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

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[Novel numerical approach for reliability-assurance of ceramics by combining self-crack-healing with proof testing](https://mdr.nims.go.jp/datasets/bcf15f56-c9e3-407c-b533-108d0fd1645c)

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Novel Numerical Approach for Reliability-assurance of Ceramics by Combining Self-Crack-Healing with Proof TestingTaiyo Maeda1, Toshio Osada2,3, Shingo Ozaki2,3,*1Graduate School of Engineering Science, Yokohama National University, Tokiwadai 79-5, Hodogaya-ku, Yokohama 240-8501, Japan2High Temperature Materials Group, Research Center for Structural Materials, National Institute for Materials Science, Sengen 1-2-1 Tsukuba, Ibaraki 305-0047, Japan3Division of System Research, Faculty of Engineering, Yokohama National University, Tokiwadai 79-5, Hodogaya-ku, Yokohama 240-8501, Japan*Correspondence: E-mail address: s-ozaki@ynu.ac.jp, Tel/Fax No.: +81-45-339-3881AbstractCeramics exhibit stochastic fracture behavior due to internal and surface defects, thereby limiting their practical use. To address this issue, a combined method of ‘self-crack-healing’ and ‘proof testing’ has been proposed. However, to efficiently implement the method, a numerical analysis is required for preliminary investigation. This study establishes a finite element analysis methodology that enables the prediction of both strength recovery due to crack healing and the scatter of strength due to microstructural features. We determine the dependence of the scatter of the three-point bending strength on the healing conditions using the Weibull distribution and demonstrate the applicability of the proposed analysis method to proof tests. Results revealed that the proposed methodology can estimate the healing conditions and proof stresses by obtaining the microstructural information and oxidation kinetics parameters of the ceramic components in advance.Keywords: Finite element analysis; Self-healing ceramics; Proof test; Oxidation kinetics; Fracture mechanics; Weibull distribution1．IntroductionCeramic materials are attracting attention owing to their lightness, specific strength, corrosive and thermal resistance etc. Therefore, they are expected to be further developed into electronic components such as multilayer capacitors [1–4], biomaterials such as dental implants[5–7], and structural components[8–11]. However, the strength of brittle ceramics is governed by fracture from the size distribution of internal [12–14] or surface defects [15,16] that are generated stochastically during manufacturing processes, such as sintering and machining, respectively. This further results in a significant variation in the fracture strength, even when the components are manufactured using the same procedure [17–25]. Therefore, it is difficult to determine the ‘endurance stress limit,’ which is the most significant aspect of the design of structural components requiring high reliability, thus limiting the practical applications. Although there have been several efforts to improve the fracture toughness value through microstructure and composition design [26–35], the practical application as structural components, where the stress acts, is still limited.A pioneering approach to determine proof stress for monotonic bending strength based on the combination of ‘crack healing ability’ and ‘proof testing’ has been proposed by Ando et al. [36]. This approach can guarantee the high-temperature reliability of ceramic ‘specimen,’ which is typically applied to oxidation-induced self-healing ceramics [37–42]. Many types of self-healing ceramics that can heal surface cracks by the oxidation of a pre-incorporated healing agent have been proposed [39,43]. The crack-healing ability can prevent damage from surface cracks that are inevitably introduced into components by the machining process. The alumina (Al2O3)/silicon carbide (SiC) composite is a self-healing ceramic suitable for high-temperature use mainly because the ceramics can ‘completely’ heal all the surface cracks introduced even by heavily machining, and this helps retain the maximum strength [44].The proof test before service is a suitable technique for removing specimens with large internal defects acting as killer defects, whose size cannot be easily detected with conventional nondestructive inspection primarily because oxidation-induced self-healing cannot heal internal defects, such as pores formed during sintering. Extensive research has been conducted on the proof test for ceramics in recent years [45–47]. In the proof test, stress must be pre-applied to the countless test specimens by setting an appropriate stress that considers the yield, cost, and service stress conditions based on strength data from the Weibull distribution of ceramics. However, evaluating the Weibull distribution of strength to determine the proof stress also requires countless destructive tests, which are costly and time-consuming. Moreover, depending on the healing condition of surface defects and size distribution of internal defects in ceramics, a scattering of fracture strength of even healed ceramics will be varied. Thus, the healing environment dependency on the strength recovery of the specimen must be evaluated in advance. However, the strength of the same specimen lot cannot be evaluated at different healing times because the evaluation requires destructive tests. In addition, the method of setting the proof stress will be different between the actual component with stress concentration parts and the test specimen with simple stress distribution. Therefore, a numerical simulation method that can withstand the preliminary validation of on actual component geometry is required.In this study, we aim to establish a novel finite element analysis (FEA)-based framework for reliability-assurance of ceramics combining ‘self-crack-healing’ with ‘proof testing’. We propose a method for determining the proof stress by predicting the strength recovery and variation of self-healing ceramics under arbitrary boundary conditions. In particular, a series of processes are simulated: the self-healing of surface cracks assumed to be introduced during the machining process of specimens; the evaluation of the bending strength variation through a three-point bending test under room temperature; and the application to the proof test. Here, the damage-healing constitutive model based on oxidation kinetics [48] and the FEA method [49,50] proposed by the authors are employed for calculating self-healing for surface cracks and the prediction of strength scatter based on the distribution of internal defects.In the following sections, the formulation of the damage-healing constitutive model and typical response characteristics are first introduced, and the analytical method for strength variation is described. Next, the FEA model and boundary conditions are described, and the dependence of strength variation due to surface and internal defects on the self-healing environment is discussed. The target material is particle-dispersed self-healing ceramics consisting of Al2O3/SiC composites with manganese (II) oxide (MnO) added as a healing activator, which shows excellent crack healing capabilities [41]. Finally, one application to a proof test is demonstrated, and the effectiveness of the proposed method is discussed.2．Damage and healing modelsIn this section, the damage-healing constitutive model and fracture mechanics model are described. Various functions and evolution laws for damage and healing phenomena are formulated based on the continuum damage model and oxidation kinetics with embedded cohesive force relationships, respectively [48]. Fig. 1(a) and 1(b) show the typical response characteristics of the damage-healing constitutive model. Here, the fracture mechanics model is used to evaluate the local fracture stress of the constitutive model based on microstructural data.2.1 Isotropic damage modelThe isotropic damage model based on fracture mechanics [51] was adopted as the basic constitutive model. The relationship between the Cauchy stress tensor σ and strain tensor ε is given by Eq. (1): , (1)where  denotes the fourth-order elastic modulus tensor. The scalar value  is the damage variable and expresses the degree of damage on a scale of 0–1, where  and  indicate the undamaged and completely damaged states, respectively. In addition, the cohesive force relation expressed in Eq. (2) is incorporated into the isotropic damage model to describe the damage process in brittle materials. , (2)where , , , and  are the cohesive force per unit area, local fracture stress, fracture energy, and crack mouth opening displacement, respectively. By transforming Eq. (2) into the same form as Eq. (1), the damage variable  is described as[51]: , (3)where  is the internal variable of the damage variable (damage history variable) and the maximum principal strain is adopted. When self-healing is unconsidered, the damage history variable  is equal to the maximum value of the maximum principal strain  in the damage history.  is the characteristic length, which corresponds to the element length in FEA. Because this study targets the bending deformation of ceramics, which are brittle materials, the local fracture stress  that is the fracture criterion is evaluated using the maximum principal stress, and the maximum principal strain at that time is considered as the strain at damage initiation, .2.2 Kinetic-based self-healing modelTo incorporate the evolution law of state variables for self-healing into the isotropic damage model, the damage variable is assumed to recover toward  over time during the healing process. In other words, the damage history variable  asymptotically approaches  with self-healing, and the damage history disappears. Based on this assumption, the damage history variable  is additively decomposed into the evolution of the maximum principal strain  and the recovery of the maximum principal strain due to self-healing  as the following equation: . (4)The evolution law of  is given by the following equation: . (5)Here, < > denotes the Macaulay brackets. In addition,  increases monotonically depending on the temperature and oxygen partial pressure during the healing process. In the case of an opening crack in Mode I, the evolution law of  is defined by the following equation: , (6)where  is the area of one side of the crack cross section (in the case of cubic elements, ), and  is the volume gain rate of oxide in the crack.  is formulated based on oxidation kinetics [43], and is given by: , (7)where  is the frequency factor,  is the volume gain due to crack filling,  is the partial pressure of oxygen,  is the partial pressure of oxygen at atmospheric pressure,  mo is the temperature-independent constant,  is the volume fraction of silicon carbide,  is the effective reactive area ratio of silicon carbide,  is the weight gain per unit volume,  is the activation energy,  is the gas constant, and  is the healing temperature. The initial value of the volume gain  in the numerical analysis is given by the following equation: , (8)where  is the initial time increment.Furthermore, the historical maximum strain , which is the criterion for damage redevelopment, must be defined because the strength at reloading after healing depends on the degree of crack filling. The evolution law of the historical maximum strain  is given by [48]:  , (9)where sgn( ) denotes the sign function.  is the correction factor for determining the strength recovery rate, and  is the strain at damage initiation after complete healing. Both parameters are obtained from the experimental results of the strength recovery. When the stiffness is fully recovered and the historical maximum strain is recovered to the strain at damage initiation, , after complete healing, the local fracture stress at the healed area is higher than that before healing . This phenomenon is called “super healing” and is one of the advantages of self-healing materials [52–54]. For the undamaged state, the initial value of  is equal to . Moreover, the model comes down to the ordinary damage model when self-healing is ignored.2.3 Fracture mechanics modelOzaki et al. [49] proposed a numerical method that can predict the stochastic fracture behavior of ceramics within the framework of the FEA by relating microstructural data (pore size, pore shape, and grain size) to parameters of the damage model via a fracture mechanics model. The method evaluates the local fracture stress , which is a parameter of the damage-healing constitutive model, based on microstructural data. This method is also used in this study to obtain the Weibull distribution of the bending strength of self-healing ceramic specimens, which are the targets of the proof tests. Here, fractures are assumed to be mode І crack propagation only, and cracks are assumed to initiate around the pores.In the fracture mechanics model, it is assumed that a circumferential initial crack exists around the oblate sphere-type pore, as shown in Fig. 2 [49]. The local fracture stress is given by the following Griffith/Irwin-type equation based on linear elastic fracture mechanics: , (10)where  is the Mode I fracture toughness value,  is the initial crack length, and  is the geometric factor. It is reported that the geometric factor of internal defects in ceramics is derived from the pore size and shape as well as the grain size (initial crack length) around the pore [55,56]. The geometric effects of circumferential cracks around oblate sphere-type pores can be summarized in Eqs. (11)–(13) [57].  (11) , (12) , (13)where  is the stress concentration factor, ,  is the pore size (major axis radius), and  is the notch root radius.In addition to the fracture toughness, respective distribution features of the pore size, pore shape (aspect ratio: short-axis radius /major-axis radius ), and grain size are used as input values to automatically evaluate the local fracture stress . The fracture energy is also evaluated from Young’s modulus  and Poisson’s ratio  using the following equation: . (14)Here, plane strain conditions are assumed.3. FEA model and analysis conditionsIn this study, crack healing (heat treatment) was simulated for specimens with an initial crack introduced on the surface (as-cracked specimen) under three levels of constant healing temperature  at atmospheric pressure. Thereafter, a three-point bending test for crack-healed specimens was conducted, and the applicability of the proposed analysis methodology to the proof test was demonstrated.The target material was an Al2O3 matrix composite containing 30 vol.% SiC particles and 0.2 vol.% MnO particles. As mentioned earlier, Al2O3/SiC/MnO composites have excellent self-healing properties [41]. Here, the number of specimens to obtain the Weibull distribution of the bending strength was set to .In the FEA, the commercial software package LS-DYNA[58] was used, and the damage-healing constitutive model based on oxidation kinetics was implemented using a user subroutine. Furthermore, the fracture mechanics model was also implemented in the subroutine, resulting in the parameters of the constitutive model being automatically set based on the microstructural data of the material as input values.3.1 FEA modelThe FEA model is shown in Fig. 3. The model was designed to reproduce a three-point bending test for as-cracked, crack-healed, and smooth specimens (no surface defects). Specimens with sizes of 3 mm × 4 mm × 22 mm (span length: 16 mm) were discretized with a one-point integral cubic element of 0.25 mm on a side. In this analysis, the jigs were not modeled, and forced displacement and constraint conditions were given to prescribed nodes (see Fig. 3(a)). Self-healing and three-point bending analyses were performed using the dynamic implicit and explicit methods, respectively. Those were switched with the restart function of the LS-DYNA.To model as-cracked specimens, two types of material parts (Materials 1 and 2) were defined [59] as shown in Fig. 3(a) and (b). Materials 1 and 2 correspond to the internal defect and surface crack parts where local fracture stress based on the fracture mechanics model is given, respectively. In Material 1, the material properties of the self-healing ceramics used in the test were input, and the probabilistic distribution of internal defects was considered. The distribution information of microstructures represented by various probability density functions was randomly assigned to each element, which set different defect information for each element (Figs. 2 and 3(c)) and enables the reproduction of variations in the fracture strength. Here, the contribution of small internal defects to fracture was ignored, because the largest internal defect (killer defect) was assumed to be the source of the fracture. Thus, within an element (integration point), there was one pore that could be the crack initiation point.In Material 2, a uniform linear crack in the width direction is assumed as shown in Fig. 3 (c). First, the geometric factor of this part was set to , and the initial crack length () was set based on the study of Ito et al. [59]. Here, we set the initial crack length to be larger than the allowable surface crack size but difficult to be detected with conventional nondestructive inspection. An allowable surface crack size is the maximum crack size that cannot be a fracture initiation point. The current local fracture stress of the surface crack part was evaluated using Eq. (10). Further, we assumed that the local fracture stress in the undamaged state is 1200 MPa in accordance with the experimental results of three-point bending test for crack-healed specimens by Osada et al. [43]. They reported the Weibull distribution for completely crack-healed specimens wherein fractures initiated anywhere other than the healed point where maximum bending stress was loaded. Therefore, we set a value that is approximately 100 MPa larger than the bending strength with a 99.9% probability of fracture (Fig. 6(b)) obtained from the experimental results [43] as the local fracture stress in the undamaged state. Based on this assumption, the damage initiation strain  was set using elastic properties. In addition, the initial values of the damage history variable  and damage variable  were also set from Eq. (3).3.2 Analysis conditions3.2.1 Material propertiesThe material properties of the Al2O3 matrix composite containing 30 vol.% SiC particles were reported [43,49]. In addition, strength test results showed that their mechanical properties are independent on the addition of MnO [41]. Table 1 shows the basic material properties. Table 2 shows the material properties related to the self-healing model based on oxidation kinetics. Regarding the material properties related to oxidation properties, Osada et al. [43] suggested that the addition of a healing activator (MnO) increases the value of the frequency factor . Therefore, the frequency factor for the target material was determined based on the experimental results of strength recovery by self-healing for specimens in which Vickers indentation was introduced by Osada et al. [41].3.2.2 Microstructural dataAs mentioned earlier, the fracture parameters of the constitutive model are set via the fracture mechanics model in this analysis method. Therefore, microstructural distribution information such as the pore size and shape and the grain size (initial crack length) are required. According to the microstructural observations of cross sections by Ozaki et al. [49], the aspect ratio of pore , grain size , and pore size  can be approximated using normal, log-normal, and power-law distributions, respectively. Therefore, these probability density functions were also employed in this study. Table 3 shows the probability density functions and their parameters.The FEA model of as-cracked specimens considering the above analytical conditions is shown in Fig. 4. The figure shows the distribution of local fracture stress  in three arbitrarily selected specimens. Random numbers were assigned to each element based on the probability density functions presented in Table 3 using the inverse function method. Further, the variation in the fracture parameters of the constitutive model was reflected in Eqs. (10)–(13), which show that the distribution of local fracture stress was different for each specimen because of the use of random numbers.4. Results and discussion4.1 Self-healing of initial cracks by heat-treatmentFirst, the results of the surface crack-healing analysis are described. Fig. 5 shows the change of the contour map of the damage variable  around the initial crack part (Material 2) over time during heat treatment. The figure corresponds to the surface portion of the specimen in the X–Y plane (Fig. 3). Bluer contours indicate smaller amounts of damage. The values of the damage variables at the initial crack are described in Section 3.1. For the healing analysis, the treatment environment adopted atmospheric pressure conditions, and the healing temperatures  were set to 800, 1000, and 1200 °C. The figure shows that crack healing progresses as the amount of oxidation reaction increases with time and the crack-healing rate increases with increasing temperature. This is due to the characteristics of the volume gain rate of oxide (Eq. (7)), which reflects that oxidation reactions tend to be accelerated at high temperatures. These phenomena correspond to the results of bending tests of crack-healed specimens under various conditions [41,43].4.2 Three-point bending test for crack-healed specimensIn this section, the analytical results of the three-point bending test for the smooth specimens are discussed, followed by the results before and after surface crack healing.Smooth specimens can be easily modeled by setting the local fracture stress in the surface crack part (Material 2) to the same conditions as those of Material 1. Fig. 6(a) shows the relationship between bending stress and deflection obtained from the three-point bending test analysis of  smooth specimens. The use of a damage model with an embedded cohesive force relationship reproduces the bending fracture behavior of a typical brittle material. The different crack initiation positions and crack propagation paths for each specimen also reproduce the scatter of bending strength. Here, to reproduce brittle fracture behavior, an FEA model consisting of elements of fine size is required, particularly in the case of materials whose local fracture stress is high similar to the current target material. Conversely, it is also difficult to conduct an analysis using an FEA model that helps reproduce stress concentration around the cracks due to enormous calculation costs. Therefore, we assumed that the force when a certain element corresponding to the origin of the crack is completely damaged corresponds to a peak value (Plots in Fig.6 (a)). Then, we evaluated the bending strength for all types of specimens (smooth, as-cracked, and crack-healed ones). Fig. 6(b) shows the Weibull distribution of the bending strength created based on the results of Fig. 6(a). The bending strength was calculated using the following equation: , (15)where  is the peak value of reaction force,  is the span length, and  and  are the width and thickness of the specimens, respectively. The cumulative fracture probability was determined using the median rank method. The cumulative fracture probability for the i-th bending strength in ascending order is given by: . (16)Fig. 6(b) also shows the experimental results of a three-point bending strength of the smooth specimens [43] () as solid lines. The Weibull modulus  and scale parameter  are  and , respectively. The results of the three-point bending test analysis for the smooth specimens are in good agreement with the experimental results (Weibull modulus  and scale parameter ). Thus, the analysis conditions for strength variation in Al2O3/SiC/MnO composites are reasonable.Next, the results of the three-point bending test analysis for the crack-healed specimens are discussed. Fig. 7(a) shows the relationship between the bending stress and deflection for healing times of , and 40 s obtained from the three-point bending test analysis. Here, one as-cracked specimen in which fractures rapidly form from the surface crack was selected. The healing conditions were the atmospheric pressure and . Specimens with healing times of  were considered as-cracked specimens. The peak value of the relationship between the bending stress and deflection increases with the healing time as the surface crack heals to an undamaged state (Fig. 5). The same fractured specimen model cannot be regenerated after a given healing time because of the use of random numbers. Therefore, the restart function of LS-DYNA[58] is used at  to replicate multiple models with the same distribution of material parameters and state variables. This enables a three-point bending test analysis of the same specimen at different healing times.Fig. 7(b) shows the Weibull distribution for  crack-healed specimens whose healing conditions are the same as those in Fig. 7(a). The Weibull distribution for the smooth specimens in Fig. 6(b) is also included. First, we focus on the result for as-cracked specimens (). All as-cracked specimens fracture from surface cracks at the same bending strength, and consequently, the slope of the Weibull distribution is vertical. In other words, surface cracks act as the fracture initiation site. Next, we focus on the results for crack-healed specimens. The scale parameter increases as the healing time increases because the crack gap volume of pre-crack placed in the part of Material 2 is gradually filled by the volume gain of oxidation products based on the kinetic model shown as Eq. (7). Here, in the damage-healing constitutive model, the strength recovery is expressed by the recovery of damage variable D  (Eq. (3)) due to an increase in strain  (Eq. (4)) with the filling of surface crack. The trends of Weibull parameters (scale parameter β and Weibull modulus m) with healing time are also discussed in Section 4.3. With the strength recovery of the part of Material 2, the fracture origin gradually shifted from the surface crack part of Material 2 to internal defects placed in the part of Material 1 as shown in Fig. 7 (b).  Fig. 7(c) shows the distribution of local fracture stress in elements around the bottom region (0.75 mm × 4 mm × 2.5 mm) of crack-healed specimens. Here, white and other color bars correspond to the frequency for the elements considering internal defects and surface cracks, respectively. Each color of the surface crack part is consistent with that in Fig. 7(b). It is confirmed that the strength of the surface crack part increases in accordance with healing, that is, around the center of the bottom where bending stress is high, the probability of the existence of elements (white bars) with lower local fracture stress compared to that of surface crack increases, and thus, fractures mainly occur from internal defects as the surface crack heals.The slope of the Weibull distribution of specimens crack-healed for and 15 s are not vertical, similar to that of as-cracked specimens, even though all specimens fracture from the surface crack (Material 2) as shown in Fig. 7 (b). This may be owing to the length of a partially healed surface crack being small (), and cases where elements, except for the surface crack part are damaged earlier than the surface crack. This causes a variation in the stress distribution state, resulting in the scatter of bending strength. In addition, there are differences in the low bending strength range between smooth specimens and completely healed specimens () even though the fractures occur from internal defects. This is because the local fracture stress in the undamaged state (i.e. completely healed state) of the surface crack part is set to 1200 MPa, as mentioned previously.4.3 Application to the proof testIn this section, we demonstrate an example of the application of the proposed FEA methodology to the proof test, referred to as a virtual experiment. In particular, specimens are sorted assuming the proof test based on the results shown in Fig. 7. In the proof test, proof stress  must be set assuming the actual working load condition. If the reference strength of the target component and safety factor corresponding to the working load condition are known,  can be determined. However, it is difficult to set appropriate safety factors for ceramic components due to the strength scatter, as mentioned previously. Therefore, the healing conditions and proof stress for ceramic components should be determined based on numerical simulation results that can estimate fracture probability under arbitrary loading conditions.Fig. 8(a) and (b) show the impact of healing time on Weibull parameters ( and ) and fracture probability from internal defects, respectively. It is confirmed that the  values significantly increased as the healing time increased, because fracture origins shifted from surface cracks to internal defects in accordance with crack-healing, as mentioned previously. Meanwhile, the  values decreased sharply from infinite to  as the healing time increased up to 15 s. Conversely, at a healing time above 15 s, wherein the fracture from internal defect appears (Fig. 8(b)), the m values increased gradually and reached a value of  corresponding to the condition that 100% specimen fracture from internal defects ().  In addition, Fig. 8(c) shows the impact of healing time on the fracture probability of healed specimens during proof tests at various proof stresses ( and 981 MPa). Here, we set proof stress conditions corresponding to 0.001, 0.01, 0.1, 1, and 10 % fracture probability of completely crack-healed specimens (), respectively, as an example. The larger the proof stress, the higher the fracture probability at the same healing time. Simultaneously, the longer the healing time, the lesser the fracture probability under the same proof stress. This means that more specimens survive during the proof test after a longer healing time, which implies the significance of the approach that combines the ‘crack healing ability’ and ‘proof testing’. Further, the figure clearly suggests that it is possible to determine healing conditions and proof stress to suffice the required yield rate. For example, to achieve 90 % yield rate after the proof test for self-healing ceramic component with machining crack and the maximum depth of 13.6 , it is likely that the proof test at an applied stress of 981 MPa after healing treatment at 1000 oC for 40 s is required. In addition, the preliminary examination for the combined method is possible through the virtual experiment, if the basic material properties (Table 1), oxidation parameters (Table 2), microstructural data (Table 3), and surface crack information of the target ceramic components are available beforehand. The proposed methodology helps estimate the healing conditions (ambient environment and time) and proof stress corresponding to the actual service conditions to suffice the production efficiency (yield rate).Much more specimens should be used for FEA because the parameters of Weibull distribution depend on the number of specimens, although the present virtual experiment was performed using  specimens referring to ordinary experiments. In fact, the proposed FEA methodology helps conduct an analysis for numerous specimens with an even lower cost and shorter time. For instance, the turbine disk used in an aircraft engine, which is known as a component that requires high reliability, -3(: standard deviation) reliable region in a normal distribution, that is strength for 0.3% fracture probability, is an index for strength evaluation. For such cases, a reliable region must be evaluated by conducting a virtual test with  specimens of the Weibull distribution.5.　ConclusionIn this study, an FEA methodology was developed to predict strength recovery and strength scatter in ceramics under arbitrary boundary conditions, and virtual experiments were conducted on a reliability assurance method for structural ceramics, which combines ‘crack healing’ and ‘proof testing’. In particular, surface crack healing (heat treatment) was simulated on specimens with an initial crack introduced on the surface. Subsequently, a three-point bending test of the crack-healed specimens was analyzed to statistically organize the strength data based on the Weibull distribution, and the applicability of the proposed analysis method to the proof test was demonstrated.It was shown that the proposed method can reproduce both crack healing and strength scattering depending on the probability distribution of internal defects. In addition, it was also shown that fractures from internal defects become dominant as the surface crack is healed. Besides, in the virtual experiment, the proof test clarified the relationship between yield rate and healing time depending on proof stress; the yield rate increased with the healing time, and this ensured strength reliability. These results suggest that the proposed FEA methodology can be used to estimate the healing conditions and proof stress to suffice the yield rate by obtaining the microstructural data and material properties of the target ceramic components in advance. This can be expanded in applications involving ceramics as structural materials.AcknowledgmentsWe would like to thank Editage (www.editage.com) for English language editing.Funding:This work was supported by project JPNP22005, commissioned by the New Energy and Industrial Technology Development Organization (NEDO). References[1] Y. 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Ozaki, Finite element analysis of fracture behavior in ceramics: Competition between artificial notch and internal defects under three-point bending, Ceram Int. 48 (2022) 36460–36468. https://doi.org/10.1016/j.ceramint.2022.08.206. Figure and table captionsFig. 1. Typical relationship between stress and strain in the damage-healing constitutive model: (a) monotonic damage process under uniaxial tension and (b) schematic of healing process.Fig. 2. Fracture mechanics model for internal defects. Here, the pore shape is oblate and R > Rb.Fig. 3. Finite element analysis model of as-cracked specimens: (a) general view (dimensions and constraint conditions), (b) enlarged view of the surrounded area in Fig. 3(a), and (c) fracture mechanics model implemented in Materials 1 and 2, respectively.Fig. 4. Contour map of local fracture stress σt in three arbitrarily selected as-cracked specimens.Fig. 5. Contour map of the damage variable during healing around initial crack portion.Fig. 6. Results of three-point bending analysis of smooth specimen models (N = 20): (a) bending stress vs. deflection curves and (b) Weibull distributions. Here, the solid line represents experimental result[43] (N = 21).Fig. 7. Results of crack-healed specimens under atmospheric pressure and TH = 1000 oC: (a) bending stress vs. deflection depending on healing time obtained by an arbitrarily selected as-cracked specimen, (b) Weibull distributions obtained after several prescribed healing times (N = 20). Here, open and closed plots mean that crack initiates from surface crack and internal defect, respectively, and (c) distribution of local fracture stress around the bottom region (0.75 mm × 4 mm × 2.5 mm) of crack-healed specimens.Fig. 8. Example of specimen sorting by proof test: (a) change over healing time of Weibull modulus m and scale parameter β, (b) change over healing time of probability of fracture from internal defects, and (c) change over healing time of fracture probability during proof test (or yield rate) under various proof stresses.Table 1. Basic material parameters (Material 1 and Material 2). Young’s modulus Poisson’s ratio Fracture toughness E [GPa] ν [-] KIC [MPa ] 398 0.21 3.8Table 2. Oxidation parameters used for Al2O3/SiC/MnO composites  Frequency factor Activation energy Reaction order  [kg2/(m4 s)] Qox [kJ/mol] mo [-] 7.85 × 10-3 161 0.835    Weight gain per unit volume gain Volume fraction Effective reactive area ratio Δρ [kg/m3] fv [-] fe [-] 1490 0.3 0.5Table 3. Microstructural parameters used for Al2O3/SiC/MnO composites  Type of distribution Mean value Standard deviation Aspect ratio A [-] Normal distribution μA = 0.6 σA = 0.1 Grain size c [mm] Log-normal distribution μc = 1.0 × 10-3 σc = 7.5 × 10-4      Type of distribution Exponent Minimum value Pore size R [mm] Power-law distribution bR = 3.0 Rmin = 1.0 × 10-3