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 sawwhere 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