Skip to main content
SpringerLink
Log in

Dimension reduction via \(\Gamma \)-convergence for soft active materials

  • Active Behavior in Soft Matter and Mechanobiology
  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

We present a rigorous derivation of dimensionally reduced theories for thin sheets of nematic elastomers, in the finite bending regime. Focusing on the case of twist nematic texture, we obtain 2D and 1D models for wide and narrow ribbons exhibiting spontaneous flexure and torsion. We also discuss some variants to the case of twist nematic texture, which lead to 2D models with different target curvature tensors. In particular, we analyse cases where the nematic texture leads to zero or positive Gaussian target curvature, and the case of bilayers.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Agostiniani V, Blass T, Koumatos K (2015) From nonlinear to linearized elasticity via \(\Gamma \)-convergence: the case of multiwell energies satisfying weak coercivity conditions. Math. Models Methods Appl. Sci. 25(01):1–38

    Article  MATH  MathSciNet  Google Scholar 

  2. Agostiniani V, Dal Maso G, DeSimone A (2015) Attainment results for nematic elastomers. Proc. R. Soc. Edinb. Sect. A 145:669–701,8

    Article  MATH  MathSciNet  Google Scholar 

  3. Agostiniani V, DeSimone A Rigorous derivation of active plate models for thin sheets of nematic elastomers. http://arxiv.org/abs/1509.07003, submitted for publication

  4. Agostiniani V, DeSimone A (2011) \(\Gamma \)-convergence of energies for nematic elastomers in the small strain limit. Contin. Mech. Thermodyn. 23(3):257–274

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Agostiniani V, DeSimone A (2012) Ogden-type energies for nematic elastomers. Int. J. Non-linear Mech. 47(2):402–412

    Article  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Aharoni H, Sharon E, Kupferman R (2014) Geometry of thin nematic elastomer sheets. Phys. Rev. Lett. 113:257801

    Article  ADS  Google Scholar 

  8. Bartels S, Bonito A, Nochetto RH Bilayer plates: model reduction, \(\Gamma \)-convergent finite element approximation and discrete gradient flow. http://arxiv.org/abs/1506.03335

  9. Bhattacharya K, DeSimone A, Hane KF, James RD, Palmstrøm CJ (1999) Tents and tunnels on martensitic films. Mater Sci Eng A 273275:685–689

    Article  Google Scholar 

  10. Bladon P, Terentjev EM, Warner M (1993) Transitions and instabilities in liquid crystal elastomers. Phys Rev E 47:R3838–R3840

    Article  ADS  Google Scholar 

  11. Ciarlet PG (2006) An introduction to differential geometry with applications to elasticity. Springer, Berlin

    Google Scholar 

  12. Conti S, DeSimone A, Dolzmann G (2002) Semisoft elasticity and director reorientation in stretched sheets of nematic elastomers. Phys Rev E 66:061710

    Article  ADS  Google Scholar 

  13. Conti S, DeSimone A, Dolzmann G (2002) Soft elastic response of stretched sheets of nematic elastomers: a numerical study. J Mech Phys Solids 50(7):1431–1451

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. DeSimone A (1999) Energetics of fine domain structures. Ferroelectrics 222:275–284

    Article  Google Scholar 

  15. DeSimone A, Dolzmann G (2002) Macroscopic response of nematic elastomers via relaxation of a class of \({\rm SO}(3)\)-invariant energies. Arch Ration Mech Anal 161(3):181–204

    Article  MATH  MathSciNet  Google Scholar 

  16. Flory PJ (1953) Principles of polymer chemistry. Baker lectures 1948. Cornell University Press, Ithaca

  17. Freddi L, Hornung P, Mora MG, Paroni R (2016) A corrected Sadowsky functional for inextensible elastic ribbons. J Elast 123(2):125–136

    Article  MATH  MathSciNet  Google Scholar 

  18. Friesecke G, James RD, Müller S (2002) A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun Pure Appl Math 55(11):1461–1506

    Article  MATH  MathSciNet  Google Scholar 

  19. Lucantonio A, Tomassetti G, DeSimone A (2016) Large-strain poroelastic plate theory for polymer gels with applications to swelling-induced morphing of composite plates. Compos Part B Eng. doi:10.1016/j.compositesb.2016.09.063

    Google Scholar 

  20. Nardinocchi P, Puntel E (2017) Unexpected hardening effects in bilayered gel beams. Meccanica. doi:10.1007/s11012-017-0635-z

    MathSciNet  Google Scholar 

  21. Petersen P (2006) Riemannian geometry, volume 171 of Graduate Texts in Mathematics, 2nd edn. Springer, New York

    Google Scholar 

  22. Sawa Y, Urayama K, Takigawa T, DeSimone A, Teresi L (2010) Thermally driven giant bending of liquid crystal elastomer films with hybrid alignment. Macromolecules 43:4362–4369

    Article  ADS  Google Scholar 

  23. Sawa Y, Ye F, Urayama K, Takigawa T, Gimenez-Pinto V, Selinger RLB, Selinger JV (2011) Shape selection of twist-nematic-elastomer ribbons. PNAS 108(16):6364–6368

    Article  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Teresi L, Varano V (2013) Modeling helicoid to spiral-ribbon transitions of twist-nematic elastomers. Soft Matter 9:3081–3088

    Article  ADS  Google Scholar 

  26. Tomassetti G, Varano V (2017) Capturing the helical to spiral transitions in thin ribbons of nematic elastomers. Meccanica. doi:10.1007/s11012-017-0631-3

    MathSciNet  Google Scholar 

  27. Urayama K (2013) Switching shapes of nematic elastomers with various director configurations. React Funct Polym 73(7):885–890 (Challenges and Emerging Technologies in the Polymer Gels)

    Article  Google Scholar 

  28. Warner M, Terentjev EM (2003) Liquid crystal elastomers. Clarendon Press, Oxford

    MATH  Google Scholar 

Download references

Acknowledgements

We thank E. Sharon for valuable discussions. We gratefully acknowledge the support by the European Research Council through the ERC Advanced Grant 340685-MicroMotility.

Author information

Authors and Affiliations

  1. SISSA, Via Bonomea 265, 34136, Trieste, Italy

    Virginia Agostiniani & Antonio DeSimone

Authors
  1. Virginia Agostiniani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antonio DeSimone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Virginia Agostiniani.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Agostiniani, V., DeSimone, A. Dimension reduction via \(\Gamma \)-convergence for soft active materials. Meccanica 52, 3457–3470 (2017). https://doi.org/10.1007/s11012-017-0630-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-017-0630-4

Keywords

Search

Navigation