《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_Finite element analysis of stresses associated with transformations

MIATERIALS ENE S ENGINEERING ELSEVIER Materials Science and Engineering A272(1999)398-409 www.elsevier.com/locate/msea Finite element analysis of stresses associated with transformations in magnesia partially stabilized zirconia C R. Chen a, * S X. Li, Q. Zhang b State Key Laboratory for Fatigue and Fracture of Materials, Institute of Metal Research, Chinese Academy of sciences Shenyang 110015, People s Republic of b Department of Materials Science, Tshinghua University, Beijing 100084, People's Republic of China Received 10 February 1999: received in revised form 8 July 1999 Abstract Anisotropy finite element analysis was performed to study stresses associated with the tetragonal precipitates and the tetragonal monoclinic transformation in Mgo partially stabilized zirconia. Stresses were assumed to be caused by the lattice misfits between the product phase and the parent phase. In the finite element calculations, the tetragonal and monoclinic precipitates were assumed to be lenticular in shape and the anisotropic elastic constants of the cubic, tetragonal and monoclinic phases were considered. The purpose of this paper is to obtain some knowledge about how stresses respond when the microstructure is changed, and how stress fields affect microstructure development. The finite element results show that increasing a/as and creasing cy/ae can create a favorable stress field to reduce the growth rate of tetragonal precipitates during the heat treatment. Stresses associated with the single tetragonal precipitate in an infinite cubic matrix are distinctly larger than stresses associated ith the homogeneously distributed tetragonal precipitates. The twin orientations within the monoclinic particle greatly affect stresses associated with the tetragonal to monoclinic transformation. o 1999 Elsevier Science S.A. All rights reserved Keywords: Finite element analysis; Transformations; Magnesia partially stabilized zirconia 1. Introduction crack induces a transformation and how the transfor- affects crack growth, it is also necessary The knowledge about stresses associated with a solid the stresses associated with the transformation transformation is becoming important for developing Stresses and strains associated with transformations advanced ceramics and shape memory alloys. Consider- can be investigated by theoretical, experimental and able experimental evidence has shown that the tetrago- numerical methods. The Eshelby theory and the nal zirconia (t-ZrO,) transformation to the monoclinic Khachaturyan theory are two well known theoretical zirconia(m-ZrO )in the region of a propagating crack methods which are widely used for studying solid trans- can be utilized to increase the fracture toughness of formations. For example, Porter [6] applied Eshelby's partially stabilized ZrO2 and composites containing transformed-inclusion method to study transformation ZrO2 particles [1-4]. For understanding how the mi- toughening in magnesia partially stabilized zirconia crostructures develop and how to control the mi- (Mg-PSZ), and Lanteri [7] applied the theory of crostructure development, it is necessary to study the Khachaturyan to explain the habit plane and precipi- stresses that are produced as the transformation pro- tate morphology observed in Mg-PSZ. Detailed ceeds. because the microstructure that results from the about the theoretical methods can be found elsewhere. solid transformation is largely determined by the stress e.g. in Refs. [8-ll]. For the experimental study of that develops as the product phase precipitates grow strain fields associated with solid transformations, high and interact [5]. For understanding how a propagating resolution electron microscopy [12] and convergent beam electron diffraction [13] have bee However, these experimental methods can only measure chrch(@imr ac cn (C.R. Chen) strains in thin samples. due to the surface stress relax 0921-5093/99/- see front matter c 1999 Elsevier Science S.A. All rights reserved. PI:s0921-5093099)00507-9

Materials Science and Engineering A272 (1999) 398–409 Finite element analysis of stresses associated with transformations in magnesia partially stabilized zirconia C.R. Chen a,*, S.X. Li a , Q. Zhang b a State Key Laboratory for Fatigue and Fracture of Materials, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110015, People’s Republic of China b Department of Materials Science, Tshinghua Uni6ersity, Beijing 100084, People’s Republic of China Received 10 February 1999; received in revised form 8 July 1999 Abstract Anisotropy finite element analysis was performed to study stresses associated with the tetragonal precipitates and the tetragonal to monoclinic transformation in MgO partially stabilized zirconia. Stresses were assumed to be caused by the lattice misfits between the product phase and the parent phase. In the finite element calculations, the tetragonal and monoclinic precipitates were assumed to be lenticular in shape, and the anisotropic elastic constants of the cubic, tetragonal and monoclinic phases were considered. The purpose of this paper is to obtain some knowledge about how stresses respond when the microstructure is changed, and how stress fields affect microstructure development. The finite element results show that increasing at /ac and decreasing ct /ac can create a favorable stress field to reduce the growth rate of tetragonal precipitates during the heat treatment. Stresses associated with the single tetragonal precipitate in an infinite cubic matrix are distinctly larger than stresses associated with the homogeneously distributed tetragonal precipitates. The twin orientations within the monoclinic particle greatly affect stresses associated with the tetragonal to monoclinic transformation. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Finite element analysis; Transformations; Magnesia partially stabilized zirconia www.elsevier.com/locate/msea 1. Introduction The knowledge about stresses associated with a solid transformation is becoming important for developing advanced ceramics and shape memory alloys. Consider￾able experimental evidence has shown that the tetrago￾nal zirconia (t-ZrO2) transformation to the monoclinic zirconia (m-ZrO2) in the region of a propagating crack can be utilized to increase the fracture toughness of partially stabilized ZrO2 and composites containing ZrO2 particles [1–4]. For understanding how the mi￾crostructures develop and how to control the mi￾crostructure development, it is necessary to study the stresses that are produced as the transformation pro￾ceeds, because the microstructure that results from the solid transformation is largely determined by the stress that develops as the product phase precipitates grow and interact [5]. For understanding how a propagating crack induces a transformation and how the transfor￾mation affects crack growth, it is also necessary to study the stresses associated with the transformation. Stresses and strains associated with transformations can be investigated by theoretical, experimental and numerical methods. The Eshelby theory and the Khachaturyan theory are two well known theoretical methods which are widely used for studying solid trans￾formations. For example, Porter [6] applied Eshelby’s transformed-inclusion method to study transformation￾toughening in magnesia partially stabilized zirconia (Mg-PSZ), and Lanteri [7] applied the theory of Khachaturyan to explain the habit plane and precipi￾tate morphology observed in Mg-PSZ. Detailed reviews about the theoretical methods can be found elsewhere, e.g. in Refs. [8–11]. For the experimental study of strain fields associated with solid transformations, high resolution electron microscopy [12] and convergent beam electron diffraction [13] have been employed. However, these experimental methods can only measure strains in thin samples. Due to the surface stress relax- * Corresponding author. E-mail address: chrch@imr.ac.cn (C.R. Chen) 0921-5093/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S0921-5093(99)00507-9

C.R. Chen et al. Materials Science and Engineering 4272(1999)398-409 ation, strains in thin samples are different from that in tribute perpendicularly to each bulk materials. For numerical study of stresses and plate, however, the crowded precipitates are strains associated with a transformation, the finite ele- parallel to each other, and the I precipitates are ment(FE)method is a powerful tool. The FE method perpendicular to the Mgo plates at most locations and has been applied to study mechanical problems associ- are parallel to the Mgo plates at some locations. Such ed with the transformations in nickel base supera distribution pattern is due to the thermal residual loys [14], shape memory alloys [15], Zro2 toughened stress field near the Mgo plates [30]. The t-m trans AlO3[16-18], partially stabilized zirconia [18, 19], and formation involves the largest strain energy, and it tetragonal zirconia polycrystals [20]. Although the Fe difficult for an entire t-ZrO2 precipitate to transform modeling of transformation is rather simple at present, as a single unit. Thus the twin-related thin plates are this method has great potential for improvement in formed in the m-ZrO2 particle to reduce the strain the analysis of a transformation energy resulting from the martensitic transformation In the partially stabilized zirconia(PSZ), a disper- [31]. The shear strain in each plate is large, however, on of metastable t-ZrO, in cubic zirconia (c-ZrO, long-range shear strains of the m-Zro2 particle are gives rise to a powerful toughening mechanism [21]. eliminated because of the twin- relation [1, 24, 27, 32 Because the toughening in PSz is achieved by the The m-ZrO2 particle takes the shape of the t-ZrO careful control of the microstructure and phase distri- precipitate. Cancellation of shear strains with pairs of bution [22] the knowledge of stresses associated with twin-related plates ensures that a m-ZrO2 particle fits the c-t transformation is helpful for conducting the into the space available for it in the matrix (i.e. the heat treatment of PSZ. The aim of the heat treatment space it occupied before the t-m transformation) is first, to develop the t-ZrO, precipitates to a condi- with the minimum possible strain [27]. The twin orien- ion of metastability so that they will transform to the tations in the m-ZI ffer from pre m-ZrO2 precipitates in the presence of a propagating precipitate, and are dependent on the local stress state crack, and second, to maximize the volume of criti- [33]. In some m-ZrO2 particles, the twinned plates are cally metastable t-ZrO2 precipitates [23]. Knowledge parallel to the short dimension of the precipitate, while bout stresses associated with the t-m transforma- in some other particles the twinned plates are perpen- tion is important for heat treatment and the applica dicular to the short dimension of the precipitates tion of PSZ. The t-m transformation in the process [26, 27 of heat treatment degrades the toughness of PSz and In this paper, the FE method is applied to stuc thus, should be avoided. Under the applied load, the stresses associated with a t-precipitate in an infinite strength of PSz is limited by the stress at which the c-ZrO2 matrix;(b) interactions between t-precipitates t-m transformation occurs [24], and this transforma-(c)stresses associated with the t-m transformation of tion is widely believed to be triggered by shear stresses a precipitate in an infinite c-ZrO2 matrix. There are [25]. The t-m transformation induced by a propagat- two features in the FE calculations: first, the elastic ing crack can greatly affect the growth rate and path anisotropy of c-Zro2, t-ZrO2 and m-ZrO2 are consid- of the crack ered; second, the t-ZrO2 and m-zrO2 pre In Mg-PSZ, the t-ZrO, precipitates are lenticular in cipitates are assumed to be lenticular in shape. The shape. The tetragonal c-axis is parallel to(100) of the purpose of this paper is to obtain some knowledge C-ZrO2 matrix and is the rotation axis of the particle, about how microstresses of a transformation depend thus three possible variants may occur [3, 26, 27]. The on microstructures and how microstresses affect the typical aspect ratio of particles is about 4: 1 [] or 5: 1 formation of the microstructure during the transfor [1, 7]. The equilibrium particle shape is controlled by mation the balance between the interfacial energy and strain computer simulation base the inzburg-Landau phenomenological theory was per- 2. Material data about Mg-PSZ formed to study the shape evolution of a t-ZrO, pre- cipitate and the microstructure development of t-ZrO2 2. 1. Geometry and crystallography orientations of precipitates. The simulation results show that a lense- t-ZrO2 and m-ZrO2 in Mg-PSZ like shape appears during growth of a t-ZrO2 particle, and upon further coarsening, the shape relaxes into a The t-ZrO2 precipitates in Mg-PSZ form as lenticu rhombus bounded by facets [28, 29]. During the heat lar particles, and the tetragonal c-axis is parallel to the treatment of Mg-PSZ, the arrangement of t-ZrO2 pre- short dimension of the particles. The cross-section of cipitates is influenced by the local stress distribution. the particle is shown as Fig. la. When viewed along In the complex PSZ stabilized by high content Mg the short axis, the shape of t-ZrO, is circular in the and low content Y,O, together, at the region far from ideal case, shown as Fig. lb; in real materials, due to the Mgo precipitate plates, the t-ZrO2 precipitates dis- the interaction between precipitates, when viewed

C.R. Chen et al. / Materials Science and Engineering A272 (1999) 398–409 399 ation, strains in thin samples are different from that in bulk materials. For numerical study of stresses and strains associated with a transformation, the finite ele￾ment (FE) method is a powerful tool. The FE method has been applied to study mechanical problems associ￾ated with the transformations in nickel base superal￾loys [14], shape memory alloys [15], ZrO2 toughened Al2O3 [16–18], partially stabilized zirconia [18,19], and tetragonal zirconia polycrystals [20]. Although the FE modeling of transformation is rather simple at present, this method has great potential for improvement in the analysis of a transformation. In the partially stabilized zirconia (PSZ), a disper￾sion of metastable t-ZrO2 in cubic zirconia (c-ZrO2) gives rise to a powerful toughening mechanism [21]. Because the toughening in PSZ is achieved by the careful control of the microstructure and phase distri￾bution [22], the knowledge of stresses associated with the ct transformation is helpful for conducting the heat treatment of PSZ. The aim of the heat treatment is first, to develop the t-ZrO2 precipitates to a condi￾tion of metastability so that they will transform to the m-ZrO2 precipitates in the presence of a propagating crack, and second, to maximize the volume of criti￾cally metastable t-ZrO2 precipitates [23]. Knowledge about stresses associated with the tm transforma￾tion is important for heat treatment and the applica￾tion of PSZ. The tm transformation in the process of heat treatment degrades the toughness of PSZ and, thus, should be avoided. Under the applied load, the strength of PSZ is limited by the stress at which the tm transformation occurs [24], and this transforma￾tion is widely believed to be triggered by shear stresses [25]. The tm transformation induced by a propagat￾ing crack can greatly affect the growth rate and path of the crack. In Mg-PSZ, the t-ZrO2 precipitates are lenticular in shape. The tetragonal c-axis is parallel to Ž100 of the c-ZrO2 matrix and is the rotation axis of the particle, thus three possible variants may occur [3,26,27]. The typical aspect ratio of particles is about 4:1 [3] or 5:1 [1,7]. The equilibrium particle shape is controlled by the balance between the interfacial energy and strain energy. A computer simulation based on the Ginzburg–Landau phenomenological theory was per￾formed to study the shape evolution of a t-ZrO2 pre￾cipitate and the microstructure development of t-ZrO2 precipitates. The simulation results show that a lense￾like shape appears during growth of a t-ZrO2 particle, and upon further coarsening, the shape relaxes into a rhombus bounded by facets [28,29]. During the heat treatment of Mg-PSZ, the arrangement of t-ZrO2 pre￾cipitates is influenced by the local stress distribution. In the complex PSZ stabilized by high content MgO and low content Y2O3 together, at the region far from the MgO precipitate plates, the t-ZrO2 precipitates dis￾tribute perpendicularly to each other; near the MgO plate, however, the crowded t-ZrO2 precipitates are parallel to each other, and the parallel precipitates are perpendicular to the MgO plates at most locations and are parallel to the MgO plates at some locations. Such distribution pattern is due to the thermal residual stress field near the MgO plates [30]. The tm trans￾formation involves the largest strain energy, and it is difficult for an entire t-ZrO2 precipitate to transform as a single unit. Thus the twin-related thin plates are formed in the m-ZrO2 particle to reduce the strain energy resulting from the martensitic transformation [31]. The shear strain in each plate is large, however, long-range shear strains of the m-ZrO2 particle are eliminated because of the twin-relation [1,24,27,32]. The m-ZrO2 particle takes the shape of the t-ZrO2 precipitate. Cancellation of shear strains with pairs of twin-related plates ensures that a m-ZrO2 particle fits into the space available for it in the matrix (i.e. the space it occupied before the tm transformation) with the minimum possible strain [27]. The twin orien￾tations in the m-ZrO2 can differ from precipitate to precipitate, and are dependent on the local stress state [33]. In some m-ZrO2 particles, the twinned plates are parallel to the short dimension of the precipitate, while in some other particles the twinned plates are perpen￾dicular to the short dimension of the precipitates [26,27]. In this paper, the FE method is applied to study: (a) stresses associated with a t-precipitate in an infinite c-ZrO2 matrix; (b) interactions between t-precipitates; (c) stresses associated with the tm transformation of a precipitate in an infinite c-ZrO2 matrix. There are two features in the FE calculations: first, the elastic anisotropy of c-ZrO2, t-ZrO2 and m-ZrO2 are consid￾ered; second, the t-ZrO2 and m-ZrO2 pre￾cipitates are assumed to be lenticular in shape. The purpose of this paper is to obtain some knowledge about how microstresses of a transformation depend on microstructures and how microstresses affect the formation of the microstructure during the transfor￾mation. 2. Material data about Mg-PSZ 2.1. Geometry and crystallography orientations of t-ZrO2 and m-ZrO2 in Mg-PSZ The t-ZrO2 precipitates in Mg-PSZ form as lenticu￾lar particles, and the tetragonal c-axis is parallel to the short dimension of the particles. The cross-section of the particle is shown as Fig. 1a. When viewed along the short axis, the shape of t-ZrO2 is circular in the ideal case, shown as Fig. 1b; in real materials, due to the interaction between precipitates, when viewed

C.R. Chen et al. Materials Science and Engineering 4272(1999)398-409 along the short axis, the shape of t-ZrO2 can be nearly cles the short dimension lies in the plane of the plates, rectangular. Because the tetragonal c-axis can be along and there is an angle between am and xe shown in Fig anyone of the (100> cube directions, there are three 2a; in some m-ZrO2 particles, the normal to the plates kinds of crystallography relation between t-ZrO2 pre- is along the short dimension of the particle, and there is cipitates and c-ZrO, matrix:[001]/[001]e, [001,/[100] an angle between cm and E, shown as Fig 2b. The b, and [001M/[010]. Typical aspect ratio of t-ZrO, precip- axis is normal to the plane of Fig. 2a and b itates is about 4 or 5. In the present FE calculations, the aspect ratio of the particles is assumed to be 4.5. 2 2. Lattice parameters The t-m transformation is a martensitic transfor- mation. After the t-ZrO2 particle is transformed to the For calculating stresses and strains caused by the m-zrO2 particle, the m-zrO2 particle takes the shape of misfits of lattice parameters between the product phase the t-ZrO2 particle, but the twin-related thin plates ar and the parent phase in Mg-PSZ, the lattice parameters formed in the m-ZrO2 particle. In some m-ZrO2 parti- of c-ZrO2, t-ZrO2 and m-zrO2 without constraints should be known. The lattice parameters of Mg-PSZ can be found in many publications, such as in Refs [6, 7, 26, 27, 34]. We selected the lattice parameters of Mg-PSZ from [26] as follows: c-ZrO2,a=0.50778nm t-ZrO2,a1=0.50803nm,c=0.51903nm. m-ZrO,: am=0.51172 nm, b 0.51770mm.c C[o01]t 0.53031nm,B=9891° Y【ol For understanding the stresses associated with the t-Zr0, precipitate in an(Mg, Y)-PSZ designed in Ref. 30]. the FE calculation is also performed for this material. This material is based on the Mg-PSZ with Dh=4.5 the molar ratio MgO/Zro2=14/86, and Y2O3 is added at the molar ratio Y,O3/ZrO,= 2/98, and some o- MgO-Al2O3 is also added. From Ref [30], the lattice parameters of c-ZrO2 and t-ZrO2 are as follows: c-ZrO,: a=0.5102 nm Fig. 1. Morphology of the t-ZrO, precipitate in Mg-PSZ:(a) the t-ZrO2:a1=0.5122nm,c1=0.5170nm cross-section, and(b) viewed along the short dimension 3. Elastic Under the local coordinate system xv=, shown in Figs. I and 2, the constitutive relation of each phase can be expressed as toi=[Celi where to) and fe) are stresses and strains, and the sequence of stress- es and strains into, and ai is [xr yy z p F, 2Ap xy]. Here [Cl, is the elastic constant matrix, and [C] is different for c-Zr02, t-ZrO2, and m- For c-Zro C12 C11 C1200 0 C12C12C1000 000C4400 where C=401. C=96 and C=56 GPa. The Fig. 2. The cross-section of the m-ZrO2 particle in Mg-PSZ:(a) constants are selected from Refs. [7, 35, 36] l/cu, and(b)aml/a, For t-ZrO

400 C.R. Chen et al. / Materials Science and Engineering A272 (1999) 398–409 along the short axis, the shape of t-ZrO2 can be nearly rectangular. Because the tetragonal c-axis can be along anyone of the Ž100 cube directions, there are three kinds of crystallography relation between t-ZrO2 pre￾cipitates and c-ZrO2 matrix: [001]t //[001]c, [001]t //[100]c and [001]t //[010]c. Typical aspect ratio of t-ZrO2 precip￾itates is about 4 or 5. In the present FE calculations, the aspect ratio of the particles is assumed to be 4.5. The tm transformation is a martensitic transfor￾mation. After the t-ZrO2 particle is transformed to the m-ZrO2 particle, the m-ZrO2 particle takes the shape of the t-ZrO2 particle, but the twin-related thin plates are formed in the m-ZrO2 particle. In some m-ZrO2 parti￾cles the short dimension lies in the plane of the plates, and there is an angle between am and xi , shown in Fig. 2a; in some m-ZrO2 particles, the normal to the plates is along the short dimension of the particle, and there is an angle between cm and zi , shown as Fig. 2b. The bm axis is normal to the plane of Fig. 2a and b. 2.2. Lattice parameters For calculating stresses and strains caused by the misfits of lattice parameters between the product phase and the parent phase in Mg-PSZ, the lattice parameters of c-ZrO2, t-ZrO2 and m-ZrO2 without constraints should be known. The lattice parameters of Mg-PSZ can be found in many publications, such as in Refs. [6,7,26,27,34]. We selected the lattice parameters of Mg-PSZ from [26] as follows: c-ZrO2, ac=0.50778 nm. t-ZrO2, at=0.50803 nm, ct=0.51903 nm. m-ZrO2: am=0.51172 nm, bm=0.51770 nm, cm= 0.53031 nm, b=98.91°. For understanding the stresses associated with the t-ZrO2 precipitate in an (Mg, Y)-PSZ designed in Ref. [30], the FE calculation is also performed for this material. This material is based on the Mg-PSZ with the molar ratio MgO/ZrO2=14/86, and Y2O3 is added at the molar ratio Y2O3/ZrO2=2/98, and some a￾MgO–Al2O3 is also added. From Ref. [30], the lattice parameters of c-ZrO2 and t-ZrO2 are as follows: c-ZrO2: ac=0.5102 nm. t-ZrO2: at=0.5122 nm, ct=0.5170 nm. 2.3. Elastic constants Under the local coordinate system xi yi zi shown in Figs. 1 and 2, the constitutive relation of each phase can be expressed as {s}i=[C]i{o}i , where {s}i and {o}i are stresses and strains, and the sequence of stress￾es and strains in{s}i and {o}i is [xi xi , yi yi , zi zi , yi zi , zi xi , xi yi] T. Here [C]i is the elastic constant matrix, and [C]i is different for c-ZrO2, t-ZrO2, and m￾ZrO2. For c-ZrO2: [C]i=Ã Ã Ã Ã Ã Æ È C11 C12 C12 0 0 0 C12 C11 C12 0 0 0 C12 C12 C11 0 0 0 0 0 0 C44 0 0 0 0 0 0 C44 0 0 0 0 0 0 C44 Ã Ã Ã Ã Ã Ç É where C11=401, C12=96, and C44=56 GPa. These constants are selected from Refs. [7,35,36]. For t-ZrO2: Fig. 1. Morphology of the t-ZrO2 precipitate in Mg-PSZ: (a) the cross-section, and (b) viewed along the short dimension. Fig. 2. The cross-section of the m-ZrO2 particle in Mg-PSZ: (a) cm//ct , and (b) am//at .

C.R. Chen et al. Materials Science and Engineering 4272(1999)398-409 Cll C12 C1300 0 xi, y and =i directions. For modeling the stress field C12 C11 C1300 0 associated with a t-ZrO2 precipitate, a lenticular shape C13C13C33000 region with the elastic constants and thermal expansion 000C4400 coefficients of the t-phase is located within a region with the elastic constants and thermal expansion coeffi 00000C66 cients of the c-phase. At a given temperature To,we assume that there is no strain in both the c-phase and where C1=327,C3=264,C12=100,C13=62,C4= t-phase. When a temperature change AT= l K is uni- 59, and C66=64 GPa. These constants are selected formly applied to the whole FE model, the shape of the from Ref [35]. By comparing th c t-phase is changed due to the anisotropic thermal ex- the t-ZrO2 is softer along the cr-axis. By comparing the ansion coefficients of the t-phase defined above. Be- Cl of t-ZrO2 with [Cl; of c-ZrO2, we know the t-ZrO, cause the t-phase region and the c-phase region are is softer than the c-zro connected at the interface ange o For m-Zro t-phase region is constrained by the c-phase region, thus the interaction between the t-phase and the c- C11 C12 C130 C15 0 phase creates a stress field within and around the t-phase. Such a stress field is the same as the stress field C12C22C230C250 associated with a t-precipitate within the c-phase CI3C23C330C350 matrix 000C440C46 After a t-ZrO2 particle is transformed to a m-ZrO2 C15C25C350C550 particle, a large shear strain is created in each thin plate 000C460C66 within the m-ZrO2 particle because the lattice parame- ter B of m-ZrO2 is not 90. However, the effect of the where C1=358, C2=426, 33 =240, C12=44, due to the twin relationship between the plates, thus the C shear strain of these plates can only affect the matrix GPa. These constants are selected from Refs. [35,37, 38]. region closely near the interface. Thus we do not con- sider shear stains in the twin-related plates of the m-ZrO, particle when calculating stresses in the matrix 3. Method for realizing lattice misfits between phases As an approximate calculation, only the anisotropy of in the fe calculation volume expansion of the particle needs to consider When the microstructure of m-ZrO, is in the form as in The stresses created by a transformation are assumed Fig. 2a, i.e. cmllct, for calculating stresses created by the to be due to the lattice misfits between the produc t-m transformation, the thermal expansion coeffi phase and the parent phase. In the FE modeling, we use cients of m-ZrO2 particles are assumed to be a thermal expansion method to realize the effect of Afx=(am cos(B-900)-aDla lattice misfits. The phases are assumed to be coherently △ connected at the interfaces. The thermal expansion coefficients of c-ZrO2 matrix are assumed to be zero, fz=(cm-a)/a fz=0, where x, y and z are axes of the When the microstructure of m-ZrO, is in the form as global coordinate system xyz. The thermal expansion in Fig. 2b, i.e. am//an, the thermal expansion coefficients coefficients of t-Zro are assumed to be. of m-ZrO, particles are assumed to be △/x=( fx=/y=(a1-a-)/a △/y=(bm-a1) where xi y, and =, are axes of the local coordinate system xy of the t-ZrO2 precipitate. These thermal △fz=( C cOS(B-90°)-c1)/a expansion coefficients are not the real thermal expan In this way, the unconstrained lattice misfits between sion coefficients of the materials. The only purpose of phases are realized when a temperature change AT defining these thermal expansion coefficients in this K is applied to the whole Fe model, and the stresses paper is to realize the shape changes of the transformed created by the t-m transformation can be obtained region within the FE model. Under the free state, if the We take the axes [100], [010) and [ool] of the c-Zro temperature T is changed AT=l K, the shape of the matrix as the global coordinate system xyz. Befo C-phase will not change, while the relative dimension performing the FE calculation, the elastic constant changes of the t-phase will be x, fr, and fz, along the matrix [C], and the thermal expansion coefficients (fJ

C.R. Chen et al. / Materials Science and Engineering A272 (1999) 398–409 401 [C]i=Ã Ã Ã Ã Ã Æ È C11 C12 C13 0 0 0 C12 C11 C13 0 0 0 C13 C13 C33 0 0 0 0 0 0 C44 0 0 0 0 0 0 C44 0 0 0 0 0 0 C66 Ã Ã Ã Ã Ã Ç É where C11=327, C33=264, C12=100, C13=62, C44= 59, and C66=64 GPa. These constants are selected from Ref. [35]. By comparing C33 with C11, we know the t-ZrO2 is softer along the ct -axis. By comparing the [C]i of t-ZrO2 with [C]i of c-ZrO2, we know the t-ZrO2 is softer than the c-ZrO2. For m-ZrO2: [C]i=Ã Ã Ã Ã Ã Æ È C11 C12 C13 0 C15 0 C12 C22 C23 0 C25 0 C13 C23 C33 0 C35 0 0 0 0 C44 0 C46 C15 C25 C35 0 C55 0 0 0 0 C46 0 C66 Ã Ã Ã Ã Ã Ç É where C11=358, C22=426, C33=240, C12=144, C13=67, C23=127, C44=99, C55=79, C66=130, C15= −26, C25=38, C35= −23, and C46= −39 GPa. These constants are selected from Refs. [35,37,38]. 3. Method for realizing lattice misfits between phases in the FE calculation The stresses created by a transformation are assumed to be due to the lattice misfits between the product phase and the parent phase. In the FE modeling, we use a thermal expansion method to realize the effect of lattice misfits. The phases are assumed to be coherently connected at the interfaces. The thermal expansion coefficients of c-ZrO2 matrix are assumed to be zero, i.e. fX=fY=fZ=0, where x, y and z are axes of the global coordinate system xyz. The thermal expansion coefficients of t-ZrO2 are assumed to be: fZi =(ct−ac)/ac fXi =fYi =(at−ac)/ac (1) where xi , yi and zi are axes of the local coordinate system xi yi zi of the t-ZrO2 precipitate. These thermal expansion coefficients are not the real thermal expan￾sion coefficients of the materials. The only purpose of defining these thermal expansion coefficients in this paper is to realize the shape changes of the transformed region within the FE model. Under the free state, if the temperature T is changed DT=1 K, the shape of the c-phase will not change, while the relative dimension changes of the t-phase will be fXi , fYi and fZi along the xi , yi and zi directions. For modeling the stress field associated with a t-ZrO2 precipitate, a lenticular shape region with the elastic constants and thermal expansion coefficients of the t-phase is located within a region with the elastic constants and thermal expansion coeffi- cients of the c-phase. At a given temperature T0, we assume that there is no strain in both the c-phase and t-phase. When a temperature change DT=1 K is uni￾formly applied to the whole FE model, the shape of the t-phase is changed due to the anisotropic thermal ex￾pansion coefficients of the t-phase defined above. Be￾cause the t-phase region and the c-phase region are connected at the interface, the shape change of the t-phase region is constrained by the c-phase region, thus the interaction between the t-phase and the c￾phase creates a stress field within and around the t-phase. Such a stress field is the same as the stress field associated with a t-precipitate within the c-phase matrix. After a t-ZrO2 particle is transformed to a m-ZrO2 particle, a large shear strain is created in each thin plate within the m-ZrO2 particle because the lattice parame￾ter b of m-ZrO2 is not 90°. However, the effect of the shear strain of these plates on the matrix is eliminated due to the twin relationship between the plates, thus the shear strain of these plates can only affect the matrix region closely near the interface. Thus we do not con￾sider shear stains in the twin-related plates of the m-ZrO2 particle when calculating stresses in the matrix. As an approximate calculation, only the anisotropy of volume expansion of the particle needs to consider. When the microstructure of m-ZrO2 is in the form as in Fig. 2a, i.e. cm//ct , for calculating stresses created by the tm transformation, the thermal expansion coeffi- cients of m-ZrO2 particles are assumed to be: DfXi =(am cos (b−90°)−at )/ac DfYi =(bm−at )/ac DfZi =(cm−at )/ac (2) When the microstructure of m-ZrO2 is in the form as in Fig. 2b, i.e. am//at , the thermal expansion coefficients of m-ZrO2 particles are assumed to be: DfXi =(am−at )/ac DfYi =(bm−at )/ac DfZi =(cm cos (b−90°)−ct )/ac (3) In this way, the unconstrained lattice misfits between phases are realized when a temperature change DT=1 K is applied to the whole FE model, and the stresses created by the tm transformation can be obtained. We take the axes [100], [010] and [001] of the c-ZrO2 matrix as the global coordinate system xyz. Before performing the FE calculation, the elastic constant matrix [C]i and the thermal expansion coefficients { f }i

02 C.R. Chen et al. Materials Science and Engineering 4272(1999)398-409 t-zrO2 are according to the Mg-PSZ [26 as: a 0.50778,a1=0.50803, and c=0.51903nm. Thus the lattice misfits are x=fy=(a-ae/ae=0.049%, z (c-ad/ac=2.216%. A three-dimensional model is alse established. A lenticular shape precipitate is assumed to be in an infinite large spherical c-ZrO, matrix. Due to the symmetry, only one-eighth of the precipitate needs -Zr 02 to be analyzed The model is shown as Fig. 3b. The FE results show that under plane strain assumption, al though the tensile o. of the matrix near the edge of precipitate is overestimated, characteristics of stresses of the model in Fig. 3a are quite similar to that of the model in Fig. 3b Fig. 4a shows the distribution of stress a. in the c-ZrO2 matrix near the t-zrO2 precipitate under plane for calculating the stresses associated with the single tate in an infinite c-ZrO, matrix: (a) the model und ssumption, and(b)the three-dimensional model. Ur,fr, z] of particles under the local coordinate system xv=, must be transformed to be under the global coordinate system xyz. The sequence of stresses and strains in o, and (e) is [xx, yy, zz, yz, zx, xy? However, in some FE software packages, such as AN OXx (MPa SYS, the sequence of stresses and strains is [xx, yy, zz, xy, yz, zx]. Therefore, when using these FE software packages, the sequence of elements in elastic constant matrix [C] should be adjusted. The detailed descriptions of the anisotropy FE analysis can be seen in our previous work [39, 40 4. Results and discussion 4.1. The single I-ZrO2 precipitate in an infinite c-ZrO 4.1.1. Characteristics of stresses associated with a t-ZrO2 precipitate in the c-ZrO2 matrix For investigating stresses associated with the single zro, precipitate in the infinite c-zro2 matrix, a Fe model is established as in Fig 3a. Constraints: u,=0 at G=-5 0 at 0 at y=0: u,=0 at y=8(8 is the thickness of the model). The 3-dimensional an isotropy element is used. There are only one layer of Fig. 4. The stresses in the c-ZrO, matrix near the single t-Zro elements along the y-direction, thus this is a plane precipitate. the lattice parameters are according to Mg-PSZ:(a)a2, strain model. The lattice parameters of c-ZrO2 and b)orr, and (c)

402 C.R. Chen et al. / Materials Science and Engineering A272 (1999) 398–409 Fig. 3. Models for calculating the stresses associated with the single t-ZrO2 precipitate in an infinite c-ZrO2 matrix: (a) the model under plane strain assumption, and (b) the three-dimensional model. t-ZrO2 are according to the Mg-PSZ [26] as: ac= 0.50778, at=0.50803, and ct=0.51903 nm. Thus the lattice misfits are fXi =fYi =(at−ac)/ac=0.049%, fZi = (ct−ac)/ac=2.216%. A three-dimensional model is also established. A lenticular shape precipitate is assumed to be in an infinite large spherical c-ZrO2 matrix. Due to the symmetry, only one-eighth of the precipitate needs to be analyzed. The model is shown as Fig. 3b. The FE results show that under plane strain assumption, al￾though the tensile szz of the matrix near the edge of precipitate is overestimated, characteristics of stresses of the model in Fig. 3a are quite similar to that of the model in Fig. 3b. Fig. 4a shows the distribution of stress szz in the c-ZrO2 matrix near the t-ZrO2 precipitate under plane Fig. 4. The stresses in the c-ZrO2 matrix near the single t-ZrO2 precipitate, the lattice parameters are according to Mg-PSZ: (a) szz, (b) sxx, and (c) sxz. =[ fXi , fYi , fZi ] T of particles under the local coordinate system xi yi zi must be transformed to be under the global coordinate system xyz. The sequence of stresses and strains in {s} and {o} is [xx, yy, zz, yz, zx, xy] T. However, in some FE software packages, such as AN￾SYS, the sequence of stresses and strains is [xx, yy, zz, xy, yz, zx] T. Therefore, when using these FE software packages, the sequence of elements in elastic constant matrix [C] should be adjusted. The detailed descriptions of the anisotropy FE analysis can be seen in our previous work [39,40]. 4. Results and discussion 4.1. The single t-ZrO2 precipitate in an infinite c-ZrO2 matrix 4.1.1. Characteristics of stresses associated with a t-ZrO2 precipitate in the c-ZrO2 matrix For investigating stresses associated with the single t-ZrO2 precipitate in the infinite c-ZrO2 matrix, a FE model is established as in Fig. 3a. Constraints: uz=0 at z=0; ux=0 at x=0; uy=0 at y=0; uy=0 at y=d (d is the thickness of the model). The 3-dimensional an￾isotropy element is used. There are only one layer of elements along the y-direction, thus this is a plane strain model. The lattice parameters of c-ZrO2 and