Abstract
Eutectic composite ceramics has a wide range of applications in the aerospace industry due to its excellent mechanical properties. The rupture stress of the materials is a subject of considerable importance. Eutectic composite ceramics primarily consist of rodshaped crystals, with a small amount of particles and preexisting defects dispersed throughout. Aligned nanomicron fibers are embedded within the rodshaped crystals. Rupture stress of a eutectic composite ceramic depends on its fracture surface energy and preexisting defects. In this study, the equivalent fracture surface energy of a eutectic ceramic composite was calculated based on its additional fracture work. Next, the effects of the preexisting defects were considered. Then, a micromechanical model of the eutectic composite ceramic was established based on its microstructural characteristics. The defects were assumed to be lamellar, and the surrounding matrix was assumed to be transversely isotropic. Using this information, the rupture stress of the eutectic ceramic composite was predicted. A comparison of the theoretical and experimental results indicated that the predicted rupture stresses corresponded with the tested data.
1. Introduction
Eutectic composite ceramics have been increasingly used due to their excellent mechanical and functional characteristics. The rupture stress of a material, an important mechanical index, greatly limits its applications. The rupture stress of a material is dependent on numerous factors, particularly its microstructural characteristics and defects. Eutectic composites ceramics are primarily comprised of rodshaped crystals formed via combustion synthesis. Aligned nanomicro fibers are embedded within the rodshaped crystals [1]. Previous experiments have shown that these cracks are propagated along the rodshaped crystals through deflection. A method of preventing these rodshaped grains has not yet been developed. As applied stress increases, interfacial debonding inevitably occurs through microcracking along the boundaries of rodshaped crystals, resulting in pullout work. The pullout work increases the fracture surface energy of the composite.
Sizes of defects and microstructures exhibit nearly identical orders of magnitude, and, in some types of eutectic composite ceramics, the matrices surrounding defects is anisotropic. Therefore, crack tip stress fields and rupture mechanisms of defects in anisotropic matrices are complex. Despite these complications, considerable progress has been made in research. Sih [2] and Willis [3] developed a way to obtain the crack tip stress fields and stress intensity factors of matrices, providing a basic method for studying the fracture mechanisms of anisotropic materials. The prediction model of eutectic composite ceramics with rodshaped crystals was built by Liu [4]. This study provided excellent resources concerning the rupture stress of eutectic composite.
The rupture stress of eutectic composite primarily depends on two factors: fracture surface energy and preexisting defects. In this study, the equivalent fracture surface energy of a eutectic ceramic composite was calculated based on its additional fracture work. Then, the effects of preexisting defects on the rupture stress were considered. In considerablysized defects, the effects of lamellar defects are much greater than the effects of spheroidal defects. Thus, only lamellar defects were analyzed in this article. Based on the microstructures of eutectic composites, the defects were simplified as elliptical cracks in an anisotropic matrix. Then, a model of the rupture stress was established and information regarding the effects of defects were obtained based on eigenstrain theory and Griffith’s fracture criterion.
2. Equivalent fracture surface energy
Eutectic composite ceramics are primarily comprised of rodshaped crystals containing nanomicron fibers, with a small amount of particles distributed throughout, as shown in Fig. 1. Since the propagation directions of cracks are different from the growth orientations of rodshaped crystals, cracks are forced to bypass rodshaped crystals, resulting in the formation of crackbridging by the rodshaped crystal in the wake of the crack tip. This crackbridging results in bridging toughening. As applied stress increases, interfacial debonding inevitably occurs via microcracking along the boundaries of rodshaped crystals, resulting in pullout. Due to the high volume fraction of rodshaped crystals in ceramics, the contribution of crackbridging to toughening and the pullout of rodshaped crystals is considered to play a predominant role in ceramic toughening.
Fig. 1SEM microstructures of the ceramic composites, primarily comprised of rodshaped crystals
When thermal residual stresses are present, rodshaped crystal pullout work can be appreciable. Residual stresses manifest as clamping forces between rodshaped crystals. These forces accumulate during final crystal separation. Additional fracture work is used to define this work during separation.
Fig. 2Pullout of a rodshaped crystal bridging a crack plane
Additional fracture work can be calculated based on its origination from a frictional clamping force. Consider a rodshaped crystal of radius $R$ that is pulled from a mating surface across the fracture plane (Fig. 2). The pullout work is given by the integral of the frictional force divded by the pullout length, $s$. The frictional stress is represented by $\tau $. The area over which this stress operates is expressed as 2$\pi Rs$ where $s$ is the instantaneous distance over which the eutectic is clamped. This length varies with the extent of pullout. Initially, the value of $s$ is represented by ${s}_{0}$ (slip length), but then decreases with the pullout distance, represented by $s={s}_{0}x$. Thus, pullout work can be expressed as:
In this equation, the frictional stress is equal to $\mu {\sigma}_{22}^{\left(0\right)}$, where $\mu $ represents the frictional coefficient, and ${\sigma}_{22}^{\left(0\right)}$ represents the transverse residual compressive stress of a rodshaped crystal determined by Liu et al. 2009 [5]. In addition, the dissociated length is associated with the position of the main crack. When the main crack is located near the rodshaped crystal, the dissociated length of its boundary will be approximately zero. When the main crack is located in the middle, the dissociated length will be approximately ${s}_{0}/2$. Thus, the average length of dissociation is approximately ${s}_{0}/4$, and the work of frictional force can be computed as:
where $\pi {R}^{2}$ represents the crosssectional area of the rodshaped crystal. By defining the volume fraction of the rodshaped crystals as ${f}_{f}$, the dissociated work per unit area can be obtained using Eq. (2):
As shown in Eq. (3), as the slip length increases, the dissociated work of the rodshaped crystal also increases. Thus, larger slip lengths are necessary to produce large amounts of more additional fracture work in a rodshaped crystal. However, slip length is limited by certain conditions, and only a relatively weak connection interface can possess a large slip length. The experimental results indicated that microslips would be observed in the weak interface of a rodshaped crystal under loading. Once reaching a certain value, the slips resulted in microinterface debonding. As the load increased, the microinterface debonding was further dissociated. Thus, the additional fracture work increased with the slip length and interface frictional shear stress.
Furthermore, the slip length should increase with the critical length (${l}_{c}$) since both are dependent on bond strength. When a rodshaped crystal is long, $l>{l}_{c}$, where $l$ is the length of the rodshaped crystal. By assuming that ${s}_{0}={l}_{c}$, ${l}_{c}$ is expressed as follow:
Here ${\sigma}_{du}$ is the breaking stress of a rodshaped crystal as determined by Ni et al. 2014 [6]. We obtain:
When the lengths of rodshaped crystals are less than the critical length, bridging still contributes to fracture work. Under these conditions, the average pullout length is likely to be on the order of $l/4$. By substituting $l$ for ${s}_{0}$ in Eq. (5), the additional fracture work per unit area can be obtained.
The toughness of a eutectic composite ceramic can be expressed as $JC=Jm+\u2206J$, where Jm represents the toughness of the matrix in a rodshaped crystal. Like the surface energy work, the fracture work scales linearly with crack length. The material toughness is defined as 2$\gamma $, where $\gamma $ is the fracture surface energy of the material. Thus, the toughness of the eutectic composite ceramic can be expressed as $2\gamma =2\gamma m+\u2206J$, where $\gamma m$ is the fracture surface energy of the matrix in a rodshaped crystal, and its equivalent fracture surface energy can be described as:
Bridging greatly affects the toughening of eutectic composite ceramics. Crack bridging leads to intergranular putout work. This effect is usually incorporated into the measured fracture surface energies of materials.
3. The rupture stress
Eutectic composites primarily consist of rodshaped crystals, with particles and small defects dispersed throughout. Consider a eutectic composite with defects consisting of threephase cells randomly and spatially distributed with an appropriate matrix volume fraction of matrix. All of these cells have the same shape and can be embedded into the equivalent medium shown in Fig. 3. In the threephase cell, the effective matrix surrounding an elliptical defect consists of uniformlydistributed rodshaped crystals and is transversely isotropic (where ${x}_{1}$ and ${x}_{3}$ comprise the basal planes), and the ellipsoid defect is defined as ${\left(X/{a}_{1}\right)}^{2}+{\left(Y/{a}_{2}\right)}^{2}+{\left(Z/{a}_{3}\right)}^{2}\le $ 1, where ${a}_{3}\ll {a}_{1}$, and ${a}_{2}$ is perpendicular to the rodshaped crystal. Let ${\sigma}^{D}$ be the effective applied stress at infinity. The boundary conditions on the crack surface are equal to:
where ${\sigma}_{3j}$ is the stress disturbance resulting from the crack. This stress disturbance can be simulated by the eigenstress derived from an equivalent ellipsoidal inclusion with eigenstrain ${\epsilon}^{*}$ as [7].
Fig. 3Threephase model of a defect
where ${C}_{pqmn}^{0}$ is a corresponding component of the fourthorder stiffness tensor of the matrix. $u={\left({a}_{1}^{2}{x}_{1}^{2}+{a}_{2}^{2}{x}_{2}^{2}+{a}_{3}^{2}{x}_{3}^{2}\right)}^{1/2}$, ${G}_{kplq}\left(x\right)={x}_{l}{x}_{q}{K}_{kp}^{1}$, ${K}_{kp}={C}_{pqmn}^{0}{x}_{i}{x}_{j}$. Using the following variable substitutions:
${G}_{kplq}$ can be expressed as ${G}_{kplq}\left(\theta ,t\right)={G}_{kplq}\left[\sqrt{1{x}_{3}^{3}}\mathrm{c}\mathrm{o}\mathrm{s}\theta ,\sqrt{1{x}_{3}^{3}}\mathrm{s}\mathrm{i}\mathrm{n}\theta ,{x}_{3}\right]$, and $dS\left(x\right)=d{x}_{3}d\theta ={\left(1{x}_{3}^{2}\right)}^{3/2}d\theta $. Thus:
and Eq. (8) can be simplified as:
As ${a}_{3}\to $ 0, by substituting Eq. (11) into Eq. (7), we can obtainand:
where ${L}_{ijmn}=\frac{1}{4\pi}{C}_{ijkl}^{0}{C}_{pqmn}^{0}{\mathrm{\Pi}}_{kplq}$ and the interaction energy can be expressed as:
Assume that the defects propagate selfsimilarly. Then the Griffith fracture criterion is equal to:
where $\gamma $ is the fracture surface energy of the material. By substituting Eqs. (13) into Eq. (14), we can obtain:
or:
Thus, the rupture stress can be expressed as:
Here:
Each component ${C}_{ijkl}^{0}$ for the stiffness tensor of matrix composed of rodshaped crystals can be determined by Li et al 2011[8].
The Al_{2}O_{3}/ZrO_{2} ceramic composites were obtained through a eutectic reaction under a high degree of undercooling. The samples were cut and grounded into rectangular bars measuring 3 mm (width)×4 mm (height)×36 mm (length) in order to determine their rupture stress.
Table 1Comparison of the theoretical and experimental results
Material number  Defects volume fraction ${f}_{p}$ (%)  Defects average diameter $a$ / μm  (experimental data) Rupture stress ${\sigma}_{cu}$ / MPa  (theoretical result) Rupture stress ${\sigma}_{cu}$ / MPa 
1  0.2  0.7  1568  1868 
2  0.8  0.8  1278  1774 
3  1.2  2.4  1256  1636 
4  2.8  1.1  1050  1115 
5  3.7  2.3  800  973 
6  8.5  1.9  680  768 
The experimental rupture stress data was measured using a threepoint bending method with a crosshead speed of 0.5 mm/min and a span of 30 mm. The volume fractions and average diameters of the defects were determined using the linear intercept method on an SEM photograph. The theoretical rupture stresses were calculated using Eq. (17). The comparison between the theoretical and tested results is shown in Table 1. The predicted rupture stresses corresponded with the tested data.
4. Conclusions
The rupture stress of the eutectic composite ceramic was determined using its fracture surface energy and preexisting defects. The equivalent fracture surface energy of the eutectic ceramic composite was computed based on its additional fracture work. The microstructure model was established based on the characteristics of eutectic composite materials with preexisting defects. Assuming a material is completely brittle, its rupture stress model containing its lamellar defects was obtained using Griffith's fracture theory. The material was assumed to be completely brittle for simplification, and the results were less than the experimental values.
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About this article
This material is based upon work supported by the National Natural Science Foundation of China under Grant No. 11272355.