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 surfaceCRITERIA 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