An overview of constitutive models for shape memory alloys

Paiva, Alberto; Savi, Marcelo Amorim

doi:https://doi.org/10.1155/MPE/2006/56876

Mathematical Problems in Engineering

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 056876 | https://doi.org/10.1155/MPE/2006/56876

An overview of constitutive models for shape memory alloys

Alberto Paiva¹and Marcelo Amorim Savi¹

Received01 Sept 2004

Revised27 Sept 2005

Accepted05 Oct 2005

Published16 May 2006

Abstract

The remarkable properties of shape memory alloys have facilitated their applications in many areas of technology. The purpose of this paper is to present an overview of thermomechanical behavior of these alloys, discussing the main constitutive models for their mathematical description. Metallurgical features and engineering applications are addressed as an introduction. Afterwards, five phenomenological theories are presented. In general, these models capture the general thermomechanical behavior of shape memory alloys, characterized by pseudoelasticity, shape memory effect, phase transformation phenomenon due to temperature variation, and internal subloops due to incomplete phase transformations.

References

R. Abeyaratne, S. J. Kim, and J. K. Knowles, “A one-dimensional continuum model for shape-memory alloys,” International Journal of Solids and Structures, vol. 31, no. 16, pp. 2229–2249, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Abeyaratne, S. J. Kim, and J. K. Knowles, “Continuum modeling of shape memory alloys,” in Mechanics of Phase Transformations and Shape Memory Alloys, pp. 59–69, ASME, New York, 1994a.
View at: Google Scholar
M. Achenbach and I. A. Müller, “A model for shape memory,” Journal de Physique, vol. 12, no. 43, pp. 163–167, 1982.
View at: Google Scholar
W. S. Anders, C. A. Rogers, and C. R. Fuller, “Vibration and low-frequency acoustic analysis of piecewise-activated adaptive composite panels,” Journal of Composite Materials, vol. 26, pp. 103–120, 1992.
View at: Google Scholar
F. Auricchio and J. Lubliner, “A uniaxial model for shape memory alloys,” International Journal of Solids and Structures, vol. 34, no. 27, pp. 3601–3618, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
F. Auricchio and E. Sacco, “A one-dimensional model for superelastic shape memory alloys with different elastic properties between austenite and martensite,” International Journal of Non-Linear Mechanics, vol. 32, no. 6, pp. 1101–1114, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
F. Auricchio, R. L. Taylor, and J. Lubliner, “Shape-memory alloys: macromodeling and numerical simulations of the superelastic behavior,” Computer Methods in Applied Mechanics and Engineering, vol. 146, no. 3-4, pp. 281–312, 1997.
View at: Publisher Site | Google Scholar
A. P. Baêta-Neves, M. A. Savi, and P. M. C. L. Pacheco, “On the fremond's constitutive model for shape memory alloys,” Mechanics Research Communications, vol. 31, no. 6, pp. 677–688, 2004.
View at: Google Scholar
A. Bertran, “Thermo-mechanical constitutive equations for the description of shape memory effects in alloys,” Nuclear Engineering and Design, vol. 74, no. 2, pp. 173–182, 1982.
View at: Publisher Site | Google Scholar
V. Birman, “Review of mechanics of shape memory alloys structures,” Applied Mechanics Review, vol. 50, no. 11, pp. 629–645, 1997.
View at: Google Scholar
Z. H. Bo and D. C. Lagoudas, “Thermomechanical modeling of polycrystalline SMAs under cyclic loading. Part III: evolution of plastic strains and two-way shape memory effect,” International Journal of Engineering Science, vol. 37, no. 9, pp. 1175–1203, 1999.
View at: Publisher Site | Google Scholar
J. G. Boyd and D. C. Lagoudas, “A thermodynamic constitutive model for the shape memory materials. Part I: the monolithic shape memory alloys,” International Journal of Plasticity, vol. 12, no. 6, pp. 805–842, 1996.
View at: Publisher Site | Google Scholar
L. C. Brinson, “One dimensional constitutive behavior of shape memory alloys: themomechanical derivation with non-constant material functions and redefined martensite internal variable,” Journal of Intelligent Material Systems and Structures, no. 4, pp. 229–242, 1993.
View at: Google Scholar
S. B. Choi, Y. M. Han, J. H. Kim, and C. C. Cheong, “Force tracking of a flexible gripper featuring shape memory alloys actuators,” Mechatronics, vol. 11, no. 6, pp. 677–690, 2001.
View at: Publisher Site | Google Scholar
R. J. Comstock Jr., T. E. Buchheit, M. Somerday, and J. A. Wert, “Modeling the transformation stress of constrained shape memory alloys single crystals,” ACTA Materialia, vol. 44, no. 9, pp. 3505–3514, 1996.
View at: Publisher Site | Google Scholar
I. Dobovsek, “On formal structure of constitutive equations for materials exhibiting shape memory effects, shape memory materials,” Materials Science Forum, vol. 327, no. 3, pp. 359–362, 2000.
View at: Google Scholar
F. Falk, “Model free-energy, mechanics and thermodynamics of shape memory alloys,” ACTA Metallurgica, vol. 28, no. 12, pp. 1773–1780, 1980.
View at: Publisher Site | Google Scholar
F. Falk, “One-dimensional model of shape memory alloys,” Archives of Mechanics, vol. 35, no. 1, pp. 63–84, 1983.
View at: Google Scholar
F. Falk and P. Konopka, “Three-dimensional Landau theory describing the martensitic transformation of shape memory alloys,” Journal de Physique, no. 2, pp. 61–77, 1990.
View at: Google Scholar
F. D. Fischer, E. R. Oberaigner, K. Tanaka, and F. Nishimura, “Transformation induced plasticity revised: an updated formulation,” International Journal of Solids and Structures, vol. 35, no. 18, pp. 2209–2227, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
F. D. Fischer and K. Tanaka, “A micromechanical model for the kinetics of martensitic transformation,” International Journal of Solids and Structures, vol. 29, no. 14-15, pp. 1723–1728, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Fremond, “Matériaux à Mémoire de Forme,” Comptes Rendus Mathematique. Academie de Sciences. Paris, vol. 304, s.II, no. 7, pp. 239–244, 1987.
View at: Google Scholar
M. Fremond, “Shape memory alloy: a thermomechanical macroscopic theory,” CISM Courses and Lectures, no. 351, pp. 3–68, Springer, New York, 1996.
View at: Google Scholar
H. Funakubo, Shape Memory Alloys, Gordon & Bleach, New York, 1987.
K. Gall, H. Sehitoglu, R. Anderson, I. Karaman, Y. I. Chumlyakov, and I. V. Kireeva, “On the mechanical behavior of single crystal Ni-Ti shape memory alloys and related polycrystalline phenomenon,” Materials Science and Engineering, vol. A317, pp. 85–92, 2001.
View at: Google Scholar
K. Gall, H. Sehitoglu, Y. I. Chumlyakov, and I. V. Kireeva, “Tension-compression asymmetry of the stress-strain response in aged single crystal and polycrystalline NiTi,” ACTA Materialia, vol. 47, no. 4, pp. 1203–1217, 1999.
View at: Publisher Site | Google Scholar
B. C. Goo and C. Lexcellent, “Micromechanics-based modeling of two-way memory effect of a single-crystalline shape-memory alloy,” ACTA Materialia, vol. 45, no. 2, pp. 727–737, 1997.
View at: Publisher Site | Google Scholar
S. Govindjee and G. J. Hall, “A computational model for shape memory alloys,” International Journal of Solids and Structures, vol. 37, no. 5, pp. 735–760, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Govindjee and E. P. Kasper, “A shape memory alloy model for uranium-niobium accounting for plasticity,” Journal of Intelligent Material Systems and Structures, vol. 8, pp. 815–826, 1997.
View at: Google Scholar
D. A. Hebda and S. R. White, “Effect of training conditions and extended thermal cycling on nitinol two-way shape memory behavior,” Smart Materials and Structures, vol. 4, no. 4, pp. 298–304, 1995.
View at: Publisher Site | Google Scholar
D. E. Hodgson and J. W. Brown, Using Nitinol Alloys, Shape Memory Applications, California, 2000.
Y. Ivshin and T. J. Pence, “A constitutive model for hysteretic phase transition behavior,” International Journal of Engineering Science, vol. 32, no. 4, pp. 681–704, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Ivshin and T. J. Pence, “A thermomechanical model for a one variant shape memory material,” Journal of Intelligent Material Systems and Structures, no. 5, pp. 455–473, 1994b.
View at: Google Scholar
P. Kloucek, D. R. Reynolds, and T. I. Seidman, “Computational modeling of vibration damping in SMA wires,” Continuum Mechanics and Thermodynamics, vol. 16, no. 5, pp. 495–514, 2004.
View at: Google Scholar | MathSciNet
P. K. Kumar, D. C. Lagoudas, and P. B. Entchev, “Thermomechanical characterization SMA actuators under cyclic loding,” in Proccedings of the ASME - International Mechanical Engineering Congress and R & D Expo, Washington, November 2003.
View at: Google Scholar
C. A. P. L. La Cava, E. P. Silva, L. G. Machado, P. M. C. L. Pacheco, and M. A. Savi, “Modeling of a shape memory preload device for bolted joints,” in Proceedings of National Congress of Mechanical Engineering (CONEM 2000 - ABCM), Rio de Janeiro, 2000.
View at: Google Scholar
S. Leclercq, G. Bourbon, and C. Lexcellent, “Plasticity like model of martensite phase transition in shape memory alloys,” Journal de Physique IV, vol. 5, pp. 513–518, 1995.
View at: Google Scholar
J. Lemaitre and J. L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990.
View at: Zentralblatt MATH
V. I. Levitas, A. V. Idesman, E. Stein, J. Spielfeld, and E. Hornbogen, “A simple micromechanical model for pseudoelastic behavior of CuZnAl alloy,” Journal of Intelligent Material Systems and Structures, no. 5, pp. 324–334, 1998.
View at: Google Scholar
C. Lexcellent, S. Leclercq, B. Gabry, and G. Bourbon, “The two-way shape memory effect of shape memory alloys: an experimental study and a phenomenological model,” International Journal of Plasticity, vol. 16, no. 10-11, pp. 1155–1168, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
C. Liang and C. A. Rogers, “One-dimensional thermomechanical constitutive relations for shape memory materials,” Journal of Intelligent Material Systems and Structures, no. 1, pp. 207–234, 1990.
View at: Google Scholar
T. J. Lim and D. L. McDowell, “Degradation of an Ni-Ti alloy during cyclic loading,” in Proceedings of 1994 North American Conference on Smart Structures and Materials, SPIE, vol. 2189, pp. 326–341, 1994.
View at: Google Scholar
Z. K. Lu and G. J. Weng, “A Self-consistent model for the stress-strain behavior of shape-memory alloy polycrystals,” ACTA Materialia, vol. 46, no. 15, pp. 5423–5433, 1998.
View at: Publisher Site | Google Scholar
L. G. Machado, Chaos in dynamical systems with shape memory and multi-degree of freedom, Master's thesis, 2002.
L. G. Machado and M. A. Savi, “Odontological applications of shape memory alloys,” Revista Brasileira de Odontologia, vol. 59, no. 5, pp. 302–306, 2002.
View at: Google Scholar
L. G. Machado and M. A. Savi, “Medical applications of shape memory alloys,” Brazilian Journal of Medical and Biological Research, vol. 36, no. 6, pp. 683–691, 2003.
View at: Publisher Site | Google Scholar
O. Matsumoto, S. Miyazaki, K. Otsuka, and H. Tamura, “Crystallography of martensitic transformation in Ti-Ni single crystals,” ACTA Metallurgica, vol. 35, no. 8, pp. 2137–2144, 1987.
View at: Publisher Site | Google Scholar
D. A. Miller and D. C. Lagoudas, “Thermomechanical characterization of NiTiCu and NiTi SMA actuators: influence of plastic strains,” Smart Materials and Structures, vol. 9, no. 5, pp. 640–652, 2000.
View at: Publisher Site | Google Scholar
I. Muller and S. Seelecke, “Thermodynamic aspects of shape memory alloys,” Mathematical and Computer Modelling, vol. 34, no. 12-13, pp. 1307–1355, 2001.
View at: Publisher Site | Google Scholar
F. Nishimura, N. Watanabe, and K. Tanaka, “Transformation lines in an Fe-based shape memory alloy under tensile and compressive stress states,” Materials Science and Engineering, vol. A22, pp. 134–142, 1996.
View at: Google Scholar
F. Nishimura, N. Watanabe, and K. Tanaka, “Stress-strain-temperature hysteresis and martensite start line in an Fe-based shape memory alloy,” Materials Science and Engineering. A-Structural Materials Properties Microstructure and Processing, vol. 238, no. 2, pp. 367–376, 1997.
View at: Google Scholar
Z. Nishiyama, Martensitic Transformation, Academic Press, New York, 1978.
M. Ortiz, P. M. Pinsky, and R. L. Taylor, “Operator split methods for the numerical solution of the elastoplastic dynamic problem,” Computer Methods of Applied Mechanics and Engineering, vol. 39, no. 2, pp. 137–157, 1983.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Otsuka and X. Ren, “Recent developments in the research of shape memory alloys,” Intermetallics, vol. 7, no. 5, pp. 511–528, 1999.
View at: Publisher Site | Google Scholar
P. M. C. L. Pacheco and M. A. Savi, “A non-explosive release device for aerospace applications using shape memory alloys,” in Proceedings of XIV Brazilian Mechanical Engineering Congress, Sao Paulo, December 1997.
View at: Google Scholar
A. Paiva, M. A. Savi, A. M. B. Braga, and P. M. C. L. Pacheco, “A constitutive model for shape memory alloys considering tensile-compressive asymmetry and plasticity,” International Journal of Solids and Structures, vol. 42, no. 11-12, pp. 3439–3457, 2005.
View at: Publisher Site | Google Scholar
J. Perkins, Shape Memory Effects in Alloys, Plenum Press, New York, 1975.
P. Prader and A. C. Kneissl, “Deformation behavior and two-way shape-memory effect of NiTi alloys,” Zeitschrift fur Metallkunde, vol. 88, pp. 410–415, 1997.
View at: Google Scholar
RAYCHEM—TYCO ELECTRONIC CORPORATION, captured on 04/29/2001, http://www.raychem.com.
O. K. Rediniotis, L. N. Wilson, D. C. Lagoudas, and M. M. Khan, “Development of a shape-memory-alloy actuated biomimetic hydrofoil,” Journal of Intelligent Material Systems and Structures, vol. 13, no. 1, pp. 35–49, 2002.
View at: Publisher Site | Google Scholar
R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, no. 28, Princeton University Press, New Jersey, 1970.
View at: Zentralblatt MATH | MathSciNet
C. A. Rogers, “Intelligent materials,” Scientific American, pp. 122–127, 1995.
View at: Google Scholar
C. A. Rogers, C. Liang, and C. R. Fuller, “Modeling of shape memory alloy hybrid composites for structural acoustic control,” Journal of Acoustic Society of America, vol. 89, no. 1, pp. 210–220, 1991.
View at: Publisher Site | Google Scholar
M. A. Savi and A. M. Braga, “Chaotic vibration of an oscillator with shape memory,” Journal of the Brazilian Society of Mechanical Science, vol. 15, no. 1, pp. 1–20, 1993.
View at: Google Scholar
M. A. Savi and A. Paiva, “Describing internal subloops due to incomplete phase transformations in shape memory alloys,” Archive of Applied Mechanics, vol. 74, no. 9, pp. 637–647, 2005.
View at: Publisher Site | Google Scholar
M. A. Savi, A. Paiva, A. P. Baêta-Neves, and P. M. C. L. Pacheco, “Phenomenological modeling and numerical simulation of shape memory alloys: a thermo-plastic-phase transformation coupled model,” Journal of Intelligent Material Systems and Structures, vol. 13, no. 5, pp. 261–273, 2002.
View at: Publisher Site | Google Scholar
J. A. Shaw and S. Kyriakides, “Thermomechanical aspects of NiTi,” Journal of the Mechanics and Physics of Solids, vol. 43, no. 8, pp. 1243–1281, 1995.
View at: Publisher Site | Google Scholar
E. P. Silva, Modelagem Mecânica De Transformações De Fase Induzidas Por Tensões Em Sólidos, Master's thesis, Department of Mechanical Engineering, University of Brasilia, Brasilia, 1995.
J. C. Simo and R. L. Taylor, “A return mapping algorithm for plane stress elastoplasticity,” International Journal for Numerical Methods in Engineering, vol. 22, no. 3, pp. 649–670, 1986.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
SINTEF, “Shape Memory Alloys in Oil Well Applications,” 1999.
View at: Google Scholar
P. Sittner and V. Novák, “Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals,” International Journal of Plasticity, vol. 16, no. 10-11, pp. 1243–1268, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
P. Sittner, V. Novák, and N. Zárubová, “Martensitic transformations in [001] Cu-Al-Zn-Mn single crystals,” ACTA Materialia, vol. 46, no. 4, pp. 1265–1281, 1998.
View at: Publisher Site | Google Scholar
P. Sittner, V. Novák, N. Zárubová, and V. Studnicka, “Stress state effect on martensitic structures in shape memory alloys,” Materials Science and Engineering, vol. A273-275, pp. 370–374, 1999.
View at: Google Scholar
A. C. Souza, E. Mamiya, and N. Zouain, “Three-dimensional model for solids undergoing stress-induced phase transformations,” European Journal of Mechanics - A Solids, vol. 17, no. 5, pp. 789–806, 1998.
View at: Google Scholar | Zentralblatt MATH
Q. P. Sun and K. C. Hwang, “Micromechanics modeling for the constitutive behavior of polycrystalline shape memory alloys—part II. Study of individual phenomenon,” Journal of the Mechanics and Physics of Solids, vol. 41, p. 19, 1993b.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Q. P. Sun and K. C. Hwang, “Micromechanics modeling for the constitutive behavior of polycrystalline shape memory alloys—part I. Derivation of general relations,” Journal of the Mechanics and Physics of Solids, vol. 41, no. 1, pp. 1–17, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Tanaka, “A thermomechanical sketch of shape memory effect: one-dimensional tensile behavior,” Materials Science Research International, vol. 18, p. 251, 1985.
View at: Google Scholar
K. Tanaka and S. Nagaki, “Thermomechanical description of materials with internal variables in the process of phase transitions,” Ingenieur—Archive, vol. 51, pp. 287–299, 1982.
View at: Google Scholar | Zentralblatt MATH
K. Tanaka, F. Nishimura, and H. Tobushi, “Phenomenological analysis on subloops in shape memory alloys due to incomplete transformations,” Journal of Intelligent Material Systems and Structures, vol. 5, pp. 387–493, 1994.
View at: Google Scholar
R. Vaidyanathan, H. J. Chiel, and R. D. Quinn, “A hydrostatic robot for marine applications,” Robotics and Autonomous Systems, vol. 30, no. 1-2, pp. 103–113, 2000.
View at: Publisher Site | Google Scholar
J. van Humbeeck, “Non-medical applications of shape memory alloys,” Materials Science and Engineering A, vol. 273-275, pp. 134–148, 1999.
View at: Publisher Site | Google Scholar
H. Warlimont, L. Delaey, R. V. Krishnan, and H. Tas, “Thermoelasticity, pseudoelasticity and the memory effects associated with martensitic transformations—part 3: thermodynamics and kinetics,” Journal of Materials Science, vol. 9, no. 9, pp. 1545–1555, 1974.
View at: Publisher Site | Google Scholar
R. J. Wasilevski, “On the nature of the martensitic transformation,” Mettalurgical Transactions, vol. 6A, pp. 1405–1418, 1975.
View at: Google Scholar
K. Williams, G. Chiu, and R. Bernhard, “Adaptive-passive absorbers using shape memory alloys,” Journal of Sound and Vibration, vol. 249, no. 5, pp. 835–848, 2002.
View at: Google Scholar
S. K. Wu and H. C. Lin, “Recent development of NiTi-based shape memory alloys in Twain,” Materials Chemistry and Physics, vol. 64, no. 2, pp. 81–92, 2000.
View at: Publisher Site | Google Scholar
S. Zhang and P. G. McCormick, “Thermodynamic analysis of shape memory phenomena—I. Effect of transformation plasticity on elastic strain energy,” ACTA Materialia, vol. 48, no. 12, pp. 3081–3089, 2000.
View at: Publisher Site | Google Scholar
S. Zhang and P. G. McCormick, “Thermodynamic analysis of shape memory phenomena—II. Modelling,” ACTA Materialia, vol. 48, no. 12, pp. 3091–3101, 2000.
View at: Publisher Site | Google Scholar
X. D. Zhang, C. A. Rogers, and C. Liang, “Modeling of two-way shape memory effect,” in Smart Structures and Material—ASME, pp. 79–90, New York, December 1991.
View at: Google Scholar

Copyright

Copyright © 2006 Alberto Paiva and Marcelo Amorim Savi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

2486

Downloads

3464

Citations