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Hemihelical local minimizers in prestrained elastic bi-strips

Abstract

We consider a double-layered prestrained elastic rod in the limit of vanishing cross section. For the resulting limit Kirchhoff rod model with intrinsic curvature, we prove a supercritical bifurcation result, rigorously showing the emergence of a branch of hemihelical local minimizers from the straight configuration, at a critical force and under clamping at both ends. As a consequence we obtain the existence of nontrivial local minimizers of the 3-d system.

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References

  1. 1.

    Agostiniani, V., DeSimone, A.: Rigorous derivation of active plate models for thin sheets of nematic elastomers. Math. Mech. Solids (2017). doi:10.1177/1081286517699991

    Google Scholar 

  2. 2.

    Agostiniani, V., DeSimone, A., Koumatos, K.: Shape programming for narrow ribbons of nematic elastomers. J. Elast. 127, 1–24 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics, vol. 34. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  4. 4.

    Bhattacharya, K., Lewicka, M., Schäffner, M.: Plates with incompatible prestrain. Arch. Ration. Mech. Anal. 221(1), 143–181 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Cicalese, M., Ruf, M., Solombrino, F.: On global and local minimizers of prestrained thin elastic rods. Cal. Var. Partial Differ. Equ. 56, 115 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Crandall, M.G., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Ration. Mech. Anal. 52, 161–180 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Kohn, R.V., O’Brien, E.: On the bending and twisting of rods with misfit. J. Elast. (2017). doi:10.1007/s10659-017-9635-4

    Google Scholar 

  9. 9.

    Kohn, R.V., Sternberg, P.: Local minimisers and singular perturbations. Proc. R. Soc. Edinb. Sect. A 111, 69–84 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Lestringant, C., Audoly, B.: Elastic rods with incompatible strain: Macroscopic versus microscopic buckling. J. Mech. Phys. Solids 103, 40–71 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Lewicka, M., Mahadevan, L., Pakzad, M.R.: The Föppl-von Karmann equations for plates with incompatible strains. Proc. R. Soc. A 467, 402–426 (2011)

    Article  MATH  Google Scholar 

  12. 12.

    Lewicka, M., Mahadevan, L., Pakzad, M.R.: Models for elastic shells with incompatible strains. Proc. R. Soc. A 470, 1471–2946 (2014)

    Article  Google Scholar 

  13. 13.

    Liu, J., Huang, J., Su, T., Bertoldi, K., Clark, D.R.: Structural transition from helices to hemihelices. PLoS ONE 9, e93183 (2014)

    Article  Google Scholar 

  14. 14.

    McMillen, T., Goriely, A.: Tendril perversion in intrinsically curved rods. J. Nonlinear Sci. 12(3), 241–281 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Schmidt, B.: Plate theory for stressed heterogeneous multilayers of finite bending energies. J. Math. Pures Appl. 88(1), 107–122 (2007)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Francesco Solombrino.

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Cicalese, M., Ruf, M. & Solombrino, F. Hemihelical local minimizers in prestrained elastic bi-strips. Z. Angew. Math. Phys. 68, 122 (2017). https://doi.org/10.1007/s00033-017-0870-0

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Mathematics Subject Classification

  • 34K18
  • 74K10
  • 49J45
  • 74B20

Keywords

  • Hemihelical minimizers
  • Prestrain
  • Bifurcation
  • Elasticity