2990 Journal of the American Ceramic Sociery-Kuo et al. Vol. 80. No. 12 approach that has been taken is to use the shrink-fit calculated residual stresses as inputs for the Liang and Hutchinson 20 push- ut model(hereafter refe The detailed mechanics of fiber debonding and sliding can best be by considering the LH model. Although the Hutchinson20 considered only phase system that consisted of fiber and matrix, their analysis can still be used to provide a qualitat ment of the effect of coating thickness on the pushout response of individual fibers. Two key equations are presented in the Lh model:(i) the load-displacement equation for a partially debonded fiber and (ii) the load-displacement equation for a In the first case, the peak stress that is experienced just before complete fiber debonding occurs, Pp(uppercase italic P indicates stress), is given as2o Po=Pp+ T Er To+μNg (b) where PR is the axial residual stress(negative for tension), To the roughness-induced(asperit )sliding stress, NR the adial residual stress, H the co interface, T; the pure Mode Il fracture energy, R the fiber radius, and E the elastic modulus of the fiber; B, and B2 are elastic properties of the composite. The term c* is given by R e where L is the thickness of the slice In this study, the calculation of elastic properties B, and B was based on the assumption of a transversely isotropic fiber and an isotropic matrix for the Al2O3 fiber system. B, and B2 were formulated following Liang and Hutchinson:26 Fig. 3. SEM micrographs of pushed-out fibers from(a) the Al,O fiber system and (b)the YAG fiber system B2 2v B, where the superscript r represents the transverse direction useful for providing a qualitative and general assessment of the assumption of an isotropic fiber(a cubic single-crystal YAG B, and B2 were formulated following Liang and Hutchinson ful measure of the interface properties. a detailed mechanistic understanding can be obtained only by using more-detailed pushout models, such as those of Liang and Hutchinson20or B1=(1-vEm+1+)E Kerans and Parthasarathay. 9 Such an approach is presented in the following sections 3) The Liang and Hutchinson Model of Fiber Pushout (3b It is postulated herein that the key effect that is associated Notably, the commonly used interfacial shear strength uses the with the coating thickness is the effect on the residual stress peak stress of the LH equation to define an instantaneous state of the model composite systems. This postulate is based debond propagation down the entire length of the fiber/matrix on the microstructural evaluation of the interfacial debonds interface. Thus, the interfacial shear strength disregards the Typical micrographs of the pushed-out fibers show that the detailed interfacial mechanics and determines an average in- debond, in all cases, is at the coating/fiber interface(Fig. 3) terfacial shear strength to quantify interfacial debonding. As a The coating is dense and almost uniform for different coating result, the interfacial shear strength parameter does not mecha- thicknesses. As a result, the debonding and sliding interfaces nistically capture the details of progressive debonding and does are the same, regardless of the coating thickness. Because the not separate the effects of the stress state from the debonding chemistry of the interphase does not change with changes in the properties coating thickness, it is reasonable to assume that the physical perties of the interface, which include the interfacial frac- qe Liang and Hutchinson20 presented an additional equation to cribe the frictional sliding stress that follows complete in- ture energy Ti the coefficient of sliding friction u, and the terfacial debonding. The frictional pushout stress, P, is giv- interface roughness, remain constant as the coating thickness en by hanges. Thus, any differences in the mechanical properties of the interface that are associated with changes in the coating thickness are due to a change in the local stress state. The =|+μ(NR-B1PR) μB1neous. Therefore, although the linear and shear-lag models are useful for providing a qualitative and general assessment of the interface, these methods do not provide an accurate or insight￾ful measure of the interface properties. A detailed mechanistic understanding can be obtained only by using more-detailed pushout models, such as those of Liang and Hutchinson20 or Kerans and Parthasarathay.19 Such an approach is presented in the following sections. (3) The Liang and Hutchinson Model of Fiber Pushout It is postulated herein that the key effect that is associated with the coating thickness is the effect on the residual stress state of the model composite systems. This postulate is based on the microstructural evaluation of the interfacial debonds. Typical micrographs of the pushed-out fibers show that the debond, in all cases, is at the coating/fiber interface (Fig. 3). The coating is dense and almost uniform for different coating thicknesses. As a result, the debonding and sliding interfaces are the same, regardless of the coating thickness. Because the chemistry of the interphase does not change with changes in the coating thickness, it is reasonable to assume that the physical properties of the interface, which include the interfacial frac￾ture energy Gi , the coefficient of sliding friction m, and the interface roughness, remain constant as the coating thickness changes. Thus, any differences in the mechanical properties of the interface that are associated with changes in the coating thickness are due to a change in the local stress state. The approach that has been taken is to use the shrink-fit calculated residual stresses as inputs for the Liang and Hutchinson20 push￾out model (hereafter referenced as the LH model). The detailed mechanics of fiber debonding and sliding can best be explained by considering the LH model. Although the analysis of Liang and Hutchinson20 considered only a two￾phase system that consisted of fiber and matrix, their analysis can still be used to provide a qualitative and rational assess￾ment of the effect of coating thickness on the pushout response of individual fibers. Two key equations are presented in the LH model: (i) the load–displacement equation for a partially debonded fiber and (ii) the load–displacement equation for a completely debonded fiber. In the first case, the peak stress that is experienced just before complete fiber debonding occurs, PP (uppercase italic P indicates stress), is given as20 PP = PR + 2S Gi Ef B2Rf D 1/2 exp z* + t0 + mNR mB1 ~exp z* − 1! (1) where PR is the axial residual stress (negative for tension), t0 the roughness-induced (asperity-induced) sliding stress, NR the radial residual stress, m the coefficient of friction at the sliding interface, Gi the pure Mode II interfacial fracture energy, Rf the fiber radius, and Ef the elastic modulus of the fiber; B1 and B2 are elastic properties of the composite. The term z* is given by z* = 2mB1S L − 1.5Rf Rf D where L is the thickness of the slice. In this study, the calculation of elastic properties B1 and B2 was based on the assumption of a transversely isotropic fiber and an istotropic matrix for the Al2O3 fiber system. B1 and B2 were formulated following Liang and Hutchinson:20 B1 = nf Em r ~1 − nf r !~Ef/Ef r !Em r + ~1 + nm r !Ef (2a) and B2 4 1−2nf B1 (2b) where the superscript r represents the transverse direction. For the YAG fiber system, B1 and B2 were based on the assumption of an isotropic fiber (a cubic single-crystal YAG fiber) and an isotropic matrix (a polycrystalline Al2O3). Again, B1 and B2 were formulated following Liang and Hutchinson:20 B1 = nfEm ~1 − nf!Em + ~1 + nm! Ef (3a) and B2 4 1−2nf B1 (3b) Notably, the commonly used interfacial shear strength uses the peak stress of the LH equation to define an instantaneous debond propagation down the entire length of the fiber/matrix interface. Thus, the interfacial shear strength disregards the detailed interfacial mechanics and determines an average in￾terfacial shear strength to quantify interfacial debonding. As a result, the interfacial shear strength parameter does not mecha￾nistically capture the details of progressive debonding and does not separate the effects of the stress state from the debonding properties. Liang and Hutchinson20 presented an additional equation to describe the frictional sliding stress that follows complete in￾terfacial debonding. The frictional pushout stress, Pl , is giv￾en by P1 = F t0 + m~NR − B1PR! mB1 G~exp zd − 1! (4) Fig. 3. SEM micrographs of pushed-out fibers from (a) the Al2O3 fiber system and (b) the YAG fiber system. 2990 Journal of the American Ceramic Society—Kuo et al. Vol. 80, No. 12