Journal of the American Ceramic Sociery-Rebillat et al. Vol 81. No 4 weak fiber/matrix interfaces, 18, 19 21, 22, 28,29 that are addressed in the push-out models. 22 28-30 It exhibits well-defined features (10) After the elastic deformation of the fiber(a-"b in Fig. 2), debonding or fiber sliding of unbonded fibers initiates at point EdI +vm)In b. In the case of unbonded fibers, the applied stress overcomes static friction. The nonlinear portion of the curve(b-c)is where r and R refer to the radius of the fiber and the matrix, characterized by a continuously decreasing slope due to pro- respectively, and t is the thickness of the sample gressive fiber debonding or the fiber overcoming static friction, Hsueh introduced the effect of Poisson s expansion of the followed by sliding. The stable growth of the interfacial crack fiber, which increases the radial compressive stresses. The in- continues up to a maximum stress at point c. Unstable debond- terfacial shear strength(for t > R)is given by the following ing of the remaining portion of bonded fiber is accompanied by equation: a sudden load drop that is observed in the decreasing portion of the curve(c-d"). Finally, the fiber is pushed out of the matrix, yielding a pseudo-plateau(d-e t== (+()会[=(=2 y 8 The average interfacial sliding stress may be estimated from plateau stre (14) where plateau is the compressive applied stress, t the embedded (12) E(1+Vm)l R In (R out model of Hsueh to the nonlinear portion of the curve("b c" in Fig. 2)through the coefficient of friction, H, the re In a more refined model. Hsueh27 added the effects of the sidual clamping stress, o and the residual axial stress in the mismatch strains between the fiber and the matrix in the radial fiber, o,(Eqs. (1H5)) and the axial directions (2) Push-Out Curves for Composites Reinforced with Treated fibers B2lexp(mt)+exp(-mt)-2] (2[exp(mt)-exp(-m)l In contrast to the classical push-out behavior shown in F 2, the push-out curves measured for the CMCs with treated A[exp(mt)+exp(mt)-2 fibers exhibit features that are not described by the interfacial aaf exp(mt)+exp(-mt)+ models. The curves(Fig 3)consist of a linear portion("a b")that corresponds to the elastic deformation of the fiber,a (13) steep load drop(b'c) that indicates the occurrence of fiber debonding, a short plateau(c) that is followed by a where B, m, and A are elastic constants and B2 is a parameter nonlinear portion("c"'d")with reversed curvature(in con- that is dependent on the thermal expansion mismatch between trast to the curves in Fig. 2), a downward small concave portion the fiber and the matrix. 27 d-e), a load drop(e-f) that coincides with fiber rotrusion,and finally a pseudo-plateau when the fiber slides IV. Results out of the matrix. As indicated above. the Hsueh model27 can- not be fitted to the nonlinear portion of the push-out curves, ( Push-Out Curves for Composites Reinforced with because of the existence of the reversed curvature. The fric- As-Received fibers tional shear stress was estimated from the plateau via Eq (15) Figure 2 shows an example of the typical stress-versus-fiber- (3)Interfacial Parameters for the end-displacement curves measured for the standard SiC/C/SiC Untreated Fiber-Reinforced Composite composite with untreated fibers. This curve is similar to those Results of the series of tests performed on composite I are obtained for other ceramic-matrix composites(CMCs)with ummarized in Table Il. previous results on the effect of 2000 1500 0 D Fig. 2. Single-fiber push-out curve measured for a SiC/C/SiC composite(sample I)reinforced with as-received NicalonTM fibers(F is the applied stress)with b = 1 r H Em Ef~1 + nm! ln S R rDJ 1/2 (10) where r and R refer to the radius of the fiber and the matrix, respectively, and t is the thickness of the sample. Hsueh26 introduced the effect of Poisson’s expansion of the fiber, which increases the radial compressive stresses. The in￾terfacial shear strength (for t >> R) is given by the following equation: ts = −sd 1 S R2 r 2 − 1DSEm Ef D coth ~at! + 2 exp~at! − exp~−at! S 2 rDH~1 + nm!F1 + S R2 r 2 − 1DSEm Ef DGFR2 ln S R r D − ~R2 − r 2 ! 2 GJ 1/2 2 (11) with a = 1 r 5 r 2 Ef + ~R2 − r 2 !Em Ef~1 + nm!FR2 ln S R r D − ~R2 − r 2 ! 2 G6 1/2 (12) In a more refined model, Hsueh27 added the effects of the mismatch strains between the fiber and the matrix in the radial and the axial directions: ts = S rm 2@exp~mt! − exp~−mt!# DSB2@exp~mt! + exp~−mt! − 2# B1 − sdHexp~mt! + exp~−mt! + A@exp~mt! + exp~−mt! − 2# B1 JD (13) where B1, m, and A are elastic constants and B2 is a parameter that is dependent on the thermal expansion mismatch between the fiber and the matrix.27 IV. Results (1) Push-Out Curves for Composites Reinforced with As-Received Fibers Figure 2 shows an example of the typical stress-versus-fiber￾end-displacement curves measured for the standard SiC/C/SiC composite with untreated fibers. This curve is similar to those obtained for other ceramic-matrix composites (CMCs) with weak fiber/matrix interfaces2,18,19,21,22,28,29 that are addressed in the push-out models.22,28–30 It exhibits well-defined features. After the elastic deformation of the fiber (‘‘a’’–‘‘b’’ in Fig. 2), debonding or fiber sliding of unbonded fibers initiates at point b. In the case of unbonded fibers, the applied stress overcomes static friction. The nonlinear portion of the curve (‘‘b’’–‘‘c’’) is characterized by a continuously decreasing slope due to pro￾gressive fiber debonding or the fiber overcoming static friction, followed by sliding. The stable growth of the interfacial crack continues up to a maximum stress at point c. Unstable debond￾ing of the remaining portion of bonded fiber is accompanied by a sudden load drop that is observed in the decreasing portion of the curve (‘‘c’’–‘‘d’’). Finally, the fiber is pushed out of the matrix, yielding a pseudo-plateau (‘‘d’’–‘‘e’’). The average interfacial sliding stress may be estimated from the plateau stress: tplateau = splateaur 2t (14) where splateau is the compressive applied stress, t the embedded length of fiber, and r the fiber radius. As discussed previously, the interfacial shear stress is also extracted by fitting the push￾out model of Hsueh to the nonlinear portion of the curve (‘‘b’’– ‘‘c’’ in Fig. 2) through the coefficient of friction, m, the re￾sidual clamping stress, sc, and the residual axial stress in the fiber, sz (Eqs. (1)–(5)).21 (2) Push-Out Curves for Composites Reinforced with Treated Fibers In contrast to the classical push-out behavior shown in Fig. 2, the push-out curves measured for the CMCs with treated fibers exhibit features that are not described by the interfacial models. The curves (Fig. 3) consist of a linear portion (‘‘a’’– ‘‘b’’) that corresponds to the elastic deformation of the fiber, a steep load drop (‘‘b’’–‘‘c’’) that indicates the occurrence of fiber debonding, a short plateau (‘‘c’’) that is followed by a nonlinear portion (‘‘c’’–‘‘d’’) with reversed curvature (in con￾trast to the curves in Fig. 2), a downward small concave portion (‘‘d’’–‘‘e’’), a load drop (‘‘e’’–‘‘f’’) that coincides with fiber protrusion, and finally a pseudo-plateau when the fiber slides out of the matrix. As indicated above, the Hsueh model27 can￾not be fitted to the nonlinear portion of the push-out curves, because of the existence of the reversed curvature. The fric￾tional shear stress was estimated from the plateau via Eq. (15). (3) Interfacial Parameters for the Untreated Fiber-Reinforced Composites Results of the series of tests performed on composite I are summarized in Table II. Previous results on the effect of Fig. 2. Single-fiber push-out curve measured for a SiC/C/SiC composite (sample I) reinforced with as-received Nicalon™ fibers (F is the applied stress). 968 Journal of the American Ceramic Society—Rebillat et al. Vol. 81, No. 4