ournal J. Am. Ceran. Soc. 85(6 1350-65(2002) Stress Rupture in Ceramic-Matrix Composites: theory and Experiment Howard G. Halverson Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 William A. Curtin Division of Engineering, Brown University, Providence, Rhode Island 02912 A micromechanically based model for the deformation, the lifetimes at 950C are greatly overpredicted. Thus, the strength, and stress-rupture life of a ceramic-matrix composite micromechanical model can be successful quantitatively but is developed for materials that do not degrade by oxidative clearly shows that the rupture life of the composite is e attack. The rupture model for a unidirectional composite tremely sensitive to the detailed mechanisms of fiber degrad incorporates fiber-strength statistics, fiber degradation with tion. The model has practical applications for extrapolating time at temperature and load, the state of matrix damage, and laboratory lifetime data and predicting life in components with the effects of fiber pullout, within a global load sharing model. evolving spatial stresses. The constituent material parameters that are required to predict the deformation and lifetime can all be obtained dependent of stress-rupture testing through quasi-static ten- L. Introduction ion tests and tests on the individual composite constituents. The model predicts the tertiary creep, the remaining composite strength, and the rupture life all of which are dependent C ERAMIC-MATRIX COMPOSITES(CMCs)are attractive materials rature h as turbine ritically on the underlying fiber-strength degradation. Sensi- combustor liners and exhaust nozzles. However, designers and tivity of the rupture life to various micromechanical parame- engineers must be able to predict the material response to applied ters is studied parametrically. To complement the model, an loads and in various environments. The quasi-static deformation extensive experimental study of stress rupture in a Nextel and tensile strength of many CMCs are well understood in terms of 610/alumina-yttria composite at temperatures of 950 and the evolution of matrix cracking and fiber failure,however, the 1050C is reported. The Larson-Miller and Monkman-Grant time-dependent properties, such as creep deformation and strength life-prediction methods are inadequate to explain the current are not as well understood at the micromechanical level, despite a data. Constituent parameters for this material system are growing body of experimental results. An important step in giving derived from quasi-static tests and literature data, and the designers the confidence to use CMCs in structural applications micromechanical model predictions are compared with mea- lies in obtaining a basic understanding of key damage mechanisms sured behavior. For a slow-crack-growth model of fiber and their effects on stress-rupture lifetime and deformation strength degradation, the lifetime predictions are shorter by Many researchers have used traditional engineering method such as the Larson-Miller (LM) and Monkman-Grant (MG one parameter, however, the model prediction of the tertiary approaches, to predict stress rupture in ceramics and composites creep and residual strength at 1050C agrees well with the under constant load. The LM approach relates the applied stress to xperimental results. For a more complex degradation model a failure parameter, 0, given by the rupture life and tertiary creep at 1050 C can be predicted quite well; however, the spread in residual strength is not, and Q=T(log tr+ C) where / is the temperature, 4, the rupture time, and C a constant Predictions for fiber composites are made by first obtaining the LM parameter versus stress for single fibers via single-fiber tests R. Kerans--contributing editor at various loads and temperatures. Then, basic mechanics is used to estimate the stress carried by the fibers in the composite and the omposite failure time is assumed to be equal to that of the Aau suppoted N 188372 Received May (1, 2001,- approved December 1& 10 individual fibers at the established stress level. This technique was sed by morscher and co-workers,to examine the stress rupture es support from the U.S. Air Force Office of Scientific of precracked Hi-Nicalon M/SiC minicomposites with BN inter Grant No. F49620-99-1-0027. from the Mechanics of omposite Materials pr faces. The predictions were accurate at low and high values of o Member, American Ceramic Society however at intermediate values. removal of the bn interface and eature
Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment Howard G. Halverson* Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 William A. Curtin* Division of Engineering, Brown University, Providence, Rhode Island 02912 A micromechanically based model for the deformation, strength, and stress-rupture life of a ceramic-matrix composite is developed for materials that do not degrade by oxidative attack. The rupture model for a unidirectional composite incorporates fiber-strength statistics, fiber degradation with time at temperature and load, the state of matrix damage, and the effects of fiber pullout, within a global load sharing model. The constituent material parameters that are required to predict the deformation and lifetime can all be obtained independent of stress-rupture testing through quasi-static tension tests and tests on the individual composite constituents. The model predicts the tertiary creep, the remaining composite strength, and the rupture life, all of which are dependent critically on the underlying fiber-strength degradation. Sensitivity of the rupture life to various micromechanical parameters is studied parametrically. To complement the model, an extensive experimental study of stress rupture in a Nextel 610/alumina–yttria composite at temperatures of 950° and 1050°C is reported. The Larson–Miller and Monkman–Grant life-prediction methods are inadequate to explain the current data. Constituent parameters for this material system are derived from quasi-static tests and literature data, and the micromechanical model predictions are compared with measured behavior. For a slow-crack-growth model of fiberstrength degradation, the lifetime predictions are shorter by two orders of magnitude. When the rupture life is fitted with one parameter, however, the model prediction of the tertiary creep and residual strength at 1050°C agrees well with the experimental results. For a more complex degradation model, the rupture life and tertiary creep at 1050°C can be predicted quite well; however, the spread in residual strength is not, and the lifetimes at 950°C are greatly overpredicted. Thus, the micromechanical model can be successful quantitatively but clearly shows that the rupture life of the composite is extremely sensitive to the detailed mechanisms of fiber degradation. The model has practical applications for extrapolating laboratory lifetime data and predicting life in components with evolving spatial stresses. I. Introduction CERAMIC-MATRIX COMPOSITES (CMCs) are attractive materials for use in high-temperature applications such as turbine combustor liners and exhaust nozzles. However, designers and engineers must be able to predict the material response to applied loads and in various environments. The quasi-static deformation and tensile strength of many CMCs are well understood in terms of the evolution of matrix cracking and fiber failure;1–5 however, the time-dependent properties, such as creep deformation and strength, are not as well understood at the micromechanical level, despite a growing body of experimental results. An important step in giving designers the confidence to use CMCs in structural applications lies in obtaining a basic understanding of key damage mechanisms and their effects on stress-rupture lifetime and deformation. Many researchers have used traditional engineering methods, such as the Larson–Miller (LM) and Monkman–Grant (MG) approaches, to predict stress rupture in ceramics and composites under constant load. The LM approach relates the applied stress to a failure parameter, Q, given by Q Tlog tr C (1) where T is the temperature, tr the rupture time, and C a constant. Predictions for fiber composites are made by first obtaining the LM parameter Q versus stress for single fibers via single-fiber tests at various loads and temperatures. Then, basic mechanics is used to estimate the stress carried by the fibers in the composite and the composite failure time is assumed to be equal to that of the individual fibers at the established stress level. This technique was used by Morscher and co-workers6,7 to examine the stress rupture of precracked Hi-Nicalon™/SiC minicomposites with BN interfaces. The predictions were accurate at low and high values of Q; however, at intermediate values, removal of the BN interface and R. Kerans—contributing editor Manuscript No. 188372. Received May 11, 2001; approved December 18, 2001. Supported by NASA Glenn Research Center, through Grant No. NAG3-2100. Author WAC also acknowledges support from the U.S. Air Force Office of Scientific Research (AFOSR), through Grant No. F49620-99-1-0027, from the Mechanics of Composite Materials program. *Member, American Ceramic Society. journal J. Am. Ceram. Soc., 85 [6] 1350–65 (2002) Feature
June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1351 its replacement with strongly bonded borosilicate glass resulted in showed the basic dependencies of the rupture time on parameters much shorter rupture lives than expected. In cases where the such as the Weibull modulus of the fiber and the slow-crad matrix is not cracked, use of the MG approach has been proposed. growth exponent of the fiber The MG approach relates the steady-state strain rate(e)to the The Coleman modell is an alternative to the slow-crack rupture time via two constants, k and D growth model for fiber rupture; in this method, the probability o iber failure is a function of time and stress and no attempt is made EL=D (2 to determine fiber strength. Ibnabdeljalil and Phoenix22developed a composite stress-rupture model for the case where the fibers DiCarlo and Yun demonstrated that MG plots of steady-state carry all the applied loads, a Coleman model was used for the fiber-rupture behavior Lamouroux et al.-developed a model for 610 fibers successfully match lifetimes obtained by Zuiker on a creep and stress rupture using a simplified model of the stress state woven Nextel 610/aluminosilicate composites of the fiber, a simple bundle model for fiber failure that neglecte Although the accuracy of engineering methods can be good, fiber pullout, and the Coleman fiber-rupture model. These model they are basically correlations between macroscopic measures of can predict rupture life and tertiary creep, and they can be extended behavior and do not contain much information about the state of to account for matrix damage, but this step has not yet beer the material during the stre attempted. However, in the Coleman model upture process. Therefore, they cannot be used to (i) predict behavior a priori;(ii)connect infinite fast-fracture strength;; hence, the ruptur the fast-fracture strength and the remaining different but related aspects of the deformation, such as the tertiary predicted at all times cannot be creep and the remaining strength, to the rupture life; and(iii) If the matrix is sufficiently stiff and/or not fully cracked provide insight into the optimization of composites, because no carries some axial stress, which leads to a spatially varying direct connection to underlying constituent properties and/or the fiber-stress profile that can decrease the rate of fiber degradation internal state of damage in the material exists. These factors also and increase composite lifetime. Many CMC materials are in- clearly limit the use of LM and MG approaches in the development tended for use at moderate stresses where the matrix cracking ()is of new composites, where changes in constituents occur regularly not saturated, (i) is dependent on the applied stress level, or(ii) as improved materials become available. Micromechanically based can evolve during constant-load testing; thus, the details of the methods to predict deformation and failure under stress-rupture matrix damage state can have a significant role in determination of onditions, as a function of the underlying behavior of the the composite lifetime onstituent materials, should be extremely useful for the design If the matrix does carry some sig of and optimization of existing materials, for the development of new load, at least initially, then its response to applied stresses must composite systems, and to complement mechanical testing and, also be considered. At low stresses, most ceramic matrixes will thus, reduce development and design costs remain uncracked, and the stresses carried by each constituent will The failure of most CMCs is concurrent with the failure of the then be controlled by the constituent creep response. The creep rate reinforcing fibers; therefore, a lifetime-prediction method should of a CMC can become a limiting factor in component design and be concerned primarily with the accumulation of fiber failure. In is, thus, an area of considerable investigation. Holmes and co- eneral, fiber failure is a function of temperature, stress, and an studied the creep rate of silicon carbide/calcium chemical interactions that occur (e.g, oxidation). For silicon aluminosilicate(SiC/CAS)and silicon carbide/silicon nitride(Sic carbide (Sic) fiber materials, where oxidation is a primary Si,N4) composites extensively. For SiC/Si,N4 composites, the concern,modeling has focused on the growth of an oxide scale on creep rate exhibited short primary and tertiary creep regions and an the fiber surface which behaves similar to a surface flaw and leads extensive secondary(steady-state) region. In the SiC/CAS com- to a time-dependent decrease in fiber strength. Lara-Curzioo posites at 1200oC, at which temperature the matrix carries only a analyzed this mechanism for a matrix-cracked composite, includ fraction of the applied stress, the stress expo t of the ite matched that of the fibers reasonably well (n= 1.3 versus n composite lifetime. Evans and co-workers i-13 considered a sim- 1.9 for Nicalon"fibers ) Composite creep due to combined fiber ilar process within growing matrix cracks wherein weakening by a', bution during rationalize some of this creep data. Stress oxide scale and subsequent failure of the exposed fibers in the redistribution during creep can be influenced by the damage state crack wake led to the growth of matrix cracks. Composite failure occurs when the remaining uncracked composite cross section however, as observed, for instance, in the work of Holmes and cannot sustain the applied load. Other mechanisms, such as the co-workers2529on SiC/Si N4 composites. They found that, when relaxation of crack-bridging stresses, because of fiber creep and the matrix was undamaged(at low applied stresses), the initial subsequent crack growth, have been discussed by Begley and composite creep rate was controlled by the transfer of stress from the creeping matrix to the noncreeping elastic fibers, as per the If environment effects can be eliminated through the use of modelot Mclean, At higher stresses, the initial loading rate coatings or oxidation- resistant constituents, fiber failure should be a function of stress and temperature only. CMCs with active matrix fracture was pronounced and composite lifetimes were relatively short. At lower rates of loading, the matrix was able to on have very limited lifetimes, therefore, we will focus on relax in creep and did not fracture, resulting in much-longer tially longer-life systems where oxidative degradation is composite lifetimes. The McLean approach was expanded by Du Specifically, we will use the slow-crack-growth model to predict fiber-damage evolution and failure and envision the appl and McMeeking" and Fabeny and Curtin to incorporate statis- cation of our models to all-oxide ceramic composites. Although tical fiber fracture and its influence on creep and rupture but not matrix damage. These works also emphasized that stress transfer evidence for any particular fiber-degradation mechanism in oxide- across the fiber/matrix interface can also be affected by matrix ceramic fibers is difficult to ascertain, the general power-law creep:30,32,33the interface shear stress along broken fibers drives form of the slow-crack-growth-rate equation lends itself to analytic creep of the matrix and a subsequent increase in the ineffective solutions and provides a relationship between the initial fiber length of broken fibers, resulting in time-dependent composite failure temperature history. Failure of the fibers in a composite under slow Given the above-described experimental and modeling back crack growth is dependent on the actual stress history experienced ground, the present paper develops the mechanics and statistics by each fiber, which is dependent on the applied load, the state of ideas needed to consider stress rupture of the fiber and composi matrix damage, and the interfacial sliding between the fibers and as a function of the matrix damage state, using the slow-crack-
its replacement with strongly bonded borosilicate glass resulted in much shorter rupture lives than expected. In cases where the matrix is not cracked, use of the MG approach has been proposed.8 The MG approach relates the steady-state strain rate (˙) to the rupture time via two constants, k and D: ˙ k tr D (2) DiCarlo and Yun8 demonstrated that MG plots of steady-state strain rate ˙ versus rupture time tr obtained from single Nextel™ 610 fibers successfully match lifetimes obtained by Zuiker9 on a woven Nextel™ 610/aluminosilicate composites. Although the accuracy of engineering methods can be good, they are basically correlations between macroscopic measures of behavior and do not contain much information about the state of the material during the stress-rupture process. Therefore, they cannot be used to (i) predict behavior a priori; (ii) connect different but related aspects of the deformation, such as the tertiary creep and the remaining strength, to the rupture life; and (iii) provide insight into the optimization of composites, because no direct connection to underlying constituent properties and/or the internal state of damage in the material exists. These factors also clearly limit the use of LM and MG approaches in the development of new composites, where changes in constituents occur regularly as improved materials become available. Micromechanically based methods to predict deformation and failure under stress-rupture conditions, as a function of the underlying behavior of the constituent materials, should be extremely useful for the design and optimization of existing materials, for the development of new composite systems, and to complement mechanical testing and, thus, reduce development and design costs. The failure of most CMCs is concurrent with the failure of the reinforcing fibers; therefore, a lifetime-prediction method should be concerned primarily with the accumulation of fiber failure. In general, fiber failure is a function of temperature, stress, and any chemical interactions that occur (e.g., oxidation). For silicon carbide (SiC) fiber materials, where oxidation is a primary concern, modeling has focused on the growth of an oxide scale on the fiber surface, which behaves similar to a surface flaw and leads to a time-dependent decrease in fiber strength. Lara-Curzio10 analyzed this mechanism for a matrix-cracked composite, including the statistics of fiber strength, and produced predictions for composite lifetime. Evans and co-workers11–13 considered a similar process within growing matrix cracks wherein weakening by oxide scale and subsequent failure of the exposed fibers in the crack wake led to the growth of matrix cracks. Composite failure occurs when the remaining uncracked composite cross section cannot sustain the applied load. Other mechanisms, such as the relaxation of crack-bridging stresses, because of fiber creep and subsequent crack growth, have been discussed by Begley and co-workers14,15 and Lewinsohn et al., 16 among others. If environment effects can be eliminated through the use of coatings or oxidation-resistant constituents, fiber failure should be a function of stress and temperature only. CMCs with active oxidation have very limited lifetimes; therefore, we will focus on the potentially longer-life systems where oxidative degradation is absent. Specifically, we will use the slow-crack-growth model to predict fiber-damage evolution and failure and envision the application of our models to all-oxide ceramic composites. Although evidence for any particular fiber-degradation mechanism in oxideceramic fibers is difficult to ascertain,17–19 the general power-law form of the slow-crack-growth-rate equation lends itself to analytic solutions and provides a relationship between the initial fiber strength and the fiber strength after some arbitrary stress and temperature history. Failure of the fibers in a composite under slow crack growth is dependent on the actual stress history experienced by each fiber, which is dependent on the applied load, the state of matrix damage, and the interfacial sliding between the fibers and the matrix. Iyengar and Curtin20 studied composite failure when the matrix was fully saturated with closely spaced cracks and showed the basic dependencies of the rupture time on parameters such as the Weibull modulus of the fiber and the slow-crackgrowth exponent of the fiber. The Coleman model21 is an alternative to the slow-crackgrowth model for fiber rupture; in this method, the probability of fiber failure is a function of time and stress and no attempt is made to determine fiber strength. Ibnabdeljalil and Phoenix22 developed a composite stress-rupture model for the case where the fibers carry all the applied loads; a Coleman model was used for the fiber-rupture behavior. Lamouroux et al.23 developed a model for creep and stress rupture using a simplified model of the stress state of the fiber, a simple bundle model for fiber failure that neglected fiber pullout, and the Coleman fiber-rupture model. These models can predict rupture life and tertiary creep, and they can be extended to account for matrix damage, but this step has not yet been attempted. However, in the Coleman model, the fibers have an infinite fast-fracture strength; hence, the rupture is not related to the fast-fracture strength and the remaining strength cannot be predicted at all times. If the matrix is sufficiently stiff and/or not fully cracked, it carries some axial stress, which leads to a spatially varying fiber-stress profile that can decrease the rate of fiber degradation and increase composite lifetime. Many CMC materials are intended for use at moderate stresses where the matrix cracking (i) is not saturated, (ii) is dependent on the applied stress level, or (iii) can evolve during constant-load testing; thus, the details of the matrix damage state can have a significant role in determination of the composite lifetime. If the matrix does carry some significant portion of the applied load, at least initially, then its response to applied stresses must also be considered. At low stresses, most ceramic matrixes will remain uncracked, and the stresses carried by each constituent will then be controlled by the constituent creep response. The creep rate of a CMC can become a limiting factor in component design and is, thus, an area of considerable investigation. Holmes and coworkers24,25 studied the creep rate of silicon carbide/calcium aluminosilicate (SiC/CAS) and silicon carbide/silicon nitride (SiC/ Si3N4) composites extensively. For SiC/Si3N4 composites, the creep rate exhibited short primary and tertiary creep regions and an extensive secondary (steady-state) region. In the SiC/CAS composites at 1200°C, at which temperature the matrix carries only a fraction of the applied stress,26 the stress exponent of the composite matched that of the fibers reasonably well (n 1.3 versus n 1.9 for Nicalon™ fibers27). Composite creep due to combined fiber and matrix creep, but without damage, was first modeled by McLean28 and can rationalize some of this creep data. Stress redistribution during creep can be influenced by the damage state, however, as observed, for instance, in the work of Holmes and co-workers25,29 on SiC/Si3N4 composites. They found that, when the matrix was undamaged (at low applied stresses), the initial composite creep rate was controlled by the transfer of stress from the creeping matrix to the noncreeping elastic fibers, as per the model of McLean.28 At higher stresses, the initial loading rate strongly influenced the creep behavior. At high rates of loading, matrix fracture was pronounced and composite lifetimes were relatively short. At lower rates of loading, the matrix was able to relax in creep and did not fracture, resulting in much-longer composite lifetimes. The McLean approach was expanded by Du and McMeeking30 and Fabeny and Curtin31 to incorporate statistical fiber fracture and its influence on creep and rupture but not matrix damage. These works also emphasized that stress transfer across the fiber/matrix interface can also be affected by matrix creep:30,32,33 the interface shear stress along broken fibers drives creep of the matrix and a subsequent increase in the ineffective length of broken fibers, resulting in time-dependent composite failure. Given the above-described experimental and modeling background, the present paper develops the mechanics and statistics ideas needed to consider stress rupture of the fiber and composite as a function of the matrix damage state, using the slow-crackgrowth model for fiber degradation. The resulting model includes fiber-strength statistics, fiber degradation with time at temperature June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1351
1352 urnal of the American Ceramic Society-Halverson and Curtin Vol. 85. No 6 prevent large-scale sintering of the materials at elevated tempera tures but small ugh to permit stress transfer between the model predicts interrelated phenomena of tertiary cree constituents by frictional shear stress. The carbon also serves to remaining composite strength, and rupture life, all of which are protect the fiber from any adverse chemical reactions during the dependent critically on the underlying fiber-strength degradation. processing of the Al,O3/Y2O3 matrix The constituent material parameters required to predict the defor To create the matrix, a slurry of Al,O, powder first was mation and lifetime predictions can be obtained independent of pressure-cast into the fiber preform. Next, a sol of Y2O3 particles stress-rupture testing through quasi-static tension tests and tests on was infiltrated into the preform, and the preform was dried at the individual composite constituents. To complement and validate 700C. After a few infiltration and drying steps, the part was fired the model, an experimental study of the stress-rupture life, creep at 1100C for -I h. Then, the infiltration/drying/firing cycle was deformation, and the associated damage modes for a unidirectional repeated until the desired density was attained. For these materials, processing was halted when the composite porosity was -20%, McDermott Technologies, Inc.(MTD), Lynchburg, VA)with which typically required 4-6 cycles. The Y2O3 reacts with the ugitive carbon interface has been conducted. This oxide/oxide Al2O3 during the firing cycle to create AlYO, and Y,Al,O CMC system should be unaffected by the oxidation, and the hence, the exact composition of the matrix was not determined unidirectional configuration permits direct comparison with the model. Using literature data for the fiber-strength degradation, (2) Mechanical Testing model predicts lifetimes that are two orders of magnitude shorter than that measured. When one parameter, the fiber-degradation The unidirectional material was used for quasi-static and stress- rate constant, is fit to the experimental results, the trends in rupture testing at three temperatures:23°,950°,and1050°C omposite lifetime with stress and temperature are well-matched Quasi-static testing was performed using a test frame(Model 880, MTS Systems, Eden Prairie, MN) with a controller (Model 458 Then, the model also predicts tertiary creep rates and remaining MTS Systems). Specimens were tabbed with 0.020 in. fully data. The measured statistical scatter in the failure times can also annealed aluminum tabs, to prevent damage from gripping. The be correlated with the scatter in the initial composite strengths tab was placed around the end of the specimen, and the specimen using the model. The success of the model in simultaneously was placed in the MTs test-frame grip. The pressure of the grip predicting several different features that are associated with the plastically deformed the aluminum to"fit "the specimen, and no deformation and failure, despite the need for a fitting parameter, dhesive was used. Grip pressures were maintained at -0.7 MPa demonstrates the power of such a micromechanically based ap- The tests were run under load control. at a rate of 180 N/s Strain proach. An alternative assumption for the fiber-degradation mech- at room temperature was measured with an extensometer(Model anism, which involves two flaw populations, results in improved 632-11B, MTS Systems). Specimen alignment was maintained fetime predictions at high temperature but poorer residual through a fixture at the grips ength and tertiary creep predictions. The implications of the A compact oven was used for the tests that were conducted at extreme sensitivity of the rupture life to the precise mechanisms of elevated temperature. The oven had four SiC resistance element fiber-strength degradation, and/or the inadequacy of the ex situ (Norton Advanced Ceramics, Worcester, MA)that heated the issue or discussion adation to the in situ behavior, is an important specimen. The oven shell was stainless steel and had nominal dimensions of 3.5 in. x 3 in. x 3 in. Fused-silica insulation Cotronics Corp, Brooklyn, NY)lined the inside of the oven, in Section ll, with a description of the experimental techniques and which reduced the nominal interior dimensions to 2.5 in. X 2 in results and a comparison of the measured rupture lives to the LM Reston, VA)controlled the SiC heating elements: one for the tw and MG models; their inadequacy motivates the subsequent mod development. In Section Ill, the analytic model for the fiber upper elements, and one for the two lower elements. Each dominated stress rupture of composites is developed. In Section controller received input from a type R(platinum/platinum- Iv, the experimental data are analyzed and compared with the rhodium(Pt/Pt-Rh)thermocouple predictions of the model. In Section V, we discuss our results by alumina-fabric insulation, for efficient heating and to hel iurther, address important issues that this work raises, and outline maintain a constant temperature. a heat shield that was attached to how the present model can be used with structural design models the oven held an extensometer(Model 621-51E, MTS Systems) for CMC components The extensometer measured strain according to the deflection of two 5 in. Al,O, rods that pass through the oven shell and contact the specimen. The entire assembly was attached to the test frame ll. Experimental Details, Results, and Predictions of at one of the posts. The extensometer, the heat shield, and the mTs Engineering Models grips were cooled by water. For high-temperature testing, the temperature was ramped at a rate of 33 C/s to the desired test We begin our discussion with the experimental results and temperature. Then, the temperature was held constant for 10 min comparisons to the Larson-Miller(LM) and Monkman-Grant before the test began, to ensure thermal equilibrium. (MG)engineering models to demonstrate that such approaches to The stress-rupture testing proceeded similarly to the quasi-static motivation for the extensive theoretical developments of Section to 660 N/s, to minimize creep effects during the initial loading ramp. When the desired load was attained, it was held constant until failure occurred. Strain data were collected throughout the (I Materio test. Some tests were stopped after specified times to determine ti The material system examined here is an oxide/oxide CMc that remaining strengths. As a test of remaining strength, the load was was produced by MTl. This material consists of Nextel 610 first returned to zero and then the specimen was ramped to failure fibers(99% Al,O3)aligned in a unidirectional configuration and at 180 N/s embedded in an alumina-yttria (Al,O,/Y2O3)matrix with gitive carbon interface. The nominal fiber volume fraction is (3) Experimental Results and Discussion 51%, and the overall composite porosity is 19%. These materials (A Virgin Specimens: Polished sections of several unidirec- have no fiber/matrix bond; this is accomplished by first coating the tional panels were examined using scanning electron microscopy fibers with a thin(80-100 nm) layer of carbon through an (SEM), and matrix cracks with a mean spacing of 40 um were immiscible-liquid coating process and then, after the matrix has visible, as shown in Fig. 1(a). The cracks likely formed to relieve been added, oxidizing the carbon to leave a small gap between the the stresses caused by the volume changes that occurred during the fibers and the matrix. This gap is intended to be large enough to sol-gel process and any thermal expansion mismatch
and load, and the effects of fiber pullout, within the wellestablished framework of the global load sharing (GLS) model.4 The model predicts interrelated phenomena of tertiary creep, remaining composite strength, and rupture life, all of which are dependent critically on the underlying fiber-strength degradation. The constituent material parameters required to predict the deformation and lifetime predictions can be obtained independent of stress-rupture testing through quasi-static tension tests and tests on the individual composite constituents. To complement and validate the model, an experimental study of the stress-rupture life, creep deformation, and the associated damage modes for a unidirectional Nextel™ 610 fiber/alumina–yttria matrix CMC (manufactured by McDermott Technologies, Inc. (MTI), Lynchburg, VA) with a fugitive carbon interface has been conducted. This oxide/oxide CMC system should be unaffected by the oxidation, and the unidirectional configuration permits direct comparison with the model. Using literature data for the fiber-strength degradation, the model predicts lifetimes that are two orders of magnitude shorter than that measured. When one parameter, the fiber-degradation rate constant, is fit to the experimental results, the trends in composite lifetime with stress and temperature are well-matched. Then, the model also predicts tertiary creep rates and remaining strength versus time in very good agreement with the experimental data. The measured statistical scatter in the failure times can also be correlated with the scatter in the initial composite strengths using the model. The success of the model in simultaneously predicting several different features that are associated with the deformation and failure, despite the need for a fitting parameter, demonstrates the power of such a micromechanically based approach. An alternative assumption for the fiber-degradation mechanism, which involves two flaw populations, results in improved lifetime predictions at high temperature but poorer residual strength and tertiary creep predictions. The implications of the extreme sensitivity of the rupture life to the precise mechanisms of fiber-strength degradation, and/or the inadequacy of the ex situ fiber-strength degradation to the in situ behavior, is an important issue of discussion. The remainder of this paper is organized as follows. We begin, in Section II, with a description of the experimental techniques and results and a comparison of the measured rupture lives to the LM and MG models; their inadequacy motivates the subsequent model development. In Section III, the analytic model for the fiberdominated stress rupture of composites is developed. In Section IV, the experimental data are analyzed and compared with the predictions of the model. In Section V, we discuss our results further, address important issues that this work raises, and outline how the present model can be used with structural design models for CMC components. II. Experimental Details, Results, and Predictions of Engineering Models We begin our discussion with the experimental results and comparisons to the Larson–Miller (LM) and Monkman–Grant (MG) engineering models to demonstrate that such approaches to life predictions are generally inadequate. This provides significant motivation for the extensive theoretical developments of Section III. (1) Material System The material system examined here is an oxide/oxide CMC that was produced by MTI. This material consists of Nextel™ 610 fibers (99% Al2O3) aligned in a unidirectional configuration and embedded in an alumina–yttria (Al2O3/Y2O3) matrix with a fugitive carbon interface. The nominal fiber volume fraction is 51%, and the overall composite porosity is 19%. These materials have no fiber/matrix bond; this is accomplished by first coating the fibers with a thin (80–100 nm) layer of carbon through an immiscible-liquid coating process and then, after the matrix has been added, oxidizing the carbon to leave a small gap between the fibers and the matrix. This gap is intended to be large enough to prevent large-scale sintering of the materials at elevated temperatures but small enough to permit stress transfer between the constituents by frictional shear stress. The carbon also serves to protect the fiber from any adverse chemical reactions during the processing of the Al2O3/Y2O3 matrix. To create the matrix, a slurry of Al2O3 powder first was pressure-cast into the fiber preform. Next, a sol of Y2O3 particles was infiltrated into the preform, and the preform was dried at 700°C. After a few infiltration and drying steps, the part was fired at 1100°C for 1 h. Then, the infiltration/drying/firing cycle was repeated until the desired density was attained. For these materials, processing was halted when the composite porosity was 20%, which typically required 4–6 cycles. The Y2O3 reacts with the Al2O3 during the firing cycle to create AlYO3 and Y3Al5O12; hence, the exact composition of the matrix was not determined. (2) Mechanical Testing The unidirectional material was used for quasi-static and stressrupture testing at three temperatures: 23°, 950°, and 1050°C. Quasi-static testing was performed using a test frame (Model 880, MTS Systems, Eden Prairie, MN) with a controller (Model 458, MTS Systems). Specimens were tabbed with 0.020 in. fully annealed aluminum tabs, to prevent damage from gripping. The tab was placed around the end of the specimen, and the specimen was placed in the MTS test-frame grip. The pressure of the grip plastically deformed the aluminum to “fit” the specimen, and no adhesive was used. Grip pressures were maintained at 0.7 MPa. The tests were run under load control, at a rate of 180 N/s. Strain at room temperature was measured with an extensometer (Model 632-11B, MTS Systems). Specimen alignment was maintained through a fixture at the grips. A compact oven was used for the tests that were conducted at elevated temperature. The oven had four SiC resistance elements (Norton Advanced Ceramics, Worcester, MA) that heated the specimen. The oven shell was stainless steel and had nominal dimensions of 3.5 in. 3 in. 3 in. Fused-silica insulation (Cotronics Corp., Brooklyn, NY) lined the inside of the oven, which reduced the nominal interior dimensions to 2.5 in. 2 in. 2 in. Two temperature controllers (Model 818S, Eurotherm, Reston, VA) controlled the SiC heating elements: one for the two upper elements, and one for the two lower elements. Each controller received input from a type R (platinum/platinum– rhodium (Pt/Pt-Rh)) thermocouple. The oven shell was surrounded by alumina-fabric insulation, for efficient heating and to help maintain a constant temperature. A heat shield that was attached to the oven held an extensometer (Model 621-51E, MTS Systems). The extensometer measured strain according to the deflection of two 5 in. Al2O3 rods that pass through the oven shell and contact the specimen. The entire assembly was attached to the test frame at one of the posts. The extensometer, the heat shield, and the MTS grips were cooled by water. For high-temperature testing, the temperature was ramped at a rate of 33°C/s to the desired test temperature. Then, the temperature was held constant for 10 min before the test began, to ensure thermal equilibrium. The stress-rupture testing proceeded similarly to the quasi-static testing. However, the load rate for the stress-rupture tests was set to 660 N/s, to minimize creep effects during the initial loading ramp. When the desired load was attained, it was held constant until failure occurred. Strain data were collected throughout the test. Some tests were stopped after specified times to determine the remaining strengths. As a test of remaining strength, the load was first returned to zero and then the specimen was ramped to failure at 180 N/s. (3) Experimental Results and Discussion (A) Virgin Specimens: Polished sections of several unidirectional panels were examined using scanning electron microscopy (SEM), and matrix cracks with a mean spacing of 40 m were visible, as shown in Fig. 1(a). The cracks likely formed to relieve the stresses caused by the volume changes that occurred during the sol–gel process and any thermal expansion mismatch. 1352 Journal of the American Ceramic Society—Halverson and Curtin Vol. 85, No. 6
June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1353 Lku 2 (b) Fig. 1. Matrix cracking in(a) a virgin specimen and (b)a specimen tested in stress rupture (B) Quasistatic and Stress-Rupture Results: Stress-strain shown in Fig. 1(b), unchanged from the virgin material. The curves for the quasi-static tension tests are shown in Fig. 2, and fiber-failure surfaces did not differ significantly in appearance their characteristic features are listed in Table I. There is a general from those tested in quasi-static tension. The stress-rupture life trend toward decreasing strength and modulus and increasing time and strain-rate data will be presented below, within the failure strain with increasing temperature. The nonmonotonic context of traditional engineering models for rupture, and again in trends are believed to be a result of specimen-to-specimen vari- Section ability, most likely a consequence of the experimental nature of the manufacturing process. Unload/reload tests have been performed (4 Predictions of Rupture Using Engineering Models at various applied loads to obtain hysteresis loops Here, we examine whether two engineering approaches tha During quasi-static testing, longitudinal splits were observed have been used in recent literature-the Larson-Miller(LM)and Iso, failure was accompanied by disintegration of the matrix near Monkman-Grant(MG)models noted in the introduction--can be the(presumed) failure plane, probably as a result of the high used to assess the behavior of the composite from the behavior of matrix porosity. Hence, fiber-pullout measurements could not be the constituent fibers accurately performed. However, observation of the failure surface indicated In regard to fiber composites, it has been suggested that the Lm that fiber fractures were not confined to one plane of the compos- te. so that cracks were deflected at the fiber/matrix interface plot of applied stress versus @(see Eq. (1) for the composite Amination of the polished edges of tested specimens demon- should be identical to that for the fibers at an appropriate stress This method is thought to be applicable when the matrix has beer strated that the matrix-crack spacing on completion of a tensile test fully cracked, so that the fibers can be assumed to carry the entire was identical to that in virgin specimens(40 um). Typical fiber failure surfaces were smooth, with no discernable fracture origin plied stress along their entire length. The LM data for the Some fiber-fracture surfaces at room temperature demonstrated derived from our stress-rupture data on the composite system mirrors: howe d temperatures are shown in Fig. 3 fibers showing was too low for accurate analysis. Composite and fiber-fracture surfaces of specimens tested at higher temperatures agreement is poor, particularly considering that the plot logarithmic time scale were similar in appearance to those tested at low temperature hat rupture damage is drive ings observed on the specimen edge were, again, -40 Hm, as stracin rate re and the nue ture lifetime n, i, fronm t. e s,e logt1+klog∈=D (3) 23°C where k and D again are constants(k s 1). Use of the measured 950°C 1050°C composite e value to predict the composite lifetime, with th constants k and d obtained from single fibers, has been proposed for composites wherein the matrix has not cracked and does carry load. For similar reasons, it should also apply when the matrix is fully cracked and does not carry any load, so that both creep and rupture are strongly fiber-dominated. Figure 4 shows a log-log 200 plot of the rupture lifetime versus the steady-state creep rate obtained for the single Nextel 610 fibers" and for the composites studied here. At a given temperature, the composite data do show a linear relationship that is consistent with Eq. (3), but the slope is substantially different from that for the individual fibers. Further more, at different temperatures, the single-fiber data almost fall 00.050.10.150.20.250.30.3504 along a common line, which indicates that Eq. (3)applies, with k Strain(%) and D independent of temperature, whereas the composite data are shifted by substantial factors, which suggests that k is independent Fig. 2. Measured quasi-static stress-strain curves at of temperature but D is strongly dependent on temperature. Such black lines)and fits at elevated temperatures using behavior has been observed previously in monolithic ceramics (colored lines)to yield in situ fast-fracture characte such as silicon nitride(see, for example, Ferber and Jenkins, 。=1060MPat950°cand1000 MPa at1050°C(cu by 0.1% Luecke er al., and Menon et al. ) At a fixed temperature, the for clarity) MG correlation between creep and rupture does apply to th
(B) Quasistatic and Stress-Rupture Results: Stress–strain curves for the quasi-static tension tests are shown in Fig. 2, and their characteristic features are listed in Table I. There is a general trend toward decreasing strength and modulus and increasing failure strain with increasing temperature. The nonmonotonic trends are believed to be a result of specimen-to-specimen variability, most likely a consequence of the experimental nature of the manufacturing process. Unload/reload tests have been performed at various applied loads to obtain hysteresis loops. During quasi-static testing, longitudinal splits were observed. Also, failure was accompanied by disintegration of the matrix near the (presumed) failure plane, probably as a result of the high matrix porosity. Hence, fiber-pullout measurements could not be performed. However, observation of the failure surface indicated that fiber fractures were not confined to one plane of the composite, so that cracks were deflected at the fiber/matrix interface. Examination of the polished edges of tested specimens demonstrated that the matrix-crack spacing on completion of a tensile test was identical to that in virgin specimens (40 m). Typical fiber failure surfaces were smooth, with no discernable fracture origin. Some fiber-fracture surfaces at room temperature demonstrated evidence of fracture mirrors; however, the proportion of such fibers showing was too low for accurate analysis. Composite and fiber-fracture surfaces of specimens tested at higher temperatures were similar in appearance to those tested at low temperature. Under stress-rupture loading conditions, the matrix-crack spacings observed on the specimen edge were, again, 40 m, as shown in Fig. 1(b), unchanged from the virgin material. The fiber-failure surfaces did not differ significantly in appearance from those tested in quasi-static tension. The stress-rupture lifetime and strain-rate data will be presented below, within the context of traditional engineering models for rupture, and again in Section IV. (4) Predictions of Rupture Using Engineering Models Here, we examine whether two engineering approaches that have been used in recent literature8 —the Larson–Miller (LM) and Monkman–Grant (MG) models noted in the introduction—can be used to assess the behavior of the composite from the behavior of the constituent fibers accurately. In regard to fiber composites, it has been suggested that the LM plot of applied stress versus Q (see Eq. (1)) for the composite should be identical to that for the fibers at an appropriate stress.8 This method is thought to be applicable when the matrix has been fully cracked, so that the fibers can be assumed to carry the entire applied stress along their entire length. The LM data for the Nextel™ fibers, as determined by Yun et al., 34 and the LM plots derived from our stress-rupture data on the composite system over a range of loads and temperatures are shown in Fig. 3. The agreement is poor, particularly considering that the plot has a logarithmic time scale. The MG approach envisions that rupture damage is driven by creep deformation. The MG relationship between the steady-state strain rate ˙ and the rupture lifetime tr is, from Eq. (2), log tr k log ˙ D (3) where k and D again are constants (k 1). Use of the measured composite ˙ value to predict the composite lifetime, with the constants k and D obtained from single fibers, has been proposed for composites wherein the matrix has not cracked and does carry load.8 For similar reasons, it should also apply when the matrix is fully cracked and does not carry any load, so that both creep and rupture are strongly fiber-dominated. Figure 4 shows a log–log plot of the rupture lifetime versus the steady-state creep rate obtained for the single Nextel™ 610 fibers8 and for the composites studied here. At a given temperature, the composite data do show a linear relationship that is consistent with Eq. (3), but the slope is substantially different from that for the individual fibers. Furthermore, at different temperatures, the single-fiber data almost fall along a common line, which indicates that Eq. (3) applies, with k and D independent of temperature, whereas the composite data are shifted by substantial factors, which suggests that k is independent of temperature but D is strongly dependent on temperature. Such behavior has been observed previously in monolithic ceramics such as silicon nitride (see, for example, Ferber and Jenkins,35 Luecke et al.,36 and Menon et al. 37). At a fixed temperature, the MG correlation between creep and rupture does apply to the Fig. 1. Matrix cracking in (a) a virgin specimen and (b) a specimen tested in stress rupture. Fig. 2. Measured quasi-static stress–strain curves at several temperatures (black lines) and fits at elevated temperatures using the present model (colored lines) to yield in situ fast-fracture characteristic fiber strengths of c 1060 MPa at 950°C and 1000 MPa at 1050°C (curves offset by 0.1% for clarity). June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1353
1354 urnal of the American Ceramic Society-Halverson and Curtin Vol. 85. No 6 Table I. Quasi-static Tension Test Results Temperature(C) Modulus(GPa) Strength(MPa) Failure strain (% Number of test 392±63 0.177±0.034 189±6 342±5 0.194±0.037 370±54 0.214±0.018 197±14 305±28 0.206±0240 composite system: a measurement of creep can be used to lo-concepts that are known to be inaccurate. A lack of correlation predict" the failure time(although Fig. 4 shows that order-of- between the composite and single-fiber data also exists, therefore, magnitude fluctuations in life exist at a fixed creep rate). However, we do not advocate the general use of these approaches to describe the shift in the mG plot with temperature limits such"predictions the high-temperature deformation and failure of ceramic compos- to each temperature of interest. Furthermore, the behavior does not ites. These facts further motivate the consideration of correlate with the single-fiber data, so that fiber data alone are chanical models for rupture insufficient to predict composite life The MG approach suggests a coupling of creep and rupture. However, a relationship that follows Eq. (3)is obtained when both IlL. Micromechanical Model of Composite Stress Rupture creep and rupture have independent power-law dependencies on the applied stress. Specifically, if the creep rate follows the relation The composite degradation and stress-rupture model proposed E=Ao while the rupture lifetime follows the relation (,= Bo here are based on an analysis of the stochastic accumulation of then one can obtain the MG form precisely, as fiber failure in the material. The matrix and the fiber/matrix interface determine, through micromechanical models, the stress state in the fibers, which governs the rate of fiber degradation, as log 4+I= log e= log(BA shown schematically in Fig. 5. Here, we begin with an analysis of the stress state on a typical fiber in the composite and th independent of any physical relationship between the two mecha- associated fiber degradation without considering the effects of nisms. If the two mechanisms are physically different, then the previously damaged fibers on the stress state. Subsequently, th "constant"log(BA)=D should be strongly dependent on temper behavior of the collection of interacting fibers in the composite is ature, despite the fact that the slope r/e is independent of temperature considered, which leads to the full model for composite damage but the rate prefactors A and B should be Armhenius-like with evolution and failure. The general approach encompasses both completely different activation energies; this dependence has been quasi-static and stress-rupture behavior quite naturally, ther observed with our composite data. Thus, the existence of an MG we begin with the quasi-static problem, because it sets the stag "correlation"at a single temperature has no implications for the the subsequent time evolution interdependence of creep and rupture in this composite system. Neither the LM approach nor the MG approach contain under- (1) Fiber Strength, Stress, and degradation lying information about damage state, nor do they provide infor- mation on the remaining strength or any other phenomena that (A) Quasi-static Behavior: Ceramic fibers are brittle materials whose strengths must be described statistically. This description is occur in the composite. Moreover, the general idea of applying commonly accomplished by assuming a single flaw population and single-fiber rupture data directly to explain composite rupture has veral incorrect implications for composite failure and rupture. failure occurring in an increment of a fiber element of length & within failure time of a single fiber at the(arbitrary) ex sifui-tested gauge an incremental stress range of o to o+ &u is given by length lo and, thus, because fiber strength is dependent on gauge length, that rupture life is dependent on the lo value used in the pA,60,8-)= single-fiber experiments. This concept also implies that failure occurs when every fiber is typically broken once within a length lo Here o is the characteristic fiber strength at gauge length lo and in the composite and that the tensile strength is simply the volume m is the Weibull modulus, which describes the statistical distribu- fraction of fibers multiplied by the typical fiber strength at length tion of the strength around oo. Following previous analyses, , 38the cumulative probability of fiber failure in a length 21, loaded at applied stress app, where gapp and matrix damage cause longitudinal-fiber stress profile o(), is given by q(amm 1) When o(=) is a constant value(app), Eq.(6) reduces to the ◆ Experiment well-known Weibull expression q(amm 1) In a unidirectional composite with a single matrix crack and a debonded, sliding fiber/matrix interface described by an interfacial Q=T(ogtr +F) shear stress T, the fiber stress is dependent on position and Eq (6) must be used to determine failure. The stress on a fiber near an isolated matrix crack located at the longitudinal position ==0 tress-rupture data and (+) measured composite st under a remote applied stress uarp is accurately modeled by a shear-lag model as follows. At the matrix-crack plane, the entire
composite system: a measurement of creep can be used to “predict” the failure time (although Fig. 4 shows that order-ofmagnitude fluctuations in life exist at a fixed creep rate). However, the shift in the MG plot with temperature limits such “predictions” to each temperature of interest. Furthermore, the behavior does not correlate with the single-fiber data, so that fiber data alone are insufficient to predict composite life. The MG approach suggests a coupling of creep and rupture. However, a relationship that follows Eq. (3) is obtained when both creep and rupture have independent power-law dependencies on the applied stress. Specifically, if the creep rate follows the relation ˙ A c while the rupture lifetime follows the relation tr B r , then one can obtain the MG form precisely, as log tr r c log ˙ log BAr/c (4) independent of any physical relationship between the two mechanisms. If the two mechanisms are physically different, then the “constant” log (BAr/c ) D should be strongly dependent on temperature, despite the fact that the slope r/c is independent of temperature but the rate prefactors A and B should be Arrhenius-like with completely different activation energies; this dependence has been observed with our composite data. Thus, the existence of an MG “correlation” at a single temperature has no implications for the interdependence of creep and rupture in this composite system. Neither the LM approach nor the MG approach contain underlying information about damage state, nor do they provide information on the remaining strength or any other phenomena that occur in the composite. Moreover, the general idea of applying single-fiber rupture data directly to explain composite rupture has several incorrect implications for composite failure and rupture. The concept implies that composite rupture occurs at the average failure time of a single fiber at the (arbitrary) ex situ-tested gauge length l0 and, thus, because fiber strength is dependent on gauge length, that rupture life is dependent on the l0 value used in the single-fiber experiments. This concept also implies that failure occurs when every fiber is typically broken once within a length l0 in the composite and that the tensile strength is simply the volume fraction of fibers multiplied by the typical fiber strength at length l0—concepts that are known to be inaccurate. A lack of correlation between the composite and single-fiber data also exists; therefore, we do not advocate the general use of these approaches to describe the high-temperature deformation and failure of ceramic composites. These facts further motivate the consideration of micromechanical models for rupture. III. Micromechanical Model of Composite Stress Rupture The composite degradation and stress-rupture model proposed here are based on an analysis of the stochastic accumulation of fiber failure in the material. The matrix and the fiber/matrix interface determine, through micromechanical models, the stress state in the fibers, which governs the rate of fiber degradation, as shown schematically in Fig. 5. Here, we begin with an analysis of the stress state on a typical fiber in the composite and the associated fiber degradation without considering the effects of previously damaged fibers on the stress state. Subsequently, the behavior of the collection of interacting fibers in the composite is considered, which leads to the full model for composite damage evolution and failure. The general approach encompasses both quasi-static and stress-rupture behavior quite naturally; therefore, we begin with the quasi-static problem, because it sets the stage for the subsequent time evolution. (1) Fiber Strength, Stress, and Degradation (A) Quasi-static Behavior: Ceramic fibers are brittle materials whose strengths must be described statistically. This description is commonly accomplished by assuming a single flaw population and using a two-parameter Weibull model, for which the probability of failure occurring in an increment of a fiber element of length z within an incremental stress range of to is given by pf , , z m m 1 0 m z l0 (5) Here, 0 is the characteristic fiber strength at gauge length l0 and m is the Weibull modulus, which describes the statistical distribution of the strength around 0. Following previous analyses,3,38 the cumulative probability of fiber failure in a length 2l, loaded at applied stress app, where app and matrix damage cause a longitudinal-fiber stress profile (z), is given by q app, l 1 exp 0 app l l m 0l0 z 0 m 1 d z d dz d (6) When (z) is a constant value ( app), Eq. (6) reduces to the well-known Weibull expression: q app, l 1 exp 2l l0 app 0 m (7) In a unidirectional composite with a single matrix crack and a debonded, sliding fiber/matrix interface described by an interfacial shear stress , the fiber stress is dependent on position and Eq. (6) must be used to determine failure. The stress on a fiber near an isolated matrix crack located at the longitudinal position z 0 under a remote applied stress app is accurately modeled by a shear-lag model as follows. At the matrix-crack plane, the entire Fig. 3. Larson–Miller plot (applied stress app versus parameter Q) for (—) single-fiber stress-rupture data and () measured composite stressrupture data. Table I. Quasi-static Tension Test Results Temperature (°C) Modulus (GPa) Strength (MPa) Failure strain (%) Number of tests 23 245 36 392 63 0.177 0.034 4 950 189 6 342 55 0.194 0.037 5 1050 191 16 370 54 0.214 0.018 3 1093 197 14 305 28 0.206 0.240 3 1354 Journal of the American Ceramic Society—Halverson and Curtin Vol. 85, No. 6