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
June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1355 10000000 ◆950°C 1000000 Fiber data Fit to Composite Data 100000 10E08100E07100E-061.00E-051.00E041.00E03 Steady-State Strain Rate(1/s) Fig. 4. Monkman-Grant plot (log(lifetime)versus log(strain rate )for fibers(indicated by dashed lines; data from DiCarlo and Yun)and the composite at various temperatures((◆)950°cand(■)1050°C load is carried by the fiber, thus, the fiber stress is given as capp/, When the fiber breaks, slip along the fiber/matrix interface occurs wherefis the fiber volume fraction. At a distance from the matrix over a fiber slip length, Is, ack, stress is transferred from the fiber to the matrix through the nterface frictional stress as fr(forl≤8) (8) that is equal to the distance at which the stress in Eq( 8)would reach a value of zero if not cut off by the far-field stress. Thus, the fiber stress, as a function of distance from a matrix crack, is given where 8 is the distance over which interfacial slip occurs and is defined ()=7(1-2.)(r0<H<b (12a) r(=) (for 8<E=D (12b) and Es Em, and Ec are the Youngs moduli of the fiber, matrix, and omposite, respectively (with Ec =Er+(1-nEm), and r is the where T=Uanp/. This stress profile and the above-described fiber radius. For 28, the fiber stresses and strains are equal and notations are illustrated in Fig. 5. Under the stress state given in the stress carried by the fiber regains the constant far-field value Eqs.(12), the probability of fiber failure q over a length 2, at the applied stress app follows from Eq(6) (10) q(,4)=1-ex-(m+1)rm(1+m-)(3) Fiber stress profile Matrix Crack matic of time-dependent flaw growth under the spatially varying stress on a fiber around a matrix crack, the crack-growth rate is dependent itial crack size and the stress acting on it
load is carried by the fiber; thus, the fiber stress is given as app/f, where f is the fiber volume fraction. At a distance z from the matrix crack, stress is transferred from the fiber to the matrix through the interface frictional stress as z app f 2z r for z (8) where is the distance over which interfacial slip occurs and is defined as r app 2f 1 f Em Ec (9) and Ef , Em, and Ec are the Young’s moduli of the fiber, matrix, and composite, respectively (with Ec fEf (1 f)Em), and r is the fiber radius. For z , the fiber stresses and strains are equal and the stress carried by the fiber regains the constant far-field value: ff app Ef Ec (10) When the fiber breaks, slip along the fiber/matrix interface occurs over a fiber slip length, ls, ls r app 2f (11) that is equal to the distance at which the stress in Eq. (8) would reach a value of zero if not cut off by the far-field stress. Thus, the fiber stress, as a function of distance z from a matrix crack, is given as z T 1 z ls for 0 z (12a) z app Ef Ec for z (12b) where T app/f. This stress profile and the above-described notations are illustrated in Fig. 5. Under the stress state given in Eqs. (12), the probability of fiber failure q over a length 2ls at the applied stress app follows from Eq. (6) as qT˜, lsT˜ 1 exp 1 m 1T˜ m 1 1 mm 1 (13) Fig. 4. Monkman–Grant plot (log (lifetime) versus log (strain rate)) for fibers (indicated by dashed lines; data from DiCarlo and Yun8 ) and the composite at various temperatures (() 950°C and () 1050°C). Fig. 5. Schematic of time-dependent flaw growth under the spatially varying stress on a fiber around a matrix crack; the crack-growth rate is dependent on both the initial crack size and the stress acting on it. June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1355
1356 urnal of the American Ceramic Society-Halverson and Curtin Vol. 85. No. 6 Here, we have introduced the dimensionless parameters Tand growth of pre-existing flaws in the fiber. The slow-crack-growth as well as the characteristic stress o rate is represented by a Paris law: T (14a) dIaS (14b) where a is the crack size and k is the stress intensity factor; B is the slow-crack-growth exponent and A is a rate constant, each of dooT+D) which generally is dependent on temperat which is a (14c) function of the current stress T, crack size a, and a geometric factor A length 21s is chosen because only fiber breaks within +l, of a K= YTa (19) matrix crack will influence tensile failure of the composite at this matrix crack(see below) lure is determined using the expression K=Klc. Thus, the Depending on the Weibull modulus m and the stiffness of the nsile strength(o, at initial crack size ai) can be related to matrix, as captured by the parameter c, there is a size a(n)and flaw strength o(n)at time r by combining eqs between fiber failure within the linearly decreasing portion of the (19)and integrating, to yield ess field and the It far-field region. For low values of mo failure in eld stress region is negligible and the probability of fiber accurately represented by (=2-c|°dr q(4)=1-exp-(m+门)产 (15) C=I5-1JAYK1-2 (20b) which is the single matrix crack"result that was derived by Thouless and Evans, as well as other researchers. For high values where r is assumed to be constant over time. Inverting Eq (20a) of mo"t, the matrix modulus is small, relative to the fibers (i.e yields the initial strength required to provide a current strength of EC FeD; thus, the length 8 is small and the far-field stress region g(n after the given stress history 1(o)on the flaw is realized very close to the matrix. Then, fiber failure will occur almost equally over the entire region and, in the limit of a=1 (corresponding to a matrix modulus of zero ), the probability of GG)=a(0-2+元pd fiber failure is given as q(,(⑦)=1 Here we have introduced a nondimensional time t and a normal ized strength o, given by which is the result obtained by Curtin for the case of saturated closely spaced matrix cracking. Thus, the value of mo"+ in Eq (22a) (13)ranges from zero to m and represents the influence of the matrix modulus on the probability of fiber failure, reducing to the single matrix crack" and"multiple matrix crack"cases in the G appropriate limits (22b) In real materials, multiple matrix cracks typically exist with some variable spacing. If two adjacent cracks are spaced by a where T is as defined previously in Eq.( 14a) distance i that is less than a fiber slip length l, then eqs If some stress history T(o)is applied to a fiber, the probability apply over a length x/2. In this case, etermine the that the fiber will fail is the probability that the initial strength of fiber failure around each matrix crack only within of the fiber is less than the initial strength given by Eq. (21) i/2, which corresponds to setting the limits of the position integral That is, for a fiber element of length 8= under some stress in Eq (6)to /=+/2. Then, the cumulative probability of failure history TO), the probability of failure over the applied stress Is increment from o(n) to o()+ So(n) is obtained by substituting Eq(21)into Eq (5) T p(t),6o(1),8-)= 86(l) ×{1+a|(m+1)-1-m(1-a) Following arguments identical to those used in the quasi-static case, the probability of failure of a fiber, with respect to time, under the stress profile of Eqs. (12)is then given by (17a) q(1,120 , xP(r)°dr for5<8(m)(17b) m(B-2) These results will be used below to determine the quasi-static +a+28-2+at(r)d composite damage evolution and quasi-static stress-strain curve up to failure. Equation (17) is the first main result of our analysis. B) Time-Dependent Behavior: We now assume that the where a change of variables from to x =1-Ehls has been fiber strength degrades with time, because of the slow crack performed. Figure 6 shows the ratio of mean fiber lifetimes for
Here, we have introduced the dimensionless parameters T˜ and , as well as the characteristic stress c: 3 T˜ T c (14a) ff z 0 f Ef Ec (14b) c 0 ml0 r 1/m 1 (14c) A length 2ls is chosen because only fiber breaks within ls of a matrix crack will influence tensile failure of the composite at this matrix crack (see below). Depending on the Weibull modulus m and the stiffness of the matrix, as captured by the parameter , there is a competition between fiber failure within the linearly decreasing portion of the stress field and the constant far-field region. For low values of mm 1 , failure in the far-field stress region is negligible and the probability of fiber failure is accurately represented by qT˜, lsT˜ 1 exp 1 m 1T˜ m 1 (15) which is the “single matrix crack” result that was derived by Thouless and Evans,38 as well as other researchers. For high values of mm 1 , the matrix modulus is small, relative to the fibers (i.e., Ec fEf ); thus, the length is small and the far-field stress region is realized very close to the matrix. Then, fiber failure will occur almost equally over the entire region and, in the limit of 1 (corresponding to a matrix modulus of zero), the probability of fiber failure is given as qT˜, lsT˜ 1 exp T˜ m 1 (16) which is the result obtained by Curtin3 for the case of saturated, closely spaced matrix cracking. Thus, the value of mm 1 in Eq. (13) ranges from zero to m and represents the influence of the matrix modulus on the probability of fiber failure, reducing to the “single matrix crack” and “multiple matrix crack” cases in the appropriate limits. In real materials, multiple matrix cracks typically exist with some variable spacing. If two adjacent cracks are spaced by a distance x that is less than a fiber slip length ls, then Eqs. (12) only apply over a length x/2. In this case, we determine the probability of fiber failure around each matrix crack only within the region x/2, which corresponds to setting the limits of the position integral in Eq. (6) to l x/2. Then, the cumulative probability of failure is qT˜, x 2 1 exp 1 m 1T˜ m 1 1 m m 1 x 2ls 1 m1 for app x 2 ls (17a) qT˜, x 2 1 exp 1 m 1T˜ m 1 1 1 x 2ls m 1 for x 2 app (17b) These results will be used below to determine the quasi-static composite damage evolution and quasi-static stress–strain curve up to failure. Equation (17) is the first main result of our analysis. (B) Time-Dependent Behavior: We now assume that the fiber strength degrades with time, because of the slow crack growth of pre-existing flaws in the fiber. The slow-crack-growth rate is represented by a Paris law: da dt AK (18) where a is the crack size and K is the stress intensity factor; is the slow-crack-growth exponent and A is a rate constant, each of which generally is dependent on temperature. K, which is a function of the current stress T, crack size a, and a geometric factor Y, is given by K YTa1/ 2 (19) Flaw failure is determined using the expression K KIc. Thus, the initial tensile strength ( i at initial crack size ai ) can be related to the flaw size a(t) and flaw strength (t) at time t by combining Eqs. (18) and (19) and integrating, to yield t i 2 C 0 t Tt dt 1/ 2 (20a) C 2 1AY2 KIc 2 (20b) where Y is assumed to be constant over time. Inverting Eq. (20a) yields the initial strength required to provide a current strength of (t) after the given stress history T(t) on the flaw: ˜ i˜t ˜t 2 0 ˜t T˜t ˜ dt ˜ 1/ 2 (21) Here, we have introduced a nondimensional time t ˜ and a normalized strength ˜, given by t ˜ tC c 2 (22a) and ˜ c (22b) where T˜ is as defined previously in Eq. (14a). If some stress history T(t) is applied to a fiber, the probability that the fiber will fail is the probability that the initial strength of the fiber is less than the initial strength given by Eq. (21). That is, for a fiber element of length z under some stress history T(t), the probability of failure over the applied stress increment from (t) to (t) (t) is obtained by substituting Eq. (21) into Eq. (5): pf t, t, z m i t m 1 l0 0 m z i t (23) Following arguments identical to those used in the quasi-static case, the probability of failure of a fiber, with respect to time, under the stress profile of Eqs. (12) is then given by qT˜, ls, t ˜ 1 exp T˜ 1 x 2 T˜ 2 0 t xT˜t dt m/ 2 dx m 1 T˜ 2 0 t 2 T˜t dt m/ 2 (24) where a change of variables from z to x 1 z/ls has been performed. Figure 6 shows the ratio of mean fiber lifetimes for 1356 Journal of the American Ceramic Society—Halverson and Curtin Vol. 85, No. 6
June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1357 o =0(high matrix modulus)to that for a= I(zero matrix As an example, for narrowly spaced matrix cracks, where x/2< modulus) under a typical constant load (T= 0.5) for various values 5, and using the normalized variables as before, the damage of m and B. For a=0, the stresses on the fiber are less severe than parameter is those for the case of a =I; hence, the fiber lives longer. For moderately high values of B, the modest stress carried by the matrix can lead to increases in the fiber lifetime of more than two orders of T magnitude. This large difference motivates a detailed analysis of in situ stresses and fiber degradation to obtain realistic predictions of m/(B-2) composite lifetime As for the quasi-static case, the effect of a finite matrix-crack x-27P-2+x70)d acing i can be introduced by appropriately changing the limits of 1-[024 tegration in Eq.(24)to yield m+ I (26) 1-exp In this expression, the subscripts"I and"ff have been added to differentiate between the two Weibull moduli and the normalized nm'iB-2 -219-2+xTr)dr stresses in rupture and in fast fracture for the two different types of flaws. In subsequent discussions, this model is termed the"two- flaw-population"model, in contrast to the"one-flaw-population model"described in Sections Ill()(A) and(B) a3-(1-a)r (2 Composite Behavior In a well-designed ceramic composite, broken fibers transfer stress to other intact fibers through friction at the fiber/matrix nterface, increasing the load on the remaining intact fibers. Here the equations for individual fiber failure discussed in Section Ill() are combined with the global load sharing(GLS) law, which is widely used to model CMCs, to yield predictions of composite behavior. In a col mechanical equilibrium must be main- for 8<-< tained at all planes pe ular to the applied loa at a matrix-crack plane, where the matrix carries no load, is a load balance between the applied stress and the stresses carried by the broken and intact fibers. In GLs, the stress previously carried by broken fibers across any given transverse plane is shared equally among all intact fibers in that plane and, thus, equilibrium is given mB-2) (1-Pe)T+pork x-7-2+|x°nr) adr'dx 1-[2) where P is the fraction of broken or sliding fibers at th matrix-crack plane, o roken their average load-carrying capacity, and T the stress on the remaining unbroken fibers The load carried for5<8)(25b) at a matrix crack by a single broken fiber is dependent on the distance- from the fiber break to the matrix crack. if the matrix This result is used below in examining overall composite failure as crack is within a fiber slip length ls of the break a function of matrix-crack spacing. Equation(25)is the second main result of our analysis (C) Multiple-Flaww Populations: It is conceivable that the o hoken=(for E=<1) flaws in a fiber that govern fast fracture are not those that govern stress rupture. For instance, creep damage may nucleate Fiber breaks that extend farther than 1 carry the full stress T and, entirely new flaws. Or, flaws that are strong under fast fracture thus, are unbroken, relative to the matrix crack of interest; this is may be susceptible to growth under stress-rupture conditions the reason why our analysis of fiber damage in previous sections while the weaker flaws that drive fast fracture do not grow was confined to the region +/, around the matrix crack. Both situations can be accommodated by considering two Now consider a"central matrix crack, at which equilibrium populations of fiber flaws: one population of weak, but time- (Eqs.(27) and (28) is to be satisfied, and the nearby surrounding independent, flaws that cause tensile fast fracture and a second matrix cracks. For a crack spacing of x, the number of surrounding matrix cracks that affect the stresses on the central matrix crack population that is initially strong enough not to be activated under fast fracture but that weaken sufficiently with time to through fiber breaks is given as 21 / Around each of these nearby matrix crack become the dominant population driving the fiber rupture. Here, a region of length +i/2 within which the probability of having a broken fiber is given by Eqs.(17),(25),or we consider such a two-flaw-population model of static and (26), depending on the loading condition. These fiber breaks are growing flaws, the former described by time-independent "fas distributed symmetrically around the matrix-crack plane, there- fracture"Weibull parameters(ocfr mer)and the latter described fore, the average distance from the central matrix crack to the by"rupture" Weibull parameters m)and slow-crack broken fibers around any other matrix crack is precisely the growth kinetic parameters C and B. Although the slow distance between the two matrix cracks. The nearby matrix cracks growth model is still being used, at the times where these flaws themselves are assumed to be evenly spaced on both sides of the become relevant to rupture the model is very similar to the central matrix crack, so that the average distance from the central Coleman model. The overall fiber failure probability is matrix crack to the other matrix cracks within /, is simply 12 simply given as q=1-(1-q1(1-42), where q and q2 Thus, the average load carried by broken fibers, broken, is the the failure probabilities for each flaw population, respectiv average of the loads carried by all the broken fibers within +l
0 (high matrix modulus) to that for 1 (zero matrix modulus) under a typical constant load (T˜ 0.5) for various values of m and . For 0, the stresses on the fiber are less severe than those for the case of 1; hence, the fiber lives longer. For moderately high values of , the modest stress carried by the matrix can lead to increases in the fiber lifetime of more than two orders of magnitude. This large difference motivates a detailed analysis of in situ stresses and fiber degradation to obtain realistic predictions of composite lifetime. As for the quasi-static case, the effect of a finite matrix-crack spacing x can be introduced by appropriately changing the limits of integration in Eq. (24) to yield q T˜, x 2 , t ˜ 1 exp T˜ 1 x 2 T˜ 2 0 t xT˜t dt m/ 2 dx m x r c 1 T˜ T˜ 2 0 t 2 T˜t dt m/ 2 for x 2 ls (25a) q T˜, x 2 , t ˜ 1 exp T˜ 1 x/2ls 1 x 2 T˜ 2 0 t xT˜t dt m/ 2 dx for x 2 (25b) This result is used below in examining overall composite failure as a function of matrix-crack spacing. Equation (25) is the second main result of our analysis. (C) Multiple-Flaw Populations: It is conceivable that the flaws in a fiber that govern fast fracture are not those that govern stress rupture. For instance, creep damage may nucleate entirely new flaws. Or, flaws that are strong under fast fracture may be susceptible to growth under stress-rupture conditions while the weaker flaws that drive fast fracture do not grow. Both situations can be accommodated by considering two populations of fiber flaws: one population of weak, but timeindependent, flaws that cause tensile fast fracture and a second population that is initially strong enough not to be activated under fast fracture but that weaken sufficiently with time to become the dominant population driving the fiber rupture. Here, we consider such a two-flaw-population model of static and growing flaws, the former described by time-independent “fastfracture” Weibull parameters ( cff, mff) and the latter described by “rupture” Weibull parameters ( cr, mr ) and slow-crackgrowth kinetic parameters C and . Although the slow-crackgrowth model is still being used, at the times where these flaws become relevant to rupture the model is very similar to the Coleman model. The overall fiber failure probability is then simply given as q 1 (1 q1)(1 q2), where q1 and q2 are the failure probabilities for each flaw population, respectively. As an example, for narrowly spaced matrix cracks, where x/2 , and using the normalized variables as before, the damage parameter is q T˜, x 2 , t ˜ 1 exp T˜r 1 x/2ls 1 x 2 T˜r 2 0 t xT˜rt dt mr/ 2 dx 1 mff 1T˜ff mff 1 1 1 x 2ls mff 1 (26) In this expression, the subscripts “r” and “ff” have been added to differentiate between the two Weibull moduli and the normalized stresses in rupture and in fast fracture for the two different types of flaws. In subsequent discussions, this model is termed the “twoflaw-population” model, in contrast to the “one-flaw-population model” described in Sections III(1)(A) and (B). (2) Composite Behavior In a well-designed ceramic composite, broken fibers transfer stress to other intact fibers through friction at the fiber/matrix interface, increasing the load on the remaining intact fibers. Here, the equations for individual fiber failure discussed in Section III(1) are combined with the global load sharing (GLS) law, which is widely used to model CMCs, to yield predictions of composite behavior. In a composite, mechanical equilibrium must be maintained at all planes perpendicular to the applied load. Equilibrium at a matrix-crack plane, where the matrix carries no load, is a load balance between the applied stress and the stresses carried by the broken and intact fibers. In GLS, the stress previously carried by broken fibers across any given transverse plane is shared equally among all intact fibers in that plane and, thus, equilibrium is given by app f 1 pfT pf broken (27) where pf is the fraction of broken or sliding fibers at the matrix-crack plane, broken their average load-carrying capacity, and T the stress on the remaining unbroken fibers. The load carried at a matrix crack by a single broken fiber is dependent on the distance z from the fiber break to the matrix crack, if the matrix crack is within a fiber slip length ls of the break: broken 2z r for z ls (28) Fiber breaks that extend farther than ls carry the full stress T and, thus, are unbroken, relative to the matrix crack of interest; this is the reason why our analysis of fiber damage in previous sections was confined to the region ls around the matrix crack. Now consider a “central” matrix crack, at which equilibrium (Eqs. (27) and (28)) is to be satisfied, and the nearby surrounding matrix cracks. For a crack spacing of x, the number of surrounding matrix cracks that affect the stresses on the central matrix crack through fiber breaks is given as 2ls/x. Around each of these nearby matrix cracks is a region of length x/2 within which the probability of having a broken fiber is given by Eqs. (17), (25), or (26), depending on the loading condition. These fiber breaks are distributed symmetrically around the matrix-crack plane; therefore, the average distance from the central matrix crack to the broken fibers around any other matrix crack is precisely the distance between the two matrix cracks. The nearby matrix cracks themselves are assumed to be evenly spaced on both sides of the central matrix crack, so that the average distance from the central matrix crack to the other matrix cracks within ls is simply ls/2. Thus, the average load carried by broken fibers, broken, is the average of the loads carried by all the broken fibers within ls June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1357
1358 urnal of the American Ceramic Society-Halverson and Curtin Vol. 85. No 6 5 100 B Fig. 6. Ratio of fiber lifetimes under a composite-like stress field(ta= OVida= 1), showing sensitivity to the matrix modulus. a =0 corresponds to a high matrix modulus(the matrix carries load away from the matrix crack), a= I corresponds to a low matrix modulus ( fibers carry the entire load (where 1= rT/(2T) of the central matrix crack, which, by the To determine composite lifetime, the initial state above-described arguments, is simply given as [(2/r)(/2)](27 time i=0 is determined using Eqs. (25)or(26)and (30). Then, x)q(Tx/2, 1), where the slip length 1 is evaluated at the stress T, time is advanced by some small time increment Ar and the fibers which acts an effective d stress on the fibers. These consid- degrade and fail according to Eq (25)or(26), yielding a new valuc of g. Then, Eq (30) 2r(L)0 2/l process is repeated until no solutions exist. In addition, the composite strength after any time I" before r*and increasing the applied stress-and, hence, the value of Tin (29)Eqs.(25)or(26) composite failure occurs. where (L)o is the length of fiber pullout just from the central crack (3)Creep Effects plane. The first term on the right-hand side represents the pull-out Creep deformation ter the distribution load from the central matrix crack; the second term represents the fibers and matrix with therefore. it has pull-out load due to the other 21/ matrix cracks; the third term stress-rupture lifetime, in the absence s some effect t creep- presents the stress carried by the fibers that remain intact. This damage mechanisms In the absence of creep, the effects of a finite expression is quite general, holding for both quasi-static and crack spacing and a nonzero matrix elastic modulus have been time-dependent conditions; it is the probability of failure q that is characterized by the parameter fE Ee, which determines the far-field fiber and matrix stresses. When the fibers and matrix dence. Because of the complicated nature of the central matri behave in a viscoelastic manner. however. the far-field fiber and crack pull-out term((L)o), Eq.(29)generally must be solved matrix stresses are determined by the equality of the strain rates, numerically. If this term is ignored-which is an assumption that not the absolute strains. That is is accurate, except when both m is small and i is quite large-then Eq(29)reduces to (32a) (30) Bum=r+ Hof (32b) where 8. is the characteristic length where the fiber and matrix strain rates are assumed to follow power-law creep relationships that are described by the parameters (31)(H, p)and(B, n), respectively. In the far-field regions, the fiber and matrix stresses must still satisfy equilibrium via the relation Equation(30) was first derived by Curtin et al. and used, along fo+(1-no with Eq.(17b), to predict quasi-static(time-independent)stress- strain deformation Equation(30) is the third main result of our Substitution of Eqs. (32)into Eq (33) yields the time evolution of analysis. the far-field fiber stress. Equation (30) relates the known applied stress to the effective stress T that acts on the unbroken fibers and the extent of d (1-nEe damage q Damage evolution in the composite corresponds to the d f Hop evolution of T and 4, through the solution of Eqs. (25)or(26)and (30)with increasing applied loading app and/or time t. When no A steady-state creep condition is attained when the far-field fiber solutions to Eq.(30)exist, equilibrium cannot be satisfied; this stress becomes constant with time. i. e. when the term in brackets condition corresponds to composite failure. The overall composite on the right-hand side of Eq (34)becomes zero. strain is the strain of the intact fibers which follows the stress As the far-field fiber stress changes via creep, the length of the profile of Eqs.(12). Thus, the present approach can be used to fiber/matrix slip length 8 will also change, thus affecting th determine the stress-strain response of a composite and the strain failure probability of the fiber and the composite strain rate. Note history under rupture conditions that the fiber stress T at each matrix -crack plane must satisfy
(where ls rT/(2)) of the central matrix crack, which, by the above-described arguments, is simply given as [(2/r)(ls/2)](2ls/ x)q(T,x/2,t), where the slip length ls is evaluated at the stress T, which acts an effective applied stress on the fibers. These considerations lead to the equilibrium condition app f qT, x 2 , t2L0 r qT, x 2 , t2ls x ls r 1 qT, x 2 , t1 2ls x T (29) where L0 is the length of fiber pullout just from the central crack plane. The first term on the right-hand side represents the pull-out load from the central matrix crack; the second term represents the pull-out load due to the other 2ls/x matrix cracks; the third term represents the stress carried by the fibers that remain intact. This expression is quite general, holding for both quasi-static and time-dependent conditions; it is the probability of failure q that is dependent on the details of the stress state and the time dependence. Because of the complicated nature of the central matrixcrack pull-out term (L0), Eq. (29) generally must be solved numerically. If this term is ignored—which is an assumption that is accurate, except when both m is small and x is quite large—then Eq. (29) reduces to app f 1 qT, x 2 , t1 T c 2 cx T (30) where c is the characteristic length:3 c r c (31) Equation (30) was first derived by Curtin et al. 5 and used, along with Eq. (17b), to predict quasi-static (time-independent) stress– strain deformation. Equation (30) is the third main result of our analysis. Equation (30) relates the known applied stress app to the effective stress T that acts on the unbroken fibers and the extent of damage q. Damage evolution in the composite corresponds to the evolution of T and q, through the solution of Eqs. (25) or (26) and (30) with increasing applied loading app and/or time t. When no solutions to Eq. (30) exist, equilibrium cannot be satisfied; this condition corresponds to composite failure. The overall composite strain is the strain of the intact fibers, which follows the stress profile of Eqs. (12). Thus, the present approach can be used to determine the stress–strain response of a composite and the strain history under rupture conditions. To determine composite lifetime, the initial fiber-damage state at time t ˜ 0 is determined using Eqs. (25) or (26) and (30). Then, time is advanced by some small time increment t ˜ and the fibers degrade and fail according to Eq. (25) or (26), yielding a new value of q. Then, Eq. (30) is used to determine a new value for T˜ and the process is repeated until no solutions exist. In addition, the composite strength after any time t* before failure (i.e., the remaining strength) can be determined by holding the time fixed at t* and increasing the applied stress—and, hence, the value of T˜ in Eqs. (25) or (26) and (30)—until composite failure occurs. (3) Creep Effects Creep deformation can alter the distribution of stresses on the fibers and matrix with time;39 therefore, it has some effect on the stress-rupture lifetime, even in the absence of explicit creepdamage mechanisms. In the absence of creep, the effects of a finite crack spacing and a nonzero matrix elastic modulus have been characterized by the parameter fEf /Ec, which determines the far-field fiber and matrix stresses. When the fibers and matrix behave in a viscoelastic manner, however, the far-field fiber and matrix stresses are determined by the equality of the strain rates, not the absolute strains. That is, ˙ m ˙f (32a) ˙ m Em B m n ˙f Ef H f p (32b) where the fiber and matrix strain rates are assumed to follow power-law creep relationships that are described by the parameters (H, p) and (B, n), respectively. In the far-field regions, the fiber and matrix stresses must still satisfy equilibrium via the relation f f 1 f m app (33) Substitution of Eqs. (32) into Eq. (33) yields the time evolution of the far-field fiber stress: d f dt 1 f EfEm Ec B app f f 1 f n H f p (34) A steady-state creep condition is attained when the far-field fiber stress becomes constant with time, i.e., when the term in brackets on the right-hand side of Eq. (34) becomes zero. As the far-field fiber stress changes via creep, the length of the fiber/matrix slip length will also change, thus affecting the failure probability of the fiber and the composite strain rate. Note that the fiber stress T at each matrix-crack plane must satisfy Fig. 6. Ratio of fiber lifetimes under a composite-like stress field (tf ( 0)/tf ( 1)), showing sensitivity to the matrix modulus. 0 corresponds to a high matrix modulus (the matrix carries load away from the matrix crack); 1 corresponds to a low matrix modulus (fibers carry the entire load). 1358 Journal of the American Ceramic Society—Halverson and Curtin Vol. 85, No. 6
June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 13 quilibrium and ot directly affected by Then, the Similarly, increasing the crack spacing x leads to overall composite rate is determined by integ the strain posites and thus prolongs the life at any fixed applied rate of the fibers er one-half of the crack However, when the stress is normalized by the fast-fracture omposite strength, the results are much less sensitive to x. Thus, the results of Fig. 7 are also largely unaffected by changes in x ∈()d Generally, a and/or x can change with time and load level. As noted earlier, o can change with time, because of creep. The matrix-crack spacing i is typically dependent on load, as more The strain rate of the fiber is a function of the local applied stress, matrix cracks form with increasing applied str thus. the which, in turn, is a function of position. Therefore, Eq. (35)can relevant x value for stress rupture at moderate stresses typically recast as will be smaller than that prevailing at the fast-fracture strength The matrix-crack spacing x can also change with time, because of slow matrix-crack growth. Such changes in a and/or x can d=+ Boa dz nfluence the predicted lifetime but can be incorporated into th present analysis by making these parameters dependent on stres and/or time. For the two-flaw-population model, the rupture here the far-field fiber stress(oa)is determined from Eq ( 34). behavior at low stresses is independent of the fast-fracture When steady-state composite creep is attained, the steady-state slip strength; hence, normalization by the fast-fracture strength is not length 8(0 can be determined by fitting the strain rate from eq useful and the lifetime can have a stronger dependence on a and/or (36) to the experimentally measured strain rate i but the general trends are similar. Here, we neglect the possible effects of creep on the interfacial behavior. For CMCs with a cracked matrix, the fiber creep near the matrix crack is typically much larger than that of the matrix, I. Predictions of Stress Rupture versus because the matrix stresses are very low. Hence, volume Experimental Results reserving fiber creep can lead to a"shrinkage" of the fiber away (1) Determination of Constitutive Material Parameters from the matrix and, thus, a decreasing interfacial shear stress T with time.'Although this effect can be incorporated into the Here, we discuss how the necessary parameters are derived theory, we do not believe it is a significant factor in the material from the experimental data obtained in Section II and from other systems here, therefore, we have neglected such effects ources. Briefly, the parameters o, C, B, and m are obtained from independent single-fiber and composite fast-fracture tensile tests. The parameters x and T are derived from observations and data (4) Predicted Trends from the quasi-static tension and hysteresis tests on the composite The present model for quasi-static and time-dependent defor- The matrix contribution a is determined from an analysis of mation, damage evolution, and failure is concisely contained in creep-deformation tests on the fibers and on the matrix material Eqs. (17),(25)or(26), and(30). The model predicts quasi-static Details are described below strength, tertiary creep, remaining strength, and rupture lifetime, in The interface frictional stress T is obtained from analysis of the terms of a few underlying micromechanical constituent material unload/reload hysteresis loop data, using the technique of Vag parameters. The constituent material parameters required to gen- gagini et al, which is applicable at elevated temperatures. Using erate stress-rupture predictions are( the single-fiber rupture temperature. Push-out tests on virgin specimens yield a value of parameters C and B, (ii) the initial fiber characteristic strength T= 33 16 MPa, which shows fairly good agreement. At fiber Weibull modulus m(which is equal to me in a two-flaw elevated temperature, the deduced interface frictional stress was -3.5 MPa but the hysteresis loops were very irregular, and noise population model), (iv) the interfacial shear stress T, (v)the in the strain signal suggests that the data are suspect. Room spacing i, and(vii) separate strength values for fiber rupture(o temperature push-out tests performed on a specimen tested at 255 m)(optional for the two-flaw-population model). The proposed MPa and 950oC for 50 h have yielded a value of T= 29 12 MPa which indicates that little, if any, degradation of the interface stres model is predictive for stress rupture because each of these due to elevated temperature stress-rupture testing occurs. There- parameters can be determined using independent experimental measurements fore, the value of T obtained at room temperature will also be used population model(Eqs. (2/ ha posite lifetime for the one-flaw. for the analyses at elevated temperature Some of the trends The single-fiber stress-rupture parameters B and C are obtained as follows. First, the parameters C and o enter into the normal- fitting experimental data on single-fiber rupture lifetimes to the slow-crack-growth model(Eq (20))and performing an appropri these parameters do not need to be studied directly. The normal ate statistical analysis. Here, we use the experimental data of Yun ized lifetime versus stress, normalized by the fast-fracture com- et al., including the initial tensile points into the fit; the resultin posite strength oulr, with varying m and B is shown in Fig. 7 for the parameter values are given in Table Il, whereas the actual case of a=0 and a single matrix crack( > 1); this situation is single-fiber experimental data and the slow-crack-growth fit for a ery different from that in the study by lyengar and Curtin."Over typical fiber are shown in Fig. 8. The"initial" fiber strengt the range of m values usually observed in ceramic reinforcing btained from fitting the Y un et al. data is not presented, becaus fibers(3 m 12), the change in lifetime is typically less than it is not relevant to the in situ fiber strength in the as-processed an order of magnitude. The lifetime is much more sensitive to B composite; the single-fiber rupture data are used only to obtain the increases in B lead to large increases in the normalized failure rate parametersβandC time, although the normalizing time does involve B. The effect of The relevant in situ fiber strength o cannot be determined from other parameters, particularly a and x, is quite minimal when the fracture mirrors in the present material; instead, we fit the stresses are normalized by gult and normalized times are consid measured quasi-static stress-strain curves and tensile strengths to ered. Decreasing the value of a leads to longer life at any fixed the(time-independent)model that was discussed in Section Ill to applied stress but also an increased fast-fracture strength. The deduce the characteristic fiber strength o and m. Recall that the increase in life is largely due to the relative decrease in the ratio of tensile strength of a composite is determined by the simultaneous pplied stress to fast-fracture strength. Thus, the results of Fig. 7 olution of two equations: the equilibrium equation(Eq. (30)and are largely unaffected by changes in a. This weak dependence on a for the composite is rather different from the results for a single fiber, shown in Fig. 6, where a has a much larger effect on life Performed by Jeff Eldridge at NASA-Glenn Research Center, Cleveland, OH
equilibrium and, thus, is not directly affected by creep. Then, the overall composite strain rate is determined by integrating the strain rate of the fibers (˙f ) over one-half of the crack spacing: ˙ c 0 x/ 2 ˙fz dz (35) The strain rate of the fiber is a function of the local applied stress, which, in turn, is a function of position. Therefore, Eq. (35) can be recast as ˙ c 0 t BTt1 z ls n dz t x/ 2 B ff n dz (36) where the far-field fiber stress ( ff) is determined from Eq. (34). When steady-state composite creep is attained, the steady-state slip length (t) can be determined by fitting the strain rate from Eq. (36) to the experimentally measured strain rate. Here, we neglect the possible effects of creep on the interfacial behavior. For CMCs with a cracked matrix, the fiber creep near the matrix crack is typically much larger than that of the matrix, because the matrix stresses are very low. Hence, volumepreserving fiber creep can lead to a “shrinkage” of the fiber away from the matrix, and, thus, a decreasing interfacial shear stress with time.23 Although this effect can be incorporated into the theory, we do not believe it is a significant factor in the material systems here; therefore, we have neglected such effects. (4) Predicted Trends The present model for quasi-static and time-dependent deformation, damage evolution, and failure is concisely contained in Eqs. (17), (25) or (26), and (30). The model predicts quasi-static strength, tertiary creep, remaining strength, and rupture lifetime, in terms of a few underlying micromechanical constituent material parameters. The constituent material parameters required to generate stress-rupture predictions are (i) the single-fiber rupture parameters C and , (ii) the initial fiber characteristic strength c (which is equal to cff in the two-flaw-population model), (iii) the fiber Weibull modulus m (which is equal to mff in a two-flawpopulation model), (iv) the interfacial shear stress , (v) the fiber/matrix modulus ratio parameter , (vi) the matrix-crack spacing x, and (vii) separate strength values for fiber rupture ( cr, mr ) (optional for the two-flaw-population model). The proposed model is predictive for stress rupture because each of these parameters can be determined using independent experimental measurements. Some of the trends in composite lifetime for the one-flawpopulation model (Eqs. (25)) with variations in the parameters are as follows. First, the parameters C and c enter into the normalizations of time (by C c 2 ) and stress (by c or ult c); therefore, these parameters do not need to be studied directly. The normalized lifetime versus stress, normalized by the fast-fracture composite strength ult, with varying m and is shown in Fig. 7 for the case of 0 and a single matrix crack (x ls); this situation is very different from that in the study by Iyengar and Curtin.20 Over the range of m values usually observed in ceramic reinforcing fibers (3 m 12), the change in lifetime is typically less than an order of magnitude. The lifetime is much more sensitive to : increases in lead to large increases in the normalized failure time, although the normalizing time does involve . The effect of other parameters, particularly and x, is quite minimal when the stresses are normalized by ult and normalized times are considered. Decreasing the value of leads to longer life at any fixed applied stress but also an increased fast-fracture strength. The increase in life is largely due to the relative decrease in the ratio of applied stress to fast-fracture strength. Thus, the results of Fig. 7 are largely unaffected by changes in . This weak dependence on for the composite is rather different from the results for a single fiber, shown in Fig. 6, where has a much larger effect on life. Similarly, increasing the crack spacing x leads to stronger composites and thus prolongs the life at any fixed applied stress. However, when the stress is normalized by the fast-fracture composite strength, the results are much less sensitive to x. Thus, the results of Fig. 7 are also largely unaffected by changes in x. Generally, and/or x can change with time and load level. As noted earlier, can change with time, because of creep. The matrix-crack spacing x is typically dependent on load, as more matrix cracks form with increasing applied stress; thus, the relevant x value for stress rupture at moderate stresses typically will be smaller than that prevailing at the fast-fracture strength. The matrix-crack spacing x can also change with time, because of slow matrix-crack growth. Such changes in and/or x can influence the predicted lifetime but can be incorporated into the present analysis by making these parameters dependent on stress and/or time. For the two-flaw-population model, the rupture behavior at low stresses is independent of the fast-fracture strength; hence, normalization by the fast-fracture strength is not useful and the lifetime can have a stronger dependence on and/or x but the general trends are similar. IV. Predictions of Stress Rupture versus Experimental Results (1) Determination of Constitutive Material Parameters Here, we discuss how the necessary parameters are derived from the experimental data obtained in Section II and from other sources. Briefly, the parameters c, C, , and m are obtained from independent single-fiber and composite fast-fracture tensile tests. The parameters x and are derived from observations and data from the quasi-static tension and hysteresis tests on the composite. The matrix contribution is determined from an analysis of creep-deformation tests on the fibers and on the matrix material. Details are described below. The interface frictional stress is obtained from analysis of the unload/reload hysteresis loop data, using the technique of Vaggagini et al., 4 which is applicable at elevated temperatures. Using the measured crack spacing of 40 m, 25 8.3 MPa at room temperature. Push-out tests† on virgin specimens yield a value of 33 16 MPa, which shows fairly good agreement. At elevated temperature, the deduced interface frictional stress was 3.5 MPa but the hysteresis loops were very irregular, and noise in the strain signal suggests that the data are suspect. Roomtemperature push-out tests performed on a specimen tested at 255 MPa and 950°C for 50 h have yielded a value of 29 12 MPa, which indicates that little, if any, degradation of the interface stress due to elevated temperature stress-rupture testing occurs. Therefore, the value of obtained at room temperature will also be used for the analyses at elevated temperature. The single-fiber stress-rupture parameters and C are obtained by fitting experimental data on single-fiber rupture lifetimes to the slow-crack-growth model (Eq. (20)) and performing an appropriate statistical analysis. Here, we use the experimental data of Yun et al., 34 including the initial tensile points into the fit; the resulting parameter values are given in Table II, whereas the actual single-fiber experimental data and the slow-crack-growth fit for a typical fiber are shown in Fig. 8. The “initial” fiber strength obtained from fitting the Yun et al.34 data is not presented, because it is not relevant to the in situ fiber strength in the as-processed composite; the single-fiber rupture data are used only to obtain the degradation-rate parameters and C. The relevant in situ fiber strength c cannot be determined from fracture mirrors in the present material; instead, we fit the measured quasi-static stress–strain curves and tensile strengths to the (time-independent) model that was discussed in Section III to deduce the characteristic fiber strength c and m. Recall that the tensile strength of a composite is determined by the simultaneous solution of two equations: the equilibrium equation (Eq. (30)) and † Performed by Jeff Eldridge at NASA–Glenn Research Center, Cleveland, OH. June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 1359