Jinfeng Yu·Xinhua Ni·Xiequan Liu·Yunwei Fu·Zhihong Du

Abstract Study of damage and fracture models to analyze the fracture mechanism of eutectic composite ceramics is of considerable importance because no accurate fracture models are available for these materials.Eutectic composite ceramics are composed of microcells with random direction.We present herein a model that predicts the damage and fracture of eutectic composite ceramics based on analysis of defect stability and the damage localization band.Firstly,given the microstructure of eutectic composite ceramics,a mesodomain and a microcell model are constructed.The local stress field in the mesodomain is then analyzed based on the interaction direct derivative estimate.Secondly,the stability of a defect around particles in a microcell is analyzed,and the stress intensity factor of an annular defect under the applied stress field and the residual stress field in the particle are calculated.The stress intensity factor of a defect is controlled by the residual stress when the defect extension is small.However,it is controlled by the applied stress when the defect extension is large.Finally,a model for the damage localization band at the crack tip is constructed based on the Dugdale-Barenblatt model.The residual intensity is the important factor affecting the length of the damage localization band.When the damage variables reach their largest value,the residual intensity and the length of the damage localization band attain their minimum value.This work provides the theoretical basis for further study of the damage mechanics of eutectic composite ceramics and guides the engineering applications of these materials.

Keywords Microcell·Interaction direct derivative estimate·Defect stability·Damage localization band

1 Introduction

Eutectic composite ceramics exhibit good mechanical properties(such as high hardness,elasticity,and toughness),low microcrack propagation velocity,and high strength at normal and high temperatures[1-3].The structure of such ceramics changes to rod-like parallel fibers after cooling at high rate,resulting in excellent mechanical properties because of the alignment of the parallel fibers in the eutectic matrix [4,5].The microstructure of eutectic composite ceramics includes the matrix, intermediates, cracks, voids, etc., making their material properties tunable. The mechanical properties of eutectic composite ceramics can be improved by using a method that can predict the behavior of intermediates,cracks,and voids,requiring the development of a damage and fracture model for such materials. To date, many theoretical models have been proposed for the damage and fracture of composite materials.For example,Huang and Zhang[6]focused on a model for the strength of fiber-reinforced composite materials. Altinkok and Koker [7] and Moorthy and Ghosh [8] concentrated on a damage model for particlereinforced composite materials. Zhao and Yu [9] focused on development of a model for the mesodamage of quasibrittle materials.Considering the interaction of microcracks,a micromechanical model was developed for microcrackweakened quasibrittle materials under complicated loads,facilitating the calculation of the constitutive response[10].Considering the effects of particle reinforcement and thermal residual stress on the damage evolution, Yao [11] focused on a micromechanical damage model for metal-matrix composites with two-phase cells. To research the competition between interface debonding and particle failure,a mechanical model was established by Charles et al.[12].A damage model has also been formulated using the Mori-Tanaka estimate of effective compliance and Eshelby tensor [13]. A mechanical model for C-SiC composites was proposed by Cheng et al. [14] in accordance with the micromechanical damage evolution mechanism of SiC fibers. The fracture strength of eutectic composite ceramics containing lamellar inclusions was researched by Ni et al.[15].However,these damage models for composites are not universal because of the complexity of the fracture mechanism and the variety of microstructures contained. Therefore, a precise mechanical model must be developed to study the damage and fracture mechanism of a given structure.

Eutectic composite materials possess some special microstructures and properties.Ni et al.[15]focused on the applied stress field and residual stress field of matrices,interphases,andfibers intheeutecticonthebasis of theinteraction direct derivative(IDD)[16].The elastic properties and fracture strength of the rod-like eutectics were investigated using a four-phase model and the constraint characteristic of the interphases[17].Du et al.[18]researched the damage strain field of a triangular symmetrical composite eutectic. An Al2O3-based eutectic with GdAlO3(GAP)as reinforcement phase was found to exhibit high flexural strength of 700 MPa and good fracture toughness of about 5-6700 MPa·m1/2[19].Nanoscale Al2O3-GAP-ZrO2powder with eutectic composition exhibited thermal expansion coefficient of 9.49×10-6K-1at 1100°C [20]. The relationship between the interphase spacing and growth rate of Al2O3-GAP-ZrO2was researched under high-temperature-gradient conditions[21].Stress intensity factor(SIF)-based fracture criteria for mixedmode interface cracks have been proposed, considering the complex stress field of an interface crack tip[22].Large-size Al2O3/Y3Al5O12(YAG)eutectic ceramic was fabricated in vacuum by the horizontal directional solidification (HDS)method, and its room-temperature mechanical properties improved after annealing [23]. Non-directionally solidified Al2O3/YAG/ZrO2showed good compressive strength at high temperature [24]. The relationship between the mechanical properties and microstructure of Al2O3/Er3Al5O12(EAG)binary eutectic ceramic prepared by the Bridgman method has also been analyzed [25]. Defects arising from the thermal coefficient mismatch between different phases are an important factor reducing the strength of eutectic composite ceramics.High residual stress causes fiber debonding and ring cracks around the matrix[26].The interphase is the transition zone between the fibers and matrix, having a strong influence on the properties of eutectic composite ceramics.The interphase is strongly bound via covalent bonds.

The research object of this paper is Al2O3-ZrO2eutectic composite ceramic with parallel fibers,prepared by combustion synthesis under high gravity [27]. We present herein a microcell structure model based on the microstructure of eutectic composite materials. Then, the local applied stress field of the microcell model is analyzed. The stability of defects around particles is obtained.Finally,a model is established to analyze the length of the damage localization band.

2 Damage and fracture model

2.1 Microcell structure model

In Al2O3-ZrO2eutectic composite ceramics(Fig.1b),fibershaped ZrO2inclusions are distributed in parallel in an Al2O3matrix in each single parallel eutectic rod. Then,several rod-like eutectics are generally arranged regularly(also called a parallel distribution),forming a microcell with ZrO2particles and other penny-shaped defects.A unidirectional arrangement of eutectic rods can be classified as a special mesodomain；these domains are compartmentalized as ellipsoidal regions to simplify computations(Fig.1a).The fiber-shaped inclusions can be observed as a parallel distribution of microcells at the microscale and the substructure of the mesodomain(Fig.1b).A two-scale mechanical structure model for Al2O3-ZrO2eutectic ceramics can be established as follows:

(1) At the microscale, each ZrO2fiber coated with interphase in the Al2O3matrix is defined as a microcell.The theoretical structure model is shown in Fig.2b.In this microcell structure model,the ZrO2fiber is elliptical in cross-section,and the interphase is a similar ellipse.

(2) At the mesoscale, the representative volume element(RVE) of the mesodomain is as shown in Fig. 2a. In this RVE of the mesodomain,eutectic rods are arranged regularly along the same direction.Some penny-shaped cracks are embedded between the eutectic rods with the crack surface perpendicular to their longitudinal axis[28,29].Other particles do not form the eutectic structure distributed around these mesodomains, which is treated as the matrix around the domain. This random distribution of mesodomains in the matrix constitutes the composite material.

Two types of failure mechanism in eutectic composite ceramics should be considered.One is microdamage in the eutectic rods,which may cause their fracture,while the other is microdefect cracks in the eutectic colonies region,which may cause the eutectic rods to separate from each other.

2.2 Stress field in the mesodomain

Fig.1 Typical meso- and micro-structures of Al2O3-ZrO2 composite ceramic. a Eutectic rods arranged in the same direction regionally； black ellipses denote a mesodomain corresponding to a zone of regularly arranged eutectic rods. b Enlargement of eutectic structure； ZrO2 fibers are arranged in the same direction in the Al2O3 matrix in the eutectic rods[27]

Fig.2 Theoretical structure models for the mesodomain and microcell.a Mesodomain model for composite ceramic and the coordinate systems,in which eutectic rods are arranged in the same direction,composing the mesodomain.The eutectic rod axis is along axis 1,while the penny-shaped defects are perpendicular to axis 2.b Microcell model in the eutectic rod；fiber inclusions coated with interphase are located in the Al2O3 matrix

The IDD estimate proposed by Zheng et al.[16]has proved to be the most accurate estimation method for predicting the effective properties of composites [17]. In this section, the effective stress field in the domain is calculated using the IDD estimate,in which a more intuitive expression to address the effect of the random distribution in space proposed by Jake and Smith[30]is introduced.For Al2O3-ZrO2eutectic composite ceramics, the ZrO2fiber diameter is approximately hundreds of nanometers while that of the eutectic rods is several millimeters[31].These conditions satisfy the hypothesis of the IDD estimate. Fibers, defects, and domains can be treated as inclusions distributed in an infinite matrix.It is therefore reasonable to use the IDD estimate to calculate the effective stiffness.

Supposethataunidirectionaltensileforceσ∞isappliedto the material.Based on the IDD estimate,the effective applied stress of the fiber and the matrix around the inclusion in the microcellstructuremodelarethenσ0andσi,givenasfollows[17]:

where

Here,subscripts“0”and“i”indicate the matrix and the fiber,H is the compliance increment of the composite material calculated using the IDD estimate, Hiis the compliance wave induced by fibers, I is a fourth-order unit tensor,Ω = C0(I -M) and Ωi= C0(I -Mi) are the intrinsic stiffness tensor for the matrix and fibers,respectively,C0is the stiffness of the matrix,M and Miare the Eshelby tensors of the matrix and fiber,respectively,and f i is the volume fraction of fibers in the eutectic.

The equivalent stiffness of a composite material with fibers can be expressed based on the four-phase model as

where S0is the flexibility of the matrix and S0=

For such a nonisotropic matrix, no simple analytical expression exists for the Eshelby tensor.Hence,the Eshelby tensor of the fiber must be obtained by numerical integration[32],having the following expression:

where i,j,p,q,k,l ∈(1,2,3).

where the numerical solution for the individual components of P can be obtained by the Gaussian integration method,is the stiffness tensor component,= R′C0R and R is the transformation matrix between the global and local coordinate systems.

In Fig. 2, considering that the ZrO2fibers exist as fibershaped inclusions in the Al2O3matrix, the stiffness of the eutectic Cfcan be determined using the equation

where Hfis calculated using the IDD estimate of Eq.(3).

where the subscripts“A,Z,P”are Al2O3matrix,ZrO2fibers,interphase,respectively.

Considering interfacial debonding of the ZrO2fibers,the compliance of the damaged eutectic is

where d1is the damage variable,with d1=0, Sdf= Sf.

The eutectic domain is composed of eutectic rods arranged in the same direction with penny-shaped cracks parallel to the eutectic axial direction.The stiffness of the eutectic domain with two types of damage Cdomconsiders the penny-shaped cracks as sheet inclusions in the eutectic domain.

where Hdomis the compliance increment due to the pennyshaped cracks.

where f c is the crack volume fraction in the eutectic domain and Mrcis the Eshelby tensor of the crack in the eutectic domain.Cdf=the effective stiffness of the damaged eutectic rods obtained by Eq.(18).

The eutectic domains are ellipsoid inclusions with a certain length-to-diameter ratio, and the inclusions placed in the matrix are composed of transformed particles. A high volume fraction of mesodomains with a random distribution in the matrix composes the entire composite. In the IDD estimate,nonspherical inclusions with different orientations should be treated as different inclusion types, resulting in complex integration for the random distribution of inclusions in real materials.To simplify this calculation,Jake and Smith[30]proved that it was sufficiently accurate to calculate the stiffness Ceby first assuming that all inclusions have the same orientation,then averaging the stiffness in accordance with the orientation probability distribution function based on the Voigt approach.Except for a portion of the reinforcing fiber composite material, the majority of such a composite material can be regarded as being formed by a random distribution of equivalent cells.The effective properties are closely related to the distribution function. The orientation distribution function of short fibers was Laplace expanded in the complex sphere by Jake and Smith[30],and the tensor representation of the orientation distribution function was obtained.In those calculations,the mean and variance of the stiffness of the materials were shown as functions of the orientation tensor.Then,the Voigt approach was applied to the material stiffness.

The effective stiffness C E of the composite has the form

where

Ceis the effective stiffness of the composite in which all the mesodomains are located in the ZrO2matrix with the same orientation as follows:

where Cdomisthestiffnessofthemesodomainobtainedusing Eq.(19).

The composite is assumed to be subjected to simple tension σ∞along the y-axis in the global coordinate system(Fig. 2a). The axis of the eutectic rods lies along axis 1,while the penny-shaped defects are perpendicular to axis 2 of the local coordinate system.The orientation of the global and local coordinate systems is shown in Fig. 2a, where α is the angle between the rod axis and the tension direction.Therefore,the applied stress in the local coordinate system is

where σ∞=(0, σ∞, 0, 0, 0, 0)T, σ g =(σ11, σ22, σ33, σ13,σ23,σ12)T.R is the transformation matrix between the global and local coordinate systems.The following expressions then apply:

while the other components of σgare 0.The stress field in the domain is then

where

where MDZis the Eshelby tensor of the domain in the ZrO2matrix.

2.3 Stability analysis of the defect around particles

In eutectic composite ceramics, particles exist between the eutectics.Unstable crack propagation of such defects around the particles is an important factor in the damage and failure of such materials.When the defect is unstable,it will propagate to the crack and finally cause failure of the material.Analysis of such particles can be considered as an inclusion problem. The elastic constants of the particles are Cp,regarded as corresponding to heterogeneous ellipsoid inclusions in a matrix composed of microcells.The plane of the defect and the two main axes of the ellipsoid lie in the same plane(Fig.3).

Fig.3 Model of a defect around particles

The equations for such an ellipsoid-shaped inclusion are as follows[28]:

Supposing that the surface forces applied on the microcell are σ∞,the applied stress field in the matrix around the inclusions in the microcell based on the IDD estimate can be obtained as follows:

where Ω f = Cf(I -MAZ)is the intrinsic stiffness tensor of the matrix.

When the internal temperature of the composite ceramic changes by ΔT,the residual stress field of the matrix in the microcell can be calculated based on Eshelby theory as follows:

where εt= (α - α0)ΔT is the increment of the thermal strain in the eutectic due to the mismatch in the coefficient of thermal expansion between the different phases.α and α0are the coefficients of thermal expansion of the eutectic and matrix,respectively.Moreover,the increment of the thermal strain in the eutectic εtcan be obtained using the equation[10]

where ωp=(I +Ω HAP)-1and ωz=(I +Ω HAZ)-1.

The force applied to the particles can be regarded as the resultant of the applied stress and the residual stress in the matrix, because the particles are surrounded by the matrix.The important factor influencing the annular crack defect around the particles is the stress perpendicular to the plane of the defect.Thus,when unidirectional tensile stress σ∞is applied to the microcell,only the applied stress σDalong the two axes is considered when analyzing the stability of the defect around the particles,as follows:

The internal and external stress fields can be expressed as

where εtis the inelastic mismatch strain of the inclusions,e.g.,thermal or plastic strain.Ω and D-Ω represent within and outside the ellipsoid-shaped inclusion.

According to the equivalent inclusion theory, the strain change caused by the inhomogeneous inclusions can be simulated by the intrinsic strain ε*of the inclusions with the same elastic tensor as the matrix.

Equation (43) is called the equivalent equation, from which ε**can be constructed as follows:

where A=(Cp-C0)-1C0, B =(Cp-C0)-1Cp.When the applied load is uniform,the strain ε1satisfies

where the methods for calculation of M and D can be found in Refs.[16,27].

If the volume fraction of the inclusions is high,the mutual influence of the inclusions cannot be ignored. The IDD estimate proposed by Zheng and Du [16] is the simplest equivalent method in terms of such considerations. Given this method,the equivalent local stress field of the inclusions and the matrix in the form of interaction can be considered as the stress field that the inclusions is embedded in the infinite matrix with the boundary condition σE.σ E includes the average effect of the mutual influence of the inclusions as follows:

where Hd= f(I + Ω H01)-1H01, Ω = C0(I - M),H01=, where f is the volume fraction of the inclusions.

When σ E is substituted into Eq.(42),the equivalent stress field of the inclusions is obtained as

where ε0can be replaced by εE= Cσ E in ε**.

The external stress field of the fibers is composed of two parts.One part is the applied stress,which includes the difference of the fiber stiffness. The other part is the residual stress caused by the intrinsic difference between the matrix and fibers.These two parts create two K fields in the annular defect. The stress intensity factor of the applied and residual stress fields can be calculated using the superposition method, and the features of the microcrack state can be described by studying the relationship between the defect length and stress intensity factor.

Fig.4 Particles under the residual stress field and applied loading

Supposing that the defect plane is vertical to axis 1 while the applied stress is along axis 1, the wave caused by the defect can be ignored, as can the dimensional limit of the defect propagation caused by the space between the inclusions.d is the semiaxis field in the propagation direction of the particles；h is the crack radius as shown in Fig.4；r is the distance between the point and the particle center； e = h/d is the dimensionless crack length.The stress intensity factor can be obtained as the superposition of the two parts,and the applied stress is obtained as follows:

The thermal residual stress is obtained as follows:

and the stress intensity factor is calculated as

The integration of Eq.(50)is shown below:

We consider the average effect of particle interactions,mainly reflecting the equivalent boundary condition σE.Taking an Al2O3-ZrO2eutectic composite ceramic as an example, the material parameters of the matrix, the fibers,and the interphases in the eutectic are E0=402 GPa,υ0=0.233, Eb= 233 GPa, υb= 0.31, and Ea=10 Em,υa=υm.Given the dimensional relationship between the fiber and the phase,the volume fraction of the phases in the composite ceramic is f p =fZ, the thickness of the phases is Δ= 1 nm, and the volume fraction of fibers is f Z = 0.3. For particles of different shapes,the relationship between equivalent applied stress/applied stress (σE/σD) and the particle content f is shown in Fig.5.

Fig.5 Equivalent average applied stress for different particle contents

Fig.6 Stress intensity factor under the residual stress

Figure 5 shows that σE/σDdecreases with increasing particle content f. The stress intensity factor becomes smaller with reduction of the particle content. Moreover, the more slender the particle, the more obvious the reduction of the applied stress. In the presence of defects around the particles, and supposing the radius of the particle to be 1 mm,the relationship between the stress intensity factor under the residual stress and the dimensionless length e of the defect is shown in Fig.6.

As shown in Fig. 6, the stress intensity factor decreases with the defect length with increasing residual stress. The rate of decrease is related to the flatness of the particle shape.Thus,if the microcracks nucleated around the particles under the residual stress reach the crack propagation condition,the propagation will stop after a certain length that changes with the flatness of the ellipsoid particles.The stress intensity factor under the residual stress and the applied stress is shown in Fig.7.

Fig.7 Stress intensity factor under the residual stress and the applied stress

As shown in Fig.7,when the defect propagation is short,the stress intensity factor increases obviously and is controlled by the residual stress. The stress intensity factor appears to increase,decrease,then increase again.This phenomenon may cause the defect to press forward and stop until the applied stress becomes sufficient,with fracture occurring as a Griffith crack.

The stress intensity factor of the defect is controlled by the residual stress when the defect extension is small,but by the applied stress for large defect extension.Elliptic particle shape is beneficial to improve the strength.

2.4 Damage localization band at crack tip

When the above-mentioned defect does not satisfy the stability condition,it will propagate to form a macroscopic crack.According to their different degrees of damage,the macrocrack tip field can be divided into three parts: the lossless area,the continuum damage area,and the damage localization band[33].On the basis of microscopic damage theory for microcracks[33],microcracks enter a period of damage localization along one direction as they extend to a certain state.This zone of damage localization is called the damage localization band before the stress drop.The damage localization band is the most heavily damaged area at the crack tip, compared with the lossless area and continuum damage area,being a prerequisite for unstable propagation of the macroscopic crack. The damage localization model shown in Fig.8[34]was established based on the theory underlying the Dugdale-Barenblatt model[35,36].The coordinate system oxy is built at the tip of the damage localization band,whose length is supposed to be l.This length is related to the far-field stress intensity factor K∞and l ?a.

Under the tensile stress σ∞,the stress intensity factor is obtained as follows:

Fig.8 Model of damage localization band at crack tip

where c=l+a.The distribution of the normal stress along the extension in the y direction is given by the following:

where r=x is the distance between the crack tip and the end of the damage localization band,0 ≤r ≤l.

Assuming that the stress distribution lies in the range of-l ≤x ≤0,it is

These conditions are reasonable because the size of the damage localization band is very small compared with the crack size, namely l ?a. Furthermore, only the stress at points close to the band are of interest, thus r ?c. When one pair of concentrated forces ~σ(ξ)dξ focuses on the point x =ξ(ξ ＜0),the normal stress along the y axis around the damage localization band can be expressed as follows:

The corresponding stress intensity factor is

Therefore,the stress intensity factor under the distributed forces in the damage zone is

Considering the damage, the stress at the point x = 0 is limited,then

Next,

For a microcell with parallel eutectic fibers, when the stress is linearly distributed in the damage localization band,the stress is given by

The fracture stress of the damaged eutectic is calculated by Ni et al.[15]as follows:

where E0is the elastic modulus of the matrix in the eutectic, Ebis the elastic modulus of the fiber inclusion in the eutectic, γ is the surface energy of the matrix in the eutectic, and υ21is the Poisson's ratio of the eutectic along axes 2 and 1. n2= [(1+υ21)Γ ln(π/fZ)]-1/2,where d is the diameter of the inclusion. A =where Γ =G12/G.G12is the shear modulus of the eutectic along axes 2 and 1.G is the effective shear modulus of the composite material. U is assumed to be (I - K Z H)-1,thus u21= U(2, 1). The maximum damage variable isThe minimum elasticity modulus isE1(1 - d1), and E1is the longitudinal elastic modulus of the undamaged eutectic rods on the basis of Eq.(13).

Supposing that the residual intensity is the fracture stress of the damaged eutectic, and the residual intensity is minimum when d1=d1max,we have

Finally,the length of the damage localization band is calculated as follows:

Under the far-field stress intensity factor K∞, the crack exhibits steady propagation.A damage localization band is generated with length of l.The band extends along the direction of the crack in the steady state. Its length is mostly influenced by the residual intensity as stated in Eq.(56)and the fracture stress of the damaged eutectic.We now use the Al2O3-ZrO2eutectic composite ceramic as an example to analyze the fracture stress. The volume fraction and fiber length are fb=0.4 and l=10 μm,respectively,and the average fiber diameter is d = 2 μm. The damage variables ared1max, and d1max.The relationship between the fracture stress of the eutectic and the length l of the fiber is shown in Fig. 9. By contrast, the relationship between the fracture stress of the eutectic and the fiber diameter is presented in Fig.10.

Fig.9 Relationship between the fracture stress of the eutectic and the length of the fiber

Fig.10 Relationship between the fracture stress of the eutectic and the fiber diameter

The fracture stress decreases with increasing length and diameter of the fiber (Figs. 9 and 10). The fracture stress also becomes smaller as the damage variable becomes larger.With d1=d1max,fracture becomes easy in the eutectic.With l ≥5d,no size dependence is observed.The comparison in Table 1 reveals that experimental data agree well with the theoretical results.The experimental results lie between the theoretical results obtained with d1=(1/3)d1maxand d1=(2/3)d1max.

Given the damage in the eutectic, the fracture stress of the eutectic decreases.A greater damage variable produces a smaller stress fracture.The reinforcing effects of the eutectic derive from the eutectic phases.With d1=d1max,the small size dependence is explained by the slight influence on the strength of the eutectic because of the existence of damage.

Table 1 Comparison of theoretical and experimental results

The length of the damage localization band attains its minimum value at d1= d1max, when the length of the steady fracture is shortest,hence entering the unstable fracture state faster.

3 Conclusions

1. A microcell model was established based on the microstructure of eutectic composite ceramics.The elastic tensor of the microcell was obtained by considering the influence of the two types of damage.Then,the local stress field was analyzed in the microcell.

2. The stability of the defect around the particles in the microcell was analyzed in accordance with the equivalent inclusion theory and the IDD estimate. The stress intensity factor of the defect under the applied stress and the residual stress was also calculated.The stress intensity factor of the defect was controlled by the residual stress under small defect propagation,but by the applied stress under large defect propagation.

3. A model for the damage localization band was established in accordance with the microscopic damage theory of the microcrack. Then, the length of this band was calculated. The length of the damage localization band was shortest when the value of the damage variable d1was maximum,entering the unstable fracture state faster.This length was also related to the far-field stress intensity factor controlled by the applied and residual stresses.Experimental data agree well with the theoretical eutectic fracture stress.

AcknowledgementThis work was supported by the National Natural Science Foundation of China(Grant 11272355).