration criteria for fiber-reinforced ceramic matrix composites 1779 equilibrium equations. Then, all of the stress compo- the shear stresses in adjoining constituents are con- nents depend on r through known shape functions tinuous. We assume the tractions on the debond crack Using the assumed stress field in the reissner functional surface vanish as shown in Fig. 2, while in the elastic and appropriate boundary/continuity conditions across zone(perfectly bonded zone) both of the radial and the region boundaries leads to a set of algebraic equations axial displacements are continuous. We note that if and linear ordinary differential equations in z only for debond crack takes place at an interface, one may the unknown coefficients of the stress fields. For a body choose between maintaining continuity of the 8N+16 equations to solve.g and N annuli, there are stress or continuity ofweighted displacement'at the composed of a core cylinder crack tip(see Ref. 8 for details). Selecting shear stress At the coarsest level, regions can be chosen to fully continuity has been shown to lead to energy con- span existing cracks and interfaces. However, in order sistency, and this condition is used throughout this to improve the accuracy of the solution for the stress study. The boundary conditions used here on the sur- fields and energy release rates, it is possible to introduce faces of the axisymmetric cell are shown in Fig. 2. At the additional annular layers in the neighborhood of inter- outer boundary of the composite cylinder(r= Im), we faces, crack tips, and the other forms of stress con- apply the traction-free boundary conditions for simpli- centrations. Although similar to mesh refinement in city, but we also consider the radial displacement finite element analysis, the refinement here is primarily boundary conditions on a limited basis, for illustrative only in the radial coordinate, refinement in the axial purposes. To obtain the proper displacement boundary coordinate is only necessary for computational tract- conditions, we proceed as follows ability in the solution and not for accuracy. The accu- racy of the ADM calculation depends on the number 1. assume a crack-free uniaxially tensile loaded com- and the location of the boundaries of these regions posite with perfectly bonded interface and a stress- Since we will deal with small finite crack extensions. we free outer surface boundary; w in Fig. 4 the general mesh structure used for 2. obtain the constant radial displacement perpend three calculations of matrix crack only, penetrated cular to the cylinder outer surface in the composite crack, and deflected crack, which we performed to using the adm etermine 3. use the radial displacement as a boundary condi represents a model composite with a 40% fiber volume tion in a matrix-cracked composite with an imper fraction and a length of 10 rf. A schematic of the fect interface. As expected, the two different additional layers and sections used for the boundary conditions do not make any difference ad=ap=0-002ra case is illustrated. In addition to the in GaGn over all range of a in the case of 1% fiber physical material boundary at r/ry=l we employ 10 volume fraction additional annular layers at r/r=0.80.9, 0.93, 0.96,0.99 0.998.1-03. 1-07. 1. 2. An additional axial section at For 40% fiber volume fraction. we notice differences is used and a section at z= ad is also required for as large as 7% in the values of energy release rates from the deflection problem he two different boundary conditions, but the bound Regarding boundary conditions, the Reissner varia- ary condition effect diminishes considerably in the final tional principle can handle mixed boundary problems so calculations for Ga Gp. In Fig. 6, the results using the that both displacements and stresses can be specified. displacement boundary condition are shown for com- General boundary conditions which should be satisfied parison, and the difference by the two boundary condi at all interfaces are as follows. The radial stresses and tions appears negligible. Regarding the loading rm=1.58/ Matrix : Addition Fiber r=0 10-080.6-0.4.2000204060.81.0 al sections Fig. 5. GalGp versus a with various crack extensions for Fig 4. General mesh structure used in the adm model

equilibrium equations. Then, all of the stress compo￾nents depend on r through known shape functions. Using the assumed stress ®eld in the Reissner functional and appropriate boundary/continuity conditions across region boundaries leads to a set of algebraic equations and linear ordinary di€erential equations in z only for the unknown coecients of the stress ®elds. For a body composed of a core cylinder and N annuli, there are 18N+16 equations to solve.8,9 At the coarsest level, regions can be chosen to fully span existing cracks and interfaces. However, in order to improve the accuracy of the solution for the stress ®elds and energy release rates, it is possible to introduce additional annular layers in the neighborhood of inter￾faces, crack tips, and the other forms of stress con￾centrations. Although similar to mesh re®nement in ®nite element analysis, the re®nement here is primarily only in the radial coordinate; re®nement in the axial coordinate is only necessary for computational tract￾ability in the solution and not for accuracy. The accu￾racy of the ADM calculation depends on the number and the location of the boundaries of these regions. Since we will deal with small ®nite crack extensions, we show in Fig. 4 the general mesh structure used for the three calculations of matrix crack only, penetrated crack, and de¯ected crack, which we performed to determine the various energy release rates. Figure 4 represents a model composite with a 40% ®ber volume fraction and a length of 10 rf. A schematic of the additional layers and sections used for the ad ˆ ap ˆ 0
002rf† case is illustrated. In addition to the physical material boundary at r=rf=1 we employ 10 additional annular layers at r=rf=0.8,0.9,0.93,0.96,0.99, 0.998,1.03,1.07,1.1and1.2. An additional axial section at z ˆ rf is used and a section at z ˆ ad is also required for the de¯ection problem. Regarding boundary conditions, the Reissner varia￾tional principle can handle mixed boundary problems so that both displacements and stresses can be speci®ed. General boundary conditions which should be satis®ed at all interfaces are as follows. The radial stresses and the shear stresses in adjoining constituents are con￾tinuous. We assume the tractions on the debond crack surface vanish as shown in Fig. 2, while in the elastic zone (perfectly bonded zone) both of the radial and the axial displacements are continuous. We note that if a debond crack takes place at an interface, one may choose between maintaining continuity of the shear stress or continuity of `weighted displacement' at the crack tip (see Ref. 8 for details). Selecting shear stress continuity has been shown to lead to energy con￾sistency, and this condition is used throughout this study. The boundary conditions used here on the sur￾faces of the axisymmetric cell are shown in Fig. 2. At the outer boundary of the composite cylinder …r ˆ rm†, we apply the traction-free boundary conditions for simpli￾city, but we also consider the radial displacement boundary conditions on a limited basis, for illustrative purposes. To obtain the proper displacement boundary conditions, we proceed as follows: 1. assume a crack-free uniaxially tensile loaded com￾posite with perfectly bonded interface and a stress￾free outer surface boundary; 2. obtain the constant radial displacement perpendi￾cular to the cylinder outer surface in the composite using the ADM; 3. use the radial displacement as a boundary condi￾tion in a matrix-cracked composite with an imper￾fect interface. As expected, the two di€erent boundary conditions do not make any di€erence in Gd/Gp over all range of  in the case of 1% ®ber volume fraction. For 40% ®ber volume fraction, we notice di€erences as large as 7% in the values of energy release rates from the two di€erent boundary conditions, but the bound￾ary condition e€ect diminishes considerably in the ®nal calculations for Gd/Gp. In Fig. 6, the results using the displacement boundary condition are shown for com￾parison, and the di€erence by the two boundary condi￾tions appears negligible. Regarding the loading Fig. 4. General mesh structure used in the ADM model. Fig. 5. Gd/Gp versus  with various crack extensions for Vf ˆ 1%. Crack de¯ection/penetration criteria for ®ber-reinforced ceramic matrix composites 1779