One Dimensional Modeling of the Shape Memory Effect

Transcription

1 odeling and Numerical Simulation o aterial Science, 3, 3, Publihed Online October 3 ( One Dimenional odeling o the Shape emory Eect Belkacem eddour *, Hamma Zedira, Hamid Djebaili Department o echanical Engineering, Univerity o Batna, Batna, Algeria Laboratory o LaSPI A, Department o Science & Technology, Univerity Abba Laghrour, Khenchela, Algeria * amum66@gmail.com Received February 6, 3; revied arch 6, 3; accepted arch 4, 3 Copyright 3 Belkacem eddour et al. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. ABSTRACT Thi paper aim to build a contitutive model intended to decribe the thermomechanical behavior o hape memory alloy. Thi behavior preent many acet, among them we have conidered the imple way o hape memory, which i one o mot important propertie o hape memory alloy. Becaue o numerou tage o thi eect, the ubject wa divided into three independent part. For each part, we built the correponding thermodynamic potential and we deduced the contitutive equation. To make thi model workable, we have developed an algorithm. The imulation wa perormed uing the NiTi a hape memory alloy. Keyword: Shape; Strain; Detwinned artenite; Region. Introduction The known behavior o conventional material ha allowed their ue in many application but the dicovery o new propertie, coming rom a ingular behavior o material known a hape memory alloy, opened a way or other application rom medical to aeropace. Thi unuual behavior ha attracted a igniicant attention o cientit and reearcher. Thereore, variou model were propoed. Thee model are baed on thermodynamic law and ramework theorie a generalized tandard material. Halphen and Nguyen [] ued by Lexcellent and Licht [], Edelen ormalim [3] ued by Tanaka and Nagaki [4]. Thee model can be claiied a ollow: ) acrocopic model: Built o the thermomechanical behavior, they are generally impler in ormulation and geared more toward engineering application. Thu, the detailed phyic o phae tranormation are uually not rigorouly addreed. ) icromechanical model are baed on the micromechanic o a ingle crytal. Chu and Jame [5]; Jame et al., [6]; Lexcellent et al., [7]; Govindjee and Hall [8]; Berveiller et al. [9]; Patoor et al. []. Generally, thi cla o model wa derived rom more undamental thermodynamic principle and the hape train o dierent martenite variant are included. In thi paper, we ocu on the eect o imple way * Correponding author. hape memory which i the irt property dicovered. The adopted approach i baed on the haring o the tudy on three ollowing tep: ) Orientation o twinned artenite; ) Heating or autenite; 3) Cooling or detwinned artenite.. ethod.. Preentation o the Subject (Figure ) The thermomechanical cycle to the memory eect o a imple hape i deined the ollowing: ) Applying a mechanical load under Temperature T lower than (Temperature o tranormation tart o artenite): The material i deormed irt elatically, ollowed by an important deormation due to twinned artenite orientation. When the load i cancelled the deormation i not ully recovered, only elatic part i recovered. ) Heating to a temperature above A (Temperature o tranormation inih o Autenite): When the temperature reache A (Temperature o tranormation tart o Autenite) the deormation begin to recover. 3) Cooling to T : Autenite begin to tranorm to artenite and inally the material i at the origin o the cycle... Deinition o Region (Figure ) The tudy will concern three region which are plane (ε, Copyright 3 SciRe.

2 B. EDDOUR ET AL. 5 Figure. Thermomechanical cycle o way memory eect. Figure. Region o tudy. σ); (ε, T) and axe T: ) Plane (ε, σ): Region o mechanical loading; ) Plane (ε, T): Region o thermal loading (heating); 3) Axe T: Region o cooling. σ : Stre o orientation tart o twinned artenite; σ : Stre o orientation inih o twinned artenite; ε : aximum deormation o orientation..3. Contitutive Equation (Figure 3) In each cae, we conider an elementary volume V V V, where V p i volume o parent phae and o incipient phae. p V V i raction o incipient phae. V i volume.3.. Region Parent phae i twinned artenite and incipient phae i detwinned artenite Free energy o Gibb: G, T, E () BT C Figure 3. Repreentative elementary volume. E : Young modulu o artenite, B, C: Coeicient to be determined by tet, : Elatic energy, E : Energy o deormation due to tranormation, BT : Free energy o phae change, C : Energy o interaction between two phae. Auming that the diipation occur during proceing Clauiu inequality can be written a ollowing: Let u write the term: G d dt G th F which i the driving orce. The diipative orce can be written: di F a a () (3) The tranormation occur when thi condition i atiied: F It will reult: B T C a a (5) Let u ue the unction: th di F (4), T, BT C a a (6) To write the evolution o the raction we ue the ollowing expreion: d, T, d dt d dt dt T dt dt (7) Ca (8) Copyright 3 SciRe.

3 6 B. EDDOUR ET AL. C a (9) The coeicient a and C can be determined uing limit value o, T, when:, T, ; ; T T; (), T, ; ; T T; () It i denoted that parameter B i identiied uing peudoelaticity tet..3.. Region Parent phae i detwinned artenite and incipient phae i Autenite Following the ame approach: G, T, EA E () B T H, T, E EA B T H a b (3) Parameter H can alo be determined by peudoelaticity tet; the coeicient a and b are deined uing limit value o, T, when:, T, ; ; T A ; (4), T, ; ; T A ; (5) The evolution o the raction o artenite can be expreed by: BT T (6) H a.3.3. Region 3 Parent phae i Autenite and incipient phae i twinned artenite G3, T, B T H (7) 3, T, B T H a b 3 3 (8) a3, b3 : Parameter to be deined by tet uing ollowing condition: 3, T, ; T ; (9).4. Application o the odel Uing K.L. Ng. and al work [], where a tenile tet wa perormed on NiTi and the ollowing parameter were determined: B.48 Pa K ; C 4.33 Pa; a.678 Pa; a 8.3 Pa; b.5 Pa; H.5 Pa; a Pa; b3.5 Pa; E 844 Pa; EA 5775 Pa; 79 K; 4 K; 7 Pa; Pa; A 96 K; A 33 K; E.464 Temperature o orientation loading T = 3 K; mechanical load 3 Pa ; 3. Reult (Figure 4-6) Dicuion The reult obtained coincide well with the experimental Figure 4. Reult o tenile tet at T = 3 C (Tet o orientation) perormed to validate the model. 3, T, ; T ; () BT T () Figure 5. Repreentation o the hape memory eect in 3D H a3 pace. Copyright 3 SciRe.

4 B. EDDOUR ET AL. 7 Figure 6. Reult o imulation. data in the cae o orientation tet, in cae o heating to make martenite changing into autenite which occur with recovering previou deormation and inally changing o martenite into autenite during the cooling. It eem that our reult are conitent with practical data o the elected material (NiTi). In each cae we ee clearly that deormation i unction o tranormation o martenite. The cycle beginning rom lower temperature reache the inal temperature with recovering the deormation caued by applied mechanical load. 4. Concluion Thi contitutive model preented in thi paper wa built uing a imple ormalim. We have divided the work into three part, and each one ha it proper particularitie ater imulating the model. The reult appear in good agreement with experimental data. Thi model can be ued in engineering ield. REFERENCES [] B. Halphen and Q. S. Nguyen, Sur le atériaux Standard Generalié, Journal de écanique, Vol. 4, 975, pp [] C. Lexcellent and C. Licht, Some Remark on the odelling o the Thermomechanical Behavior o Shape emory Alloy, Journal de Phyique IV, Vol., No. C4, 99, pp. C4-35-C4-39. [3] D. C. Edelen, On the Characterization o Fluxe in Nonlinear Irreverible Thermodynamic, International Journal o Engineering Science, Vol., No. 5, 974, pp [4] K. Tanaka and S. Nagaki, A Thermomechanical Decription o aterial with Internal Variable in the Proce o Phae Tranormation, Ingenieur-Archiv, Vol. 5, No. 5, 98, pp Copyright 3 SciRe.

5 8 B. EDDOUR ET AL. [5] C. Chu and R. D. Jame, Analyi o icrotructure in Cu-4.%Al-3.9%Ni by Energy inimization, Proceeding o the ICOAT-95, Vol. 5, No. C8, 995, pp. C8-43-C8-49. [6] R. D. Jame, R. V. Kohn and T. W. Shield, odeling o Branched Needle icrotructure at the Edge o a artenite Laminate, Proceeding o the ICOAT-95, Vol. 5, No. C8, 995, pp. C8-53-C8-59. [7] C. Lexcellent, B. C. Goo, Q. P. Sun and J. Bernardint, Characterization, Thermomechanical Behaviour and icromechanical-baed Contitutive odel o Shape-emory Cu-Zn-Al Single Crytal, Acta aterialia, Vol. 44, No. 9, 996, pp [8] S. Govindjee and G. J. Hall, A Computational odel or Shape emory Alloy, International Journal o Solid and Structure, Vol. 37, No. 5,, pp [9]. Berveiller, E. Patoor and. Buion, Thermomechanical Contitutive Equation or Shape emory Alloy, Journal de Phyique IV, Vol., No. C4, 99, pp. C4-387-C [] E. Patoor, A. Eberhardt and. Berveiller, On icromechanic o Thermoelatic Phae Tranition, Proceeding o Platicity 93: The 4th International Sympoium on Platicity and It Application, Baltimore, 9-3 July 993. [] K. L. Ng and Q. P. Sun, Stre-Induced Phae Tranormation and Detwinning in NiTi Polycrytalline Shape emory Alloy Tube, echanic o aterial, Vol. 38, No. -, 6, pp Copyright 3 SciRe.