《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_j.1151-2916.1997.tb03088.x[1]

ournal J. Am Ceram Soc, 80 [8]20413-55(1997) Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites Triplicane A. Parthasarathy* T and Ronald J Kerans United States Air Force Wright Laboratory Materials Directorate, Wright-Patterson AFB, Ohio 45433-7817 Potential effects of interfacial roughness in ceramic com- MPa of interfacial shear stress of frictional sliding. 9 Such posites were studied using a model that included the pro- hbe fibera increasing contribution of roughness with rela- understanding before one can intelligently select or design the atrix interface. A parametric approach was used te Initially, the source of the radial clamping stress had been study interfacial roughness in conjunction with other pa- assumed to be mostly due to the residual stresses that result rameters such as the strength, radius, and volume fraction from the thermal mismatch between the fiber and the matrix of the fiber. The progressive roughness contribution during Now, two more factors have been identified as being equally initial fiber/matrix sliding caused a high effective coeffi- important in determining the sliding resistance. First, the path cient of friction, as well as an increased clamping stress of the interfacial debond crack can cause roughness of the which led to rapidly changing friction with increasing sliding surfaces, which can significantly increase the sliding debond length Calculated effects implied a potentially sig resistance. Second, the compliance of any interfacial layer pre nificant contribution to the behavior of real composite sys- sent can significantly affect how the mismatch strains, which tems and the necessity for explicit consideration in the in- result from differential thermal expansion or roughness, are terpretation of experimental data to understand composite accommodated. The significance of the second factor is onl behavior correctlY. In a tension test, the poisson's contrac- beginning to be recognized and is discussed elsewhere, 20, 21 tion of the fiber may negate the effects of roughness, allow whereas the former has received more attention and is the ing an"effective constant shear stress"()approximation. subject of the present work. In particular, interfacial roughness This was evaluated using a piecewise linear approximation may have a significant role in the design of oxidation-resistant fiber/matrix interface control, which is the key technological posite stress-strain behavior; for the Nicalon/SiC system, challenge to the use of ceramic composite, 2,23 based on oxide be obtained from fiber pushout tests and/or matrix crack coatings are being investigated (for a brief review, see Ker- ans2). The focus of these investigations is the achievement of sufficiently low shear strength to allow impinging matrix L. Introduction cracks to deflect along the interface and bypass fibers without fiber failure. this characteristic is essential. However the to- phy of the interfacial fracture surfaces may differ greatly those of conventional interphases(mostly carbon, some interface debonding and sliding 1-3 Those details are, in turn, BN). This will necessitate explicit attention to interfacial determined by the toughness of the interface, the sliding fric- roughness in the design of the fiber coating system, which, in tion, and the stress state. The resistance to sliding per unit area turn, requires a thorough understanding of the phenomena that of the interface is usually considered either to be a constant or are involved The relative magnitude of the effect of roughness was shown to obey the Coulomb friction law, i. e, the product of the co- to be high, using the fiber pushback or"seating drop.mea- efficient of friction and the radial clamping stress across the fiber/matrix interface or, more accurately, the debonding crack surements.8, 24 Initial modeling of the effect of roughness was faces. Within this framework, fiber pushout and pullout exper ments have been used to measure the interfacial friction coef- facial roughness of amplitude h results in a mismatch strain of ficient and radial clamping stress in composites. -l8 However, h/R, where R is the fiber radius(Fig. 1). This roughness attempts to extend these one or two parameters to a detailed induced strain simply adds to the thermal mismatch strain. This understanding of the influence on macroscopic fracture behav or have had limited success. Qualitative inferences have be the fiber and matrix surfaces mate completely made and frictional shear stresses in the range of 2-50 MPa nal position) and (ii) when the fiber slides, relative to the ma lave been correlated with"good"composites. However, com trix, through a distance greater than a characteristic half-period posites with excellent properties have been obtained with >100 of the roughness, the fiber develops the mismatch strain hIRe Experimental work has shown that this approach reasonably captures the major aspects of actual behavior, 6 however, the choice of the value of h that properly describes a real surface is Evans--contributing editor ambiguous. Recent works27-30 have clarified this somewhat; however, more work is required all these developments have addressed the effect of rough ness on the sliding of a fiber along its entire length, relative to ript No. 191659. Received July 29. 1996: approved Materia s Directorate sliding. However, to model the fracture behavior of a compos- the matrix, as in a pushout or pullout test during""steady-state unde Air Force Contract No. F-33615-91-C-5663 4 ite, it is important to consider the effect of roughness on a tWith UES, Inc, Dayton, OH 454 debond crack as it propagates along the fiber/matrix interface 2043

Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites Triplicane A. Parthasarathy*,† and Ronald J. Kerans* United States Air Force Wright Laboratory Materials Directorate, Wright-Patterson AFB, Ohio 45433–7817 Potential effects of interfacial roughness in ceramic com￾posites were studied using a model that included the pro￾gressively increasing contribution of roughness with rela￾tive fiber/matrix displacement during debonding of the fiber/matrix interface. A parametric approach was used to study interfacial roughness in conjunction with other pa￾rameters such as the strength, radius, and volume fraction of the fiber. The progressive roughness contribution during initial fiber/matrix sliding caused a high effective coeffi￾cient of friction, as well as an increased clamping stress, which led to rapidly changing friction with increasing debond length. Calculated effects implied a potentially sig￾nificant contribution to the behavior of real composite sys￾tems and the necessity for explicit consideration in the in￾terpretation of experimental data to understand composite behavior correctly. In a tension test, the Poisson’s contrac￾tion of the fiber may negate the effects of roughness, allow￾ing an ‘‘effective constant shear stress’’ (
) approximation. This was evaluated using a piecewise linear approximation to the progressive roughness model in an analysis of com￾posite stress–strain behavior; for the Nicalon/SiC system, the effective
value was lower than the values that would be obtained from fiber pushout tests and/or matrix crack spacings. I. Introduction THE major aspects of the mechanical behavior of ceramic composites are determined by the details of fiber/matrix interface debonding and sliding.1–3 Those details are, in turn, determined by the toughness of the interface, the sliding fric￾tion, and the stress state. The resistance to sliding per unit area of the interface is usually considered either to be a constant or to obey the Coulomb friction law, i.e., the product of the co￾efficient of friction and the radial clamping stress across the fiber/matrix interface or, more accurately, the debonding crack faces. Within this framework, fiber pushout and pullout experi￾ments have been used to measure the interfacial friction coef￾ficient and radial clamping stress in composites.4–18 However, attempts to extend these one or two parameters to a detailed understanding of the influence on macroscopic fracture behav￾ior have had limited success. Qualitative inferences have been made and frictional shear stresses in the range of 2–50 MPa have been correlated with ‘‘good’’ composites. However, com￾posites with excellent properties have been obtained with >100 MPa of interfacial shear stress of frictional sliding.19 Such phenomenological studies need to be extended to a mechanistic understanding before one can intelligently select or design the best material for the desired application. Initially, the source of the radial clamping stress had been assumed to be mostly due to the residual stresses that result from the thermal mismatch between the fiber and the matrix. Now, two more factors have been identified as being equally important in determining the sliding resistance. First, the path of the interfacial debond crack can cause roughness of the sliding surfaces, which can significantly increase the sliding resistance. Second, the compliance of any interfacial layer pre￾sent can significantly affect how the mismatch strains, which result from differential thermal expansion or roughness, are accommodated. The significance of the second factor is only beginning to be recognized and is discussed elsewhere,20,21 whereas the former has received more attention and is the subject of the present work. In particular, interfacial roughness may have a significant role in the design of oxidation-resistant fiber/matrix interface control, which is the key technological challenge to the use of ceramic composites in high-temperature structural applications. Many approaches20,22,23 based on oxide coatings are being investigated (for a brief review, see Ker￾ans20). The focus of these investigations is the achievement of sufficiently low shear strength to allow impinging matrix cracks to deflect along the interface and bypass fibers without fiber failure; this characteristic is essential. However, the to￾pography of the interfacial fracture surfaces may differ greatly from those of conventional interphases (mostly carbon, some BN). This will necessitate explicit attention to interfacial roughness in the design of the fiber coating system, which, in turn, requires a thorough understanding of the phenomena that are involved. The relative magnitude of the effect of roughness was shown to be high, using the fiber pushback or ‘‘seating drop’’ mea￾surements.8,24 Initial modeling of the effect of roughness25 was based on a simple approximation that proposed that an inter￾facial roughness of amplitude h results in a mismatch strain of h/Rf , where Rf is the fiber radius (Fig. 1). This roughness￾induced strain simply adds to the thermal mismatch strain. This model assumes that (i) the roughness is nonaxisymmetric (thus, the fiber and matrix surfaces mate completely only at the origi￾nal position) and (ii) when the fiber slides, relative to the ma￾trix, through a distance greater than a characteristic half-period of the roughness, the fiber develops the mismatch strain h/Rf . Experimental work has shown that this approach reasonably captures the major aspects of actual behavior;26 however, the choice of the value of h that properly describes a real surface is ambiguous. Recent works27–30 have clarified this somewhat; however, more work is required. All these developments have addressed the effect of rough￾ness on the sliding of a fiber along its entire length, relative to the matrix, as in a pushout or pullout test during ‘‘steady-state’’ sliding. However, to model the fracture behavior of a compos￾ite, it is important to consider the effect of roughness on a debond crack as it propagates along the fiber/matrix interface A. G. Evans—contributing editor Manuscript No. 191659. Received July 29, 1996; approved January 8, 1997. Research was performed, in part, at the Wright Laboratory Materials Directorate under U.S. Air Force Contract No. F-33615-91-C-5663. *Member, American Ceramic Society. † With UES, Inc., Dayton, OH 45432. J. Am. Ceram. Soc., 80 [8] 2043–55 (1997) Journal 2043

2044 Journal of the American Ceramic Society-Parthasarathy and Kerans Vol. 80. No 8 Malix fibcr R Fig. 1. Schematic sketch illustrating how the effect of interfacial roughness can be modeled as a radial misfit between the fiber and matrix. away from the plane of the matrix crack. In this situation, the composites that are implied by the maximum misfit stresses fiber slides, relative to the matrix, to different extents along the have received some attention, explicit consideration of the im length of the fiber(Fig. 2). For a fiber pulling out of a matrix plications of the model for specific ceramic composites has not under axial loading, the interfacial normal stress in the un- been considered in any detail. The purpose of this work is to slipped region ahead of the crack tip(region I of Fig. 2)is examine the predicted effects of progressive roughness in more determined by the residual stresses and, to a minor degree detail in selected ceramic composites and the magnitude of the ferences in Poissons ratios and the appli errors that are expected from applying smooth-interface as- region Ill, the normal stress is the sum of the residual stresses sumptions to real composites with rough interfaces and, hence, and the stresses that result from the full effect of the topo- the utility of conventional treatments. Debond length at fiber aphical misfit. Region II extends, with incre isfit fre fracture is probably the single parameter that is most illustra- plicates analysis and results in the questions of tis>on com the crack tip to the beginning of region Ill. This regi tive of composite behavior, because it has a key role in deter- mining the matrix crack spacing, fiber failure sites, apparent ance. For the scenario in region Il, the fracture and sliding fiber strength, pullout length, and total energy dissipation. Un- etween the matrix and the fiber has been handled in detail for fortunately, debond length is rarely measurable. Average fric- simple form of roughness by Parthasarathy et al. 3 Although tion along the fiber is less instructive about composite behav- nat treatment leads to a set of solutions that is suitable for ior; however, it is both intuitively and quantitatively directly nalyses of pullout and pushout data, the solutions are not well related to the measured quantity of force applied to the fiber ted corporation into higher-level-design analyses, be- Consequently, this evaluation of the significance of roughness ause of the additional complications illustrated in Fig. 2 and effects is focused on consideration of debond length and fric- discussed above. It also is not readily apparent how progressivetion oughness will affect particular composite systems. The obv The effects of roughness during progressive debonding were ous first order issue relates to the magnitude of progressive examined by calculating the predicted behavior for rough in- oughness effects: are they significant compared to the constant terfaces using an extension of the treatment of Parthasarathy et roughness region Ill, or can they be neglected entirely? Second, al a copy of the code that is thus developed and used in this are there simplifying approximations that provide adequate re- work can be obtained by writing to the authors. Thes are compared with the predictions of the constant Some of the implications for metal-matrix composites have model 25 where a smooth interface is assumed but th en evaluated by Marshall et al. Although effects in ceramic ness makes a constant contribution to the radial Debur」( Gridgis F Fig. 2. Illustration of the effect of interfacial roughness during progressive debonding away from a matrix crack in a composite under tension. Three different regions-labeled L. Il. and lll. as sh an be envisioned. (See text for details

away from the plane of the matrix crack. In this situation, the fiber slides, relative to the matrix, to different extents along the length of the fiber (Fig. 2). For a fiber pulling out of a matrix under axial loading, the interfacial normal stress in the un￾slipped region ahead of the crack tip (region I of Fig. 2) is determined by the residual stresses and, to a minor degree, by differences in Poisson’s ratios and the applied axial stress. In region III, the normal stress is the sum of the residual stresses and the stresses that result from the full effect of the topo￾graphical misfit. Region II extends, with increasing misfit from the crack tip to the beginning of region III. This region com￾plicates analysis and results in the questions of first impor￾tance. For the scenario in region II, the fracture and sliding between the matrix and the fiber has been handled in detail for a simple form of roughness by Parthasarathy et al.31 Although that treatment leads to a set of solutions that is suitable for analyses of pullout and pushout data, the solutions are not well suited for incorporation into higher-level-design analyses, be￾cause of the additional complications illustrated in Fig. 2 and discussed above. It also is not readily apparent how progressive roughness will affect particular composite systems. The obvi￾ous first order issue relates to the magnitude of progressive roughness effects: are they significant compared to the constant roughness region III, or can they be neglected entirely? Second, are there simplifying approximations that provide adequate re￾sults? Some of the implications for metal–matrix composites have been evaluated by Marshall et al.11 Although effects in ceramic composites that are implied by the maximum misfit stresses have received some attention, explicit consideration of the im￾plications of the model for specific ceramic composites has not been considered in any detail. The purpose of this work is to examine the predicted effects of progressive roughness in more detail in selected ceramic composites and the magnitude of the errors that are expected from applying smooth-interface as￾sumptions to real composites with rough interfaces and, hence, the utility of conventional treatments. Debond length at fiber fracture is probably the single parameter that is most illustra￾tive of composite behavior, because it has a key role in deter￾mining the matrix crack spacing, fiber failure sites, apparent fiber strength, pullout length, and total energy dissipation. Un￾fortunately, debond length is rarely measurable. Average fric￾tion along the fiber is less instructive about composite behav￾ior; however, it is both intuitively and quantitatively directly related to the measured quantity of force applied to the fiber. Consequently, this evaluation of the significance of roughness effects is focused on consideration of debond length and fric￾tion. The effects of roughness during progressive debonding were examined by calculating the predicted behavior for rough in￾terfaces using an extension of the treatment of Parthasarathy et al.;31 a copy of the code that is thus developed and used in this work can be obtained by writing to the authors. These results are compared with the predictions of the constant roughness model,25 where a smooth interface is assumed but the rough￾ness makes a constant contribution to the radial clamping Fig. 1. Schematic sketch illustrating how the effect of interfacial roughness can be modeled as a radial misfit between the fiber and matrix. Fig. 2. Illustration of the effect of interfacial roughness during progressive debonding away from a matrix crack in a composite under tension. Three different regions—labeled I, II, and III, as shown—can be envisioned. (See text for details.) 2044 Journal of the American Ceramic Society—Parthasarathy and Kerans Vol. 80, No. 8

August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2045 tress. The effect of roughness on interfacial friction is consid where ered and discussed first; this is followed by a presentation of the calculated effects on debond length, either as a function of (1+x) fiber stress or at fiber fracture. Interactions between rough f1 x(1+x) 2 nd other variables that have been examined by a parametric evaluation are included in the discussion. Next, the validity of g tg 1(1-x) the constant shear stress(T) approximation is examined by (4) for t g the possibility of extracting and using a single value for T to predict debond lengths as a function of fiber stress Finally, the calculated fiber stress-debond length relationship was used in the model of Curtin, 32 to predict composite stress- 2 strain behavior and ultimate strengths in the presence of inter- facial roughness. The results are compared with predictions 十 -1(1-x)′ hat have been obtained using the same approach in a conven tional manner using a constant shear stress T with values that would be derived from conventional methods such as the fiber ='=n/ pushout or pullout tests (7) The case of an interface crack propagating away from a matrix crack is shown schematically in Fig. 2. The elastic misfit is assumed to cause uniform radial strains in the fiber and the matrix, summing to the magnitude of the misfit. As 2μ'b1 discussed above, the debonded length comprises two different (9) egions of misfit behavior In region Ill, the fiber has slid past the matrix by a distance greater than the half-period, d. The 2uL'Be roughness is assumed to be nonaxisymmetric, and there is no d’=tan (10) position of fit between the fiber and the matrix, except in the original position; hence, there is no further change in misti ith additional sliding. The effects of roughness in this large a=y+ocn+σ (11) displacement region can be modeled as suggested in earlier works, 18,25 with u as the friction coefficient and the quantity Ge AaAT+(/r) as a constant mismatch strain(no effect of Re sliding distance on clamping stress), where AaAT is the con- tribution from the thermal mismatch. Under these conditions (13) region II will have a constant length, but moves with the debond crack tip, whereas region IlI will increase in length In the above equations, B. ar, bu, and Eh are functions of the the debond crack progresses elastic properties of the matrix and fiber, as well as the fiber Within region Il, the roughness-induced clamping stress is volume fraction, as defined by Hutchinson and Jensen. G is been described elsewhere that derives expressions that relate residual axial stress in the fiber, and om, a residual stress the length of the debond crack as a function of the fiber stress rameter whose magnitude gives the fiber stress at which the The model includes the effects of the thermal mismatch, the Poisson's contraction of the fiber exactly cancels the residua Poisson's contraction of the fiber. the axial residual strain in clamping stress during fiber pullout. When the roughness am- litude is zero, x l and the equations then reduce to those computational tractability is such that the misfit increases lin- same equations were used, with x=0 and the radial clampino than the half-period d and is constant thereafter. Approxima- +(h/R). The specifics of the calculations are discussed in more tions in the model require that the roughness amplitude h and detail in the Appendix od d be much smaller than the fiber rad Using the approach above, the debond length was calculated, result of the model is that, over region Il, an effective friction as a function of fiber stress, for different sets of constituent coefficient, u, replaces the friction coefficient, u, between the parameters. For all the calculations, the boundary conditions fiber and matrix. The effective friction coefficient is given by that corresponded to that of a multifiber pullout were used(no radial displacement at the matrix surface; type ll as defined by u+ tan 6 Hutchinson and Jensen), because this corresponds to the ten- (1) sile testing of a composite. Because of the lack of reliable data on the Poisson's ratio of the fibers, and to simplify the calcu- where tan 0= h/d for the sawtooth roughness shape that is lations. the fibers and matrices were assumed to have the same assumed in the model with the requirement that tan 0< 1/u Poissons ratio. The debond length at a fiber stress that equals The roughness-induced clamping stress at any point increase he fiber strength was considered to be the parameter of most linearly with increasing relative displacement between the fiber interest. It must be noted that this debond length at the fiber and the matrix. The final equations relate the debond length, L, fracture stress is not the same as the mean pullout length that to the remote fiber stress, Oa, and fiber displacement, u, in is measured at the composite fracture surface, however, the two terms of the properties of the fiber, matrix, and interface. These re closely related through a prefactor that is dependent on the relations can be described in terms of two functions, f, and f2 Weibull modulus of the fiber strength. I It also is noteworthy that the fiber stress at fiber fracture is not the same as that ( (2) which is measured in a fiber test that usually has a gauge length of 1 in. the effective gauge length for the fiber failure within a composite can be much lower. The best estimates of fiber fracture stress in composites may be those which are obtaine

stress. The effect of roughness on interfacial friction is consid￾ered and discussed first; this is followed by a presentation of the calculated effects on debond length, either as a function of fiber stress or at fiber fracture. Interactions between roughness and other variables that have been examined by a parametric evaluation are included in the discussion. Next, the validity of the constant shear stress (
) approximation is examined by studying the possibility of extracting and using a single value for
to predict debond lengths as a function of fiber stress. Finally, the calculated fiber stress–debond length relationship was used in the model of Curtin1,32 to predict composite stress– strain behavior and ultimate strengths in the presence of inter￾facial roughness. The results are compared with predictions that have been obtained using the same approach in a conven￾tional manner using a constant shear stress
with values that would be derived from conventional methods such as the fiber pushout or pullout tests. II. Approach The case of an interface crack propagating away from a matrix crack is shown schematically in Fig. 2. The elastic misfit is assumed to cause uniform radial strains in the fiber and the matrix, summing to the magnitude of the misfit. As discussed above, the debonded length comprises two different regions of misfit behavior. In region III, the fiber has slid past the matrix by a distance greater than the half-period, d. The roughness is assumed to be nonaxisymmetric, and there is no position of fit between the fiber and the matrix, except in the original position; hence, there is no further change in misfit with additional sliding. The effects of roughness in this large￾displacement region can be modeled as suggested in earlier works,18,25 with as the friction coefficient and the quantity T + (h/Rf ) as a constant mismatch strain (no effect of sliding distance on clamping stress), where T is the con￾tribution from the thermal mismatch. Under these conditions, region II will have a constant length, but moves with the debond crack tip, whereas region III will increase in length as the debond crack progresses. Within region II, the roughness-induced clamping stress is not constant. This case has been treated in the model that has been described elsewhere31 that derives expressions that relate the length of the debond crack as a function of the fiber stress. The model includes the effects of the thermal mismatch, the Poisson’s contraction of the fiber, the axial residual strain in the fiber, and the roughness at the fiber/matrix interface. The simple form of roughness to which the analysis is restricted by computational tractability is such that the misfit increases lin￾early as the displacement increases for displacements of less than the half-period d and is constant thereafter. Approxima￾tions in the model require that the roughness amplitude h and half-period d be much smaller than the fiber radius. A key result of the model is that, over region II, an effective friction coefficient,
, replaces the friction coefficient, , between the fiber and matrix. The effective friction coefficient is given by
= + tan  1 − tan  (1) where tan  h/d for the sawtooth roughness shape that is assumed in the model with the requirement that tan  < 1/. The roughness-induced clamping stress at any point increases linearly with increasing relative displacement between the fiber and the matrix. The final equations relate the debond length, l, to the remote fiber stress, a, and fiber displacement, u, in terms of the properties of the fiber, matrix, and interface. These relations can be described in terms of two functions, f1 and f2: u =
 Eb f1 (2) a − f + = f2 (3) where f1 = 4
g − 1 1 − x 2 +
g − gx − 2 x
1 + x exp −
1 + xz
2  −
g + gx − 2 x
1 − x exp −1
1 − xz
2  (4) f2 = −
g − gx − 2 2x exp −
1 + xz
2  +
g + gx − 2 2x exp −1
1 − xz
2  (5) z
= l (6) x =
1 − 4
2 12 (7) g =  (8)  = 2
b1 Rf (9) 
= tan 
2
BEm EbRf 2 (10)  = + fo + + Ro (11) = 2
EbGc Rf 12 (12) f + = a1fa + fo + (13) In the above equations, B, a1, b1, and Eb are functions of the elastic properties of the matrix and fiber, as well as the fiber volume fraction, as defined by Hutchinson and Jensen.33 Gc is the interface toughness, f the fiber volume fraction, fo + the residual axial stress in the fiber, and Ro a residual stress pa￾rameter whose magnitude gives the fiber stress at which the Poisson’s contraction of the fiber exactly cancels the residual clamping stress during fiber pullout. When the roughness am￾plitude is zero, x 1 and the equations then reduce to those derived earlier25,33,34 for smooth fibers. Thus, for region III, the same equations were used, with x 0 and the radial clamping stress set to that which results from a mismatch strain of T + (h/Rf ). The specifics of the calculations are discussed in more detail in the Appendix. Using the approach above, the debond length was calculated, as a function of fiber stress, for different sets of constituent parameters. For all the calculations, the boundary conditions that corresponded to that of a multifiber pullout were used (no radial displacement at the matrix surface; type II as defined by Hutchinson and Jensen33), because this corresponds to the ten￾sile testing of a composite. Because of the lack of reliable data on the Poisson’s ratio of the fibers, and to simplify the calcu￾lations, the fibers and matrices were assumed to have the same Poisson’s ratio. The debond length at a fiber stress that equals the fiber strength was considered to be the parameter of most interest. It must be noted that this debond length at the fiber fracture stress is not the same as the mean pullout length that is measured at the composite fracture surface; however, the two are closely related through a prefactor that is dependent on the Weibull modulus of the fiber strength.1 It also is noteworthy that the fiber stress at fiber fracture is not the same as that which is measured in a fiber test that usually has a gauge length of 1 in.; the effective gauge length for the fiber failure within a composite can be much lower. The best estimates of fiber fracture stress in composites may be those which are obtained August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2045

2046 Journal of the American Ceramic Society--Parthasarathy and Kerans Vol. 80. No 8 rom fracture mirrors that are observed in post-mortem frac cludes with a discussion on the design implications for oxida- tography; however, such data is not always available, mal assun motions mand ( Efects of Roughness on Friction IlL. Composite Systems The frictional contribution that is actually measured in tests such as the pushout test is the total friction, an average T value The composite systems that are being considered seriously is then extracted, neglecting the progressive contribution for practical applications can be classified into two group roughness and often assuming a uniform stress distribution non-oxide fiber-based systems and oxide fiber-based systems. However, during progressive debonding, the frictional stress is There is substantially more data in the literature for non-oxide highly nonuniform. The frictional stress that is experienced at reinforced systems. SiC composites reinforced with NicalonTM an arbitrary location on a sliding fiber is illustrated in Fig 3 fiber(Nippon Carbon, Tokyo, Japan) have been the subjects of Poissons effect is neglected for illustration purposes. When numerous investigations into interface, constituent, and com any point on the fiber is in its original position, the normal posite properties, although few studies have been on exactly (clamping)stress is simply the residual stress and the relevant milar materials. This system also is likely to be one in which coefficient of friction is H'(Eq. (1); thus, the friction stress is roughness effects will be larger. because of the stiff matrix μ'[A(△a△D], where A( equal to BE) is a collection of elastic Consequently, the Nicalon-fiber-reinforced SiC matrix com- posite system has been selected as the principal system for ing point of origin on the matrix side of the interface, the local consideration in the present stud clamping stress increases linearly with the displacement, u,as e The roughness amplitude that has been reported in the lit- (u/d)[A(M/R)), where d is the half-period. The rate of increase ture for Nicalon-based composites ranges from -2.5 nm on in clamping stress is linear with e(equal to h/d). When u=d, the as-received fibers to 20-30 nm, which is inferred from the maximum misfit is attained and remains unchanged with indirect measurements. It is believed that, in composites hav further displacement. At that same displacement, the coeffi- ing an interfacial coating, the roughness amplitude can be as cient of friction becomes u, the smooth-fiber value. The fric- high as the coating thickness itself. Thus, for the parametric tion stress for subsequent sliding of the point is then study, the roughness amplitude was varied from zero(smooth H([AAT+(h/R Ii. This value may be higher or lower than fiber)to 30 nm. The fiber strength of Nicalon is usually re- the value at zero displacement, depending on the magnitudes of ported in the range of 2-2.5 GPa(based on a gauge length of u' and h and, thus, on the magnitudes of the roughness param- I in. ) however, processing is expected to weaken the fiber to eters h and d Other shapes of roughness that lead to a nonlinear some extent. The work by Prewo 6 on Nicalon/LASIll co increase in misfit with displacement will result in a nonlinear posites suggests that the fiber strength degrades to -1.7 GPa increase of the clamping stress until u d. The change in after processing in this particular system. Heredia et al. re- frictional stress with relative fiber/matrix displacement can be ported in-situ strengths of 2.2 GPa in a Nicalon/carbon com- expected to be smoother for real surfaces, as illustrated in Fig posite system, Fiber strengths in the range of 1-2 GPa were 3. Although the model captures the key aspects, the magnitude sed in the study of the error that is introduced by the simple form of roughness is unknown V. Results and Discussion A reasonable quantity to consider in seeking insight is the average friction stress along the fiber; this is defined as the total The control set of parameters that were used in this study load on the fiber(less that required for debond initiation) di (listed in Table I) were selected based on the available infor- vided by the debond area. As a fiber begins to slide, the sliding mation and best estimates. One or more of the parameters were length increases as the interfacial crack tip, or the front of then varied systematically, to study the effect of the variable on slippage initiation, propagates down the interface. The dis- friction and debond length placement of the fiber, relative to its original position, increases This section begins with a discussion on the effects of rough- from zero with distance from the crack tip. Similarly, the misfit ness on friction(through defining an average T value); its rel increases as the distance from the crack tip increases until the evance to the measurement of roughness effects during fiber boundary between regions II and Ill is attained Until the slid- pushout tests is presented. This is followed by calculations of ing length exceeds the region Il length, the average friction debond lengths, as functions of roughness parameters, and stress monotonically increases. When the sliding length ex- evaluation of roughness effects relative to other parameters ceeds the region Il length, all additional sliding lengths are Finally, the accuracy of the constant-T approximation is evalu ated, and an alternate piecewise linear approximation is used to predict the composite stress-strain behavior. The paper con- friction H. A zone of region II moves with the crack i oa becomes an ever-smaller fraction of the contribution to the te Table I. Listing of the Control Set of Parameters Used for the Different Composite Systems Studied 密密宽 SCSa/ SCSa/ Fiber modulus(GPa) dulls(GPa) Axial/radial modulus Fiber poisson's ratio 0.l Fiber radius, R(um) Roughness amplitude, h(nm) Roughness period, 2d (um) Friction coefficient Interface toughness, G.(J/m2) Thermal mismatch strain,△a△T(x10-3) Fiber strength(GPa) Fiber volume fraction, f 0.45

from fracture mirrors that are observed in post-mortem frac￾tography; however, such data is not always available, making reasonable assumptions mandatory. III. Composite Systems The composite systems that are being considered seriously for practical applications can be classified into two groups: non-oxide fiber-based systems and oxide fiber-based systems. There is substantially more data in the literature for non-oxide reinforced systems. SiC composites reinforced with Nicalon™ fiber (Nippon Carbon, Tokyo, Japan) have been the subjects of numerous investigations into interface, constituent, and com￾posite properties, although few studies have been on exactly similar materials. This system also is likely to be one in which roughness effects will be larger, because of the stiff matrix. Consequently, the Nicalon-fiber-reinforced SiC matrix com￾posite system has been selected as the principal system for consideration in the present study. The roughness amplitude that has been reported in the lit￾erature for Nicalon-based composites ranges from ∼2.5 nm on the as-received fiber35 to 20–30 nm, which is inferred from indirect measurements.12 It is believed that, in composites hav￾ing an interfacial coating, the roughness amplitude can be as high as the coating thickness itself. Thus, for the parametric study, the roughness amplitude was varied from zero (smooth fiber) to 30 nm. The fiber strength of Nicalon is usually re￾ported in the range of 2–2.5 GPa (based on a gauge length of 1 in.); however, processing is expected to weaken the fiber to some extent. The work by Prewo36 on Nicalon/LASIII com￾posites suggests that the fiber strength degrades to ∼1.7 GPa after processing in this particular system. Heredia et al.37 re￾ported in-situ strengths of 2.2 GPa in a Nicalon/carbon com￾posite system. Fiber strengths in the range of 1–2 GPa were used in the study. IV. Results and Discussion The control set of parameters that were used in this study (listed in Table I) were selected based on the available infor￾mation and best estimates. One or more of the parameters were then varied systematically, to study the effect of the variable on friction and debond length. This section begins with a discussion on the effects of rough￾ness on friction (through defining an average
value); its rel￾evance to the measurement of roughness effects during fiber pushout tests is presented. This is followed by calculations of debond lengths, as functions of roughness parameters, and evaluation of roughness effects relative to other parameters. Finally, the accuracy of the constant-
approximation is evalu￾ated, and an alternate piecewise linear approximation is used to predict the composite stress–strain behavior. The paper con￾cludes with a discussion on the design implications for oxida￾tion-resistant composites. (1) Effects of Roughness on Friction The frictional contribution that is actually measured in tests such as the pushout test is the total friction; an average
value is then extracted, neglecting the progressive contribution of roughness and often assuming a uniform stress distribution. However, during progressive debonding, the frictional stress is highly nonuniform. The frictional stress that is experienced at an arbitrary location on a sliding fiber is illustrated in Fig. 3; Poisson’s effect is neglected for illustration purposes. When any point on the fiber is in its original position, the normal (clamping) stress is simply the residual stress and the relevant coefficient of friction is
(Eq. (1)); thus, the friction stress is
[ (T)], where (equal to BEm) is a collection of elastic constants. As the point is displaced relative to the correspond￾ing point of origin on the matrix side of the interface, the local clamping stress increases linearly with the displacement, u, as (u/d)[ (h/Rf )], where d is the half-period. The rate of increase in clamping stress is linear with  (equal to h/d). When u d, the maximum misfit is attained and remains unchanged with further displacement. At that same displacement, the coeffi￾cient of friction becomes , the smooth-fiber value. The fric￾tion stress for subsequent sliding of the point is then { [T + (h/Rf )]}. This value may be higher or lower than the value at zero displacement, depending on the magnitudes of
and h and, thus, on the magnitudes of the roughness param￾eters h and d. Other shapes of roughness that lead to a nonlinear increase in misfit with displacement will result in a nonlinear increase of the clamping stress until u d. The change in frictional stress with relative fiber/matrix displacement can be expected to be smoother for real surfaces, as illustrated in Fig. 3. Although the model captures the key aspects, the magnitude of the error that is introduced by the simple form of roughness is unknown. A reasonable quantity to consider in seeking insight is the average friction stress along the fiber; this is defined as the total load on the fiber (less that required for debond initiation) di￾vided by the debond area. As a fiber begins to slide, the sliding length increases as the interfacial crack tip, or the front of slippage initiation, propagates down the interface. The dis￾placement of the fiber, relative to its original position, increases from zero with distance from the crack tip. Similarly, the misfit increases as the distance from the crack tip increases until the boundary between regions II and III is attained. Until the slid￾ing length exceeds the region II length, the average friction stress monotonically increases. When the sliding length ex￾ceeds the region II length, all additional sliding lengths are under region III conditions: maximum misfit and coefficient of friction . A zone of region II moves with the crack tip but becomes an ever-smaller fraction of the contribution to the total Table I. Listing of the Control Set of Parameters Used for the Different Composite Systems Studied Property System (fiber/matrix/coating) Nicalon/ SiC/ carbon Nicalon/ MAS/ carbon Sapphire/ YAG/ hibonite Nextel/ alumina/ monazite SCS6/ glass SCS6/ RBSN Fiber modulus (GPa) 200 200 465 380 415 415 Matrix modulus (GPa) 400 75 300 400 60 100 Axial/radial modulus 1 1 1.08 1 1 1 Fiber Poisson’s ratio 0.15 0.15 0.25 0.2 0.2 0.2 Fiber radius, Rf (m) 8 8 75 6 71 71 Roughness amplitude, h (nm) 20 20 200 25 25 25 Roughness period, 2d (m) 0.3 0.5 60 1 10 10 Friction coefficient, 0.05 0.04 0.1 0.2 0.1 0.15 Interface toughness, Gc (J/m2 ) 2 2 5 2.5 5 1 Thermal mismatch strain, T (× 10−3) 1 0 0 0.5 0.5 0 Fiber strength (GPa) 1.5 1.5 1.5 1.5 2.5 2.5 Fiber volume fraction, f 0.4 0.45 0.3 0.4 0 0.4 2046 Journal of the American Ceramic Society—Parthasarathy and Kerans Vol. 80, No. 8

August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2047 Friction stress nal Fort。r From Urstaliny toringo on for Ks引 6-3(hal" pcriod; Rrs danl Strcss Relative Fiber/Matrix Displaceme 8) Fig. 3. Schematic sketch of the effect of progressive roughness on the interfacial friction stress at a selected point on the fiber friction stress. the total friction stress asymptotically measured T approaches the The calculated average friction stress, including Poisson effect, is shown plotted for the control set in Fig. 4(a), for progressive model predictions. Theg. 5(b, al t by Eldridge in a SCSa/RI et al 38 Their results are shown in F along with the lower amplitude of roughness in Fig 4(b), and for two different progressive roughness periods in Figs. 4(c), and(d). In all cases, an equivalent smooth model clearly helps rationalize the dependence of T on the fiber with a clamping stress equal to the maximum(region Ill) maximum load of the cyclic push-in tests clamping stress of the rough fiber is plotted for comparison The average friction stress that is plotted in Fig. 4(a)retains the same basic character as the schematic in Fig 3, with the fea-(2) Effects of Roughness on Debond Length tures smoothed by the distributed nature of the transitions along The effect of the interfacial roughness amplitude h on the the length of sliding fiber. The roughness amplitude has the debond length was studied as a function of the fiber stress. Thi expected effect on the amplitude of the peak in friction stress fect for the Nicalon/Sic comBo ite system is shown in Fig. 6 Fig. 4(b)). Note that because d is held constant, 0 decreases as The behavior at low fiber stresses and debond lengths is showr the roughness decreases, and, hence, the initial slope also is in Fig. 6(a), and the behavior at stresses near the fiber strength decreased In Figs. 4(c)and(d), the amplitude is held constant(taken as 1.5 GPa)is shown in Fig. 6(b). It is clear that rough and the period is varied, which leads to even more- interesting ness decreases the debond length significantly for a given fiber behavior. As the period is decreased, the slope and the am stress. When the roughness amplitude is >10 nm, the debond tude of the effect both increase. It is evident from Figs. 4(c)and length decreases rapidly for a fiber stress of 1.5 GPa; at 30 nm )that the rough-fiber friction can be either lower or higher the debond length is <150 um than the smooth-fiber case and that the smooth -fiber value can As shown in Fig. 6(a), there are two distinct regions to each be asymptotically approached from either above or below. This curve for nonzero roughness amplitudes, for reasons discussed ather remarkable result means that the nature and even exis earlier. The first is the portion of the debonding that is the tence of a pushback seating drop 8, 28 will be dependent on the development of region Il. The transition occurs at the inception shape (or 0), not just the amplitude h of the roughness. The of region Ill, and, for the remainder of the debonding process, predicted seating drop is proportional to the difference in there is no change in the length of (and contribution from) values between the maximum constant roughness model (slid region II: the entire increase in debonded length is in the form ing friction)and the progressive roughness model; note that of increasing region Ill. It is evident that the region lll contri- this varies with the debond length, which can be taken to be the bution is dominant for roughness amplitudes of up to 10 nm. specimen thickness in fiber pushout/pushback tests. Physically, For larger roughnesses, region lI has a significant effect on the the thickness dependence comes from the fact that, at the bot- total debond length until fiber fracture tom of the seating drop, the fraction of the fiber that is fully The development of two different regions has a consequence seated in its original position decreases as the thickness do in the relationship between matrix crack spacing and pullout lengths. At lower stresses, the debond lengths are shorter than Fiber pushback tests have shown a significant seating drop in what a constant roughness model (which assumes region Ill the SCS6/glass system. 2 The predictions for this system along the entire fiber) would have predicted. This should lead shown in Fig. 5(a), in which the rough-fiber case is plotted with to relatively small matrix crack spacing; howeve the constant roughness approximation for two cases: one with and using this behavior at low stresses to predict the pullout minimum clamping(residual stress only, or no roughness)and lengths(which are dependent on debond lengths at stresses the other with maximum clamping stress(residual stress+ near fiber fracture)will underestimate the pullout length. The roughness-induced clamping stress, as in region III). The rough debond length at high stresses is dominated by region Ill be- fiber does indeed start at almost the same frictional stress havior, which is assumed in the constant roughness model that for a smooth fiber and gradually increases to the maximum Consequently, the relationship between matrix crack spacing value(rough-fiber)at large debond lengths. Thus, for speci- and pullout length will be significantly different than that pre- mens mm thick, a significant load drop will be predicted in dicted using a constant roughness model or, for similar reasons, seating drop experiments. The predicted dependence of the a constant T model

friction stress; hence, the total friction stress asymptotically approaches the region III bound. The calculated average friction stress, including Poisson’s effect, is shown plotted for the control set in Fig. 4(a), for a lower amplitude of roughness in Fig. 4(b), and for two different periods in Figs. 4(c), and (d). In all cases, an equivalent smooth fiber with a clamping stress equal to the maximum (region III) clamping stress of the rough fiber is plotted for comparison. The average friction stress that is plotted in Fig. 4(a) retains the same basic character as the schematic in Fig. 3, with the fea￾tures smoothed by the distributed nature of the transitions along the length of sliding fiber. The roughness amplitude has the expected effect on the amplitude of the peak in friction stress (Fig. 4(b)). Note that because d is held constant,  decreases as the roughness decreases, and, hence, the initial slope also is decreased. In Figs. 4(c) and (d), the amplitude is held constant and the period is varied, which leads to even more-interesting behavior. As the period is decreased, the slope and the ampli￾tude of the effect both increase. It is evident from Figs. 4(c) and (d) that the rough-fiber friction can be either lower or higher than the smooth-fiber case and that the smooth-fiber value can be asymptotically approached from either above or below. This rather remarkable result means that the nature and even exis￾tence of a pushback seating drop18,28 will be dependent on the shape (or ), not just the amplitude h of the roughness. The predicted seating drop is proportional to the difference in
values between the maximum constant roughness model (slid￾ing friction) and the progressive roughness model; note that this varies with the debond length, which can be taken to be the specimen thickness in fiber pushout/pushback tests. Physically, the thickness dependence comes from the fact that, at the bot￾tom of the seating drop, the fraction of the fiber that is fully seated in its original position decreases as the thickness de￾creases. Fiber pushback tests have shown a significant seating drop in the SCS6/glass system.24 The predictions for this system are shown in Fig. 5(a), in which the rough-fiber case is plotted with the constant roughness approximation for two cases: one with minimum clamping (residual stress only, or no roughness) and the other with maximum clamping stress (residual stress + roughness-induced clamping stress, as in region III). The rough fiber does indeed start at almost the same frictional stress as that for a smooth fiber and gradually increases to the maximum value (rough-fiber) at large debond lengths. Thus, for speci￾mens 10 nm, the debond length decreases rapidly for a fiber stress of 1.5 GPa; at 30 nm, the debond length is <150 m. As shown in Fig. 6(a), there are two distinct regions to each curve for nonzero roughness amplitudes, for reasons discussed earlier. The first is the portion of the debonding that is the development of region II. The transition occurs at the inception of region III, and, for the remainder of the debonding process, there is no change in the length of (and contribution from) region II; the entire increase in debonded length is in the form of increasing region III. It is evident that the region III contri￾bution is dominant for roughness amplitudes of up to 10 nm. For larger roughnesses, region II has a significant effect on the total debond length until fiber fracture. The development of two different regions has a consequence in the relationship between matrix crack spacing and pullout lengths. At lower stresses, the debond lengths are shorter than what a constant roughness model (which assumes region III along the entire fiber) would have predicted. This should lead to relatively small matrix crack spacing; however, interpreting and using this behavior at low stresses to predict the pullout lengths (which are dependent on debond lengths at stresses near fiber fracture) will underestimate the pullout length. The debond length at high stresses is dominated by region III be￾havior, which is assumed in the constant roughness model. Consequently, the relationship between matrix crack spacing and pullout length will be significantly different than that pre￾dicted using a constant roughness model or, for similar reasons, a constant
model. Fig. 3. Schematic sketch of the effect of progressive roughness on the interfacial friction stress at a selected point on the fiber. August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2047

Journal of the American Ceramic Society-Parthasarathy and Kerans Vol. 80. No 8 Progressive roughness h=20nm:2d-=0.3m 8 Progressive rough h=2nm:2d=0.3m 可6 Constant roughness 10 Constant Roughness Model (Smooth same clamping stress) 00.]0.20.3040.506呀O 00.l020.3040.50.6 Debond Length(mm) Debond Length(mm) 40 Pros Progressive roughne Roughness h=20nm:2d=1.5 园30 h=20 nm: 2d=0.6 um 0 E 10 Constant Roughness Model. 10 Constant Roughness 0 0 00.【0.20.3040.50.6 0010.20.304050.6 Debond Length(mm) Debond length(mm) (d) Fig. 4. For the Nicalon/SiC system, the progressive roughne redicts the average T value(see text for definition) to be higher than that predicted by a constant roughness model(Fig 4(a)). Lowering the hess amplitude, h, only diminishes the difference, not the trend(Fig. 4(b). However, as the roughness period is increased, the trend is reversed(Figs. 4(c)and(d).(See text for implications on seating drop during push-back ( Interaction with Other Parameters doubling the fiber strength from I GPa to 2 GPa increases the debond length by a factor of-4, whereas doubling h from 0.01 Because several different constituent and composite proper- mm to 0.02 mm causes an increase in the debond length by a ties can affect debond length, it is important to know how the factor of only -2 influences from other factors compare with that from rough- Fiber diameters can vary significantly ness. To estimate the relative effects of parameters other than Nicalon-fiber diameter can vary by a factor of 2. The fiber 30m由mr订mbd1hea到 ignificant effect on the Is u varies as the ratio of roughness amplitude to fiber radius. This volume fraction of the fiber. interface toughness interfacial could be one source of the wide variation in measured inter- friction coefficient, and fiber radius. Of these factors, only fiber facial properties. Now, when the fiber diameter varies, it is strength and fiber radius had effects that were significant, com- possible that roughness either scales with fiber size, because of pared to the effect of roughness amplitude; the effects of these processing artifacts, or is independent of fiber size, equal to two parameters are presented and discussed below coating thickness, for example. If the roughness is independent The strength of Nicalon fiber has been measured and re- of fiber diameter, then the misfit stress also will vary by a ported to be in a range of 2-2.5 GPa In processed composites, factor of 2 amongst different fibers. The conse the strengths of the fiber can be lower because of processing substantial, as illustrated in Fig. &(a). The effect of fibe effects. In one study, 6 the strength was estimated from fracture is far more than that of the roughness amplitude. He mirrors on fibers in a composite to be -1. 7 GPa. Depending on when the roughness amplitude and period scale with the processing cycle, the strength can probably vary in the radius, then there is no significant effect(Fig. 8(b)) range of 1-2 GPa. In addition, there also is potential fiber From Fig. 8, it is clear that a distribution of fiber diameters strength degradation from environmental exposure. Figure 7 an cause a distribution in debond lengths and, thus, pullout shows the effect of varying the fiber strength over the range of lengths in a composite, provided that the roughness paramete 1-2 GPa, as a function of the roughness amplitude. generally do not scale with the fiber diameter. The work of Benoit er a/ 39 speaking, there is a fairly strong effect of fiber stress through experimentally explored the above-mentioned effect in the out the range of roughnesses. For example, at h=0.02 mm Nicalon/MAS-L system. Their results are shown, plotted with

(3) Interaction with Other Parameters Because several different constituent and composite proper￾ties can affect debond length, it is important to know how the influences from other factors compare with that from rough￾ness. To estimate the relative effects of parameters other than roughness, the debond lengths were calculated as a function of roughness amplitude for different values of fiber strength, Pois￾son’s ratio of the fiber (which is usually not well known), volume fraction of the fiber, interface toughness, interfacial friction coefficient, and fiber radius. Of these factors, only fiber strength and fiber radius had effects that were significant, com￾pared to the effect of roughness amplitude; the effects of these two parameters are presented and discussed below. The strength of Nicalon fiber has been measured and re￾ported to be in a range of 2–2.5 GPa. In processed composites, the strengths of the fiber can be lower because of processing effects. In one study,36 the strength was estimated from fracture mirrors on fibers in a composite to be ∼1.7 GPa. Depending on the processing cycle, the strength can probably vary in the range of 1–2 GPa. In addition, there also is potential fiber strength degradation from environmental exposure. Figure 7 shows the effect of varying the fiber strength over the range of 1–2 GPa, as a function of the roughness amplitude. Generally speaking, there is a fairly strong effect of fiber stress through￾out the range of roughnesses. For example, at h 0.02 mm, doubling the fiber strength from 1 GPa to 2 GPa increases the debond length by a factor of ∼4, whereas doubling h from 0.01 mm to 0.02 mm causes an increase in the debond length by a factor of only ∼2. Fiber diameters can vary significantly; for example, the Nicalon-fiber diameter can vary by a factor of 2. The fiber radius may be expected to have a significant effect on the debond length, because the roughness-induced misfit stress varies as the ratio of roughness amplitude to fiber radius. This could be one source of the wide variation in measured inter￾facial properties. Now, when the fiber diameter varies, it is possible that roughness either scales with fiber size, because of processing artifacts, or is independent of fiber size, equal to coating thickness, for example. If the roughness is independent of fiber diameter, then the misfit stress also will vary by a factor of 2 amongst different fibers. The consequences can be substantial, as illustrated in Fig. 8(a). The effect of fiber radius is far more than that of the roughness amplitude. However, when the roughness amplitude and period scale with the fiber radius, then there is no significant effect (Fig. 8(b)). From Fig. 8, it is clear that a distribution of fiber diameters can cause a distribution in debond lengths and, thus, pullout lengths in a composite, provided that the roughness parameters do not scale with the fiber diameter. The work of Benoit et al.39 experimentally explored the above-mentioned effect in the Nicalon/MAS-L system. Their results are shown, plotted with Fig. 4. For the Nicalon/SiC system, the progressive roughness model predicts the average
value (see text for definition) to be higher than that predicted by a constant roughness model (Fig. 4(a)). Lowering the roughness amplitude, h, only diminishes the difference, not the trend (Fig. 4(b)). However, as the roughness period is increased, the trend is reversed (Figs. 4(c) and (d)). (See text for implications on seating drop during push-back tests.) 2048 Journal of the American Ceramic Society—Parthasarathy and Kerans Vol. 80, No. 8

August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites SCSG/Glass- Single fiber Pushout SCSG/ RBSN. Single Fiber Pushout onstant Roughness Model Constant roughness conStant Toughness.inditEd stress) 6 Smooth Fiber(n4 RcHEhness Debond Lcngth (mm) 0 (a) Maximum Load (N) Fig. 5. For large-fiber-diameter systems, such as(a)SCS6/glass and(b)SCS6-RBSN, the progressive roughness model that the average value will change as the debond increases from that of a smooth fiber with no roughness to that predicted by a roughness model Fig. 5(b), experimental evidenc idge et al.3 on SCS/RBSN) for such an effect of debond length(or load on the measured average T value is shown for comp or these systems, the smooth transition is such that a pronounced seating drop is predicted for specimen of thicknesses up to at least 5 mm, in contrast with the small-fiber-diameter Nicalon/SiC system(Fig. 4), where the seating drop is not predicted for reasonable choice of roughness parameters. These are consistent with experimental findings(see text) the progressive roughness model predictions, in Fig. 9. Al- mm(0. 2-0.3 mm is typical), then the fiber stress at a debond though the correlation is not definitive, the trend favors the length of 0. 3 mm from the plot gives the peak fiber stress in the constant roughness amplitude model pushout test. If this peak stress is used to calculate a constant T value, it will yield a value of 40 MPa. However, this value will vary with the specimen thickness, as shown in Fig. 10(a) (4 Constant-T Approximation If one uses this T value to predict the debond length in a The obvious advantages of simplifying assumptions has led multifiber pullout test(composite fracture), one obtains debond to the development and use of the constant- approximation. lengths that are far from the real value, as shown in Fig. 10(c); Although this has been helpful as a simple approximation, its note that Fig. 10(c)represents a plot of debond length versus accuracy has always been in doubt. The validity of this ap- bridging fiber stress. A similar problem occurs for a frictionally and using it to predict the debond length during mults st lading fiber that is completely debonded, as shown in Fig proximation has been examined by studying the consequence f measuring a constant T value from a single-fiber pushout tes o(b); however, the range of T that is obtained is much na ullout, as in a composite. Figure 10(a) shows the debond The values of T in Fig. 10(c)are those that would be obtained est(Nicalon/SiC), calculated using the progressive roughness cured at the corresponding point on the curve. Because matrix del. If the specimen thickness during a pushout test is 0.3 cracking usually occurs at small strains (0.05%0.01%), the T Nicalon/Sic 1.2 04;:1 日 ′btm (6 2505007501000 50010015002000 Fiber Stress(MPa) Fiber stress (MPa) ig. 6. Calculated roughness on debond in a Nicalon/SiC composite, plotted as a function of fiber stress. At low fiber stresses(Fig. 6( has slid, relative to the x, by distances less than the period of roughness, as in region Il(Fig. 2) At higher stresses, a of the plot is observed, corresponding to the initiation of region Ill ( Fig. 2). At high stresses( Fig. 6(b), most of the fiber is that roughness decreases the debond length significantly

the progressive roughness model predictions, in Fig. 9. Al￾though the correlation is not definitive, the trend favors the constant roughness amplitude model. (4) Constant-
Approximation The obvious advantages of simplifying assumptions has led to the development and use of the constant-
approximation. Although this has been helpful as a simple approximation, its accuracy has always been in doubt. The validity of this ap￾proximation has been examined by studying the consequence of measuring a constant
value from a single-fiber pushout test and using it to predict the debond length during multifiber pullout, as in a composite. Figure 10(a) shows the debond length as a function of the fiber stress during a fiber pushout test (Nicalon/SiC), calculated using the progressive roughness model. If the specimen thickness during a pushout test is 0.3 mm (0.2–0.3 mm is typical), then the fiber stress at a debond length of 0.3 mm from the plot gives the peak fiber stress in the pushout test. If this peak stress is used to calculate a constant
value, it will yield a value of 40 MPa. However, this value will vary with the specimen thickness, as shown in Fig. 10(a). If one uses this
value to predict the debond length in a multifiber pullout test (composite fracture), one obtains debond lengths that are far from the real value, as shown in Fig. 10(c); note that Fig. 10(c) represents a plot of debond length versus bridging fiber stress. A similar problem occurs for a frictionally sliding fiber that is completely debonded, as shown in Fig. 10(b); however, the range of
that is obtained is much nar￾rower. The values of
in Fig. 10(c) are those that would be obtained from matrix crack spacing if the cracking were to have oc￾curred at the corresponding point on the curve. Because matrix cracking usually occurs at small strains (0.05%–0.01%), the
Fig. 6. Calculated effect of the interfacial roughness on debond lengths in a Nicalon/SiC composite, plotted as a function of fiber stress. At low fiber stresses (Fig. 6(a)), much of the fiber has slid, relative to the matrix, by distances less than the period of roughness, as in region II (Fig. 2). At higher stresses, a transition in the slope of the plot is observed, corresponding to the initiation of region III (Fig. 2). At high stresses (Fig. 6(b)), most of the fiber is in region III. It is clear that roughness decreases the debond length significantly. Fig. 5. For large-fiber-diameter systems, such as (a) SCS6/glass and (b) SCS6-RBSN, the progressive roughness model predicts that the average
value will change as the debond length increases from that of a smooth fiber with no roughness to that predicted by a constant roughness model. In Fig. 5(b), experimental evidence (Eldridge et al.38 on SCS6/RBSN) for such an effect of debond length (or load on the fiber) on the measured average
value is shown for comparison. For these systems, the smooth transition is such that a pronounced seating drop is predicted for specimens of thicknesses up to at least 5 mm, in contrast with the small-fiber-diameter Nicalon/SiC system (Fig. 4), where the seating drop is not predicted for reasonable choice of roughness parameters. These are consistent with experimental findings (see text). August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2049

Journal of the American Ceramic Society--Parthasarathy and Kerans Vol. 80. No 8 L.5 .0I0U20.D30 Rudius fum) Roughness Anplitude (um) Fig 9. In the Nicalon/MAS-L system, the effect of fiber radius or Fig. 7. Fiber strength degradation during processing ca the pullout length has been measured experimentally. Dependence or debond lengths significantly, as shown in this plot for the fiber radius is shown, compared with the prediction of the model ystem. Effect of the maximum fiber stress(strength) is obs (debond length), for two situations: one where the roughness param- ge at lower roughness amplitudes(as-received Nicalon eters are held constant, and another where they scale with the fiber nificant even at a roughness amplitude of 0.03 um(the radius. Predictions using the constant roughness parameters match the reported roughness amplitude measured indirectly from composite be. trend of experimental data better. See text for possible implications. havior). model is obtained by using a piecewise linear fit, as indicated ralue thus extracted will have a tendency to be rather high. in Fig. 11(a). However, this will involve multiple parameters From Fig. 10(c), even a very high matrix cracking strain would (rather than one)to define the debond length-fiber stress rela- yield a T value of -60 MPa at a fiber stress of 500 MPa tionship (composite stress of 200 MPa); this is significantly higher than what would be obtained using fiber pushout tests in the sliding Ld=a。+m1or(foro。2)(14) define a unique value of T, as determined from experimental tests; this is consistent with the data of Eldridge et al, 38 which Although not especially elegant, it allows a reasonable com- are shown in Fig. 5(b). However, Fig. 10(c)also shows that it romise between accuracy and computational tractability F toler be possible to choose a constant T value that will yield a ure I(b)shows how a piecewise linear fit also may be used to describe frictional sliding of completely debonded fiber/matrix a range of fiber stress of interest, say 0-2 GPa. If such an interfaces. In this plot, the slip length over which the fiber approach were to be used, a protocol for choosing an effective stress is transferred to the matrix is plotted against fiber stress T value would be required. The best choice for T would be one that predicts the composite behavior well. The model of Cur- is equal to zero tin, for composite stress-strain behavior was used to evalu The above-mentioned approximation was examined for its ate the choice of an effective T valu utility in predicting the composite behavior. The relationship between slip length and fiber stress can be used to estimate the (5) Composite Stress-Strain Behavior tensile stress-strain behavior of a composite using the analysis To use the analysis of Curtin, one needs a relationship be- of Curtin. ,32 This analysis focuses on the composite behavior tween the fiber stress and the debond length or slip length. We after matrix cracking with the fiber/matrix interface completely suggest that a good approximation to the progressive roughness debonded. The matrix contribution is neglected, so the curve is roughness 1.2 100- 08 0+R2 00010020030.04 00001000200030.004 Ruugluness Amplitude (un) Roughness Amplitude/Fiber radius Fig 8. Fiber radius has a large effect on the debond length(Fig. 8(a)); however, if the period of the roughness is scaled with changes in roughness, and the debond length and roughness amplitude are normalized with respect to the fiber radius, there is a negligible effect of fiber radius(Fig. 8(b)) Thus, if roughness parameters scale with the fiber radius, then there is no effect, which is consistent with intuition

value thus extracted will have a tendency to be rather high. From Fig. 10(c), even a very high matrix cracking strain would yield a
value of ∼60 MPa at a fiber stress of 500 MPa (composite stress of 200 MPa); this is significantly higher than what would be obtained using fiber pushout tests in the sliding regime (Fig. 10(b)). It is evident that tests that sample different portions of the curve will give different values for
in any constant-
approximation. It is clear from Figs. 10(a)–(c) that it is almost impossible to define a unique value of
, as determined from experimental tests; this is consistent with the data of Eldridge et al.,38 which are shown in Fig. 5(b). However, Fig. 10(c) also shows that it may be possible to choose a constant
value that will yield a tolerable fit to the debond length–fiber stress relationship over a range of fiber stress of interest, say 0–2 GPa. If such an approach were to be used, a protocol for choosing an effective
value would be required. The best choice for
would be one that predicts the composite behavior well. The model of Cur￾tin1,32 for composite stress–strain behavior was used to evalu￾ate the choice of an effective
value. (5) Composite Stress–Strain Behavior To use the analysis of Curtin, one needs a relationship be￾tween the fiber stress and the debond length or slip length. We suggest that a good approximation to the progressive roughness model is obtained by using a piecewise linear fit, as indicated in Fig. 11(a). However, this will involve multiple parameters (rather than one) to define the debond length–fiber stress rela￾tionship: Ld = o + m1f (for o 2) (14) Although not especially elegant, it allows a reasonable com￾promise between accuracy and computational tractability. Fig￾ure 11(b) shows how a piecewise linear fit also may be used to describe frictional sliding of completely debonded fiber/matrix interfaces. In this plot, the slip length over which the fiber stress is transferred to the matrix is plotted against fiber stress. For this case, the slip length in Eq. (14), Ls, replaces Ld and ° is equal to zero. The above-mentioned approximation was examined for its utility in predicting the composite behavior. The relationship between slip length and fiber stress can be used to estimate the tensile stress–strain behavior of a composite using the analysis of Curtin.1,32 This analysis focuses on the composite behavior after matrix cracking with the fiber/matrix interface completely debonded. The matrix contribution is neglected, so the curve is Fig. 8. Fiber radius has a large effect on the debond length (Fig. 8(a)); however, if the period of the roughness is scaled with changes in roughness, and the debond length and roughness amplitude are normalized with respect to the fiber radius, there is a negligible effect of fiber radius (Fig. 8(b)). Thus, if roughness parameters scale with the fiber radius, then there is no effect, which is consistent with intuition. Fig. 9. In the Nicalon/MAS-L system, the effect of fiber radius on the pullout length has been measured experimentally. Dependence on fiber radius is shown, compared with the prediction of the model (debond length), for two situations: one where the roughness param￾eters are held constant, and another where they scale with the fiber radius. Predictions using the constant roughness parameters match the trend of experimental data better. (See text for possible implications.) Fig. 7. Fiber strength degradation during processing can affect the debond lengths significantly, as shown in this plot for the Nicalon/SiC system. Effect of the maximum fiber stress (strength) is observed to be large at lower roughness amplitudes (as-received Nicalon) and is sig￾nificant even at a roughness amplitude of 0.03 m (the maximum reported roughness amplitude measured indirectly from composite be￾havior). 2050 Journal of the American Ceramic Society—Parthasarathy and Kerans Vol. 80, No. 8

August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2051 ssentially that of an already microcracked composite--that is, 0.5L the stress-strain behavior during a test run after an initial load (for o2)(17 rough the volume fraction, f, of fibers and the fraction of roken fibers, q A fiber strength g. of 2 GPa with a Weibull modulus m of 4, over a reference gauge length Lo of 25.4 mm, was used for (-q) go (15) the calculations. In Fig. 12(a), the stress-strain curves that were obtained using the piecewise linear fit are shown for a smooth nd g is given by fiber and one with a roughness amplitude h of 20 nm. The plots include debond lengths at the ultimate stress. in units of fiber diameters and millimeters. The fiber roughness enhances the ultimate strength of the composite, provided that the roughne does not affect the fiber strength(e. g, through stress raisers where o is the fiber strength that is measured over a reference This might rationalize the observations of Naslain et al. 19 who gauge length Lo and m is the Weibull modulus. In Eq (16), Ls found that the composite strength increased as the shear is the distance over which slip occurs at a fiber stress of o, as strength of the interface increased, which, in turn, could be shown in Fig. 11(b)and described by Eq (14)(with o =0 attributed to microcracking of the interphase which causes According to Curtin, 32 the average stress ob(supported by the increased interfacial roughness. Thus, roughness may have a broken fibers)is the fiber stress for a slip length of 0.5Ls. Thus, beneficial effect if it does not affect the fiber properties. .4 NicxlunSsiC-Siuule Fiber Puslluut 0.8 AicaloniSIC-Single Fiter sliding 0.3 06 (Peak load) shang with 0.2 0.4+2m Progressiv E0.1 (b=20;m 0.2 G +(Sliding Load)! o2 0 43 01000200030004000 1500300045006000 Fiber Stress(MPa) fiber stress(Mfa) 0.4 Nicalon'SK--Multinhtr tension 0.3 Progressive τ(MPa) 0.2 (Matrix 0. 4 0 1000200030XX Fiber stress(MPa) ig. 10. Constant shear stress(T) approximation is evaluated for its validity of use as a unique parameter, the value obtained for T using a fiber shout test is dependent on whether the peak stress(shown in Fig. 10(a)or plateau stress(frictional sliding, shown in Fig. 10(b)) is used. From Figs. 10(a)and (b), it also is observed that the T value thus extracted is dependent on the specimen thickness (equal to the debond/sliding length) From Fig. 10(c), it is observed that the T value obtained using the matrix crack spacing during a composite tensile test also is a function of the stress at which the crack spacing (debond length x 2)is measured. Because the fiber stress during matrix cracking is typically low, the extracted T value will have a tendency to be high(T= 60 MPa at 500 MPa, a composite stress of 200 MPa)

essentially that of an already microcracked composite—that is, the stress–strain behavior during a test run after an initial load￾ing to the strain that is required for complete matrix micro￾cracking. In this state, the composite is loaded with no notable features in the stress–strain curve until it begins to bend over, as a result of fiber failures, until the ultimate strength is at￾tained. Thereafter, the remaining intact fibers continue to fail as the composite stress decreases. As per Curtin’s analysis, the remote applied stress, app, is related to the stress on unbroken fibers, f , and the stress on broken (and sliding-out) fibers, b, through the volume fraction, f, of fibers and the fraction of broken fibers, q: app = f
1 − qf + qb] (15) and q is given by q = 2Ls Lo
f o m (16) where o is the fiber strength that is measured over a reference gauge length Lo and m is the Weibull modulus. In Eq. (16), Ls is the distance over which slip occurs at a fiber stress of f , as shown in Fig. 11(b) and described by Eq. (14) (with o 0). According to Curtin,1,32 the average stress b (supported by the broken fibers) is the fiber stress for a slip length of 0.5Ls. Thus, b = 0.5Ls m1
for  2) (17) A fiber strength o of 2 GPa with a Weibull modulus m of 4, over a reference gauge length Lo of 25.4 mm, was used for the calculations. In Fig. 12(a), the stress–strain curves that were obtained using the piecewise linear fit are shown for a smooth fiber and one with a roughness amplitude h of 20 nm. The plots include debond lengths at the ultimate stress, in units of fiber diameters and millimeters. The fiber roughness enhances the ultimate strength of the composite, provided that the roughness does not affect the fiber strength (e.g., through stress raisers). This might rationalize the observations of Naslain et al.,19 who found that the composite strength increased as the shear strength of the interface increased, which, in turn, could be attributed to microcracking of the interphase, which causes increased interfacial roughness. Thus, roughness may have a beneficial effect if it does not affect the fiber properties. Fig. 10. Constant shear stress (
) approximation is evaluated for its validity of use as a unique parameter; the value obtained for
using a fiber pushout test is dependent on whether the peak stress (shown in Fig. 10(a)) or plateau stress (frictional sliding, shown in Fig. 10(b)) is used. From Figs. 10(a) and (b), it also is observed that the
value thus extracted is dependent on the specimen thickness (equal to the debond/sliding length). From Fig. 10(c), it is observed that the
value obtained using the matrix crack spacing during a composite tensile test also is a function of the stress at which the crack spacing (∼debond length × 2) is measured. Because the fiber stress during matrix cracking is typically low, the extracted
value will have a tendency to be high (
≈ 60 MPa at 500 MPa, a composite stress of 200 MPa). August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2051

Journal of the American Ceramic Society-Parthasarathy and Kerans Vol. 80. No 8 目E 2010 4000 6000 20004000 6000 Fiber Stress (MPa Fiber Stress (MPa) Fig. 11. Piecewise linear approximation, suggested for the relationship(a) between the debond length and the fiber stress for the case of progressive debonding along a rough and(b) for the case of a post-debond fiber sliding relative to the matrix. 1800 s1200 h=20 nm 600 h=o nm 0 0010.020.03 Strain 1800 I(MPu) 1400,3(9 60 l200 077(18 Amm(d) 600 9(127 Peak I oad: =40-140 MPa Sliding Load: t=25-40 MPa) 0.0l 0.02 0.03 Strai ig. 12. Composite stress-strain predicted the model along with the slip length, ( ), at the ultimate stress. In Fig. 12(a), the ecewise linear approximation of Fig. ll(b) strength, whereas in Fig. 12(b), the constant shear stress approximation used. In the inset shown in Fig. 12(b), the range of T value tha measured using different methods is shown (from Fig. 10), all estimates T will overpredict the ultimate strength of the h= 20 nm of Fig. 12(a)

Fig. 11. Piecewise linear approximation, suggested for the relationship (a) between the debond length and the fiber stress for the case of progressive debonding along a rough interface and (b) for the case of a post-debond fiber sliding relative to the matrix. Fig. 12. Composite stress–strain predicted using the model of Curtin,1 along with the slip length, 〈l〉, at the ultimate stress. In Fig. 12(a), the piecewise linear approximation of Fig. 11(b) is used to predict the ultimate strength, whereas in Fig. 12(b), the constant shear stress approximation is used. In the inset shown in Fig. 12(b), the range of
value that would be measured using different methods is shown (from Fig. 10); all estimates of
will overpredict the ultimate strength of the h 20 nm prediction of Fig. 12(a). 2052 Journal of the American Ceramic Society—Parthasarathy and Kerans Vol. 80, No. 8