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 accompanying 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 composites, by incorporating ®bers in ceramic matrices, however, not only exploit their attractive high-temperature 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 additional 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 propagates 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 composite, 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 composite after FMC, however, is not that extensive. Firstly in early 1970's, from the consideration of the maximum shear stress at the ®ber±matrix interface, Aveston and Kelly [6] (AK) proposed the fundamental 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 random 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 nonlinear portion (A±B) is complex because of the diculty 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 interrelationship 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 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 eects 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 transferred from the ®ber to the matrix by interfacial shear stress [6]: dF dz 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 maximum value of ti, named interfacial shear strength tu, depends on the nature of bonding at the interface. The general criterion for determining the interface type is established as: tu 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 interface is considered as weakly-bonded. For an unbonded interface, ti is equal to the constant 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 sa Vf ÿ 2tf z r 4 sm z 2tf Vf Vm z r 5 ti z 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 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 statistically 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 statistical 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 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 dependent 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 Ld) are determined by the following expressions [12]: sf zs1 f " Vm Vf s1 m ÿ2tf Ld r # exp ÿ r z ÿ Ld r 8 sm zs1 m ÿ " s1 m ÿ Vf Vm 2tf Ld r # exp ÿ r z ÿ Ld r 9 ti z 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 4EcGm VmEmEfj , where Gm is the shear modulus of matrix and j 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 residual 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 interface with zero coecient of friction as estimated by BEH [12], tBEH d 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 coecient of friction, 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 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 dierent models as: sBHE d 2Vf EfEcGd VmEmr r (BEH[12]) 13 sACK d Vf 4EfGd r r (ACK[14], LI[13]) 14 sHJ d 1 c1 EmGd r r (HJ[10]) 15 where c1 is a constant calculated from the material parameters. Despite the dierent expressions, these equations are consistent with the fact that the interfacial properties are controlled primarily by the interfacial debond energy Gd. The larger the interfacial debond energy, the larger is the debond stress. All the above expressions of sd have not considered the eect of interfacial frictional stress. In a composite with a bonded interface of non-zero coecient of friction, sd can be de®ned from the shear-lag theory as [12] sd 2VfEc rVmEm td 16 where td is available from equation (12). For a frictionally-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) dier 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 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 propagates 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 interaction does not in¯uence the debond length. Using an energy balance approach, LI [13] theoretically 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 following relationship: Ld r sa 2tfVf 1 ÿ VfEf Ec VmEm Ec sd sa 2 " #1=2 8 < : 9 = ; 18 This relationship diers from the results of the force balance approach in terms of the appearance of an extra oset term of (sa/tf) 1/2 [13]. Although the measurements of interfacial properties such as tf and Gd are facilitated by the availability 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 diculty 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 nonlinear 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, borosilicate (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 carbon±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 asfabricated 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 parameters 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±displacement 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 debonding 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) provided 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 dierent 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 transmission 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 $a is the coecient of thermal expansion (25±5008C). %The composition for F glass is 76% SiO2±16% B2O3±8% K2O, obtained from Corning Glass Works, Corning, NY Fig. 3. A cross section of the SiC ®ber reinforced glass composite displaying uniformly distributed ®bers. Table 2. Parameters and properties for SCS-6 ®ber reinforced borosilicate glass composite Fiber volume fraction, Vf 0.12 Elastic modulus of composite, Ec (GPa) 97.3$ 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
SUN and SINGH: MULTIPLE MATRIX CRACKING B5651 B58 1502 100 B5-18 0 0.2 0.6 0.8 Displacement(mm) Fig. 5. The load-displacement curves for a composite loaded to different stress levels(for example, B5 18 denotes the sample B5 loaded at 18 N). between the fiber and matrix at the interface which 4. RESULTS AND DISCUSSION led to debonding as shown in Fig. 6. Each white band corresponds to a fiber-matrix interfacial 4. 1. Development of multiple matrix cracks and debonding with a matrix crack running across in the crack saturation middle of the band. The debond length was thus Four-point flexure tests were performed on the measured as half of the length of the white band. Sic fiber-reinforced transparent borosilicate glass After each load-unload cycle, the number and pos- composites. The first matrix crack occurs at a stress ition of matrix cracks, and debond length around of 85 MPa. A close look at the matrix under an each crack surface were recorded and measured by optical microscope revealed that more than one photographing and video recording method. The crack was generated which also included an incom- agnification of the pictures in Fig. 6 was about 16. letely propagated crack. This behavior was also We did not observe any time dependent growth observed by Dutton [21]. Therefore, an evidence of of the debond crack within a short time of our ex- first load drop in the load-displacement curve may periments in which specimens were loaded to a pre- not represent the real FMC point. After FMC, the determined level of stress and then immediately stress distribution along the fiber length but in the unloaded for measuring the debond length. No matrix phase varied from zero at the matrix crack change in the debond length was observed as a surfaces to a new pre-crack value. Subsequent for function of time after unloading. We understand mation of new cracks always occurred at progress- that glasses are prone to stress corrosion cracking ively higher stress levels because the flaw size is but we have not seen any growth of the debond smaller than that for the previous matrix cracks length during our experiments and therefore have This preference of crack for a larger flaw demon- not considered in the model calculations strated that the matrix cracking stress can vary over Although the continuity of the composite is dis- a range of values, which was also discussed by rupted by the multiple matrix cracks, the externally Zok [7] and Curtin [9]. The stiffness of the compo- applied stress is still calculated as the stress on the site which is indicated by the slope of the linear outermost tension surface of a rectangular specimen portion of each curve in Fig. 5 decreased with the suming that the elastic beam theory for the generation of additional new matrix cracks flexure model(MIL-STD-1942A)is valid Figure 7 shows the sequence and position of the Experiments were conducted on a total of 10 matrix cracks generated because of the increasing samples of the same fiber volume fraction. Debond load. The numbers on the dashed lines from I to 16 length measurement techniques were applied to in Fig. 7 indicate the sequence of matrix crack gen- only five of them. All of them showed a similar eration. The magnitude of the load/stress, written behavior. These data on the number of cracks and horizontally along the various crack numbers, indi- crack spacing from the 10 samples were used for cates the load/ stress at which these cracks were cre- the statistical distribution of crack density and ated. For example, the first three cracks 1, 2 and 3 crack spacing were detected at a load of 18 N( 85 MPa,corre-
between the ®ber and matrix at the interface which led to debonding as shown in Fig. 6. Each white band corresponds to a ®ber±matrix interfacial debonding with a matrix crack running across in the middle of the band. The debond length was thus measured as half of the length of the white band. After each load±unload cycle, the number and position of matrix cracks, and debond length around each crack surface were recorded and measured by photographing and video recording method. The magni®cation of the pictures in Fig. 6 was about 16. We did not observe any time dependent growth of the debond crack within a short time of our experiments in which specimens were loaded to a predetermined level of stress and then immediately unloaded for measuring the debond length. No change in the debond length was observed as a function of time after unloading. We understand that glasses are prone to stress corrosion cracking but we have not seen any growth of the debond length during our experiments and therefore have not considered in the model calculations. Although the continuity of the composite is disrupted by the multiple matrix cracks, the externally applied stress is still calculated as the stress on the outermost tension surface of a rectangular specimen assuming that the elastic beam theory for the ¯exure model (MIL-STD-1942A) is valid. Experiments were conducted on a total of 10 samples of the same ®ber volume fraction. Debond length measurement techniques were applied to only ®ve of them. All of them showed a similar behavior. These data on the number of cracks and crack spacing from the 10 samples were used for the statistical distribution of crack density and crack spacing. 4. RESULTS AND DISCUSSION 4.1. Development of multiple matrix cracks and crack saturation Four-point ¯exure tests were performed on the SiC ®ber-reinforced transparent borosilicate glass composites. The ®rst matrix crack occurs at a stress of 85 MPa. A close look at the matrix under an optical microscope revealed that more than one crack was generated which also included an incompletely propagated crack. This behavior was also observed by Dutton [21]. Therefore, an evidence of ®rst load drop in the load±displacement curve may not represent the real FMC point. After FMC, the stress distribution along the ®ber length but in the matrix phase varied from zero at the matrix crack surfaces to a new pre-crack value. Subsequent formation of new cracks always occurred at progressively higher stress levels because the ¯aw size is smaller than that for the previous matrix cracks. This preference of crack for a larger ¯aw demonstrated that the matrix cracking stress can vary over a range of values, which was also discussed by Zok [7] and Curtin [9]. The stiness of the composite which is indicated by the slope of the linear portion of each curve in Fig. 5 decreased with the generation of additional new matrix cracks. Figure 7 shows the sequence and position of the matrix cracks generated because of the increasing load. The numbers on the dashed lines from 1 to 16 in Fig. 7 indicate the sequence of matrix crack generation. The magnitude of the load/stress, written horizontally along the various crack numbers, indicates the load/stress at which these cracks were created. For example, the ®rst three cracks 1, 2 and 3 were detected at a load of 18 N (85 MPa, correFig. 5. The load±displacement curves for a composite loaded to dierent stress levels (for example, B5- 18 denotes the sample B5 loaded at 18 N). 1662 SUN and SINGH: MULTIPLE MATRIX CRACKING
SUN and SINGH: MULTIPLE MATRIX CRACKING matrix debond 43 a) Cracks# 1, 2, 3 at 18N b)Cracks# 1, 2, 3, 4 at 23N 2-1=2-00 c) Cracks#f 1, 2, 3, 4, 10 at 28N d) Cracks#1,2,3,4,10,12at33N e)Matrix crack saturation and associated debond length at 58N Fig. 6. Optical micrographs showing the progression of debonding (patterns)with increasing applied load Inside pin Inside pin N/MPa) :}16 46/217 28/132 Fig. 7. Evolution and distribution of matrix cracks on the tensile face of the specimen
Fig. 6. Optical micrographs showing the progression of debonding (patterns) with increasing applied load. Fig. 7. Evolution and distribution of matrix cracks on the tensile face of the specimen. SUN and SINGH: MULTIPLE MATRIX CRACKING 1663
SUN and SINGH: MULTIPLE MATRIX CRACKING 100 1门2 Error Bar 06185109132156179217245307 Applied Stress(MP Fig. 8. A dependence of matrix crack density on the applied stress. sponding to the first matrix cracking), and cracks distribution of the normalized crack spacing. For 15, 16 were generated at a load of 46 N(217 MPa, Fig. 9, the scatter of the crack spacing is from corresponding to the saturation of matrix cracking). 0.66mm to 2.53 mm, and the probability ranged No more matrix cracks were generated between the from 1. 2 to 19.8%. Figure 9 is obtained from a sum inside two pins after the saturation load of 46 N. of data from 10 samples with 164 cracks(7 samples Figure 8 shows the influence of the applied stress with 16 cracks, 2 samples with 17 cracks, and I on the crack density. The crack density was sample with 18 cracks). The predicted matrix crack obtained as the ratio of the number of cracks at a spacing calculated from equation (7)with statistical certain stress level to the total number of cracks at modification is about 0.65 mm by taking the inter- saturation. The saturation stress is about twice as face as a frictional case with a value of tr equal much as the FMC stress for this composite. This 30 MPa [22]. The experimental average crack range of stress is much larger than Zok's model [7, spacing value is much higher than the calculated in which the saturation stress was about 30% crack spacing value because firstly the real interface higher than the FMC stress. This difference can be of the composite shows debonding rather than explained by the fact that Zok,s model is valid purely frictional coupling, and secondly the statisti- Inder the assumption of a frictionally-coupled cal modification parameter a may not be equal to fiber-matrix interface. But, the current composite 1.34 in the case of a partially-bonded and debonded shows a higher interfacial shear strength (a large interface. In the present study, a is experimentally debond initiation stress aa)as shown in Table 2. found to be 2, and the overlap of debonded zone ccording to equation(17). the saturation stress for was not commonly observed in those composites a bonded interface will be higher than the friction- The non-uniformly distributed cracks implies that ally-coupled interface(aa>slide) provided other the matrix cracks and crack spacing are dependent roperties remain the same. Another explanation on the distribution of flaw and flaw size in the from Weitsman and Zhu [8] and LI [13] is that the matrix. amount of mechanical energy absorbed during the multiple fracture will be larger because of the appearance of an additional interfacial debond energy Ia term for a bonded interface. Therefore, the saturation stress will be higher because of the ontribution from interfacial debonding Another interesting phenomenon was the non- 8 uniformity of matrix crack distribution. The crack =10 spacing did not follow equation(7)strictly. The ex perimental average value from 10 specimens was E bout 1. I mm. The ratio of a particular crack spa- cing to the average crack spacing was taken as the normalized crack spacing. The statistical distri 聊] bution of matrix crack spacing was then obtained 06070.91.11.2151.82.3 by sorting the crack spacing, counting the number of their appearance, and calculating their prob- ability of occurrence. Figure 9 shows the statistical Fig 9 Histogram of matrix crack spacing
sponding to the ®rst matrix cracking), and cracks 15, 16 were generated at a load of 46 N (217 MPa, corresponding to the saturation of matrix cracking). No more matrix cracks were generated between the inside two pins after the saturation load of 46 N. Figure 8 shows the in¯uence of the applied stress on the crack density. The crack density was obtained as the ratio of the number of cracks at a certain stress level to the total number of cracks at saturation. The saturation stress is about twice as much as the FMC stress for this composite. This range of stress is much larger than Zok's model [7], in which the saturation stress was about 30% higher than the FMC stress. This dierence can be explained by the fact that Zok's model is valid under the assumption of a frictionally-coupled ®ber±matrix interface. But, the current composite shows a higher interfacial shear strength (a large debond initiation stress sd) as shown in Table 2. According to equation (17), the saturation stress for a bonded interface will be higher than the frictionally-coupled interface (sa>sslide) provided other properties remain the same. Another explanation from Weitsman and Zhu [8] and LI [13] is that the amount of mechanical energy absorbed during the multiple fracture will be larger because of the appearance of an additional interfacial debond energy Gd term for a bonded interface. Therefore, the saturation stress will be higher because of the contribution from interfacial debonding. Another interesting phenomenon was the nonuniformity of matrix crack distribution. The crack spacing did not follow equation (7) strictly. The experimental average value from 10 specimens was about 1.1 mm. The ratio of a particular crack spacing to the average crack spacing was taken as the normalized crack spacing. The statistical distribution of matrix crack spacing was then obtained by sorting the crack spacing, counting the number of their appearance, and calculating their probability of occurrence. Figure 9 shows the statistical distribution of the normalized crack spacing. For Fig. 9, the scatter of the crack spacing is from 0.66 mm to 2.53 mm, and the probability ranged from 1.2 to 19.8%. Figure 9 is obtained from a sum of data from 10 samples with 164 cracks (7 samples with 16 cracks, 2 samples with 17 cracks, and 1 sample with 18 cracks). The predicted matrix crack spacing calculated from equation (7) with statistical modi®cation is about 0.65 mm by taking the interface as a frictional case with a value of tf equal to 30 MPa [22]. The experimental average crack spacing value is much higher than the calculated crack spacing value because ®rstly the real interface of the composite shows debonding rather than a purely frictional coupling, and secondly the statistical modi®cation parameter a may not be equal to 1.34 in the case of a partially-bonded and debonded interface. In the present study, a is experimentally found to be 2, and the overlap of debonded zones was not commonly observed in those composites. The non-uniformly distributed cracks implies that the matrix cracks and crack spacing are dependent on the distribution of ¯aw and ¯aw size in the matrix. Fig. 8. A dependence of matrix crack density on the applied stress. Fig. 9. Histogram of matrix crack spacing. 1664 SUN and SINGH: MULTIPLE MATRIX CRACKING
SUN and SINGH: MULTIPLE MATRIX CRACKING 12 o.- Crack Al …Crak E +一 Crack#10 已号 05010015020 250300350 plied Stress(MPa) Fig 10. The dependence of debond length on the applied stress different matrix cracks 4.2. Interfacial debonding and debond length and finally overlapped with their neighbors, as The initiation and propagation of interfacial shown in Fig. 6(e) bonding were the most prominent phenomena accompanying the multiple matrix cracking which length as a function of the external stress for differ were visually observed in this investigation. Figure 6 ent matrix cracks illustrated in Fig. 10. A close- shows the debonded regions (lengths) on either side to-linear relationship between the debond length and applied stress is observed before the matrix of the matrix cracks at five different load levels. crack saturation which is quite similar to the trend The measured FMC stress for this composite with a predicted by ACK [6], BHE [12], HJ [10] and LI[13] fiber volume fraction of 12% is 85 MPa, and the in- models. But, a suppression of debond length is also ial debond stress obtained from a fiber pushout observed before the saturation of the matrix crack how similar Therefore, upon FMC, the additional load which is debond lengths because they are from the same thrown onto the bridging fibers locally around the region of the sample and show similar crack spa- matrix crack is sufficiently large to initiate the lowever, cracks #9 and 12 differ greatly from debond as well. This is the reason why debonding other cracks in measured values of the debond was observed at the interface when the FMC length. This unusual behavior results from a non- occurred. This observation is even true for an uniform crack spacing and stress distribution along incompletely propagated crack #4 in Fig. 6(a). Such the length of the bend specimen after a large displa a weakly-bonded interface is desired for toughening cement upon matrix cracking brittle matrix materials. As expected, the debond Figure 11 shows a comparison of the calculated propagated further with an increase of applied load and measured dependencies of debond length on 12 #1Fd=1.2J/ * -Crack #2 Crack 30MPa C#2 Energy approach 少象=譬 = 100 200 250 Applied Stress(MP Fig. 11. Experimental and theoretical dependencies of debond length on the applied stress
4.2. Interfacial debonding and debond length The initiation and propagation of interfacial debonding were the most prominent phenomena accompanying the multiple matrix cracking which were visually observed in this investigation. Figure 6 shows the debonded regions (lengths) on either side of the matrix cracks at ®ve dierent load levels. The measured FMC stress for this composite with a ®ber volume fraction of 12% is 85 MPa, and the initial debond stress obtained from a ®ber pushout test [22] is about 55 MPa, as shown in Table 2. Therefore, upon FMC, the additional load which is thrown onto the bridging ®bers locally around the matrix crack is suciently large to initiate the debond as well. This is the reason why debonding was observed at the interface when the FMC occurred. This observation is even true for an incompletely propagated crack #4 in Fig. 6(a). Such a weakly-bonded interface is desired for toughening brittle matrix materials. As expected, the debond propagated further with an increase of applied load and ®nally overlapped with their neighbors, as shown in Fig. 6(e). The experimental data of measured debond length as a function of the external stress for dierent matrix cracks are illustrated in Fig. 10. A closeto-linear relationship between the debond length and applied stress is observed before the matrix crack saturation which is quite similar to the trend predicted by ACK [6], BHE [12], HJ [10] and LI [13] models. But, a suppression of debond length is also observed before the saturation of the matrix cracking. The cracks #1, 2, 3 and 10 show similar debond lengths because they are from the same region of the sample and show similar crack spacing. However, cracks #9 and 12 dier greatly from other cracks in measured values of the debond length. This unusual behavior results from a nonuniform crack spacing and stress distribution along the length of the bend specimen after a large displacement upon matrix cracking. Figure 11 shows a comparison of the calculated and measured dependencies of debond length on Fig. 10. The dependence of debond length on the applied stress for six dierent matrix cracks. Fig. 11. Experimental and theoretical dependencies of debond length on the applied stress. SUN and SINGH: MULTIPLE MATRIX CRACKING 1665
SUN and SINGH: MULTIPLE MATRIX CRACKING 6 FTTTTTTTTTTTTTTTTTTTTTTTTTTTT 04 岛 ⊥uLuL 200250300 Applied Stress(MPa) Fig 12. Interrelationships among average block size, interfacial debonding, and the applied stress the applied stress. The calculated values from both The interactions between these two processes are force balance approach and energy balance clearly observed in all experiments of this study approach are close to each other for the short Figure 12 illustrates these two dynamic processes debond lengths, but deviate a lot for the long upon increase of the external stress. The generation debond length corresponding to the evolution of of new matrix cracks is dependent on both the multiple matrix cracks. The model from energy bal- interfacial properties such as tr and ta and distri- ance approach fits the data much better than the bution of flaws in the matrix. For matrix cracking force balance approach for the debond length in the to occur, the size of the uncracked matrix block case of multiple matrix cracking. This difference can must ss In e explained by the fact that two conditions are matrix due to load transfer from fiber to matrix can necessary for interfacial debonding to form and occur up to the matrix cracking stress. Saturation propagate. Firstly, the fiber/matrix interfacial shear in the matrix cracking occurs when the matrix stress stress must be greater than the shear strength of the cannot rise sufficiently because of the complete interface. Secondly, the work done by the externally propagation of the debonded zones from either applied stress must supply a sufficient energy for sides of the uncracked matrix blocks the debonding process. In most cases, a higher Both the force and energy balance approaches stress level is required for the second criterion. In have not considered the saturation phenomena in our case, for a small debond length, the frictional the multiple matrix cracking situation which is cor dissipation energy plays little role compared with monly seen in fiber-reinforced ceramic composites the interfacial debond energy. But, with the increase Therefore, based on the knowledge of the inter- of debond length, the frictional energy dissipation relationship between the interaction of matrix increases while the debond energy term remains the cracks and debond length, the saturation stress as same. The frictional energy dissipation can not be for a composite displaying multiple matrix cracking ignored after a certain extension of the debond can provide an upper bound for the application of length which then contributes partly to the suppres- these models. Because the model from energy bal ion of the debond length for a long debond length, ance approach provides a better fit to the exper as observed in Figs 10 and 11 imental data, especially for a long debond length as shown in Fig. ll, equation(18)is used for deter- 4.3. Interaction between matrix cracks and interfacial mining the saturation stress as This saturation debonding, and determination of stress as is the critical applied stress corresponding The saturation of matrix cracks for a composite to the maximum average debond length, which is In ith weakly-bonded interface is determined by the quation(18), Oa is substituted by os, and La by 2, interaction of two dynamic processes without con- where Les can be statistically approximated as az sidering the delamination effect and z is obtained from equation (7). It needs to be because of the generation of new cracks md x equal to 1.34[7, 9], especially for a composite show- (b)longitudinal growth of the interfacial ing debonding. Rearranging equation (18)then because of the increased stresses in fibers gives
the applied stress. The calculated values from both force balance approach and energy balance approach are close to each other for the short debond lengths, but deviate a lot for the long debond length corresponding to the evolution of multiple matrix cracks. The model from energy balance approach ®ts the data much better than the force balance approach for the debond length in the case of multiple matrix cracking. This dierence can be explained by the fact that two conditions are necessary for interfacial debonding to form and propagate. Firstly, the ®ber/matrix interfacial shear stress must be greater than the shear strength of the interface. Secondly, the work done by the externally applied stress must supply a sucient energy for the debonding process. In most cases, a higher stress level is required for the second criterion. In our case, for a small debond length, the frictional dissipation energy plays little role compared with the interfacial debond energy. But, with the increase of debond length, the frictional energy dissipation increases while the debond energy term remains the same. The frictional energy dissipation can not be ignored after a certain extension of the debond length which then contributes partly to the suppression of the debond length for a long debond length, as observed in Figs 10 and 11. 4.3. Interaction between matrix cracks and interfacial debonding, and determination of saturation stress The saturation of matrix cracks for a composite with weakly-bonded interface is determined by the interaction of two dynamic processes without considering the delamination eect: (a) size-reduction of the uncracked block of matrix because of the generation of new cracks, and (b) longitudinal growth of the interfacial debond because of the increased stresses in ®bers. The interactions between these two processes are clearly observed in all experiments of this study. Figure 12 illustrates these two dynamic processes upon increase of the external stress. The generation of new matrix cracks is dependent on both the interfacial properties such as tf and td and distribution of ¯aws in the matrix. For matrix cracking to occur, the size of the uncracked matrix block must be suciently large so that the stress in the matrix due to load transfer from ®ber to matrix can occur up to the matrix cracking stress. Saturation in the matrix cracking occurs when the matrix stress cannot rise suciently because of the complete propagation of the debonded zones from either sides of the uncracked matrix blocks. Both the force and energy balance approaches have not considered the saturation phenomena in the multiple matrix cracking situation which is commonly seen in ®ber-reinforced ceramic composites. Therefore, based on the knowledge of the interrelationship between the interaction of matrix cracks and debond length, the saturation stress ss for a composite displaying multiple matrix cracking can provide an upper bound for the application of these models. Because the model from energy balance approach provides a better ®t to the experimental data, especially for a long debond length as shown in Fig. 11, equation (18) is used for determining the saturation stress ss. This saturation stress ss is the critical applied stress corresponding to the maximum average debond length, which is half of the matrix crack spacing LCS. Thus, in equation (18), sa is substituted by ss, and Ld by Lcs 2 , where Lcs can be statistically approximated as az', and z' is obtained from equation (7). It needs to be mentioned that the parameter a is not necessarily equal to 1.34 [7, 9], especially for a composite showing debonding. Rearranging equation (18) then gives the saturation stress ss as Fig. 12. Interrelationships among average block size, interfacial debonding, and the applied stress. 1666 SUN and SINGH: MULTIPLE MATRIX CRACKING