Journal Am. Crra SeN、8721261-67(2004 Mullite/Alumina Mixtures for Use as Porous Matrices in Oxide Fiber Composites Hiroki Fujita, George Jefferson, Robert M. McMeekings and Frank W. Zok*t Materials Department, Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, California 93106 Department of Aeronautics and Astronautics. Air Force Institute of Technology, Wright Patterson Air Force Base, Ohio 45433 Weakly bonded particle mixtures of mullite and alumina are tolerance under fiber-dominated loadings. These offsetting effects assessed as candidate matrixes for use in porous matrix suggest the existence of an optimum in matrix properties at which ceramic composites. Conditions for the deflection of a matrix a prescribed balance of properties is attained. However. the crack at a fiber- matrix interface are used to identify the relationships between matrix structure and composite performance combinations of modulus and toughness of the fibers and the are understood presently at only a rudimentary level. Conse- natrix for which damage-tolerant behavior is expected to quently, the pathway to optimization remains ill defined occur in the composite. Accordingly, the present study focuses on the modulus and toughness of the particle mixtures, as well connections between composition, microstructure, and mechanical as the changes in these properties following aging at elevated properties of a candidate family of porous matrices, with emphasis temperature comparable to the targeted upper- use tempera aggregates are presented, assessed, and calibrated. The exper- addressed elsewhere. 2 The work stems from concurrent activities imental and modeling results are combined to predict the on the development of all-oxide CFCCs for use in future gas critical aging times at which damage tolerance is lost because turbine systems. Because of the interest in mullite/alumina of sintering at the particle junctions and the associated changes fibers for use in these applications. the present article focuses in mechanical properties. For an aging temperature of 1200C, specifically on mullite/alumina matrixes. These matrices are both the critical time exceeds 10 000 h for the mullite-rich mixtures. chemically compatible with the fibers and exhibit mechanical characteristics that make them attractive for use in ccCS. A L. Introduction perspective based on the mechanics of crack deflection is used to guide the experimental measurements and provide a framework for T interpreting the property values. Consideration is given to the USE of porous matrixes to enable damage tolerance in stability of the properties following long-term exposure at elevated ontinuous-fiber ceramic composites( CFCCs) has emerged as temperature, comparable to the targeted upper use temperature for paradigm in high-performance materials. The concept o* we present a review of the mechanics of crack deflection at a tide CFCS obviates the need for fiber coatings to affect crack deflection thereby providing opportunities for lower-cost manufacturing relative to that of conventional coated-fiber systems Furthermore fiber-matrix interface. The results are used to identify the critical on selection of all-oxide constituents, the long-term durability combination of matrix modulus and matrix toughness needed for requirements in the targeted high-temperature applications can be deflection and to motivate the subsequent mechanical measure- ments. The nature of the candidate matrix system, the processing Although the porous matrix concept offers new opportunities route, and the measurement procedures are described. Results for modulus and toughness are presented, along with predictions of for the development of damage-tolerant CFCCs, it also presents these properties from models of bonded-particle aggregates. Fi. hallenges in the design and synthesis of microstructures that meet nally, we address the implications on crack deflection in CFCCs. in the absence of fiber coatings, the matrix must be sufficiently aging time at which the crack deflection condition is no longer yet retain adequate strength to ensure acceptable off-axis proper atisfied and thus the damage tolerance is lost. ties. In principle, the combination of properties can be tailored through changes in the state of the matrix. For instance, improve- ments in the interlaminar strength and off-axis in-plane strength l. Mechanics of Crack Deflection can be obtained by reducing the matrix porosity: however, these mprovements come at the expense of a reduction in the damage As a minimum requirement for damage tolerance in CFCCs cracks in the matrix must either arrest at or deflect into the fiber/matrix interface rather than penetrate into the fibers. The conditions that satisfy this requirement are obtained from the He T A. Panthasarathy-contnbuting editor and Hutchinson diagram shown in Fig. 1. Crack arrest or deflection is predicted to occur when the ratio of interface toughness, Ii, to fiber toughness, T falls below the corresponding ratio of energy release rates. G /G, associated with deflection into Manuscript No 10088 Received March 28, 2003: approved September I the boundary and penetration into the fiber. The critical ratio is under Contract No, F49620-02-1-0128, monitored by Dr B. L Lee, as well controlled by the elastic mismatch parameter from NGK Insulator Ican Ceramic Society Department of Mechanical and Environmental Engineering -(E+
Joumal of the American Ceramic Sociery-Fujira er al Vol 87, No. 2 alumina precursor liquids in conjunction with either pure mullite particles or mullite/alumina particle mixtures. The latter approach has the advantage that it does not compromise the contiguity of the particle network and should yield a n e, But L 0.6 Crack Penetration it comes at the expense of additional steps in material processing 二 beyond that associate ted with particle mixtures alone. The present Crack Deflection paper explores the former topology ( based on particle mixtures) the latter is developed elsewhere. Porous mullite-alumina compacts were fabricated using a pres sure infiltration technique, The compositions ranged from pure mullite to 60% mullite 40%6 alumina by volume, The compo- sitions are henceforth designated by the percentages of mullite and alumina, i.e., 80M/20A denotes 80% mullite 20% alumina. A small number of pure alumina specimens were fabricated also, fo the purpose of assessing the efficacy of the mullite network in inhibiting shrinkage associated with sintering of alumina in the 0.0 0.2 0.4 0.6 0.8 1.0 mixed compacts. The mullite particulates were mullite(MU-107 Elastic mismatch parameter, a Showa Denko KK. Tokyo. Japan). The size distribution was rather broad. characterized by a mean diameter d I um and Fig. 1. Conditions for crack deflection at a fiber-matrix interface(solid diameters at the 1Oth and 90th percentiles of the distribution of d,o line). Experimental measurements from this study are superimposed 0. 4 um and don= 2.3 um, respectively. The alumina particu- symbols). The latter assume a toughness ratio w= T/n- I and fiber lates(AKP-50, Sumitomo Chemical, Tokyo, Japan), had size properties I= 15 J/m"and Er= 260 GPa distribution characterized by dav =-0.2 um, do= 0. I um, and dog=0.3 um. An aqueous slurry containing the powder mixture was prepared and vacuum-infiltrated into a mold with dimensions where E is the plane strain modulus, and the subscripts f and m 10 mm X 50 mm x 90 mm. The green compacts were dried and refer to fiber and matrix, respectively. For porous matrix systems, sintered at 900 C for 2 h to promote partial sintering at the particle c takes on relatively high values (0. 5), and hence the allowabl toughness ratio is also relatively high. In the systems of present mens and the specimens given a 2 h heat treatment at 1200"C To assess the stability of the microstructure under conditions that e, mullite/alumina, and hence the nature of bonding at the mimic the targeted service conditions for oxide CFCCs, some test fiber-matrix interface is similar to that between particles in the specimens were given additional aging treatments of 100 h or matrix. Consequently, the interface toughness is expected to be 1000 h at 1200'C, proportional to the matrix toughness: I=olm where o is a porosity, P, of the compacts was determined from mea- efficiency of particle packing along the fiber surfaces. it is pertinent densities of the constituent powders needed for calculat- expected that I,< Im such that o< 1. For conservative design. o is taken to be I The critical combination of matrix and fiber properties are obtained in the following way: For a 20. the energy release rate Even after the longest aging treatments, the powders had not ratio, G /G, can be represented by an approximate empirical sintered together and remained free flowing. The measurements revealed a very slight increase in the density of the pure mullite powder, from 3 14 to 3. 24 g/cm, as the aging time increased from This equation is an exact representation of the Hbe range l ti(2) 2-1000 h. The associated mechanism is unknown,although the rI1 values are in broad agreement with those reported in the litera- ture. In contrast, the alumina powder density remained ur hanged over the same range of aging time, at 4.01 t 0.01 g/cm 0,95 On setting G /G, =I /r in Eq(2)and combining the result The Youngs modulus of the fired compacts was measured using strain-gauged flexure specimens, 5 mm x 5 mm X 50 mm with Eq.(I), the deflection condition is re-expressed as follows long loaded in four-point bending. The inner and outer loading Here the Poissons ratios of the fiber and the matrix are assumed to be the same. Consequently, the plane strain modulus ratio Er/Em obtained from work of fracture measurements on chevron-notched can be replaced with Youngs modulus ratio Ep/Em) bend specimens with the same dimensions, but loaded in three-point bending and instrumented with an LVDT to measure ∑≡0.13 (3) load-point displacement. In all cases, the crack grew stably across the specimen cross section. The fracture surfaces were examined Here 2 is a nondimensional parameter that characterizes the subsequently via scanning electron microscopy (SEM) propensity for crack deflection. It identifies the critical combina Henceforth. the symbol is used to denote properties of the tion of TTr and En/Er needed to cause deflection TOUS C In light of this prediction. the subsequent experimental study ely: the subscripts A and M are used to denote pure focuses on the matrix properties Im and Em, with a view to and pure mullite; and the absence of a subscript implies a assessing whether the crack-deflection condition in Eq. (3)is mIx satisfied when the matrix is combined with the specified fiber, notably Nextel 720TM(3M Corp, St Paul, MN) IV. Pure Mullite Il. Materials and Test Procedures (1) Experimental Results The variation in the porosity of the pure mullite compacts with Two generic matrix topologies have previously been aging time is plotted in Fig. 2. Despite the slight change in powder The first involves the use of particle mixtures of m density, the porosity of the compacts remains essentially constant alumina and relies on the more rapid sintering kinetics over the entire range of aging times, within 0. 2% of its initial to bond the network together. The second is based on
February 2004 Mullite/Alumina Mixtures for Use as Porous Matrices in Oride Fiber Composites Aging time 0h.2h e 言 L 0.1 Alumina content.×A(%) Aging time, t(h) 1.02 Fig 3. Effects of aging time on Youngs modulus and toughne (b mullite. Power law fits of the experimental data yield exponents 90M/10A 0.03 and 0.28= 0.03 for modulus and toughness, respectively 100M with those predicted for surface diffusion controlled sintering BOM/20A 0.143ando.286 ≥o6. where r is sintering time. I, is a reference time and n is a constant both Ig and n depend on the transport mechanism. Sintering Alumina 70M/0A models yield values of n=3 for vapor transport, n= 5 for lattice diffusion, and n=7 for surface diffusion 094 In turn, for monosized spherical particles with small junctions 60M/40A scales linearly with junction radius in accordance with / segregate (a/R < I, the Youngs modulus of a bonded particle ag 0.92 Aging time, t(h) E Fig. 2. (a) Effects of composition and aging time on compact porosity. where Ep and v are the Youngs modulus and Poisson's ratio of the The data for O h and 2 h are virtually identical to one another and ar particles: z is the particle coordination number (-6 for random indistinguishable on the graph. (b) Results in (a), normalized by the initial packing): D is the relative packing density: E is a numerical (green) porosity at the same composition. The solid line for alumina in the parameter: and g(v) is given by inset was obtained by dilatometry. the symbols are based on the Archimedes measurements, and all other lines are simply curve fits through g()=(1= For values of v in the range0≤v≤0.25.8(v)=1±0.004the dependence on v is extremely small and is subsequently neglected Both the modulus and the toughness of the mullite comp This result, with E=I was derived by Walton, using Hertzian bly with aging g time, by a factor of -3-4(Fig 3). contact mechanics to describe junction stiffness and assuming a These property changes indicate sintering at the particle junctions, uniform aggregate strain. A more rigorous model that accounts for but the absence of porosity change implies a mechanism that does finite junction size and multiparticle interactions yields results in the domain of small junctions(a/R s 0.3)that are consistent with include transport of matter from surface sources via either surface Eq (5)when E is taken to be -076(simulation of modulus has diffusion, lattice diffusion, or vapor transport. The dominant been performed for two limiting cases: assuming that the torsional mechanical measurements in the manner described in the follow- of results are fit well by Eq. (5). using 5=0.65 and 0.88, Ing section. respectively. The average of these, E=0. 76 is used in the present SEM observations of the fracture surfaces reveal only a small work). Combining Eqs. (4)and (5) and taking g(v)= I yields the number of well-defined broken junctions(Figs. 4(a) and (b)), time-dependence of the modulus These are somewhat more prevalent after the longest agi because of the increase in junction size. The small number g time Junctions indicates that fracture normally occurs"cleanl r of such E nly"without E=0.76127 (7) appreciable crack meandering, suggesting a low fracture energy. It also precludes direct measurement of junction size. The relationship between toughness and junction radius for a bonded particle nte has been obtnined herical simulations of fracture, using a technique based on the discrete When the junction radius, a, is much smaller than the partic le element method(DEM). Details of the method and its implemen radius,R, junction growth owing to sintering follows a power law tation are presented elsewhere. Briefly, a computer-simulated of the form: 3. 1-4 sintering algorithm is used to generate a random three-dimensional array of spherical particles with a prescribed junction radius. The simulated aggregates consist of1000 particles. A crack is defined by a plane separating particles that have had the junction
Journal of the American Ceramic Society--Fujita et al Vol. 87. No. 2 OOM/OA 100M 0M4 60M40A Fig 4. Fracture surfaces of compacts that cover the range of alumina content, from O% (a, b) to 40%(c, d). following a 1000 h aging at 1200 C. Examples of fractured particle junctions are highlighted by the arrows between them broken. The simulation proceeds by incrementally where Ti is the intrinsic junction toughness. The results of the increasing the remote displacement while allowing the junctions at simulations ( Fig. 5)are well described by the crack tip to lail he junction stress reaches a critical bining Eqs. (4) and (9) yields the time dependence of the (10) rr=12(aR)2 predictions of modulus and toughness are compared with rimental measurements in Fig. 4. The best correlations are by taking n= 7 in Eqs. (7)and (10). The inference is that growth occurs predominantly by surface diffusion. For this mechanism, the reference time is given by 0.1 =1126Dy where k is Boltzman's constant, T is absolute temperature. 8, is the 0.2 030.40.5 effective surface thickness, D. is the surface diffusion coefficient. Relative junction size, a/R y is the surface energy (I J/m"), and n is the molecular volume (2.2x 1028 m for mullite Fig. 5. Results of fracture simulations based on the discrete element A further assessment of the models was made in the following way: The numerical values of the various parameters for mullite
February 200-4 Mullite/Alumina Mixtures for Use as Porous Matrices in Oxide Fiber Composites 265 are known, with the exception of the junction toughness T, and the effective surface diffusivity 8.Ds. The diffusivity was inferred from the modulus data by equating the proportionality constant ing t obtained from a In power law fit of the experimental measure- ments(Fig. 3) with the value predicted by Eq. (7). This yields Ds=3.8 x 10 m /s. The junction toughness was then aou ●100h 口1000h btained by equating the proportionality constant obtained from a 2/7 power law fit of the toughness measurements with the value predicted by Eq. (9). making use of the diffusivity inferred from the modulus measurements. The result is I a 3 J/m". This value is only slightly higher than the surface energy contribution, T= 2y-2J/m, and well below the toughness of bulk polycrystalline mullite, I'a 20-30 J/m. The latter toughness has significant contributions from crack tilting and twisting and associated step formation when cracks propagate by transgranular cleavage This mechanism is not operative at the particle junctions because the particles themselves are single crystals, and the junctions between them are essentially grain boundaries Alumina content, x(%) V. Mullite/Alumina Mixtures (l) Experimental Results Aging time The physical properties of the mullite/alumina mixtures are o 2 h ummarized in Fig. 2. The initial porosity exhibits a shallow 100h 口1000h 山pm20)h minimum porosity is only -2% lower than that of pure mullite. For moderate alumina content, Xa s 0.2, the porosity remains extremely stable during aging. within 0. 5% of P. For higher 日 agIn transition occurs near the expected percolation threshold, i.e., for 3 monosized spheres with random dense packing, the threshold is predicted to be at -0.25. Interestingly, even beyond the thresh old(.>0.25). the porosity change occurs remarkably slowly For example, for XA =0.4, the porosity reduction is%% of P. after a 1000 h aging. By comparis re alumina attains this porosity reduction after only 10 min at the same temperature Alumina content.×A(° of Fig. 2(b)). The results demonstrate that the mullite netw effective in inhibiting densification even for relatively cts of composition and aging time on (a) Young's modulus amounts of sinterable phase, significantly beyond the percolation Fig. and (b) toughness. Solid lines are model predictions (Eqs. (16)and (18)). threshold with A=2. n=3 and v=I For each aging treatment, the modulus increases approximatel nearly with X.(Fig. 6(a) at a rate characterized empirically by E= EM(I+ BXA) mullite-alumina mixtures. Here, three junction types are present alumina-alumina (AA). alumina-mullite(AM), and mullite-mullite where B 2-3. The increase is caused in part to the difference in (MM). The modeling approach involves an averaging of the particle moduli: EM-200 GPa vs. EA =400 GPa. But a pertinent junction property, weighted by the number fraction of the rule-of-mixtures prediction based on modulus difference alone associated junction type, to obtain the corresponding aggregate yields a lower modulus increase notably assumed to be the same size and arranged randomly in the mixture. E (13) A rudimentary statistical analysis yields the junction fractions. f: The disparity is attributed to the unction sizes at the alumina-alumina and alumina-mullite contacts, relative to mullite f=(1-XA)2 mullite, because of the more rapid features are incorporated in the model presented in the following fAM=2XA(1-XA) (15c) section. Similar empirical trends are obtained in toughness, char s modulus of acterized by has been derived following an approach similar to that of Walton' T=TM+YXA) (14) for monophase aggregates, making appropriate changes to reflect differences in junction characteristics: (i) The description of The fracture surfaces of the mixed particle compacts are similar paricles on either side of the junction. For dissimilar particle where y =2 junction stiffness is modified to account for the moduli of the to those of pure mulllite in that there is little evidence of broken junctions. the junction stiffness in the Hertzian limit is proportional to 2M(I+ A) where A- EA/EM (ii) The junction stiffness is junctions even after the longest aging time. Some examples are further proportional to the junction radius. In light of the relative shown in Figs. 4(c) and (dx sintering rates of alumina and mullite, the radii of the three (iil)The (2) Modeling junction types are expected to rank as ax 7dan7ye thunctionge The preceding models for junction growth and associated of the areas of the alumina-alumina and mullite-mullite junctions
Journal of the American Ceramic Society--Fujita er al Vol. 87. No. 2 V(+m/ where n is a junction area ratio, defined by n- and the alumina-mullite (".(iv) The modulus is determined from the unction stiffness, weighted by the fraction of the three Using the arithmetic aver types, given by Eq.(15). Both the arithmetic mean harmonic mean are considered Using the arithmetic mean of the junction stiffness, the resulti 1+y ggregate modulus is predicted to be =XAVn+2X(I-XAl X E 1+n)/2A - XAA m+2XA(1-XA) V2(1+A 3)+=xy The models were implemented in the following way. The modulus (16) EM of the pure mullite aggregate was obtained from Eg. (7) additive terms on the right side represent the contribu calibrated according to the procedures outlined above. Based on alumina-alumina, alumina-mullite, and mullite-mullite the known elastic moduli of the particles, A-2 In comparing Eq respectively. Analogously. the modulus prediction ob- (15)to the experimental data, the remaining unknown variable, n was used as a fitting parameter. The results of this fitting are on the harmonic mean of the junction stiffness is: shown in Fig. 6(a). The procedure yields n-3+ I for all aging E+2x0-x0(2+=8 times. This value is qualitatively consistent with the expectation that the alumina-containing junctions sinter more rapidly than those with only mullite, and hence their junctions are larger. A (17) predictions with the experimental me pare the toughness model similar procedure was used to ce A preliminary assessment of these results has been made the value of n inferred from the modulus measurements was used arough comparisons with numerical simulations based on the in Eq(18). The remaining unknown variable, y, was used for discrete element method. The nature of the simulations is essen- fitting(Fig. 6(b)). The procedure yields a junction toughness ratio tially the same as those used for determining toughness of particle y= 10=0.3. This result suggests that the nd that the increase chesses of the ggregates, with the exception that the aggregate now consists of two particle types with elastic properties given by those of alumina in aggregate toughness with alumina content is mainly caused by nd mullite (A-2). In the present simulations, all junctions are the increase in the average junction area. The quality of the fits to assumed to be of the same size(n= 1). Figure 7 shows the results both the modulus and the toughness data for all three aging times for two sets of simulations, for junction radii a/R= 0.053 and appears satisfactory, especially in light of the scatter in the 0.313. The results are presented in terms of a modulus parameter experimental data. Furthermore, it is recognized that the scatter hat varies from O for X0 to I for Xa= I. Also shown are the precludes determination of the fitting parameters n and y with predictions from the analytical models(Eqs. (16)and(17)).Over Can eels hoe ange of XA, the predictions from the two analytical entration of the stiffer particles approaches zero(XA-0 in VI. Implications for Crack Deflection his case), the arithmetic mean method vields more accurate esults. By contrast, in the limit where Xa- l, the harmonic mean Both the experimental results and the model predictions have method is more accurate. For the range of compositions considered een used to calculate the variation in the crack deflection preferred and is used in subsequent calculations mean method is parameter, S. with aging time, assuming the mixtures to be An analogous analytical model has been developed for the The results are plotted on Fig. 8. Over the entire range of toughness of a mixed aggregate. Following Eq (9). the contribu- parameter values considered in this study, the crack deflection tion from each junction type is assumed to be proportional to condition is predicted to be satisfied: that is, 2>1.The T(alR) weighted by the corresponding number fraction (Eq (15) experimental results are also plotted on Fig. I, demonstrating that The junction toughnesses are denoted TAA, TMM and T AM: the alumina to mullite toughness ratio is defined as y TAA/TMM 1.0 闪 100M/OA OM/10A DEM:aR=0.053,2=6.15 80M/20A o8“0“ DEM: a/R=0313z=68 ∠70M30A Arithmetic mean(Eqn. 16 60M/40A Harmonic mean(Eqn. 17) 0.6 DMOA 0.4 口9oM10A 80M20A O 70M/0A ◆60M40A 02 Aging time, t (h) Alumina content, x f aging time and composition on the crack deflection fiber properties Tr= 15 J/m"and E,=260 Numerical simulations and analytical results( Egs. (16) and lines are editions. Extrapolations of the predictions erical monosized alumina I yield the mes, Ie, at which crack penetration is ullite particles (A=2 and n= 1) occur
February 200-4 Mullite/Alumina Mixtures for Use as Porous Matrices in Oxide Fiber Composite the property combinations lie within the domain of crack deflec References tion in the He and Hutchinson' diagram. These results reaffirm mullite-alumina mixtures as good candidate matrixes for use in C.G.Levi..Y,Yang.B,. Dalgleish, F. w. Zok, and A Gi. Evans, "Processing and Performance of an AIl-Oxide Ceramic Composite!".Ann. Ceram. Soc. 8I oa on extrapolation of the predictions in Fig 8 to 2-0-1. the oxide composites, even with rather high alumina content. M. A Mattoni and F w. Zok. to be published. M. A Mistloni, I. Y. Yang. C G. Levi, and F w. Zok."Eflects of Matrix Porosity ange from 4000 h for 60M/40A to 60 000 h for pure inL. Ceran. SN. 84 omparable to the targeted service times for CFCC comp A Heathcote, x.Y. Gong, J. Y: Yang U. Ramamurty and F. W.Zok, "In-Plane Based on the differences in t. matrixes with lower Mechanical Properties of an Al-Oxide Ceramic Composite. "J. An Cerum. Sor, 82 contents would be preferred 1012721-30(1999 A. V. Carelli. H Fujita, J. Y, Yang, and F. w. Zok, "Effects of Thermal Aging on the Mechanical Properties of a Porous Matrix Ceramic Compcsite. "J, Am, Cerant G. Levi. F. W. Zok. J. Y. Yang. M. Mattoni, and J. P. A. Lofvandet Microstructural Design of Stable Porous Matrices for All-Oxide Ceramic Compos- ites," Z Mletallkd, 90 1121 1037. Mullite-alumina mixtures exhibit physical and mechanical char M-Y He and J W. Hutchinson, "Crack Deflection at an Interface betwee acteristics that make them attractive for use as porous matrixes in Dissimilar Elastic Materials. "Int. J. Soitds S9/991 LK251911053-6701989 Density of Composite Powder ' H. Fujita. G Jefferson, C. G. Levi, and F. w. Zok. "On the Use of an Alumina for alumina concentrations <20%, and combinations of modulus Precursor for Controlling the Mechanical Properties of Porous Mullite/Alumina and toughness that should lead to deflection of a matrix crack at a Am, Ceran Soc. in review uASTM Designation C 20-92. Book of ASTM Standards, American Society for fiber-matrix interface. The models of mechanical properties of Testing and Materials, West Conshohocken. PA. particle aggregates presented here coupled with the crack deflec- I Ceramic Souree h. The American Ceramie Society. Westerville. OH, 1990 tion parameter provide a means of extrapolating the experimental G. Tattersall and G. Tappin. "The Work of Fracture and Its Measurement in data to longer aging times, for the purpose of determining the M F, Ashby,"A First Report on Sintering Diagrams. "Acta Metall. 22 131 critical times at which damage tolerance will be lost. With 275-89(1974) knowledge of the activation energies of the sintering mechanisms. " R. M. German,"Fundamentals of Sintering": pp. 260)-69 in Ceramics and the models could be extended to make such predictions for other Glases, Engineered Materials Handtok4ASM Intemational, Materials Park,OH Temperatures. IK. Walton. "The Effective Elastic Moduli of a Random Packing of Spheres,J Design of porous matrixes requires consideration of other Mech. Phyx Solids, 35 [21213-26 (1987 roperties, including shear and interlaminar strength, as well as G. Jefferson.G. K. Haritos, and R. M. Mc Meeking. " The Elastic Response of a Cohesive Aggregate: A Discrete Element Model with Coupled Particle Interaction. mpressive strength. These generally increase as the matrix is Mech. Phys. Solids,50 (12)2539-75(2002) strengthened. As a result. among the mixtures considered in this G. jefferson and R. M. McMeeking, to be published. study, those with higher alumina contents would be preferred. in I B. R Lawn, Fracture of Brittle Solids. and Ed. Cambridge Solid State Science contrast to the mullite-rich mixtures selected based on crack Series. Cambridge University Press, 1993 R. M. German, Particle Packing Characterixtics, Metal Powder deflection considerations. These issues are currently being ederation Princeton. NJ, 1989. aK. L Johnson, Centet Mechanies. Cambridge University Press. 1985
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