ournal Am,Ceam.Sox.,8s161350-65(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. lacksburg. 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 composit micromechanical model can be successful quantitatively but developed for materials that do not degrade by oxidative clearly shows that the rupture life of the composite is ex attack, The rupture model for a unidirectional composite nsitive to the detailed mechanisms of fiber degrada incorporates fiber-strength statistics, fiber degradation with odel has practical applications for extrapolating time at temperature and load, the state of matrix damage, and 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 independent of stress-rupture testing through quasi-static ten L. Introduction sion tests and tests on the individual composite constituents. The model predicts the tertiary creep, the remaining composite trength, and the rupture life, all of which are dependent C ERAMIC-MATRIX COMPOSITES (CMCs) are attractive materials for use in high-temperature applicati critically on the underlying fiber-strength degradation Sensi 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.l-s however,the 1050.C 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 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 mechanisn sured behavior. For a slow-crack-growth model of fiber and their effects on stress-rupture lifetime and deformation trength degradation, the lifetime predictions are shorter by Many researchers have used traditional engineering methods, two orders of magnitude. When the rupture life is fitted with 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 1050 C agrees well with the under constant load. The LM approach relates the applied stress to experimental results. For a more complex degradation model, a failure parameter, @. 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 t,+ C) where T is the temperature, I, 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 test R, Kerans-contributins editor 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 88.72 Received May 11. 2001: approved December 18. 2001 individual fibers at the established stress level. This technique was used by Morscher and co-workers"to examine the stress rupture 4 suppor from the U S. Air Force O压和m的 of precracked Hi-NicalonTM/SiC minicomposites with BN inter faces. The predictions were accurate at low and high values of 2: however, at intermediate values, removal of the bn interface an Feature
June 200 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 uch as the Weibull modulus of the fiber and the slow -crack atrix is not cracked, use of the MG approach has been propose The MG approach relates the steady-state strain rate (e)to the The Coleman model 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 of fiber failure is a function of time and stress and no attempt is made Et=D (2 to determine fiber strength. Ibnabdeljalil and Phoenixdeveloped 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 tiber-rupture behavior. Lamouroux et al.developed a model for 610 fibers successfully match lifetimes obtained by Zuiker on a of the fiber, a simple bundle model for fiber failure that neglected woven Nextel 610/aluminosilicate composites Although the accuracy of engineering methods can be good fiber pullout, and the Coleman fiber-rupture model. These models they are basically correlations between macroscopic measures of to account for a e life and tertiary creep, and they can be extended can predict rupt behavior and do not contain much information about the state of matrix damage, but this step has not yet been he material during the stress-rupture process. Therefore, they attempted. However, in the Coleman model. the fibers have an annot be used to (i) predict behavior a priori; (i)connect infinite fast-fracture strength; hence, the rupture is not related to different but related aspects of the deformation, such as the tertiary predicted at all timer th and the remaining strength cannot be creep and the remaining strength, to the rupture life: and (iii) If the matrix is sufficiently stiff and/or not fully cracked. it rovide insight into the optimization of composites, because ne direct connection to underlying constituent properties and/or the fiber-stress profile that can decrease the rate of fiber degrada a carries some axial stress, which leads to a spatially vary 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(i)is of new composites, where changes in constituents occur regularly ot saturated, (ii) is dependent on the applied stress level, or (iii) 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 constituent materials, should be extremely useful for the design If the matrix does carry some significant portion of the applied 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 hus, 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 en 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 is, thus, an area of considerable investigation. Holmes and co- general, fiber failure is a function of temperature, stress, and any workers2. 2 studied the creep rate of silicon carbide/calcium chemical interactions that occur (e.g, oxidation). For silicon arbide (SiC) fiber materials, where oxidation is a primary SiaN.) composites extensively. For SiC/Si N composites, the oncern, 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 1200C, at which temperature the matrix carries only a analyzed this mechanism for a matrix-cracked composite, includ- fraction of the applied stress. the stress exponent of the compos g the statistics of fiber strength, and produced predictions for composite lifetime. Evans and co-workers- considered a sim 1.9 for Nicalon fibers:"). Composite creep due to combined fiber ilar process within growing matrix cracks wherein weakening by nd matrix creep, but without damage, was first modeled by McLean" and can 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 occurs when the remaining uncracked composite cross section cannot sustain the applied load. Other mechanisms, such as the co-workers2529 on SiC/Si, N, 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 co-workers.and Lewinsohn er al., i6 among others he creeping matrix to the noncreeping elastic fibers, as per the If environment effects can be eliminated through the use of model of McLean. At higher stresses, the initial loading rate coatings or oxidation-resistant constituents, fiber failure should be strongly influenced the creep behavior. At high rates of loading a function of stress and temperature only. CMCs with active matrix fracture was pronounced and composite lifetimes were oxidation have very limited lifetimes; therefore, we will focus on relatively short. At lower rates of loading, the matrix was able to lax in creep and did not fracture, resulting in much-longer absent. Specifically, we will use the slow-crack-growth model to and McMeeking 0 and Fabeny and Curtin" to incorporate statis- dict fiber-damage evolution and failure and envision the app cation of our models to all-oxide ceramic composites, Although ical fiber fracture and its influence on creep and rupture but not matrix damage. These works also emphasized that stress transfe evidence for any particular fiber- degradation mechanism in oxide- ceramic fibers is difficult to ascertain. 7-9 the general power-law solutions and provides a relationship between the initial fiber reep of the matrix and a subsequent increase in the ineffective form of the slow-crack-growth-rate equation lends itself to analytic length of broken fibers, resulting in time-dependent composite strength and the fiber strength after some arbitrary stress and failure temperature history. Failure of the fibers in a composite under slow Giv modeling back- crack growth is dependent on the actual stress history experienced ground cs and statistics by each fiber, which is dependent on the applied load, the state of and composite matrix damage, and the interfacial sliding between the fibers and as a function of the matrix damage state, using the slow- crack the matrix. lyengar and Curtin studied composite failure when growth model for fiber degradation. The resulting model includes the matrix was fully saturated with closely spaced cracks and fiber-strength statistics, fiber degradation with time at temperature
1352 Journal of the American Ceramic Society-Halverson and Curtin Vol. 85. No. 6 and load, and the effects of fiber pullout, within the well- prevent large-scale sintering of the materials at elevated tempera established framework of the global load sharing (GLS) model. ures but small enough to permit stress transfer between the The model predicts interrelated phenomena of tertiary cree constituents by frictional shear stress. The carbon also serves to dependent critically on the underlying fiber-strength degradation. processing of the AL,O,/Y,O, mall mical reactions during the remaining composite strength, and rupture life, all of which are 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 Y2O, 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 700"C. After a few infiltration and drying steps, the part was fired the model, an experimental study of the stress-rupture life, creep at 1 100C 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, Nextel 610 fiber/alumina-yttria matrix CMC(manufactured by processing was halted when the composite porosity was% McDermott Technologies, Inc (MTD), Lynchburg. VA) with a which typically required 4-6 cycles. The Y,O, reacts with the fugitive carbon interface has been conducted, This oxide/oxide Al, O, 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, the (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 composite lifetime with stress and temperature are well-matched Then, the model also predicts tertiary creep rates and remaining MTS Systems, Eden Prairie, MN) with a controller(Model 458, data. The measured star ery good agreement with the experimental MTS Systems ). Specimens were tabbed with 0.020 in. fully annealed aluminum tabs, to prevent damage from gripping. The be correlated with the scatter in the initial composite strength predicting several different features that are associated with the ps s placed in the MrS test-frame grip. The pressure of the grip using the model. The success of the model in simultaneously ically deformed the aluminum to"fit"the specimen, and no adhesive was used. Grip pressures were maintained at -0.7 MPa deformation and failure, despite the need for a fitting parameter. The tests were run under load control. at a rate of 180 N/s Strain demonstrates the power of such a micromechanically based ap- proach. An alternative assumption for the fiber-degradation mech- 63211B. MTS Systems). Specimen alignment was maintained lifetime predictions at high temperature but poorer residual rough a fixture at the grips. strength and tertiary creep predictions. The implications of A compact oven was used for the tests that were conducted at extreme sensitivity of the rupture life to the precise mechanisms elevated temperature. The oven had four SiC resistance elements fiber-strength degradation, and/or the inadequacy of the ex sittt (Norton Advanced Ceramics, Worcester, MA) that heated the specimen. The oven shell was stainless steel and had nominal fiber-strength degradation to the in situ behavior, is an important dimensions of 3.5 in. x 3 in.X 3 in. Fused-silica insulation issue of discussion The remainder of this paper is organized as follows. We Cotronics Corp, Brooklyn, NY) lined the inside of the oven in Section Il, 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 X 2 in. Two temperature controllers (Model 818S, Eurotherm d MG models; their inadequacy motivates the subsequent model Reston, VA)controlled the SiC heating elements: one for the two 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 predictions of the modeL. In Section V, we discuss our results by alumina-fabric insulation, for efficient heating and to help further, 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 model the oven held an extensometer(Model 621-5IE. 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 Il. Experimental Details, Results, and Predictions of at one of the posts. The extensometer, the heat shield, and the MTs Engineering models We begin our discussion with the experimental results and temperature. Then, the temperature was held constant for 10 min omparisons 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 life predictions are generally inadequate. This provides significant testing. However, the load rate for the stres motivation for the extensive theoretical developments of Section to 660 N/s, to minimize creep effects during the initial loading III ramp. When the desired load was attained. it was held constant until failure occurred. Strain data were collected throughout the (1) Material System test. Some tests were stopped after specified times to determine the The material system examined here is an oxide/oxide CMC that remaining strengths. As a test of remaining strength, the load was vas produced by MTl. This material consists of Nextel 610 first returned to zero and then the specimen was ramp fibers(>99% Al,O,)aligned in a unidirectional configuration and at 180 N/s embedded in an alumina-yttria (AL,O,Y, O,) matrix with a fugitive 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. I(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
June 2002 Stress Rupture in Ceramic-Matrir Composite teory and Experiment Fig. I. 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. I(b), unchanged from the virgin material.The curves for the quasi-static tension tests are shown in Fig. 2, and tiber-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 var Section Iv bility, 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. (4) Predictions of Rupture Using Engineering Models examine whether two engineering approaches that g 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 the(presumed) failure plane, probably as a result of the high matrix porosity. Hence, fiber-pullout measurements could not b used to assess the behavior of the composite from the behavior of performed. However, observation of the failure surface indicated he constituent fibers accurately that fiber fractures were not confined to one plane of the compos In regard to fiber composites, it has been suggested that the LM plot of applied stress versus @(see Eq.())for the composite ite, so that cracks were deflected at the fiber/matrix interface. should be identical to that for the fibers at an appropriate stress Examination of the polished edges of tested specimens demon- strated that the matrix-crack spacing on completion of a tensile test This method is thought to be applicable when the matrix has been was identical to that in virgin specimens(40 um). Typical fiber fully cracked, so that the fibers can be assumed to carry the entire failure surfaces were smooth, with no discernable fracture origin applied stress along their entire length. The LM data for the Some fiber-fracture surfaces at room temperature demonstrated Nextel"fibers, as determined by Yun et al., and the LM plots derived fro evidence of fracture mirrors: however, the proportion of such a range of loads and temperatures are shown in Fig. 3. The fiber-fracture surfaces of specimens tested at higher temperatures agreement is poor, particularly considering that the plot has a were similar in appearance to those tested at low temperature logarithmic time scale. Under stress-rupture loading conditions, the matrix-crack spac The MG approach envisions that rupture damage is driven by gs observed on the specimen edge were, again, 40 um, as creep deformation. The MG relationship between the steady-state strain rate t and the rupture lifetime f, is, from Eq (2) logt+klog∈=D where k and D again are constants(k= 1). Use of the measured 1050°C composite e 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. 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 fibers and for the composites studied here. At a given te ure, the composite data do show 100 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 0.1502 030.3504 long a common line, which indicates that Eq. (3)applies, with k and D independent of temperatur where data are shifted by substantial fa hich suggests that k is indep Fig. 2. Measured quasi-static stress-strain curves at several temperatures of temperature but D is strongly dependent on temperature. Such black lines) and fits at elevated behavior has been observed previously in monolithic ceramics such as silicon nitride(see, for example, Ferber and Jenkins, S 0.=1060 MPa at 950 C and 1000 MPa at 1050 C (curves offset by 0.1% Luecke et al.,and Menon et al. "). At a fixed temperature, the for cla MG corre n creep and rupture do to t
Journal of the American Ceramic Society-Halverson and Curtin Vol. 85. No 6 Table L. Quasi-static Tension Test Results Modulus(GPa) Strength(MPa Failure strain (5) Number of tests 245± 392±63 0.177±0.034 4 189±6 0.194±0.037 1050 191±16 370±54 0.214±0.018 1093 197±14 305+28 0.206±0.240 composite system: a measurement of creep can be used to accurate. A lack of correlation "predict"the failure time(although Fig. 4 shows that order-of- between the composi fiber data also exists: therefore magnitude fluctuations in life exist at a fixed creep rate). However, we do not advocate th e of these approaches to describe the shift in the MG plot with temperature limits such"predictions the high-temperature ind failure of ceramic compos to each temperature of interest. Furthermore, the behavior does not ites. These facts further motivate the consideration of microme relate 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 rut However, a relationship that follows Eq. (3)is obtained when IlL. Micromechanical Model of Composite Stress Rupture reep 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 t,= Bo cre 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 log∈=lg(BA") state in the fibers, which governs the rate of fiber degradation, as (4) shown schematically in Fig. 5. Here, we begin with an analysis of 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 th previously damaged fibers on the stress state. Subsequently, the "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 rc is independent of temperature considered, which leads to the full model for composite damage but the rate prefactors A and B should be Arrhenius-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; therefore. observed with our te data. Thus. the existence of an MG 1G we begin with the quasi-static problem, because it sets the stage for 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 ying information about damage state, nor do they provide infor- (A) Quasi-static Behavior: Ceramic fibers are brittle materials remaining strength or any other phenomena that whose strengths must be described statistically. This description is occur in the composite. Moreover, the general idea of applying single-fiber rupture data directly to explain composite rupture has commonly accomplished by assuming a single flaw population and using a tw eter Weibull model, for which the probability of several incorrect implications for composite failure and rupture. failure occurring in an increment of a fiber element of length 8z within The concept implies that composite rupture occurs at the average failure time of a single fiber at the(arbitrary)ex situt-tested gauge incremental stress range of o to o 8o 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 P(,b0,82) single-fiber experiments. This concept also implies that failure occurs when every fiber is typically broken once within a length lo Here, oo 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 lengt tion of the strength around o. Following previ ses 3. 38 the cumulative probability of fiber failure in a length 21, loaded at pplied stress Uno, where U nn and matrix damage cause a 10000 longitudinal-fiber stress profile o(z), is given by q(0=,D)=1 1000 叫 do(z) do'dz do(6) When o(z) is a constant value app), Eq.(6) reduces to the well-known Weibull expression (m,D)=1 n a unidirection a single matrix crack and a 25 33 debonded, sliding fiber/matrix interface described by an interfacial Q=T( t,+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 Fig. 3. Larso - rupture data and/unn versus parameter @) for isolated matrix crack located at the longitudinal position z =0 plot(applied stress (- single-fibe measured composite stress- under a remote applied stress app is accurately modeled by a rupture data. shear-lag model as follows. At the matrix- crack plane. the entire
June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment 355 10000000 1050°C 1000000 Fiber Data - Fit to Composite Data 100000 10000 1.00E081.00E-071.00E061.00E05100E041.00E03 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 vanous temperatures(◆)950Cand(1050°C load is carried by the fiber; thus, the fiber stress is given as cann f. When the fiber breaks, slip along the fiber/matrix interface occurs where f is the fiber volume fraction. At a distance the matrix over a fiber slip length. I rack, stress is transferred from the fiber to the through the interface frictional stress as . 2Tf (11) 27z 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 where 8 is the distance over which interfacial slip occurs and is as er stress. as a function of distance z from a matrix crack is given defined as o(3)=71-1)(ok≤6 rooo[(I-DE Tf E (2)=0mE < (12b) and E Em, and Ec are the Youngs moduli of the fiber, matrix, and ively (with EC=Er+(1-DEm, and r is the where T- App/ f. This stress profile and the above-described fiber radiu 28, the fiber stresses and strains are equal and notations are illustrated in Fig. 5. Under the stress state given in the stress 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 oapp follows from Eq(6)as E (10) -(1+ma"+)(13) Fiber stress profile 2T Er E Matrix Crack 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 de on both the initial crack size and the stress acting on it
Journal of the American Ceramic Sociery -Halverson and Curtin Vol. 85. No 6 1356 Here, we have introduced the dimensionless parameters T and a, 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 da (14b) crack size and K is the stress intensity factor; p is G(z=0) growth exponent and a is a rate constant, each of ly is dependent on temperature. K, which is a (14c) of the current stress T, crack size a, and a geometric factor Y, is given by A length 2I, is chosen because only fiber breaks within t/, of a K=YTa matrix crack will influence tensile failure of the composite at this Flaw failure is determined using the expression K=Kle. Thus, the Depending on the Weibull modulus m and the stiffness of the initial tensile strength(o, at initial crack size a, )can be related to matrix crack(see below) matrix,as captured by the parameter a, there is a competition the flaw size afn) and flaw strength o() at time r by combining Egs to vield between fiber failure within the linearly decreasing portion of the stress field and the constant far-field region. For low values of 1B-2 failure in the far-field stress region is negligil probability of fiber failure is accurately represented by o(n=o8-2-c T(r)adr q(7,()=1-exp-(m+1 C=(5-1)AFK2 (20b) which is thesingle matrix crack" result that was derived by Thouless and Evans, as well as other researchers, For high values here y ed to be constant over time. Inverting Eq.(20a) of ma"*, the matrix modulus is small, relative to the fibers (i. e. yields the trength required to provide a current strength of Ee RE, thus, the length 8 is small and the far-field stress region a(t)after n stress history T(n 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= d =o(-+1(?)di (21) ( corresponding to a matrix modulus of zero), the probability of fiber failure is given as q(7,(7)=1-exp(-7) (16 Here, we have introduced a nondimensional time i and a normal- th given by which is the result obtained by Curtin for the case of saturated, (22a) closely spaced matrix cracking. Thus, the value of mo" in E (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 (22b) In real materials, multiple matrix cracks typically exist with If two adjacent crack paced by a where T is as defined previously in Eq (14a) distance i that is less than a fiber slip length 1,, then Eqs. (12)only apply over a length 2. In this case, we determine the probability that the fiber will fail is the probability that the initia of fiber failure around each matrix crack only within the region of the fiber is less than the initial strength given by Eq.(21) i2, which corresponds to setting the limits of the position integral That is, for a fiber element of length 8z under some stress in Eq. (6)to /=t/2. Then, the cumulative probability of failure history /(n), the probability of failure over the applied stress increment from o(n) to o(n)+ So(n) is obtained by substituting (t) p(or(,.bo(),6x)= 3 {1+a(m+1) Following arguments identical to those used in the quasi-static he probability of failure of a fiber, with respect to time. r the stress profile of Eqs.(12)is then given by fonm)<5<4)(1a) x-79-2+x"T(r)dr results will be used below to determine the +a"2+a7 te damage evolution and quasi-static stress- failure. Equation(17) is the first main result of o (B) Time-Dependent Behav or: We now assun where a change of variables from z to x= I-al. has been er strength degrades with time, because of the performed. Figure 6 shows the ratio of mean fiber lifetimes for
June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment a=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 6. 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 hose for the case of a= I hence the fiber lives longer. For moderately high values of B, the modest stress camied by the matrix can lead to increases in the fiber lifetime of more than two orders of expl-7 magnitude, This large difference motivates a detailed analysis of in situ stresses and fiber degradation to obtain realistic predictions of JIt8-2) composite lifetime. As for the quasi-static case, the effect of a finite ma 2782+xT(r)dr' atrix-crac pacing x can be introduced by appropriately changing the limits of integration in Eq. (24) to yield l+ qlr.a -expl-7 In this expression, the subscripts "I"and"ff" have been added -2) differentiate between the two Weibull moduli and the normalized 1-7-2+rTr)dr stresses in rupture and in fast fracture for the two different types of flaws. In subsequent discussions. this model is termed the"two- model"described in Sections II(I)A) and (B/.population (1-a)T (2) Composite Behavior In a well-designe nterface, increasing the load on the remaining intact fibers. Here x78-2+ane)dr he equations for individual fiber failure discussed in Section III(/ are combined with the global load sharing(GLS)law, which is widely used to model CMCs, to yield predictions of composite behavior. In a for 8<x<I tained at all planes perpendicular to the applied load. Equilibrium at a matrix-crack plane, where the matrix carries no load, is a load (25a) en the applied stress and the stresses carried by the Oken n and intact fibers. In GLS, the stress previously carried by fibers across any given transverse plane is shared equally 引, among all intact fibers in that plane and, thus, equilibrium is given mAB-2) =(1-PoT+ Poken where Pr is the fraction of broken or sliding fibers at the matrix-crack plane, broken their average load-carrying capacit (25b) 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 his result is used below in examining overall composite crack is within a fiber slip length /, of the break a function of matrix-crack spacing, Equation(25) is th main result of our analysis (28) (C) Multiple-Flaw Populations: It is conceivable that the Oeken =-(for <4) flaws in a fiber that govern fast fracture are not those that Fiber breaks that extend farther than 1, carry the full stress T and govern stress rupture. For instance, creep damage may nucleate thus, are unbroken, relative to the matrix crack of interest: this is entirely new flaws. Or flaws that are strong under fast fracture hile he usceptible to growth under stress-rupture conditions was confined to the region tl, around the matrix crackSections weaker flaws that drive fast fracture do not grow 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 ndependent, 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 cracks is a region of length +/2 within which the become the dominant population driving the fiber rupture. Here, 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"fast- distributed symmetrically around the matrix-crack plane: there fracture"Weibull parameters(oe, m)and the latter described fore, the average distance from the central matrix crack to the by "rupture" Weibull parameters (oer m,)and slow-crack- broken fibers around any other matrix crack is precisely the growth kinetic parameters C and B. Although the slow-crack 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 then matrix crack to the other matrix cracks within I, is simply 1/2. simply given as q=1-(1-41(1-q2), where q, and qa are Thus, the average load carried by broken fibers, Broken, is the the failure probabilities for each flaw population, respectively average of the loads carried by all the broken fibers within +l
1358 Journal of the American Ceramic Society-Halverson and Curtin Vol. 85. No. 6 000 m=5 m=10 B Fig. 6. Ratio of fiber lifetimes under a composite-like stress field (ida=0)i,a= 1)), showing sensitivity to the matrix mo c=0 corresponds 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 L= rT/(2T))of the central matrix crack, which, by the To determine composite lifetime. the initial fiber-damage state above-described arguments, is simply given as ((2T/r)( 2)1(24, at time i=0 is determined using Eqs. (25)or(26)and (30).Then, x )g(T. x/2. n), where the slip length I, is evaluated at the stress T, time is advanced by some small time increment Ar and the fibers which acts an effective applied stress on the fibers. These consid- degrade and fail according to Eq (25)or(26), yielding a new value erations lead to the equilibrium condition ocess is repeated until no solutions exist. In addition, the Tl oo. q. Then, Eq(30) is used to determine a new value for Tand the composite strength after any time r* before failure (i.e,the remaining strength) can be determined by holding the time fixed at 2/ r*and increasing the applied stress-and, hence, the value of Tin Egs. (25)or (26)and(30)--until composite failure occurs (L)o is the length of fiber pullout just from the central crack (3) Creep Efects The first term on the right-hand side represents the pull-out Creep deformation can alter the distribution of stresses on the load from the central matrix crack; the second term represents the fibers and matrix with time. 9 therefore, it has some effect on the pull-out load due to the other 2l matrix cracks; the third term stress-rupture lifetime, even in the absence of explicit creep represents 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 and crack spacing and a nonzero matrix elastic modulus have been time-dependent conditions: it is the probability of f hat is characterized by the parameter a=fE/Ee, which determines the dependent on the details of the stress state and far-field fiber and matrix stresses. When the fibers and matrix dence. Because of the complicated nature of the 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 curate, except when both m is small and x is quite large-then Eq(29)reduces to ∈m=∈ (32a) 72--)+m (30) (32b) where 8. is the characteristic length where the fiber and matrix strain rates are assumed to follow (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 far+(I-f)om=gapp with Eq (17b), to predict quasi-static (time-independent)stress- train 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-fiele stress Equation (30) relates the known applied stress Japp to the effective stress T that acts on the unbroken fibers and the extent of do,(I-f)EEm Ln hage q. Damage evolution in the composite corresponds to the lution of Tand q, through the solution of Eqs.(25)or(26) and (30)with increasing applied loading app and/or time r. When no A steady-state creep condition is attained when the far-field fiber 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 the 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 satis
June 2002 Stress Rupture in Ceramic-Matrix Composites: Theory and Experiment equilibrium and, thus, is not directly affected by creep. Then, the Similarly, increasing the crack spacing x leads to stronger com- verall composite strain rate is determined by integrating the strain posites and thus prolongs the life at an lied stress rate of the fibers(E)over one-half of the crack spacing 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 ∈:=ez)dz Generally, a and/or x can change with time and load level. As noted earlier, a can change with time, because of creep. The matrix-crack spacing x is typically dependent on load, as more The strain rate of the fiber is a function of the local applie matrix cracks form with increasing applied stress: thus,the which, in turn, is a function of position. Therefore, Eq. (35)ca 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 i can also change with time, because of dz+ Bo/t dz resent analysis by making these parameters dependent on stress and/or time. For the two-flaw-population model, the rupture where the far-field fiber stress(o) 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(r) can be determined by fitting the strain rate from Ec useful and the lifetime can have a stronger dependence on a and/or (6)to the experimentally measured strain rate x 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. IV. Predictions of Stress Rupture versus because the matrix stresses are very low. Hence, volume Experimental Results preserving 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 Il and from other systems here; therefore, we have neglected such effect sources. 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 The interface frictional stress T is obtained from analysis of the unload/reload hysteresis loop data, using the technique of Vag- terms of a few underlying micromechanical constituent material gagini et al., which is applicable at elevated temperatures. Using parameters. The constituent material parameters required to gen- the measured crack spacing of 40 um, T=25+8.3 MPa at room erate stress-rupture predictions are(i) 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 or (which is equal to ufr in the two-flaw-population model), (iii)the T-33+ 16 MPa, which shows fairly good agreement. At iber Weibull modulus m(which is equal to m in a two-flaw population model).(iv) the interfacial shear stress T,(v)the -3.5 MPa but the hysteresis loops were very irregular, and n s. p elevated temperature, the deduced interface frictional stress was tiber/matrix modulus ratio parameter a,(vi) the matrix-crack in the strain signal suggests that the data are suspect. Room pacing x, and (vi) 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 950 C for 50 h have yielded a value of T=29 12 MPa, model is predictive for stress rupture because each of these which indicates that little, if any, degradation of the interface stress arameters can be determined using independent experimenta measurements fore, the value of T obtained at room temperature will also be used Some of the trends in composite lifetime for the one-flaw for the analyses at elevated temperature. The single-fiber stress-rupture parameters B and C are obtained opulation model(Eqs. (25))with variations in the parameters are by fitting experimental data on single-fiber rupture lifetimes to the as follows. First, the parameters C and o enter into the normal slow-crack-growth model (Eq. (20) and performing an appropri ations of time(by Co2)and stress(by o, or ult " og): therefore, ate statistical analysis. Here, we use the experimental data of Yur hese parameters do not need to be studied directly. The normal- et aL., including the initial tensile points into the fit; the resulting ized lifetime versus stress, normalized by the fast-fracture com- parameter values are given in Table II, whereas the actual posite strength cult with varying m and B is shown in Fig. 7 for the single-fiber experimental data and the slow-crack-growth fit for a case of a=0 and a single matrix crack (x > 1); this situation is very different from that in the study by lyengar and Curtin. Over typical fiber are shown in Fig. 8. The"initial"fiber strength obtained from fitting the Yun et al.data is not presented, because ibers(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 composite; the single-fiber rupture data are used only to obtain the an order of magnitude. The lifetime is much more sensitive to B: degradation-rate parameters B and C. increases in p lead to large increases in the normalized failure time. although the normalizing time does involve B: The effect of fracture mirrors in the present material; instead, we fit the stresses are normalized by out and normalized times are consid- the(time-independent) model that was discussed in Section Ill to ered, Decreasing the value of a leads to longer life at any fixed deduce the characteristic fiber strength g and m. Recall that the applied stress but also an increased fast-fracture strength. The tensile strength of a composite is determined by the simultaneous 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 ution of two equations: the equilibrium equation(Eq (30)and are largely unaffected by changes in a. This weak dependence or 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