Simulation of Microscopic Fracture Behavior in Nanocomposite Ceramic Tool Materials

: In this paper, the microstructures of nanocomposite ceramic tool materials are represented through Voronoi tessellation. A cohesive element model is established to perform the crack propagation simulation by introducing cohesive elements with fracture criteria into microstructure models. Both intergranular and transgranular cracking are considered in this work. The inﬂuences of nanoparticle size, microstructure type, nanoparticle volume content and interface fracture energy are analyzed, respectively. The simulation results show that the nanoparticles have changed the fracture pattern from intergranular mode in single-phase materials to intergranular–transgranular–mixed mode. It is mainly the nanoparticles along grain boundaries that have an impact on the fracture pattern change in nanocomposite ceramic tool materials. Microstructures with smaller nanoparticles, in which there are more nanoparticles dispersed along matrix grain boundaries, have higher fracture toughness. Microstructures with a nanoparticle volume content of 15% have the most obvious transgranular fracture phenomenon and the highest critical fracture energy release rate. A strong interface is useful for enhancing the fracture toughness of nanocomposite ceramic tool materials.


Introduction
Nanocomposite ceramic tool materials possess excellent mechanical properties, along with nanoparticles that have great specific surface area, high chemical activity and large dispersivity [1,2].Fracture is a primary failure mechanism for nanocomposite ceramic tool materials, just like ordinary ceramic materials.Grain boundaries (GBs), nanoparticles and thermal expansion mismatch between matrix grains and nanoparticles may lead to crack deflection, pinning and bridging, which tend to improve the fracture energy of materials [3].Therefore, investigating the fracture behavior of nanocomposite ceramic tool materials in microscale contributes to the exploration of the relationship between microstructures and macroscopic properties, like fracture strength and toughness.
Due to the heterogeneous microstructures and the mechanical property anisotropies of constituent phases, the microscopic fracture behavior of ceramic tool materials cannot be reflected by means of continuum mechanics, which are suitable for predicting the average response of heterogeneous materials.Therefore, a proper approach is needed to perform the cracking simulation of polycrystalline materials; which does not only characterize microstructural heterogeneities, but also tracks the microscopic damage process, such as crack initiation, propagation and coalescence [4].The cohesive element model is an approach that could achieve both objectives.
Dugdale [5] conducted the earliest studies on the cohesive element theory and found that cohesive tractions along the fracture process zone were constant for metallic materials under the condition of ideal elastic-plasticity.Xu and Needleman [6] carried out the dynamic crack growth simulation of isotropic elastic solids based on the cohesive element model they proposed previously.Wang et al. [7] proposed a computational framework for the impact fracture of laminated glass based on the cohesive element model, which was applied to the impact fracture simulation of a laminated glass plate.The simulation results were found to be in agreement with the experimental ones.Geng et al. [8] conducted numerical simulations on two-dimensional standard concrete specimens under axial compression and tension based on the cohesive element theory.Their results indicated that the proposed method exhibited exceptional capabilities in predicting initial stiffness, uniaxial strength and corresponding strain.Xu et al. [9] studied the interface cracking mechanism of calcium-magnesium-alumino-silicates (CMAS)-corroded thermal barrier coatings based on the cohesive element model.Their theoretical model validated the spalling phenomena of thermal barrier coatings caused by CMAS corrosion at high temperatures.Zavattieri and Espinosa [10,11] developed a grain level micromechanical model of brittle materials in the context of cohesive elements to model crack initiation and propagation.Simulations of wave propagation experiments on alumina are used to illustrate the model capabilities in the framework of experimental measurements.Zhou, Tomar and Zhai [4,12] investigated the dynamic fracture process of Al 2 O 3 /TiB 2 microstructures and discussed the effects of heterogeneous phases on crack paths.Sun et al. [13] investigated the fracture behavior of graphene-toughened Al 2 O 3 ceramic materials and characterized the Al 2 O 3 /graphene interface properties using the cohesive element method.The effects of phase interface bonding strength and graphene content on fracture behavior were analyzed and some toughening mechanisms induced by nano-graphene, such as crack bridging, crack branching, microcracks and pull-out of graphene, were discussed.
This paper mainly deals with microscopic cracking simulations based on nanocomposite ceramic tool materials.Although many studies have been carried out regarding the microscopic cracking behavior of both single phase ceramic tool materials and composite ceramic tool materials reinforced with microsized particles, little attention has been paid to the work on the microscopic fracture mechanisms of nanocomposite ceramic tool materials.It is meaningful for the optimization of mechanical properties to investigate the microscopic fracture behavior of nanocomposite ceramic tool materials, and to analyze the effects of microstructure heterogeneities such as nanoparticle distribution and the nanoparticle size on cracking paths.
In this paper, a micromechanical model of Al 2 O 3 /SiCn nanocomposite is developed by embedding cohesive elements with fracture criteria into microstructures represented using the Voronoi tessellation.Then crack propagation modeling is conducted in order to explore the effects of microstructural morphologies, like nanoparticle size, microstructure type, nanoparticle volume content and interface fracture energy, on fracture patterns and mechanical responses.

The Cohesive Element Model for Microscopic Cracking Simulation 2.1. Traction-Separation Law
In the cohesive element model, it is assumed that crack surfaces carry cohesive tractions that resist normal separation (T n ) and tangential sliding (T t ) before cracking [14].The relationship between cohesive traction (T) and cracking displacement (δ) can be described using the traction-separation law.
In this work, the linear traction-separation law for brittle materials (Figure 1) is utilized to characterize the properties of cohesive elements in ceramic tool materials.When the cohesive traction increases to a maximum value T max , the damage process begins.As the traction gradually reduces to zero, the cohesive interface will fail completely and reach the maximum displacement (δ max ).The area under the traction-separation curve equals the interface fracture energy Γ.It is obvious that the traction-separation law can be determined by two parameters: the maximum interface traction T max and the interface fracture energy Γ.This makes the cohesive element model very attractive for practical applications.
the interface fracture energy .It is obvious that the traction-separation law can be determined by two parameters: the maximum interface traction Tmax and the interface fracture energy .This makes the cohesive element model very attractive for practical applications.

Development of the Cohesive Element Model
In this work, all the crack propagation simulation jobs are performed based on the microstructures of ceramic tool materials, which are represented by a Voronoi tessellation.Many studies have shown that the grain level structures of polycrystalline materials can be modeled very well by the Voronoi tessellation [15], and the influences of microstructure heterogeneities and neighboring elements on grain shapes can be reflected by the Voronoi tessellation.This feature gives the Voronoi tessellation the ability to quantitatively describe the characteristics of microstructures, such as the volume content, the size, the shape and the orientation of each composition phase.Due to the high computational efficiency and the simplicity of programming, the Voronoi tessellation has been widely used to characterize microstructures of polycrystalline materials [16][17][18].
In order to conduct the microscopic fracture simulation, a cohesive element model should be established by embedding cohesive elements with cracking law among adjacent volume elements using the Voronoi tessellation.Therefore, the Voronoi tessellation should be meshed with triangular volume elements first.In contrast with the simulation procedures for single phase ceramic tool materials described in [19], nanoparticles should be generated after the microstructure models are divided by triangular elements.In order to conduct the simulation of intergranular and transgranular cracking in nanocomposites, rectangular cohesive elements with cracking law should be embedded among all triangular volume elements to implement the arbitrary extension of the crack.The process of generating the Voronoi tessellation, the nanoparticles and the cohesive elements is carried out through programing with MATLAB 2020.The simulation procedures are illustrated in Figure 2.
Figure 3a shows the microstructure model represented with the Voronoi tessellation, in which each Voronoi polygon represents a matrix grain.And Figure 3b is the Voronoi diagram after embedding cohesive elements and triangular elements.It is observed from Figure 3b that white triangles indicate volume elements of the matrix and blue triangles denote volume elements of nanoparticles.Cohesive elements in matrix grains and those along matrix GBs are represented by yellow and red quadrilaterals, respectively.Cohesive elements in nanoparticles and along interfaces are characterized by green and black quadrilaterals, respectively.For presentation purposes, the thickness of cohesive elements is magnified in Figure 3b.

Development of the Cohesive Element Model
In this work, all the crack propagation simulation jobs are performed based on the microstructures of ceramic tool materials, which are represented by a Voronoi tessellation.Many studies have shown that the grain level structures of polycrystalline materials can be modeled very well by the Voronoi tessellation [15], and the influences of microstructure heterogeneities and neighboring elements on grain shapes can be reflected by the Voronoi tessellation.This feature gives the Voronoi tessellation the ability to quantitatively describe the characteristics of microstructures, such as the volume content, the size, the shape and the orientation of each composition phase.Due to the high computational efficiency and the simplicity of programming, the Voronoi tessellation has been widely used to characterize microstructures of polycrystalline materials [16][17][18].
In order to conduct the microscopic fracture simulation, a cohesive element model should be established by embedding cohesive elements with cracking law among adjacent volume elements using the Voronoi tessellation.Therefore, the Voronoi tessellation should be meshed with triangular volume elements first.In contrast with the simulation procedures for single phase ceramic tool materials described in [19], nanoparticles should be generated after the microstructure models are divided by triangular elements.In order to conduct the simulation of intergranular and transgranular cracking in nanocomposites, rectangular cohesive elements with cracking law should be embedded among all triangular volume elements to implement the arbitrary extension of the crack.The process of generating the Voronoi tessellation, the nanoparticles and the cohesive elements is carried out through programing with MATLAB 2020.The simulation procedures are illustrated in Figure 2.
Figure 3a shows the microstructure model represented with the Voronoi tessellation, in which each Voronoi polygon represents a matrix grain.And Figure 3b is the Voronoi diagram after embedding cohesive elements and triangular elements.It is observed from Figure 3b that white triangles indicate volume elements of the matrix and blue triangles denote volume elements of nanoparticles.Cohesive elements in matrix grains and those along matrix GBs are represented by yellow and red quadrilaterals, respectively.Cohesive elements in nanoparticles and along interfaces are characterized by green and black quadrilaterals, respectively.For presentation purposes, the thickness of cohesive elements is magnified in Figure 3b.
The addition of cohesive elements among volume elements has a negative effect on the stiffness of the FEM model.The density of cohesive elements increases as finite elements are refined, which will intensify the reduction of stiffness [12].However, this issue can be avoided by carefully choosing a lower bound of finite element size and the constitutive parameters of elements.On the other hand, for the purpose of ensuring computational accuracy, there should be several cohesive elements distributed in the cohesive zone, which imposes an upper limit of finite element size.According to the reference [12], in order to ensure simulation accuracy and calculation convergence, the finite element size h takes the form: where E = E for plane stress and E = E/ 1 − ν 2 for plane strain.E and ν are the elasticity modulus and Poisson's ratio, respectively.λ cr = δ 0 /δ max is the critical effective displacement jump.
Substituting the constitutive parameters of cohesive elements into Equation (1), the range of finite element size is specialized to 0.0016 µm ≤ h ≤ 2.54 µm.In our work, the finite element size of h = 0.073 µm is adopted, which is reasonable for achieving accurate modeling.The addition of cohesive elements among volume elements has a negative effect on the stiffness of the FEM model.The density of cohesive elements increases as finite elements are refined, which will intensify the reduction of stiffness [12].However, this issue can be avoided by carefully choosing a lower bound of finite element size and the constitutive parameters of elements.On the other hand, for the purpose of ensuring computa-   The addition of cohesive elements among volume elements has a negative effect on the stiffness of the FEM model.The density of cohesive elements increases as finite elements are refined, which will intensify the reduction of stiffness [12].However, this issue can be avoided by carefully choosing a lower bound of finite element size and the constitutive parameters of elements.On the other hand, for the purpose of ensuring computa-

Implementation of FEM Simulation
The microscopic fracture problems depicted in Figure 4 can be solved with the cohesive element model.An initial crack added in the simulation model aims to form the stress concentration, which can form regular crack paths under loading and is convenient for analyzing the influences of microstructures on cracking behaviors.In this work, the initial crack is set along grain boundaries and its length is equivalent to grain size.The effect of the initial crack on simulation results can be ignored, since the length of the initial crack is smaller than 1/10 of the microstructure model's length.The constant velocity load is imposed symmetrically at the top and bottom edges of the model.The corresponding velocity load can be obtained using the expression v = .ε•h, while the constant strain rate is .ε = 5 × 10 3 and h is the microstructure height.The total simulation time is 0.5 µs.
where  =  for plane stress and  = /(1 −  ) for plane strain.E and  are the elasticity modulus and Poisson's ratio, respectively. =  / is the critical effective displacement jump.
Substituting the constitutive parameters of cohesive elements into equation ( 1), the range of finite element size is specialized to 0.0016 μm ≤ ℎ ≤ 2.54 μm.In our work, the finite element size of h = 0.073 μm is adopted, which is reasonable for achieving accurate modeling.

Implementation of FEM Simulation
The microscopic fracture problems depicted in Figure 4 can be solved with the cohesive element model.An initial crack added in the simulation model aims to form the stress concentration, which can form regular crack paths under loading and is convenient for analyzing the influences of microstructures on cracking behaviors.In this work, the initial crack is set along grain boundaries and its length is equivalent to grain size.The effect of the initial crack on simulation results can be ignored, since the length of the initial crack is smaller than 1/10 of the microstructure model's length.The constant velocity load is imposed symmetrically at the top and bottom edges of the model.The corresponding velocity load can be obtained using the expression  =  • ℎ, while the constant strain rate is  = 5 × 10 and h is the microstructure height.The total simulation time is 0.5 μs.The linear decaying cohesive law (Figure 1) is used to characterize the properties of cohesive elements.In this work, all modeling calculations are based on Al2O3/SiCn nanocomposite ceramic tool materials.The material parameters for various elements are listed in Table 1.The linear decaying cohesive law (Figure 1) is used to characterize the properties of cohesive elements.In this work, all modeling calculations are based on Al 2 O 3 /SiC n nanocomposite ceramic tool materials.The material parameters for various elements are listed in Table 1.

Influence of Microstructure Type
In nanocomposite ceramic tool materials, nanoparticle size is under nanoscale and matrix grain size is under microscale or sub-microscale.Therefore, nanoparticles distribute in matrix grains as well as along GBs.According to the distribution of nanoparticles in matrix phases, nanocomposite ceramic tool materials can be divided into three microstructure types, which are intragranular, intergranular and intragranular/intergranular structures.In order to explore the influences of nanoparticle distribution on mechanical properties, models of the intergranular, intragranular and intragranular/intergranular microstructures for Al 2 O 3 /SiC n have been established, as shown in Figure 5.The average size of Al 2 O 3 matrix grains is 1 µm, while the average size of nano SiC is d nano = 70 nm and the volume content of nano SiC is V nano = 10%.
in matrix phases, nanocomposite ceramic tool materials can be divided into three microstructure types, which are intragranular, intergranular and intragranular/intergranular structures.In order to explore the influences of nanoparticle distribution on mechanical properties, models of the intergranular, intragranular and intragranular/intergranular microstructures for Al2O3/SiCn have been established, as shown in Figure 5.The average size of Al2O3 matrix grains is 1 μm, while the average size of nano SiC is  = 70 nm and the volume content of nano SiC is  = 10%.The cracking paths of microstructure models are shown in Figure 6.It can be observed that transgranular cracking is predominant in the intergranular structure (Figure 6a), and the crack is deflected continuously into Al2O3 grains by SiC nanoparticles located along matrix GBs.For the intragranular structure (Figure 6b), only intergranular cracking happens.The transgranular fracture in the intragranular/intergranular structure is less remarkable than that in the intergranular structure (Figure 6c).As for Al2O3/SiCn nanocomposite ceramic tool materials, SiC nanoparticles and Al2O3 matrix grains usually bond tightly together and no glass phases exist.Many researchers have observed this phenomenon by means of high resolution TEM. Figure 7 displays the interface between the Al2O3 matrix and the SiC nanoparticle in Al2O3/SiCn [20].In addition, S. Jiao [21] concluded that the fracture surface energy of interfaces between Al2O3 and SiC in Al2O3/SiCn is higher than that of Al2O3 grains at home temperature, according to experimental results and empirical formulae.The high-strength interfaces between nanoparticles and matrix grains have a pinning effect on the intergranular crack.At matrix GBs without nano SiC particles, the intergranular cracking is predominant.When the intergranular crack encounters nano SiC located along GBs, it is liable to be deflected into Al2O3 matrix grains, and transgranular cracking occurs.As the crack travels along GBs without nanoparticle distribution, the intergranular cracking continues until it is deflected into matrix grains again by another nanoparticle located at GBs. Figure 8 shows the SEM morphologies of Al2O3-based nanocomposite ceramic tool materials, in which typical cleavage steps caused by transgranular fractures are clearly observed, and nanoparticles located at GBs are indicated by red arrows [22].In the case of the intragranular structure (Figure 6b), the intergranular cracking will not be hindered by nanoparticles located in matrix grains.Therefore, the transgranular fracture hardly arises.The cracking paths of microstructure models are shown in Figure 6.It can be observed that transgranular cracking is predominant in the intergranular structure (Figure 6a), and the crack is deflected continuously into Al 2 O 3 grains by SiC nanoparticles located along matrix GBs.For the intragranular structure (Figure 6b), only intergranular cracking happens.The transgranular fracture in the intragranular/intergranular structure is less remarkable than that in the intergranular structure (Figure 6c).As for Al 2 O 3 /SiC n nanocomposite ceramic tool materials, SiC nanoparticles and Al 2 O 3 matrix grains usually bond tightly together and no glass phases exist.Many researchers have observed this phenomenon by means of high resolution TEM. Figure 7 displays the interface between the Al 2 O 3 matrix and the SiC nanoparticle in Al 2 O 3 /SiC n [20].In addition, S. Jiao [21] concluded that the fracture surface energy of interfaces between Al 2 O 3 and SiC in Al 2 O 3 /SiC n is higher than that of Al 2 O 3 grains at home temperature, according to experimental results and empirical formulae.The high-strength interfaces between nanoparticles and matrix grains have a pinning effect on the intergranular crack.At matrix GBs without nano SiC particles, the intergranular cracking is predominant.When the intergranular crack encounters nano SiC located along GBs, it is liable to be deflected into Al 2 O 3 matrix grains, and transgranular cracking occurs.As the crack travels along GBs without nanoparticle distribution, the intergranular cracking continues until it is deflected into matrix grains again by another nanoparticle located at GBs. Figure 8 shows the SEM morphologies of Al 2 O 3 -based nanocomposite ceramic tool materials, in which typical cleavage steps caused by transgranular fractures are clearly observed, and nanoparticles located at GBs are indicated by red arrows [22].In the case of the intragranular structure (Figure 6b), the intergranular cracking will not be hindered by nanoparticles located in matrix grains.Therefore, the transgranular fracture hardly arises.Based on the energy equilibrium theory of cracking systems [23], as a crack propagates for the length of c, the critical fracture energy release rate G C can be expressed as: where U S is the fracture energy dissipated during facture, which can be obtained through simulation.
The critical fracture energy release rate G C is usually regarded as a quantitative description of the fracture resistance of materials.The relationship between fracture toughness K IC and the critical fracture energy release rate G c is given by: Figure 11 shows the calculation results of G C during the fracture process.The G C of the intergranular structure is higher than that of the intragranular/intergranular structure, and the intragranular structure exhibits the lowest level of G C .Since the fracture surface energy of matrix grains is above that of matrix GBs (shown in Table 1), the transgranular fracture will consume more energy than the intergranular fracture during the cracking process.Therefore, if taking no consideration of residual stress and merely analyzing the effect of interface bonding strength, nanoparticles located along GBs dominate in the fracture mode transformation from intergranular to transgranular cracking, and in the improvement of fracture toughness for nanocomposite ceramic tool materials.

Influence of Nanoparticle Size
Figure 9 shows two intergranular microstructure models with different nanoparticle sizes (d nano = 150 nm in Figure 9a and d nano = 70 nm in Figure 9b).The two models have the same matrix morphologies, with the average matrix grain size d matrix = 1µm and the volume content of nanoparticles V nano = 10%.
the intergranular structure is higher than that of the intragranular/intergranular structure, and the intragranular structure exhibits the lowest level of GC.Since the fracture surface energy of matrix grains is above that of matrix GBs (shown in Table 1), the transgranular fracture will consume more energy than the intergranular fracture during the cracking process.Therefore, if taking no consideration of residual stress and merely analyzing the effect of interface bonding strength, nanoparticles located along GBs dominate in the fracture mode transformation from intergranular to transgranular cracking, and in the improvement of fracture toughness for nanocomposite ceramic tool materials.

Influence of Nanoparticle Size
Figure 9 shows two intergranular microstructure models with different nanoparticle sizes ( = 150 nm in Figure 9a and  = 70 nm in Figure 9b).The two models have the same matrix morphologies, with the average matrix grain size  = 1 μm and the volume content of nanoparticles  = 10%.The cracking paths of the two microstructure models after FEM calculation are shown in Figure 10.For the single phase ceramic tool materials, the main fracture mode is basically intergranular cracking due to weak grain boundaries, while transgranular cracking is more liable to occur in metals with strong grain boundaries [24][25][26].Compared with the intergranular cracking in single phase ceramic tool materials, the fracture mode of nanocomposite ceramic tool materials has changed to the mixed mode of intergranular and transgranular cracking.And the transgranular cracking is more obvious in the microstructure with smaller nanoparticles ( = 70 nm).Nanoparticles located along GBs represent strong obstacles to crack propagation due to their higher mechanical properties.
The intergranular crack tends to be deflected into matrix phases when meeting nanoparticles at GBs.The cracking paths of the two microstructure models after FEM calculation are shown in Figure 10.For the single phase ceramic tool materials, the main fracture mode is basically intergranular cracking due to weak grain boundaries, while transgranular cracking is more liable to occur in metals with strong grain boundaries [24][25][26].Compared with the intergranular cracking in single phase ceramic tool materials, the fracture mode of nanocomposite ceramic tool materials has changed to the mixed mode of intergranular and transgranular cracking.And the transgranular cracking is more obvious in the microstructure with smaller nanoparticles (d nano = 70nm).Nanoparticles located along GBs represent strong obstacles to crack propagation due to their higher mechanical properties.The intergranular crack tends to be deflected into matrix phases when meeting nanoparticles at GBs.
The results of G C for the intragranular, intergranular and intragranular/intergranular microstructures with two different sizes of nanoparticles (150 nm and 70 nm) are summarized in Figure 11.It can be found that microstructure models with smaller nanoparticles generally exhibit higher levels of G C , and the characteristics of G C resulting from different nanoparticle sizes are most remarkable in the intergranular microstructure.As for nanocomposite with the same content of nanoparticles, the toughening effect of microstructures with smaller nanoparticles is improved.That is because microstructures with smaller nanoparticles have nanoparticles that are more widely dispersed along GBs, which results in more significant effects on crack pinning and deflection into matrix grains.Crack deflection will absorb extra fracture energy, and the transgranular cracking consumes more energy compared with the intergranular cracking during the fracture process.The results of GC for the intragranular, intergranular and intragranular/intergranular microstructures with two different sizes of nanoparticles (150 nm and 70 nm) are summarized in Figure 11.It can be found that microstructure models with smaller nanoparticles generally exhibit higher levels of GC, and the characteristics of GC resulting from different nanoparticle sizes are most remarkable in the intergranular microstructure.As for nanocomposite with the same content of nanoparticles, the toughening effect of microstructures with smaller nanoparticles is improved.That is because microstructures with smaller nanoparticles have nanoparticles that are more widely dispersed along GBs, which results in more significant effects on crack pinning and deflection into matrix grains.Crack deflection will absorb extra fracture energy, and the transgranular cracking consumes more energy compared with the intergranular cracking during the fracture process.
In addition, based on the grain growth theory, nanoparticles located along matrix GBs have a pinning influence on the migration of matrix GBs.The relationship between average matrix grain size and nanoparticle size is given as follows [27]: where  is the average matrix grain size,  is the nanoparticle size and  is the volume content of nanoparticles.Smaller nanoparticles have a stronger pinning effect on matrix GB migration during the sintering procedure, which will lead to a refinement of matrix grains in microstructures.Cheng [28] observed this phenomenon by fabricating nanocomposite ceramic tool materials with different sizes of nanoparticles at the same volume content.It is known that grain refinement contributes to improving the fracture strength of materials, since the length of inherent cracks in ceramic materials is commonly the same as the maximum grain size, and materials containing shorter inherent cracks exhibit higher fracture strength based on the Griffith fracture theory [3,29].From the above, conclusions can be drawn that smaller nanoparticles are useful for the enhancement of the fracture strength and toughness of nanocomposite ceramic tool materials.

Influence of Nanoparticle Volume Content
Figure 12 shows microstructure models for nanocomposite with Vnano = 3%, 5%, 10%, 15%, 20% and 30%, respectively.The microstructures have the same matrix morphology, with  = 1 μm and  = 70 nm.The cracking paths for the microstructures are shown in Figure 13.When the volume In addition, based on the grain growth theory, nanoparticles located along matrix GBs have a pinning influence on the migration of matrix GBs.The relationship between average matrix grain size and nanoparticle size is given as follows [27]: where D is the average matrix grain size, R is the nanoparticle size and V f is the volume content of nanoparticles.Smaller nanoparticles have a stronger pinning effect on matrix GB migration during the sintering procedure, which will lead to a refinement of matrix grains in microstructures.Cheng [28] observed this phenomenon by fabricating nanocomposite ceramic tool materials with different sizes of nanoparticles at the same volume content.It is known that grain refinement contributes to improving the fracture strength of materials, since the length of inherent cracks in ceramic materials is commonly the same as the maximum grain size, and materials containing shorter inherent cracks exhibit higher fracture strength based on the Griffith fracture theory [3,29].From the above, conclusions can be drawn that smaller nanoparticles are useful for the enhancement of the fracture strength and toughness of nanocomposite ceramic tool materials.The cracking paths for the microstructures are shown in Figure 13.When the volume content of nanoparticles (V nano ) is lower (3%~10%), the intergranular cracking dominates.The microstructure with 15 vol% nanoparticles exhibits obvious transgranular cracking.As V nano arrives at 20%, nanoparticle agglomeration does not arise and the transgranular cracking appears at some sites of the crack path, although V nano is at a higher level and the distances among nanoparticles are relatively small.Some nanoparticles agglomerate together when V nano reaches 30%.The main crack propagates from some region with weaker properties at the bottom of the microstructure model, rather than from the initial crack.The transgranular cracking still exists at some sites of the crack path, but is at a relatively lower level.The results of GC are plotted in Figure 14.As depicted in Figure 14, GC firstly shows an increase, and then a decrease trend as the nanoparticle content increases.GC reaches the peak at Vnano = 15%, which indicates that the fracture resistance is at its highest.Many researchers carried out theoretical and experimental investigations on the effect of the volume content of nano SiC on the fracture toughness.Niihara et al. [30] found that the fracture toughness of Al2O3/SiCn reached the maximum with 5 vol% SiC through experiments, The results of G C are plotted in Figure 14.As depicted in Figure 14, G C firstly shows an increase, and then a decrease trend as the nanoparticle content increases.G C reaches the peak at V nano = 15%, which indicates that the fracture resistance is at its highest.Many researchers carried out theoretical and experimental investigations on the effect of the volume content of nano SiC on the fracture toughness.Niihara et al. [30] found that the fracture toughness of Al 2 O 3 /SiC n reached the maximum with 5 vol% SiC through experiments, as depicted in Figure 15.Levin et al. [31] investigated the relationship between the fracture toughness of Al 2 O 3 /SiC n and the SiC volume content, and obtained similar results to Niihara.Conclusions can be drawn from the modeling results in this paper that the microstructure with 15 vol% SiC has the maximum fracture toughness, which is obviously higher than the results published by other researchers.The primary reason for this difference is that the residual stress caused by the thermal expansion mismatch between nano SiC and matrix Al 2 O 3 is not taken into account in our model.As the nanoparticle content is at a relatively higher level, there are more nanoparticles in the matrix grains, and the area of residual tensile stress (the thermal expansion coefficient of the Al 2 O 3 matrix is higher than that of the SiC nanoparticle) increases as well.Cracking happens easily under the residual tensile stress.That is why the predicted nano SiC content with the maximum fracture toughness is higher in this work compared with other researchers' results.toughness of Al2O3/SiCn and the SiC volume content, and obtained similar results to Niihara.Conclusions can be drawn from the modeling results in this paper that the microstructure with 15 vol% SiC has the maximum fracture toughness, which is obviously higher than the results published by other researchers.The primary reason for this difference is that the residual stress caused by the thermal expansion mismatch between nano SiC and matrix Al2O3 is not taken into account in our model.As the nanoparticle content is at a relatively higher level, there are more nanoparticles in the matrix grains, and the area of residual tensile stress (the thermal expansion coefficient of the Al2O3 matrix is higher than that of the SiC nanoparticle) increases as well.Cracking happens easily under the residual tensile stress.That is why the predicted nano SiC content with the maximum fracture toughness is higher in this work compared with other researchers' results.The dispersion of nanoparticles in this work is well controlled by programming.For instance, nanoparticles' dispersion remains excellent in the microstructure as nano SiC content is 20 vol% (Figure 13e), while the agglomeration usually occurs as the nano SiC content arrives at 10% and higher in sintering experiments.Compared with the sintering toughness of Al2O3/SiCn and the SiC volume content, and obtained similar results to Niihara.Conclusions can be drawn from the modeling results in this paper that the microstructure with 15 vol% SiC has the maximum fracture toughness, which is obviously higher than the results published by other researchers.The primary reason for this difference is that the residual stress caused by the thermal expansion mismatch between nano SiC and matrix Al2O3 is not taken into account in our model.As the nanoparticle content is at a relatively higher level, there are more nanoparticles in the matrix grains, and the area of residual tensile stress (the thermal expansion coefficient of the Al2O3 matrix is higher than that of the SiC nanoparticle) increases as well.Cracking happens easily under the residual tensile stress.That is why the predicted nano SiC content with the maximum fracture toughness is higher in this work compared with other researchers' results.The dispersion of nanoparticles in this work is well controlled by programming.For instance, nanoparticles' dispersion remains excellent in the microstructure as nano SiC content is 20 vol% (Figure 13e), while the agglomeration usually occurs as the nano SiC content arrives at 10% and higher in sintering experiments.Compared with the sintering The dispersion of nanoparticles in this work is well controlled by programming.For instance, nanoparticles' dispersion remains excellent in the microstructure as nano SiC content is 20 vol% (Figure 13e), while the agglomeration usually occurs as the nano SiC content arrives at 10% and higher in sintering experiments.Compared with the sintering experiments, more excellent nanoparticle dispersion in our work may be another reason why the predicted nanoparticle volume content with optimum toughening effect is relatively higher.Under the condition of excellent nanoparticle dispersion, fracture toughness increases as the nanoparticle content rises.All of the above analysis leads up to the conclusion that excellent dispersion procedures contribute to improving the fracture toughness of materials.

Influence of Interface Fracture Energy
In nanocomposite ceramic tool materials, there exist not only GBs among matrix grains, but also interfaces between nanoparticles and matrix grains.Interface bonding strength, matrix GB strength and matrix grain strength will affect the macroscopic strength and fracture toughness of materials through their effects on cracking paths.Interface bonding strength between matrix grains and nanoparticles can vary from different processing methods and conditions, and many investigations have shown that it has a significant effect on the fracture mode.Due to the small grain size and complicated microstructure morphologies, it is very difficult to measure the interface bonding strength by means of experiments.In this section, the influence of the interface bonding strength between nanoparticles and matrix grains on cracking paths and fracture toughness will be discussed.
In actual ceramic tool materials, the strength of nanoparticles is usually higher than that of matrix grains in order to obtain a good toughening effect.Therefore, only the influence of the fracture energy proportions among the interface, matrix grain and matrix GB on the fracture mode is taken into account in this work.Three levels of the proportions of the fracture energy of the interface Γ in f , the matrix grain Γ grain and the matrix GB Γ gb (Γ in f : Γ grain : Γ gb = 3 : 2 : 1, Γ in f : Γ grain : Γ gb = 2 : 3 : 1 and Γ in f : Γ grain : Γ gb = 1 : 3 : 2) are considered.These values indicate strong, intermediate and weak interfaces, respectively.Figure 16 represents three different microstructure models, which are the intergranular, intragranular/intergranular and intragranular structures.The average size of the matrix grain and the average nanoparticle size are 1 µm and 150 nm, respectively.The volume content of nanoparticles is equal to 10%.experiments, more excellent nanoparticle dispersion in our work may be another reason why the predicted nanoparticle volume content with optimum toughening effect is relatively higher.Under the condition of excellent nanoparticle dispersion, fracture toughness increases as the nanoparticle content rises.All of the above analysis leads up to the conclusion that excellent dispersion procedures contribute to improving the fracture toughness of materials.

Influence of Interface Fracture Energy
In nanocomposite ceramic tool materials, there exist not only GBs among matrix grains, but also interfaces between nanoparticles and matrix grains.Interface bonding strength, matrix GB strength and matrix grain strength will affect the macroscopic strength and fracture toughness of materials through their effects on cracking paths.Interface bonding strength between matrix grains and nanoparticles can vary from different processing methods and conditions, and many investigations have shown that it has a significant effect on the fracture mode.Due to the small grain size and complicated microstructure morphologies, it is very difficult to measure the interface bonding strength by means of experiments.In this section, the influence of the interface bonding strength between nanoparticles and matrix grains on cracking paths and fracture toughness will be discussed.
In actual ceramic tool materials, the strength of nanoparticles is usually higher than that of matrix grains in order to obtain a good toughening effect.Therefore, only the influence of the fracture energy proportions among the interface, matrix grain and matrix GB on the fracture mode is taken into account in this work.Three levels of the proportions of the fracture energy of the interface  , the matrix grain  and the matrix GB    Figure 17 shows the cracking paths of all microstructures with different fracture energy proportions.It can be seen from Figure 17(a1) that transgranular cracking occurs in the intergranular microstructures with strong interfaces ( :  :  = 3: 2: 1 ).Since the interface fracture energy is higher than that of the matrix grain, the intergranular crack is liable to extend into the matrix when encountering nanoparticles located at GBs. Comparing Figure 17(a1) with Figure 18 ( :  :  = 4: 2: 1), it can be concluded that the higher the interface bonding strength, the stronger the pinning and deflecting effect of Figure 17 shows the cracking paths of all microstructures with different fracture energy proportions.It can be seen from Figure 17(a1) that transgranular cracking occurs in the intergranular microstructures with strong interfaces (Γ in f : Γ grain : Γ gb = 3 : 2 : 1).Since the interface fracture energy is higher than that of the matrix grain, the intergranular crack is liable to extend into the matrix when encountering nanoparticles located at GBs. Comparing Figure 17(a1) with Figure 18 (Γ in f : Γ grain : Γ gb = 4 : 2 : 1), it can be concluded that the higher the interface bonding strength, the stronger the pinning and deflecting effect of nanoparticles at GBs on the intergranular crack, and the more remarkable the transgranular fracture phenomenon.In Figure 17(b1), where the interfaces are stronger than GBs but weaker than matrix grains (Γ in f : Γ grain : Γ gb = 2 : 3 : 1), the intergranular crack tends to surround nanoparticles as it encounters those located at GBs, and continue travelling along GBs when meeting matrix GBs.Similar fracture patterns appear in the intergranular microstructures with weaker interfaces (as shown in Figure 17(c1) with Γ in f : Γ grain : Γ gb = 1 : 3 : 2).As for the intragranular/intergranular microstructures, transgranular cracking exists in the model with strong interfaces (Figure 17(a2)).Due to relatively fewer nano ticles located at GBs in the intragranular/intergranular microstructure, the transgran fracture phenomenon is less pronounced compared with that in the intergranular m structure with the same material parameters.In microstructures with interme  :  :  = 2: 3: 1 and  :  :  = 1: 3: 2 (Figure 17(b2,c2)), where the i faces are weaker than matrix grains, cracks are liable to propagate around nanopar lying at GBs if they encounter them.And the intergranular fracture continues onc crack meets the matrix GB.
In the case of the intragranular microstructures, the intergranular cracking is dominant for the three fracture energy proportions (Figure 17(a3-c3)), which indi that nanoparticles distributed in matrix grains have little pinning and deflecting effe the intergranular crack.
Figure 19 shows the results of GC for microstructures with different inte strengths.It can be found that for the same microstructure type, materials with st interfaces ( :  :  = 3: 2: 1) exhibit the highest level of GC, as well as the hig fracture toughness.This is in good agreement with the simulation results, that the remarkable transgranular fracture can be observed in microstructures  :  :  = 3: 2: 1.In models with weak interfaces ( :  :  = 1: 3: 2), alth a few transgranular cracks arise, GC is the lowest due to the low interface fracture en From the above, strong interfaces ( :  :  = 3: 2: 1) seem to have a positive e on the fracture toughness of materials, while weak interfaces ( :  :  = 1: 3: 2) be disadvantageous for material toughening.The results of GC are in accordance wit results of cracking paths presented in Figure 17.The results of GC in the intergran models are relatively higher than other microstructures with the same interface bon strength.Compared with other microstructure types, GC in intragranular microstruc is at the lowest level no matter how strong the interface bonding effect is, since nano ticles in this matrix have negligible impact on the transition of the intergranular crac to the mixed mode of intergranular and transgranular cracking.Actually, nanocomp ceramic materials prepared using the sintering method primarily possess the intragr lar/intergranular microstructure.Thus, in actual nanocomposite ceramic tool mate strong interfaces have a significant effect on the fracture mode transition to the m mode of intergranular and transgranular cracking, which is advantageous for th hancement of fracture toughness of materials.As for the intragranular/intergranular microstructures, transgranular cracking still exists in the model with strong interfaces (Figure 17(a2)).Due to relatively fewer nanoparticles located at GBs in the intragranular/intergranular microstructure, the transgranular fracture phenomenon is less pronounced compared with that in the intergranular microstructure with the same material parameters.In microstructures with intermediate Γ in f : Γ grain : Γ gb = 2 : 3 : 1 and Γ in f : Γ grain : Γ gb = 1 : 3 : 2 (Figure 17(b2,c2)), where the interfaces are weaker than matrix grains, cracks are liable to propagate around nanoparticles lying at GBs if they encounter them.And the intergranular fracture continues once the crack meets the matrix GB.
In the case of the intragranular microstructures, the intergranular cracking is predominant for the three fracture energy proportions (Figure 17(a3-c3)), which indicates that nanoparticles distributed in matrix grains have little pinning and deflecting effect on the intergranular crack.
Figure 19 shows the results of G C for microstructures with different interface strengths.It can be found that for the same microstructure type, materials with strong interfaces (Γ in f : Γ grain : Γ gb = 3 : 2 : 1) exhibit the highest level of G C , as well as the highest fracture toughness.This is in good agreement with the simulation results, that the most remarkable transgranular fracture can be observed in microstructures with Γ in f : Γ grain : Γ gb = 3 : 2 : 1.In models with weak interfaces (Γ in f : Γ grain : Γ gb = 1 : 3 : 2), although a few transgranular cracks arise, G C is the lowest due to the low interface fracture energy.From the above, strong interfaces (Γ in f : Γ grain : Γ gb = 3 : 2 : 1) seem to have a positive effect on the fracture toughness of materials, while weak interfaces (Γ in f : Γ grain : Γ gb = 1 : 3 : 2) may be disadvantageous for material toughening.The results of G C are in accordance with the results of cracking paths presented in Figure 17.The results of G C in the intergranular models are relatively higher than other microstructures with the same interface bonding strength.Compared with other microstructure types, G C in intragranular microstructures is at the lowest level no matter how strong the interface bonding effect is, since nanoparticles in this matrix have negligible impact on the transition of the intergranular cracking to the mixed mode of intergranular and transgranular cracking.Actually, nanocomposite ceramic materials prepared using the sintering method primarily possess the intragranular/intergranular microstructure.Thus, in actual nanocomposite ceramic tool materials, strong interfaces have a significant effect on the fracture mode transition to the mixed mode of intergranular and transgranular cracking, which is advantageous for the enhancement of fracture toughness of materials.Yu [32] modeled the cracking extension in Al2O3/TiB2 composite ceramic tool materials and investigated the relation between Al2O3/TiB2 interface bonding strength and cracking behavior.He found that high interface bonding strength (higher than that of matrix grains) would result in the transition of the fracture mode from purely intergranular cracking to a mix of intergranular and transgranular cracking, as well as the enhancement of the fracture toughness of materials.Zhai [4] studied the influences of interface bonding strength on the cracking behavior of Al2O3/TiB2 composite ceramic materials and reached the conclusion that strong interfaces contribute to enhancing the critical energy release rate of microstructures.These results agreed well with our conclusion.
It should be noted that the conclusion drawn from the numerical results-that weak interfaces are disadvantageous for enhancing the fracture toughness of materials-is based on the premise of the simulation model, in which the interfaces between second phases and matrix phases are all weakly bonded and an intergranular main crack is easily formed under an applied load.However, things may be different in materials with weakinterface-bonding particles independently distributed through the strong interface network.On one hand, weak interfaces are liable to fracture to form microcrack zones under an applied load, which can facilitate microcrack toughening.On the other hand, strong interfaces are able to prevent the coalescence of the main crack and microcracks, and the material strength will not be reduced [33,34].This is the toughening mechanism for the weak-interface-toughening materials, which is not dealt with in this paper.

Conclusions
The macroscopic mechanical properties of ceramic tool materials are determined by microstructures.In order to reduce the brittleness and enhance the fracture toughness of materials, it is necessary to carry out a simulation in microscale and investigate the strengthening and toughening mechanisms in essence.In this work, the cohesive element model of nanocomposite ceramic tool materials is built up by embedding cohesive elements with fracture criteria into microstructures represented with the Voronoi tessellation.Then, the microscopic fracture behavior simulation is carried out to explore the relationship between microstructural morphologies and material responses.This research Yu [32] modeled the cracking extension in Al 2 O 3 /TiB 2 composite ceramic tool materials and investigated the relation between Al 2 O 3 /TiB 2 interface bonding strength and cracking behavior.He found that high interface bonding strength (higher than that of matrix grains) would result in the transition of the fracture mode from purely intergranular cracking to a mix of intergranular and transgranular cracking, as well as the enhancement of the fracture toughness of materials.Zhai [4] studied the influences of interface bonding strength on the cracking behavior of Al 2 O 3 /TiB 2 composite ceramic materials and reached the conclusion that strong interfaces contribute to enhancing the critical energy release rate of microstructures.These results agreed well with our conclusion.
It should be noted that the conclusion drawn from the numerical results-that weak interfaces are disadvantageous for enhancing the fracture toughness of materials-is based on the premise of the simulation model, in which the interfaces between second phases and matrix phases are all weakly bonded and an intergranular main crack is easily formed under an applied load.However, things may be different in materials with weak-interfacebonding particles independently distributed through the strong interface network.On one hand, weak interfaces are liable to fracture to form microcrack zones under an applied load, which can facilitate microcrack toughening.On the other hand, strong interfaces are able to prevent the coalescence of the main crack and microcracks, and the material strength will not be reduced [33,34].This is the toughening mechanism for the weak-interfacetoughening materials, which is not dealt with in this paper.

Conclusions
The macroscopic mechanical properties of ceramic tool materials are determined by microstructures.In order to reduce the brittleness and enhance the fracture toughness of materials, it is necessary to carry out a simulation in microscale and investigate the strengthening and toughening mechanisms in essence.In this work, the cohesive element model of nanocomposite ceramic tool materials is built up by embedding cohesive elements with fracture criteria into microstructures represented with the Voronoi tessellation.Then, the microscopic fracture behavior simulation is carried out to explore the relationship between microstructural morphologies and material responses.This research provides some mean-ingful investigation results for the design and development of nanocomposite ceramic tool materials, which aims to optimize the tools' fracture toughness.The effect of microstructure type on the cracking patterns and critical energy release rate G C is analyzed.It shows that the existence of nanoparticles changes the fracture mode from the intergranular cracking of single phase ceramic tool materials into mixed intergranular/transgranular cracking.Nanoparticles located at matrix GBs play a predominant role in the fracture mode transition of nanocomposite, while nanoparticles located in matrix grains have little effect on the intergranular cracking.The influence of nanoparticle size on microscopic fracture behavior is also discussed.Under the condition of the same nanoparticle content, the microstructure with smaller nanoparticles possesses more remarkable transgranular cracking and higher fracture resistance.The relationship between G C and nanoparticle content is also examined.Finally, the influence of interface bonding strength between nanoparticles and matrix grains on cracking patterns and fracture toughness is investigated, resulting in the conclusion that strong interfaces contribute to improving the fracture toughness of nanocomposite ceramic tool materials.

Figure 3 .
Figure 3. Cohesive element model.(a) Before the addition of cohesive elements.(b) After the addition of cohesive elements.

Figure 6 .
Figure 6.Crack growth paths of different microstructure models.Figure 6. Crack growth paths of different microstructure models.

Figure 6 .
Figure 6.Crack growth paths of different microstructure models.Figure 6. Crack growth paths of different microstructure models.

Figure 6 .
Figure 6.Crack growth paths of different microstructure models.

Figure 6 .
Figure 6.Crack growth paths of different microstructure models.

Figure 11 .
Figure 11.Comparison of G C for microstructures with different nanoparticle sizes.

Figure 13 .
Figure 13.Crack growth paths of microstructures with different nanoparticle content.

Figure 13 .
Figure 13.Crack growth paths of microstructures with different nanoparticle content.

Figure 14 .
Figure 14.GC versus nanoparticle volume content (simulation results in this work).

Figure 15 .
Figure 15.Relationship between the fracture toughness of Al2O3/SiCn and the volume content of SiC (published by Niihara et al.) [24].

Figure 14 .
Figure 14.G C versus nanoparticle volume content (simulation results in this work).

Figure 14 .
Figure 14.GC versus nanoparticle volume content (simulation results in this work).

Figure 15 .
Figure 15.Relationship between the fracture toughness of Al2O3/SiCn and the volume content of SiC (published by Niihara et al.) [24].

Figure 15 .
Figure 15.Relationship between the fracture toughness of Al 2 O 3 /SiC n and the volume content of SiC (published by Niihara et al.) [24].

Lubricants 2023 ,
11,  x FOR PEER REVIEW 15 of 19 nanoparticles at GBs on the intergranular crack, and the more remarkable the transgranular fracture phenomenon.In Figure17(b1), where the interfaces are stronger than GBs but weaker than matrix grains ( :  :  = 2: 3: 1), the intergranular crack tends to surround nanoparticles as it encounters those located at GBs, and continue travelling along GBs when meeting matrix GBs.Similar fracture patterns appear in the intergranular microstructures with weaker interfaces (as shown in Figure17(c1) with  :  :  = 1: 3: 2).

Figure 17 .
Figure 17.Crack paths in different types of microstructures with different fracture energy proportions.Figure 17.Crack paths in different types of microstructures with different fracture energy proportions.

Figure 17 .
Figure 17.Crack paths in different types of microstructures with different fracture energy proportions.Figure 17.Crack paths in different types of microstructures with different fracture energy proportions.

Figure 19 .
Figure 19.GC in different types of microstructures with different fracture energy proportions.

Figure 19 .
Figure 19.G C in different types of microstructures with different fracture energy proportions.

Table 1 .
Material properties for the microstructure of Al2O3/SiCn.

Table 1 .
Material properties for the microstructure of Al 2 O 3 /SiC n .