《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_THE GENERATION OF MULTIPLE MATRIX CRACKING AND FIBER±MATRIX INTERFACIAL DEBONDING IN A GLASS COMPOSITE

Acta mater. VoL 46, No 5, pp. 1657-1667, 1998 8y丿 Pergamon Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PI!:Sl359-6454(97)00347-9 1359-6454/98s19.00+0.00 THE GENERATION OF MULTIPLE MATRIX CRACKING AND FIBER-MATRIX INTERFACIAL DEBONDING IN A GLASS COMPOSITE YONGJIAN SUN and RAJ N SINGHT Department of Materials Science and Engineering, University of Cincinnati, P O. Box 210012, Cincinnati. OH 45221-0012. U.S.A 7 Received 17 June 1997, accepted 10 September 1997) Abstract-The phenomena of multiple matrix cracking and fiber-matrix interfacial debonding are directly observed and analyzed in a transparent SiC fiber-reinforced borosilicate glass composite. Under a certain load, the number of matrix cracks and the length of interfacial debonding are directly measured because of the availability of this transparent glass composite. These observations on the interfacial debonding accom- nying the multiple matrix cracking, and their comparison with the theoretical analyses of the acking and debonding behaviors are presented. The micromechanics of matrix cracking, interfacial debonding, and associated matrix crack saturation/saturation stress are also analyzed. C 1998Acta Metallurgica Inc. 1 INTRODUCTION Ceramic materials exhibit superior mechanical increased because of the reinforcing fibers [2/are stress. ultimate strength. and work of fracture properties at high temperature. But, their use as Most research workers focused their interests on structural components is severely limited because of the study of first matrix cracking stress, ultimate their brittleness. Fiber-reinforced ceramic compo- strength of composites, and the significant role of sites, by incorporating fibers in ceramic matrices he interface on these two properties [2-5]. The however, not only exploit their attractive high-tem- study of the initial non-linear behavior of the com- perature strength but also enhance their toughness. posite after FMC. however, is not that extensive A typical load-displacement curve for a ceramic Firstly in early 1970s, from the consideration of composite subjected to tensile loading parallel to the maximum shear stress at the fiber-matrix inter- the fiber direction is shown in Fig. 1 [ At th face, Aveston and Kelly [6(Ak) proposed the fun- point A, the first matrix crack initiates when the damental concepts and relationships among matrix matrix reaches its elastic limit. If the fibers and crack spacing, interfacial debonding (or sliding) matrix are weakly-bonded or frictionally coupled, ength, and interfacial shear stress of a continuous he matrix crack propagates transversely across the fiber-reinforced composite. Since then Zok and fibers thus creating bridged-fibers to carry ad- Spearing [7 developed a model for multiple matrix ditional load. When this happens, the matrix crack crack spacing showing both the periodic and ran- deflects at the fiber/coating/matrix interfaces dom cracking patterns. Weitsman and Zhu [8] because of the interfacial debonding and sliding the increase of the applied load, more matrix TTTTTTTTTTTTTTTTTTTTTTTH cracks are generated and interfacial debonding pro- agates with a larger sliding zone. This process Fig.I. Point B indicates the saturation of the 3 80 leads to a non-linear curve between a and b in matrix cracking behavior. Beyond point B, the curve shows additional non-linear behavior until the onset of fiber failures and fiber pull-out at point C. This point C represents the strength of the com- posite, at which the breakage of a large number of ⊥⊥ fibers leads to decreased load-bearing capacity of 00.5 3.03.54.0 the composite. The first matrix cracking (FMC) Displacement (mm) Fig. 1. A typical load-displacement curve for a Sic fiber To whom all correspondence should be addressed

THE GENERATION OF MULTIPLE MATRIX CRACKING AND FIBER±MATRIX INTERFACIAL DEBONDING IN A GLASS COMPOSITE YONGJIAN SUN and RAJ N. SINGH{ Department of Materials Science and Engineering, University of Cincinnati, P.O. Box 210012, Cincinnati, OH 45221-0012, U.S.A. (Received 17 June 1997; accepted 10 September 1997) AbstractÐThe phenomena of multiple matrix cracking and ®ber±matrix interfacial debonding are directly observed and analyzed in a transparent SiC ®ber-reinforced borosilicate glass composite. Under a certain load, the number of matrix cracks and the length of interfacial debonding are directly measured because of the availability of this transparent glass composite. These observations on the interfacial debonding accom￾panying the multiple matrix cracking, and their comparison with the theoretical analyses of the matrix cracking and debonding behaviors are presented. The micromechanics of matrix cracking, interfacial debonding, and associated matrix crack saturation/saturation stress are also analyzed. # 1998 Acta Metallurgica Inc. 1. INTRODUCTION Ceramic materials exhibit superior mechanical properties at high temperature. But, their use as structural components is severely limited because of their brittleness. Fiber-reinforced ceramic compo￾sites, by incorporating ®bers in ceramic matrices, however, not only exploit their attractive high-tem￾perature strength but also enhance their toughness. A typical load±displacement curve for a ceramic composite subjected to tensile loading parallel to the ®ber direction is shown in Fig. 1 [1]. At the point A, the ®rst matrix crack initiates when the matrix reaches its elastic limit. If the ®bers and matrix are weakly-bonded or frictionally coupled, the matrix crack propagates transversely across the ®bers thus creating bridged-®bers to carry ad￾ditional load. When this happens, the matrix crack de¯ects at the ®ber/coating/matrix interfaces because of the interfacial debonding and sliding. With the increase of the applied load, more matrix cracks are generated and interfacial debonding pro￾pagates with a larger sliding zone. This process leads to a non-linear curve between A and B in Fig. 1. Point B indicates the saturation of the matrix cracking behavior. Beyond point B, the curve shows additional non-linear behavior until the onset of ®ber failures and ®ber pull-out at point C. This point C represents the strength of the com￾posite, at which the breakage of a large number of ®bers leads to decreased load-bearing capacity of the composite. The ®rst matrix cracking (FMC) stress, ultimate strength, and work of fracture are increased because of the reinforcing ®bers [2]. Most research workers focused their interests on the study of ®rst matrix cracking stress, ultimate strength of composites, and the signi®cant role of the interface on these two properties [2±5]. The study of the initial non-linear behavior of the com￾posite after FMC, however, is not that extensive. Firstly in early 1970's, from the consideration of the maximum shear stress at the ®ber±matrix inter￾face, Aveston and Kelly [6] (AK) proposed the fun￾damental concepts and relationships among matrix crack spacing, interfacial debonding (or sliding) length, and interfacial shear stress of a continuous ®ber-reinforced composite. Since then Zok and Spearing [7] developed a model for multiple matrix crack spacing showing both the periodic and ran￾dom cracking patterns. Weitsman and Zhu [8] Acta mater. Vol. 46, No. 5, pp. 1657±1667, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S1359-6454(97)00347-9 1359-6454/98 $19.00 + 0.00 Fig. 1. A typical load±displacement curve for a SiC ®ber {To whom all correspondence should be addressed. reinforced borosilicate glass composite. 1657

1658 SUN and SINGH: MULTIPLE MATRIX CRACKING mployed an energy criterion to analyze the stress- redistributed in the cracked area. The stress is trans- strain relationship for the initial non-linear curve ferred from the fiber to the matrix by interfacial A-B). Curtin [9]. using the statistical approach, shear stress [61 theoretically analyzed the evolution f multip matrix cracking and related the crack spacing to the dF 2Vrti faw distribution. Most of their analyses were done by assuming an interface loosely coupled where dF is the stress transferred from fiber to y friction. Hutchinson and Jenssen [10(HJ), matrix over a distance dz along the fiber, and ti is Marshall [I], and Budiansky, Evans and the shear stress acting at the interface. The may Hutchison [12 (BER)worked on the phenomenon mum value of ti, named interfacial shear strength of debonding using an axisymetric cylindrical model tu, depends on the nature of bonding at the inter in which the debonding was treated as mode 2 face. The general criterion for determining the inte interface fracture. From a force balance approach, face type is established Ip among debond length, crack opening displacement, and f(tf, td) applied stress. Li and Mura(LD[13] developed a where tr is the frictional interfacial stress and ta is similar model based on an energy approach and the model II debond shear stress. If ta is equal to verified the model using a steel fiber-reinforced cement system. However, the analysis of the non ero, the interface is unbonded and if td > tr, the near portion (A-B) is complex because of the interface is strongly-bonded. Otherwise, the inter- difficulty in measuring debond or sliding length in face is considered as weakly-bonded. most ceramic composites. For an unbonded interface, ti is equal to the con- The objective of this research is a study of the stant frictional stress tr, and the stress distribution evolution of matrix cracks and their relationship to in fiber (od and matrix (om) in the sliding area is interfacial debonding. By measuring the debond determined by the following equations(Fig. 2) length and matrix crack spacing directly, the inter- relationship among the multiple matrix cracking, ar(2)= interfacial debonding, and external stress is analyzed n2()) 2. THEORETICAL BACKGROUND t1(2)=t Fibers with a failure strain larger than the matrix material are desired in a continuous fiber-reinforced a characteristic sliding length z can be derived composites to create crack-bridging fibers which from equation(5). At this location(2). the recovery the mat nt the catastrophic failure. When strain of matrix stress from the crack surface reaches the matrix reaches its ultimate value. the stress strength of matrix material, omu composite reaches the FMC stress. Aveston, Cooper and Kelly [14](ack)derived an expression for the FMC stress, aFMc for a long steady-state matrix crack using the fracture mechanics approach This well-known relation, derived by AK [6]. and by assuming that the interfacial fracture energy implies that a small increase in the external load is very small, and the fiber-matrix interfacial sliding generates conditions for further matrix cracking to behavior is characterized by a constant shear stress occur simultaneously throughout the composite with a spacing between 2 and 2z because omu is a 6Er) material property of the matrix 6]. Zok and FFMC E (1 Spearing [7] proposed, from the steady state strain where v, E, I denote the volume fraction, the at the initial matrix cracking stress is bounded by elastic modulus, and the matrix fracture energy, 22 to 42. The average matrix crack spacing is stat respectively. The subscripts f, m, and c denote the istically derived to be az(a=1.34)[9].However, fiber, matrix, and composite, respectively, and r is in pract acking stress is dependent the fiber radius. MCE [15] and BHE [16] followed a on the iaws which are inevitable in most ceramic similar approach, incorporating the effects of crack and glass matrix materials. A range of values for length, interfacial debonding energy, and residual matrix cracking stress instead of a single value of stress on the fmc stress amu is expected to lead to multiple matrix cracking After the FMC, a portion of the load that was in a practical matrix material. Therefore, the stat once shared by the matrix is thrown onto the fibers. istical consideration of multiple matrix cracking The stresses in the fibers and matrix are locally very important [9]

employed an energy criterion to analyze the stress± strain relationship for the initial non-linear curve (A±B). Curtin [9], using the statistical approach, theoretically analyzed the evolution of multiple matrix cracking and related the crack spacing to the ¯aw distribution. Most of their analyses were done by assuming an interface loosely coupled by friction. Hutchinson and Jenssen [10] (HJ), Marshall [11], and Budiansky, Evans and Hutchison [12] (BEH) worked on the phenomenon of debonding using an axisymetric cylindrical model in which the debonding was treated as mode 2 interface fracture. From a force balance approach, they theoretically derived a relationship among the debond length, crack opening displacement, and applied stress. Li and Mura (LI) [13] developed a similar model based on an energy approach and veri®ed the model using a steel ®ber-reinforced cement system. However, the analysis of the non￾linear portion (A±B) is complex because of the diculty in measuring debond or sliding length in most ceramic composites. The objective of this research is a study of the evolution of matrix cracks and their relationship to interfacial debonding. By measuring the debond length and matrix crack spacing directly, the inter￾relationship among the multiple matrix cracking, interfacial debonding, and external stress is analyzed. 2. THEORETICAL BACKGROUND Fibers with a failure strain larger than the matrix material are desired in a continuous ®ber-reinforced composites to create crack-bridging ®bers which can prevent the catastrophic failure. When strain in the matrix reaches its ultimate value, the stress in composite reaches the FMC stress. Aveston, Cooper and Kelly [14] (ACK) derived an expression for the FMC stress, sFMC for a long steady-state matrix crack using the fracture mechanics approach and by assuming that the interfacial fracture energy is very small, and the ®ber±matrix interfacial sliding behavior is characterized by a constant shear stress tf, sFMC ˆ 6EfV2 f tfE2 cGm E2 mVmr  1=3 …1† where V, E, G denote the volume fraction, the elastic modulus, and the matrix fracture energy, respectively. The subscripts f, m, and c denote the ®ber, matrix, and composite, respectively, and r is the ®ber radius. MCE [15] and BHE [16] followed a similar approach, incorporating the e€ects of crack length, interfacial debonding energy, and residual stress on the FMC stress. After the FMC, a portion of the load that was once shared by the matrix is thrown onto the ®bers. The stresses in the ®bers and matrix are locally redistributed in the cracked area. The stress is trans￾ferred from the ®ber to the matrix by interfacial shear stress [6]: dF dz ˆ 2Vf ti r …2† where dF is the stress transferred from ®ber to matrix over a distance dz along the ®ber, and ti is the shear stress acting at the interface. The maxi￾mum value of ti, named interfacial shear strength tu, depends on the nature of bonding at the inter￾face. The general criterion for determining the inter￾face type is established as: tu ˆ Maximum of …tf; td† …3† where tf is the frictional interfacial stress and td is the model II debond shear stress. If td is equal to zero, the interface is unbonded, and if td >> tf, the interface is strongly-bonded. Otherwise, the inter￾face is considered as weakly-bonded. For an unbonded interface, ti is equal to the con￾stant frictional stress tf, and the stress distribution in ®ber (sf) and matrix (sm) in the sliding area is determined by the following equations (Fig. 2): sf…z† ˆ sa Vf ÿ 2tf z r …4† sm…z† ˆ 2tf Vf Vm   z r   …5† ti…z† ˆ tf …6† A characteristic sliding length z' can be derived from equation (5). At this location (z'), the recovery of matrix stress from the crack surface reaches the strength of matrix material, smu, z0 ˆ Vmsmur 2Vf tf …7† This well-known relation, derived by AK [6], implies that a small increase in the external load generates conditions for further matrix cracking to occur simultaneously throughout the composite with a spacing between z' and 2z' because smu is a material property of the matrix [6]. Zok and Spearing [7] proposed, from the steady state strain energy release rate approach, that the crack spacing at the initial matrix cracking stress is bounded by 2z' to 4z' . The average matrix crack spacing is stat￾istically derived to be az' (a = 1.34) [9]. However, in practice, the matrix cracking stress is dependent on the ¯aws which are inevitable in most ceramic and glass matrix materials. A range of values for matrix cracking stress instead of a single value of smu is expected to lead to multiple matrix cracking in a practical matrix material. Therefore, the stat￾istical consideration of multiple matrix cracking is very important [9]. 1658 SUN and SINGH: MULTIPLE MATRIX CRACKING

SUN and SINGH: MULTIPLE MATRIX CRACKING 1659 Frictional Sliding Debonding Tip ■ Fiber Matrix (b) (MPa) De bonded zone Bonded zone 800· 400 200 Distance From Crack Surface (z) Fig. 2.(a)Schematic of interfacial debonding and fiber pullout. (b) Stress distributions in the fiber, matrix and at the interface For a partially bonded and debonded interface, matrix far from the debonded region, respectively the stress distribution in the cracked area is depen- p is the shear-lag parameter determined by the dent on the debond length Ld, which is determined following formula by the interfacial shear strength ty. Figure 2(a) p2 4ECG shows a cylindrical unit of composite with a debonded interface of length Ld. The stresses in the debonded region(0 <2< La follow equations (4- where Gm is the shear modulus of matrix and (6). According to the shear-lag solution given by BHE [16, the stresses beyond the debonded region 2In Vr+Vm(3-vr (z>Ld) determined by the folle exp ons The above expressions are fundamentally identical (8) along the interface from the matrix crack surface to the far field beyond the debonded region is show in Fig. 2(b). The interfacial shear stress is shown to am()=0m-o have its maximum value at the debonded tip (=La if ta is larger than t (9) The mode II debond shear stress, ta is equal to he interfacial shear strength tu for a bonded inter- face with zero coefficient of friction as estimated by r(z)= BEH [12] (BHE[2(11) where of and am are the stresses in fiber and

For a partially bonded and debonded interface, the stress distribution in the cracked area is depen￾dent on the debond length Ld, which is determined by the interfacial shear strength tu. Figure 2(a) shows a cylindrical unit of composite with a debonded interface of length Ld. The stresses in the debonded region (0 < z < Ld) follow equations (4)± (6). According to the shear-lag solution given by BHE [16], the stresses beyond the debonded region (z>Ld) are determined by the following expressions [12]: sf…z†ˆs1 f ‡ " Vm Vf   s1 m ÿ2tf Ld r # exp ÿ r…z ÿ Ld† r  …8† sm…z†ˆs1 m ÿ " s1 m ÿ Vf Vm   2tf Ld r # exp ÿ r…z ÿ Ld† r  …9† ti…z†ˆ r 2   " Vm Vf   s1 m ÿ2tf Ld r # exp ÿ r…z ÿ Ld† r  …10† where s1 f and s1 m are the stresses in ®ber and matrix far from the debonded region, respectively. r is the shear-lag parameter determined by the following formula r2 ˆ 4EcGm VmEmEfj , where Gm is the shear modulus of matrix and j ˆ ÿ 2 ln Vf ‡ Vm…3 ÿ Vf† 2V2 m , The above expressions are fundamentally identical for AK [6], BEH [12], HJ [10], and Sutcu and Hillig [17] (SH) models without considering the re￾sidual stresses in composite. The stress distribution along the interface from the matrix crack surface to the far ®eld beyond the debonded region is shown in Fig. 2(b). The interfacial shear stress is shown to have its maximum value at the debonded tip (z = Ld) if td is larger than tf. The mode II debond shear stress, td is equal to the interfacial shear strength tu for a bonded inter￾face with zero coecient of friction as estimated by BEH [12], tBEH d ˆ  4GmGd rj s (BHE[12]) …11† Fig. 2. (a) Schematic of interfacial debonding and ®ber pullout. (b) Stress distributions in the ®ber, matrix and at the interface. SUN and SINGH: MULTIPLE MATRIX CRACKING 1659

SUN and SINGH: MULTIPLE MATRIX CRACKING where Id is the interfacial debond energy. For a under a certain force equilibrium, and that the bonded interface with non-zero coefficient of fric- stress redistribution due to further multiple matrix tion, sH [17 correlated the debonding energy Id to cracking, matrix-crack interaction, and debond in the interfacial debond strength t and frictional teraction does not influence the debond length shear stress tr by the following expressio Using an energy balance approach, LI [13] theoreti- cally analyzed the influence of the dissipation d-trs /4GmIa (SH[17D (12) energy of friction Gs and interfacial debonding energy GD on the relationship between debond length and external stress, and obtained the follow The external stress required to initiate the interfacial ing relationship debonding, namely the debond initiation stress ad can be determined by various expressions derived Vrer vm Emad from different models as Ec VEmE(BEH[I2D (13) This relationship differs from the results of the force balance approach in terms of the appearance tErra (ACK[14,Ln3D(14) of an extra offset term of(oa/t)4[13] Although the measurements of interfacial ?vail- (H[10]) (15)ability of new techniques [18-20], the above theoretical relationships between debond length and where c is a constant calculated from the material external stress have not been closely examined by parameters. Despite the different expressions. these experiments because of the difficulty in measuring equations are consistent with the fact that the inter- the debond length. With the increase of debond facial properties controlled primarily by the length along the fiber/matrix interface, will the interfacial debond energy Ia. The larger the inter trix generate more cracks? What is the critical facial debond energy, the larger is the debond condition for the saturation of matrix cracking? tress.All the above expressions of ad have not When does the debonding occur during failure of considered the effect of interfacial frictional stress the composite? Are those models based on the force In a composite with a bonded interface of non-zero balance approach and energy balance approach coefficient of friction, oa can be defined from the valid in real composites? Answers to these questions derstanding of micro-mechanics associated with the initial non- 2VrEc (16 linear behavior after the first matrix cracking of a ceramic-matrix composite (region AB in Fig. 1). In his paper, an attempt is made to answer these where t is available from equation (12). For a fric tionally-coupled interface, ta is reduced to to. and questions ad is reduced to slide, a stress required to initiate an interfacial sliding [12]. Equations(12) and(16) 3. EXPERIMENTAL PROCEDURE differ from equations (11) and(13H(15) mainly because SH [17 considered the contribution of 3.1. Specimen preparation shear strain to elastic strain energy upon the In order to obtain a transparent composite, bor- appearance of frictional shear stress. silicate(F)glass( Corning Glass Works, NY) was In order to establish the relationship betw een chosen as the matrix material. SiC (SCS-6)fibers debond length and externally applied stress, two (Textron Specialty Materials, Lowell, MA)were fundamental approaches have been employed: the used as a continuous fiber reinforcement. This force balance approach [10-12] and energy balance fiber was made 37-m-diameter carbon pproach [13]. According to the force balance core on which the 50-Hm-thick silicon carbide was approach, the debond length Ld is related to the deposited. Additional 3-um-thick carbon and car- applied stress aa by the following equation [12]. on-silicon coat were deposited making an overall fiber diameter of 142 um. The mechanical (7)properties of SCS-6 SiC fiber and F glass are listed in Table I The debond initiates when a is equal to od. With The composites were made by Tape Casting/ an increase of the external stress, the debond propa- Binary Sintering (TCBS) method that resulted in gates until the interfacial shear stress at the debond high density and good transparency [1. The as- tip is equal or less than the interfacial shear fabricated sample was ground with a diamond strength tu. The application of a force balance wheel and polished on an ar mplies that any cylindrical unit of composite is machine. It was finally cut with a diamond saw

where Gd is the interfacial debond energy. For a bonded interface with non-zero coecient of fric￾tion, SH [17] correlated the debonding energy Gd to the interfacial debond strength td and frictional shear stress tf by the following expression: td ÿ tf ˆ  4GmGd rj s (SH[17]) …12† The external stress required to initiate the interfacial debonding, namely the debond initiation stress sd, can be determined by various expressions derived from di€erent models as: sBHE d ˆ 2Vf  EfEcGd VmEmr r (BEH[12]) …13† sACK d ˆ Vf  4EfGd r r (ACK[14], LI[13]) …14† sHJ d ˆ 1 c1  EmGd r r (HJ[10]) …15† where c1 is a constant calculated from the material parameters. Despite the di€erent expressions, these equations are consistent with the fact that the inter￾facial properties are controlled primarily by the interfacial debond energy Gd. The larger the inter￾facial debond energy, the larger is the debond stress. All the above expressions of sd have not considered the e€ect of interfacial frictional stress. In a composite with a bonded interface of non-zero coecient of friction, sd can be de®ned from the shear-lag theory as [12] sd ˆ 2VfEc rVmEm   td …16† where td is available from equation (12). For a fric￾tionally-coupled interface, td is reduced to tf, and sd is reduced to sslide, a stress required to initiate an interfacial sliding [12]. Equations (12) and (16) di€er from equations (11) and (13)±(15) mainly because SH [17] considered the contribution of shear strain to elastic strain energy upon the appearance of frictional shear stress. In order to establish the relationship between debond length and externally applied stress, two fundamental approaches have been employed: the force balance approach [10±12] and energy balance approach [13]. According to the force balance approach, the debond length Ld is related to the applied stress sa by the following equation [12], Ld r ˆ VmEm VfEc   sa ÿ sd 2tf …17† The debond initiates when sa is equal to sd. With an increase of the external stress, the debond propa￾gates until the interfacial shear stress at the debond tip is equal or less than the interfacial shear strength tu. The application of a force balance implies that any cylindrical unit of composite is under a certain force equilibrium, and that the stress redistribution due to further multiple matrix cracking, matrix±crack interaction, and debond in￾teraction does not in¯uence the debond length. Using an energy balance approach, LI [13] theoreti￾cally analyzed the in¯uence of the dissipation energy of friction GS and interfacial debonding energy GD on the relationship between debond length and external stress, and obtained the follow￾ing relationship: Ld r ˆ sa 2tfVf 1 ÿ VfEf Ec ‡ VmEm Ec sd sa  2 " #1=2 8 < : 9 = ; …18† This relationship di€ers from the results of the force balance approach in terms of the appearance of an extra o€set term of (sa/tf) 1/2 [13]. Although the measurements of interfacial proper￾ties such as tf and Gd are facilitated by the avail￾ability of new techniques [18±20], the above theoretical relationships between debond length and external stress have not been closely examined by experiments because of the diculty in measuring the debond length. With the increase of debond length along the ®ber/matrix interface, will the matrix generate more cracks? What is the critical condition for the saturation of matrix cracking? When does the debonding occur during failure of the composite? Are those models based on the force balance approach and energy balance approach valid in real composites? Answers to these questions are essential for a better understanding of the micro-mechanics associated with the initial non￾linear behavior after the ®rst matrix cracking of a ceramic±matrix composite (region AB in Fig. 1). In this paper, an attempt is made to answer these questions. 3. EXPERIMENTAL PROCEDURE 3.1. Specimen preparation In order to obtain a transparent composite, bor￾osilicate (F) glass (Corning Glass Works, NY) was chosen as the matrix material. SiC (SCS-6) ®bers (Textron Specialty Materials, Lowell, MA) were used as a continuous ®ber reinforcement. This ®ber was made using a 37-mm-diameter carbon core on which the 50-mm-thick silicon carbide was deposited. Additional 3-mm-thick carbon and car￾bon±silicon coatings were deposited making an overall ®ber diameter of 142 mm. The mechanical properties of SCS-6 SiC ®ber and F glass are listed in Table 1. The composites were made by Tape Casting/ Binary Sintering (TCBS) method that resulted in high density and good transparency [1]. The as￾fabricated sample was ground with a diamond wheel and polished on an automatic polishing machine. It was ®nally cut with a diamond saw. 1660 SUN and SINGH: MULTIPLE MATRIX CRACKING

SUN and SINGH: MULTIPLE MATRIX CRACKING Table 1. Mechanical properties of SiC fiber and F glass Materials Elastic modulus(GPa) Strength(GPa) Failure strain (% SCS-6 SiC fibre 0.8-1.0 423[21 F glass+ 0.056 ta is the coefficient of thermal expansion(25-500.C). +The composition for F glass is 76% Sioz-16% B203-8%K2O, obtained from Corning Glass Works, Corning. NY Fig. 4. From the load-displacement curve, the debond initiation point (point A in Fig. 4), the peak load (point B)at which the interfacial debond ing has propagated through the sample thickness, and the load drop (point C) corresponding to the beginning of a steady-state interfacial sliding can be measured. The debond initiation stress ad at point A was about 55 MPa [22]. The frictional sliding stress tr determined from the load value at point C was about 30 MPa [22]. The debond energy Ia can then be obtained from equations(12)and(16) vided that ad and tr are available. Therefore the interfacial properties of these Sic fiber-reinforced 1 00um borosilicate glass composites, as listed in Table 2, were determined in this way from the fiber pushout Fig.3. A cross section of the SiC fiber reinforced glass load-displacement curves composite displaying uniformly distributed fibers 3.3. Matrix cracking and debond length measurement The sample had a surface dimension of techniques 50mm x 3.3 mm and a thickness of about 1.4 mm. The sample was stressed using an Instron testing Figure 3 shows the well-distributed fibers from the machine in the four point flexure mode. The outer ross section of a composite. The material par- span was 40 mm and the inner span was 20 mm meters and mechanical properties of composite are For the purpose of finding the evolution of multiple listed in Table 2 matrix cracks and interfacial debonding, the sample as loaded to several different levels of stress/load 3. 2. Measurement of interfacial properties t a crosshead speed of 0.2 mm/min and then unloaded. The loading curves during a typical test are shown in Fig. 5. The changes in the light trans- debond initiation stress and frictional sliding stress mission pattern(white band) on either side of a were measured using the fiber pushout technique. matrix crack were observed. It was believed that the Fiber pushout tests were conducted using a Micro white band was caused by the light scattering and Measure Machine(Process Equipment Company, absorption at the debonded portion of the fiber OH)at a loading rate of 10 N/min. The pushout matrix interface. The change in the white band was probe, made of tungsten carbide, was about 100 Am attributed to the process of relative displacement by pushing a fully dense bulk alumina plate of 5mm in thickness. The real fiber displacement uring a pushout test was obtained after subtracting the machine compliance. A typical load-displace- Thickne t curve for a fibi per pushout test is shown in Table 2. Parameters and properties for SCS-6 fiber reinforced Fiber volume fraction, Vr 0.12 modulus of composite. Ec(GPa) 2 Matrix)(MPa/m) 0.77D2 First matrix cracking stress, FMc (MPa) d o/m) 1.2±0.3[22] Debond initiation stress, ad(MPa) 55+5122 Fig. 4. A typical load-displacement curve during a fiber cAlculated by rule of mixture

The sample had a surface dimension of 50 mm 3.3 mm and a thickness of about 1.4 mm. Figure 3 shows the well-distributed ®bers from the cross section of a composite. The material par￾ameters and mechanical properties of composite are listed in Table 2. 3.2. Measurement of interfacial properties The interfacial properties such as interfacial debond initiation stress and frictional sliding stress were measured using the ®ber pushout technique. Fiber pushout tests were conducted using a Micro Measure Machine (Process Equipment Company, OH) at a loading rate of 10 N/min. The pushout probe, made of tungsten carbide, was about 100 mm in diameter. The machine compliance was measured by pushing a fully dense bulk alumina plate of 5 mm in thickness. The real ®ber displacement during a pushout test was obtained after subtracting the machine compliance. A typical load±displace￾ment curve for a ®ber pushout test is shown in Fig. 4. From the load±displacement curve, the debond initiation point (point A in Fig. 4), the peak load (point B) at which the interfacial debond￾ing has propagated through the sample thickness, and the load drop (point C) corresponding to the beginning of a steady-state interfacial sliding can be measured. The debond initiation stress sd at point A was about 55 MPa [22]. The frictional sliding stress tf determined from the load value at point C was about 30 MPa [22]. The debond energy Gd can then be obtained from equations (12) and (16) pro￾vided that sd and tf are available. Therefore, the interfacial properties of these SiC ®ber-reinforced borosilicate glass composites, as listed in Table 2, were determined in this way from the ®ber pushout load±displacement curves. 3.3. Matrix cracking and debond length measurement techniques The sample was stressed using an Instron testing machine in the four point ¯exure mode. The outer span was 40 mm and the inner span was 20 mm. For the purpose of ®nding the evolution of multiple matrix cracks and interfacial debonding, the sample was loaded to several di€erent levels of stress/load at a crosshead speed of 0.2 mm/min and then unloaded. The loading curves during a typical test are shown in Fig. 5. The changes in the light trans￾mission pattern (white band) on either side of a matrix crack were observed. It was believed that the white band was caused by the light scattering and absorption at the debonded portion of the ®ber± matrix interface. The change in the white band was attributed to the process of relative displacement Table 1. Mechanical properties of SiC ®ber and F glass Materials Elastic modulus (GPa) Strength (GPa) Failure strain (%) a$ (10ÿ6 /8C) SCS-6 SiC ®bre 400 3.4 0.8±1.0 4.23 [21] F glass% 56 0.056 0.1 4.25 $a is the coecient of thermal expansion (25±5008C). %The composition for F glass is 76% SiO2±16% B2O3±8% K2O, obtained from Corning Glass Works, Corning, NY Fig. 3. A cross section of the SiC ®ber reinforced glass composite displaying uniformly distributed ®bers. Table 2. Parameters and properties for SCS-6 ®ber reinforced borosilicate glass composite Fiber volume fraction, Vf 0.12 Elastic modulus of composite, Ec (GPa) 97.3$ Matrix porosity (%) 1±2 [1] KIc (Matrix) (MPa/Zm) 0.77 [21] First matrix cracking stress, sFMC (MPa) 90 [1] Ultimate strength (composite), scu (MPa) 440 [1] Interfacial frictional stress, tf (MPa) 3023 [22] Interfacial debonding energy, Gd (J/m2 ) 1.220.3 [22] Debond initiation stress, sd (MPa) 5525 [22] $Calculated by rule of mixture Fig. 4. A typical load±displacement curve during a ®ber pushout test. SUN and SINGH: MULTIPLE MATRIX CRACKING 1661