Analytical Modeling for Stress-Strain Curve of a Porous NiTi

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1 Ying Zhao Minoru Taya Department of Mechanical Engineering, Univerity of Wahington, Box , eattle, WA Analytical Modeling for tre-train Curve of a orou NiTi Two model for predicting the tre-train curve of porou NiTi under compreive loading are preented in thi paper. orou NiTi hape memory alloy i conidered a a compoite compoed of olid NiTi a matrix and pore a incluion. Ehelby equivalent incluion method and Mori-Tanaka mean-field theory are employed in both model. Two type of pore connectivity are invetigated. One i cloed cell (model ); the other i where the pore are interconnected to each other forming an open-cell microtructure (model 2). We alo conider two different hape of pore, pherical and ellipoidal. The tre-train curve of porou hape memory alloy with pherical pore and ellipoidal pore are compared. It i found that the ellipoidal hape aumption i more reaonable than the aumption of pherical pore. Comparion of the tre-train curve of the two model how that ue of open-cell microtructure (model-2) make the prediction more agreeable to the experimental reult of porou NiTi whoe microtructure exhibit opencell microtructure. DOI: 0.5/ Introduction Over the lat two decade hape memory alloy MA have attracted great interet in variou application ranging from aeropace and naval tructure 2 to urgical intrument, medical implant and fixture 3,4. The ue of MA ha promoted extenive reearche on developing MA contitutive model. Among MA, NiTi alloy ha been ued mot extenively due to it large flow tre and hape memory effect ME. Mot recently, porou NiTi attracted an increaing attention a a poible application to medical implant device and high energy aborption tructural material and potential material for urface cooling. The progre in both manufacturing and characterization of the porou NiTi MA ha been reported by a number of reearcher. A hort review of the exited proceing method i preented here. Li et al. 5,6 fabricated porou NiTi MA by combution ynthei method, the tre-train curve in their work exhibit brittle behavior. Li et al. 7 alo fabricated the porou NiTi by powder intering; it how that the there i no tre plateau in the tre-train curve and the material i till brittle. ome tudie 8,9 report uperelatic behavior of the foam in compreion, but tree are low due to the high poroity. Lagouda et al. 0 ued the HI hot iotatic pre method and the tre-train curve in their work howing brittle behavior. Recently, we proceed the porou NiTi by the park plama intering method, and the pecimen exhibit large uperelatic loop with high tre flow and high ductility. In order to make an optimum deign of the microtructure of the porou MA, it i important to contruct a imple, yet accurate model to decribe it microtructure-mechanical behavior relation. Thu far, there are few analytical tudie focued on the porou MA, particularly no analytical model for the porou MA with open poroity. Therefore, in thi paper two model are introduced. The porou NiTi i treated a a compoite with olid NiTi a it matrix and pore a the incluion. If a porou NiTi can be Contributed by the Applied Mechanic Diviion of AME for publication in the JOURNAL OF ALIED MECHANIC. Manucript received April 7, 2005; final manucript received January 7, Review conducted by M. R. Begley. Dicuion on the paper hould be addreed to the Editor, rof. Robert M. McMeeking, Journal of Applied Mechanic, Department of Mechanical and Environmental Engineering, Univerity of California-anta Barbara, anta Barbara, CA , and will be accepted until four month after final publication of the paper itelf in the AME JOURNAL OF ALIED MECHANIC. viewed a a pecial or ellipoidal incluion cae of a particular reinforced compoite, then one can contruct a micromechanical model baed on Ehelby equivalent incluion method 2 with Mori-Tanaka mean-field theory 3. In the firt model model, the pore in the porou NiTi are cloed, i.e., pore are not connected to each other. In the econd model model 2, pore are aumed to be connected to each other, i.e., open poroity microtructure. Effect of pore with different geometrie pherical and ellipoidal are alo tudied in both model. There are two advantage of thee model. Firt, it i a impler model. We need le input data, and the input data for our model are taken from the experimental tre-train curve of the olid NiTi, i.e., no need to manipulate type of matenite variant, their orientation, etc., often ued by other model. econd, in the literature, there exit no model that treated porou NiTi with the open-cell tructure. The model propoed in thi paper i the firt in treating the open-cell tructure within the framework of Ehelby model. In the following, we hall tate firt two contitutive model, model ec. 2 and model 2 ec. 3 and dicu the reulting tre-train curve they predict. Thi repone will be compared to previou experimental reult,, and concluion will be preented. 2 Model-: tre-train Curve of orou NiTi With Cloed ore The tre-train curve of a MA with uperelatic grade i aumed to be compoed of four tage; ee Fig.. The firt tage I i a linear elatic tage, with the matrix of 00% autenite. The econd tage II i a tre-induced matenitic tranformation tage; in thi tage, a volume fraction of the autenite decreae from 00% to 0% while that of the martenite increae from 0% to 00%, continuouly. The third tage III i a tage with the matrix of 00% matenite. Therefore, the third tage i an linear elatic tage. The fourth tage VI i the autenite tranformation tage, in which the volume fraction of martenite change from 00% to 0% while that of the autenite in the matrix increae from 0% to 00%., Mf, A, and Af are the critical tree between the above four tage, the upercript denote porou and ubcript M,, A, and A f denote martenite tart, martenite finih, autenite tart, and autenite finih, repectively. E and E Mf are the moduli of the firt and third tage and E T i the Journal of Applied Mechanic Copyright 2007 by AME MARCH 2007, Vol. 74 / 29

2 due to exitence of pore, repectively. kl i the average train diturbance in the matrix due to the pore p. * ij i the fictitiou eigentrain, which ha nonvanihing component in the domain p. Here, we introduce ** kl * T kl kl 2 For the entire domain D, the following relation alway hold: 0 ij C m 0 ijkl kl 3 Following the Mori-Tanaka mean filed theory, the average tre diturbance in the matrix, m i given by Fig. Linearized four-tage tre-train curve of porou NiTi tangent modulu of the econd and fourth tage. It i noted that the lope of the forth tage i the ame a that of the econd tage. In thi paper, we ue the tre-train curve of the olid NiTi i.e., without pore a a reference. 2. Ehelby Model for a orou MA With Cloed ore. Conider an Ehelby model for a porou NiTi with cloed pore ubjected to applied tre 0 ij where the matrix i olid with uniform martenite tranformation train T ij Fig. 2 a. The uniform tranformation train T ij i aumed only for the econd and fourth tage, while for the firt and third tage, T ij 0. In the Ehebly model, an infinite elatic body D, which contain pherical or 0 ellipoidal pore p, i ubjected to a uniform tre 33 a hown in Fig. 2. A far a the tre field i concerned, the model of Fig. 2 a i equivalent to that of Fig. 2 b where the uniform tranformation train in the matrix i removed from and added with minu ign to the pore domain. Thu, the preent problem i reduced to the inhomogeneou incluion problem where the elatic tiffne tenor C m ijkl i homogeneou in the entire domain D, Fig. 2 b 4. The Ehelby inhomogeneou incluion problem with Mori- Tanaka mean field theory provide the total tre field given by 0 ij + C m ijkl e 0 kl + kl + kl * kl T kl C m ijkl 0 kl + kl + kl ** kl C p ijkl 0 kl + kl + kl where C m ijkl and C ijkl are the elatic tiffne tenor of matrix and pore, repectively. and kl are tre and train diturbance Fig. 2 The Ehelby model for a porou MA: a the problem of pore embedded in the NiTi matrix with tiffne C m ijkl and tranformation train T ij, which can be converted to equivalent * incluion problem b, where i the fictitiou eigentrain, which i unknown. m C m ijkl kl 4 The train diturbance i related to ** mn a ** kl klmn mn 5 The requirement that integration of the tre diturbance over the entire body D vanihe, leading to kl f klmn ** mn ** kl 6 where klmn i Ehelby tenor for pore incluion and it exact value are given in Appendix and f p i the volume fraction of pore, i.e., poroity of the MA. A ubtitution of Eq. 3 6 into ** provide a olution for kl ** ij f C m ijkl klmn I klmn + C ijkl f klmn + f C mnt C m 0 tpq I mnpq pq where I i identity matrix of 6 6. In thi paper boldface ymbol,c,i, are fourth-order tenor, and they are converted to 6 6 matrix form for facilitation of calculation. Thee boldface ymbol with ubcript hould be written a not boldface with ubcript when they are calar component. However, to avoid confuion, we keep boldface ymbol. A the tiffne of the pore i zero, C ijkl 0; thu, Eq. 7 i given a ** mn klmn I klmn C f ijkl By etting C ijkl 0, Eq. become m ij 9 Next, we hall conider the train energy denity of the inhomogeneou incluion problem of Fig. 2 b, which i given by 5 W mi 2 0 ij 0 ij + 2 f 0 ij * ij 2 f T 0 Let u call W mi a microcopic train energy denity. It i noted in Eq. 0 that W mi i valid for all three tage, i.e., for the firt and third tage T ij 0, wherea for the econd tage, all three term on the right-hand ide of Eq. 0 are nonvanihing. 2.. tiffne of Firt and Third tage. The equivalency of train energy denity of porou MA for the firt and third tage can be derived from Eq. 0 with T ij 0, which i et equal to the train energy denity of a porou MA with it elatic tiffne c tenor C ijkl, where c refer to compoite, a poroity i a pecial cae of compoite. 2 C c ijkl 0 ij 0 kl 2 C m ijkl 0 ij 0 kl + 2 f 0 * ij where C c ijkl and C m ijkl are the elatic compliance of the compoite and the matrix material olid NiTi, repectively. ince only the nonvanihing component of 0 ij i Fig. 2, Eq. i reduced to 292 / Vol. 74, MARCH 2007 Tranaction of the AME

3 f 2E c 2E m 2 * where E c and E m are the Young moduli of the compoite and the matrix, repectively. In the firt tage, E c i E, and E m i the Young modulu of autenite E A. Combining Eq. 2 with Eq. 2 and 8 with T ij 0, the Young modulu of the porou NiTi in the firt tage, E normalized by E A i given a E 3 E A + f where i a parameter, a function of poroity f and hape of the porou incluion H 33 + H H a f p H i a 6 6 matrix, which i given a H ijmn kl I ijkl C klmn 4b Young modulu of the porou NiTi at the third tage, E Mf can be obtained in the ame manner a the above, E E M + f 4c 2..2 Critical tree and Mf. Under the uniaxial tre along the x 3 -axi 0, the tranformation train T ij i aumed to be uniform with the following component: T ij T T T 000 T 5 where T i the tranformation train along x 3 -axi. The tre diturbance i obtained from Eq. 9. The change of the total potential energy of the inhomogeneou incluion of the problem of Fig. 2 b, U, due to the change in tranformation train T ij i given by 6 U T ij f 0 ij f 6 The work done by the applied tre Q cauing infiniteimal tranformation train T i where Q f T 7 i the tre of the matrix at the onet of tre-induced martenite tranformation, at the beginning of the econd tage of the olid NiTi, the upercript denote olid material, and the ubcript M denote matenitic tart tranformation. ince U + Q0, we obtain 0 + f 33 8 f ubtituting 33 0 and 0 from Eq. 9 into Eq. 8, we obtain f The martenitic tranformation finih critical tre Mf obtained in the ame manner f 9a can be 9b 2..3 tiffne of econd tage. Refer to Fig. 3 a, Young modulu E of a MA with tranformation T i etimated by E T E A + T T E M E A 20 where E A, E M are the Young modulu of 00% autenite and T 00% martenite phae, repectively, Fig. 3 a, and Mf i the Fig. 3 a tre-train curve of porou or olid ample i or, b tre-train curve of olid NiTi maximum tranformation train, and it i given by T 2 E M Equation 20 i valid for both dene and porou MA; thu, we can rewrite Eq. 20 uing Eq. 2 a E i i E A i i E M i i T 22 /E M E A i where the upercript i denote i olid or porou. In order to obtain the lope of the linearized econd tage of compreive tre-train curve of a porou NiTi, we conider the equivalency of train energy denity. In addition, in the cae of the econd tage, we evaluate the macrocopic train energy denity of a porou NiTi graphically from Fig. 3 a, i.e., area encloed by the curve. W ma i given by W ma 2 M + 0 T + 0 M E AM E 23 where i the martenitic tart tranformation tre of porou MA, 0 i an applied tre, T i the train correponding to 0. ince there i no tranformation train in pore, the tranformation Journal of Applied Mechanic MARCH 2007, Vol. 74 / 293

4 train for porou MA T i the uniform tranformation train in the matrix, i.e., olid NiTi, T T T 24 The above macrocopic train energy denity i et equal to the microcopic tain energy denity W mi defined by Eq. 0, where the olution for * ij i obtained from Eq. 7 a * kl T kl klmn I klmn C f mnij m 0 25 In Eq. 25, the firt term on the right repreent the tranformation in the olid NiTi matrix; the econd term come from the interaction between pore and applied tre. ubtituting Eq. 25 into Eq. 0, the microcopic train energy denity, W mi i given by W mi 2 0 ij 0 ij + 2 f 0 2 T ij kl I ijkl f kl ince the porou NiTi i ubjected to uniaxile load, i.e., T, and tranformation train given by T T T T 000 T, and the pore are aumed to be pherical; thu, Eq. 26 can be reduced to W mi f 0 2 T Z 3333 f 0 27 where Z 3333 i a component of a 6 6 matrix, Zkl I ijkl. i Ehelby tenor for pherical and ellipoidal incluion, repectively. 0 i the macrocopic train of the porou MA, and it i related to applied tre 0 a E AM ubtituting Eq. 28 into Eq. 27, the microcopic train energy denity W mi of porou NiTi i finally reduced to W mi E AM 2 f T Z 3333 E AM 29 where E AM i the Young modulu of olid matrix MA with T. By equating the macrocopic train energy denity W ma of Eq. 23 W mi of Eq. 29, and uing Eq. 22 with i, we obtained the following algebraic equation of econd-order T a: F T 2 + G T + H 0 30 F 0 +, G M + 0, E H 0 2 2, f Z 3333, E f E, 2f 3 E The olution of T, which correpond to the econd kink point D in Fig. 3 a, i given by T G + G 2 4FH 32 2F The tangent modulu of the porou MA i the lope of the econd tage of the tre-train curve hown in Fig., thu, E T can be expreed in term of tranformation train T and the tree, 0 and a E T 0 33 T 2.2 Unloading Curve. During unloading, the porou MA material undergoe revere tranformation martenite phae to autenite phae. Before the applied tre reache to a critical value A, the matrix MA remain 00% martenite phae firt tage of the unloading tre-train curve in the modeling curve. When the applied tre i decreaed to A, revere tranformation tart and it finihe when the tre reache another critical value Af ; thereafter, the porou MA material remain 00% autenite. Therefore, the lope of the firt and third tage of the unloading curve i the Young modulu of the 00% martenite and 00% autenite phae, repectively. The lope of the fourth tage i the ame a that of the loading curve in the econd tage. Therefore, the Young moduli of the unloading curve are related to thoe of the loading curve a E A E E T u E T 34a 34b E Af E 34c where E u T i the lope of the econd tage of the unloading curve. The upercript u denote unloading, wherea thoe without upercript are the lope of loading curve. The autenite tart and finih tranformation tree of porou MA, A and Af are related to the correponding tree of the olid NiTi by where A and Af A f A Af f Af 35a 35b are autenite tart and finih tranformation tree of the olid NiTi, repectively. Firt, we aume that the olid NiTi matrix i iotropic with oion ratio A M Model 2: tre-train Curve of orou NiTi With Open ore Here we hall dicu the Ehelby model for a porou MA with open pore where pore are interconnected Fig. 4, where unknown fictitiou eigentrain i in pore * ij and in pore 2 *2 ij will be determined by Ehelby method. Initially, we ditinguih the elatic contant C ijkl and eigentrain * aociated with pore and thoe with pore 2, but later we will et thoe identical to each other a two adjacent pore of the ame hape, and orientation hould have the ame C ijkl and * ij. There are two tep to obtain the eigentrain. The firt tep i to find the eigentrain * ij and the diturbed tre ij in for an infinite body containing ellipoidal or pherical pore and ubjected to uniaxial applied tre 0. In the firt problem, the other pore 2 i not conidered for obtaining * ij, but the interaction between the pore i taken into account by Mori-Tanaka mean field theory. The econd tep i to find the eigentrain *2 ij and diturbed tre 2 ij in 2, where the interaction between and 2 i taken into account. After obtaining * ij and *2 ij, we take the average of thee eigentrain to repreent the eigentrain for the two interconnected pore. The tranformation train T i zero in the firt and third tage, ince there i no tranformation occurring in thee two tage, wherea it i not zero in the econd and fourth tage. That i the ame a in ec / Vol. 74, MARCH 2007 Tranaction of the AME

5 Fig. 4 a Ehelby model for interconnected pore in NiTi matrix, which i converted to equivalent incluion problem b 3. Computation of Average Eigentrain in Open ore. In thi model, we will imulate the tre-train curve of a porou MA with open poroity where two kind of pore hape pherical and ellipoidal are ued. 3.. olution of Eigentrain in Without 2, ** ij. The Ehelby model for finding ** ij i the ame a the in model ; therefore, we lit only the final olution ** kl klmn I klmn C f mnij m olution of Eigentrain in 2 by Accounting Interaction Between and 2, **2 ij. In thi tep, the diturbed tre in 2 i obtained in term of **2 ij, which i unknown thu far. In the firt problem, diturbed tre outide the end of the, ij i expreed in term of t ij. Then the total tre in 2 vanihe t 0 ij + 2 ij **2 ij + ij 0 37 where 2 ij **2 **2 ij i the elf-tre in 2 induced by eigentrain The diturbed tre ij i given by the formula of Hill-Walpole- Mura jump condition 7 where ij ij C ijkl C pqmn ** mn M kp n q n l + ** kl 38a M kp kp nknp 2 38b and where n i i the ith component of an unit vector outer normal to the incluion and i given by n 00 38c It i noted in Fig. 4 that the interconnection between and 2 i at the equator region with it normal bae vector pointing to x -axi. With Eq. 38c, Eq. 38a yield ij ij + C ij C mn ** mn M + ** 39 where M E The applied tre i given a 0 ij tre in 2 in term of eigenrain **2 ij i given by 2 ij f C m **2 ijkl klmn I klmn mn 42 By ubtituting Eq. 39, 4, and 42 into Eq. 37, we obtain eigentrain in 2 **2 ij a **2 ** mn mnr r 43a where mnr mnpq I mnpq C pqr M + I pqr / f 43b We take average of the eigentrain in and 2 to repreent the eigentrain of all interconnected pore ** mn 2 ** mn + **2 mn 2 ** mnr + I mnr r 44 Once the average eigentrain ** mn i obtained, we can ue the firt model to calculate the moduli and critical tree. Therefore, we lit only the final ueful for the moduli and critical tree in the following. 3.2 Elatic Moduli at tage I and III. ince there i no tranformation at tage I and III, the Young modulu of each tage can be obtained by equivalency of energy denity a in model from Eq.. Therefore, the Young modulu of tage I i given a E 45a E A +D 3333 f and that of tage III i E 45b E M +D 3333 f where D 3333 i a component of a 6 6 matrix D klr, which i given a D klr A klmn B mnpq C pqr 46 2 f where A klmn I klmn + klmn, B mnpq mnpq I mnpq, C pqr C pqr. 3.3 Tangent tiffne of tage II and IV. To obtain E T, firt we have to obtain the tranformation train T. The tranformation train i obtained in the ame manner a that in model T G G2 4G G G where p G + f p 0 48a G 2 f p 0 E A f D 3333 p 0 2 p + p 0 E A 2 48b Table Mf Input data given by olid NiTi reference curve Af E A E M Mf 420 Ma 780 Ma 450 Ma 75 Ga 3 Ga Journal of Applied Mechanic MARCH 2007, Vol. 74 / 295

6 Fig. 6 Microtructure of 3% poroity pecimen 3.4 Critical tre. The four critical tree are obtained in the ame manner a that in model. They are given by f 5a f A f A 5b 5c Af f Af 5d It i noted here that upercript and denote olid and porou MA, repectively. Fig. 5 Comparion of the experimental data to prediction by the preent two model and G 3 2 E A p p 0 + f p D 3333 p 0 2 E A E A E M p /E M 48c 49a 4 Dicuion We ue Ti-50.9 at. %Ni a MA, and the experimental data of the tre-train curve of olid NiTi under compreive loading i made into piecewie linear wire four tage, Fig. 3 a. Table how the input data of the piecewie linearized tre-train curve of olid NiTi to imulate the tre-train curve of 3% poroity NiTi pecimen. Figure 5 how the tre-train curve predicted by model and 2. Table 2 lit the value of the critical tree and Young p 2 M + p p 0 E 49b Therefore, the tangent modulu, E T i given in term of tranformation train a E T 0 50 T Table 2 Comparion of the critical tree and Young moduli of experimental data and prediction by two model E Ga E T Ga E Mf Ga Experimental Ellipoidal pore Model pherical pore Model Ellipoidal pore Model pherical pore Model Fig. 7 tre-train curve predicted by ellipoidal and pherical open-cell model 296 / Vol. 74, MARCH 2007 Tranaction of the AME

7 moduli predicted by two model, a well a the experiment. The comparion how that the imulation have reaonable good agreement with the experimental data. It i noted that the tre-train curve of the ellipoidal porehape aumption i cloer to the experimental curve than that of the pherical pore-hape aumption. Thi indicate that the ellipoidal pore hape i more likely the realitic pore hape in the pecimen ued for the experiment. Figure 6 how the microtructure of the porou NiTi pecimen. Although, from thi figure we can ee that the realitic pore hape i not ellipoidal, but the ellipoidal i cloer than the pherical to the realitic hape. Therefore, the model with ellipoidal pore hape predict more accurately. Comparing the tre-train curve predicted by Model and 2, we can ee that model 2 i more accurate than model. Thi i becaue model 2 take into account the interaction between two adjacent pore by auming they are interconnecting to each other. Figure 6 upport thi aumption that pore are indeed interconnected. Model 2 can take account for the interaction between the interconnected pore; therefore, the prediction by model 2 give rie to cloer agreement with the experiment. Figure 7 how how the poroity f influent tre-train curve predicted by model 2, the open-cell model. It i found in Fig. 7 that when the poroity i 0%, the ellipoidal and pherical hape have almot ame prediction; but once when the poroity get larger and larger, the difference between the two hape increae dramatically. 5 Concluion Two model predicting the tre-train curve of porou MA ubjected to compreive load are preented. ore are treated a eparate individual in model, wherea they are interconnected to each other in model 2. Both model explain the experimental data reaonably well. Model 2, which can take account for the interaction among interconnected pore, provide better prediction than model in predicting the experimental data of the Young moduli of porou NiTi. Acknowledgment The preent work wa upported by a grant from ONR-MURI project N via Univerity of California at an Diego, where I i rofeor. Nemat-Naer. The program monitor at ONR i Dr. R. Baroum. Reference Lagouda, D. C., trelec, J. K., Yen, J., and Khan, M. A., 200, Intelligent Deign Optimization of a hape-memory-alloy-actuated Reconfigurable Wing, roc. IE, 3984, pp Garner, L. N., Wilon, L. J., Lagouda, D. C., and Redinioti, O. K., 2000, Development of a hape Memory Alloy Actuated Biomimetic Vehicle, mart Mater. truct., 9, pp Martynova, I., korohod, V., and olonin,., 99, hape Memory and uperelaticity Behavior of orou Ti-Ni Material, J. hy. IV, 99, pp. C4/ 42 C4/ Goncharuk, N. V., Martynova, I., Naidenova, I. F., and korokhod, O. R., 992, uperelaticity and hape Memory of intered orou Titanium Nickelide, orohk. Metall. Kiev, 4, pp Li, Y. H., Rong, L. J., and Li, Y. Y., 998, orou NiTi Alloy repared From Elemental owder intering, J. Mater. Re., 3, pp Li, Y. H., Rong, L. J., Luo, X. H., and Li, Y. Y., 2000, Microtructure and uperelaticity of orou NiTi Alloy, ci. China, er. E: Technol. ci., E-42, pp Li, B. Y., Rong, L. J., and Li, Y. Y., 998, Tranformation Behavior of intered orou NiTi Alloy, Metall. Mater. Tran. A, 30A, pp Yuan, B., Chung, C. Y., and Zhu, M., 2004, Microtructure and Martenitic Tranformation Behavior of orou NiTi hape Memory Alloy repared by Hot Iotaic reing roceing, Mater. ci. Eng., A, 382, pp Li, B. Y., Rong, L. J., Gjuanter, V. E., and Li, Y. Y., 2000, orou NiTI hape Memory Alloy roduced by Two Different Method, Z. Metallkd., 9, pp Lagouda, D. C., Entchev,. B., and Vandygriff, E. C., 2000, Modeling of Thermomehcanical Repone of orou hape Memory Alloy, roc. IE, 3992, pp Zhao, Y., Taya, M., Kang, Y.., and Kawaaki, A., 2005, Compreive Behavior of orou NiTi hape Memory Alloy, Acta Mater., 53 2, pp Ehelby, J. D., 957, The Determination of the Elatic Field of an Ellipoidal Incluion, and Related roblem, roc. R. oc. London, er. A, 3, pp Tanaka, K., and Mori, T., 973, Average tre in Matrix and Average Elatic Energy of Material With Mifitting Incluion, Acta Metall., 2, pp Mura, T., 987, Micromechanic of Defect in olid, 2nd ed., Martinu Nijhoff, Dordrecht, pp Taya, M., and Chou, T. W., 98, On Two Kind of Ellipoidal Inhomogeneitie in an Infinite Elatic Body: An Application to a Hybrid Compoite, Int. J. olid truct., 36, pp Taya, M., and Mura, T., 98, On tiffne and trength of an Aligned hort- Fiber Reinforced Compoite Containing Fiber-End Crack Under Uniaxial Applied tre, AME J. Appl. Mech., 48, pp Arenault, R. J., and Taya, M., 987, Thermal Reidual tre in Metal Matrix Compoite, Acta Metall., 35, pp Journal of Applied Mechanic MARCH 2007, Vol. 74 / 297