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