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Meccanica

, Volume 53, Issue 15, pp 3759–3777 | Cite as

A shearable and thickness stretchable finite strain beam model for soft structures

  • Liwen He
  • Jia Lou
  • Youheng Dong
  • Sritawat Kitipornchai
  • Jie Yang
Article
  • 110 Downloads

Abstract

Soft materials and structures have recently attracted lots of research interests as they provide paramount potential applications in diverse fields including soft robotics, wearable devices, stretchable electronics and biomedical engineering. In a previous work, an Euler–Bernoulli finite strain beam model with thickness stretching effect was proposed for soft thin structures subject to stiff constraint in the width direction. By extending that model to account for the transverse shear effect, a Timoshenko-type finite strain beam model within the plane-strain context is developed in the present work. With some kinematic hypotheses, the finite deformation of the beam is analyzed, constitutive equations are deduced from the theory of finite elasticity, and by employing the standard variational method, the equilibrium equations and associated boundary conditions are derived. In the limit of infinitesimal strain, the new model degenerates to the classical extensible and shearable elastica model. The corresponding incremental equilibrium equations and associated boundary conditions are also obtained. Based on the new beam model, analytical solutions are given for simple deformation modes, including uniaxial tension, simple shear, pure bending, and buckling under an axial load. Furthermore, numerical solution procedures and results are presented for cantilevered beams and simply supported beams with immovable ends. The results are also compared with the previously developed finite strain Euler–Bernoulli beam model to demonstrate the significance of transverse shear effect for soft beams with a small length-to-thickness ratio. The developed beam model will contribute to the design and analysis of soft robots and soft devices.

Keywords

Finite strain Soft materials Hyperelastic Bending-to-stretching transition Shearable 

Notes

Acknowledgements

The work described in this paper was fully supported by the Australian Research Council Grant under Discovery Project scheme (DP160101978). The authors are very grateful for these financial supports. Dr. He and Dr. Lou are also grateful for the support from National Natural Science Foundation of China (Grant Nos. 11602118 and 11602117) and also sponsored by K.C. Wong Magna Fund in Ningbo University.

References

  1. 1.
    Antman SS (1974) Kirchhoff’s problem for nonlinearly elastic rods. Q Appl Math 32:221–240MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Attard MM (2003) Finite strain—beam theory. IJSS 40:4563–4584zbMATHGoogle Scholar
  3. 3.
    Attard MM, Hunt GW (2008) Column buckling with shear deformations—a hyperelastic formulation. Int J Solids Struct 45:4322–4339CrossRefzbMATHGoogle Scholar
  4. 4.
    Attard MM, Kim M-Y (2010) Lateral buckling of beams with shear deformations—a hyperelastic formulation. Int J Solids Struct 47:2825–2840CrossRefzbMATHGoogle Scholar
  5. 5.
    Auricchio F, Carotenuto P, Reali A (2008) On the geometrically exact beam model: a consistent, effective and simple derivation from three-dimensional finite-elasticity. Int J Solids Struct 45:4766–4781CrossRefzbMATHGoogle Scholar
  6. 6.
    Beatty MF (1987) Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues—with examples. ApMRv 40:1699–1734ADSGoogle Scholar
  7. 7.
    Betsch P, Steinmann P (2002) Frame-indifferent beam finite elements based upon the geometrically exact beam theory. Int J Numer Meth Eng 54:1775–1788MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carrera E (2002) Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch Comput Methods Eng 9:87–140MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Carrera E, Giunta G, Petrolo M (2011) Beam structures: classical and advanced theories. Wiley, HobokenCrossRefzbMATHGoogle Scholar
  10. 10.
    Chan BQY, Low ZWK, Heng SJW, Chan SY, Owh C, Loh XJ (2016) Recent advances in shape memory soft materials for biomedical applications. ACS Appl Mater Interfaces 8:10070–10087CrossRefGoogle Scholar
  11. 11.
    Gaharwar AK, Peppas NA, Khademhosseini A (2014) Nanocomposite hydrogels for biomedical applications. Biotechnol Bioeng 111:441–453CrossRefGoogle Scholar
  12. 12.
    Ge Q, Qi HJ, Dunn ML (2013) Active materials by four-dimension printing. Appl Phys Lett 103:131901ADSCrossRefGoogle Scholar
  13. 13.
    Gladman AS, Matsumoto EA, Nuzzo RG, Mahadevan L, Lewis JA (2016) Biomimetic 4D printing. Nat Mater 15:413–418ADSCrossRefGoogle Scholar
  14. 14.
    He L, Lou J, Dong Y, Kitipornchai S, Yang J (2018) Variational modeling of plane-strain hyperelastic thin beams with thickness stretching effect. Acta Mech.  https://doi.org/10.1007/s00707-018-2258-4 CrossRefGoogle Scholar
  15. 15.
    He L, Lou J, Du J, Wang J (2017) Finite bending of a dielectric elastomer actuator and pre-stretch effects. IJMS 122:120–128Google Scholar
  16. 16.
    He L, Yan S, Li B, Zhao G, Chu J (2013) Adhesion model of side contact for an extensible elastic fiber. IJSS 50:2659–2666Google Scholar
  17. 17.
    Hong S, Sycks D, Chan HF, Lin S, Lopez GP, Guilak F, Leong KW, Zhao X (2015) 3D printing of highly stretchable and tough hydrogels into complex, cellularized structures. Adv Mater 27:4035–4040CrossRefGoogle Scholar
  18. 18.
    Ibrahimbegović A (1995) On finite element implementation of geometrically nonlinear Reissner’s beam theory: three-dimensional curved beam elements. Comput Methods Appl Mech Eng 122:11–26ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Irschik H, Gerstmayr J (2009) A continuum mechanics based derivation of Reissner’s large-displacement finite-strain beam theory: the case of plane deformations of originally straight Bernoulli-Euler beams. Acta Mech 206:1–21CrossRefzbMATHGoogle Scholar
  20. 20.
    Irschik H, Gerstmayr J (2011) A continuum-mechanics interpretation of Reissner’s non-linear shear-deformable beam theory. Math Comput Model Dyn Syst 17:19–29MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jeong J-W, Shin G, Park SI, Yu KJ, Xu L, Rogers JA (2015) Soft materials in neuroengineering for hard problems in neuroscience. Neuron 86:175–186CrossRefGoogle Scholar
  22. 22.
    Kempaiah R, Nie Z (2014) From nature to synthetic systems: shape transformation in soft materials. J Mater Chem B 2:2357–2368CrossRefGoogle Scholar
  23. 23.
    Khoo ZX, Teoh JEM, Liu Y, Chua CK, Yang S, An J, Leong KF, Yeong WY (2015) 3D printing of smart materials: a review on recent progresses in 4D printing. Virtual Phys Prototyp 10:103–122CrossRefGoogle Scholar
  24. 24.
    Kim S, Laschi C, Trimmer B (2013) Soft robotics: a bioinspired evolution in robotics. Trends Biotechnol 31:287–294CrossRefGoogle Scholar
  25. 25.
    Lu T, Huang J, Jordi C, Kovacs G, Huang R, Clarke DR, Suo Z (2012) Dielectric elastomer actuators under equal-biaxial forces, uniaxial forces, and uniaxial constraint of stiff fibers. Soft Matter 8:6167–6173ADSCrossRefGoogle Scholar
  26. 26.
    Lubbers LA, van Hecke M, Coulais C (2017) A nonlinear beam model to describe the postbuckling of wide neo-Hookean beams. J Mech Phys Solids 106:191–206ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Magnusson A, Ristinmaa M, Ljung C (2001) Behaviour of the extensible elastica solution. IJSS 38:8441–8457zbMATHGoogle Scholar
  28. 28.
    Majidi C (2014) Soft robotics: a perspective—current trends and prospects for the future. Soft Robot 1:5–11CrossRefGoogle Scholar
  29. 29.
    Mata P, Oller S, Barbat A (2007) Static analysis of beam structures under nonlinear geometric and constitutive behavior. Comput Methods Appl Mech Eng 196:4458–4478ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nachbagauer K, Pechstein AS, Irschik H, Gerstmayr J (2011) A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation. Multibody SysDyn 26:245–263CrossRefzbMATHGoogle Scholar
  31. 31.
    Ogden RW (1997) Non-linear elastic deformations. Courier Corporation, ChelmsfordGoogle Scholar
  32. 32.
    Reddy JN (2006) Theory and analysis of elastic plates and shells. CRC Press, Boca RatonGoogle Scholar
  33. 33.
    Reissner E (1972) On one-dimensional finite-strain beam theory: the plane problem. Zeitschrift für angewandte Mathematik und Physik: ZAMP 23:795–804ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Rogers JA, Someya T, Huang Y (2010) Materials and mechanics for stretchable electronics. Science 327:1603–1607ADSCrossRefGoogle Scholar
  35. 35.
    Rudykh S, Bhattacharya K (2012) Snap-through actuation of thick-wall electroactive balloons. Int J Non-Linear Mech 47:206–209CrossRefGoogle Scholar
  36. 36.
    Rus D, Tolley MT (2015) Design, fabrication and control of soft robots. Nature 521:467–475ADSCrossRefGoogle Scholar
  37. 37.
    Simo JC (1985) A finite strain beam formulation. The three-dimensional dynamic problem. Part I. CMAME 49:55–70ADSzbMATHGoogle Scholar
  38. 38.
    Simo JC, Vu-Quoc L (1991) A geometrically-exact rod model incorporating shear and torsion-warping deformation. Int J Solids Struct 27:371–393MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Son D, Lee J, Qiao S, Ghaffari R, Kim J, Lee JE, Song C, Kim SJ, Lee DJ, Jun SW (2014) Multifunctional wearable devices for diagnosis and therapy of movement disorders. Nat Nanotechnol 9:397–404ADSCrossRefGoogle Scholar
  40. 40.
    Song J (2015) Mechanics of stretchable electronics. Curr Opin Solid State Mater Sci 19:160–170ADSCrossRefGoogle Scholar
  41. 41.
    Suo Z (2012) Mechanics of stretchable electronics and soft machines. MRS Bull 37:218–225CrossRefGoogle Scholar
  42. 42.
    Ventola CL (2014) Medical applications for 3D printing: current and projected uses. Pharm Ther 39:704Google Scholar
  43. 43.
    Yükseler RF (2015) A theory for rubber-like rods. Int J Solids Struct 69:350–359CrossRefGoogle Scholar
  44. 44.
    Zeng W, Shu L, Li Q, Chen S, Wang F, Tao XM (2014) Fiber-based wearable electronics: a review of materials, fabrication, devices, and applications. Adv Mater 26:5310–5336CrossRefGoogle Scholar
  45. 45.
    Zhao X, Suo Z (2007) Method to analyze electromechanical stability of dielectric elastomers. Appl Phys Lett 91:061921ADSCrossRefGoogle Scholar
  46. 46.
    Zupan E, Saje M, Zupan D (2013) On a virtual work consistent three-dimensional Reissner–Simo beam formulation using the quaternion algebra. AcMec 224:1709–1729MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mechanics and Engineering ScienceNingbo UniversityNingboPeople’s Republic of China
  2. 2.School of Mechanics and EngineeringSouthwest Jiaotong UniversityChengduPeople’s Republic of China
  3. 3.School of Civil EngineeringUniversity of QueenslandBrisbaneAustralia
  4. 4.School of EngineeringRMIT UniversityBundooraAustralia

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