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Taiyo Maeda, [Toshio Osada](https://orcid.org/0000-0003-1539-9264), [Shingo Ozaki](https://orcid.org/0000-0003-3450-6774)

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[Reliability evaluation scheme for ceramics based on defect size distribution inversely estimated from standardized tests](https://mdr.nims.go.jp/datasets/8a8538c4-343f-4861-8c17-f268986b9cdb)

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Reliability evaluation scheme for ceramics based on defect size distribution inversely estimated from standardized testsReceived: 16 January 2025 Revised: 18 April 2025 Accepted: 6 May 2025DOI: 10.1111/jace.20660RESEARCH ARTICLEReliability evaluation scheme for ceramics based on defectsize distribution inversely estimated from standardized testsTaiyo Maeda1 Toshio Osada2,3 Shingo Ozaki2,31Graduate School of Engineering Science,Yokohama National University,Yokohama, Japan2High-Reliability Heat-ResistantMaterials Group, Research Center forStructural Materials, National Institute forMaterials Science, Tsukuba, Japan3Division of System Research, Faculty ofEngineering, Yokohama NationalUniversity, Yokohama, JapanCorrespondenceShingo Ozaki, Division of SystemResearch, Faculty of Engineering,Yokohama National University,Yokohama, Japan.Email: s-ozaki@ynu.ac.jpFunding informationNew Energy and Industrial TechnologyDevelopment Organization, Grant/AwardNumber: JPNP22005AbstractCeramic components fracture stochastically, resulting in scatter and size depen-dence of the strength. Therefore, the strength properties obtained from astandardized test cannot be directly used to determine the design strength ofthe arbitrary-shaped components under different boundary conditions for brittleceramics. In this study, we propose a novel strength evaluation scheme where aninversely analyzed defect distribution is used as a common indicator to evalu-ate the reliability of brittle ceramic components. First, the defect distribution ofthe target lot is inversely estimated as a common indicator based on experimen-tal results of the bending strengths using the swarm intelligence optimizationmethod. The scatter of the bending strength tested under different boundary con-ditions is then predicted using the estimated defect distribution. The analyzeddefect distribution was found to be consistent with the observed distributionand can be used to predict other experimental results. The versatility and limita-tions of the scheme were discussed by examining the selection of a standardizedstrength test for optimization and the impact of model discretization on theinverse estimation of defect distribution. The results suggest that the proposedscheme can be applied to evaluate the strength scatter of components witharbitrary shapes and under arbitrary boundary conditions.KEYWORDSdefects, extreme value statistics, inverse analysis, strength analysis, Weibull plot1 INTRODUCTIONCeramics are used as structural materials in automobileand aircraft engine components because of their lightweight and excellent heat and corrosion resistance.1–5 Inrecent years, there has been increased research on theadditive manufacturing of ceramics and its applicationto complex-shaped components.6–11 However, ceramicsare brittle and highly susceptible to defects, resulting inThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided theoriginal work is properly cited.© 2025 The American Ceramic Society.scatter and size dependency on the strengths of thosecomponents.12–15 For components of ductile materialssuch as metals, various strength indices such as ultimatetensile strength and 0.2% proof stress (or yield stress)obtained from “standardized tests” can be directly appliedto the reliability design of components under actualworking loads. By contrast, this conventional reliabilitydesign scheme is not applicable to components of brit-tle materials. The strengths of ceramic components ofJ Am Ceram Soc. 2025;108:e20660. wileyonlinelibrary.com/journal/jace 1 of 14https://doi.org/10.1111/jace.20660https://orcid.org/0009-0007-4552-7251https://orcid.org/0000-0003-1539-9264https://orcid.org/0000-0003-3450-6774mailto:s-ozaki@ynu.ac.jphttp://creativecommons.org/licenses/by/4.0/https://wileyonlinelibrary.com/journal/jacehttps://doi.org/10.1111/jace.20660http://crossmark.crossref.org/dialog/?doi=10.1111%2Fjace.20660&domain=pdf&date_stamp=2025-05-12MAEDA et al. 2 of 14F IGURE 1 Schematics of strength evaluation schemes for ceramic components: (A) conventional scheme, (B) proposed scheme.different sizes and shapes can also be evaluated to someextent using Weibull statistics and the concept of effectivevolume (Figure 1A), assuming that the Weibull modulusm, a statistical material constant, is constant regardlessof the effective volume.14,16–19 However, since the Weibullmodulus shows a large deviation depending on the num-ber of test specimens,14,15,20 it is difficult to determine thetrue value of m with a realistic number of tests. Further-more, since the m-values can vary depending on effectivevolume,14,15 the assumption thatm= constant inWeibull’ssize effect law cannot hold. In addition, it is difficultto evaluate the effective volume of a complex-shapedcomponent under actual load conditions, and the strengthindex estimated based onWeibull’s size effect law does notalways guarantee the reliability of components as shown inFigure 1A. From a cost standpoint, evaluating the strengthreliability of target components using strength tests withmany components under service conditions is not realistic.Thus, novel strategies are required to correlate the resultsof standardized strength tests with the strength evaluationof components to advance the reliable design of ceramics.Ozaki et al.15,21–23 proposed a numerical simulationmethod for alumina fine ceramics to predict the scatterof the bulk strength of a component from microstructuralinformation, such as the distribution of pore size, poreaspect ratio, and grain size (referred to as “forward anal-ysis” in this study). These studies predict the bulk strengthscatter based on defect distribution data obtained frommicrostructural observations of the target material andalso the strength scatter undermultiple test conditions.15,22Various studies have reported a strong correlation betweenstrength scatter in ceramics and the distribution of defectsgenerated in the manufacturing process.12–15,17–19,21–31 Inparticular, Andreasen reported the prediction method offracture statistics for components with diverse shapesand boundary conditions by correlating Weibull distribu-tions based on test results with extreme value statisticaldistributions of crack sizes.18 Cook and DelRio reportedattempts to estimate flaw populations from strength testresults and demonstrated its effectiveness.19,28 These pre-vious studies implicitly suggested that microstructuraldistribution features such as defect distribution, ratherthan strength distribution obtained from specific tests, areconsidered appropriate as common indicators for reliabil-ity design strategies for brittle components of the samelot.Thus, in this study, we propose a novel strength evalu-ation scheme based on the new assumption that the dis-tribution feature of the defect is constant, rather than theconventional assumption that them-value, an indicator ofstrength distribution, is constant, as shown in Figure 1B.The proposed scheme can evaluate the strength scatter ofcomponents with diverse shapes and boundary conditionsby using the inversely estimated defect distribution fromstandardized strength test results as a universal distribu-tion indicator. This is difficult to achieve using the conven-tional scheme. Therefore, this will be a useful method forperforming the reliability design of ceramic components 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License3 of 14 MAEDA et al.more efficiently and accurately. To demonstrate thescheme, the following topics are addressed in this study:1. Particle swarm optimization (PSO),32 a type of swarmintelligence optimization method, is applied to the for-ward analysis method (microstructural information→strength distribution)15,21–23 to inversely estimate thedefect distribution (equivalent crack length distribu-tion) based on the results of standardized bending testsfor alumina specimens.2. The effectiveness of the proposed scheme for predictingthe strength scatter of differently sized specimens underdifferent loading conditions (other forward analyses) isdiscussed using the estimated equivalent crack lengthdistribution.The generalized extreme value (GEV) distribution,33a type of extreme value statistical model, is adoptedfor the mathematical description of significant equiva-lent cracks distribution because only large defects thatcould be fracture origins are the focus. Extreme valuestatistics have been widely applied in metal fatigue anal-ysis, and the relationship between fatigue strength andinclusion size is discussed.34–38 Studies have also been con-ducted on ceramics, relating pore size and cracks to staticstrength.18,39–412 TARGETMATERIAL ANDSTANDARDIZED TESTSIn this study, AS999 (Ferrotec Material Technologies Cor-poration, Japan), sintered plates of high-purity alumina,were used as the target material. Reported dataset15 forfour types of bending tests with different effective volumesobtained by varying the external and internal span lengthswere used to demonstrate the proposed scheme. Thedatasets include two types of three-point bending tests(3pb-S: external span length of 16 mm; 3pb-L: externalspan length of 30mm) and two types of four-point bendingtests (4pb-S: external span length of 30 mm; 4pb-L: exter-nal span length of 60 mm) with internal span lengths of10 mm. Following the Japanese Industrial Standard (JIS R1601), the surfaces of the specimens were mirror-polished,and the bending tests were conducted. The experimentalsetup comprised a bending testing machine (AG-X plus,10 kN, Shimadzu Corporation) and attached bend test jigs.Here, bending tests were performed at a crosshead speedof 0.5 mm/s at room temperature. The specimens and testgeometries are listed in Table 1. All the specimens used inthe bending tests were fabricated from the same lot. ModeI fracture toughness KIC of AS999 was reported as 4.0 MPam0.5.15F IGURE 2 Analysis model of bending test with the fracturemechanics model implemented: (A) specimen and test geometries,(B) schematic for determining the fracture load PE corresponding tothe i-element of the specimen.Bending strength σB was obtained using Equation (1):𝜎B =3𝑃B(𝐿o − 𝐿i)2𝑏ℎ2, (1)where PB is the peak value of the jig reaction force, Lo isthe external span length, Li is the internal span length, bis the width of the specimen, and h is the thickness of thespecimen (Figure 2A).The bending strength of each specimen was organizedusing the following two-parameter Weibull distribution(Equation 2)42:𝐹(𝜎B) = 1 − exp{−(𝜎B𝛽)𝑚}, (2)wherem and β are theWeibull modulus and scale parame-ter, respectively. Cumulative fracture probability F(σB) was 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons LicenseMAEDA et al. 4 of 14TABLE 1 Dimension of the test specimen, span length, and strength properties of the bending test.15Specimen geometry Test geometry Strength propertyTest type b [mm] h [mm] Lo [mm] Li [mm] m [–] β [MPa] 𝝈𝟎.𝟎𝟎𝟕%𝑩𝝈𝟗𝟗.𝟗𝟗%𝑩3pb-S 4 3 16 0 14.1 412.6 208.9 484.63pb-L 30 14.5 383.9 198.3 448.64pb-S 30 10 14.9 352.1 185.0 409.84pb-L 60 18.9 338.3 203.8 381.3Note: The Mode I fracture toughness KIC of AS999 was reported as 4.0 MPa m0.5.15calculated using themedian rankmethod. Under each testcondition, the number of specimens was N = 30.The bending test results obtained under these condi-tions are listed in Table 1. Specifically, Weibull modulusm and scale parameter β determined using maximumlikelihood estimation are listed. In addition, bendingstrengths for 0.007% and 99.99% cumulative fracture prob-ability 𝜎0.007%𝐵 and 𝜎99.99%𝐵 corresponding to the weakest(ascending rank: 1) and strongest (ascending rank: 10 000)strength assumingN= 10 000, respectively, are also shown.They were estimated from m and β in each test con-dition and Equation (2). The fracture origins of all testspecimens were confirmed to be internal defects contain-ing unsintered grains and not surface flaws caused bymachining.153 FORWARD ANALYSIS OFSTRENGTH SCATTERThis section presents the forward analysis model for thebending tests. In addition, the application of extreme valuestatistics to describe equivalent cracks is explained.3.1 Analysis model of bending testsTo simulate the bending tests, the specimens were dis-cretized with cubic elements on one side he, as shown inFigure 2A. When a load P is applied, the bending stressoccurring in each element σE is given as Equation (3):𝜎E =⎧⎪⎨⎪⎩𝑌E(𝐿o−𝐿i)4𝐼𝑃𝑌E(𝐿o∕2−𝑋E)2𝐼𝑃forfor𝑋E < 𝐿i∕2𝐿i∕2 ≤ 𝑋E < 𝐿o∕2, (3)where I is the moment of inertia of the area, XE is the dis-tance from the center cross-section of the specimen to anarbitrary element (calculation point), and YE is the dis-tance from the neutral plane. In what follows, evaluationpoints of bending stresses σE were selected at the center ofeach element.Because the strength scatter in brittle ceramic com-ponents originates from differences in the location offracture origins and differences in local strength owingto defect distribution, it is necessary to evaluate thelocal fracture stress. To evaluate the local fracture stressσt around the intrinsic defects, we adopted the follow-ing Griffith/Irwin-type equation based on linear elasticfracture mechanics for the fracture mechanics model(Equation 4):𝜎t =𝐾IC√𝜋𝑎e, (4)where ae is the equivalent crack length corrected forshape effects for cracks of various shapes and lengths.As mentioned earlier, this study considered only internaldefects because all specimens were fractured from defectscontaining unsintered grains.15We explain the forward analysis, in which the scatter ofthe bending strength is numerically determined using thestress evaluation and fracture mechanics models given byEquations (3) and (4). First, the inverse function method43is applied to the appropriate cumulative distribution func-tion of the equivalent crack length to generate randomnumbers for each element. Each equivalent crack lengthae is then transformed into a local fracture stress σt viathe fracture mechanics model (Equation 4). This enablesthe generation of a specimen model with distributed localfracture stresses. Next, we evaluate the applied load Pwhen the bending stress σE calculated using Equation (3)exceeds the local fracture stress σt for each element, asshown in Figure 2B. The applied load when σE exceedsσt in each element is taken as PE. The minimum PEwithin all elements is considered the peak load PB of thetarget specimen under test, and the bending strength σBis calculated using Equation (1). By performing the sameprocedure for multiple specimen models, the strengthscatter can be evaluated using a two-parameter Weibulldistribution (Equation 2). The bending stress at the initialfracture and the bending strength (maximum bendingstress) obtained from the bending test were assumed tobe the same because the fine-grained alumina exhibitslimited R-curve behavior.44 In addition, the bending stress 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License5 of 14 MAEDA et al.distribution was assumed to be similar to that of fracturebecause the fine-grained alumina exhibits minimal plasticbulk deformation at room temperature.453.2 Distribution function of equivalentcrack lengthBecause the largest equivalent crack in an element (blockmaxima) can be a candidate for the fracture origin ofeach element, we focused only on the largest equivalentcracks in each element (see Figure 2A). Therefore, theGEVmodel was employed to statistically describe the distribu-tion characteristics of a large equivalent crack length ae.In the GEV model, all the data of interest are divided intoblocks of equal size, and the maximum values of eachblock are treated. In the analysis, the blocks correspondto elements. The cumulative distribution function G andprobability density function g of the GEV distribution aregiven by Equations (5) and (6)33,46:𝐺(𝑎e) =⎧⎪⎨⎪⎩exp{−[1 + 𝜉(𝑎e − 𝜇𝜎)]−1∕𝜉}exp{−exp[−(𝑎e − 𝜇𝜎)]} forfor𝜉 ≠ 0𝜉 = 0,(5)𝑔(𝑎e) =⎧⎪⎪⎨⎪⎪⎩exp{−[1 + 𝜉(𝑎e − 𝜇𝜎)]−1∕𝜉} 1𝜎{1 + 𝜉(𝑎e − 𝜇𝜎)}−1∕𝜉−1exp{−exp[−(𝑎e − 𝜇𝜎)]} 1𝜎exp(−𝑎e − 𝜇𝜎) forfor𝜉 ≠ 0𝜉 = 0, (6)where μ, σ, and ξ are the location, scale, and shapeparameters, respectively. The shape parameter ξ is closelyrelated to the tail behavior of its probability densityfunction. In particular, when the distribution is a long-tailed type with no maximum value, it is called theFréchet distribution in the case of ξ > 0. By con-trast, a short-tailed type with a finite upper endpointis called a negative Weibull distribution. Moreover, itcorresponds to the Gumbel distribution in the case ofξ = 0.33,464 INVERSE ANALYSIS OF THEEQUIVALENT CRACK LENGTHDISTRIBUTIONThis section describes a method for the inverse estimationof the equivalent crack length distribution correspondingto block maxima data using the standardized test resultsshown in Figure 1B. The objective function was set basedon the Weibull plot indices obtained from the bendingtests shown in Table 1, and the parameters of the GEVmodel that can reproduce the strength scatter were opti-mized. In this study, PSO32 was employed among variousoptimization methods.PSO is a heuristic optimization method based on thecollective behavior of a flock of animals such as fish andbirds. This method is widely used in various fields owingto the simplicity of the algorithm, smaller number of con-trol parameters, and high convergence to a solution.47–51In the PSO, many particles (search individuals), whichare candidate solutions, are placed in the solution spaceto search for an optimal solution. The migration paths ofthe particles in the dimensional space corresponding toparameters of the GEV model (μ, σ, and ξ) are determinedbased on the experience and knowledge of each particleand the entire flock during the optimization process. Theoptimization process involves forward analysis using thecurrent parameters of the GEV model possessed by eachparticle to evaluate the fitness of the experimental results.The details of the PSO algorithm (Figures S1 and S2), theresponse surface of the objective function (Figures S3 andS4), the effects of particle population (Figures S5 and S6),and the parameter settings (Table S1) are described in theSupporting Information Materials.In PSO, the fitness level of each particle having (μ, σ,ξ)-values is evaluated using an objective function. In thisstudy, Equation (7) was adopted as the objective func-tion (Error) to ensure that the properties of the Weibullplot obtained from the experiment and simulation areconsistent.𝐸𝑟𝑟𝑜𝑟 =𝑛ref∑𝑖=1⎧⎪⎨⎪⎩(ln 𝜎0.007%Exp − ln 𝜎0.007%Simln 𝜎0.007%Exp)2+(ln 𝜎99.99%Exp − ln 𝜎99.99%Simln 𝜎99.99%Exp)2⎫⎪⎬⎪⎭ , (7) 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons LicenseMAEDA et al. 6 of 14F IGURE 3 Effect of number of specimens on scatter in (A)Weibull modulusm, (B) scale parameter β. Three different elementsizes are used for the examination. The test condition in thesimulation is the 3pb-S shown in Table 1. The input materialproperties and microstructural information are those reported by Itoet al.15 Each plot and its error bar show the mean value and theminimum and maximum values, respectively, for 100 simulations.where σ0.007% and σ99.99% are the strengths for0.007% and 99.99% cumulative fracture probability,corresponding to the weakest and strongest strength inN = 10 000 specimens, respectively. The subscripts “Exp”and “Sim” denote values obtained from experiment andsimulation, respectively. As described in Section 2, σ0.007%and σ99.99% are calculated using the Weibull distributionparameters m and β. Because the strengths of the higherand lower cumulative fracture probabilities stronglyreflect the characteristics of the Weibull distributionparameters, we used the upper- and lower-limit strengthsfor the objective function. Here, the population of equiv-alent cracks that could be fracture origins was assumedto be the same in the range of stress levels from σ0.007% toσ99.99%. nref in Equation (7) is the number of reference test(experimental) types.The Weibull distribution parameters obtained usinginsufficient specimens showed dispersion in both exper-iments and simulations. In these cases, the fitness levellacks stability and reliability. Therefore, the forwardstrength analysis in the optimization was conducted usinga sufficient number of specimens, which ensured thestability of the Weibull distribution parameters in thisstudy. Figure 3A,B shows the dispersion of the Weibullmodulusm and scale parameter β depending on the num-ber of specimens in the simulation, respectively, whosetest type is 3pb-S in Table 1. Here, the fracture tough-ness KIC = 4.0 MPa m0.5 and microstructural information(internal defect distribution) of AS999 reported by Itoet al.15 were input, and the discretized size per side of ele-ments in the simulation was set to he = 0.125, 0.25, and0.5 mm. The maximum, minimum, and mean values ofthe Weibull modulus and the scale parameter are shownby error bars, where 100 simulations were performed foreach number of specimens. The Weibull modulus m con-verged to almost the same value regardless of elementsize he, whereas the scale parameter β tended to increaseas the element size increased. This is due to the differ-ence in stress evaluation points: as in the ordinary finiteelement method, the larger the element size, the smallerthe evaluated bending stress value (Equation 3). However,independent of element size, both the Weibull modu-lus and the scale parameter became more stable as thenumber of specimens increased. Therefore, N = 10 000specimens were used for forward analysis in the PSO opti-mization because of the stability of theWeibull distributionparameters and computational cost.5 RESULTS AND DISCUSSION5.1 Estimation of equivalent cracklength distribution and reproducibility ofstrength scatterFirst, the GEV distribution of equivalent crack lengthswhen the block size is one element under the element sizehe = 0.25 mm was estimated based on the results of the3pb-S strength test (nref = 1). The search ranges for theGEV model parameters were μ = 0–30 µm, σ = 0–10 µm,and ξ = −0.5–0.5. The fracture toughness KIC in Equa-tion (4), required for the PSO optimization analysis, wasset to 4.0 MPa m0.5.15Figure 4A,B shows the inversely estimated equivalentcrack length distributions and measured equivalentcrack length data based on microstructural observationsusing the probability density and Gumbel probabilitypaper, respectively. Here, the estimation of the equivalentcrack length distribution was performed five times. The 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License7 of 14 MAEDA et al.F IGURE 4 Estimated equivalent crack length distributionsbased on experimental strength data under 3pb-S, and 4224datapoints of equivalent crack length converted from the results ofmicrostructural observation15: (A) probability density; (B) Gumbelprobability paper.TABLE 2 Parameters of the estimated equivalent crack lengthdistributions shown in Figure 4.Label Location μ [µm] Scale σ [µm] Shape ξ [–]Sim. 1 6.8 7.1 −0.03Sim. 2 11.3 5.5 0.02Sim. 3 21.5 2.2 0.18Sim. 4 22.8 1.8 0.22Sim. 5 8.7 6.6 −0.02observation-based data in the figures were converted toequivalent crack lengths by referring to the distributionsof pore size (major pore radius), pore aspect ratio (minorpore radius/major pore radius), and grain size for the samelot of AS999 reported by Ito et al.15 In addition, the blockmaxima data of equivalent crack lengths for 4224 elementsof the same size as those in the optimization analysiswere generated for comparison. Table 2 lists the estimatedequivalent crack length distribution parameters (μ, σ, andF IGURE 5 Comparison of Weibull plots betweenexperimental and simulation results for N = 10 000 specimensunder 3pb-S using estimated equivalent crack length distributionsshown in Figure 4. Dashed lines represent the 90% confidenceinterval of experimental results based on the maximum likelihoodestimation of the Weibull distribution.ξ). The estimated parameters showed large variations forthe performed numbers (five times) and did not perfectlyagree with the observed distribution. In particular, bothpositive and negative values were estimated for the shapeparameter ξ, resulting in large differences in the tailbehaviors of the distribution, as shown in Figure 4B.Nevertheless, the distribution properties are broadlyconsistent for cumulative probabilities within the limitedrange from approximately 90% to 99.9% (see Figure 4B).Figure 5 shows a comparison of the Weibull plots ofthe bending strengths for the 3pb-S test obtained from theexperiment (N = 30) and forward analysis (N = 10 000),which uses the inversely estimated equivalent crack lengthdistributions shown in Figure 4. Here, the element size inthe forward analysis was the same as in the optimizationanalysis: he = 0.25 mm. The dashed lines correspond tothe 90% confidence intervals of the experimental results.As seen from Figure 5, the results of all the forwardanalyses using the estimated distributions agree well withthe experimental result, within the range of cumulativefracture probabilities ranging from approximately 10% to90%. By contrast, the lower strength properties stronglyreflect the characteristics of the input equivalent cracklength distributions. For distributions with a positiveshape parameter ξ, that is, no maximum equivalentcrack length, the Weibull plot becomes a straight-linedistribution (Sim. 3 and Sim. 4). For distributions with anegative shape parameter ξ, the Weibull plot becomes acurve-type with a lower limit of strength (Sim. 1 and Sim.5). Table 3 lists the Weibull distribution parameters of theforward analysis results shown in Figure 5. The Weibull 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons LicenseMAEDA et al. 8 of 14TABLE 3 Weibull distribution parameters of the results of theforward analyses shown in Figure 5.Label m [–] β [MPa]Sim. 1 14.0 413.1Sim. 2 14.2 413.0Sim. 3 14.2 412.2Sim. 4 14.0 412.3Sim. 5 14.1 413.0distribution parameters have almost identical values,although the lower strength properties differ widely.This means that the Weibull distribution parameters arelargely influenced by the strength data with a cumulativefracture probability of approximately 10% to 90%. In thecase of test condition 3pb-S, the equivalent crack lengthsto reproduce the experimental strengths with cumulativefracture probabilities from approximately 10% to 90% cor-respond to approximately 90% to 99.9% of the cumulativeprobabilities in Figure 4B, and hence the distributionproperties are consistent in this region (N = 30).5.2 Effect of reference test data oninverse estimationA key to the reliable design of components is the accu-rate estimation of the defect distribution over a wide sizerange because the size of the defects that can becomefracture origins depends on the size and shape of the com-ponent and the boundary condition. In Section 5.1, it wasconfirmed that the estimated equivalent crack length dis-tributions agree with the microstructural observation dataonly in the range of equivalent crack lengths correspond-ing to the candidate of fracture origins in the referencetest (strength test results used for the inverse analysis).Therefore, in this section, the experimental results ofstrength tests for 3pb-S and 4pb-S listed in Table 1 weresimultaneously used as reference tests for a single esti-mation of equivalent crack length distribution (nref = 2).This is expected to allow a wider range of equivalentcrack lengths as candidates for fracture origins to beestimated.3pb-Swas selected as one of the reference tests because ithas the smallest effective volume, that is, the small equiva-lent crack could be a candidate fracture origin, among thefour test conditions listed in Table 1. 4pb-S was selectedbecause it is a test condition of JIS and has a large effectivevolume. Here, the element size was set to he = 0.25 mm,as in Section 5.1. The search ranges of the GEV modelparameters were set to μ = 0–30 µm, σ = 0–10 µm, andξ = −0.5–0.5.F IGURE 6 Estimated equivalent crack length distributionsbased on experimental strength data under 3pb-S and 4pb-S, and4224 datapoints of equivalent crack length converted from theresults of microstructural observation15: (A) probability density; (B)Gumbel probability paper.TABLE 4 Parameters of the estimated equivalent crack lengthdistributions shown in Figure 6.Label Location μ [µm] Scale σ [µm] Shape ξ [–]Sim. 1 17.1 3.5 0.10Sim. 2 15.7 3.9 0.08Sim. 3 14.0 4.4 0.06Sim. 4 16.3 3.7 0.09Sim. 5 15.3 4.0 0.08Figure 6 shows a comparison of inversely estimatedequivalent crack length distributions andmeasured equiv-alent crack length data obtained from microstructuralobservations. Figure 6A,B shows the probability densityand Gumbel probability paper, respectively. Estimation ofthe equivalent crack length distribution was performedfive times. Table 4 lists the estimated equivalent cracklength distribution parameters (μ, σ, and ξ). Comparedwith the results obtained using only 3pb-S as the referencetest (see Section 5.1), the estimated equivalent crack lengthdistribution showed less scatter and better agreement with 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License9 of 14 MAEDA et al.F IGURE 7 Comparison of Weibull plots between experimental and simulation results for N = 10 000 specimens under (A) 3pb-S; (B)3pb-L; (C) 4pb-S; and (D) 4pb-L using estimated equivalent crack length distributions shown in Figure 6. Dashed lines represent the 90%confidence interval of experimental results based on the maximum likelihood estimation of the Weibull distribution.the observation-based data. These results indicate that it ismore effective to use strength test results under multipleconditions as reference tests, particularly a combinationof those such that the fracture origin size is widely dis-tributed. Hence, to inversely estimate equivalent cracklength distribution more efficiently from a test cost stand-point, it is effective to use reference test conditions with assmall and as large effective volumes as possible. As a result,the uniqueness of the inversely estimated distributions andthe agreementwith the true value of the distributionwouldbe guaranteed not only on the tail side but also on thesmaller side.The accuracy of the strength data of the reference testsalso significantly affects the accuracy of the inversely esti-mated equivalent crack length distribution. The reliabilityof theWeibull distribution parameters is highly affected bythe number of specimens, particularly when the numberof specimens is small,14,20 as shown in Figure 3. Therefore,to improve the accuracy of the estimated equivalent cracklength distributions, it is ideal to increase the number ofspecimens in the standardized test to obtain highly reliablestrength data.Figure 7 shows comparisons of the Weibull plots ofthe bending strengths for the 3pb-S, 3pb-L, 4pb-S, and4pb-L tests obtained from the experiments (N = 30)and forward analyses (N = 10 000), which use theinversely estimated equivalent crack length distributionsshown in Figure 6. Here, the element size in the forwardanalyses was the same as in the optimization analy-sis: he = 0.25 mm. The dashed lines correspond to the90% confidence intervals of the experimental results. Theresults of the forward analyses reasonably predicted notonly the strength scatter of 3pb-S and 4pb-S, which werethe target of optimization but also the scatter under dif-ferent test conditions (3pb-L and 4pb-L) with differentsizes and loading conditions. It is suggested that theproposed strength evaluation schemewhich inversely esti-mates the equivalent crack length distribution of the targetlot from the standardized strength test and evaluates thestrength scatter of components with diverse shapes andboundary conditions using it as an input parameter, iseffective.The proposed scheme allows the direct evaluation of thestrength scatter of the target components using FEA15,21–23without the need to analyze the size dependency onthe strengths of the ceramic components or to evaluatethe effective volume in advance. In addition, the lowerstrength limit for a large number of components can be 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons LicenseMAEDA et al. 10 of 14F IGURE 8 Estimated equivalent crack length distributionsbased on experimental strength data under 3pb-S and 4pb-S withhe = 0.125, 0.25, and 0.5 mm, and equivalent crack lengthdistributions of 8- and 64-year AMS obtained from the distributionestimated with he = 0.125 mm: (A) probability density; (B) Gumbelprobability paper.evaluated by performing a forward analysis using the esti-mated equivalent crack length distribution, as shown inFigure 7.5.3 Impact of model discretizationThe effect of element discretization in the proposedscheme is discussed. Because the GEV model employedin this study was based on the “block maxima concept”,the correspondence between the element sizes of theoptimization and forward analyses should be noted.Figure 8A,B shows the inversely estimated equivalentcrack length distributions in the probability density andthe Gumbel probability paper, respectively, where the ele-ment size was set to he = 0.125, 0.25, and 0.5 mm. Theoptimization targets were the 3pb-S and 4pb-S test results,and the search ranges for the GEVmodel parameters wereμ= 0–30 µm, σ= 0–10 µm, and ξ=−0.5–0.5. The estimatedequivalent crack length distribution with he = 0.25 mmcorresponds to the result of Sim. 1 in Figure 6. The figuresalso show the equivalent crack length distributions of 8-and 64-year annual maximum series (AMS) with 8- and64-times block sizes (0.253 mm3 and 0.53 mm3) obtainedfrom the distribution estimated with he = 0.125 mm. Here,the distribution function Gn(x) of the n-year AMS can beexpressed using that of reference (1-year) AMS G(x) by the“statistics of extremes” as Equation (8)52:𝐺𝑛(𝑥) = [𝐺(𝑥)]𝑛. (8)Figure 8 confirms the estimated distribution shifts to thelarger side for larger element size he. This reflects the factthat, as the block size (element volume) increases, largerequivalent cracks can exist within that volume. Comparingthe distributions of n-year AMS obtained from the esti-mated distribution with he = 0.125 mm (hereafter referredto as n-year AMS) with the directly estimated distributionwith respective element sizes, the distribution of the 8-yearAMS and the directly estimated distribution is almost iden-tical for he = 0.25 mm. In contrast, for he = 0.5 mm, thedistribution of the 64-year AMS is slightly smaller thanthe directly estimated distribution, particularly in the tailregion. This is due to the accuracy of the calculated bend-ing stress distribution (Equation 3). In general, the finerthe discretization, the more accurately the stress distribu-tion is evaluated in the bending test analysis as describedin Section 4. Therefore, even if equal local fracture stressesare distributed to the bottom elements between the innerspans in a four-point bending test analysis, where the cal-culation points are at the center of respective elements,the bulk strength of the specimens with larger elementsizes will be higher. In this scheme, the estimated equiv-alent crack length distribution with he = 0.5 mm wasintended to reproduce the experimental strength scatterusing the element size used in the optimization rather thanthe actual equivalent crack length distribution. Thus, theoptimized distributionwithhe= 0.5mmwas slightly largerthan the 64-year AMS, whereas the strength distribu-tion characteristics were consistent with the experimentalresults.Figure 9A–C shows the results of the forward analy-sis of the bending tests for N = 10 000 specimens usingthe inversely estimated equivalent crack length distribu-tion with he = 0.125 mm and the equivalent crack lengthdistributions of 8- and 64-year AMS shown in Figure 8,respectively. Here, the element size in the forward analysisis 0.125 mm, 0.25 mm (8-year AMS), and 0.5 mm (64-yearAMS), respectively. The figures also show the experimentalresults listed in Table 1. As shown in Figure 8, the equiv-alent crack length distribution differs depending on the 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License11 of 14 MAEDA et al.F IGURE 9 Comparison of Weibull plots betweenexperimental and simulation results using (A) estimated equivalentcrack length distribution with he = 0.125 mm, (B) equivalent cracklength distribution of 8-year AMS, and (C) equivalent crack lengthdistribution of 64-year AMS shown in Figure 8. The element size inthe forward analysis is he = 0.125, 0.25, and 0.5 mm, respectively.Dashed lines represent the 90% confidence interval of theexperimental results.element size in the optimization; however, the results ofthe forward analyses in Figure 9A reproduce the strengthdistribution in the experiment well for all test conditions.The same tendency is confirmed for the results of the for-ward analyses in Figure 9B using the equivalent cracklength distribution of 8-year AMS. However, the resultsof the forward analyses in Figure 9C using the equivalentcrack length distribution of 64-year AMS tend to be higherthan the experimental results, reflecting the results shownin Figure 8. It is believed that 64-year AMS can also be usedfor forward analysis with he = 0.5 mm, provided that thestress distribution is limited (e.g., low-stress gradient andsimple tensile cases). Ito et al.15 reported that the resultsof the forward analyses for the same test conditions withhe = 0.125, 0.25, and 0.5 mm showed that the strengthscatter with he = 0.125 and 0.25 mmwere almost identical.The impact of model discretization in the proposedscheme can be summarized as follows:1. To inversely estimate the equivalent crack length dis-tribution, adopting an element size that can accuratelyreproduce the stress distribution (e.g., the bendingstress distribution) in the standardized test is desir-able. Then, the estimation results (he ≤ 0.25 mm in thiscase) are generally consistent with the actual equivalentcrack length distribution characteristics of the targetmaterial, as shown in Figure 6.2. When the equivalent crack length distribution obtainedin item 1 is applied as input parameters to a bound-ary value problem with arbitrarily shaped componentsand under arbitrary loading conditions, n-year AMScan be used to handle discretization arbitrarily n timeslarger than the element size used in the optimization,provided that the stress distribution in the model isappropriate.6 OUTLOOKIn this study, only internal defectswere considered becauseall specimens used the reference tests that fractured fromthem. However, it is difficult to eliminate surface flawsin actual components by machining due to productiv-ity aspects, particularly in large components, resultingin a mixture of internal and surface defects as fractureorigins.53–55 The proposed scheme assumes that the defectscausing fracture can be organized in the same population.Therefore, it is important to conduct screening based onfracture surface observations on the specimens used in thestandardized tests and classify them into types of fractureorigins (e.g., internal or surface, pores or inclusions). Inthe case of standardized tests in which the fracture ori-gins are both internal and surface defects, it can be handled 15512916, 2025, 9, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.20660 by National Institute For, Wiley Online Library on [03/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons LicenseMAEDA et al. 12 of 14by conducting inverse estimation of the defect distributionseparately for each, and inputting the inversely estimatedinternal and surface defect distributions for each internaland surface element when predicting the strength scatterof the actual components. Here, it should be noted thatthe definition of the unit block from which the maximumvalue is extracted may be different when organizing inter-nal and surface defects using the GEV model. In the caseof internal defects, it is the largest per volume, whereas inthe case of surface defects, it is the largest per area. Tomakethe proposed scheme more versatile, we should verify thispoint in the future.Meanwhile, it is also important to estimate “the allow-able surface crack size”, which is the surface crack size thatdoes not affect the strength performance of the component,and to eliminate in advance any component with surfacecracks larger than the allowable surface crack size. Usingthe proposed scheme, one predicts the strength scatter ofcomponents with diverse shapes and boundary conditionsbased on the results of a standardized strength test. Hence,by subsequently comparing the relationship between thestrength scatter due to internal defects and the fracturestrength due to surface crack, it could also be possible toevaluate the allowable surface crack size.567 CONCLUSIONIn this study,we proposed a strength evaluation scheme forthe reliability design of brittle ceramic components. Theeffectiveness of the proposed schemewas demonstrated bycomparing the results of the microstructural observationswith the equivalent crack length distribution obtained byPSO estimation. It was also confirmed that the inverselyestimated equivalent crack length distribution could beused to predict the strength scatter for the specimens ofthe four types of bending tests with different sizes andunder loading conditions. The versatility and caveats of thescheme were discussed by examining the selection of thestandardized strength (reference) tests and the impact ofmodel discretization on the estimation of the equivalentcrack length distribution. Even for brittle ceramic compo-nents that exhibit scatter and size dependency of strength,it is possible to predict the strength scatter under differ-ent boundary conditions using microstructural properties,such as equivalent crack length distribution, as a com-mon indicator. In this study, the GEV distribution wasemployed as the probability density function describingthe defect distribution of ceramics. Still, other distribu-tions such as log-normal and power-law types can be used.Furthermore, the optimization method can be substitutedfor algorithms such as genetic and reinforcement learningalgorithms.However, because the inverse estimation performanceof defect distribution depends on the objective function,a review of the objective functions will be considered infuture research. In addition, the proposed scheme assumesa single population of large defects inside the compo-nents. However, they may differ between the near-surfaceand interior regions in cases of large components andadditivemanufacturingmaterials. The effect of such differ-ences in microstructural properties on strength propertiesshould be considered. Moreover, in this study, althoughthe validity of the proposed scheme was verified for onlyone type of alumina, the applicability to other ceram-ics such as silicon nitride57–60 should be examined in thefuture. Here, we believe that the proposed scheme canbe applied to other brittle materials, particularly whenthe forward analysis can predict the results of the stan-dardized tests of the target materials. We will addressthese issues to generalize the proposed scheme in thefuture.ACKNOWLEDGMENTSThis article is based on results obtained from a project,JPNP22005, commissioned by the New Energy and Indus-trial Technology Development Organization (NEDO).ORCIDTaiyoMaeda https://orcid.org/0009-0007-4552-7251ToshioOsada https://orcid.org/0000-0003-1539-9264ShingoOzaki https://orcid.org/0000-0003-3450-6774REFERENCES1. OhnabeH,Masaki S, OnozukaM,Miyahara K, Sasa T. Potentialapplication of ceramic matrix composites to aero-engine com-ponents. 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See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttps://doi.org/10.1016/j.ceramint.2013.08.093https://doi.org/10.1016/j.ceramint.2013.08.093https://doi.org/10.1115/1.4010337https://doi.org/10.1115/1.4010337https://doi.org/10.1016/S0927-0507(06)13004-2https://doi.org/10.1016/S0927-0507(06)13004-2https://doi.org/10.1016/S0955-2219(00)00137-0https://doi.org/10.1016/S0955-2219(00)00137-0https://doi.org/10.1111/j.1151-2916.1964.tb12994.xhttps://doi.org/10.1111/j.1151-2916.1964.tb12994.xhttps://doi.org/10.1007/978-1-4471-3675-0https://doi.org/10.1007/978-1-4471-3675-0https://doi.org/10.1016/j.eswa.2011.02.075https://doi.org/10.1016/j.eswa.2011.02.075https://doi.org/10.1016/j.rser.2015.08.007https://doi.org/10.1016/j.rser.2015.08.007https://doi.org/10.1016/j.pnucene.2018.11.003https://doi.org/10.1016/j.pnucene.2018.11.003https://doi.org/10.1016/j.jksuci.2020.10.016https://doi.org/10.1016/j.jksuci.2020.10.016https://doi.org/10.1016/j.energy.2022.124848https://doi.org/10.1016/j.energy.2022.124848https://doi.org/10.7312/gumb92958https://doi.org/10.1016/j.dental.2011.12.005https://doi.org/10.1016/j.jeurceramsoc.2017.03.018https://doi.org/10.1016/j.dental.2017.03.004https://doi.org/10.1016/j.dental.2017.03.004https://doi.org/10.1016/j.ceramint.2022.08.206https://doi.org/10.1016/j.ceramint.2022.08.206https://doi.org/10.1111/j.1151-2916.1991.tb06812.xhttps://doi.org/10.1111/j.1151-2916.1991.tb06812.xhttps://doi.org/10.1007/BF00356067https://doi.org/10.1111/j.1151-2916.1995.tb09092.xhttps://doi.org/10.1016/j.actamat.2011.03.023https://doi.org/10.1016/j.actamat.2011.03.023https://doi.org/10.1111/jace.20660 Reliability evaluation scheme for ceramics based on defect size distribution inversely estimated from standardized tests Abstract 1 | INTRODUCTION 2 | TARGET MATERIAL AND STANDARDIZED TESTS 3 | FORWARD ANALYSIS OF STRENGTH SCATTER 3.1 | Analysis model of bending tests 3.2 | Distribution function of equivalent crack length 4 | INVERSE ANALYSIS OF THE EQUIVALENT CRACK LENGTH DISTRIBUTION 5 | RESULTS AND DISCUSSION 5.1 | Estimation of equivalent crack length distribution and reproducibility of strength scatter 5.2 | Effect of reference test data on inverse estimation 5.3 | Impact of model discretization 6 | OUTLOOK 7 | CONCLUSION ACKNOWLEDGMENTS ORCID REFERENCES SUPPORTING INFORMATION