A three-dimensional constitutive model for shape memory alloys. (English) Zbl 1271.74344

Summary: Shape memory alloys (SMAs) are materials that, among other characteristics, have the ability to present high deformation levels when subjected to mechanical loading, returning to their original form after a temperature change. Literature presents numerous constitutive models that describe the phenomenological features of the thermomechanical behavior of SMAs. The present paper introduces a novel three-dimensional constitutive model that describes the martensitic phase transformations within the scope of standard generalized materials. The model is capable of describing the main features of the thermomechanical behavior of SMAs by considering four macroscopic phases associated with austenitic phase and three variants of martensite. A numerical procedure is proposed to deal with the nonlinearities of the model. Numerical simulations are carried out dealing with uniaxial and multiaxial single-point tests showing the capability of the introduced model to describe the general behavior of SMAs. Specifically, uniaxial tests show pseudoelasticity, shape memory effect, phase transformation due to temperature change and internal subloops due to incomplete phase transformations. Concerning multiaxial tests, the pure shear stress and hydrostatic tests are discussed showing qualitatively coherent results. Moreover, other tensile-shear tests are conducted modeling the general three-dimensional behavior of SMAs. It is shown that the multiaxial results are qualitative coherent with the related data presented in the literature.


74M05 Control, switches and devices (“smart materials”) in solid mechanics
74A20 Theory of constitutive functions in solid mechanics
Full Text: DOI


[1] Aguiar, R.A.A., Savi, M.A., Pacheco, P.M.C.L.: Experimental and numerical investigations of shape memory alloy helical springs. Smart Mater. Struct. (2010). doi: 10.1088/0964-1726/19/2/025008
[2] Auricchio F., Reali A., Stefanelli U.: A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity. Int. J. Plast. 23, 207–226 (2007) · Zbl 1105.74031 · doi:10.1016/j.ijplas.2006.02.012
[3] Baêta-Neves A.P., Savi M.A., Pacheco P.M.C.L.: On the Fremond’s constitutive model for shape memory alloys. Mech. Res. Commun. 31(6), 677–688 (2004) · Zbl 1098.74519 · doi:10.1016/j.mechrescom.2004.06.007
[4] Brocca M., Brinson L.C., Bazant Z.P.: Three-dimensional constitutive model for shape memory alloys based on microplane model. J. Mech. Phys. Solids 50, 1051–1077 (2002) · Zbl 1027.74002 · doi:10.1016/S0022-5096(01)00112-0
[5] Fremond, M.: Shape memory alloy: a thermomechanical macroscopic theory. CISM Courses and Lectures, 351, New York (1996)
[6] Gall K., Sehitoglu H.: The role of texture in tension-compression asymmetry in polycrystalline Ni–Ti. Int. J. Plast. 15, 69–92 (1999) · Zbl 1056.74525 · doi:10.1016/S0749-6419(98)00060-6
[7] Grabe, C., Bruhns, O.T.: Tension/torsion tests of pseudoelastic, polycrystalline NiTi shape memory alloys under temperature control. Mater. Sci. Eng. A (2007). doi: 10.101111111116/j.msea.2007.03.117
[8] Jackson, C.M., Wagner, H.J., Wasilewski, R.J.: 55-Nitinol–the alloy with a memory: its physical metallurgy, properties, and applications. NASA-SP-5110 (1972)
[9] Kalamkarov A.L., Kolpakov A.G.: Analysis, Design and Optimization of Composite Structures. Wiley, Chichester (1997) · Zbl 0936.74002
[10] Lagoudas D.C.: Shape Memory Alloys: Modeling and Engineering Applications. Springer, New York (2008) · Zbl 1225.74005
[11] Lagoudas D.C., Entchev P.B., Popov P., Patoor E., Brinson L.C., Gao X.: Shape memory alloys, part II: modeling of polycrystals. Mech. Mater. 38, 430–462 (2006) · doi:10.1016/j.mechmat.2005.08.003
[12] Lemaitre J., Chaboche J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990) · Zbl 0743.73002
[13] Levitas V.I., Preston D.L., Lee D.-W.: Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations III. Alternative potentials, critical nuclei, kink solutions, and dislocation theory. Phys. Rev. B 68, 134201 (2003) · doi:10.1103/PhysRevB.68.134201
[14] Machado L.G., Savi M.A.: Medical applications of shape memory alloys. Braz. J. Med. Biol. Res. 36(6), 683–691 (2003)
[15] Manach P., Favier D.: Shear and tensile thermomechanical behavior of near equiatomic NiTi alloy. Mater. Sci. Eng. A 222, 45–47 (1997) · doi:10.1016/S0921-5093(96)10510-4
[16] Matsumoto O., Miyazaki S., Otsuka K., Tamura H.: Crystallography of martensitic-transformation in Ti–Ni single crystals. ACTA Metall. 35(8), 2137–2144 (1987) · doi:10.1016/0001-6160(87)90042-3
[17] McNaney J.M., Imbeni V., Jung Y., Papadopoulos P., Ritchie R.O.: An experimental study of the superelastic effect in a shape-memory Nitinol alloy under biaxial loading. Mech. Mater. 35, 969–986 (2007) · doi:10.1016/S0167-6636(02)00310-1
[18] Monteiro P.C.C. Jr, Savi M.A., Netto T.A., Pacheco P.M.C.L.: A phenomenological description of the thermomechanical coupling and the rate-dependent behavior of shape memory alloys. J. Intell. Mater. Syst. Struct. 20(14), 1675–1687 (2009) · doi:10.1177/1045389X09341199
[19] Ortiz M., Pinsky P.M., Taylor R.L.: Operator split methods for the numerical solution of the elastoplastic dynamic problem. Comput. Methods Appl. Mech. Eng. 39, 137–157 (1983) · doi:10.1016/0045-7825(83)90018-X
[20] Otsuka K., Ren X.: Recent developments in the research of shape memory alloys. Intermetallics 7, 511–528 (1999) · doi:10.1016/S0966-9795(98)00070-3
[21] Paiva, A., Savi, M.A.: An overview of constitutive models for shape memory alloys. Math. Probl. Eng. Article ID 56876, 1–30 (2006) · Zbl 1196.74151
[22] Paiva A., Savi M.A., Braga A.M.B., Pacheco P.M.C.L.: A constitutive model for shape memory alloys considering tensile-compressive asymmetry and plasticity. Int. J. Solids Struct. 42(11–12), 3439–3457 (2005) · Zbl 1127.74027 · doi:10.1016/j.ijsolstr.2004.11.006
[23] Panico M., Brinson L.C.: A three-dimensional phenomenological model for martensite reorientation in shape memory alloys. J. Mech. Phys. Solids 55(11), 2491–2511 (2007) · Zbl 1171.74037 · doi:10.1016/j.jmps.2007.03.010
[24] Patoor E., Lagoudas D.C., Entchev P.B., Brinson L.C., Gao X.: Shape memory alloys, part I: general properties and modeling of single crystals. Mech. Mater. 38, 391–429 (2006) · doi:10.1016/j.mechmat.2005.05.027
[25] Popov P., Lagoudas D.C.: A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite. Int. J. Plast. 23, 1679–1720 (2007) · Zbl 1127.74008 · doi:10.1016/j.ijplas.2007.03.011
[26] Rockafellar R.T.: Convex Analysis. Princeton Press, Princeton, New Jersey (1970) · Zbl 0193.18401
[27] Sittner P., Hara Y., Tokuda M.: Experimental-study on the thermoelastic martensitic-transformation in shape-memory alloy polycrystal induced by combined external forces. Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 26(11), 2923–2935 (1995) · doi:10.1007/BF02669649
[28] Savi M.A., Paiva A., Baêta-Neves A.P., Pacheco P.M.C.L.: Phenomenological modeling and numerical simulation of shape memory alloys: a thermo-plastic-phase transformation coupled model. J. Intell. Mater. Syst. Struct. 13(5), 261–273 (2002) · doi:10.1177/104538902761696742
[29] Savi M.A., Paiva A.: Describing internal subloops due to incomplete phase transformations in shape memory alloys. Arch. Appl. Mech. 74(9), 637–647 (2005) · Zbl 1119.74515 · doi:10.1007/s00419-005-0385-6
[30] Schroeder T.A., Wayman C.M.: The formation of martensite and the mechanism of the shape memory effect in single crystals of Cu–Zn alloys. ACTA Metall. 25, 1375 (1977) · doi:10.1016/0001-6160(77)90069-4
[31] Shaw J.A., Kyriades S.: Thermomechanical aspects of Ni–Ti. J. Mech. Phys. Solids 43(8), 1243–1281 (1995) · doi:10.1016/0022-5096(95)00024-D
[32] Souza A.C., Mamiya E., Zouain N.: Three-dimensional model for solids undergoing stress-induced phase transformations. Eur. J. Mech. A Solids 17, 789–806 (1998) · Zbl 0921.73024 · doi:10.1016/S0997-7538(98)80005-3
[33] Wang Y.F., Yue Z.F., Wang J.: Experimental and numerical study of the superelastic behaviour on NiTi thin-walled tube under biaxial loading. Comput. Mater. Sci. 40(2), 246–254 (2007) · doi:10.1016/j.commatsci.2006.12.004
[34] Zaki, W., Moumni, Z.: A three-dimensional model of the thermomechanical behavior of shape memory alloys. J. Mech. Phys. Solids (2007). doi: 10.1016/j.jmps.2007.03.012 · Zbl 1171.74032
[35] Zhang, X.D., Rogers, C.A., Liang, C.: Modeling of two-way shape memory effect. Smart Struct. Mater. ASME, pp. 79–90 (1991)
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