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 shearlag 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 applications 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 plasticity 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 oxidative attack, leads to decreasing composite tensile strength and decreasing fiber pullout on the fracture surface [1–3]. In this paper, we investigate the tendency towards the embrittlement phenomenon associated purely with increased interfacial shear stresses. Failure in ceramic-matrix composites is traditionally 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 standard 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 “premature” 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 failure. Dollar and Steif [9] considered a two-dimensional 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 factor 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 enhancement at the fiber surface. The stress enhancements were then used in a Whitney–Nuismer criterion [12], wherein the stress is averaged over some characteristic 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 neglects 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 criterion 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 dependent on gage length due to a spectrum of strengthlimiting 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 surface flaws and the spatial variation of the stress intensity. 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 probability of fiber failure within the slip region is then determined. Composite damage evolution and ultimate failure are then calculated within the GLS model, wherein broken fibers transfer their loads equally 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 “standard” 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 ceramic composites. The remainder of this paper is organized as follows. 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 probability 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 discussion of our results and comparisons with experiments. 2. MODEL COMPOSITE AND FIBER STRESSES We consider a unidirectional fiber-reinforced composite 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 Poisson’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 finiteelement (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 friction 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 displacement 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) continuous 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 friction 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 composite constituents are listed in Table 1. For completeness, 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 Dutton et al. Figure 2 shows the calculated axial stress distribution szz(r, z) versus radial position r at varying distances 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, normalized 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 independent 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 Nicalon 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 follows 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 independent of material properties. The composite stress state was also calculated utilizing 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 cooling 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 radial 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 composites under Coulomb friction along with our previous 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 interface and in the crack plane, respectively, for the Coulomb 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 predicted 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 concentrations occur. The numerical results suggest that the fiber stress state is largely independent of the material constitutive properties. In fact, for a single fiber subject to a constant shear stress on its surface, the stress equations for equilibrium in cylindrical coordinates can be written as 1 r ∂ ∂r (rsrr) 1 ∂srz ∂z 2sqq r 5 0 (1a) and
XIA and CURTIN: CERAMIC- MATRIX COMPOSITES on)+=0 G==b1+b2x+∑中)e+,(4a) In terms of displacement, the equations can be expressed as where b, and b2 are constants, and p (r) is a complex function of J(r). Since p (r) equals a constant at 0, (r) may be expressed in the form of a con a2ar+2(1-varar+)+(1-2v82=0 stant plus a function of r For the sake of simplifi (lc) cation we assume that the series in equation(4a)can be repl plied by a single exponential in : We choose a logar- ithmic function based on the solution of stresses in a homogeneous elastic body due to a line forces. Thus d-ar+r)+(1-2vrar+a/ (d e approximate equation(4a) G==b1+b2 +2(1-V)==0, tc+ c2 1- where u and w are the displacements in the r and 2 directions, respectively. Differentiating equation(Id) where Ci, c, and B are constants. Since the by r and =, and substituting equation(Ic)into equ- of the left term over the fiber area is the average ation(Id), we obtain the biharmonic equation b,= T, where T=capp is the 1a)a2「a/a,aal stress in the matrix crack plane. Also, b2=-TlLs at :=Is, where I (2)(4b)is then rewritten as The general solution of equation(2)is (Ao+B)A3+A4-)+φ(r)( Be)+o(r(e-r+ Be2) where a is a constant. Equation(5)indicates that the axial stress in the slip zone is the superposition of a linearly decreasing term that is exactly that obtained 1()=A1(k1p)+B1H1(k1r) by the simple shear-lag model, and an additional term accounting for the stress concentrations that depends only on t and not on the applied stress. Note that the shear-lag model can be obtained if we choose a 01 between the al FE model and the general form of equation (5) ()=A2R1(k2r)+B2S1(k2r) are shown in Figs 3, 4 and 6 for several different stress levels. interfacial shear stresses and interface models. Equation (5)fits the FE results extremely A, and B,(=0-4),and kI and k2, are constants, well for SiC/SiC for the values a= 1.32 and B= JI(k r)and Y(kr) are Bessel functions of the first 10, and for SiC/glass composites if a= 1.50 and and second kind of order one, respectively; and Ri(r) B=10. The analytical model works nearly as well and S, (r)are functions of, (r)and Y,(r), respectively. for the Coulomb friction model, although there are Since u is bounded at r=0, and as 3-o, Bo-BA= some slight differences due to the rapidly varying rad 0. Integrating equation(Ic) with respect to r and = ial stress at the interface in the Coulomb friction case we can obtain an expression for w. Now, applying the Therefore, equation (5)can be considered as an accur- undary conditions shown in Fig. 1, we may obtain ate, approximate, analytical solution to the stress field an accurate solution for stresses in the form of a ser- o_(r, =)in the fiber. Equation(5) is the first main The axial stress o(r, =) can be expressed result of this p
4884 XIA and CURTIN: CERAMIC-MATRIX COMPOSITES 1 r ∂ ∂r (rsrz) 1 ∂szz ∂z 5 0. (1b) In terms of displacement, the equations can be expressed as ∂2 w ∂z∂r 1 2(12n) ∂ ∂r S ∂u ∂r 1 u r D 1 (122n) ∂2 u ∂z2 5 0 (1c) and ∂ ∂z S ∂u ∂r 1 u r D 1 (122n)S 1 r ∂2 w ∂r 1 ∂2 w ∂r2 D (1d) 1 2(12n) ∂2 w ∂z2 5 0, where u and w are the displacements in the r and z directions, respectively. Differentiating equation (1d) by r and z, and substituting equation (1c) into equation (1d), we obtain the biharmonic equation F ∂ ∂r S 1 r 1 ∂ ∂r D 1 ∂2 ∂z2GF ∂ ∂r S u r 1 ∂u ∂r D 1 ∂2 u ∂z2G 5 0. (2) The general solution of equation (2) is u 5 (A0r 1 B0/r)(A3 1 A4z) 1 f1(r)(e2k1 z (3) 1 B3ek 1z ) 1 f2(r)(e2k2z 1 B4ek 2z ), where f1(r) 5 A1J1(k1r) 1 B1Y1(k1r) and f2(r) 5 A2R1(k2r) 1 B2S1(k2r); Aj and Bj (j 5 0–4), and k1 and k2, are constants; J1(k1r) and Y1(k1r) are Bessel functions of the first and second kind of order one, respectively; and R1(r) and S1(r) are functions of J1(r) and Y1(r), respectively. Since u is bounded at r 5 0, and as z→`, B0|B4 5 0. Integrating equation (1c) with respect to r and z, we can obtain an expression for w. Now, applying the boundary conditions shown in Fig. 1, we may obtain an accurate solution for stresses in the form of a series. The axial stress szz(r, z) can be expressed as szz 5 b1 1 b2z 1 O ` i 5 1 Fi (r) e2k1 z , (4a) where b1 and b2 are constants, and Fi (r) is a complex function of J1(r). Since Fi (r) equals a constant at r 5 0, Fi (r) may be expressed in the form of a constant plus a function of r. For the sake of simplifi- cation we assume that the series in equation (4a) can be replaced by a single logarithmic function multiplied by a single exponential in z. We choose a logarithmic function based on the solution of stresses in a homogeneous elastic body due to a line forces. Thus, we approximate equation (4a) as szz 5 b1 1 b2z (4b) 2tHc1 1 c2 lnF12S r RD 2 GJ e2b(z/R) , where c1, c2 and b are constants. Since the integral of the left term over the fiber area is the average stress in the fiber at z 5 0, we must have c1 5 c2 and b1 5 T, where T 5 sapp/f is the average stress in the matrix crack plane. Also, b2 5 2T/ls at z 5 ls, where ls 5 rT/2t is the slip length. Equation (4b) is then rewritten as szz 5 TS12z ls D2atH1 1 lnF12S r RD 2 GJ e2b(z/R) , (5) where a is a constant. Equation (5) indicates that the axial stress in the slip zone is the superposition of a linearly decreasing term that is exactly that obtained by the simple shear-lag model, and an additional term accounting for the stress concentrations that depends only on t and not on the applied stress. Note that the shear-lag model can be obtained if we choose a 5 0 in equation (5). Comparisons between the numerical FE model and the general form of equation (5) are shown in Figs 3, 4 and 6 for several different stress levels, interfacial shear stresses and interface models. Equation (5) fits the FE results extremely well for SiC/SiC for the values a 5 1.32 and b 5 10, and for SiC/glass composites if a 5 1.50 and b 5 10. The analytical model works nearly as well for the Coulomb friction model, although there are some slight differences due to the rapidly varying radial stress at the interface in the Coulomb friction case. Therefore, equation (5) can be considered as an accurate, approximate, analytical solution to the stress field szz(r, z) in the fiber. Equation (5) is the first main result of this paper
XIA and CURTIN: CERAMIC- MATRIX COMPOSITES 4885 3. STRESS INTENSITIES ON FIBER FLAWS AND FIBER FAILURE PROBABILITY Simple shear lag model The stresses found in Section 2 can drive the propa.F a 3.1.S gation of pre-existing crack-like flaws in the fibers Here, we use the axial stress field o(r, a within the fibers as an applied field acting on such flaws. Con- istent with the axisymmetric geometry, we envision for simplicity, annular cracks located at the fiber sur- face and extending inwards and denote the length of any particular annular crack as a. We treat the fiber single fiber not embedded in the 0 For small flaw sizes a<R(see below). the tmmmommmmmmmmmm metric nature of the problem is not important and so 0 we can utilize the standard integral equation for stress intensities at the tip of an edge crack in a plate under Distance from matrix crack, z, microns an opening pressure, given by Fig. 7. Stress intensity factor K versus distance latrIx or fiber faw SiC/SIC sites(mo 600 MPa, T= 200 ME bols. F fe results solid line: from analytic equation(5); dashed line: shear-lag prediction. we denote as K(, a, Oanm, t) to explicitly show all of where p(x)is the opening pressure on the crack, and the important dependencies. Near the matrix crack F(la)is a dimensionless function given in Tada et plane ==0, the enhanced stress field is seen to cause 1. [13] that can be approximated well an increased K relative to that expected from the shear-lag model(which yields a K versus of con- stant slope). Smaller flaws naturally have a smaller K Fxla)=-0.1603-+0.7353 but show a larger fractional increase in K since the smaller flaws experience the enhanced stress field -0.716 0.098 1-0.0604(a/(7) over a larger portion of their length. Analytic progress +1.3 Substituting equation (5)into equation(6)and carry ing out some of the trivial integrals leads to Here, p(r)=o(r, =)is the pressure acting on a flaw at longitudinal location z K=Y There are several assumptions made above in com- puting K. First, the flaws are assumed to be annular Second, we neglect the influence of the flaw itself on 2ar (8) the stress field by using the flaw-free stress field. This superposition should be accurate if the flaw is small Third. we assume that the fiber surface is a free sur {1+ln[2(u/R)-(va/R)]}F(y) face. However we shall show below that in the sim- plified shear-lag case, all of these factors disappear and one is left with fracture based on the standard Weibull failure model. Thus, these assumptions are Y= 1. 12 for free-edge flaws. The rer in equation(8) depends only on the nor the general case where the field o(r, =) is non-trivial crack size a/R and so is independent of but have a small impact on the final results. The dif- app and t. Thus, this integral can be performed only ferences we find between the finite-element and once as a function of a/R, rendering all subsequent shear-lag models will be predominantly due to the calculations of K versus = a, oap and t very easy to difference in local stress fields and not due to the mild evaluate. The results obtained from equation(8)are assumptions made regarding flaw propagation shown in Fig. 7, and are nearly indistinguishable from With the stress field o(, =)as found above by the those derived using the FE model data directly. Since FE method, we can then calculate the stress intensity computing the evolving fiber damage in a composite factor K using equation(6). Figure 7 shows a sample must be done as a continuous function of the applied of the results for the stress intensity as a function of stress, it is inconvenient and unwieldy to rely on the longitudinal position z and annular faw size a, which numerical results alone. Hend reduction of the
XIA and CURTIN: CERAMIC-MATRIX COMPOSITES 4885 3. STRESS INTENSITIES ON FIBER FLAWS AND FIBER FAILURE PROBABILITY 3.1. Stress intensities The stresses found in Section 2 can drive the propagation of pre-existing crack-like flaws in the fibers. Here, we use the axial stress field szz(r, z) within the fibers as an applied field acting on such flaws. Consistent with the axisymmetric geometry, we envision, for simplicity, annular cracks located at the fiber surface and extending inwards and denote the length of any particular annular crack as a. We treat the fiber with flaw as a single fiber not embedded in the matrix. For small flaw sizes a¿R (see below), the axisymmetric nature of the problem is not important and so we can utilize the standard integral equation for stress intensities at the tip of an edge crack in a plate under an opening pressure, given by K 5 2 √πaE a 0 dx p(x)F(x/a) √12(x/a) 2 , (6) where p(x) is the opening pressure on the crack, and F(x/a) is a dimensionless function given in Tada et al. [13] that can be approximated well as F(x/a) 5 20.1603S x aD 5 1 0.7353S x aD 4 20.7161S x aD 3 20.0988S x aD 2 20.0604S x aD (7) 1 1.3. Here, p(r) 5 szz(r, z) is the pressure acting on a flaw at longitudinal location z. There are several assumptions made above in computing K. First, the flaws are assumed to be annular. Second, we neglect the influence of the flaw itself on the stress field by using the flaw-free stress field. This superposition should be accurate if the flaw is small. Third, we assume that the fiber surface is a free surface. However, we shall show below that, in the simplified shear-lag case, all of these factors disappear and one is left with fracture based on the standard Weibull failure model. Thus, these assumptions are critical for constructing a tractable methodology in the general case where the field szz(r, z) is non-trivial but have a small impact on the final results. The differences we find between the finite-element and shear-lag models will be predominantly due to the difference in local stress fields and not due to the mild assumptions made regarding flaw propagation. With the stress field szz(r, z) as found above by the FE method, we can then calculate the stress intensity factor K using equation (6). Figure 7 shows a sample of the results for the stress intensity as a function of longitudinal position z and annular flaw size a, which Fig. 7. Stress intensity factor K versus distance z from the matrix crack, for fiber flaw sizes a of 0.1–0.9 µm, in the SiC/SiC composites (sapp 5 600 MPa, t 5 200 MPa). Symbols: FE results; solid line: from analytic equation (5); dashed line: shear-lag prediction. we denote as K(z, a, sapp, t) to explicitly show all of the important dependencies. Near the matrix crack plane z 5 0, the enhanced stress field is seen to cause an increased K relative to that expected from the shear-lag model (which yields a K versus z of constant slope). Smaller flaws naturally have a smaller K but show a larger fractional increase in K since the smaller flaws experience the enhanced stress fields over a larger portion of their length. Analytic progress can be made using the fitted form for the axial stress. Substituting equation (5) into equation (6) and carrying out some of the trivial integrals leads to K 5 YTS12z ls D√πa 22at e2b(z/R) π √πaE 1 0 (8) {1 1 ln[2(ya/R)2(ya/R) 2 ]}F(y) √12y2 dy, where Y 5 1.12 for free-edge flaws. The remaining integral in equation (8) depends only on the normalized crack size a/R and so is independent of z, sapp and t. Thus, this integral can be performed only once as a function of a/R, rendering all subsequent calculations of K versus z, a, sapp and t very easy to evaluate. The results obtained from equation (8) are shown in Fig. 7, and are nearly indistinguishable from those derived using the FE model data directly. Since computing the evolving fiber damage in a composite must be done as a continuous function of the applied stress, it is inconvenient and unwieldy to rely on the numerical results alone. Hence, our reduction of the
4886 XIA and CURTIN: CERAMIC- MATRIX COMPOSITES problem to a set of accurate stress fields [equation to the fiber strength oo. Equation(Il)is not exact 5)] and concomitant accurate stress intensities versus because it neglects a subtle conditional aspect of the flaw size a [equation( 8)] is a major advantage in pro- failure probability, but we show below that the error ceeding with a full statistical attack on fiber and com- this causes for the shear-lag model is generally negli posite failure gible. For a given type of fiber whose strength has 3. 2. Fiber failure probability been characterized via equation (9), and assuming that the flaws are all of the same type(e.g, edge Fiber strength is a statistical quantity since it is cracks with identical y factor), we thus have th governed by the propagation of pre-existing flaws or underlying flaw distribution of equation(Il) acks in the fiber. Typically, statistical fiber strength Now consider failure of fibers inside the composite, distributions are measured directly by performing sin- wherein the stress is spatially non-uniform. A fiber gle-fiber tension tests on a collection of fibers at a will fail at position at applied stress capp if there is common gage length L In such a test, the fiber stress a flaw of size a at that position satisfying is uniform across the fiber cross-section and uniform along the length of the fiber within the gage section. K(=,a,O甲pt)=Kc (12) The strength data obtained from such a test are usu- ally characterized in terms of a Weibull probability Let us denote the flaw size a satisfying equation (12) failure of a fiber of length L at stress o is given by as ae), where we suppress the dependence of this a failed at the current stress level while faws smalle PAo, L)=I than a()will survive at this stress level. The prob- ability of failure in the increment of length dz around this position is then simply equation(11)with a where oo is the characteristic (63.2% probability matrix crack plane, the total probability of fiber d a LEVel) fiber strength at gage length Lo and m is the vival is the product of the survival probability in each le eibull modulus characterizing the spread in the dis- of the increments de within the length =o, tribution of strengths at any gage length At the root of fiber failure as characterized by eq ation(9)is the distribution of flaws. For a spatially P=g2门0 uniform applied field as pertains in the single-fiber strength tests, fracture mechanics relates the applied stress o, flaw size a and crack-tip stress intensity fac- Converting the product of exponentials into the tor k as K= Yova Failure occurs when K= exponential of the integral over = we obtain the asso- where Kie is the mode I fracture toughness or critical ciated cumulative probability of failure within k< stress intensity factor for the fiber, or at the applied =o as stress P=0,a)=1-exp-2 Loa(=) where Gie= Kf /E is the critical energy releas Equation(14)is the second main result of this paper or fracture energy. For fiber strengths greater the a which will be used extensively below to determine GPa and K values less than 4 MPa m 2 fiber failure in a composite. In the shear-lag limit, the stress field across the sizes are smaller I um, justifying the assumption fiber radius is constant so that the field then only var a<R made earlier in using equations(6)and(7) Substituting equation(10) for the flaw ies with [equation(3)with a=0]. With the stress failure stress o into the Weibull expression of equ ndependent of r, the stress intensity on a flaw of size a at location is simply K(, a, Oapp, T)=Y(I ation(9), and considering an incremental length fllyVTa within the slip zone of length 8=[(I L=d=, we obtain the probability of finding a flaw of size larger than a in an increment d as DEmEcl(Ropp/2r), and thus the appropriate critical crack size is a(=)=[K, /YVT(1-/L )1?.Substituting nis result for a(=)into equation(14)and performing (1) the integration for=o≤6 in the slip 1(15)
4886 XIA and CURTIN: CERAMIC-MATRIX COMPOSITES problem to a set of accurate stress fields [equation (5)] and concomitant accurate stress intensities versus flaw size a [equation (8)] is a major advantage in proceeding with a full statistical attack on fiber and composite failure. 3.2. Fiber failure probability Fiber strength is a statistical quantity since it is governed by the propagation of pre-existing flaws or cracks in the fiber. Typically, statistical fiber strength distributions are measured directly by performing single-fiber tension tests on a collection of fibers at a common gage length L. In such a test, the fiber stress is uniform across the fiber cross-section and uniform along the length of the fiber within the gage section. The strength data obtained from such a test are usually characterized in terms of a Weibull probability distribution, wherein the cumulative probability of failure of a fiber of length L at stress s is given by Pf(s, L) 5 12expF2 L L0 S s s0 D m G, (9) where s0 is the characteristic (63.2% probability level) fiber strength at gage length L0 and m is the Weibull modulus characterizing the spread in the distribution of strengths at any gage length. At the root of fiber failure as characterized by equation (9) is the distribution of flaws. For a spatially uniform applied field as pertains in the single-fiber strength tests, fracture mechanics relates the applied stress s, flaw size a and crack-tip stress intensity factor K as K 5 Ys√πa. Failure occurs when K 5 KIc, where KIc is the mode I fracture toughness or critical stress intensity factor for the fiber, or at the applied stress s 5 KIc YÎπa 5 1 Y! EGIc πa , (10) where GIc 5 K2 Ic/E is the critical energy release rate or fracture energy. For fiber strengths greater than 2 GPa and KIc values less than 4 MPa m1/2, the flaw sizes are smaller 1 µm, justifying the assumption a¿R made earlier in using equations (6) and (7). Substituting equation (10) for the flaw size a versus failure stress s into the Weibull expression of equation (9), and considering an incremental length L 5 dz, we obtain the probability of finding a flaw of size larger than a in an increment dz as P(a, dz) 5 12expF2dz L0 S a0 a D m/2G, (11) where a0 5 (KIc/Ys0) 2 is the flaw size corresponding to the fiber strength s0. Equation (11) is not exact because it neglects a subtle conditional aspect of the failure probability, but we show below that the error this causes for the shear-lag model is generally negligible. For a given type of fiber whose strength has been characterized via equation (9), and assuming that the flaws are all of the same type (e.g., edge cracks with identical Y factor), we thus have the underlying flaw distribution of equation (11). Now consider failure of fibers inside the composite, wherein the stress is spatially non-uniform. A fiber will fail at position z at applied stress sapp if there is a flaw of size a at that position satisfying K(z, a, sapp, t) 5 KIc. (12) Let us denote the flaw size a satisfying equation (12) as a(z), where we suppress the dependence of this a on sapp and t. Flaws larger than a(z) will have already failed at the current stress level while flaws smaller than a(z) will survive at this stress level. The probability of failure in the increment of length dz around this position z is then simply equation (11) with a replaced by a(z). In a finite region uzu,z0 around a matrix crack plane, the total probability of fiber survival is the product of the survival probability in each of the increments dz within the length z0, Ps(z0, s) 5 P dz expF2dz L0 S a0 a(z) D m/2G. (13) Converting the product of exponentials into the exponential of the integral over z, we obtain the associated cumulative probability of failure within uzu, z0 as Pf(z0, s) 5 12expF22E z 0 0 dz L0 S a0 a(z) D m/2G. (14) Equation (14) is the second main result of this paper, which will be used extensively below to determine fiber failure in a composite. In the shear-lag limit, the stress field across the fiber radius is constant so that the field then only varies with z [equation (3) with a 5 0]. With the stress independent of r, the stress intensity on a flaw of size a at location z is simply K(z, a, sapp, t) 5 YT(12 z/ls)√πa within the slip zone of length d 5 [(12 f)Em/fEc](Rsapp/2t), and thus the appropriate critical crack size is a(z) 5 [KIc/Y√πT(12z/ls)]2 . Substituting this result for a(z) into equation (14) and performing the integration over z, for z0#d in the slip zone, leads to Pf(z0, T) 5 12expH2 1 m 1 1S T sc D m 1 1 [1 (15)
XIA and CURTIN: CERAMIC- MATRIX COMPOSITES Simple shear lag model we have used the de Oapp-90oMPa RT/2T, have referenced Pr to the peak stress T rather 0.25 than the applied stress fT, and have introduced the characteristic fiber strength Oapp=BOOMP: IA(n 700MPa that arises naturally in the analysis. o is the fiber 0 strength at a characteristic gage length 8.=Ro/t Note that equation(15)is independent of Kle and of (b) Distance from matrix crack, zo, Hm y details regarding the nature of the flaws aside om their final role in establishing a Weibull distri- bution for the failure strength. The result of equation 0.6 Present model (15)can be compared with an exact result, derived Simple shear lag model by Curtin et al. 14] without any recourse to the flaw sizes, given by 。 1-(1(17 02 Equation(15)differs from equation(17) only in the final term of the argument of the exponential, where 30 50 a power of m+ I has become m. While the exact Distance from matrix crack, Zo, Hm result can be obtained for the special shear-lag case, results for the more general case where a(=)cannot matrix crack in SiC/SiC composite be obtained analytically are much more complicated of m= 4;(b) fiber Weibull modulus Thus, below we consider the approximate result of 000 MPa at Lo 25.4 mm, G quation(14)in all cases and the corresponding m2, t= 50 MPa) approximate result of equation(15)in the shear-lag limit, for consistency in the probabilistic analysis. region of the fiber near the matrix crack plane are Using equations (8)and(12)to obtain a(=), and insufficient to substantially increase the failure prob then carrying through the integration of equation(14), ability. The failure probability is dominated by the we obtain the failure probability as a function of the contributions further away from the matrix crack length =o considered around the matrix crack. The plane, where the stresses are smaller but over a much probability depends on all of the underlying material larger region(the entire slip length). For larger fiber properties(t, m, Oo, Lo, R, Klc)but, from the shear- Weibull modulus [Fig. 8(b)], the probability of failure ag limit result of equation(15), it is clear that the rises rapidly at small distances and the local stress major dependencies are on m and o. [with l,= enhancements make a significant contribution to the (1/2)8.T/o )] Figure 8(a) and(b) shows the pre- failure probability. Failure further from the crack dicted probability of failure versus length =o for sev- plane at lower stress is greatly suppressed and hence eral different cases and corresponding results the probability of failure quickly becomes inde- btained using the shear-lag result of equation(15), pendent of position =o. For high m, there is a signi where the fiber strength is taken as oo=1000 MPa. cant difference in the failure probability between the For small fiber Weibull modulus [Fig 8(a), the fail- shear-lag and stress concentration models. The ure probability increases gradually with distance enhanced probability of failure due to the stress con- away from the matrix crack and there is very little centrations at high m is a main result of this paper, difference in probability between the simple shear-lag and will be shown below to lead to reduced composite model and the model with stress concentrations. For tensile strength and reduced fiber pullout on the frac low m, the enhanced stresses in a very small local ture surface
XIA and CURTIN: CERAMIC-MATRIX COMPOSITES 4887 2(12z0/ls) m 1 1 ]J, where we have used the definitions of a0 and ls 5 RT/2t, have referenced Pf to the peak stress T rather than the applied stress fT, and have introduced the characteristic fiber strength sc 5 S sm 0 tL0 R D 1/(m 1 1) (16) that arises naturally in the analysis. sc is the fiber strength at a characteristic gage length dc 5 Rsc/t. Note that equation (15) is independent of KIc and of any details regarding the nature of the flaws aside from their final role in establishing a Weibull distribution for the failure strength. The result of equation (15) can be compared with an exact result, derived by Curtin et al. [14] without any recourse to the flaw sizes, given by Pf(z0) 5 12expH2 1 m 1 1S T sc D m 1 1 [12(1 (17) 2z0/ls) m]J. Equation (15) differs from equation (17) only in the final term of the argument of the exponential, where a power of m 1 1 has become m. While the exact result can be obtained for the special shear-lag case, results for the more general case where a(z) cannot be obtained analytically are much more complicated. Thus, below we consider the approximate result of equation (14) in all cases and the corresponding approximate result of equation (15) in the shear-lag limit, for consistency in the probabilistic analysis. Using equations (8) and (12) to obtain a(z), and then carrying through the integration of equation (14), we obtain the failure probability as a function of the length z0 considered around the matrix crack. The probability depends on all of the underlying material properties (t, m, s0, L0, R, KIc) but, from the shearlag limit result of equation (15), it is clear that the major dependencies are on m and sc [with ls 5 (1/2)(dcT/sc)]. Figure 8(a) and (b) shows the predicted probability of failure versus length z0 for several different cases and corresponding results obtained using the shear-lag result of equation (15), where the fiber strength is taken as s0 5 1000 MPa. For small fiber Weibull modulus [Fig. 8(a)], the failure probability increases gradually with distance away from the matrix crack and there is very little difference in probability between the simple shear-lag model and the model with stress concentrations. For low m, the enhanced stresses in a very small local Fig. 8. Cumulative probability of fiber failure Pf(z0, T) as a function of distance z0 from matrix crack in SiC/SiC composite: (a) fiber Weibull modulus of m 5 4; (b) fiber Weibull modulus m 5 20 (f 5 0.3, s0 5 1000 MPa at L0 5 25.4 mm, GIc 5 15 J/m2 , t 5 50 MPa). region of the fiber near the matrix crack plane are insufficient to substantially increase the failure probability. The failure probability is dominated by the contributions further away from the matrix crack plane, where the stresses are smaller but over a much larger region (the entire slip length). For larger fiber Weibull modulus [Fig. 8(b)], the probability of failure rises rapidly at small distances and the local stress enhancements make a significant contribution to the failure probability. Failure further from the crack plane at lower stress is greatly suppressed and hence the probability of failure quickly becomes independent of position z0. For high m, there is a signifi- cant difference in the failure probability between the shear-lag and stress concentration models. The enhanced probability of failure due to the stress concentrations at high m is a main result of this paper, and will be shown below to lead to reduced composite tensile strength and reduced fiber pullout on the fracture surface
4888 XIA and CURTIN: CERAMIC- MATRIX COMPOSITES 4. COMPOSITE TENSILE STRENGTH fiber failure probability distribution, which in turn is 4.1. Single matrix crack simply the normalized derivative of the cumulative failure probability PA=, T) with respect to position e are interested in the transition from tough to = henc brittle We thus first focus on the failure a single matrix crack. If the composite fails when the first matrix crack appears, then it will dPe, T) be linearly elastic to failure and will not show the PA(=0 (20) "tough"behavior that occurs when multiple matrix cracks arise. We thus imagine that only a single matrix crack exists and study the ultimate tensile PA(=,)d=, strength of such a system The high stress fields on the fil natrix crack induce preferential failure near the matrix crack. But, Fig. 8(a)and(b) shows that the where the second equality follows after an integration probability of failure increases with increasing =o and by parts. Introducing the slip length Is, we can then so the fibers do not break precisely at the crack plane. write Here we consider fiber failure to occur anywhere within the slip region around the matrix crack; i.e we fix =0=8. We neglect the possible failure at ape=[l-PkEo, DIT+-PAEo, TT:. (21) locations in the"far-field"region of the composite for which failure is much less likely, as demonstrated by the fact that the cumulative failure probabillty The maximum value of osp versus t gives the ulti- ( Fig. 8) becomes nearly constant at larger distances mate strength of the composite. The average fiber pul In fact. failure further from the matrix crack lout upon complete failure is obtained from equation plane than a distance of the order of d. has no effect (20) at very large T on composite strength, as recognized in many pre- Figure 9 shows the predicted tensile strength versus vious models of failure interfacial shear resistance t for low and high values Now consider a single matrix crack and a spectrum of the fiber Weibull modulus. alo long with the predic- load sharing "wherein the unbroken fibers equally equation(19)and prior equations. The underlying fiber strength parameters oo=1000 MPa and Lo Then, in the matrix crack plane, the unbroken fibers 25.4 mm have been fixed for all cases. For low Weib- nal applied stress per fiber, Apl/f, due to the damage. obtained from the shear-lag model and the stim t u) will carry some stress T that is higher than the nomi- ull moduli, the strength scaling of o Mechanical equilibrium in the plane of the matrix o are generally preserved out to fairly large values ack dictates that the average applied load appf of [200 MPa. This is consistent with the behavior equals the sum of the average stress carried by the unbroken fibers plus a contribution due to the average residual bridging or"pullout" stresses o, carried by 1000 the fibers that are broken away from the crack plane Equilibrium can then be expressed as =1=Pon了+Po(18)喜 where =0 =8 is the slip length within which fibers g are considered to break. The average pullout stress o is the average axial fiber stress at the matrix crack plane due to a fiber break at the mean break position G=15m2 (which is also the mean pullout length of the broken fibers), and can be written as Fig. 9. Ultimate tensile strength versus interfacial shear I for SiC/SiC composites with a single matrix crack an high and low values of fiber weibull moduli and several of fiber nerev The mean break position is the first moment of the
4888 XIA and CURTIN: CERAMIC-MATRIX COMPOSITES 4. COMPOSITE TENSILE STRENGTH 4.1. Single matrix crack We are interested in the transition from tough to brittle behavior. We thus first focus on the failure associated with a single matrix crack. If the composite fails when the first matrix crack appears, then it will be linearly elastic to failure and will not show the “tough” behavior that occurs when multiple matrix cracks arise. We thus imagine that only a single matrix crack exists and study the ultimate tensile strength of such a system. The high stress fields on the fibers around the matrix crack induce preferential failure near the matrix crack. But, Fig. 8(a) and (b) shows that the probability of failure increases with increasing z0 and so the fibers do not break precisely at the crack plane. Here we consider fiber failure to occur anywhere within the slip region around the matrix crack; i.e., we fix z0 5 d. We neglect the possible failure at locations in the “far-field” region of the composite, for which failure is much less likely, as demonstrated by the fact that the cumulative failure probability (Fig. 8) becomes nearly constant at larger distances. In fact, failure further away from the matrix crack plane than a distance of the order of dc has no effect on composite strength, as recognized in many previous models of failure. Now consider a single matrix crack and a spectrum of broken fibers in the slip region. We assume “global load sharing” wherein the unbroken fibers equally share the load shed by all of the broken fibers [4, 5]. Then, in the matrix crack plane, the unbroken fibers will carry some stress T that is higher than the nominal applied stress per fiber, sapp/f, due to the damage. Mechanical equilibrium in the plane of the matrix crack dictates that the average applied load sapp/f equals the sum of the average stress carried by the unbroken fibers plus a contribution due to the average residual bridging or “pullout” stresses sp carried by the fibers that are broken away from the crack plane. Equilibrium can then be expressed as sapp f 5 [12Pf(z0, T)]T 1 Pf(z0, T)sp, (18) where z0 5 d is the slip length within which fibers are considered to break. The average pullout stress sp is the average axial fiber stress at the matrix crack plane due to a fiber break at the mean break position z ¯ (which is also the mean pullout length of the broken fibers), and can be written as sp 5 2tz ¯ R . (19) The mean break position z ¯ is the first moment of the fiber failure probability distribution, which in turn is simply the normalized derivative of the cumulative failure probability Pf(z, T) with respect to position z. Hence, z ¯ 5 1 Pf(z0, T) E z 0 0 dz zdPf(z, T) dz (20) 5 z02 1 Pf(z0, T) E z 0 0 Pf(z, T) dz, where the second equality follows after an integration by parts. Introducing the slip length ls, we can then write sapp f 5 [12Pf(z0, T)]T 1 z ¯ ls Pf(z0, T)T. (21) The maximum value of sapp versus T gives the ultimate strength of the composite. The average fiber pullout upon complete failure is obtained from equation (20) at very large T. Figure 9 shows the predicted tensile strength versus interfacial shear resistance t for low and high values of the fiber Weibull modulus, along with the predictions of the simple shear-lag model, as obtained from equation (19) and prior equations. The underlying fiber strength parameters σ0 5 1000 MPa and L0 5 25.4 mm have been fixed for all cases. For low Weibull moduli, the strength scaling of suts~t1/(m 1 1) obtained from the shear-lag model and the scaling of sc are generally preserved out to fairly large values of t<200 MPa. This is consistent with the behavior Fig. 9. Ultimate tensile strength versus interfacial shear stress t for SiC/SiC composites with a single matrix crack and for high and low values of fiber Weibull moduli and several values of fiber fracture energy (f 5 0.3, s0 5 1000 MPa, L0 5 24.5 mm)
