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 composites were studied using a model that included the progressively increasing contribution of roughness with relative fiber/matrix displacement during debonding of the fiber/matrix interface. A parametric approach was used to study interfacial roughness in conjunction with other parameters such as the strength, radius, and volume fraction of the fiber. The progressive roughness contribution during initial fiber/matrix sliding caused a high effective coefficient of friction, as well as an increased clamping stress, which led to rapidly changing friction with increasing debond length. Calculated effects implied a potentially significant contribution to the behavior of real composite systems and the necessity for explicit consideration in the interpretation of experimental data to understand composite behavior correctly. In a tension test, the Poisson’s contraction of the fiber may negate the effects of roughness, allowing an ‘‘effective constant shear stress’’ () approximation. This was evaluated using a piecewise linear approximation to the progressive roughness model in an analysis of composite 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 friction, 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 coefficient 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 experiments have been used to measure the interfacial friction coefficient 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 behavior 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, composites 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 present 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 Kerans20). 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 topography 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’’ measurements.8,24 Initial modeling of the effect of roughness25 was based on a simple approximation that proposed that an interfacial roughness of amplitude h results in a mismatch strain of h/Rf , where Rf is the fiber radius (Fig. 1). This roughnessinduced 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 original position) and (ii) when the fiber slides, relative to the matrix, 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 roughness 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 composite, 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 unslipped 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 topographical misfit. Region II extends, with increasing misfit from the crack tip to the beginning of region III. This region complicates analysis and results in the questions of first importance. 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, because 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 obvious 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 results? 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 implications 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 assumptions 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 illustrative of composite behavior, because it has a key role in determining the matrix crack spacing, fiber failure sites, apparent fiber strength, pullout length, and total energy dissipation. Unfortunately, debond length is rarely measurable. Average friction along the fiber is less instructive about composite behavior; 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 friction. The effects of roughness during progressive debonding were examined by calculating the predicted behavior for rough interfaces 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 roughness 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 considered 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 interfacial roughness. The results are compared with predictions that have been obtained using the same approach in a conventional 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 largedisplacement 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 contribution 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 linearly as the displacement increases for displacements of less than the half-period d and is constant thereafter. Approximations 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 = 4g − 1 1 − x 2 + g − gx − 2 x1 + x exp −1 + xz 2 − g + gx − 2 x1 − x exp −11 − xz 2 (4) f2 = − g − gx − 2 2x exp −1 + xz 2 + g + gx − 2 2x exp −11 − xz 2 (5) z = l (6) x = 1 − 4 2 12 (7) g = (8) = 2b1 Rf (9) = tan 2BEm 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 parameter 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 amplitude 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 tensile testing of a composite. Because of the lack of reliable data on the Poisson’s ratio of the fibers, and to simplify the calculations, 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 fractography; 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 composite 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 composite system has been selected as the principal system for consideration in the present study. The roughness amplitude that has been reported in the literature 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 having 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 reported 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 composites suggests that the fiber strength degrades to ∼1.7 GPa after processing in this particular system. Heredia et al.37 reported in-situ strengths of 2.2 GPa in a Nicalon/carbon composite 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 information 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 roughness on friction (through defining an average value); its relevance 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 evaluated, and an alternate piecewise linear approximation is used to predict the composite stress–strain behavior. The paper concludes with a discussion on the design implications for oxidation-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 corresponding 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 coefficient of friction becomes , the smooth-fiber value. The friction 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 parameters 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) divided 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 displacement 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 sliding length exceeds the region II length, the average friction stress monotonically increases. When the sliding length exceeds 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 features 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 amplitude 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 existence 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 (sliding 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 bottom of the seating drop, the fraction of the fiber that is fully seated in its original position decreases as the thickness decreases. 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 specimens <3 mm thick, a significant load drop will be predicted in seating drop experiments. The predicted dependence of the measured value (average ) on the debond length (or load/ stress on the fiber) has recently been validated experimentally in a SCS6/RBSN system using a cyclic push-in test by Eldridge et al.38 Their results are shown in Fig. 5(b), along with the progressive model predictions. The progressive roughness model clearly helps rationalize the dependence of on the maximum load of the cyclic push-in tests. (2) Effects of Roughness on Debond Length The effect of the interfacial roughness amplitude h on the debond length was studied as a function of the fiber stress. This effect for the Nicalon/SiC composite system is shown in Fig. 6. The behavior at low fiber stresses and debond lengths is shown in Fig. 6(a), and the behavior at stresses near the fiber strength (taken as 1.5 GPa) is shown in Fig. 6(b). It is clear that roughness decreases the debond length significantly for a given fiber stress. When the roughness amplitude is >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 contribution 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 behavior, which is assumed in the constant roughness model. Consequently, the relationship between matrix crack spacing and pullout length will be significantly different than that predicted 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