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

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[Finite Element Analysis of <i>R</i>‐Curve Behavior in Ceramics Using the Damage Model Based on the Cohesive‐Zone Relationship](https://mdr.nims.go.jp/datasets/a090a9ae-4fa4-4c51-8d1a-44ec3da3e64e)

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Finite Element Analysis of R‐Curve Behavior in Ceramics Using the Damage Model Based on the Cohesive‐Zone RelationshipInternational Journal of Ceramic Engineering & ScienceRESEARCH ARTICLEFinite Element Analysis of R-Curve Behavior in CeramicsUsing the Damage Model Based on the Cohesive-ZoneRelationshipMostafizur Rahman1,2 Taiyo Maeda1 Toshio Osada3,4 Shingo Ozaki3,41Graduate School of Engineering Science, Yokohama National University, Yokohama, Japan 2Department of Mechanical Engineering, Chittagong Universityof Engineering & Technology (CUET), Chattogram, Bangladesh 3High-Reliability Heat-Resistant Materials Group, Research Center for Structural Materials,National Institute for Materials Science, Tsukuba, Ibaraki, Japan 4Division of System Research, Faculty of Engineering, Yokohama National University,Yokohama, JapanCorrespondence: Shingo Ozaki (s-ozaki@ynu.ac.jp)Received: 16 June 2025 Revised: 15 August 2025 Accepted: 1 September 2025Funding: This work is based on results obtained from a project, JPNP22005, commissioned by the New Energy and Industrial Technology DevelopmentOrganization (NEDO). Part of this study was supported by Grant-in-Aid for Scientific Research [Grant (B) 22H01357], Japan Society for the Promotion of Science(JSPS).Keywords: ceramics | damage model | finite element analysis | R-curve behavior | toughnessABSTRACTThe evaluation of the R-curve behavior of ceramics, which is characterized by an increase in crack resistance with crackpropagation, is crucial for advancing their implementation in engineering applications that require high reliability. In this study,weinvestigated the applicability of a finite element analysis (FEA) approach that implements a continuum damage model embeddedwith a cohesive-zone relationship to predict crack occurrence and the subsequent increase in crack resistance (toughness) ofceramics. Specifically, by employing a compliance-based method, the R-curve behavior was systematically examined under abending load to assess the impacts of fracture stress and toughness on diverse chevron-notched specimens. The output criticalstress intensity factors were found to increase with the crack length, eventually converging nearly to the input fracture toughness.Subsequently, the stable crack growth behavior obtained from the FEA and experiment under a three-point bending test ofhigh-purity alumina was compared. A consistent result was confirmed in the force–displacement relationships. Furthermore,the R-curve behavior of the target material could be indirectly evaluated using the present approach. The results support theeffectiveness of the present approach, highlighting the quantitative assessment of not only crack initiation but also R-curvebehavior under arbitrary boundary conditions.1 IntroductionCeramics have recently garnered attention for structural applica-tions owing to their exceptional mechanical properties, includinglightness, thermal resistance, and high specific strength [1–4].These properties, coupled with their resistance to wear andcorrosion, make ceramics ideal for a wide range of applications,such as aerospace, energy generation, automotive, and electronics[5–11]. However, to ensure reliability and safety in sophisti-cated engineering applications, comprehensively investigatingthe fracture behavior of ceramics is crucial. This investigationis key to optimizing the microstructure and shape of ceramicsto enhance their fracture resistance and achieve controlledstable crack growth [12–14]. This is important, as rapid andThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properlycited.© 2025 The Author(s). International Journal of Ceramic Engineering & Science published by Wiley Periodicals LLC on behalf of American Ceramic Society.International Journal of Ceramic Engineering & Science, 2025; 7:e70025https://doi.org/10.1002/ces2.700251 of 14https://doi.org/10.1002/ces2.70025https://orcid.org/0009-0009-2212-2205https://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://doi.org/10.1002/ces2.70025http://crossmark.crossref.org/dialog/?doi=10.1002%2Fces2.70025&domain=pdf&date_stamp=2025-09-17catastrophic crack growth can result in sudden failure, whichdecreases the reliability and integrity of ceramic structuralcomponents [15, 16].Fracture toughness, especially the R-curve behavior, which ischaracterized by an increase in crack resistance with crackpropagation, is important for assessing the resistance of ceramicsto the crack growth rate. Although the fracture toughness offersa single-value measure of this resistance, the R-curve provides amore dynamic and comprehensive perspective; it illustrates theevolution of the fracture resistance with increasing crack length[17]. Toughening mechanisms, such as crack deflection, crackbridging, and interlocking, contribute to the enhancement offracture resistance, resulting in a rising-type R-curve [18]. Becheret al. [19, 20] investigated these mechanisms and the improve-ments in the fracture toughness of alumina- and mullite-basedceramics.Although previous experimental approaches explored the micromechanisms responsible for toughening, they often lacked adetailed analysis of the R-curve behavior. Despite the significanceof the R-curve analysis, studying these behaviors in ceramicspresents several challenges, including the need for precise exper-imental techniques to measure the stress intensity factor (SIF)and accurately track the crack growth. Advances in experimentalmethods, such as digital image correlation (DIC) and in situmechanical testing, have simplified the study ofR-curve behavior,as demonstrated by Grutzik et al. [21]. They analyzed the R-curvebehavior in glass- and Si3N4-based ceramics under four-pointbending (4PB) test conditions.However, many early attempts have primarily focused on improv-ing fracture stress and toughness by optimizing microstruc-tural features, manufacturing techniques, and testing methods.To evaluate the R-curve behavior and promote the applica-tion of ceramics, integrating the experimental and numericalapproaches is essential. Finite element analysis (FEA) approachis one of the leading candidates of simulation methods forassessing not only crack initiation but also following the crackgrowth under arbitrary boundary conditions. Furthermore, FEAenables a detailed analysis of stress distribution and SIF withcrack growth through the fractured surface and ahead of thecrack tips. Thus, it provides a deeper insight into the fractureprocess, which is difficult to achieve experimentally alone,especially in complex geometries. In addition, FEA facilitatesthe evaluation of stable crack growth and enables the explo-ration of the influence of the diverse conditions on the R-curvebehavior.In this study, the effectiveness of the FEA approach to predictcrack occurrence, followed by an increase in the crack resistance(toughness) of ceramics, was investigated. In the FEA approach, acontinuum damagemodel with an embedded cohesive-zone rela-tionship was implemented [22]. Specifically, theR-curve behaviorof diverse chevron-notched specimens subjected to bendingloads was examined, which was designed to induce stable crackgrowth. First, R-curve analyses for various fracture properties,specimen geometries, and test conditions were systematicallyperformed using the compliance-based method. Subsequently,the experimental results of three-point bending (3PB) tests usingchevron-notched alumina ceramics were comparedwith the FEAFIGURE 1 Typical stress–strain relationship in the isotropic consti-tutive damage model [23–25]. The fracture stress is assumed to decreaseexponentially with the crack growth.simulations. Then, using the FEA results corresponding to theexperiment, we investigated the feasibility of the approach toexamine the R-curve behavior in detail, even with systems inwhich experimental measurements related to stable crack growthbehavior are difficult.2 Continuum Damage ModelAn isotropic damagemodel based on fracturemechanics [22] wasadopted to analyze the R-curve behavior of brittle ceramics. Thestress–strain relationship between the Cauchy stress tensor σ andinfinitesimal strain tensor ε can be written as follows:𝝈 = (1 − 𝐷) 𝐜 ∶ 𝜺, (1)where c and D represent the fourth-order elastic modulus tensorand damage variable, respectively. The damage variable D rangesfrom0 to 1, indicating the degree of damage, with 0 correspondingto the undamaged state and 1 corresponding to the perfectlyfractured state, as shown in Figure 1 [23–25].To describe the damage process in brittle materials, the cohesiveforce-crack opening relationship (Equation 2) was incorporatedinto the damage model:𝜎 = 𝜎t exp(−𝜎t𝐺f𝑤), (2)where σ is the cohesive force per unit area, σt is the local fracturestress, Gf is the fracture energy, and w is the crack openingdisplacement. Equation (2) can be rearranged in the same form asEquation (1) [22]. Thus, the damage variable D can be expressedas follows:𝐷(𝜅) = 1 −𝜅0𝜅exp{−𝜎tℎe𝐺f(𝜅 − 𝜅0)}, (3)2 of 14 International Journal of Ceramic Engineering & Science, 2025 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 LicenseFIGURE 2 FEA model for 3PB and 4PB test: (a) generalized view of dimensions and constraint conditions; (b) dimensions and constraintconditions from the right side view of (a); (c) a close-up view around the chevron notch area. Here, the model is symmetric in the z-direction. As shownin the close-up view around the damaged part in the notch tip, the crack propagates through the center Part (III), as indicated by the red elements.where he the characteristic length (i.e., the length in the crackopening direction of the unit element used in the FEA) [26]. κis the damage history variable, which is the internal variable ofthe damage variable and corresponds to themaximum equivalentstrain in loading history. As this study was focused on ModeI fracture behavior of ceramics, the local fracture stress σt (thefracture criterion) was evaluated on the basis of the maximumprincipal stress. Further, the equivalent strain when the maxi-mumprincipal stress reaches the local fracture stress was adoptedfor κ0 [23–25].3 FEA of R-Curve Behavior3.1 FEAModelThe FEA of the R-curve behavior and stable crack growth wasperformed using chevron-notched specimen models, followingthe American Society for Testing andMaterials (ASTM) standard[27]. The commercial software package LS-DYNA [28] was usedin the FEA, and the damage model was implemented using theuser subroutine umatXX. FEA was performed using the dynamicexplicit method and mass scaling. In this study, the applicabilityof FEA based on the continuum damage model to the R-curvebehavior of ceramicswas examined under various geometries andloading conditions, as well as the material properties of chevron-notched specimens. The FEA models used for the bending testanalyses are shown in Figure 2 and their dimensions are listed inTable 1. Here, according to the ASTM standard, configurations A,D, and B were used for the 4PB and 3PB tests, respectively. Thebending tests were performed by applying forced displacementsand constraint conditions to specific nodes, as shown in Figure 2a.Figure 2b shows a cross-sectional view of the notched area in thex-positive direction (Figure 2a). The model was halved using thex–y plane shown (Figure 2a) as the symmetry plane to reduce thecomputational cost.The FEAmodel consisted of three parts: Part (I), the smooth part;Part (II), the lower side of the notch; and Part (III), the lowercenter of the notch. Although the same Young’s modulus andPoisson’s ratio were input for all parts, the linear elastic modelwas applied as the material model in Parts (I) and (II). This isbecause fracture did not occur at these locations owing to thestress concentration at the notch tip. The damage model wasapplied to Part (III). Figure 2c shows the area around the notchtip viewed from the positive z-direction (Figure 2a). Tied contactswere used as the nodal coordinates did not overlap at the interfacebetween Parts (I) and (II) [28]. In addition, the notch tip wasmodeled with a curvature formed assuming drilling machining,which enabled the behavior of stable crack growth from the centerof the notch tip to be reproduced in the experiments (Section 5).To reproduce a stable crack growth in FEA, the element size,particularly in the crack propagation zone, is important. In thisstudy, the most focused was Part (III) with the damage model,featuring a unit element size of 5.95 µm (=he). The mesh sizes inParts (I)–(III) were different, as shown in Figure 2.3 of 14 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 LicenseTABLE 1 Specimen and test geometries of three-point bending (3PB) and four-point bending (4PB) tests.Configuration B [mm] W [mm] a0 [mm] Si [mm] So [mm]ASTM A 3.2 4.2 0.87 20 40ASTM B 6.4 6.4 2.54 0 20, 30, 40ASTM D 3.2 4.2 1.47 20 40Abbreviation: ASTM, American Society for Testing and Materials.TABLE 2 Basic material properties of alumina ceramic (AS999).Young’s modulus Poisson’s ratio DensityE [GPa] ν [−] ρ [kg/m3]380 0.24 3950The basic material properties of the alumina-based ceramics arelisted in Table 2. Two variations of fracture properties, localfracture stress σt and fracture toughness KIC, were prescribedas follows: Type (i) σt = 890 MPa with KIC = 4.0, 8.0, and12.0 MPa m0.5; and Type (ii) σt = 450 MPa with KIC = 3.0, 6.0,and 9.0 MPa m0.5. These properties were approximately identicalto those of alumina reinforced with silicon carbide particles,alumina reinforced with silicon carbide whiskers, nacre-like alu-mina with silicon carbide whiskers, pure alumina, and nacre-likealumina [29–32].3.2 Evaluation of R-Curve BehaviorIn this study, a compliance-basedmethodwas adopted to evaluatethe R-curve behavior of ceramics. The R-curve represents theSIF obtained as a function of crack length in the stable fractureprocess of a notched specimen [33]. The critical SIF beforefull fracture is termed as the Mode I fracture toughness, KIC[34]. Here, the fracture toughness can be determined using themaximum value of the bending load, geometric condition of thespecimen, crack size, and rate of change of the dimensionlesscompliance with the crack length, without information on theR-curve [35]. The SIF for mode I, KI, under the plane straincondition is expressed as follows [34, 36]:𝐾I =√𝐸𝐺I1 − 𝜈2, (4)where E is the Young’s modulus, ν is the Poisson’s ratio, and GI isthe energy release rate. When a bending load p is applied in the3PB or 4PB test on a chevron-notched specimen (Figure 3a), theenergy release rate for the chevron crack (Figure 3b) is given by[36]𝐺I =𝑝22𝑏d𝐶d𝑎, (5)where b is the crack-tip width, C is the compliance, and a is thecrack length. On the basis of the geometry of the chevron notch(Figure 3b), the crack-tip width b is given by𝑏 =𝑎 − 𝑎0𝑎1 − 𝑎0𝐵, (6)FIGURE 3 Schematic of chevron-notched specimen in analysis: (a)3PB and 4PB test model; (b) cross-sectional shape of test specimen. 3PB,three-point bending; 4PB, four-point bending.where B is the specimen width, a0 is the pre-crack length to thetip, and a1 is the pre-crack length at the edge. Therefore, KI canbe written by arranging Equations (4–6) as follows:𝐾I =𝑝𝐵√𝑊√12𝛼1 − 𝛼0𝛼 − 𝛼0d𝐶′d𝛼, (7)whereW is the thickness of a specimen, C′ (=CBE/(1 − ν2)) is thedimensionless compliance, and α (=a/W) is the dimensionlesscrack length. In the FEA, the crack length is the sum of theelement lengths in the direction of crack growth, where theelement damage variable D > 0.05.4 FEA Results of R-Curve BehaviorWe investigated the R-curve behavior of ceramics by inducing a>3-mm-length crack through the chevron notch part by utilizingthe prescribed fracture properties. Furthermore, the occurrence4 of 14 International Journal of Ceramic Engineering & Science, 2025 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 LicenseFIGURE 4 FEA results from fracture toughness KIC [MPa m0.5] variation under the 3PB test for ASTM B specimens: (a) change in dimensionlesscompliance with dimensionless crack length; (b) change in reaction force with dimensionless crack length; (c) change in reaction force withdimensionless compliance; (d) change in reaction force with change in dimensionless factor. Here, the outer span length (So) is 20, 30, and 40 mm,and fracture stress (σt) of 890 MPa is maintained constant for all cases.of stable crack growth during the fracture process of ceramicsusing FEA was evaluated.4.1 Chevron-Notched ASTMModel for 3PB TestFigure 4 shows the FEA results obtained by 3PB tests (outerspan lengths So = 20, 30, and 40 mm) following the ASTMB configuration (Table 1). Here, the input fracture propertieswere of Type (i). Figure 4a shows the relationship betweenthe dimensionless compliance and dimensionless crack length.Regardless of the fracture toughness values, the dimension-less compliance increased in the same manner as the cracklength, leading to similar crack growth. The obtained behaviorresembled the reported experimental results for coarse-grainedalumina obtained by adopting several test approaches [37,38]. Conversely, the compliance depends on the outer spanlength So, influenced by the effective stress concentration area.This consistency suggests the validity of the fracture behaviorsobtained by FEA and calculated using the compliance-basedmethod.Furthermore, the change in the reaction force with the dimen-sionless crack length indicated that the force increased graduallywith the crack growth and even decreased progressively afterthe crack reached a certain region in the chevron notch area,as shown in Figure 4b. This is because the material no longershowed resistance to crack growth as the crack growth rateincreased. Notably, the higher the fracture toughness value andthe smaller the outer span length, the larger the peak reactionforce, which was confirmed to induce almost the same level ofcrack length. Figure 4c,d illustrates the changing tendencies ofdifferent peak reaction forces with respect to the changes in thedimensionless compliance and dimensionless factor, respectively.Here, the dimensionless factor is denoted as the ratio betweenthe increments of dimensionless compliance and dimensionless5 of 14 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 LicenseFIGURE 5 FEA results from fracture toughness KIC [MPa m0.5]variation under the 3PB test for ASTM B specimens: (a) comparisonof force–CMOD and crack growth length–CMOD relationships; (b)comparison of R-curves. Here, the outer span length (So) is 20, 30, and40 mm, and fracture stress (σt) of 890 MPa is maintained constant for allcases. CMOD, crack mouth opening displacement.crack length (=∆C′/∆α). The reaction force initially increasedwith dimensionless compliance and thereafter decreased in alikely exponential manner, following almost the same line whenthe fracture toughness was the same, even when the outer spanlength was varied.Figure 5a shows the variations in the reaction force and crackgrowth length with respect to the crack mouth opening displace-ment (CMOD; Figure 2c). Even after crack initiation, the forceincreased gradually up to a peak with the CMOD and decreasedstably after the peak force as the crack growth rate increasedgradually. As the crack initiation force of ceramics is dominatedby the local fracture stress [39], cracks initiate at the sameCMODand reaction force, even for a different fracture toughness.FIGURE 6 FEA results from fracture toughness KIC [MPa m0.5]variation under the 3PB test for ASTM B specimens: (a) comparison offorce—CMOD and crack growth length—CMOD; (b) comparison of R-curves. Here, the outer span length (So) is 20, 30, and 40mm, and fracturestress (σt) of 450 MPa has been kept fixed for all cases. CMOD, crackmouth opening displacement.Furthermore, the crack initiation force depending on the testcondition was estimated to be 44, 29, and 21 N for outer spanlengths of 20, 30, and 40 mm, respectively. However, the crackgrowth rate is significantly influenced by the fracture toughness;the higher the fracture toughness, the smaller the crack growthlength with respect to the CMOD. Furthermore, the crack growthrate was identical regardless of the outer span length when theinput fracture toughness was the same. These fracture behaviorsare reflected in theR-curve (Figure 5b). Initially, the SIF increasedwith the crack growth and converged to almost the same values asthe respective input fracture toughness after the crack reached acertain length in the chevron notch. Mesh size discretizationmaycause a slight discrepancy between the input fracture toughnessvalue and the output SIF from the FEA.6 of 14 International Journal of Ceramic Engineering & Science, 2025 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 LicenseFIGURE 7 FEA results from fracture toughness KIC [MPa m0.5]variation under 4PB test: (a) comparison of force–CMOD and crackgrowth length–CMOD relationships; (b) comparison of R-curves. Here,the inner span length (Si) and the outer span length (So) are 20 and40 mm, respectively, and the fracture stress (σt) of 890 MPa is maintainedconstant for all cases. CMOD, crack mouth opening displacement.The shapes and slopes of the R-curve obtained by FEA weresignificantly dependent on the input fracture energyGf calculatedfrom the fracture toughness KIC and basic material proper-ties (Table 2). This trend was experimentally demonstrated byconsidering several grades of ceramics [40–43]. Therefore, theeffects of the fracture properties on the fracture and R-curvebehavior (Figure 6) were investigated. The effect of the variationin fracture properties (i and ii) can be illustrated by comparingFigures 5a and 6a. The crack initiation force, peak reactionforce, and crack growth behavior were significantly dependenton the input fracture properties. The crack initiation forceswere estimated to be 24, 15, and 11 N for outer span lengthsof 20, 30, and 40 mm, respectively, for fracture property type(ii) (Figure 6a).Output SIFs converged and nearly equalized to input values atKIC = 3.0 and 6.0 MPa m0.5 (Figure 6b). These two behaviorsFIGURE 8 FEA results from fracture toughness KIC [MPa m0.5]variation under 4PB test: (a) comparison of force–CMOD and crackgrowth length–CMOD relationships; (b) comparison of R-curves. Here,the inner span length (Si) and the outer span length (So) is 20 and 40mm,respectively, and the fracture stress (σt) of 450MPa ismaintained constantfor all cases. ASTM, American Society for Testing and Materials; CMOD,crack mouth opening displacement.appeared as flat-type R-curves, similar to those reported by Bleiseand Steinbrech [44], who adopted coarse-grained alumina toexperimentally produce long cracks. In contrast, the output SIFdid not converge and was slightly higher than the input valueat KIC = 9.0 MPa m0.5. This behavior can be treated as a rising-type R-curve [41, 45, 46], which was experimentally observed inhighly toughened bioinspired nacre-like alumina [31] and siliconnitride [45], the fracture stress and toughness values of whichwere measured to be almost identical to the values input intoFEA. These findings highlight the effectiveness of the presentFEA approach for analyzing the fracture and R-curve behaviorsof ceramics.In situ experimental observations, including crack deflection,crack branching, bridging, microcracking, and delamination,7 of 14 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 Licensewere conducted in the single-edge notched bending test, whichcaused the rising-type R-curve behavior. This phenomenon wasalso observed by many researchers when investigating improve-ments in the mechanical properties of different ceramic-basedbrittle materials [47–53]. Therefore, the causes of the rising-typebehavior in FEA indirectly reflected the effects of the experi-mental toughening mechanisms, that is, crack deflection, crackbranching, bridging, microcracking, and delamination. Thesetoughening mechanisms significantly affect crucial fractureparameters, such as reactive force and dimensionless compliance,under variations in fracture properties. Hence, the integrationof the changes in the fracture parameters and their substitutioninto Equation (7) reflect the flat and rising-typeR-curve behaviorsaccording to the fracture properties.4.2 Chevron-Notched ASTMModel for 4PB TestThe fracture and R-curve behaviors using the fracture properties(i) and (ii) under the 4PB test adopting configurations A andD (Table 1) were evaluated to examine the occurrence of stablecrack growth. The only distinction between these configurationswas the pre-crack length, whereas the other geometries andconstraint conditions were identical to those shown in Figure 2.Figure 7 shows the relationships of force and crack growth lengthwith the CMOD and the corresponding R-curve behavior forboth configurations A and D for fracture property type (i). Thedifference in the pre-crack length between the configurationscaused variations in the crack initiation force, peak force, andcrack growth length, as shown in Figure 7a. The crack initiationforce was estimated to be 15 and 11 N for configurations A andD, respectively. In addition, the slopes of the force and CMODcurves changed owing to the difference in the stress concentrationarea through the chevron notch. The peak force induced inconfiguration A (shorter pre-crack length) was estimated to behigher than that in configuration D (longer pre-crack length)to generate the same level of CMOD. Furthermore, the changesin the crack growth length with the CMOD of configuration Awere greater than those of configuration D. To investigate theconvergence and coherency, these results were reflected in the R-curve behavior (Figure 7b). The SIFs increased with increasingcrack length and converged when the crack reached a certainlength into the chevron notch. Type (i) fracture properties outputa flat R-curve behavior [44] from both configurations under the4PB test conditions.The impacts of the fracture properties (Types (i) and (ii)) areshown in Figures 7 and 8 for both configurations. Collectively,these results correspond to stable crack growth in the stablefracture process of ceramics, which is similar to the stable crackgrowth and corresponding R-curve behavior of the ZrB2/SiCceramic composite reported by Lugovy et al. [54] at roomtemperature using the V-notched specimens in the 4PB test.4.3 Comparison of R-Curve Behaviors Between3PB and 4PB TestsFigure 9 illustrates the changes in SIF with the crack growthlength between the 3PB test (ASTMB) and 4PB test (ASTMA andD) under fracture property type (i). Noteworthy, the relationshipsFIGURE 9 Comparison of R-curve behaviors between 3PB and 4PBtests adopting fracture properties type (i). Here, the fracture toughnessKIC[MPa m0.5] was varied, whereas the fracture stress (σt) of 890 MPa wasmaintained constant for all cases. ASTM, American Society for Testingand Materials.between SIF and crack growth length were independent of speci-men geometries and test conditions, indicating a similar trend ofstress distribution over the fractured surface and ahead of cracktip. Furthermore, this type of independence was also observedfor fracture property type (ii) (Figure 10). However, a slightdeviation was observed in the 4PB tests at KIC = 9.0 MPa m0.5,probably owing to the difference in specimen geometries andrandom stress–toughness trade-off (Figure 10). Moreover, in thiscase, the peaks of the SIF corresponding to the maximum crackgrowth length were approximately identical for the specimengeometries. The SIF was quantified with the crack growth length,depending on the nature of the fracture toughness of the ceramics(Figures 9 and 10). These results highlight the effectiveness ofthe proposed FEA approach for evaluating the fracture initiationand subsequent R-curve behavior of ceramics under diverse testconditions that induce stable crack growth.5 Comparison Between FEA and Experiment5.1 ExperimentHigh-purity alumina AS999 (Ferrotec Material TechnologiesCorporation, Japan) was used in the experiment. It should8 of 14 International Journal of Ceramic Engineering & Science, 2025 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 LicenseFIGURE 10 Comparison of R-curve behaviors between 3PB and4PB tests adopting fracture properties type (i). Here, the fracture tough-nessKIC [MPam0.5]was varied,whereas the fracture stress (σt) of 450MPawas maintained constant for all cases. ASTM, American Society forTesting and Materials.be noted that the distribution of the local relative density ofAS999 is narrow, with an average value of approximately 0.98[32]. Hence, the microstructure of AS999 is standard densealumina and cannot be a variable in the crack propagation inchevron-notched specimens. The specimen for the 3PB test wasfabricated and machined by duplicating the ASTM B geometries(Table 1); however, the geometry deviated slightly from thestandard owing to machining accuracy issues. Figure 11a,b showsclose-up views around the chevron notch area (an enlarged viewof “C” in Figure 12b) and the fractured surface of ASTM Bspecimen, respectively. A 3PB test for the ASTM B specimenunder So = 40 mm was conducted.Figure 12a,b shows the experimental apparatus and specimensetup, respectively. The experimental setup comprised a bendingtesting machine (AG-X plus, 10 kN, Shimadzu Corporation,Japan), a digital microscope (VHX-6000, Keyence Corporation,Japan), and two microphones. Two microphones were attachedto both ends of the specimen with rubber bands to fix them(Figure 12b). The sounds detected by the microphones wereamplified by a preamplifier and analyzed for acoustic emis-sion (AE) using continuous-wave memory [55], which was theoriginal AE analysis equipment. To evaluate the stable crackFIGURE 11 ASTM B specimen model for 3PB test experimentshowing (a) a close-up view around the chevron notch, an enlarged viewof “C” in Figure 12b; (b) the fractured surface.FIGURE 12 Experimental setup for 3PB test: (a) overall view withbending testingmachine, digital microscope withmonitor, and AE devicewith monitor; (b) setting of the experimental specimen, an enlarged viewof “B” in (a).9 of 14 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 Licensegrowth, a single-3PB test was performed at a crosshead speedof 0.001 mm/min at room temperature. In addition, the pre-crack area (Figure 11) was photographed at 15 s intervals, theshortest interval possible with the equipment function using adigital microscope. The images were used to measure the CMODusing DIC analysis software (VIC-2D, Laser Measurement Co.Ltd., Japan).5.2 Results and DiscussionDirectly evaluating the R-curve behavior from the results of the3PB tests using the ASTM B specimen is impossible owing to theinvisibility of crack growth under the above experimental setup.Therefore, using the FEA results corresponding to the specimenused in the experiment with fracture properties of AS999, weattempted to evaluate the evolution of SIF with crack growth.Then, the occurrence of stable crack growth was determined. Thegeometry of the specimen in the FEA reproduced that shown inFigure 11.As mentioned previously, in the present FEA approach, two frac-ture parameters (local fracture stress, σt, and fracture toughness,KIC) are crucial to analyze the fracture and R-curve behavior ofceramics. For the comparison, σt = 544 MPa and KIC = 3.4, 3.5,and 3.7 MPa m0.5 of AS999 were used. Here, as the bulk strengthof ceramics showed scatter, AE analysis was used to detect theforce at the crack initiation of the target specimen during the 3PBtest, and the local fracture stress σt was determined in an inverseanalyticalmanner. TheKIC valueswere obtained from single-edgenotched beam (SENB) tests conducted on 20 specimens of thesame lot material. Ten specimens were tested at a span length of30 mm, and the remaining 10 specimens were tested at 18 mm.The three levels of measured fracture toughness values,KIC = 3.4,3.5, and 3.7MPam0.5, correspond to the lower, average, and upperlimits, respectively.First, a comparison with the experimental results is discussed.Figure 13a illustrates the force–CMOD relationship for the ASTMB specimen at So = 40 mm. The peak force from the FEA usingthe upper and lower KIC was slightly higher and lower than thatof the experiment, respectively; the average KIC showed betteragreement with the experiment. More importantly, under thesetest conditions, even after crack initiation, the force increasedgradually with the CMOD. Furthermore, after the peak, theforce decreased stably with the CMOD, indicating stable crackgrowth. Interestingly, in both the FEA and experiment, aftercrack initiation and peak force, the changes in the reaction forcewith the CMOD were progressive and stable. This confirmed astable crack growth under this test condition. The present FEAapproach using the continuum damage model confirmed thatthe fracture behavior after the peak force could be appropriatelypredicted.Additionally, Figure 13b shows the variation in crack growthlength with the CMOD obtained from the FEA results. Even aftercrack initiation and peak force, the crack growth length withthe CMOD increased progressively and stably. This confirmed astable crack growth under the test conditions. The crack growthwas observed to increase up to CMOD ≈12 µm and reached alength of ∼3.5 mm into the chevron notch.FIGURE 13 Comparison between experimental and FEA resultsusing the ASTM B specimen under outer span length, So = 40 mm:(a) force–CMOD relationships; (b) CMOD–crack growth length relation-ships. Here, FEAs were conducted by maintaining the fracture stressconstant at σt = 544 MPa with the variation of fracture toughness (lowerlimit, average, and upper limit). ASTM, American Society for Testing andMaterials; CMOD, crack mouth opening displacement.Figure 14 shows the R-curve behavior of AS999 obtained fromFEA and the compliance-based method. The results revealedthat the SIF increased with the crack growth and eventuallyconverged to approximately the same values as the respectiveinput fracture toughness after the crack reached a certain lengthin the chevron notch. In addition to this result, the consistencyof the force–CMOD relationship (Figure 13a) supported theeffectiveness of the present FEA approach for evaluating theR-curve behavior of ceramics. Therefore, the detailed R-curvebehavior can be evaluated simultaneously using experimental10 of 14 International Journal of Ceramic Engineering & Science, 2025 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 LicenseFIGURE 14 Variation in stress intensity factor with crack growthlength using the ASTMB specimen under outer span length, So = 40mm.Here, FEAswere conducted bymaintaining the fracture stress constant atσt = 544MPa and varying the fracture toughness (lower limit, average, andupper limit). ASTM, American Society for Testing and Materials.and FEA results, even with experimental systems in which crackpropagation cannot be measured owing to certain limitations.Furthermore, as the R-curve behavior almost converged toinput fracture toughness values, the experimental crack growthbehavior was also reasonably estimated using the FEA approach(Figure 13b).One of the features of FEA is the visualization of the stress andstrain fields and the distribution of the state variables. Figure 15shows the contour plots of the maximum principal stress anddamage variable D at Part (III) (Figure 2c) corresponding tothe three crack growth processes during the 3PB test with anaverage KIC. Figure 15a–c corresponds to crack growth lengthsof approximately 0.5, 1.5, and 2.5 mm, respectively ((i), (ii),and (iii) in Figures 13 and 14). The figure confirms that thecrack gradually propagated from the tip of the chevron notchat the center of the specimen in the width direction to thebottom side. High maximum principal stresses were constantlyapplied at the crack tip, and the change in damage variable Dcorresponded to it, resulting in the reproduction of stable crackgrowth.Hence, the present FEA approach overcomes the limitation ofmeasuring the crack growth in ASTM chevron-notched speci-mens. Furthermore, by fitting the fracture parameters requiredfor FEA analysis, it can be connected not only to the evaluationof fracture behaviors but also to the feasibility study of stablecrack growth of the same ceramics under arbitrary boundaryconditions. Another application involves evaluating the perfor-mance of self-healing ceramics. For example, when evaluatingthe repeated healing performance of self-healing ceramics, intro-ducingmillimeter-sized cracks stably and repeatedly is necessary.Osada et al. [39] experimentally analyzed bone-like MAX-phaseself-healing ceramics using a wedge-splitting test. In such cases,the FEA approach can be applied to the feasibility of stable crackFIGURE 15 Contour plots of the maximum principal stress anddamage variable D at Part (III) in Figure 2c corresponding to three crackgrowth processes during the 3PB test with averageKIC. (a)–(c) correspondto crack growth lengths of approximately 0.5, 1.5, and 2.5mm, respectively((i), (ii), and (iii) in Figures 13 and 14).growth and to evaluate the strength and toughness recovery afterrepeated cracking and healing.6 ConclusionIn this study, an FEA approach for the fracture and R-curvebehavior of ceramics was investigated using a continuum dam-age model embedded with a cohesive-zone relationship. The11 of 14 25783270, 2025, 5, Downloaded from https://ceramics.onlinelibrary.wiley.com/doi/10.1002/ces2.70025 by National Institute For, Wiley Online Library on [23/09/2025]. 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 LicenseFEA of bending tests for chevron-notched specimens undervarious fracture properties and test geometries was conducted.Different boundary conditions resulted in different reaction forceresponses. However, for the same fracture toughness, the crackgrowth behavior (R-curves) was consistent regardless of thetest conditions, and the output SIF of mode I converged tothe input fracture toughness values as the cracks propagated.Furthermore, the 3PB test results for the chevron-notched AS999specimens were compared with the FEA results. The overallagreement revealed that the force-CMOD-crack growth lengthrelationships under diverse test conditions can be predicted byfitting the required fracture parameters, even with systems inwhich experimental measurements related to the R-curve andstable crack growth behavior are difficult.It is important to note that the present damage model wasformulated on the basis of cohesive zone embedded modeling,combined with a regularization technique applied to the stress–strain curve. As a result, the model’s response during crackpropagation within a single element is independent of the FEAmesh density. However, the damage initiation criterion is basedon the maximum principal stress, which is sensitive to meshdensity in regions of the stress concentration. Consequently,similar to conventional FEA, the influence of stress concentrationon damage initiation exhibits slight dependence on crack length.We believe that if an FEAmodel is prepared to accurately capturestress gradients consistent with experimental results, the presentanalysis approach can reasonably simulate crack progression,regardless of mesh density and boundary conditions.Collectively, the systematic investigation, including the compar-ison between FEA and experiment, supported the effectivenessof the present approach for evaluating the fracture and R-curvebehavior of ceramics under arbitrary boundary conditions. Byusing the present FEA approach combined with the experiment,we can evaluate the R-curve behavior in detail, even with systemsin which experimental measurements related to stable crackgrowth behavior are difficult. Furthermore, the present FEAapproach can also be applied to feasibility studies on stable crackgrowth in actual components to ensure reliability under serviceconditions.AcknowledgmentsWe would like to thank Dr. Kenta Goto, Dr. Hideaki Nishikawa, Dr.Hisashi Yamawaki, and Mr. Takuma Kohata at the National Institute forMaterials Science (NIMS) for their support in the system development,AE analysis, and microstructural observations. Acknowledgement alsogoes to Mr. Hayato Ono and Mr. Haruki Okuma, Graduate School ofEngineering Science, Yokohama National University, for their efforts todevelop the FEAmodel and Python program and to measure the fracturetoughness value of AS999.References1. H. Ohnabe, S. Masaki, M. Onozuka, K. Miyahara, and T. 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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.2013.01.041https://doi.org/10.1111/jace.12292https://doi.org/10.1080/17436753.2018.1471443https://doi.org/10.2320/matertrans.I-MRA2007850 Finite Element Analysis of R-Curve Behavior in Ceramics Using the Damage Model Based on the Cohesive-Zone Relationship 1 | Introduction 2 | Continuum Damage Model 3 | FEA of R-Curve Behavior 3.1 | FEA Model 3.2 | Evaluation of R-Curve Behavior 4 | FEA Results of R-Curve Behavior 4.1 | Chevron-Notched ASTM Model for 3PB Test 4.2 | Chevron-Notched ASTM Model for 4PB Test 4.3 | Comparison of R-Curve Behaviors Between 3PB and 4PB Tests 5 | Comparison Between FEA and Experiment 5.1 | Experiment 5.2 | Results and Discussion 6 | Conclusion Acknowledgments References