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 givesthe 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 bal￾ance approach ®ts the data much better than the force balance approach for the debond length in the case of multiple matrix cracking. This di€erence 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 sucient 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 suppres￾sion 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 con￾sidering the delamination e€ect: (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 distri￾bution of ¯aws in the matrix. For matrix cracking to occur, the size of the uncracked matrix block must be suciently 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 suciently 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 com￾monly seen in ®ber-reinforced ceramic composites. Therefore, based on the knowledge of the inter￾relationship 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 bal￾ance approach provides a better ®t to the exper￾imental data, especially for a long debond length as shown in Fig. 11, equation (18) is used for deter￾mining 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 show￾ing 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