Advertisement

Computational Mechanics

, Volume 54, Issue 5, pp 1315–1329 | Cite as

A new SMA shell element based on the corotational formulation

  • P. Bisegna
  • F. Caselli
  • S. Marfia
  • E. SaccoEmail author
Original Paper

Abstract

Aim of this paper is to develop a new shape memory alloy (SMA) facet-shell finite element accounting for material and geometric nonlinearities. A corotational formulation is exploited, able to filter out large rigid-body motions from the element transformation. Accordingly, a geometrically linear core-element is employed, along with a SMA constitutive model formulated in the small strain framework. In particular, in accordance with the formulation of the classical thin shell theory, a plane-stress SMA model accounting for the pseudo-elastic as well as the shape memory effect is adopted. The time integration of the evolutive equation is performed developing a step-by-step backward-Euler numerical procedure. A highly efficient implementation of the corotational machinery is used, endowed with a fully consistent tangent stiffness. Applications are carried out for assessing the performances of the developed computational procedure and to investigate on some interesting engineering examples. The numerical results show the effectiveness of the proposed shell element, whose simplicity makes it attractive for the design of new advanced SMA-based devices undergoing significant configuration changes during their operation.

Keywords

Shape memory alloy Shells  Large displacements and rotations Corotational formulation 

Notes

Acknowledgments

The financial supports of PRIN 2009, project “Multi-scale modelling of materials and structures” CUP n. H31J1100021001, and PRIN 2010-11, project “Advanced mechanical modeling of new materials and technologies for the solution of 2020 European challenges” CUP n. F11J12000210001 are gratefully acknowledged.

References

  1. 1.
    Mohd Jani J, Leary M, Subic A, Gibson M (2014) A review of shape memory alloy research, applications and opportunities. Mater Des 56:1078–1113. doi: 10.1016/j.matdes.2013.11.084 CrossRefGoogle Scholar
  2. 2.
    Lagoudas D (2008) Shape memory alloys: modeling and engineering applications. Springer, New YorkGoogle Scholar
  3. 3.
    Reese S, Christ D (2008) Finite deformation pseudo-elasticity of shape memory alloys—Constitutive modelling and finite element implementation. Int J Plast 24(3):455–482. doi: 10.1016/j.ijplas.2007.05.005
  4. 4.
    Evangelista V, Marfia S, Sacco E (2010) A 3D SMA constitutive model in the framework of finite strain. Int J Numer Methods Eng 81(6):761–785. doi: 10.1002/nme.2717
  5. 5.
    Arghavani J, Auricchio F, Naghdabadi R (2011) A finite strain kinematic hardening constitutive model based on Hencky strain: general framework, solution algorithm and application to shape memory alloys. Int J Plast 27(6):940–961. doi: 10.1016/j.ijplas.2010.10.006
  6. 6.
    Arghavani J, Auricchio F, Naghdabadi R, Reali A (2011) An improved, fully symmetric, finite-strain phenomenological constitutive model for shape memory alloys. Finite Elem Anal Des 47(2):166–174. doi: 10.1016/j.finel.2010.09.001
  7. 7.
    Teeriaho J-P (2013) An extension of a shape memory alloy model for large deformations based on an exactly integrable Eulerian rate formulation with changing elastic properties. Int J Plast 43:153–176. doi: 10.1016/j.ijplas.2012.11.009
  8. 8.
    Marfia S, Sacco E (2007) Analysis of SMA composite laminates using a multiscale modelling technique. Int J Numer Methods Eng 70(10):1182–1208. doi: 10.1002/nme.1916
  9. 9.
    Artioli E, Marfia S, Sacco E, Taylor RL (2012) A nonlinear plate finite element formulation for shape memory alloy applications. Int J Numer Methods Eng 89(10):1249–1271. doi: 10.1002/nme.3285 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hartl DJ, Lagoudas DC (2009) Constitutive modeling and structural analysis considering simultaneous phase transformation and plastic yield in shape memory alloys. Smart Mater Struct 18(10):104017. doi: 10.1088/0964-1726/18/10/104017
  11. 11.
    Boyd J, Lagoudas D (1996) A thermodynamic constitutive model for the shape memory alloy materials. Part I The monolithic shape memory alloy. Int J Plast 12(6):805–842. doi: 10.1016/S0749-6419(96)00030-7
  12. 12.
    Souza AC, Mamiya EN, Zouain N (1998) Three-dimensional model for solids undergoing stress-induced phase transformations. Eur J Mech A Solids 17(5):789–806. doi: 10.1016/S0997-7538(98)80005-3
  13. 13.
    Auricchio F, Petrini L (2004) A three dimensional model describing stress-temperature induced solid phase transformations: solution algorithm and boundary value problems. Int J Numer Methods Eng 61(6):807–836. doi: 10.1002/nme.1086
  14. 14.
    Evangelista V, Marfia S, Sacco E (2009) Phenomenological 3D and 1D consistent models for shape-memory alloy materials. Comput Mech 44(3):405–421. doi: 10.1007/s00466-009-0381-8
  15. 15.
    Nour-Omid B, Rankin CC (1991) Finite rotation analysis and consistent linearization using projectors. Comput Methods Appl Mech Eng 93(3):353–384. doi: 10.1016/0045-7825(91)90248-5 CrossRefzbMATHGoogle Scholar
  16. 16.
    Felippa CA, Haugen B (2005) A unified formulation of small-strain corotational finite elements: I. Theory. Comput Methods Appl Mech Eng 194(21–24):2285–2335. doi: 10.1016/j.cma.2004.07.035
  17. 17.
    Rankin CC (2006) Application of linear finite elements to finite strain using corotation. In: AIAA paper No. AIAA-2006-1751, 47th AIAA/ASME/ASCE/ASC structures, structural dynamics, and materials conference. Newport, Rhode IslandGoogle Scholar
  18. 18.
    Areias P, Garção J, Pires EB, Infante Barbosa J (2011) Exact corotational shell for finite strains and fracture. Comput Mech 48(4):385–406. doi: 10.1007/s00466-011-0588-3 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Caselli F, Bisegna P (2014) A corotational flat triangular element for large strain analysis of thin shells with application to soft biological tissues. Comput Mech. doi: 10.1007/s00466-014-1038-9 MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gal E, Levy R (2006) Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element. Arch Comput Methods Eng 13(3):331–388. doi: 10.1007/BF02736397 CrossRefzbMATHGoogle Scholar
  21. 21.
    Battini J-M, Pacoste C (2006) On the choice of the linear element for corotational triangular shells. Comput Methods Appl Mech Eng 195(44–47):6362–6377. doi: 10.1016/j.cma.2006.01.007
  22. 22.
    Alsafadie R, Battini J-M, Somja H, Hjiaj M (2011) Local formulation for elasto-plastic corotational thin-walled beams based on higher-order curvature terms. Finite Elem Anal Des 47(2):119–128. doi: 10.1016/j.finel.2010.08.006
  23. 23.
    Mostafa M, Sivaselvan MV, Felippa CA (2013) Reusing linear finite elements in material and geometrically nonlinear analysis—Application to plane stress problems. Finite Elem Anal Des 69:62–72. doi: 10.1016/j.finel.2013.02.002
  24. 24.
    Caselli F, Bisegna P (2013) Polar decomposition based corotational framework for triangular shell elements with distributed loads. Int J Numer Methods Eng 95(6):499–528. doi: 10.1002/nme.4528 MathSciNetCrossRefGoogle Scholar
  25. 25.
    Simo JC, Taylor RL (1986) A return mapping algorithm for plane stress elastoplasticity. Int J Numer Methods Eng 22(3):649–670. doi: 10.1002/nme.1620220310
  26. 26.
    de Souza Neto EA, Peric D, Owen DRJ (2008) Computational methods for plasticity: theory and applications. Wiley, ChichesterCrossRefGoogle Scholar
  27. 27.
    Spurrier RA (1978) Comment on “singularity-free extraction of a quaternion from a direction-cosine matrix”. J Spacecr Rockets 15(4):255–255. doi: 10.2514/3.57311 CrossRefGoogle Scholar
  28. 28.
    Souza AC, Mamiya EN, Zouain N (1998) Three-dimensional model for solids undergoing stress-induced phase transformations. Eur J Mech A Solids 17(5):789–806. doi: 10.1016/S0997-7538(98)80005-3 CrossRefzbMATHGoogle Scholar
  29. 29.
    Batoz J-L, Bathe K-J, Ho L-W (1980) A study of three-node triangular plate bending elements. Int J Numer Methods Eng 15(12):1771–1812. doi: 10.1002/nme.1620151205
  30. 30.
    Jeyachandrabose C, Kirkhope J, Ramesh Babu C (1985) An alternate explicit formulation for the DKT plate-bending element. Int J Numer Methods Eng 21(7):1289–1293. doi: 10.1002/nme.1620210709
  31. 31.
    Felippa CA (2003) A study of optimal membrane triangles with drilling freedoms. Comput Methods Appl Mech Eng 192(16–18):2125–2168. doi: 10.1016/S0045-7825(03)00253-6 CrossRefzbMATHGoogle Scholar
  32. 32.
    Merzouki T, Duval A, Ben Zineb T (2012) Finite element analysis of a shape memory alloy actuator for a micropump. Simul Model Pract Theory 27:112–126. doi: 10.1016/j.simpat.2012.05.006 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Civil Engineering and Computer ScienceUniversity of Rome “Tor Vergata”RomeItaly
  2. 2.Department of Civil and Mechanical EngineeringUniversity of Cassino and Southern LazioCassinoItaly

Personalised recommendations