Mechanical properties of zirconia-titanium composites

Transcription

1 International Journal of Materials Siene and Appliations 24; 3(5): Published online September 3, 24 ( doi:.648/j.ijmsa ISSN: (Print); ISSN: (Online) Mehanial properties of zironia-titanium omposites Hideaki Tsukamoto Graduate Shool of Engineering, Nagoya Institute of Tehnology, Nagoya, Japan address: a.jp To ite this artile: Hideaki Tsukamoto. Mehanial Properties of Zironia-Titanium Composites. International Journal of Materials Siene and Appliations. Vol. 3, No. 5, 24, pp doi:.648/j.ijmsa Abstrat: This study aims to investigate mehanial properties of ZrO 2 -Ti omposites. The samples have been fabriated by spark plasma sintering (SPS). The experimental results demonstrate that a small amount of ZrO 2 an enhane Ti matrix in hardness and tensile strength, while a large amount of ZrO 2 degrade the tensile strength and dutility beause of high extent of formation of titanium oxide. The mehanial properties of ZrO 2 -Ti omposites an be related to thermo-mehanial behavior of ZrO 2 /Ti funtionally graded materials (FGMs). In the FGMs, titanium oxide has also formed, whih an largely influene thermo-mehanial behavior of the FGMs. Keywords: ZrO 2 -Ti Composites, Titanium Oxide, Funtionally Graded Materials (FGMs), Spark Plasma Sintering (SPS). Introdution ZrO 2 -Ti omposites inluding ZrO 2 /Ti funtionally graded materials (FGMs) are attrative in totally different fields of both bio-medial and aerospae industries. Beause of good bioompatibility of Ti due to its non-toxiity and non-rejetion to the human body, Ti has been of onsiderable interest in medial fields, whih an be used for surgial implements and implants. ZrO 2 -dispersed Ti matrix omposites have been of remarkable use in biomedial fields []. Besides this, ZrO 2 -dispersed Ti matrix omposites have been also expeted to be high-temperature materials in aerospae and automobile industries. ZrO 2 has superior thermal and mehanial properties, whih are effetive as thermal barrier funtions [2]. One of superior mehanial properties of ZrO 2 is high frature toughness due to stress-indued transformation (from tetragonal to monolini rystal strutures under stress onditions), whih an be used in enhanement of frature toughness of other eramis and eramis matrix omposites [3-6]. Mehanial properties of ZrO 2 -Ti omposites have been of interest in many industries. In the past deade, ZrO 2 -Ti omposites have been of high potential for appliations to high temperature strutures inluding thermal barrier oatings (TBCs) in aerospae and automobile industries. Partiularly the funtionally graded (FG) strutures of ZrO 2 -Ti systems is potential as super high-temperature TBCs. The FGMs are advaned multi-phase omposites that are engineered to have a smooth spatial variation of material onstituents. The advantages of FGMs as an alternative to two dissimilar materials (eramis and metal) joined diretly together inlude smoothing of thermal stress distributions aross the layers, minimization or elimination of stress onentrations and singularities at the interfae orners and inrease in bonding strength [6-9]. Several studies have been onduted investigating the phase transformation of ZrO 2 aused by thermal stresses generated in FGMs [, ]. Teng et al [] showed that only α-ti, tetragonal t-zro 2, and monolini m-zro 2 were found in various Ti-ZrO 2 omposites after annealing from 4 to 65 o C. It was reported that the volume fration of m-zro 2 inreased with Ti ontent in the Ti-ZrO 2 omposites, while the interfaial stresses, arising from the plasti deformation of Ti and the thermal expansion mismath of Ti and ZrO 2, were the driving fores for the phase transformation from t-zro 2 to m-zro 2. The interfaial reation between Ti and ZrO 2 in FGMs has been also studied so far [2]. Thermodynami analysis is essential for hemial design of Ti-ZrO 2 omposites [, 2]. It was reported that the standard Gibbs free energy of Ti 2 O plays an important role in the thermodynami assessment of Ti ZrO 2 system, in whih the formation of Ti 2 O is easier than that of the other titanium oxides [3]. It was reported by Lin and Lin [2] that in ZrO 2 /Ti FGMs the mirostrutures and reation produts of Ti and ZrO 2 depends on Ti/ZrO 2 ratios. It is well known that mehanial properties of ZrO 2 an be enhaned by inorporating Ti. Weber et al. [4] reported that

2 International Journal of Materials Siene and Appliations 24; 3(5): sintered ZrO 2 ruibles ontaining 5 at% Ti showed superior strength and thermal shok resistane. Arias [5] also found that the thermal shok resistane and strength of ZrO 2 with 5 mol % Ti were inreased due to the inhabitane of grain growth of ZrO 2 by Ti during the sintering at 85 o C. In their researh, they onsidered that these phenomenon were related to dissolution of oxygen atom into Ti to form titanium oxides without taking into aount possible reations between Ti and ZrO 2. Spark plasma sintering (SPS) is haraterized by an eletri soure of diret pulsed urrent, whih is similar to that of an eletri disharge mahine. The diret pulsed urrent of SPS an be ontinuously applied from start to finish of the proess. SPS is advantageous for sintering of metals and eramis, and has been also used for onsolidation of metalli glass powder, hemial reation in organi ompounds, rystal growth of materials suh as CoSb 3, joining of steel, fabriation of Al alloy with fine rystal, failitation of FGMs [6]. In this study, mehanial properties of miro-vikers hardness and tensile behaviour of ZrO 2 -Ti omposites fabriated by SPS have been investigated. The effets of amount of ZrO 2 in Ti matrix on the mehanial properties have been examined systematially to effetively design suh kinds of the omposites. The results an be also of onsiderable use to understand thermo-mehanial behaviour of ZrO 2 /Ti FGMs based on a mean-field miromehanis-based model [7]. 2. Experimental Proedures 2.. Fabriation The prepared powders were Ti powder with the average diameter of 45 µm and ZrO 2 stabilized by 3 mol % Y 2 O 3 and mol % Y 2 O 3 with the average diameter of 26 nm. The SPS was onduted in vauum at 4 o C under the uniaxial pressure of 3 MPa with the time duration of 2 minutes. The sintered samples have a diameter of 2mm and a thikness of 9mm for ZrO 2 -Ti omposites and the FGMs. The omposites, in whih ZrO 2 (3mol %Y 2 O 3 ) partiles are distributed uniformly, ontain ZrO 2 stabilized by 3 mol % Y 2 O 3, whose volume fration ranges from to 3%. The FGM sample has the FGM part with a 3 mm thikness and Ti substrate part with a 6mm thikness. The FGM part onsists of 6 layers with different ompositions suh as, 2, 4, 6, 8 and vol% of ZrO 2 with an equal thikness (.5mm) for eah layer resulting in suh a linear step-wised ompositional gradient pattern. For the FGMs, two kinds of powders of ZrO 2 stabilized with 3 mol % Y 2 O 3 and mol % Y 2 O 3 were used. The FGMs were also sintered under the above onditions using SPS. voltage of 5 kv. The omposition in the samples was identified using the SEM energy-dispersive X-ray spetrosopy (EDS, JED-22). The mirosope samples were prepared by embedding into the resin. The XRD investigation was performed with Cu Kα radiation using the Rigaku XRD meter. The 2θ was measured from o to 9 o in steps of.2 o with a san speed of 2 o /min. Miro-Vikers hardness tests were onduted with Shimazdu Miro-Vikers Tester. Nanoindentation tests were performed with a U982A Agilent nano indenter G2 (Agilent Teh. In., USA), whose resolutions of the loading and displaement of the systems are 5nN and less than. nm, respetively. A Berkovih diamond indenter was used. The thermal drift rate was less than nm/s. A strain rate was set at.5/s. All the tests were onduted at room temperature (R.T.). Tensile tests were onduted with an Instron-type mahine with the strain rate of -5 /se. The tensile speimen with the dimensions is shown in Fig.. The speimen was ut from the sintered omposite samples. The strain was measured using strain gages attahed to both sides of the tensile speimens Figure. Tensile test speimen for Ti-ZrO 2 omposites fabriate by SPS. 3. Experimental and Simulation Results 2.2. Charaterization A mirostruture observation was onduted using optial mirosope (OM) and sanning eletron mirosope (SEM, JSM-59LV, JEOL, Japan) with an aeleration Figure 2. Miro Vikers hardness of Ti-ZrO2 omposites (a)range of volume of ZrO2 is to % (b)range of volume of ZrO2 is to 3%.

3 262 Hideaki Tsukamoto. Mehanial Properties of Zironia-Titanium Composites The fabriated ZrO 2 -Ti omposites were investigated on hardness and tensile behavior. Fig. 2 shows Miro Vikers hardness of ZrO 2 -Ti omposites. It is seen that small amount (less than 5 vol. %) of ZrO 2 an inrease hardness of Ti matrix drastially. In the range of higher than vol. % of ZrO 2, hardness of ZrO 2 -Ti omposites inreases more gradually with inreasing volume fration of ZrO 2 ompared to less than 5 vol. % of ZrO 2. Muh attention will be paid to the omposites with small amount of ZrO 2 reinfored Ti omposites beause of high effetiveness of ZrO 2 addition into Ti in enhanement of mehanial properties suh as hardness of Ti. Figure 3. Tensile stress-strain urves of Ti-ZrO 2 omposites. Next let us look at tensile behavior of ZrO 2 -Ti omposites. Fig. 3 shows the tensile stress-strain urves of the omposites. It is seen that up to vol. % of ZrO 2 ontent, tensile strength inreases with inreasing volume fration of ZrO 2, while the elongation of the omposites derease drastially at higher ontent of ZrO 2 than vol. %. Now the data has been summarized in Fig. 4. It is seen that even.2 vol % ZrO 2 derease elongation by a half, whih implies that frature toughness and dutility of the omposites drastially dereases with inreasing ZrO 2 ontent. It is inevitable to have high aution to redution of dutility due to ZrO 2 addition in Ti matrix in design of suh omposites. Aording to the work by Lin and Lin [2], ZrO 2 reats with Ti to form Ti oxide, whih obviously makes ZrO 2 -Ti omposites brittle. Young s modulus of the omposites slightly inreases with inreasing ZrO 2 ontent. Fig. 5 shows SEM images (low and high magnifiations) and EDS analysis data of ZrO 2 -Ti omposites with, 2 and 3 vol. % of ZrO 2. It is seen in Fig. 5 (a) that ZrO 2 seems to disperse uniformly in Ti matrix beause there is no gathering of ZrO 2 partiles, whih is supported by EDS analysis data showing onstant ratios of elements anywhere in the image. From SEM images and EDS data in Fig. 5 (b) and (), there is some lak of uniformity in Zr, Ti and O distributions and it is onsidered that Ti-Zr-O ompounds exist in the omposites, whih may ause redution of dutility and inrease of brittleness of the omposites. Figure 5. SEM images and EDS analysis data for Ti-ZrO 2 omposites (a) Ti- vol. % ZrO 2 omposites(b) Ti-2 vol. % ZrO 2 omposites () Ti-3 vol. % ZrO 2 omposites Figure 4. Features in tensile behavior of Ti-ZrO 2 omposites Ultimate tensile strength Elongation at failure () Young s modulus. So far we have been looking at mehanial properties of ZrO 2 -Ti omposites, in whih ZrO 2 are distributed uniformly in Ti matrix. Next let us examine ZrO 2 /Ti FGMs.

4 International Journal of Materials Siene and Appliations 24; 3(5): Fig. 6 shows shemati illustration of an ZrO 2 (3 mol % Y 2 O 3 ) /Ti FGM sample with an OM image and an SEM image with EDS analysis data providing element (Zr, Ti and O) distributions. It an be seen that the ZrO 2 (3 mol % Y 2 O 3 ) /Ti FGM samples were suessfully fabriated without any defets and raks. FGM parts have step-wised linear gradation patterns, but diffusion during fabriating proess made ompositional gradation smoother. Based on the EDS examination, Zr and Ti vary gradually from one side to another side, while O is distributed relatively uniformly in the FGM part, whih means that Ti possibly makes an oxidation with O oming from ZrO 2 [,2]. Figure 6. Sample illustration and OM image of ZrO 2 (3 vol. Y 2O 3)/Ti FGMs and SEM image with EDS data of ompositional distributions. (a)sample illustration and OM image of ZrO 2 (3 vol. Y 2O 3)/Ti FGMs (b)sem image ()EDS map for distribution of Zr (d) EDS map for distribution of Ti (e)eds map for distribution of O. Nanoindentation elasti modulus and hardness of ZrO 2 (3 mol % Y 2 O 3 ) /Ti FGMs are shown in Fig. 8. Nanoindentation indenter is very small ompared to miro-vikers one, whih make it possible to probe elasti modulus and hardness of eah onstituent phase suh as Ti or ZrO 2. It is seen that hardness and elasti modulus of Ti in the FGM part is as high as those of ZrO 2. Hardness and elasti modulus of Ti in the substrate far from the FGM part are muh lower than those of Ti in the FGM part. These may be attributed to presene of titanium oxide in Ti phase in the FGM part. Figure 7. MiroVikers hardness distribution in ZrO 2 (3mol % Y 2O 3)/Ti FGMs. Figure 8. Nanoindentation elasti modulus and hardness of ZrO 2/Ti FGMs. (a) Elasti modulus (b) Hardness. Fig. 9 shows X-ray diffration pattern for the ZrO 2 surfae layer in ZrO 2 (3 mol % Y 2 O 3 ) /Ti FGMs. It is seen that the peaks for tetragonal rystal struture appears learly and a small peak for monolini rystal struture ould be deteted. Figure 9. X-ray diffration pattern of ZrO 2 surfae in the FGM part. Next we examine the stress states just after SPS fabriation based on a mean-field miromehanis-based analysis. Brief desription of the mean-filed miromehanis model used in the urrent study is given in Appendix. The detail of the model is shown in the work by Tsukamoto [7]. Thermo-mehanial boundary onditions the FGM experiene during the fabriation proesses and the material property data of both Ti and ZrO 2 used in the alulation are also desribed in Appendix. Fig. shows the simulation results for in-plane and out-of plane stresses in ZrO 2 phase and Ti phase in the FGMs. It is seen that in-plane stresses in ZrO 2 phase in the FGMs are tensile, while out-of-plane stresses n ZrO 2 phase in the FGMs are ompressive. It is seen that both in-plane and out-of-plane stresses in Ti phase in the FGMs are tensile, in whih in-plane tensile stresses are higher than out-of-plane stresses. Suh in-plane stresses may ause rak generations and

5 264 Hideaki Tsukamoto. Mehanial Properties of Zironia-Titanium Composites propagations in Ti phase ontaining titanium oxides in the FGMs. (a) (b) 25 o C 5 o C o C Stress in ZrO2 / GPa Figure. Miro-stresses in ZrO 2 phase and Ti phase in the FGMs. (a)miro stress in ZrO 2 phase (b)miro stress in Ti phase. Figure. OM image of ZrO 2 ( mol % Y 2O 3)/ Ti FGMs and X-ray diffration pattern of ZrO 2 surfae. (a)om image (b)x-ray diffration pattern. Fig. shows OM image of mirostruture and XRD pattern for ZrO 2 surfae layer in ZrO 2 ( mol% Y 2 O 3 )/Ti FGMs. The distintive feature of mirostruture of ZrO 2 ( mol% Y 2 O 3 )/Ti FGMs inludes the fat that there are a lot of raks around the 2%ZrO 2 / 8%Ti and %Ti layers in ZrO 2 ( mol% Y 2 O 3 )/Ti FGM samples. All these raks are perpendiular to the ZrO 2 surfae, existing around Ti layer in the FGM part. It is expeted that in-plane residual stresses in the Ti (ontaining titanium oxides) layer ause suh rak generations and propagations in ZrO 2 ( mol% Y 2 O 3 )/Ti FGMs, in whih toughening by stress-indued transformation of ZrO 2 is onsidered to hardly our in ZrO 2 5 In-plane - Out-of-plane Position/ m 25 o C 5 o C o C In-plane Stress in Ti / GPa 2 Out-of-plane Position/ m ( mol% Y 2 O 3 ) beause most of ZrO 2 ( mol% Y 2 O 3 ) were already transformed from tetragonal to monolini strutures during the fabriation proess as seen in the XRD pattern in Fig. (b). The volume expansion due to the ZrO 2 transformation during the ooling proesses in fabriation also auses large stresses in the materials, whih leads to breakage of suh FGMs. 4. Disussion ZrO 2 -Ti omposites are promising materials in bio-medial, aerospae as well as automobile industries. In ZrO 2 -Ti omposites, behavior of atom of oxygen is a key fator to understand mehanial properties of the omposites. In the fabriation proess, some of oxygen atoms are onsidered to reat with Ti, and form titanium oxide, whih may ontribute to enhanement of hardness of Ti phase [7]. Mehanial properties of ZrO 2 -Ti omposites are highly affeted by formation of titanium oxide in Ti matrix. However, XRD data with vol.% ZrO 2 dispersed Ti matrix omposites ould not show any peak for titanium oxide as desribed in the previous work by Tsukamoto et al. [7]. Hardness drastially inreases with inreasing ZrO 2 ontent up to vol. % of ZrO 2, whih make onsider that the ompression strength of the omposites may inrease in the similar way to hardness. Even very small amount of ZrO 2 ontribute to forming Ti oxide, whih leads to enhanement of hardness as well as brittleness of the omposites as shown in Fig. 4. For ZrO 2 /Ti FGMs Ti phase in the FGM part showed high nanoindentation hardness and elasti modulus as seen in Fig.8. It is onsidered that small amount of ZrO 2 an inrease hardness of Ti in the FGMs as well beause of formation of titanium oxides, in whih O may ome from ZrO 2. Aording to the work by Lin and Lin [2], Ti reated with and was mutually soluble in ZrO 2, resulting in the formation of a-ti(o, Zr), Ti 2 ZrO, and/or TiO. These oxygen ontaining phases extrated oxygen ions from ZrO 2, whereby oxygen-defiient ZrO 2 was generated. For relatively small Ti/ ZrO 2 ratios, less than 3 mol % Ti, TiO were formed, and for 5 mol % Ti, lamellar type of Ti 2 ZrO preipitated in a-ti(zr,o) without TiO. In the present study on ZrO 2 -Ti omposites, some zironium-titanium-oxide and α-ti (Zr,O) an form in the omposites. In ase of ZrO 2 ( mol% Y 2 O 3 )/Ti FGMs, a lot of large raks perpendiular to the surfae an be observed as seen in Fig. (a), whih an be aused by in-plane stresses in Ti phase, as seen in miromehanis-based simulation results shown in Fig., applied to Ti layer made brittle by titanium oxide in the FGMs desribed above and due to least of effet of toughening by stress-indued transformation with non Y 2 O 3 -ontaining ZrO 2 in both ZrO 2 phase and Ti (with titanium oxides) phase. In addition, transformation of ZrO 2 ( mol% Y 2 O 3 ) during the fabriation proess auses large strains and stresses in the materials, whih an lead to rak generations and propagations and the frature of the materials. Aordingly, ZrO 2 ( mol% Y 2 O 3 )/Ti FGMs

6 International Journal of Materials Siene and Appliations 24; 3(5): annot be soundly fabriated under the present thermo-mehanial onditions by SPS. Doping of Y 2 O 3 is important to prevent the frature in the ZrO 2 /Ti FGMs during the fabriation proesses. Consequently, ZrO 2 -Ti omposites are very attrative in biomedial, aerospae as well as automobile industries beause of their superior bioompatibility and thermo-mehanial properties. Therefore, it is neessary to understand mehanial properties of ZrO 2 -Ti omposites aurately based on examination of their mirostrutures and oxide produts. From view points of hardness (ompressive properties), ZrO 2 is very effetive to improve mehanial properties of Ti matrix, but for tensile properties, ZrO 2 -Ti omposites exhibit muh brittleness even though ZrO 2 -Ti omposites ontain very small amount of ZrO 2. Suh properties highly influene design proesses for ZrO 2 -Ti omposites and also ZrO 2 /Ti FGMs. The urrent study on mehanial behavior of ZrO 2 -Ti omposites is onsidered to suggest guiding prinipals for engineering suh ZrO 2 -Ti omposites and ZrO 2 /Ti FGMs. 5. Conlusions Mehanial properties of ZrO 2 -Ti omposites were investigated on hardness and tensile properties. The samples were fabriated using SPS methods. The experimental results demonstrated that a very small amount (less than.5 vol. %) of ZrO 2 an enhane Ti matrix in hardness and tensile strength, while an even relatively small amount (around vol. %) of ZrO 2 degrade the tensile strength and dutility beause of high extent of formation of titanium oxide. Data of mehanial properties of ZrO 2 -Ti omposites an be of importane to design suh omposites used for biomedial and aerospae as well as automobile strutures and understand thermo-mehanial behavior of ZrO 2 /Ti FGMs. In the FGMs, titanium oxide also formed in Ti phase aording to examination on miro-vikers and nanoindentation analysis data, whih an largely influene thermo-mehanial behavior of the FGMs. A mean-field miromehanis-based analysis desribed that after the fabriation proesses, tensile stresses are present in Ti layer in the FGMs. For ZrO 2 ( mol% Y 2 O 3 )/Ti FGMs, after fabriation proesses, a lot of raks were observed in Ti layer in the FGMs, whih may be attributed to formation of titanium oxide in Ti layer, generation of residual stresses due to differene in thermal expansion oeffiients between Ti and ZrO 2, and a volume hange of ZrO 2 with no Y 2 O 3 by transformation aompanied with rystal struture hanges during the fabriation proesses, whih also leads to less transformation toughening by suh ZrO 2 partiles. Appendix A mean-filed miromehanis-based model is shown here to examine the experimental results. The model used in the study is based on the work by Tsukamoto [7]. Marosopially homogeneous omposites with spherial partiles are onsidered for a building blok of an FG TBC plate. The building bloks are assumed to be subjet to balaned bi-axial plane stresses. Here, the metal phase is assumed to be matrix and erami phase is partiles, whih are indiated by subsript and, respetively, while the inversion of the relation of matrix and partiles an be easily handled in the similar way. The inelasti deformation of onstituents of the omposites inlude the reep with the p strain, ε, the plasti deformation with the strain, ε, and the diffusional mass transport along the metal-erami d interfae with the eigen strain of the partile, ε. The in in in-plane miro-stresses,, and out-of-plane out out miro-stresses,, in eah phase an be written as follows [6, 7], in out = 2( β = 2( β + / 3γ 2 / 3γ ) + 3f β ( α + 2f γ {( ε ε ) + ε d ε p } 4f γ ) + 3f β ( α α ) θ {( ε ε ) + ε d ε p } α ) θ for the metal matrix ((indiated by subsript ), and in out = 2( β = 2( β + / 3γ ) 3f β ( α 2 / 3 2f γ γ ) 3f β ( α + 4f γ {( ε ε ) + ε d ε p } α ) θ {( ε ε ) + ε d ε p } α ) θ (A) (A2) (A3) (A4) for the erami partile (indiated by subsript ). Here, f and f are the volume fration, α and α are the oeffiinet of thermal expansion, and is a maro-stress due to balaned bi-axial loadings. β, β, γ, γ, β and γ are miromehanial onstants depending on the elasti onstants and volume fration of eah phase, whih were given in the work by Tsukamoto [7] In this analysis, plasti and reep deformations are supposed to obey the assoiated flow rule in whih both deformation potentials are taken equal to the von Mises-type yield funtion. Plasti deformation of metal phase is assumed to be expressed by the Swift s equation: where eq p eq n p = a( + ε ) (A5) a, and np are onstants. eq is the flow stress of metals. When the reep deformation of eah phase is assumed to be ontrolled by grain-boundary diffusion (Coble reep), the onstitutive equation is expressed as follows, ω D eq gb gb eq εɺ oble = C (A6) 3 ktd C is the geometri onstant (~6), D gb the grain boundary diffusivity, ω gb the grain boundary width, Ω the volume of a diffusing atom and k the Boltsman s onstant. The Ω

7 266 Hideaki Tsukamoto. Mehanial Properties of Zironia-Titanium Composites inelasti strain ε by mass transport along the interfae between metal and erami phases is expressed as follows [7], d eq ω D Ω d int int int eq εɺ eq = C 3 (A7) ktd p int C is the material onstant derived from miromehanial onsiderations, ω int the interfae width for diffusion, D int the interfaial diffusivity and Ω the volume of diffusing atom. Therefore, when onsidering the omposites under plane-stress onditions, the onstitutive equations an be derived as, ɺ (z,t) = e p { S (z) + S (z,t)} p(d) d { εɺ (z,t) α(z) θɺ (z,t) ε (z,t) ε (z,t)} (A8) ɺ (z,t) is the plane stress rate, εɺ (z,t) the plane strain rate, S e (z) the overall plane-stress elasti ompliane, S p (z, t) the overall plane-stress plasti ompliane, α (z) the overall in-plane thermal expansion oeffiient, εɺ p(d) (z,t) the overall plasti strain rate due to the differene between reep abilities of eah phase and interfaial diffusion, and d εɺ (z,t) the overall reep strain rate. The details for mathematial expressions of these funtions are given in the work by Tsukamoto [7]. ɺ (z,t) given in Eq. (A8) an be inorporated with the lamination theory. The material property data of both Ti and ZrO 2 used in the alulation are given in Table. The sample size is the same as that of real samples, for whih the thikness is 9 mm (inluding 3 mm thikness for the FGM part and 6mm thikness for the substrate part). The thermo-mehanial boundary onditions are assumed as follows: at the beginning, the samples being sintered is ompated in die (fully onstraint in both in-plane and out-of-plane deformations) at 4 o C. Under this ondition, there is no stress in the materials. Then temperature goes down to room temperature (25 o C) under the mehanial boundary onditions of fully onstraint deformation. After that, at R.T., all the mehanial onstraints are relieved (mehanial onstraint is free.). Table. Material property data of Ti and ZrO 2 used in the alulation. Ti ZrO 2 Young s modulus /GPa 6 2 Poisson s ratio.32.3 CTE / -6 K Thermal ondutivity/ Wm - K Speifi heat / Jkg - K Density / kgm [Coble reep parameters] D gb (pre-exp. Term) W gb / m 3 s Ativation energy / Jmol Atomi volume / m Grain size / m [Plastiity parameter] a/ MPa n p Referenes [] A.W. Nugroho, G. Leadbeater, I.J. Davies, Proessing of a porous titanium alloy from elemental powders using a solid state isothermal foaming tehnique, J. Mater. Si Mater Med, 2, vol. 2, pp [2] T. Lidong, L. Wenhao. Residual stress analysis of Ti-ZrO 2 thermal barrier graded materials, Mater Design, 22, vol.23, pp [3] R.C. Gravie, R.H. Hannink, R.T. Pasoe, Cerami steels, Nature, 975, vol. 258, pp [4] P.M. Kelly, L.R.F. Rose, The martensiti transformationin eramis- its role in transformation toughening, Prog Mater Si, 22, vol. 47, pp [5] H. Tsukamoto, Miromehanial modeling of transformation toughening in multi-phase omposites enrihed with zironia partiles, Comp Mater Si, 2, vol. 48, pp [6] H. Tsukamoto, Design against frature of funtionally graded thermal barrier oatings using transformation toughening, Mater Si Eng A, 2, vol. 527, pp [7] H.Tsukamoto, Analytial method of inelasti thermal stresses in a funtionally graded material plate by a ombination of miro- and maromehanial approahes, Compos Part B-Eng, 23, vol. 34, pp [8] K. Kokini, J. DeJonge, S. Rangaraj, B. Beardsley, Thermal shok of funtionally graded thermal barrier oatings with similar thermal resistane, Surf Coat Teh, 22, vol. 54, pp [9] Z. Han, B. Xu, H. Wang, S. Zhou, A omparison of thermal shok behavior between urrently plasma spray and supersoni plasma spray CeO 2 Y 2 O 3 ZrO 2 graded thermal barrier oatings, Surf Coat Teh, 27, vol. 2, pp [] L.D. Teng, F.M. Wang, W.C. Li, Thermodynamis and mirostruture of Ti-ZrO 2 metal-erami funtionally graded materials, Mater Si Eng A, 2, vol. 293, pp [] L.D. Teng, W.C. Li, F. M. Wang, Effet of Ti ontent on the martensiti transformation in zironia for Ti-ZrO 2 omposites, J Alloy Compd, 2, vol. 39, pp [2] K.L. Lin, C.C. Lin, Reation between titanium and zironia powders during sintering at 5 o C, J Am Ceram So, 27, vol. 9, [3] T.A. Shaedler, O. Fabrihnaya, C.G. Levi, Phase equilibria in the TiO 2 -YO.5 -ZrO 2, J Euro Ceram So, 28, vol. 28, pp [4] B.C. Weber, H.J. Garrett, F.A. Mauer, M.A. Shwartz, Observations on the stabilization of zironia, J Am Ceram So, 956, vol. 39, pp [5] A. Arias, Thermal shok resistane of zironia with 5 mol %Titanium, J Am Ceram So, 966, vol. 49, pp

8 International Journal of Materials Siene and Appliations 24; 3(5): [6] M. Omori, Sintering, onsolidation, reation and rystal growth by the spark plasma system (SPS), Mater Si Eng A, 2, vol. 287, pp [7] H. Tsukamoto, T. Kunimine, M. Yamada, H. Sato, Y.Watanabe, Mirostruture and mehanial properties of Ti-ZrO 2 omposites fabriated by spark plasma sintering, Key Eng Mat, 22, vol. 52, pp