《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_TOUGH-TO-BRITTLE TRANSITIONS IN CERAMIC-MATRIX

C MATERIALIA Pergamon Acta mater.48(200048794892 www.elsevier.com/locate/actamat TOUGH-TO-BRITTLE TRANSITIONS IN CERAMIC-MATRIX COMPOSITES WITH INCREASING INTERFACIAL SHEAR STRESS Z XIA and wA curtins Division of Engineering, Brown University, Providence, RI 02912, USA Received 11 May 2000: received in revised form 18 July 2000: accepted 18 July 2000) Abstract-The possibility of decreasing ultimate tensile strength associated with increasing fiber/matrix terfacial sliding is investigated in ceramic-matrix composites. An axisymmetric finite-element model is sed to calculate axial fiber stresses versus radial position within the slipping region arour Impinging matrix crack as a function of applied stress and interfacial sliding stress t. The stress fields, showing an nhancement at the fiber surface, are then utilized as an effective applied field acting on annular flaws at the fiber surface, and a mode I stress intensity is calculated as a function of applied stress, interface t and flaw size. The total probability of failure due to a pre-existing spectrum of flaws in the fibers is then determined nd utilized within the Global Load Sharing model to predict fiber damage evolution and ultimate failure For small fiber Weibull moduli (m=4), the local stress enhancements are insufficient to preferentially drive failure near the matrix crack. Hence, the composite tensile strength is weakly affected and follows the shear- moduli(m=20), the composite is found to weaken beyond about t= 50 MPa and exhibit reduced fiber allout, both leading to an apparent embrittlement and showing substantial differences compared with the shear-lag model Literature experimental data on an SiC fiber/glass matrix system are compared with the predictions. 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Keywords: Composites; Interface; Mechanical properties; Theory modeling; Computer simulation 1 INTRODUCTION propagation through both matrix and fibers alike Ceramic-matrix composites(CMCs)aI There are some data suggesting that increasing candidates for high-temperature struct interfacial shear stress. even in the absence of oxidat- cations owing to their ability to deform ive attack, leads to decreasing composite tensile ith applied load, leading to notch-insensitive strength and decreasing fiber pullout on the fracture trength behavior. The non-linear behavior stem surface [1-3]. In this paper, we investigate the tend- from the formation of matrix cracks that circumvent ency towards the embrittlement phenomenon associa- the fibers and debond along the fiber/matrix interface ted purely with increased interfacial shear stresses Subsequent sliding between the fibers and matrix Failure in ceramic-matrix composites is tradition- deformation around each matrix crack and permits radial stress variations across the fiber surface. These additional matrix cracking at other remote locations. models predict that the tensile strength scales as The cumulative matrix cracking causes irreversible rl/om +n) where m is the fiber Weibull modulus, and strain and acts in many respects like continuum plas- so increases with increasing fiber/matrix interfacial ticity in a metal. Many current CMC materials are sliding stress t 14, 51. The general validity of the stan- embrittled, or do not show extensive non-linear dard model has been confirmed by cyclic fatigue behavior, at elevated temperatures due to oxidative experiments on CMCs [6-8]. In cyclic fatigue, the degradation of the critical interface between the fibers fiber/matrix interface undergoes wear that is attended and matrix. The formation of interfacial oxides by an independently measurable decrease in th increases the interfacial shear stress and prevents the interfacial shear stress t and hence a decrease in the debonding necessary to prevent continuous crack tensile strength. Such strength losses during fatigue have been measured and correlated with the decrease in T. However, with increasing t the axial fiber stress s To whom all correspondence should be addressed. Fax: becomes strongly dependent upon radial position +4018631157 within the fiber, with a high stress at the fiber surface 1359-6454100/520.00@ 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved PI:S1359-6454(00)00291-3

Acta mater. 48 (2000) 4879–4892 www.elsevier.com/locate/actamat TOUGH-TO-BRITTLE TRANSITIONS IN CERAMIC-MATRIX COMPOSITES WITH INCREASING INTERFACIAL SHEAR STRESS Z. XIA and W. A. CURTIN* Division of Engineering, Brown University, Providence, RI 02912, USA ( Received 11 May 2000; received in revised form 18 July 2000; accepted 18 July 2000 ) Abstract—The possibility of decreasing ultimate tensile strength associated with increasing fiber/matrix interfacial sliding is investigated in ceramic-matrix composites. An axisymmetric finite-element model is used to calculate axial fiber stresses versus radial position within the slipping region around an impinging matrix crack as a function of applied stress and interfacial sliding stress t. The stress fields, showing an enhancement at the fiber surface, are then utilized as an effective applied field acting on annular flaws at the fiber surface, and a mode I stress intensity is calculated as a function of applied stress, interface t and flaw size. The total probability of failure due to a pre-existing spectrum of flaws in the fibers is then determined and utilized within the Global Load Sharing model to predict fiber damage evolution and ultimate failure. For small fiber Weibull moduli (m<4), the local stress enhancements are insufficient to preferentially drive failure near the matrix crack. Hence, the composite tensile strength is weakly affected and follows the shear￾lag model predictions, which show a monotonically increasing strength with increasing t. For larger Weibull moduli (m<20), the composite is found to weaken beyond about t 5 50 MPa and exhibit reduced fiber pullout, both leading to an apparent embrittlement and showing substantial differences compared with the shear-lag model. Literature experimental data on an SiC fiber/glass matrix system are compared with the predictions.  2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Composites; Interface; Mechanical properties; Theory & modeling; Computer simulation 1. INTRODUCTION Ceramic-matrix composites (CMCs) are excellent candidates for high-temperature structural appli￾cations owing to their ability to deform non-linearly with applied load, leading to notch-insensitive strength behavior. The non-linear behavior stems from the formation of matrix cracks that circumvent the fibers and debond along the fiber/matrix interface. Subsequent sliding between the fibers and matrix against an interfacial sliding resistance t localizes the deformation around each matrix crack and permits additional matrix cracking at other remote locations. The cumulative matrix cracking causes irreversible strain and acts in many respects like continuum plas￾ticity in a metal. Many current CMC materials are embrittled, or do not show extensive non-linear behavior, at elevated temperatures due to oxidative degradation of the critical interface between the fibers and matrix. The formation of interfacial oxides increases the interfacial shear stress and prevents the debonding necessary to prevent continuous crack * To whom all correspondence should be addressed. Fax: 1401 863 1157. 1359-6454/00/$20.00  2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S13 59-6454(00)00291-3 propagation through both matrix and fibers alike. There are some data suggesting that increasing interfacial shear stress, even in the absence of oxidat￾ive attack, leads to decreasing composite tensile strength and decreasing fiber pullout on the fracture surface [1–3]. In this paper, we investigate the tend￾ency towards the embrittlement phenomenon associa￾ted purely with increased interfacial shear stresses. Failure in ceramic-matrix composites is tradition￾ally studied with a shear-lag model that neglects the radial stress variations across the fiber surface. These models predict that the tensile strength scales as t1/(m 1 1), where m is the fiber Weibull modulus, and so increases with increasing fiber/matrix interfacial sliding stress t [4, 5]. The general validity of the stan￾dard model has been confirmed by cyclic fatigue experiments on CMCs [6–8]. In cyclic fatigue, the fiber/matrix interface undergoes wear that is attended by an independently measurable decrease in the interfacial shear stress t and hence a decrease in the tensile strength. Such strength losses during fatigue have been measured and correlated with the decrease in t. However, with increasing t the axial fiber stress becomes strongly dependent upon radial position within the fiber, with a high stress at the fiber surface

4880 XIA and CURTIN: CERAMIC- MATRIX COMPOSITES For fibers whose strength is controlled by surface Aa employed by Dutton et al. is essentially a fitting flaws, which predominate among existing commercial parameter; i.e., there is no independent mechanics cri- fibers, the enhanced axial stress can drive "prema- terion or physical"flaw"criterion for choosing any ture"fiber and composite failure. The quantitative particular value. Also, by using a unique fiber success of the shear-lag models has led many workers strength, their failure prediction did not consider the to neglect the possibility of stress enhancements a well-established fact that brittle fiber strength is a priori, and a detailed analysis has not been made to statistical quantity and hence that strength is depen- spectrum of strength Several workers have addressed the issue of fiber limiting flaws distributed throughout a typical brittle stress concentrations and have considered the possible fiber effects of the stress concentrations on composite fail- In the present work, we investigate the influence ure.Dollar and Steif [9] considered a two-dimen- of local stress concentrations on fiber and composite sional model with an interface governed by Coulomb failure through a combination of numerical studies to friction, and predicted stress concentrations versus obtain detailed fiber stress distributions, fracture friction coefficient u using integral equation methods. mechanics to connect stresses to the stress intensities Their results showed that the stress concentration fac- that drive pre-existing crack growth, and statistical tor decreases with increasing applied stress. Zhu and analysis to accurately determine the probability that Weitsman [10] employed an approximate numerical the fiber will fail given its intrinsic spectrum of sur- method to solve for the fiber and matrix stresses face flaws and the spatial variation of the stress inten- around a matrix crack, and showed fiber-surface sity. Specifically, an axisymmetric finite-element stress enhancements qualitatively similar to those model is used to calculate axial fiber stresses versus found by Dollar and Steif. Zhu and Weitsman used radial position within the slipping region around an these stresses directly in a pointwise Weibull strength impinging matrix crack as a function of applied stres model to predict fiber failure and composite stress- and interfacial sliding stress t. These stress fields are ain behavior for one particular composite, Nicalon then utilized as an effective applied field acting on SiC fibers in a calcium aluminosilicate(CAS)glass annular flaws at the fiber surface to determine the matrix. Little effect on composite strength was found, mode I stress intensity versus applied stress, interface system. More recently, Dutton et al. [ll] performed ability of fiber failure within the slip region is ther detailed calculations of the stress state around a determined. Composite damage evolution and ulti- matrix crack under two conditions, (1)with a fixed mate failure are then calculated within the gls debond region having zero interfacial shear stress and model, wherein broken fibers transfer their loads equ 2)in the absence of any interface debonding, for the ally to all other surviving fibers in the cross-section. particular case of Sigma SiC fibers in a glass matrix. The results are then compared with the"standard In the former case, a strong stress enhancement at the shear-lag strength model consisting of the same gLS tip of the debond crack was found and in the latter model but without the stress enhancements within case the complex singularity around the undeflected the fibers matrix crack was manifest as a strong stress enhance- The main results are as follows. For small fiber ment at the fiber surface. The stress enhancements Weibull moduli (m=4), the stress enhancements were then used in a Whitney-Nuismer criterion [12], which are confined to a small region around the wherein the stress is averaged over some character- matrix crack, are insufficient to preferentially drive stic distance Aa, and this averaged stress was then failure near the matrix crack as compared with failure compared with a deterministic fiber stress to assess further from the matrix crack, where the stress is whether or not composite failure occurred. Dutton et smaller but the lengths experiencing the lower stress al obtained good agreement of this theory with their are much longer. Hence, the composite tensile experiments for fibers having a high Weibull modulus strength is only weakly affected and increases nearly (m= 23)and a single characteristic length Aa for monotonically with increasing t, following the"stan- samples at three different fiber volume fractions In dard" shear-lag prediction of uts Tm + I) out to omparison, the standard Global Load Sharing(GLS) large t values. For larger Weibull moduli (m=20), model [4, 51, based on the shear-lag model that neg. the local stresses are sufficient to cause preferential lects the local stress concentrations, overpredicts the fiber failure and a weakened composite beyond about composite strengths by a factor of about two. Each T=50 MPa. Coupled with reduced fiber pullout, of these studies is insightful but not complete. Zhu there is thus an apparent embrittlement in this regime and Weitsman used a point-stress criterion for failure, Overall, the standard shear-lag GLS model appears to which is not strictly correct since failure is driven by be robust for low Weibull moduli m and low interface finite-size flaws larger than the lengths over which t but there is an apparent transition to more brittle the stress is spatially varying, and did not perform behavior when both m and t increase. The possibility any parametric studies over a range of constituent and of such a mechanism transition with increasing m was interface material properties. The characteristic length also postulated by Dutton et al. as a means of rationa-

4880 XIA and CURTIN: CERAMIC-MATRIX COMPOSITES For fibers whose strength is controlled by surface flaws, which predominate among existing commercial fibers, the enhanced axial stress can drive “prema￾ture” fiber and composite failure. The quantitative success of the shear-lag models has led many workers to neglect the possibility of stress enhancements a priori, and a detailed analysis has not been made to date. Several workers have addressed the issue of fiber stress concentrations and have considered the possible effects of the stress concentrations on composite fail￾ure. Dollar and Steif [9] considered a two-dimen￾sional model with an interface governed by Coulomb friction, and predicted stress concentrations versus friction coefficient m using integral equation methods. Their results showed that the stress concentration fac￾tor decreases with increasing applied stress. Zhu and Weitsman [10] employed an approximate numerical method to solve for the fiber and matrix stresses around a matrix crack, and showed fiber-surface stress enhancements qualitatively similar to those found by Dollar and Steif. Zhu and Weitsman used these stresses directly in a pointwise Weibull strength model to predict fiber failure and composite stress– strain behavior for one particular composite, Nicalon SiC fibers in a calcium aluminosilicate (CAS) glass matrix. Little effect on composite strength was found, which we shall attribute below to the low value of t and the low fiber Weibull modulus in this particular system. More recently, Dutton et al. [11] performed detailed calculations of the stress state around a matrix crack under two conditions, (1) with a fixed debond region having zero interfacial shear stress and (2) in the absence of any interface debonding, for the particular case of Sigma SiC fibers in a glass matrix. In the former case, a strong stress enhancement at the tip of the debond crack was found and in the latter case the complex singularity around the undeflected matrix crack was manifest as a strong stress enhance￾ment at the fiber surface. The stress enhancements were then used in a Whitney–Nuismer criterion [12], wherein the stress is averaged over some character￾istic distance Da, and this averaged stress was then compared with a deterministic fiber stress to assess whether or not composite failure occurred. Dutton et al. obtained good agreement of this theory with their experiments for fibers having a high Weibull modulus (m 5 23) and a single characteristic length Da for samples at three different fiber volume fractions. In comparison, the standard Global Load Sharing (GLS) model [4, 5], based on the shear-lag model that neg￾lects the local stress concentrations, overpredicts the composite strengths by a factor of about two. Each of these studies is insightful but not complete. Zhu and Weitsman used a point-stress criterion for failure, which is not strictly correct since failure is driven by finite-size flaws larger than the lengths over which the stress is spatially varying, and did not perform any parametric studies over a range of constituent and interface material properties. The characteristic length Da employed by Dutton et al. is essentially a fitting parameter; i.e., there is no independent mechanics cri￾terion or physical “flaw” criterion for choosing any particular value. Also, by using a unique fiber strength, their failure prediction did not consider the well-established fact that brittle fiber strength is a statistical quantity and hence that strength is depen￾dent on gage length due to a spectrum of strength￾limiting flaws distributed throughout a typical brittle fiber. In the present work, we investigate the influence of local stress concentrations on fiber and composite failure through a combination of numerical studies to obtain detailed fiber stress distributions, fracture mechanics to connect stresses to the stress intensities that drive pre-existing crack growth, and statistical analysis to accurately determine the probability that the fiber will fail given its intrinsic spectrum of sur￾face flaws and the spatial variation of the stress inten￾sity. Specifically, an axisymmetric finite-element model is used to calculate axial fiber stresses versus radial position within the slipping region around an impinging matrix crack as a function of applied stress and interfacial sliding stress t. These stress fields are then utilized as an effective applied field acting on annular flaws at the fiber surface to determine the mode I stress intensity versus applied stress, interface t and flaw size. Using the spectrum of flaw sizes that are obtained from single-fiber tension tests, the prob￾ability of fiber failure within the slip region is then determined. Composite damage evolution and ulti￾mate failure are then calculated within the GLS model, wherein broken fibers transfer their loads equ￾ally to all other surviving fibers in the cross-section. The results are then compared with the “standard” shear-lag strength model consisting of the same GLS model but without the stress enhancements within the fibers. The main results are as follows. For small fiber Weibull moduli (m<4), the stress enhancements, which are confined to a small region around the matrix crack, are insufficient to preferentially drive failure near the matrix crack as compared with failure further from the matrix crack, where the stress is smaller but the lengths experiencing the lower stress are much longer. Hence, the composite tensile strength is only weakly affected and increases nearly monotonically with increasing t, following the “stan￾dard” shear-lag prediction of suts~t1/(m 1 1) out to large t values. For larger Weibull moduli (m<20), the local stresses are sufficient to cause preferential fiber failure and a weakened composite beyond about t 5 50 MPa. Coupled with reduced fiber pullout, there is thus an apparent embrittlement in this regime. Overall, the standard shear-lag GLS model appears to be robust for low Weibull moduli m and low interface t but there is an apparent transition to more brittle behavior when both m and t increase. The possibility of such a mechanism transition with increasing m was also postulated by Dutton et al. as a means of rationa-

XIA and CURTIN- CERAMIC-MATRIX COMPOSITES 4881 lizing the failure of the shear-lag GLs model when matrix in the damage state described above, we use applied to their data in spite of many prior successful an axisymmetric concentric cylinder model and th applications of the shear-lag GLS model to other cer- finite-element method. The fiber is a central cylinder amic composites of radius r with center line at r=0. The matrix is The remainder of this paper is organized as fol- an annulus of inner radius R and outer radius ows. In Section 2, we present the model used to Router = R(1 +f 2. Due to symmetry in the determine local fiber stresses around a matrix crack geometry and the boundary conditions, the finite nd show a spectrum of results for the axial fiber element(FE) calculations are performed only on a stress that can be collapsed into an accurate analytic quarter-section of the composite as shown in Fig form. In Section 3, we use the stress fields to calculate 1(a). In the debond zone, the interface sliding stress intensities acting upon putative annular flaws behavior is characterized by two models: a constant n the surface of a fiber and derive the failure prob- interfacial shear stress model and the Coulomb fric ability for a fiber containing a statistical distribution tion model. In the constant t model, a constant force of flaws in the presence of a spatially varying stress in the =-direction is added on each interface node to intensity factor. In Section 4, we use the GLS model create constant interfacial shear stress t [Fig. I(b)] to calculate composite tensile strength for a CMc across nearly the entire debonded region. The fiber with a single matrix crack and for multiple matrix node at the matrix crack plane ==0 has fixed dis cracking, and compare our results with the"standard" placement u:=0 required by symmetry, however. At shear-lag GLS model. Section 5 contains further dis- the tip of the debond crack, there can be a singularity session of our results and comparisons with experi- in the stress fields. Although this singularity may play some role in driving fiber failure. it is not the focus of our attention here. Hence, we self-consistently 2. MODEL COMPOSITE AND FIBER STRESSES determine the debond length by making r(=)continu- ous across the debond crack tip. For the Coulomb We consider a unidirectional fiber-reinforced com- friction model, the interface shear stress in the slip posite consisting of cylindrical fibers of radius R, zone is t=-HOrr for or <0(radial compression) Youngs modulus Er and Poisson's ratio ve, embedded and zero otherwise, where u is the coefficient of fric in a matrix material of Young's modulus em and Pois- tion and or is the radial stress in the interface. ahead son's ratio Vm. The fiber volume fraction is denoted of the debond zone perfect interface adhesion is f. A uniform axial stress Oapp is applied to the system. assumed. Below, we shall compare the predictions of We model the situation in which there is a single these two models under conditions where the shear matrix crack perpendicular to the fiber axis and to the stress along the interface is similar in the two cases applied load, located at the plane denoted ==0. The Although it is difficult to model the singular matrix crack causes debonding and sliding at the behavior of the stress field in the neighborhood of the fiber/matrix interface. as discussed below crack tip by using linear interpolation, very fine To determine the stress states in the fibers and meshes were used to limit inaccuracy. The resulting uniform u Fiber Matrix Matrix interface iform Debond iip Debond rix cra Fig. 1. Schematic illustrations of (a) the finite-element model and boundary conditions and(b)details of the fine mesh near the debond zone. indicating how a constant interfacial t is establishe

XIA and CURTIN: CERAMIC-MATRIX COMPOSITES 4881 lizing the failure of the shear-lag GLS model when applied to their data in spite of many prior successful applications of the shear-lag GLS model to other cer￾amic composites. The remainder of this paper is organized as fol￾lows. In Section 2, we present the model used to determine local fiber stresses around a matrix crack and show a spectrum of results for the axial fiber stress that can be collapsed into an accurate analytic form. In Section 3, we use the stress fields to calculate stress intensities acting upon putative annular flaws on the surface of a fiber and derive the failure prob￾ability for a fiber containing a statistical distribution of flaws in the presence of a spatially varying stress intensity factor. In Section 4, we use the GLS model to calculate composite tensile strength for a CMC with a single matrix crack and for multiple matrix cracking, and compare our results with the “standard” shear-lag GLS model. Section 5 contains further dis￾cussion of our results and comparisons with experi￾ments. 2. MODEL COMPOSITE AND FIBER STRESSES We consider a unidirectional fiber-reinforced com￾posite consisting of cylindrical fibers of radius R, Young’s modulus Ef and Poisson’s ratio nf, embedded in a matrix material of Young’s modulus Em and Pois￾son’s ratio nm. The fiber volume fraction is denoted f. A uniform axial stress sapp is applied to the system. We model the situation in which there is a single matrix crack perpendicular to the fiber axis and to the applied load, located at the plane denoted z 5 0. The matrix crack causes debonding and sliding at the fiber/matrix interface, as discussed below. To determine the stress states in the fibers and Fig. 1. Schematic illustrations of (a) the finite-element model and boundary conditions and (b) details of the fine mesh near the debond zone, indicating how a constant interfacial t is established. matrix in the damage state described above, we use an axisymmetric concentric cylinder model and the finite-element method. The fiber is a central cylinder of radius R with center line at r 5 0. The matrix is an annulus of inner radius R and outer radius Router 5 R(1 1 f) 1/2. Due to symmetry in the geometry and the boundary conditions, the finite￾element (FE) calculations are performed only on a quarter-section of the composite as shown in Fig. 1(a). In the debond zone, the interface sliding behavior is characterized by two models: a constant interfacial shear stress model and the Coulomb fric￾tion model. In the constant t model, a constant force in the z-direction is added on each interface node to create constant interfacial shear stress t [Fig. 1(b)] across nearly the entire debonded region. The fiber node at the matrix crack plane z 5 0 has fixed dis￾placement uz 5 0 required by symmetry, however. At the tip of the debond crack, there can be a singularity in the stress fields. Although this singularity may play some role in driving fiber failure, it is not the focus of our attention here. Hence, we self-consistently determine the debond length by making t(z) continu￾ous across the debond crack tip. For the Coulomb friction model, the interface shear stress in the slip zone is t 5 2msrr for srr,0 (radial compression) and zero otherwise, where m is the coefficient of fric￾tion and srr is the radial stress in the interface. Ahead of the debond zone perfect interface adhesion is assumed. Below, we shall compare the predictions of these two models under conditions where the shear stress along the interface is similar in the two cases. Although it is difficult to model the singular behavior of the stress field in the neighborhood of the crack tip by using linear interpolation, very fine meshes were used to limit inaccuracy. The resulting

4882 XIA and CURTIN: CERAMIC- MATRIX COMPOSITES mesh typically contains a number of nodes varying from 7200 to 36.000. The mesh size in the radial direction near the fiber interface and the matrix crack plane is typically set to 0. I um. The mesh size in the longitudinal direction is typically set to 0.25 um but 2000 studies with a mesh size of 0. 1 um have been found to produce essentially identical results. The specific 2 materials studies here are Sic/SiC and Sic (Sigma)glass composites. The properties of the com- posite constituents are listed in Table 1. For complete ness, we have also considered the multilayer structure of the coated Sigma fiber by including both carbon and TiB2 outer layers, with the properties given in Table I taken from Dutton et al. [11. In the SiC (Sigma)glass system, the matrix cracks are observed to penetrate to the inner C/SiC interface of the Sigma fiber, and in our modeling the"matrix crack" also Fig. 2. Axial stress o.(s, r)versus radial position in the Nica- penetrates to this same interface, at which sliding lon Sic fiber /Sic matrix system for various distances from occurs. As a check on our numerical procedures and the matrix crack at ==0(Oapp=600 MPa, t= 50 MPa) accuracy, we performed the FE analysis for some geometries identical to those of Dutton et al. That is, we fixed the debond length a priori, used Coulomb =30AP&t=02 our finite-element method and meshing scheme. Our 18. sics dace 600MPa, t= results for the interfacial region showed open and closed zones in very good agreement with those 51.6 reported by Dutton et al. for various fiber volume fractions. Thus, our numerical scheme is comparabl in accuracy to the variational method utilized by Dut-3 ton et al Figure 2 shows the calculated axial stress distri- ution o_(r, =)versus radial position r at varying dis- 31 tances from the matrix crack at ==0, for the0.9 SiC/SiC composite with a constant interfacial shear stress of 50 MPa and an applied stress of 600 MPa. 0.7 In the matrix crack plane, a high stress concentration fron liber center. ra occurs near the fiber/matrix interface which appears to be diverging at r=R. There is an associated Fig. 3. Normalized stress o(ryT, where T=andf,in the reduction in the stress below the average value T crack plane ==0 as predicted by the FE analytical model as functions of r and t, for both SiC/SiC and app near the fiber center. The magnitude of the Siciglass composites using the constant interfacial shear stress stress concentration decreases rapidly away from the model, for a variety of applied loads and t values crack plane(=>0) and becomes uniform in the fiber at distances larger than R. Figure 3 shows the axial stress o_(,==0) in the matrix crack plane, nor- R-O. I um, =), versus is shown in Fig 4 for similar malized by the average fiber stress T=capp at cases. In spite of the different material properties, the 0, as a function of r/R for both the SiC/SiC and normalized length of the non- linear region is nearly iC/glass composites. In such a normalized form, it the same, and the axial stress in the linear region fol- is evident that the fiber stresses are largely inde- lows the form o==7(1-=lls )that obtains from pendent of the elastic properties of the constituents. the simple shear-lag model. This suggests that the The axial stress just inside the interface, o( Table 1. Thermoelastic parameters of SiC/SiC and SiC/glass [111 Parameter SiC(Nicalon) fiber SiC matrix SiC ( Sigma 1240) 7047 glass matrix Carbon coating TiB, coating E(GPa) 14 Gc O

4882 XIA and CURTIN: CERAMIC-MATRIX COMPOSITES mesh typically contains a number of nodes varying from 7200 to 36,000. The mesh size in the radial direction near the fiber interface and the matrix crack plane is typically set to 0.1 µm. The mesh size in the longitudinal direction is typically set to 0.25 µm but studies with a mesh size of 0.1 µm have been found to produce essentially identical results. The specific materials studies here are SiC/SiC and SiC (Sigma)/glass composites. The properties of the com￾posite constituents are listed in Table 1. For complete￾ness, we have also considered the multilayer structure of the coated Sigma fiber by including both carbon and TiB2 outer layers, with the properties given in Table 1 taken from Dutton et al. [11]. In the SiC (Sigma)/glass system, the matrix cracks are observed to penetrate to the inner C/SiC interface of the Sigma fiber, and in our modeling the “matrix crack” also penetrates to this same interface, at which sliding occurs. As a check on our numerical procedures and accuracy, we performed the FE analysis for some geometries identical to those of Dutton et al. That is, we fixed the debond length a priori, used Coulomb friction, and calculated the resulting stress fields using our finite-element method and meshing scheme. Our results for the interfacial region showed open and closed zones in very good agreement with those reported by Dutton et al. for various fiber volume fractions. Thus, our numerical scheme is comparable in accuracy to the variational method utilized by Dut￾ton et al. Figure 2 shows the calculated axial stress distri￾bution szz(r, z) versus radial position r at varying dis￾tances z from the matrix crack at z 5 0, for the SiC/SiC composite with a constant interfacial shear stress of 50 MPa and an applied stress of 600 MPa. In the matrix crack plane, a high stress concentration occurs near the fiber/matrix interface which appears to be diverging at r 5 R. There is an associated reduction in the stress below the average value T 5 sapp/f near the fiber center. The magnitude of the stress concentration decreases rapidly away from the crack plane (z>0) and becomes uniform in the fiber at distances larger than z<R. Figure 3 shows the axial stress szz(r, z 5 0) in the matrix crack plane, nor￾malized by the average fiber stress T 5 sapp/f at z 5 0, as a function of r/R for both the SiC/SiC and SiC/glass composites. In such a normalized form, it is evident that the fiber stresses are largely inde￾pendent of the elastic properties of the constituents. The axial stress just inside the interface, szz(r 5 Table 1. Thermoelastic parameters of SiC/SiC and SiC/glass [11] SiC (Sigma 1240) Parameter SiC (Nicalon) fiber SiC matrix 7047 glass matrix Carbon coating TiB2 coating fiber R (µm) 7.7 50 E (GPa) 200 400 325 50 90 140 n 0.12 0.2 0.2 0.2 0.11 0.1 a (1026 /°C) 2.9 4.6 4.23 5.4 3.0 7.0 GIC (J/m2 ) 15 5–25 15 Fig. 2. Axial stress szz(z, r) versus radial position r in the Nica￾lon SiC fiber/SiC matrix system for various distances z from the matrix crack at z 5 0 (sapp 5 600 MPa, t 5 50 MPa). Fig. 3. Normalized stress szz(z, r)/T, where T 5 sapp/f, in the matrix crack plane z 5 0 as predicted by the FE model and analytical model as functions of r and t, for both SiC/SiC and SiC/glass composites using the constant interfacial shear stress model, for a variety of applied loads and t values. R20.1 µm, z), versus z is shown in Fig. 4 for similar cases. In spite of the different material properties, the normalized length of the non-linear region is nearly the same, and the axial stress in the linear region fol￾lows the form szz(z) 5 T(12z/ls) that obtains from the simple shear-lag model. This suggests that the stress concentration in the slip region can be

XIA and CURTIN- CERAMIC- MATRIX COMPOSITES 488 Constant friction 70 辆60 0 ormalized distance from crack plane, z/R Fig. 4. Axial fiber stress o(-, r) near the fiber surface(at Fig. 5. Shear stress distributions along the fiber surface pr dicted with the FE model. with the interfacial stress transfe r=R-O. I um)as predicted with FE and analytical models for characterized by a constant t and by the Coulomb friction lav Oapp=600 MPa, /=0.4,t=90 MPa or u=0.8; SiC/glass: Sic/glass: oan=550 MPa,/=0.31, t= 40 MPa or u= additional applied radial stress of 100 MPa) 0.3 with an additional applied radial stress of 100 MPa) described with a general expression that is inde pendent of material properties The composite stress state was also calculated 12 izing the Coulomb frictional law for the interfa shear stress. In such cases, residual thermal stresses 5 must be included to obtain the full radial Residual stresses are induced due to composite coo- SiC/SiC and 500oC for SiC/glass. Besides, since the i C/glass thermal residual stress is low in SiC/glass, an 2 additional radial stress is applied in the SiC/glass to z0.9 create higher frictional stress. Such an enhanced rad ial stress can be motivated physically by asperity sperity contact across rough sliding interfaces, 0.8 although we have not modeled this process precisel Figure 5 shows the shear stress distribution along the fiber/matrix interface in SiC/SiC and SiC/glass com-Fig.6.Axial stress O_(=, r)in the matrix crack posites under Coulomb friction along with our pre- plane :=0 versus dicted with FE and analytical vious constant t results where t is chosen to equal to models for constar ress and Coulomb friction Coulomb frictional stress at :=0.5R. Generally, the (SiC/glass with radial stress of 100 MPa) nterfacial shear stress distributions are different between these two models. The debond length for the Coulomb friction is larger than for the constant t. but matrix crack plane where the high stress concen- this is primarily due merely to the presence of the trations occur. residual axial thermal stress that is not included in the The numerical results constant t calculation. In addition, the interface can state is largely independent of the material constitut be open near the matrix crack and/or the debond tip if ive properties. In fact, for a single fiber subject to a the value of the radial stress is too low(for example, constant shear stress on its surface, the stress equa- o,<40 MPa for SiC/glass). Figures 4 and 6 show the tions for equilibrium in cylindrical coordinates can be axial stress distribution along the fiber/matrix inter written as face and in the crack plane, respectively, for the Cou- lomb friction and constant t models. There is an o region of about I um in the Sic/glass interface near the matrix crack, in which the axial stress in the fiber is reduced slightly. In spite of this, the agreement between the two models is good, particularly near the and

XIA and CURTIN: CERAMIC-MATRIX COMPOSITES 4883 Fig. 4. Axial fiber stress szz(z, r) near the fiber surface (at r 5 R20.1 µm) as predicted with FE and analytical models for constant shear stress and Coulomb friction models. (SiC/SiC: sapp 5 600 MPa, f 5 0.4, t 5 90 MPa or m 5 0.8; SiC/glass: sapp 5 550 MPa, f 5 0.31, t 5 40 MPa or m 5 0.3 with an additional applied radial stress of 100 MPa). described with a general expression that is inde￾pendent of material properties. The composite stress state was also calculated util￾izing the Coulomb frictional law for the interfacial shear stress. In such cases, residual thermal stresses must be included to obtain the full radial stresses. Residual stresses are induced due to composite coo￾ling from a processing temperature of 1000°C for SiC/SiC and 500°C for SiC/glass. Besides, since the thermal residual stress is low in SiC/glass, an additional radial stress is applied in the SiC/glass to create higher frictional stress. Such an enhanced rad￾ial stress can be motivated physically by asperity– asperity contact across rough sliding interfaces, although we have not modeled this process precisely. Figure 5 shows the shear stress distribution along the fiber/matrix interface in SiC/SiC and SiC/glass com￾posites under Coulomb friction along with our pre￾vious constant t results where t is chosen to equal to Coulomb frictional stress at z>0.5R. Generally, the interfacial shear stress distributions are different between these two models. The debond length for the Coulomb friction is larger than for the constant t, but this is primarily due merely to the presence of the residual axial thermal stress that is not included in the constant t calculation. In addition, the interface can be open near the matrix crack and/or the debond tip if the value of the radial stress is too low (for example, srr,40 MPa for SiC/glass). Figures 4 and 6 show the axial stress distribution along the fiber/matrix inter￾face and in the crack plane, respectively, for the Cou￾lomb friction and constant t models. There is an open region of about 1 µm in the SiC/glass interface near the matrix crack, in which the axial stress in the fiber is reduced slightly. In spite of this, the agreement between the two models is good, particularly near the Fig. 5. Shear stress distributions along the fiber surface pre￾dicted with the FE model, with the interfacial stress transfer characterized by a constant t and by the Coulomb friction law (SiC/SiC: sapp 5 400 MPa, f 5 0.4, t 5 49 MPa or m 5 0.4; SiC/glass: sapp 5 550 MPa, f 5 0.31, t 5 40 MPa or m 5 0.3 with an additional applied radial stress of 100 MPa). Fig. 6. Axial stress distributions szz(z, r) in the matrix crack plane z 5 0 versus r/R as predicted with FE and analytical models for constant shear stress and Coulomb friction (SiC/glass with an additional radial stress of 100 MPa). matrix crack plane where the high stress concen￾trations occur. The numerical results suggest that the fiber stress state is largely independent of the material constitut￾ive properties. In fact, for a single fiber subject to a constant shear stress on its surface, the stress equa￾tions for equilibrium in cylindrical coordinates can be written as 1 r ∂ ∂r (rsrr) 1 ∂srz ∂z 2sqq r 5 0 (1a) and