Skip to main content
SpringerLink
Log in

A Beam Finite Element for Nonlinear Analysis of Shape Memory Alloy Devices

  • Chapter
New Trends in Thin Structures: Formulation, Optimization and Coupled Problems

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 519))

Abstract

A large displacement finite rotation beam finite element formulation for shape memory alloy structural analysis is proposed. The Reissner-Mindlin beam model is considered in the total Lagrangian form. A reference configuration macroscopic constitutive model with internal variables is adopted for the evaluation of the stress components acting on the beam cross section. The computation of stress resultants and couples is performed iteratively using an algorithm that grants cross section equilibrium given material strain measures.

The Author acknowledges the ESF S3T EUROCORES Programme’ sMARTeR: Shape Memory Alloys to Regulate Transient Responses in civil engineering’ for partial financial support during a research stay in 2008 at the University of California, Berkeley, where he initiated this work.

The Author acknowledges the financial support of the ESF S3T EUROCORES Programme ‘SMARTeR: Shape Memory Alloys to Regulate Transient Responses in civil engineering’.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • R. Abraham, J.E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis and Applications. Ad dins on-Wesley, 1983.

    Google Scholar 

  • S.S. Antman. The theory of rod. In Handbuch der Physik. Springer, Berlin, 1972.

    Google Scholar 

  • S.S. Antman. Kirchhoff problem for nonlinearly elastic rods. Quart. J. Appl. Math., 23(3): 221–240, 1974.

    MathSciNet  Google Scholar 

  • S.S. Antman. Ordinary differential equations of linear eastciity i: Foundations ot the theories of non-linearly elastic rods and shells. Arch. Rat. MEch. Anal, 61(4): 307–351, 1976a.

    MATH  MathSciNet  Google Scholar 

  • S.S. Antman. Ordinary differential equations of linear eastciity ii: Existency and regularity theory for conservative boundary value problems. Arch. Rat. MEch. Anal, 61(2): 353–393, 1976b.

    MATH  MathSciNet  Google Scholar 

  • J.H. Argyris. An excursion into large rotations. Comput. Meth. Appl. Mech. Engrg., 32:85–155, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  • F. Auricchio and L. Petrini. Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations. Int. J. Num. Meth. Engrg., 55:1255–1284,2002.

    Article  MATH  MathSciNet  Google Scholar 

  • F. Auricchio and L. Petrini. A three-dimensional model describing stress-temperature induced solid phase transformations, part i: solution algo-rithm and boundary value problems. Int. J. Num. Meth. Engrg., 61: 807–8F, 2004a.

    Article  MATH  MathSciNet  Google Scholar 

  • F. Auricchio and L. Petrini. A three-dimensional model describing stress-temperature induced solid phase transformations, part ii: thermome-chanical coupling and hybrid composite applications. Int. J. Num. Meth. Engrg., 61:716–737, 2004b.

    Article  MATH  MathSciNet  Google Scholar 

  • F. Auricchio and R.L. Taylor. Two material models for cyclic plasticity: non-linear kinematic hardening and generalized plasticity. Int. J. Plasticity, 11:65–98, 1995.

    Article  MATH  Google Scholar 

  • F. Auricchio, P. Carotenuto, and A. Reali. On the geometrically exact beam model: A consistent, effective and simple derivation from three-dimensional finite-elasticity. International Journal of Solids and Structures, 45:4766–4781, 2008.

    Article  MATH  Google Scholar 

  • Y. Bellouard. Oshape memory alloys for microsystems: A review from a material research perspective. Materials Science and Engineering: A, 481–482:582–589, 2008.

    Article  Google Scholar 

  • M.L. Boubakar, S. Moyne, C. Lexcellent, and Ph. Boisse. SMA pseudoelastic finite strains. Theory and numerical applications. Journal of Engineering Materials and Technology, 121:44–47, 1999.

    Article  Google Scholar 

  • G. Bourbon, C. Lexcellent, and S. Leclercq. Modelling of the non isothermal cyclic behaviour of a polycristalline CuZnAl shape memory alloy. Journal de Physique, C8–5:221–226, 1995.

    Google Scholar 

  • C. Bouvet, S. Calloch, and C. Lexcellent. Mechanical Behavior of a Cu-Al-Be Shape Memory Alloy Under Multiaxial Proportional and Loadings. Transaction of ASME, 124:112–124,2002.

    Google Scholar 

  • A. Cardona and M. Geradin. A beam finite element non-linear theory with finite rotations. Int. J. Num. Meth. Engrg., 26:2403–2438, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  • D. Christ and S. Reese. Finite deformation pseudo-elasticity of shape memory alloys constitutive modelling and finite element implementation. International Journal of Solids and Structures, 46:3694–3709, 2009.

    Article  MATH  Google Scholar 

  • B.D. Coleman and M.E. Gurtin. J. Chem. Phys., 47:5977613, 1967.

    Article  Google Scholar 

  • M. Crisfield and G. Jelenic. Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite element implementation. Proceedings of the Royal Society of London. Series A Mathematical Physical and Engineering Sciences, 455:1125–1147, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  • M.A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures. John Wiley & Sons, Ltd, 1996.

    Google Scholar 

  • T.W. Duerig, K.N. Melton, and J.L. Proft. Wide hysteresis shape memory alloys. In T.W. Duerig, K.N. Melton, D. Stökel, and C.M. Wayman, editors, Engineering aspects of shape memory alloys, pages 130–136, 1990a.

    Google Scholar 

  • T.W. Duerig, K.N. Melton, D. Stökel, and C.M. Wayman, editors. Engineering aspects of shape memory alloys, 1990b. Butterworth-Heinemann.

    Google Scholar 

  • T.W. Duerig, D. Stökel, and A. Keeley. Actuator and work production devices. In T.W. Duerig, K.N. Melton, D. Stökel, and C.M. Wayman, editors, Engineering aspects of shape memory alloys, pages 181–194, 1990c.

    Google Scholar 

  • B. Fedelich and G. Zanzotto. One-dimensional quasistatic nonisothermal evolution of shape-memory material inside the hysteresis loop. Continuum Mechanics and Thermodynamic, 3:251–276, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Fremond. Shape memory alloy, a thermomechanical macroscopic theory. In CISM courses and lectures: shape memory alloys, volume 351, pages 1–68. Springer Wien New York, 1996.

    Google Scholar 

  • H. Funakubo. Shape memory alloys. Gordon and Breach Science Publishers, 1987. Translated from the Japanese by J.B. Kennedy.

    Google Scholar 

  • M. Geradin and A. Cardona. Flexible multibody dynamics: a finite element approach. John Wiley & Sons, Ltd, 2001.

    Google Scholar 

  • S. Govindjee and E.P. Kasper. Computational aspects of one-dimensional shape memory alloy modeling with phase diagrams. Computer Methods in Applied Mechanics and Engineering, 171:309–326, 1999.

    Article  MATH  Google Scholar 

  • A. Ibrahimbegovic. On the choice of finite rotation parameters. Comp. Meth. Appl. Mech. Engrg., 149:49–71, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Ibrahimbegovic, F. Frey, and I. Kozar. Computational aspects of vector-like parametrization of three-dimensional finite rotations. Int. J. Num. Meth. Engrg., 38:3653–3673, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Jelenic and M. Crisfield. Geometrically exact 3d beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comp. Meth. Appl. Mech. Engrg., 171:141–171, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Konya, M. Maxin, and K.C. Wright. New embolization coil containing a nitinol wire core: Preliminary in vitro and in vivo experiences. Journal of vascular and Interventional Radiology, 12:869–977, 2001.

    Article  Google Scholar 

  • G.V. Kurdjumov and G. Sachs. Z. Phys., 64:325, 1930.

    Article  Google Scholar 

  • D.C. Lagoudas and Z. Ding. Modeling of the thermoelectric heat transfer in shape memory alloy actuators: transient and multiple cycle solutions. International Journal of Engineering Science, 33:2345–2364, 1995.

    Article  MATH  Google Scholar 

  • D.C. Lagoudas, P.B. Entchev, P. Popov, E. Patoor, L.C. Brinson, and X. Gao. Shape memory alloys, part ii: Modeling of polycrystals. Mechanics of Materials, 38:430–462,2006.

    Article  Google Scholar 

  • C. Lexcellent and H. Tobushi. Internal loops in pseudoelastic behavior of Ti-Ni shape memory alloys: experiment and modelling. Meccanica, 30: 459–466,1995.

    Article  Google Scholar 

  • C. Liang and C.A. Rogers. Design of shape memory alloy actuators. Journal of Intelligent Material Systems and Structures, 8:303–313, 1997.

    Article  Google Scholar 

  • J. Lubliner. Plasticity theory. Macmillan, 1990.

    Google Scholar 

  • J. Lubliner and F. Auricchio. Generalized plasticity and shape-memory alloys. Int. J. Solids Structures, 33(7): 991–1003, 1996.

    Article  MATH  Google Scholar 

  • J. Makinen. Total lagrangian reissner’s geometrically exact beam element without singularities. Int. J. Num. Meth. Engrg., 70:1009–1048,2007.

    Article  MathSciNet  Google Scholar 

  • P.Y. Manach and D. Favier. Shear and tensile thermomechanical behavior of equiatomic NiTi alloy. Materials Science and Engineering, A222:45–57, 1997.

    Article  Google Scholar 

  • J.E. Marsden and T.J.R. Hughes. Mathematical Foundations of Elasticity. Prentice-Hall, 1983.

    Google Scholar 

  • J.E. Marsden and T.S. Ratiu. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer, 1999.

    Google Scholar 

  • P. Mata, S. Oller, and A.H. Barbat. Static analysis of beam structures under nonlinear gemetric and constitutive behavior. Computer Methods in Applied Mechanics and Engineering, 196:4458–4478, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  • N.B. Morgan. Medical shape memory alloy applicationsthe market and its products. Materials Science and Engineering A, 378:16–23, 2004.

    Article  Google Scholar 

  • Z. Moumni, W. Zaki, and Q.S. Nguyen. Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys. International Journal of Plasticity, 24:614–645,2008.

    Article  MATH  Google Scholar 

  • R.W. Ogden. Non-Linear Elastic Deformations. Ellis Horwood, 1984.

    Google Scholar 

  • L. Orgéas and D. Favier. Non-symmetric Tension-Compression Behavior of NiTi Alloy. Journal de Physique IV, C8:605–610,1995.

    Google Scholar 

  • K. Otsuka and K. Shimizu. Pseudoelasticity and shape memory effects in alloys. International Metals Reviews, 31:93–114, 1986.

    Google Scholar 

  • E. Patoor, A. Eberhardt, and M. Berveiller. Micromechanical modelling of superelasticity in shape memory alloys. Journal de Physique IV, Cl–6: 277–292, 1996.

    Google Scholar 

  • E. Patoor, D. Lagoudas, P.B. Entchev, L. C. Brinson, and X. Gao. Shape memory alloys, part i: General properties and modeling of single crystals. Mechanics of Materials, 38:391–429, 2006.

    Article  Google Scholar 

  • B. Peultier, T. Ben Zineb, and E. Patoor. Macroscopic constitutive law of shape memory alloy thermomechanical behaviour, application to structure computation by fern. Mechanics of Materials, 38:510–524, 2006.

    Article  Google Scholar 

  • S. Reese and D. Christ. Finite deformation pseudo-elasticity of shape memoryalloys constitutive modelling and finite element implementation. International Journal of Plasticity, 24:455–482, 2008.

    Article  MATH  Google Scholar 

  • S.M. Russel and A.R. Pelton, editors. Proceedings of the Third International Conference on Shape Memory and Superelastic Technologies, Asilomar. CA, 2000.

    Google Scholar 

  • S.M. Russel and A.R. Pelton, editors. Proceedings of the Fourth International Conference on Shape Memory and Superelastic Technologies, Asilomar, CA, 2003.

    Google Scholar 

  • J.C. Simo. A finite strain beam formulation, the three-dimensional dynamic problem, part i. Comp. Meth. Appl. Mech. Engrg., 49:55–70, 1985.

    Article  MATH  Google Scholar 

  • J.C. Simo. Topics on the numerical analysis and simulation of plasticity. In P.G. Ciarlet and J.L. Lions, editors, Handbook of numerical analysis, volume III. Elsevier Science Publisher B.V., 1999.

    Google Scholar 

  • J.C. Simo and T.J.R. Hughes. Computational inelasticity. Springer-Verlag, 1998.

    Google Scholar 

  • J.C. Simo and L. Vu-Quoc. A three-dimensional finite-strain rod model. part ii: computational aspects. Comp. Meth. Appl. Mech. Engrg., 58: 79–116, 1986.

    Article  MATH  Google Scholar 

  • J.C. Simo and L. Vu-Quoc. On the dynamics in space of rods undergoing large motions — a geometrically exact approach. Comp. Meth. Appl. Mech. Engrg., 66:125–161, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  • A.C. Souza, E.N. Mamiya, and N. Zouain. Three-dimensional model for solids undergoing stress-induced phase transformations. 17:789–806, 1998.

    MATH  Google Scholar 

  • R. L. Taylor. FEAP — A Finite Element Analysis Program, Ver. 8.2 User Manual, 2008. URL www.ce.berkeley.edu/feap.

    Google Scholar 

  • W. van Moorleghem, P. Besselink, and D. Aslanidis, editors. S MST-Europe 1999, Antwerp Zoo, Belgium, 1999. CD-ROM only.

    Google Scholar 

  • C.M. Wayman. Shape memory and related phenomena. Progress in Material science, 36:203–224, 1992.

    Article  Google Scholar 

  • P. Wollants, J.R. Roos, and L. Delaey. Thermally-and stressed-induced thermoelastic martensitic transformations in the reference frame of equilibrium thermodynamics. Progress in Materials Science, 37:227–288, 1993.

    Article  Google Scholar 

  • Wael Zaki and Ziad Moumni. A three-dimensional model of the thermo-mechanical behavior of shape memory alloys. Journal of the Mechanics and Physics of Solids, 55:2455–2490, 2007.

    Article  MATH  Google Scholar 

  • O.C. Zienkiewicz and R.L. Taylor. The finite element method: Its basis & fundamentals. Elsevier, 2005.

    Google Scholar 

  • O.C. Zienkiewicz and R.L. Taylor. The Finite element method, volume II. Butterworth-Heinemann, New York, fifth edition, 2000.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, University of Rome ‘Tor Vergata’, Rome, Italy

    Edoardo Artioli

  2. Department of Structural Mechanics, University of Pavia, Pavia, Italy

    Ferdinando Auricchio

  3. IMATI, Institute of Applied Mathematics and Information Technology, National Research Council, Pavia, Italy

    Edoardo Artioli & Ferdinando Auricchio

  4. Department of Civil and Environmental Engineering, University of California, Berkeley, CA, USA

    Robert L. Taylor

Authors
  1. Edoardo Artioli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ferdinando Auricchio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robert L. Taylor
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of São Paulo, São Paulo, Brazil

    Paulo De Mattos Pimenta

  2. Leibniz University of Hanover, Hanover, Germany

    Peter Wriggers

Rights and permissions

Reprints and permissions

Copyright information

© 2010 CISM, Udine

About this chapter

Cite this chapter

Artioli, E., Auricchio, F., Taylor, R.L. (2010). A Beam Finite Element for Nonlinear Analysis of Shape Memory Alloy Devices. In: De Mattos Pimenta, P., Wriggers, P. (eds) New Trends in Thin Structures: Formulation, Optimization and Coupled Problems. CISM International Centre for Mechanical Sciences, vol 519. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0231-2_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-0231-2_3

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-7091-0230-5

  • Online ISBN: 978-3-7091-0231-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation