Criteria for Progressive Interfacial Debonding with Friction in Fiber-reinforced Ceramic Composites

Criteria for progressive debonding at the fiber/mattix interface with friction along the debonded interface are considered for fiber-reinfoxed ceramic composites. The energy-based criterion is adopted to analyze the debond length, the crack-opening displacement, and the displacement of the composite due to interfacial debonding. The analytical solutions are identical to those obtained from the mismatch-strain criterion, in which interfacial debonding is assumed to occur when the mismatch in the axial strain between the fiber and the matrix reaches a critical value. Furthermore, the mismatch-strain criterion is found to bear the same physical meaning as the strength-based criterion. INTRODUCTION Bridging of matrix cracks by fibers, which debond from and slip frictionally against the matrix, is an importanz toughening mechanism in fiber-reinforced ceramic composites [1,2]. To analyze the tougiiening effect, a criterion for progressive debonding at the fiberhanix interface accompanied by friction along the debonded interface is required. The loading stress on the fiber to initiate deboilding (or the debond stress for a frictionless interface), ad, has been analyzed by using either the energy-based [3-61 or the strength-based criterion [7-91. The effect of constant friction along the debonded interface on progressive debonding was analyzed recently by Nair [lo] using the energy-based criterion and by Budiansky et al. [ 113 using the strength-based criterion. It is noted that rehement is required in Nair's analysis regarding the work done by load. An altanative debonding criterion was proposed recently in which debonding is assumed to occur when the mismatch in the axial strain between the fiber and the matrix reaches a critical value [12]. Based on this assumption, the solutions for progressive debonding have been obtained [13]. A question is raised as to whether the solutions obtained from the three debonding criteria mentioned above agree with each other. The purpose of the present study is to address the above question. First, using the energybased criterion, solutions for progressive debonding with a constant friction along the debonded interface are obtained by modifying Nair's analysis [lo]. These solutions are then compared to those obtained from the mismatch-strain criterion. Finally, the physical meaning of the approach using the strength-based criterion is examined and compared to the mismtch-strain criterion. THE ENERGY-BASED CRITERION A unidirectional composite subjected to a tensile load in the direction parallel to the fiber axis is considered Matrix cracking occurs perpendicular to the loading direction and is bridged by intact fibers, which exert a bridging stress, 00, to oppose mk-opening. This problem can be modeled by using a representative volume element shown in Fig. 1. A fiber with a radius, a, is located at the center of a coaxial cylindrical shell of matrix with an outer radius, by such that a2/@ corresponds to the volume fraction of fibers, Vf, in the composite pig. la). m e n the interface remains bonded, the composite is subjected to a tensile stress, VfOa and has a displacement, Ubonded, in the axial direction (Fig. lb). In the presence of interfacial debonding, the bridging fiber is subjected to a tensile stress, ob, and the matrix is stress-free at the crack surface (Fig. IC). Interfacial debonding and sliding occur along a length, h, with a frictional stress, z, and the end of the debonding zone and the crack surface are located at z=O and z=h, respectively. The half crack-opening displacement, ~ 0 , is defined by the relative displacement between the fiber and the matrix at the crack surface (Fig. IC). Also, compared to the composite without interfacial debonding (Fig. lb), the composite with interfacial debonding has an additional displacement, Zldebond, in the loading direction (Fig. IC). Udebond . 40 . I Fig. 2. A representative volume element for the fiber bridging problem: (a) prior to loading, (b) loading without interfacial debonding, and (c) loading with interfacial debonding. The half cracking opening displacement, w, and the displacement of the composite due to interfacial debonding, Udebond, are also shown. in the fiber and t he matrix When the interface is bonded, the equilibrium axial stresses in the fiber and the matrix, af and (1) om, satisfy both the equilibrium and the continuity conditions, such that Vf Of + VmOm = Vf 00 where V, (=l-Vf) is the volume fraction of the matrix, and Ef and Em are Young's moduli of the fiber and the matrix, respectively. Combination of Eqs. (1) and (2) yields Of = Vf Ef 00 (for bonded interface) (3a) EC am= V E f o 0 (for bonded interface) (3b) EC where E, = VfEf + VmEm. For a frictional interface, both of and om can be approximated to be independent of the radial coordinate [4,5], and Eq. (1) is satisfied. The axial stresses in the fiber and the matrix at the end of the debond length, afd and omd, can be obtained from the stress transfer equation, such that Omd =2 h V f ~ (4b) avm Solutions of Ofd and a d are contingent upon the determination of h. With constant friction, the axial stress distributions in the fiber and the matrix, of and %, along the debond length are