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A three-dimensional description of shape memory alloy thermomechanical behavior including plasticity

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Abstract

This paper introduces a novel three-dimensional constitutive model that describes the thermomechanical behavior of shape memory alloys (SMAs). The model is developed within the framework of continuum mechanics and the standard generalized materials. Four macrocospic phases are considered associated with austenite and three variants of martensite. Phase transformations can be induced by temperature, volumetric and deviatoric strains. The description of plasticity is also of concern assuming kinematic and isotropic hardening. Numerical simulations are carried out showing that the proposed model is able to capture the general thermomechanical behavior of SMAs for uniaxial and multiaxial tests, even in situations with complex thermomechanical loading paths.

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Abbreviations

B + :

Thermodynamic force related to the volume fraction due to positive detwinned martensite

B − :

Thermodynamic force related to the volume fraction due to negative detwinned martensite

B A :

Thermodynamic force related to the volume fraction due to austenite

\(E_{ijkl}\) :

Elastic tensor

\(E_{ijkl}^{A} , E_{ijkl}^{M}\) :

Elastic tensor for autenitic and martensitic phases

G :

Shear modulus

H :

Kinematic hardening modulus

\(H^{A} ,H^{M}\) :

Kinematic hardening modulus of austenite and martensite

\(I_{\varTheta }\) :

Indicator function associated with the convex set Θ

\(I_{\pi }\) :

Indicator function associated with the convex π

\(I_{\chi }\) :

Indicator function of the convex set χ

\(I_{f}\) :

Indicator function associated with yield surface (classical plasticity)

K :

Isotropic plastic modulus

\(L_{0}^{ + } ,L_{{}}^{ + }\) :

Parameters related to the critical stress due to positive detwinned martensite phase

\(L_{0}^{ - } ,L_{{}}^{ - }\) :

Parameters related to the critical stress due to negative detwinned martensite phase

\(L_{0}^{A} ,L_{{}}^{A}\) :

Parameters related to the critical stress due to the austenitic phase

\(q_{j}\) :

Heat flux

\(r_{ij}\) :

Second-order tensor defined from the loading history

T :

Temperature

T A :

Temperature above with the austenitic phase is stable

T M :

Temperature below with the martensitic phase is stable

T 0 :

Reference temperature in a stress-free state

T F :

Reference temperature to determine the yield strength at high temperatures

\(X_{ij}\) :

Thermodynamic force associated with the plastic strain tensor

Y :

Thermodynamic force associated with the isotropic hardening

\(Z_{ij}\) :

Thermodynamic force associated with the kinematic hardening tensor

\(\alpha\) :

Parameter that controls the stress–strain hysteresis loop height

\(\alpha_{ijkl}^{h}\) :

Fourth-order tensor that controls the stress–strain hysteresis loop width

\(\alpha_{N}^{h} ,\alpha_{s}^{h}\) :

Normal and shear components of the tensor \(\alpha_{ijkl}^{h}\)

\(\varLambda^{A} ,\varLambda^{M}\) :

Phase transformation temperature functions related to austenite and martensite

\(\varLambda ,\varLambda^{\aleph }\) :

Phase transformation temperature functions

\(\beta^{ + }\) :

Volume fraction related to the positive detwinned martensite

\(\beta^{ - }\) :

Volume fraction related to the negative detwinned martensite

\(\beta^{A}\) :

Volume fraction referring to austenite

\(\beta^{M}\) :

Volume fraction related to the twinned martensite

\(\beta_{s}^{ + } ,\beta_{s}^{ - }\) :

Volume fraction associated with the starting of phase transformation process

\(\varepsilon_{ij}\) :

Total strain tensor

\(\hat{\varepsilon }_{ij}\) :

Deviatoric strain tensor

\(\varepsilon_{ij}^{e}\) :

Elastic strain tensor

\(\varepsilon_{ij}^{p}\) :

Plastic strain tensor

\(\varepsilon_{ij}^{t}\) :

Phase transformation strain tensor

\(\vartheta\) :

Internal variable associated with isotropic hardening

\(\varsigma_{ij}\) :

Internal variable associated with kinematic hardening

\(\rho\) :

Density

\(\sigma_{ij}\) :

Stress tensor

\(\hat{\sigma }_{ij}\) :

Deviatoric stress tensor

\(\sigma^{Y}\) :

Yield stress

\(\sigma_{Y}^{A} ,\sigma_{Y}^{M}\) :

Yield stress of the austenitic and martensitic phases

\(\sigma_{A,i}^{Y} ,\sigma_{A,f}^{Y}\) :

Yield stress at temperature T A and T F

\(\gamma\) :

Plastic multiplier

\(\eta^{ + } ,\eta^{ - } ,\eta^{A}\) :

Parameter associated with the internal dissipation (positive detwinned martensite, negative detwinned martensite and austenite)

\(\eta^{I}\) :

Parameter that defines the coupling between the phase transformation and isotropic hardening

\(\eta_{ij}^{k}\) :

Parameter that defines the coupling between the phase transformation and kinematic hardening

\(\lambda ,\mu\) :

Lamé coefficients

\(\lambda^{A} ,\lambda^{M}\) :

Lamé coefficients referring to austenite and martensite

\(\mu^{A} ,\mu^{M}\) :

Lamé coefficients related to austenite and martensite

\(\varPhi\) :

Pseudo-potential of dissipation

\(\varPhi^{M}\) :

Mechanical part of the pseudo-potential of dissipation

\(\varPhi^{H}\) :

Mechanical part of the pseudo-potential of dissipation

\(\psi\) :

Helmholtz free energy density

\(\psi^{ + } ,\psi^{ - } ,\psi^{A} ,\psi^{M}\) :

Helmholtz free energy density of isolated phases (positive detwinned martensite, negative detwinned martensite, austenite and twinned martensite)

Γ :

Equivalent strain field

υ :

Poisson’s ratio

\(\upsilon^{A} ,\upsilon^{M}\) :

Poisson’s ratio associated with austenite and martensite

\(\varOmega_{ij}\) :

Tensor related to the thermal expansion coefficients

\(\varOmega_{ij}^{A} ,\varOmega_{ij}^{M}\) :

Tensor related to the thermal expansion coefficients of the austenite and martensite

τ :

Set related to subdifferential associated with the convex set π (tetrahedron)

\(\daleth\) :

Set related to subdifferential associated with the convex set χ

References

  1. Aguiar RAA, Savi MA, Pacheco PMCL (2010) Experimental and numerical investigations of shape memory alloy helical springs. Smart Mater Struct 19(2):025008

    Article  Google Scholar 

  2. Andani MT, Elahinia M (2014) A rate dependent tension–torsion constitutive model for superelastic nitinol under non-proportional loading; a departure from von Mises equivalency. Smart Mater Struct 23:015012

    Article  Google Scholar 

  3. Andani MT, Alipour A, Elahinia M (2013) Coupled rate-dependent superelastic behavior of shape memory alloy bars induced by combined axial-torsional loading: a semi-analytic modeling. J Intell Mater Syst Struct 24(16):1995–2007

    Article  Google Scholar 

  4. Arghavani J, Auricchio F, Naghdabadi R, Reali A, Sohrabpour S (2010) A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings. Int J Plast 26(7):976–991

    Article  MATH  Google Scholar 

  5. Auricchio F, Reali A, Stefanelli U (2007) A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity. Int J Plast 23:207–226

    Article  MATH  Google Scholar 

  6. Baêta-Neves AP, Savi MA, Pacheco PMCL (2004) On the Fremond’s constitutive model for shape memory alloys. Mech Res Commun 31:677–688

    Article  MATH  Google Scholar 

  7. Bandeira EL, Savi MA, Monteiro PCC Jr, Netto TA (2006) Finite element analysis of shape memory alloy adaptive trusses with geometrical nonlinearities. Arch Appl Mech 76(3–4):133–144

    Article  MATH  Google Scholar 

  8. Bessa WM, De Paula AS, Savi MA (2013) Adaptive fuzzy sliding mode control of smart structures. Eur Phys J Spec Top 222(7):1541–1551

    Article  Google Scholar 

  9. Boyd J, Lagoudas DC (1996) A thermodynamical constitutive model for shape memory materials, part I: the monolithic shape memory alloy. Int J Plast 12(6):805–842

    Article  MATH  Google Scholar 

  10. Brocca M, Brinson LC, Bazant ZP (2002) Three-dimensional constitutive model for shape memory alloys based on microplane model. J Mech Phys Solids 50:1051–1077

    Article  MATH  Google Scholar 

  11. Chapman C, Eshghinejad A, Elahinia M (2011) Torsional behavior of NiTi wires and tubes: modeling and experimentation. J Intell Mater Syst Struct 22:1239–1248

    Article  Google Scholar 

  12. Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal 13(1):167–178

    Article  MathSciNet  MATH  Google Scholar 

  13. Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. J Chem Phys 47(2):597–613

    Article  Google Scholar 

  14. De Paula AS, dos Santos MVS, Savi MA, Bessa WM (2014) Controlling a shape memory alloy two-bar truss using delayed feedback method. Int J Struct Stabil Dyn 14(8):1440032

    Article  MathSciNet  Google Scholar 

  15. De Paula AS, Savi MA, Lagoudas DC (2012) Nonlinear dynamics of a SMA large-scale space structure. J Braz Soc Mech Sci Eng XXXIV:401–412

    Article  Google Scholar 

  16. Eringen AC (1980) Mechanics of continua. Robert E. Krieger, New York

    MATH  Google Scholar 

  17. Fremond M (1996) Shape memory alloy: a thermomechanical macroscopic theory. CISM Courses and Lectures, New York, vol 351, pp 3–68

  18. Fischer FD, Reisner G, Werner E, Tanaka K, Cailletaud G, Antretter T (2000) A new view on transformation induced plasticity. Int J Plast 16:723–748

    Article  MATH  Google Scholar 

  19. Grabe C, Bruhns OT (2008) Tension–torsion tests of pseudoelastic, polycrystalline NiTi shape memory alloys under temperature control. Mater Sci Eng A 481:109–113

    Article  Google Scholar 

  20. Hartl DJ, Chatzigeorgiou G, Lagoudas DC (2010) Three-dimensional modeling and numerical analysis of rate-dependent irrecoverable deformation in shape memory alloys. Int J Plast 26(10):1485–1507

    Article  MATH  Google Scholar 

  21. Kalamkarov AL, Kolpakov AG (1997) Analysis, design and optimization of composite structures. Wiley, New York

    MATH  Google Scholar 

  22. Khalil W, Saint-Sulpice L, ArbabChirani S, Bouby C, Mikolajczak A, Ben Zineb T (2013) Experimental analysis of Fe-based shape memory alloy behavior under thermomechanical cyclic loading. Mech Mater 63:1–11

    Article  Google Scholar 

  23. Lagoudas DC (2008) Shape memory alloys: modeling and engineering applications. Springer, New York

    MATH  Google Scholar 

  24. Lagoudas DC, Hartl D, Chemisky Y, Machado LG, Popov P (2011) Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys. Int J Plast 32:155–183

    Google Scholar 

  25. Leblond JB, Devaux J, Devaux JC (1989) Mathematical Modeling of Transformation Plasticity in Steels I: Case of Ideal-Plastic Phases. Int J Plast 5(6):551–572

    Article  Google Scholar 

  26. Lemaitre J, Charboche JL (1990) Mechanics of solid materials. Cambridge University Press, Cambridge

    Book  Google Scholar 

  27. Machado LG, Savi MA (2003) Medical applications of shape memory alloys. Braz J Med Biol Res 36(6):302–306

    Article  Google Scholar 

  28. Manach P, Favier D (1997) Shear and tensile thermomechanical behavior of near equiatomic NiTi alloy. Mater Sci Eng A222:45–57

    Article  Google Scholar 

  29. McNaney JM, Imbeni V, Jung Y, Papadopoulos P, Ritchie RO (2007) An experimental study of the superelastic effect in a shape-memory nitinol alloy under biaxial loading. Mech Mater 35:969–986

    Article  Google Scholar 

  30. Mehrabi R, Kadkhodaei M (2013) 3D phenomenological constitutive modeling of shape memory alloys based on microplane theory. Smart Mater Struct 22(2):025017

    Article  Google Scholar 

  31. Mehrabi R, Kadkhodaei M, Andani MT, Elahinia M (2015) Microplane modeling of shape memory alloy tubes under tension, torsion, and proportional tension–torsion loading. J Intell Mater Syst Struct 26(2):144–155

    Article  Google Scholar 

  32. Monteiro PCC Jr, Savi MA, Netto TA, Pacheco PMCL (2009) 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

    Article  Google Scholar 

  33. Muller I, Xu H (1991) On the pseudo-elastic hysteresis. Acta Metall Mater 39(3):263–271

    Article  Google Scholar 

  34. Oliveira SA, Savi MA, Kalamkarov AL (2010) A three-dimensional constitutive model for shape memory alloys. Arch Appl Mech 80(10):1163–1175

    Article  MATH  Google Scholar 

  35. Oliveira SA, Savi MA, Santos IF (2014) Uncertainty analysis of a one-dimensional constitutive model for shape memory alloy thermomechanical description. Int J Appl Mech 6(6):1450067

    Article  Google Scholar 

  36. Ortiz M, Pinsky PM, Taylor RL (1983) Operator split methods for the numerical solution of the elastoplastic dynamic problem. Comput Methods Appl Mech Eng 39:137–157

    Article  MathSciNet  MATH  Google Scholar 

  37. Paiva A, Savi MA (2006) An overview of constitutive models for shape memory alloys. Math Probl Eng 2006:1–30. Article ID 56876

  38. Paiva A, Savi MA, Braga AMB, Pacheco PMCL (2005) A constitutive model for shape memory alloys considering tensile-compressive asymmetry and plasticity. Int J Solids Struct 42(11–12):3439–3457

    Article  MATH  Google Scholar 

  39. Panico M, Brinson LC (2007) A three-dimensional phenomenological model for martensite reorientation in shape memory alloy. J Mech Phys Solids 55(11):2491–2511

    Article  MathSciNet  MATH  Google Scholar 

  40. Popov P, Lagoudas DC (2007) A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite. Int J Plast 23:1679–1682

    Article  MATH  Google Scholar 

  41. Rockafellar RT (1970) Convex analysis. Princeton Press, New Jersey

    Book  MATH  Google Scholar 

  42. Saleeb AF, Dhakal B, Padula SA, Gaydosh DJ (2013) Calibration of a three-dimensional multimechanism shape memory alloy material model for the prediction of the cyclic attraction character in binary NiTi alloys. J Intell Mater Syst Struct 24(1):70–88

    Article  Google Scholar 

  43. Santos BC, Savi MA (2009) Nonlinear dynamics of a nonsmooth shape memory alloy oscillator. Chaos Solitons Fractals 40(1):197–209

    Article  Google Scholar 

  44. Savi MA, Braga AMB (1993) Chaotic vibration of oscillator with shape memory. J Braz Soc Mech Sci 15(1):1–20

    Google Scholar 

  45. Savi MA, Paiva A, Baeta-Neves AP, Pacheco PMCL (2002) Phenomenological modeling and numerical simulation of shape memory: a thermo-plastic-phase transformation coupled model. J Intell Mater Syst Struct 3(5):261–273

    Article  Google Scholar 

  46. Savi MA, Pacheco PMCL, Braga AMB (2002) Chaos in a shape memory two-bar truss. Int J Non-Linear Mech 37(8):1387–1395

    Article  MATH  Google Scholar 

  47. Savi MA, Paiva A (2005) Describing internal subloops due to incomplete phase transformations in shape memory alloys. Arch Appl Mech 74(9):637–647

    Article  MATH  Google Scholar 

  48. Savi MA, De Paula AS, Lagoudas DC (2011) Numerical investigation of an adaptive vibration absorber using shape memory alloys. J Intell Mater Syst Struct 22(1):67–80

    Article  Google Scholar 

  49. Savi MA, Nogueira JB (2010) Nonlinear dynamics and chaos in a pseudoelastic two-bar truss. Smart Mater Struct 19(11):1–11, (Article 1150222010)

    Article  Google Scholar 

  50. Savi MA (2015) Nonlinear dynamics and chaos of shape memory alloy systems. Int J Non-Linear Mech 70:2–19

    Article  Google Scholar 

  51. Schroeder TA, Wayman CM (1977) The formation of martensite and the mechanism of the shape memory effect in single crystals of Cu–Zn alloys. Acta Metall 25:1375, (Article ID 1150222010)

    Article  Google Scholar 

  52. Shaw JA, Kyriades S (1995) Thermomechanical aspects of Ni–Ti. J Mech Phys Solids 43(8):1243–1281

    Article  Google Scholar 

  53. Silva LC, Savi MA, Paiva A (2013) Nonlinear dynamics of a rotordynamic nonsmooth shape memory alloy system. J Sound Vib 332(3–4):608–621

    Article  Google Scholar 

  54. Simo JC, Hugles TJR (1998) Computational inelasticity. Springer, New York

    MATH  Google Scholar 

  55. Sitnikova E, Pavlovskaia E, Wiercigroch M, Savi MA (2010) Vibration reduction of the impact system by an SMA restraint: numerical studies. Int J Non-Linear Mech 45(9):837–849

    Article  Google Scholar 

  56. Sittner P, Heller L, Pilch J, Sedlak P, Frost M, Chemisky Y, Duval A, Piotrowski B, Ben Zineb T, Patoor E, Auricchio F, Morganti S, Reali A, Rio G, Favier D, Liu Y, Gibeau E, Lexcellent C, Boubakar L, Hartl D, Oehler S, Lagoudas DC, Van Humbeeck J (2009) Roundrobin SMA Modeling. In: ESOMAT 2009-8th European symposium on martensitic transformations. doi:10.1051/esomat/200908001

  57. Sittner P, Hara Y, Tokuda M (1995) Experimental study on the thermoelastic martensitic transformation in shape memory alloy polycrystal induced by combined external forces. Metall Mater Trans 26A:2923–2935

    Article  Google Scholar 

  58. Sittner P, Takakura M, Tokuda M (1997) Shape memory effects under combined forces. Mater Sci Eng A 234:216–219

    Article  Google Scholar 

  59. Souza AC, Mamiya EN, Zouain N (1998) Three-dimensional model for solids undergoing stress-induced phase transformations. Eur J Mech A Solids 17:189–806

    Article  MATH  Google Scholar 

  60. Tobushi H, Iwanaga N, Tanaka K, Hori T, Sawada T (1991) Deformation behavior of Ni–Ti shape memory alloy subjected to variable stress and temperature. Contin Mech Thermodyn 3:79–93

    Article  Google Scholar 

  61. Tokuda M, Ye M, Takakura M, Sittner P (1999) Thermomechanical behavior of shape memory alloy under complex loading conditions. Int J Plast 15(2):223–239

    Article  MATH  Google Scholar 

  62. Tokuda M, Ye M, Takakura M, Sittner P (1998) Calculation of mechanical behaviors of shape memory alloy under multi-axial loading conditions. Int J Mech Sci 40(2–3):227–235

    Article  MATH  Google Scholar 

  63. Wang YF, Yue ZF, Wang J (2007) Experimental and numerical study of the supereslastic behavior on NiTi thin-walled tube under biaxial loading. Comput Mater Sci 40(2):246–254

    Article  Google Scholar 

  64. Zaki W, Moumni Z (2007) A three-dimensional model of the thermomechanical behavior of shape memory alloys. J Mech Phys Solids. doi:10.1016/j.jmps.2007.03.012

    MATH  Google Scholar 

  65. Zhang XD, Rogers CA, Liang C (1991) Modeling of two-way shape memory effect. Smart Struct Mater ASME 24:79–90

    Google Scholar 

  66. Zhou B (2012) A macroscopic constitutive model of shape memory alloy considering plasticity. Mech Mater 48:71–81

    Article  Google Scholar 

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Acknowledgments

The authors would like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES and FAPERJ and through the INCT-EIE (National Institute of Science and Technology—Smart Structures in Engineering) the CNPq and FAPEMIG. The Air Force Office of Scientific Research (AFOSR) is also acknowledged.

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Authors and Affiliations

  1. Department of Mechanical Engineering, CEFET/RJ, Rio de Janeiro, RJ, 20.271.110, Brazil

    Sergio A. Oliveira

  2. Center for Nonlinear Mechanics, COPPE - Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, P.O. Box 68.503, Rio de Janeiro, RJ, 21.941.972, Brazil

    Marcelo A. Savi & Nestor Zouain

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  1. Sergio A. Oliveira
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Correspondence to Marcelo A. Savi.

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Oliveira, S.A., Savi, M.A. & Zouain, N. A three-dimensional description of shape memory alloy thermomechanical behavior including plasticity. J Braz. Soc. Mech. Sci. Eng. 38, 1451–1472 (2016). https://doi.org/10.1007/s40430-015-0476-4

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