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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 crack propagation, crack-gap-filling, and recovery behavior of mechanical properties in oxidation-induced self-healing ceramics](https://mdr.nims.go.jp/datasets/cd8b822a-dc1e-4ae8-a8f9-808b0eca2f8c)

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TF_Template_Word_Windows_2013Finite element analysis of crack propagation, crack-gap-filling, and recovery behavior of mechanical properties in oxidation-induced self-healing ceramicsMostafizur Rahman1,2, Taiyo Maeda1, Toshio Osada3 and Shingo Ozaki3,4*1 Graduate School of Engineering Science, Yokohama National University, Tokiwadai 79-5, Hodogaya-ku, Yokohama 240-8501, Japan; rahman-mostafizur-tc@ynu.jp, maeda-taiyo-rp@ynu.jp2 Department of Mechanical Engineering, Chittagong University of Engineering & Technology (CUET), Chattogram-4349, Bangladesh; mostafiz_rasel64@cuet.ac.bd 3 High-Reliability Heat-Resistant Materials Group, Research Center for Structural Materials, National Institute for Materials Science, Sengen 1-2-1 Tsukuba, Ibaraki 305-0047, Japan; OSADA.Toshio@nims.go.jp4 Division of System Research, Faculty of Engineering, Yokohama National University, Tokiwadai 79-5, Hodogaya-ku, Yokohama 240-8501, Japan; s-ozaki@ynu.ac.jp4*Correspondence: E-mail: s-ozaki@ynu.ac.jp; Tel.: +81-45-339-3881AbstractThe oxidation-induced self-healing of cracks is an attractive function for the application of ceramics in high-temperature structural components requiring high reliability. To further optimize materials or components for practical applications, the development of numerical simulation techniques is of importance. In this study, we examined crack growth, crack-gap-filling by oxide, and re-cracking behaviors in chevron-notched specimens under various load and temperature conditions by adopting a finite element analysis (FEA) approach incorporating a damage-healing constitutive model based on fracture mechanics and oxidation kinetics. Furthermore, by implementing the mechanical properties and oxidation kinetic parameters of reported self-healing ceramics composites into the FEA, we examined the effects of the composition and composite structure on the cracking and healing behaviors. Crack-gap-filling simulations suggested that the damage variables gradually decreased from the crack tip, and the minimum healing time was determined by the time required for the complete filling of the element at the crack mouth with the largest crack opening width. Furthermore, the recovery of the stiffness and strength could be successfully reproduced after complete healing with a reasonable healing temperature and time. The proposed FEA approach could also contribute to estimating the minimum healing time required at various temperatures to heal a given damage for various composites.Keywords: Self-healing ceramics, Finite element analysis, Constitutive model, Oxidation-kinetics, Chevron notch1. IntroductionCeramic materials, known for their outstanding mechanical properties, including high-temperature resistance, light weight, and high specific strength, have become indispensable in engineering applications such as aerospace, electronics, advanced manufacturing, and healthcare (Auerkari, 1996; Huet et al., 2011; Levine et al., 2002; Purohit et al., 2015). However, owing to their brittleness, even the initiation of small cracks significantly reduces their strength, thereby limiting their application to structural components requiring high reliability. Among the various structural ceramics, self-healing ceramics have the potential to provide a groundbreaking solution for extending applications in high-temperature structural components (Greil, 2020; Takahashi et al., 2011; van der Zwaag, 2007) because they can autonomically repair surface cracks to recover the strength of damaged components at high temperatures (Ando and Takahashi, 2002; Liu and Ando, 2004; Nakao et al., 2007). Self-healing ceramics are designed biomimetically, drawing inspiration from the natural healing process of bones in living organisms (Osada et al., 2020b). The concept of self-healing ceramics was originally proposed by Ando et al. (1995) for mullite/silicon carbide (SiC) composites in 1995. Self-healing ceramics that are commonly designed to heal cracks with oxidation products by pre-incorporating healing agents and healing activators are categorized as oxidation-induced self-healing ceramics (Greil, 2020). Since then, many researchers have aimed to design new composites capable of healing cracks at higher speeds or lower temperatures, proposing various types of oxidation-induced self-healing ceramics using different composites by varying the matrices (alumina (Ando et al., 2004; Chlup et al., 2008; Osada et al., 2007), mullite (Ando et al., 2001, 2005), trisilicon tetranitride (Si3N4) (Hu et al., 2019; Yao et al., 2001), titanium aluminum carbide (Ti2AlC) (Li et al., 2012; Song et al., 2012; Yang et al., 2012), healing agents (SiC (Lee et al., 2005; Liu and Ando, 2004; Nakao et al., 2008; Osada et al., 2007, 2020a; Takahashi et al., 2006), titanium carbide (TiC) (Chen et al., 2020; Yoshioka et al., 2016), Ti2AlC (Boatemaa et al., 2018)), healing activators (Manganese(II) oxide (MnO) (Osada et al., 2017a), magnesium oxide (MgO) (Osada et al., 2017a)), sintering additives (MgO (Chen et al., 2020), titanium diboride (TiB2) (Chen et al., 2020), and yttrium oxide (Y2O3) (Lee et al., 2005; Yao et al., 2001)). Despite many early attempts, self-healing ceramics have not yet been implemented practically as high-temperature materials. A major problem arises from the difficulty of proving that cracks of various sizes introduced into components during service conditions will be autonomically healed, as well as the challenge of quantitatively assessing damage evolution and healing kinetics to ensure through-life reliability. To overcome these challenges, it has become indispensable to establish new evaluation methods and numerical simulation techniques incorporating damage evolution and self-healing kinetics under service conditions. Particularly, the development of numerical simulation techniques will help to optimize the self-healing performance of oxidation-induced composites by reducing the need for experimental trial-and-error strategies.As a new evaluation method, Sloof et al. (2016) proposed an in situ four-dimensional observation technique that combines synchrotron X-ray tomography with high-temperature mechanical testing by inserting a wedge into a chevron-notched specimen adopting MAX phase ceramics. They demonstrated that this method not only reproduced stable crack growth at high temperatures, even in brittle ceramics but also enabled the in situ observation of crack-gap filling behaviors by oxidation products. The method using chevron-notched specimens is also ideal for evaluating repeated crack-healing phenomena (Osada et al., 2020b). Meanwhile, there have been a few empirical or theoretical kinetic models for crack-gap filling by incorporating various healing agents such as SiC (Osada et al., 2009), niobium aluminum carbide (Nb2AlC) (Stumpf et al., 2018), and Ti (Boatemaa et al., 2018) into different matrices. Among them, Osada et al. (2020a) proposed a universal kinetic model of strength recovery applicable to oxidation-induced self-healing ceramics with healing agents. Importantly, this model could be helpful for the detailed design and microstructure optimization of self-healing ceramics, as it incorporates microstructural details, crack geometry, the volume fraction of healing agents, crack-propagation path, and kinetic parameters for the parabolic oxidation rate of healing agents. Furthermore, Ozaki et al. (2016, 2021) proposed a damage-healing constitutive model within the framework of finite element analysis (FEA), incorporating the universal kinetic model of strength recovery (Osada et al., 2020a). It has been demonstrated that the FEA framework enables the prediction of strength recovery behaviors by healing cracks introduced by Vickers indentation under various healing conditions, including different healing temperatures, times, and oxygen partial pressures, specifically in alumina/SiC composites with various volume fractions of SiC (Maeda, et al., 2024; Nakamura et al., 2017; Ozaki et al., 2021). However, most of the previous research focused on demonstrating the healing ability of composites considering small Vickers cracks. Research on the healing behavior of long cracks extended by stable crack growth or on repeated self-healing evaluations for various types of self-healing ceramic composites is still limited in the literature. Therefore, this study aims to analyze stable crack growth and its healing behaviors in chevron-notched specimens adopting various oxide-matrix-based self-healing ceramics pre-incorporated with numerous healing agents under various damage-healing conditions, with a specific focus on long cracks. The kinetic parameters (activation energy and frequency factor), which were inversely optimized in our previous research (Rahman et al., 2023) for the parabolic oxidation rate of healing agents, were used in this study. The three-point bending (3 PB) of the chevron-notched American Society for Testing and Materials (ASTM) specimen (ASTM International, 2001), which can induce stable crack growth up to millimeter-sized crack lengths without sudden and rapid fracture, was adopted in the present FEA to reproduce arbitrary damage magnitudes. Subsequently, we demonstrated the effectiveness of the proposed FEA approach through crack-gap-filling and strength recovery simulations for several damage conditions under arbitrary boundary conditions. The FEA approach could also contribute to estimating the minimum healing time required for a given damage and healing environment for various composites. 2. Constitutive model As mentioned earlier, to promote the practical implementation of self-healing ceramics in high-temperature environments, the development of numerical simulation techniques capable of describing crack growth behaviors and healing kinetics has been indispensable for frequent analysis. To do that, FEA incorporating the continuum damage constitutive models or cohesive zone models (Ponnusami et al., 2015; Shojaei et al., 2014; Wang and Li, 2015) could be used to predict and describe the damage and crack growth behaviors under arbitrary boundary conditions. In addition, both damage and healing processes within the framework of continuum damage-healing constitutive model reflecting the thermodynamics and thermokinetics for self-healing materials were mathematically formulated by researchers at different times (Alfredsson and Stigh, 2004; Barbero et al., 2005; Darabi et al., 2012, 2013; Shojaei et al., 2015; Voyiadjis et al., 2011, 2012; Voyiadjis and Kattan, 2014).  Subsequently, Ozaki et al. (2016, 2021) proposed a damage-healing constitutive models based on the oxidation-kinetics for self-healing ceramics which are described in this section. In this model, the evolution law of self-healing was incorporated into a continuum damage model to represent the recovery of mechanical properties and effects of the healing environment, such as healing temperature, time, and oxygen partial pressure. Firstly, we describe the constitutive damage model based on fracture mechanics, and next the evolution law of state variables in self-healing processes.2.1 Stress–strain relationshipThe stress–strain relationship is given as follows: ,  where σ, ε, c, and D are the Cauchy stress tensor, small-strain tensor, fourth-order elastic-coefficient tensor, and damage variable, respectively. Values of D ranging from 0 to 1 correspond to the undamaged and perfectly damaged states, as shown schematically in Fig. 1(a). (:) represents a second-order scalar product. The concrete function of the damage variable is expressed as follows (Kurumatani et al., 2016): .  Here,  is the state variable with respect to damage and healing, and we adopted the equivalent strain . When self-healing is not considered, the damage history variable  is equal to the maximum value of the equivalent strain in the damage history.  is the equivalent strain at damage initiation, and he is the characteristic length (Oliver, 1989) (corresponding to the characteristic length of the element in the FEA). Further, σt is the maximum principal stress corresponding to the magnitude of local fracture stress, and Gf is the fracture energy. 2.2 Evolution laws for the state variableIn this section, we briefly explain the incorporation of the evolution laws of self-healing into the constitutive model. To incorporate the self-healing behavior into the damage model, the state variable κ is additively decomposed into the equivalent strain (damage) part  and self-healing part  as follows: ,  where the evolution of the damaged part is given by the following equation: .  Here,  denotes Macaulay brackets. Fig. 1(b) schematically shows that the oxidative products, such as SiO2 (s) and TiO2 (s) fill the crack gap completely or partially depending on the healing conditions and recover the damaged mechanical properties. To represent this phenomenon, the self-healing part  is assumed to be a monotonically increasing function that describes the process . From the oxidation-kinetics-based model of strength recovery (Osada et al., 2020a; Ozaki et al., 2021), the evolution law of  is as follows:   Here, the volume gain Vh owing to oxidation is as follows: ,  and the volume gain rate  is also given as . Ae is the area of one side of a fractured surface in an element, fV is the volume fraction of the healing agents, fE is the exposure ratio that is determined by the crack-propagation direction, and is the weight gain per unit-volume gain. Furthermore,  and  are the frequency factor and the activation energy for parabolic oxidation of the healing agents, respectively; Th is the healing temperature; R is the gas constant;  is the oxygen partial pressure;  is the standard oxygen partial pressure; m is the temperature-independent constant for the healing reaction in the N2-O2 mixed gas; th is the oxidation time (healing time); and n indicates the rate-controlling oxidation mechanism. Additionally, the maximum equivalent strain  obtained previously must be reevaluated to describe the dependence of the strain at damage initiation after healing on the degree of self-healing (Ozaki et al., 2021). The evolution law of  is as follows: ,  where sgn( ) represents the sign function.  is the parameter affecting the strength-recovery rate. For the self-healing state,  gradually approaches  following Eq.  as shown in Fig. 1(b). The initial value of   for the undamaged state corresponds to . For more details on the damage and loading criteria, see Ozaki et al. (2021). Additionally, Osada et al. (2020a) and Rahman et al. (2023) provide a comprehensive formulation of the oxidation-kinetics-based model for strength recovery and the kinetic parameters for target composites, respectively. For further information on healing behaviors based on oxidation kinetics, as reflected in Eqs. (5)-(7) (Osada et al. 2020a, Ozaki et al. 2021, Maeda et al. 2024), please refer to Appendix A.3. FEA model and boundary conditionsIn this study, a 3 PB test was conducted on a chevron-notched ASTM specimen to demonstrate the proposed FEA approach. Fig. 2 shows the geometry of the specimen based on ASTM model (ASTM International, 2001). The dimensions of the specimen are as follows: thickness of 6.4 mm, distance of 40 mm between the loading nodes, chevron-notch height of 3.86 mm, and pre-crack length of 2.54 mm. Fixed nodes were employed instead of jigs for loading and providing support. Here, symmetry is in the z-direction as shown in Fig. 2(b), that is, a specimen width of 3.2 mm was considered to reduce computational costs. In addition, the damage-healing constitutive model was applied exclusively to the central part of the notch area (part I and II) because crack propagation is controlled by the chevron notch. The element size of the tensile surface in the central part was a fine cubic mesh of 17.8543 μm, which was used to express the stress concentration effectively. The surrounding (part III) was considered to be a linear elastic body with the same elastic properties.The commercial software package LS-DYNA and its related user subroutine (ANSYS, 2021) were used to perform the damage-healing analysis. In addition, the dynamic-explicit method based on the central-difference method and dynamic-implicit method based on the Newmark β method were adopted for damage and healing analysis, respectively. The restart function was used to switch the analysis from explicit to implicit and vice versa. The strain increments of the six components are provided as arguments in the user subroutine umat XX for the dynamic explicit method. The loading criterion is then assessed, and the stress is calculated using Eq. (1). Internal variables related to damage and healing are evaluated based on the current strain increments and historical variables from the previous time step. Since we assume that the deformation of each element is small and that rotation can be ignored in the subsequent analysis, the co-rotational rate, which ensures objectivity for stress rate, was not used. Similarly, in the user subroutine utan XX for the dynamic implicit method, the strain increments of the six components are provided as arguments. A tangential modulus (material Jacobian) is then calculated according to the loading criterion and the updated internal variables, evaluated using the current strain increments and historical variables from the previous time step. LS-DYNA by default implements an iterative scheme of the BFGS quasi-Newton method, an incremental-iterative numerical algorithm, to ensure convergence in solving the nonlinear equilibrium equations.The analysis comprised three stages: loading-unloading, healing, and reloading. During the loading-unloading stage, a prescribed forced velocity of 0.5 mm/s was applied to loading nodes in the downward direction, as shown in Figs. 2(a) and 2(b), to achieve targeted crack length and crack-mouth opening displacement (CMOD) of the notch mouth. A crack was initiated and propagated through the central part, as highlighted by the red element shown in Fig. 2(c), as an example with a surface-crack length of 155 µm. Different surface-crack lengths and CMODs can be made in this way to perform the subsequent healing analyses. In the healing stage, the prescribed damage is healed, and the soundness state is obtained. This process was conducted by varying the holding temperature, holding time, and at standard oxygen partial pressure of 21 kPa. Here, as in the previously reported experiments (Ando et al., 2001, 2002; Osada et al., 2020a) heating and cooling processes were considered in the healing analysis. The recovery of the soundness state and mechanical properties depends on the degree of healing, which can be full or partial. After healing, the reloading stage was analyzed to confirm the recovery of the stiffness and strength.Here, the present damage model was formulated based on cohesive-zone modeling. Additionally, the regularization technique, often referred to as the Crack Band method, is adopted based on stress-strain curves that depend on mesh and element characteristics (Bazant and Oh, 1983; Jirasek and Bauer, 2012). As demonstrated in Kurumatani et al. (2016), the response of the damage model during crack propagation within a single element does not depend on the FEA mesh density. Similarly, the healing behaviors for a given crack size are also independent of mesh density. However, since we adopted the maximum principal stress as the criterion for damage initiation, the representation of stress concentration due to mesh density affects the 'crack initiation' behavior, as observed in conventional finite element methods. For reference, examples of the impact of mesh density are shown in Appendix B.4. Kinetic parameters and mechanical properties of target self-healing ceramic composites The mechanical properties and kinetic parameters of the self-healing ceramics used in the FEA are summarized in Table 1. Alumina/SiC, alumina/TiC, and mullite/SiC composites were used as the target composites in this study. The mechanical properties of each composite were derived from previous studies (Ando et al., 2001, 2002, 2004, 2005; Lee et al., 2005; Osada et al., 2020a; Ozaki et al., 2022; Yoshioka et al., 2016), utilizing indentation and bending tests. Additionally, the kinetic parameters,  and , for the volume gain of the oxidation products for each composite, based on Wagner’s parabolic model (n=2), are listed in Table 1 (Rahman et al., 2023). These parameters were determined through inverse analysis, incorporating both experimental and theoretical strength recoveries. Here, ‘AS15P’ represents the alumina matrix reinforced with 15 vol.% silicon-carbide particles, ‘MS15P’ indicates the mullite matrix reinforced with 15 vol.% silicon-carbide particles, ‘AT15P’ is the alumina matrix reinforced with 15 vol.% titanium-carbide particles, and ‘AS30P0.2MnO’ represents the alumina matrix reinforced with 30 vol.% silicon carbide doped with healing activator of 0.2 vol.% manganese oxide (MnO). In this study, we assumed that AS30P and AS30P0.2MnO have the same mechanical properties except their healing abilities. Other parameters regarding microstructure and oxidation of healing agent are shown in Table 2. Hence, the volume fractions of the healing agents, fV, in the respective composites were 0.15, 0.2, and 0.3. The role of the oxygen partial pressure in self-healing cannot be disregarded, and a standard oxygen partial pressure of 21 kPa was used. Detailed information on the kinetic parameters is provided in Refs. (Osada et al., 2009, 2017b, 2020a). 5. Results and discussion5.1 Force-CMOD curves and damage behaviors of chevron-notched specimensPrior to the crack-healing simulation, we simulated force-CMOD curves and crack growth behaviors for each target composite by adopting chevron-notched ASTM specimens in the FEA as shown in Fig. 3. As CMOD increased, the force-CMOD curves of all target composites linearly increased up to crack initiation at the chevron notch tip. Even after crack initiation, the force gradually increased until it reached a peak with stable crack growth into the chevron notch. After reaching the peak force, the force gradually decreased as the crack growth rate increased. The crack growth behavior in the notched specimens was markedly different from the experimental and simulation results under common bending tests of self-healing ceramics (Ozaki et al., 2020, 2021, 2022) in which a sudden drop of force and rapid failure occurred after reaching the peak force. Instead, the behaviors resembled the reported experimental results of bone-like self-healing ceramics from mechanical testing by inserting a wedge into a chevron-notched specimen (Osada et al., 2020b), implying that this method could also achieve stable crack growth up to millimeter-sized crack lengths. Meanwhile, differences between the tested composites were mainly manifested at the crack initiation force (Fc) and peak force (Fmax). Fc increased with the tensile strength (t ) of the composites (Table 1) since Fc was determined by t obtained experimentally by bending tests. In addition, Fmax was determined by botht and the fracture toughness (KIC) of the composites (Table 1). Note that the mechanical properties such as Young’s modulus, Poisson ratio, fracture stress, and fracture energy of each composite as listed in Table 1 were incorporated into the FEA to conduct the damage analysis. Thus, in this damage constitutive model, AT15P showed almost the same Fc compared to MS20P because of the practically same t, but exhibited higher Fmax due to the higher KIC than MS20P (Fig. 3(a)). Furthermore, AS30P, with the highest t and KIC among the target composites, showed the largest Fc and Fmax, and the lowest crack growth rate (Fig. 3(b)). Hence, this numerical analysis can reasonably simulate the full damage and crack growth behaviors of target composites, as shown in Fig. 3, with the variation of mechanical properties as listed in Table 1. Further, aiming to control the crack size and investigate the influence of the size on  healing behaviors, we set four different types of damage conditions as follows: Damage 1: controlling different CMODs to achieve the same crack length (106 µm); Damage 2: deforming until the same CMOD (2 µm); Damage 3: deforming until the force corresponded to 80% of the Fmax of each target composite; Damage 4: deforming until the force corresponded to the Fmax of each target composite. Unloading was performed for all damage conditions immediately after achieving the targeted damage conditions. The deformation and damage behaviors under Damage 1 to 4 are shown in Figs. 4(a)-(d), respectively. Therefore, by varying mechanical properties, the damage and crack growth behaviors can be evaluated, and the required crack size can be produced, as shown in Fig. 4. In the following section, we evaluated the healing behaviors of the composites for various crack lengths obtained by the four damage conditions (Fig. 4).5.2 Healing behavior of crack introduced into chevron notchFig. 5 shows the time-series snapshots of the healing process at 1473 K, 1573 K, and 1673 K for AS30P with a crack length of 765 µm introduced by Damage 4. The figure shows the effects of several healing temperatures and times on the damage state of elements during the healing process at a cross-sectional area (center part) approximately 1 mm in length from the tip of the chevron notch. The blue elements represent the undamaged state (D = 0) and the red elements represent the damaged state (D =1.0).The figure clearly shows that the damaged region gradually disappears from the crack tip and finally recovers to a sound state within the entire notch. The minimum times for complete healing (thmin) at 1473 K, 1573 K, and 1673 K were estimated as 87.28 h, 43.17 h, 23.17 h, respectively, under Damage 4. Furthermore, the healing rate changed according to the healing temperature, mirroring actual phenomena observed in experiments. Previous studies have reported that AS30P takes 0.17 h at 1573 K to completely heal the surface crack (100 m) introduced by Vickers indentation, both experimentally (Osada et al., 2020a) and by FEA (Ozaki et al., 2021). Here, the simulation results suggested that higher temperatures and longer times were required for complete healing under the present conditions due to the extended crack length and extremely severe damage conditions. The damage-healing modeling based on the reported experimental self-healing behaviors for minute cracks introduced by the Vickers indentation method (Ando et al., 2001, 2002, 2004, 2005; Osada et al., 2020a) will be helpful to simulate the repeated crack-healing of long cracks in notched specimens, as reported by Sloof et al. (2016) and Osada et al. (2020b), and to simulate the minimum healing condition for the cracks.The thickness gain behavior of the oxidation products at the damaged elements prescribed by Eq.  and the decreasing trend of the crack width due to gap filling by oxidation products in each damaged element during the healing process at 1473 K for AS30P are shown in Figs. 6(a) and 6(b), respectively. Fig. 6(b) confirms the details of crack-gap-filling behavior by the oxidation products under different time conditions through simulation. Furthermore, the deformation in the elements and crack-healing in real ceramics are shown in Fig. 6(c). The thickness of the oxidation products was modeled to increase as the healing temperature and time increased, according to Wagner’s parabolic model, as shown in Fig. 6(a). The oxidation rate was calculated using the kinetic parameters for each composite obtained from the experiment described in Table 1. Note that oxidation kinetic parameters other parameters related to the microstructural features and self-healing kinetics of each composite, as listed in Tables 1 and 2, were incorporated into the FEA. The thicknesses of the oxidation products for 15 h, 30 h, 60 h, and 90 h were calculated to be 0.54 m, 0.77m, 1.08m, and 1.33 m, respectively, for AS30P at 1473 K (Fig. 6(a)). Meanwhile, the crack width introduced by Damage 4 in AS30P decreased as the distance from the notch tip increased, and the maximum crack width of the elements was observed to be approximately 1.31 m at the notch tip. The crack width of each element decreased as the thickness of the oxidation products increased, and finally, the crack width of all elements reached zero within 90 h. Consequently, by reflecting such a gap-filling behavior, the damage variable, D, decreased to 0 from the crack tip. Finally, the value of D recovered to 0 even in the element at the crack mouth (notch tip) where the opening width was the maximum within 90 h (Fig. 6(d)). The filling behavior was similar to the tip-to-mouth filling model proposed by Osada et al. (2020a). The minimum healing time, thmin, to reduce the D of all elements on the crack to zero was determined by the complete filling time of the element with the maximum crack width from these results. For example, thmin for AS30P under Damage 4 was estimated to be approximately 87.28 h at 1473 K. This demonstration for the decrease in the crack width and damage variable with respect to the distance from the notch tip is shown at arbitrarily chosen time conditions of 0 h, 15 h, 30 h, 60 h, and 90 h, respectively. Thus, it was concluded that the proposed FEA approach could reasonably represent the healing behavior depending on the crack size and geometry as well as the environment. In addition, it also enabled continuous simulation of the cracking and healing processes.5.3 Minimum healing time for various damage conditions and compositesFig. 7(a) shows the maximum crack width corresponding to the crack length for various composites. The maximum crack width is crucial in determining thmin. Here, the crack length was controlled by different types of damage conditions mentioned in section 5.1. As shown in the figure, AS30P and AS15P, with higher t and KIC, show larger maximum crack widths than MS20P and MS15P, which have lower t and KIC. Fig. 7(b) and 7(c) summarize the effects of the crack length and healing temperature, respectively, on the thmin required for complete healing in various composites calculated by the proposed FEA approach. In Fig. 7(b), thmin is estimated to increase as the crack length increases for all composites. Furthermore, thmin is estimated to decrease significantly with an increase in the parabolic oxidation rate depending on the kinetic parameters and volume fraction of the healing agents for each composite (shown in Tables 1 and 2). thmin also becomes shorter as the parabolic oxidation rate increases with an increase in the healing temperature, as shown in Fig. 7(c).It was revealed that thmin followed the order of TiC-healing agent (alumina) < SiC-healing agent with MnO (alumina) << SiC-healing agent (Mullite) < SiC-healing agent (alumina). Additionally, in the case of the SiC-type healing agents reinforced with alumina and mullite, thmin can be rated in the order of AS30P0.2MnO << MS20P < MS15P < AS30P < AS15P, suggesting that the selection of the healing activator and matrix together with the healing agent is also important in designing ceramics that can achieve the rapid self-healing of surface cracks. The effect of kinetic parameters, healing temperature, and O₂ partial pressure on volume gain and crack-gap filling can be described by the evolution law as shown in Eqs. (5)-(7) and Appendix A. Consequently, the time and temperature required to heal a given damage can be estimated for each composite using the proposed approach and compared among different composites under service conditions, reflecting their healing kinetics. It is important to note that a verification of the specific functions for the evolution laws should be conducted by performing a quantitative comparison with relevant experimental results under the same boundary conditions. However, there are currently few reports on experiments determining self-healing time/rate (or the minimum healing time and recovery of mechanical properties with healing temperature and time) for arbitrary crack sizes in oxidation-induced self-healing ceramics, and further development in this area is anticipated.5.4 Loading and reloading curves after healingSubsequently, we analyzed the recovery of the fracture strength and stiffness after complete healing of all composites, as shown in Figs. 8, 9, and 10. Three types of analyses were performed: loading, loading-unloading, and reloading. Here, “loading” signifies the full damage analysis of virgin specimens, “loading-unloading” makes targeted damages, as shown in Fig. 4, and “reloading” is used for damage analysis after complete and without crack healing. For comparison, the figures also show the results of the reloading analysis without healing. Fig. 8 shows the comparison of loading, loading-unloading, reloading without healing, and reloading after healing under aforementioned four damage conditions considering AS30P composite as an example. Furthermore, the magnitudes of the target damages in Figs. 9 and 10 correspond to those shown in Figs. 4(a) and 4(d), respectively. Here, in these figures, the results under Damage 2 and 3 as shown in Figs. 4(b) and 4(c) are not shown to compare the recovery of mechanical properties considering all composites. The differences in the fracture properties of the composites, as listed in Table 1, resulted in variations in the force-CMOD-crack growth length behavior and crack-initiation force. As shown in the figures, the virgin and completely healed specimens exhibited almost identical behaviors, indicating that the damage-healing constitutive model was effective. The reloading analysis without healing, which cannot clearly differentiate the effect of healing on the recovery of stiffness, is not shown in Fig. 9 because the damage is small, as shown in Fig. 4(a). However, the difference in the reloading analyses without healing and after complete healing is shown in Figs. 10. Stiffness significantly decreased without healing, whereas it was fully regained after complete healing. A comparison of Figs. 9 and 10 combined with Fig. 7 also shows that the time required for complete healing varied depending on the size of the damage. Using the proposed FEA approach, the recovery of the stiffness and strength can be successfully reproduced after complete recovery with a reasonable healing temperature and time. The recovery was incomplete in the cases of partial healing. As shown in the above demonstration, using the proposed FEA approach, strength analysis could be performed under the actual loading conditions in which components and members will be used, followed by healing analysis and reloading analyses in a service environment. This is beneficial for the selection of an appropriate self-healing ceramic composite for implementation in high-temperature environments, considering the congruous mechanical properties, oxidation kinetic parameters, microstructural features, and self-healing kinetics. However, the analysis of damage-healing processes was conducted independently, without considering the interaction between damage and healing phenomena. To perform simultaneous damage-healing analyses under stressed conditions, the coupling effect of damage and healing, in addition to the creep phenomenon, should be considered in the evolution laws.6. ConclusionsThis study presents a comprehensive analysis of the damage-healing behaviors of previously reported self-healing ceramic composites under several damage conditions by implementing a damage-healing constitutive model with optimized kinetic parameters. The present FEA approach could reasonably simulate the damage recovery behavior during the healing process corresponding to the healing temperature, regardless of the damage magnitude under arbitrary boundary conditions. Hence, this approach could also contribute to estimating the minimum healing time required at various temperatures to heal a given damage for various composites. Moreover, the effects of the crack length and width, along with the kinetic parameters, on the gap-filling behavior during crack-healing were easily discernible using this approach. Because FEA can estimate the magnitude of damage in a service environment according to the mechanical and geometric properties of materials and components, strength analysis can be coupled with healing behavior. The sensitivity of self-healing to the recovery of damage and the mechanical response can also be examined using this approach. Notably, this approach helps users to discuss and determine the appropriate combination of mechanical properties, kinetic parameters, and microstructure in the self-healing components they wish to design. Collectively, these findings will be beneficial for advancing the design of next-generation self-healing ceramic composites used in high-temperature applications and for exploring the characteristics of crack-healing behaviors.In this study, we adopted a formulation where the internal state variable is additively decomposed into a damage component and a healing component. This ensures consistency in the constitutive modeling of healing energy in relation to the energy dissipated during crack formation. It should be noted that not all chemical reaction energy dissipated during self-healing contributes to crack-gap filling. Therefore, a more detailed discussion grounded in thermodynamics would be required, especially when crack propagation and self-healing occur simultaneously.Disclosure statementAuthors declare that they have no potential conflict of interest. AcknowledgmentsWe would also like to thank Editage (www.editage.com) for English language editing.FundingThis study was supported by a Grant-in-Aid for Scientific Research [Grant No. (B) 22H01357], the Japan Society for the Promotion of Science (JSPS), Japan.References Alfredsson, K.S., Stigh, U., 2004. Continuum damage mechanics revised: A principle for mechanical and thermal equivalence. Int. J. Solids Struct. 41, 4025–4045. Ando, K., Chu, M.-C., Tsuji, K., Hirasawa, T., Kobayashi, Y., Sato, S., 2002. Crack healing behaviour and high-temperature strength of mullite/SiC composite ceramics. J. Eur. Ceram. Soc. 22, 1313-1319.Ando, K., Furusawa, K., Chu, M.-C., Hanagata, T., Tsuji, K., Sato, S., 2001. Crack-Healing Behavior Under Stress of Mullite/Silicon Carbide Ceramics and the Resultant Fatigue Strength. J. Am. Ceram. Soc. 84, 2073-2078.Ando, K., Furusawa, K., Takahashi, K., Sato, S., 2005. 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J. Mater. Chem. 22, 8304–8313. Yao, F., Ando, K., Chu, M.-C., Sato, S., 2001. Static and cyclic fatigue behaviour of crack-healed Si3N4/SiC composite ceramics. J. Eur. Ceram. Soc. 21, 991–997. Yoshioka, S., Boatemaa, L., Zwaag, S. van der, Nakao, W., Sloof, W.G., 2016. On the use of TiC as high-temperature healing particles in alumina-based composites. J. Eur. Ceram. Soc. 36, 4155–4162. Appendix A. Formulation of oxidation kinetics model of crack healingIn this appendix, we review the detailed formulation of the crack healing model based on oxidation kinetics (Osada et al., 2020a). The model proposed by Osada et al. outlines a universal approach to crack healing and strength recovery through oxidation kinetics, demonstrating its applicability to a wide range of composites and matrices incorporating various healing agents. In oxidation-induced self-healing ceramics, healing agents (HAs) undergo oxidation to form healing products (HPs), which facilitate crack-gap filling and subsequent strength recovery. The generalized formulation of the chemical reaction of the healing agents is expressed as follows: . The volume gains resulting from the oxidation-induced chemical reactions correspond to weight gains, as illustrated schematically in Fig. 1(b). Consequently, the isothermal weight gain can be expressed as a function of healing temperature, time, oxygen partial pressure, and oxidation kinetic parameters. Thus, the general form of the isothermal weight gain formula is as follows: , where  is the oxidation rate constant,  is the holding healing time, and n denotes the rate-controlling oxidation mechanism. For the oxidation of SiC as a healing agent, n = 2 (diffusion-controlled weight gain) was employed, which indicates the Wagner’s parabolic weight gain model (Osada et al., 2020a), i.e., . Meanwhile, n = 2 was also considered for the TiC healing agent oxidation. Next, the rate constant can be derived by considering the effect of the oxygen partial pressure as follows: , where  and are activation energy and frequency factor of healing agent-oxidation, respectively; is the healing temperature; is the oxygen partial pressure;  is the standard oxygen partial pressure; R is the gas constant; and m = 0.835 is a temperature-independent constant experimentally determined for the healing reaction in the N2-O2 mixed gas (Osada et al., 2020a). Hence, the weight gain for healing-agent oxidation can be described as:  . Furthermore, the volume gain during the isothermal oxidation reaction can be converted from the weight gain as follows: , where (reactive area fraction) = and Ae is the area of one side of the fractured surfaces. The volume gained resulting from the self-healing reaction depends on the availability of free surface of the unreacted healing agents on the fractured surfaces. The volume fraction of the healing agents and the crack propagation path determine the actual reactive area fraction of the healing agents. In general, the volume fraction of the healing agents, ranging from 0.15 to 0.30, was utilized in alumina and mullite matrix in this study. On the other hand, the exposure ratio,  was assumed to be 0.5 when the crack propagates along the interface between the matrix and healing agents, as the healing agents are located on one side of the fractured surface as shown in Fig. 1. In this study,  was assumed to be 0.5 for SiC healing-agent oxidation with an alumina-and mullite-based matrices. Furthermore,  was also assumed to be 0.5 for TiC healing-agent oxidation with an alumina-based matrix. Hence, considering the characteristics of fractured surface and healing agent, the volume gain can be written as follows: . Furthermore, is the weight gain per unit volume gain from healing-agent oxidation. For SiC oxidation,can be written as: , where MSiC, , , and  are the molar mass and molar density of SiC and SiO2, respectively. Furthermore,  can be calculated in a similar manner for TiC healing agent oxidation by applying the following general formula: , where , are the molar mass and molar density of the ith healing product, respectively, and  and  are the molar mass and molar density of healing agents, respectively. Finally, the volume gain rate, is also given as follows: , Moreover, the initial voluem gain,  is given by: , where  is the time interval in the numerical calculation.Appendix B. Mesh Density Impact on Simulation This appendix demonstrates the impact of mesh density on damage-healing behaviors and the convergence of mechanical responses, using the arbitrarily selected AT30P composite. The mesh density in the center part (I), as shown in Fig. 2, was refined by reducing the tensile surface size of the unit element to 5.9514 µm, approximately three times smaller than the default mesh size used in this study. Three analyses—loading, loading-unloading, and reloading—were conducted under Damage 4 conditions. Under these conditions, the estimated crack lengths were 765 µm and 822 µm for mesh sizes of 17.8543 µm and 5.9514 µm, respectively. This slight variation in crack length was attributed to stress concentrations during loading in the center part (I) shown in Fig. 2.Meanwhile, the minimum time required to fully heal the prescribed damage at a temperature of 1073 K was estimated to be 0.4758 h for the larger mesh size and 0.3774 hours for the smaller mesh size. Despite these differences, the virgin and completely healed AT30P composites exhibited almost identical behaviors. For the larger mesh density, the peak force shown in Fig. 11 was slightly lower compared to the smaller mesh density depicted in Fig. 10, despite both cases incorporating the same mechanical properties listed in Table 1.It is important to note that the present damage-healing model was formulated based on cohesive zone embedded modeling and the regularization technique for the stress-strain curve. Therefore, the response of the damage model during crack propagation in one element is independent of the FEA mesh density. Additionally, the healing behavior for a given crack size is also unaffected by mesh density. However, we used the maximum principal stress, which depends on the mesh density of the stress concentration area, for the damage initiation criterion. Consequently, as with typical FEA, the stress concentration's effect on damage initiation is slightly dependent on crack length and minimum healing time. We believe that if an FEA model is prepared to represent sufficient stress gradients in line with experimental results, the present analysis scheme can reasonably simulate damage progression and healing behavior, regardless of mesh density and boundary conditions.Figure and table captionsFigure 1. Schematic of stress vs. strain relationship during cracking and crack-gap filling by oxidation assumed in the damage-healing constitutive model. (a) damage process; (b) recovery of mechanical properties due to self-healing. Figure 2. FEA model of the symmetric chevron-notched ASTM specimen under the 3 PB test: (a) FEA model; (b) schematic of the model to clearly show the chevron notch area; (c) a close-up view around the chevron notch area. Here, the model is symmetric in the z-direction. As shown in the close-up view around the damaged part in the notch tip, the crack propagates through the center part, as indicated by the red elements.Figure 3. FEA results of all tested composites adopting the symmetric chevron-notched ASTM specimen under the 3 PB test: (a) relationships between force and CMOD; (b) relationships between crack growth length and CMOD. Here, Fc denotes the crack initiation force. Figure 4. Prescribed damage conditions for healing: (a) same crack length (106 µm) with different CMODs; (b) same CMOD (2 µm) with different crack lengths; (c) different crack lengths and CMODs corresponding to 80% of Fmax of the respective composites; (d) different crack lengths and CMODs corresponding to the Fmax of the respective composites.Figure 5. Time-series contour plots of the healing process of the damaged part of the AS30P composite. Here, the healed damage corresponds to Damage 4 shown in Fig. 4(d). The healing temperatures are 1473 K, 1573 K, and 1673 K and the oxygen partial pressure is 21 kPa. Note that the blue elements show the undamaged state and red elements show the fully damaged state. Each snapshot corresponds to close-up views of the cross-section of the center (part I), approximately 1 mm from the notch tip. Figure 6. Detailed results of healing behavior for the crack geometry corresponding to Damage 4 in Fig. 4(d). Here, the composite is AS30P, and the healing temperature is 1473 K. (a) variation in the thickness gain of oxidation products with healing; (b) decrease in the crack width due to gap filling by oxidation products in damaged elements on the crack; (c) comparison of crack opening and filling by oxidation products in real ceramics and an element in FEA; (d) decrease in the damage variable in damaged elements on cracks. Figure 7. Minimum healing time estimated by FEA: (a) relationship between the crack length and maximum crack width at the notch tip; (b) variation in the minimum healing time with crack length under healing temperature of 1273 K; (c) variation in the minimum healing time with the healing temperature for Damage 1. Figure 8. FEA results of the AS30P composite under four damage conditions. The figure shows the relationships between the force-CMOD-crack growth length from loading, loading-unloading, reloading after healing, and reloading without healing to confirm the recovery of mechanical properties. The healing times at the healing temperature of 1873 K for Damages 1, 2, 3, and 4 were 0.6202, 1.0771, 2.1146, and 8.1681 h, respectively. Figure 9. Damage analysis of the composites to confirm the recovery of strength and stiffness. Here, the crack geometry corresponds to Damage 1 shown in Fig. 4(a). The healing temperature was 1573 K and the healing time was 10 h. Figure 10. Damage analysis of the composites to confirm the recovery of strength and stiffness. Here, the crack geometry corresponds to Damage 4 shown in Fig. 4(d). The healing temperature was 1873 K and the healing time was 90 h.Figure 11. Damage analysis of AT30P composite to confirm the recovery of strength and stiffness. Here, the crack geometry corresponds to Damage 4 condition. The healing temperature and time were set to be 1073 K and 2 h, respectively. The minimum time required to completely heal the damage was estimated to be 0.3774 h.Table 1 Mechanical properties and oxidation kinetic parameters of self-healing ceramic composites. Name Mechanical Properties Oxidation kinetic parameters  Young’s modulus Poissonratio Fracture stress Fracturetoughness Fractureenergy Frequencyfactor Activation energy  E [GPa] ν [-]  [MPa]   [N/mm]   AS15P 390 0.23 908.0 4.0 0.03886 9.5 138.0 AS30P 398 0.22 979.0 4.0 0.03826 8.9 136.0 AS30P0.2MnO 398 0.22 979.0 4.0 0.03826 7.6 156.0 MS15P 385 0.24 700.1 2.8 0.01919 6.1 114.5 MS20P 385 0.24 516.3 2.8 0.01919 6.0 113.0 AT15P 395 0.27 530.0 4.0 0.03755 9.0 101.0 AT30P 398 0.25 600.0 4.0 0.03769 5.2 101.0Table 2 Other parameters related to the microstructural features and self-healing kinetics of composites. For SiC and TiC oxidation, n = 2 was adopted corresponding to Wagner’s parabolic model. 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