Skip to main content
SpringerLink
Log in

Universal Deformations of Incompressible Nonlinear Elasticity as Applied to Ideal Liquid Crystal Elastomers

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

Liquid crystal elastomers are rubber-like solids with liquid crystalline mesogens (stiff, rod-like molecules) incorporated either into the main chain or as a side chain of the polymer. These solids display a range of unusual thermo-mechanical properties as a result of the coupling between the entropic elasticity of rubber and the orientational phase transitions of liquid crystals. One of these intriguing properties is the soft behavior, where it is able to undergo significant deformations with almost no stress. While the phenomenon is well-known, it has largely been examined in the context of homogenous deformations. This paper investigates soft behavior in complex inhomogeneous deformations. We model these materials as hyperelastic, isotropic, incompressible solids and exploit the seminal work of Ericksen, who established the existence of non-trivial universal deformations, those that satisfy the equations of equilibrium in every hyperelastic, isotropic, incompressible solid. We study the inflation of spherical and cylindrical balloons, cavitation and bending.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. The closely related topic of “universal relations” is reviewed in [39].

  2. We also have \(p_{1}=1\) in the original Bladon-Terentjev-Warner model but we do not require it here.

  3. See [43, 51] for detailed studies using a model with non-ideality.

  4. Homogeneous deformations satisfy the equilibrium equation trivially.

  5. Ericksen identified the case \(d=0\), and the generalization was independently proposed by Kingbell and Shield [26] and Singh and Pipkin [41]. This family also shows that it is possible to have inhomogeneous deformations where the strain invariants are constants [21].

  6. with the terminology SMLMS denoting that beam goes through regions \(S\), \(M\), \(L\), \(M\), \(S\) from bottom (\(X_{1} =-W\)) to top (\(X_{1} = W\)), etc.

References

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

    Article  Google Scholar 

  2. Alexander, H.: Tensile instability of initially spherical balloons. Int. J. Eng. Sci. 9, 151–160 (1971)

    Article  Google Scholar 

  3. Ambulo, C.P., Burroughs, J.J., Boothby, J.M., Kim, H., Shankar, M.R., Ware, T.H.: Four-dimensional printing of liquid crystal elastomers. ACS Appl. Mater. Interfaces 9, 37332–37339 (2017)

    Article  Google Scholar 

  4. Baardink, G., Cesana, P.: Torsion of a liquid crystal elastomer cylinder. Working notes: private communication (2019)

  5. Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. A 306, 557–611 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Biggins, J.S., Terentjev, E.M., Warner, M.: Semisoft response of nematic elastomers to complex deformations. Phys. Rev. E 78, 041704 (2008)

    Article  Google Scholar 

  7. Biggins, J.S., Warner, M., Bhattacharya, K.: Supersoft elasticity in polydomain nematic elastomers. Phys. Rev. Lett. 103, 037802 (2009)

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Cesana, P., Plucinsky, P., Bhattacharya, K.: Effective behavior of nematic elastomer membranes. Arch. Ration. Mech. Anal. 218, 863–905 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Dacorogna, B.: Direct Methods in the Calculus of Variations. Springer, New York (2008)

    MATH  Google Scholar 

  13. de Gennes, P.-G.: Réflexions sur un type de polymères nématiques. C. R. Acad. Sci., Sér. B 281, 101–103 (1975)

    Google Scholar 

  14. De Pascalis, R.: The semi-inverse method in solid mechanics: theoretical underpinnings and novel applications (2010)

  15. DeSimone, A., Dolzmann, G.: Material instabilities in nematic elastomers. Physica D 136, 175–191 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Ericksen, J.L.: Deformations possible in every isotropic, incompressible, perfectly elastic body. Z. Angew. Math. Phys. 5, 466–489 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fan, W., Wang, Z., Cai, S.: Rupture of polydomain and monodomain liquid crystal elastomer. Int. J. Appl. Mech. 08, 1640001 (2016)

    Article  Google Scholar 

  19. Farre-Kaga, H.J., Saed, M.O., Terentjev, E.M.: Dynamic pressure sensitive adhesion in nematic phase of liquid crystal elastomers. Adv. Funct. Mater. 32(12), 2110190 (2022)

    Article  Google Scholar 

  20. Finkelmann, H., Kock, H.-J., Rehage, G.: Investigations on liquid crystalline polysiloxanes. 3. Liquid crystalline elastomers – a new type of liquid crystalline material. Macromol. Rapid Commun. 2, 317–322 (1981)

    Article  Google Scholar 

  21. Fosdick, R.: Modern Developments in the Mechanics of Continua, pp. 109–127 (1966)

    Google Scholar 

  22. Fridrikh, S.V., Terentjev, E.M.: Polydomain-monodomain transition in nematic elastomers. Phys. Rev. E 60(2), 1847–1857 (1999)

    Article  Google Scholar 

  23. Gent, A.N., Lindley, P.B.: Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A 249, 195–205 (1959)

    Article  Google Scholar 

  24. Giudici, A., Biggins, J.S.: Giant deformations and soft-inflation in LCE balloons. Europhys. Lett. 132, 36001 (2020)

    Article  Google Scholar 

  25. He, Q., Zheng, Y., Wang, Z., He, X., Cai, S.: Anomalous inflation of a nematic balloon. J. Mech. Phys. Solids 142, 104013 (2020)

    Article  MathSciNet  Google Scholar 

  26. Klingbeil, W.W., Shield, R.T.: On a class of solutions in plane finite elasticity. Z. Angew. Math. Phys. 17, 489–511 (1966)

    Article  MathSciNet  Google Scholar 

  27. Kundler, I., Finkelmam, H.: Strain-induced director reorientation in nematic liquid single crystal elastomers. Macromol. Rapid Commun. 16, 679–686 (1995)

    Article  Google Scholar 

  28. Küpfer, J., Finkelmann, H.: Nematic liquid single crystal elastomers. Macromol. Rapid Commun. 12, 717–726 (1991)

    Article  Google Scholar 

  29. Lanzoni, L., Tarantino, A.M.: The bending of beams in finite elasticity. J. Elast. 139, 91–121 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lee, V., Bhattacharya, K.: Actuation of cylindrical nematic elastomer balloons. J. Appl. Phys. 129, 114701 (2021)

    Article  Google Scholar 

  31. Mihai, L.A., Goriely, A.: Instabilities in liquid crystal elastomers. Mater. Res. Soc. Bull. 46, 784–794 (2021)

    Article  MATH  Google Scholar 

  32. Petridis, L., Terentjev, E.: Nematic-isotropic transition with quenched disorder. Phys. Rev. E 74, 051707 (2006)

    Article  MathSciNet  Google Scholar 

  33. Rivlin, R.S.: Large elastic deformations of isotropic materials. I. Fundamental concepts. Philos. Trans. R. Soc. Lond. A 240, 459–490 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rivlin, R.S.: Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure, homogeneous deformation. Philos. Trans. R. Soc. Lond. A 240, 491–508 (1948)

    Article  MathSciNet  Google Scholar 

  35. Rivlin, R.S.: Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Philos. Trans. R. Soc. Lond. A 845, 173–195 (1949)

    MathSciNet  MATH  Google Scholar 

  36. Rivlin, R.S., Rideal, E.K.: Large elastic deformations of isotropic materials. III. Some simple problems in cyclindrical polar co-ordinates. Philos. Trans. R. Soc. Lond. A 240, 509–525 (1948)

    Article  MATH  Google Scholar 

  37. Rivlin, R.S., Rideal, E.K.: Large elastic deformations of isotropic materials. V. The problem of flexure. Proc. R. Soc. Lond. A 195, 463–473 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rivlin, R.S., Rideal, E.K.: Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Philos. Trans. R. Soc. Lond. A 242, 173–195 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  39. Saccomandi, G.: Universal results in finite elasticity. In: Fu, Y.B., Ogden, R.W. (eds.) Nonlinear Elasticity: Theory and Applications. London Mathematical Society Lecture Note Series, pp. 97–134. Cambridge University Press, Cambridge (2001)

    Chapter  MATH  Google Scholar 

  40. Saed, M.O., Elmadih, W., Terentjev, A., Chronopoulos, D., Williamson, D., Terentjev, E.M.: Impact damping and vibration attenuation in nematic liquid crystal elastomers. Nat. Commun. 12, 6676 (2021)

    Article  Google Scholar 

  41. Singh, M., Pipkin, A.C.: Note on Ericksen’s problem. Z. Angew. Math. Phys. 16(5), 706–709 (1965)

    Article  Google Scholar 

  42. Tajbakhsh, A., Terentjev, E.: Spontaneous thermal expansion of nematic elastomers. Eur. Phys. J. B 6, 181–188 (2001)

    Google Scholar 

  43. Tokumoto, H., Zhou, H., Takebe, A., Kamitani, K., Kojio, K., Takahara, A., Bhattacharya, K., Urayama, K.: Probing the in-plane liquid-like behavior of liquid crystal elastomers. Sci. Adv. 7, abe9495 (2021)

    Article  Google Scholar 

  44. Treloar, L.R.G.: The Physics of Rubber Elasticity, 3rd edn. Oxford University Press, Oxford (2005)

    Google Scholar 

  45. Urayama, K., Kohmon, E., Kojima, M., Takigawa, T.: Polydomain–monodomain transition of randomly disordered nematic elastomers with different cross-linking histories. Macromolecules 42, 4084–4089 (2009)

    Article  Google Scholar 

  46. Ware, T.H., McConney, M.E., Wie, J.J., Tondiglia, V.P., White, T.J.: Voxelated liquid crystal elastomers. Science 347, 982–984 (2015)

    Article  Google Scholar 

  47. Warner, M., Terentjev, E.M.: Liquid Crystal Elastomers. Oxford University Press, Oxford (2003)

    Google Scholar 

  48. White, T.J., Broer, D.J.: Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers. Nat. Mater. 14, 1087–1098 (2015)

    Article  Google Scholar 

  49. Yakacki, C.M., Saed, M., Nair, D.P., Gong, T., Reed, S.M., Bowman, C.N.: Tailorable and programmable liquid-crystalline elastomers using a two-stage thiol-acrylate reaction. RSC Adv. 5, 18997–19001 (2015)

    Article  Google Scholar 

  50. Yavari, A.: Universal deformations in inhomogeneous isotropic nonlinear elastic solids. Proc. R. Soc. A, Math. Phys. Eng. Sci. 477, 20210547 (2021)

    MathSciNet  Google Scholar 

  51. Zhou, H., Bhattacharya, K.: Accelerated computational micromechanics and its applications to nematic elastomers. J. Mech. Phys. Solids 153, 104470 (2021)

    Article  Google Scholar 

  52. Zubov, L., Karyakin, M.: Nonlinear deformations of a cylindrical pipe with pre-stressed thin coatings. Math. Mech. Solids 27, 1703–1720 (2022)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful for the financial support of the US Air Force Office of Scientific Research through the MURI Grant No. FA9550-16-1-0566.

Author information

Authors and Affiliations

  1. Saint-Gobain Competency Research Laboratory, Northborough, MA, 01532, USA

    Victoria Lee

  2. California Institute of Technology, Pasadena, CA, 91125, USA

    Kaushik Bhattacharya

Authors
  1. Victoria Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kaushik Bhattacharya
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both authors were involved in all aspects of the work.

Corresponding author

Correspondence to Kaushik Bhattacharya.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Dedicated to the memory of Jerald L. Ericksen, an original thinker and inspiring teacher

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was conducted while Victoria Lee was affiliated with the California Institute of Technology.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, V., Bhattacharya, K. Universal Deformations of Incompressible Nonlinear Elasticity as Applied to Ideal Liquid Crystal Elastomers. J Elast (2023). https://doi.org/10.1007/s10659-023-10018-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10659-023-10018-9

Keywords

Mathematics Subject Classification

Search

Navigation