We compute effective energies of thin bilayer structures composed of soft nematic elastic liquid crystals in various geometrical regimes and functional configurations. Our focus is on elastic foundations composed of an isotropic layer attached to a nematic substrate where order-strain interaction results in complex opto-mechanical instabilities activated via coupling through the common interface. Allowing out-of-plane displacements, we compute Gamma-limits for vanishing thickness which exhibit spontaneous stress relaxation and shape-morphing behaviour. This extends the plane strain modelling of Cesana and Leon Baldelli [Math. Models Methods Appl. Sci. (2018) 2863-2904], and shows the asymptotic emergence of fully coupled active macroscopic nematic foundations. Subsequently, we focus on actuation and compute asymptotic configurations of an active plate on nematic foundation interacting with an applied electric field. From the analytical standpoint, the presence of an electric field and its associated electrostatic work turns the total energy non-convex and non-coercive. We show that equilibrium solutions are min-max points of the system, that min-maximising sequences pass to the limit and, that the limit system can exert mechanical work under applied electric fields.
Keywords: Liquid crystals, nematic elastomers, linearized elasticity, bi-layers, Gamma-convergence
@article{COCV_2022__28_1_A36_0,
author = {Cesana, Pierluigi and Baldelli, Andr\'es A. Le\'on},
title = {Gamma-convergence results for nematic elastomer bilayers: relaxation and actuation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022029},
mrnumber = {4438712},
zbl = {1491.82021},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022029/}
}
TY - JOUR AU - Cesana, Pierluigi AU - Baldelli, Andrés A. León TI - Gamma-convergence results for nematic elastomer bilayers: relaxation and actuation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022029/ DO - 10.1051/cocv/2022029 LA - en ID - COCV_2022__28_1_A36_0 ER -
%0 Journal Article %A Cesana, Pierluigi %A Baldelli, Andrés A. León %T Gamma-convergence results for nematic elastomer bilayers: relaxation and actuation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022029/ %R 10.1051/cocv/2022029 %G en %F COCV_2022__28_1_A36_0
Cesana, Pierluigi; Baldelli, Andrés A. León. Gamma-convergence results for nematic elastomer bilayers: relaxation and actuation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 36. doi: 10.1051/cocv/2022029
[1] , and , Attainment results for nematic elastomers. Proc. Roy. Soc. Edinburgh A 145 (2015) 669–701. | MR | Zbl | DOI
[2] and , Gamma-convergence of energies for nematic elastomers in the small strain limit. Continu. Mech. Thermodyn. 23 (2011) 257–274. | MR | Zbl | DOI
[3] and , Dimension reduction for soft active materials via Gamma-convergence. Meccanica 52 (2017) 3457–3470. | MR | Zbl | DOI
[4] and , Rigorous derivation of active plate models for thin sheets of nematic elastomers. Math. Mech. Solids 25 (2020) 1804–1830. | MR | Zbl | DOI
[5] , and , Shape programming for narrow ribbons of nematic elastomers. J. Elasticity 127 (2017) 1–24. | MR | Zbl | DOI
[6] and , Photomechanical coupling in photoactive nematic elastomers. J. Mech. Phys. Solids 144 (2020) 104115. | MR | DOI
[7] , , , , , , , , , , , , , , , , , and , PETSc Users Manual. Tech. Rep. ANL-95/11 - Revision 3.8, Argonne National Laboratory (2017).
[8] , , and , Efficient management of parallelism in object oriented numerical software libraries, in , and (editors), Modern Software Tools in Scientific Computing. Birkhäuser Press (1997) 163–202. | Zbl | DOI
[9] and , Fine phase mixtures as minimizers of energy, in Analysis and Continuum Mechanics. Springer (1989), pp. 647–686.
[10] and , Frank energy for nematic elastomers: a nonlinear model. ESAIM: COCV 21 (2015) 372–377. | MR | Zbl | Numdam
[11] and , Wrinkles as the result of compressive stresses in an annular thin film. Commun. Pure Appl. Math. 67 (2014) 693–747. | MR | Zbl | DOI
[12] , Microstructure of martensite: why it forms and how it gives rise to the shape-memory effect. Oxford University Press (2003). | MR | Zbl | DOI
[13] , and , Transitions and instabilities in liquid crystal elastomers. Phys. Rev. E 47 (1993) R3838. | DOI
[14] , PhD Thesis (2009).
[15] , Relaxation of multiwell energies in linearized elasticity and applications to nematic elastomers. Arch. Ratl. Mech. Anal. 197 (2010) 903–923. | MR | Zbl | DOI
[16] , Nematic elastomers: Gamma-limits for large bodies and small particles. SIAM J. Math. Anal. 43 (2011) 2354–2383. | MR | Zbl | DOI
[17] , , , and , Exact constructions in the (non-linear) planar theory of elasticity: from elastic crystals to nematic elastomers. Arch. Ratl. Mech. Anal. 237 (2020) 383–445. | MR | Zbl | DOI
[18] and , Strain-order coupling in nematic elastomers: equilibrium configurations. Math. Models Methods Appl. Sci. 19 (2009) 601–630. | MR | Zbl | DOI
[19] and , Quasiconvex envelopes of energies for nematic elastomers in the small strain regime and applications. J. Mech. Phys. Solids 59 (2011) 787–803. | MR | Zbl | DOI
[20] and , Variational modelling of nematic elastomer foundations. Math. Models Methods Appl. Sci. 28 (2018) 2863–2904. | MR | Zbl | DOI
[21] , and , Effective behavior of nematic elastomer membranes. Arch. Ratl. Mech. Anal. 218 (2015) 1–43. | MR | Zbl | DOI
[22] , vol. 1 of Three-dimensional elasticity. Elsevier (1988). | Zbl
[23] , and , Semi-soft elasticity and director reorientation in stretched sheets of nematic elastomers. Phys. Rev. E 60 (2002) 61710-1-8.
[24] , and , Soft elastic response of stretched sheets of nematic elastomers: a numerical study. J. Mech. Phys. Solids 50 (2002) 1431–1451. | MR | Zbl | DOI
[25] , An introduction to -convergence, volume 8 of Progress in Nonlinear Differential Equations and their Applications. Springer Science+Business Media, LLC (1993). | MR | Zbl
[26] and , vol. 23 of The physics of liquid crystals, Clarendon Press, Oxford (1993). | DOI
[27] , Energy minimizers for large ferromagnetic bodies. Arch. Ratl. Mech. Anal. 125 (1993) 99–143. | MR | Zbl | DOI
[28] , Hysteresis and imperfection sensitivity in small ferromagnetic particles. Meccanica 30 (1995) 591–603. | MR | Zbl | DOI
[29] , Energetics of fine domain structures. Ferroelectrics 222 (1999) 275–284. | DOI
[30] and , Macroscopic response of nematic elastomers via relaxation of a class of -invariant energies. Arch. Ratl. Mech. Anal. 161 (2002) 181–204. | MR | Zbl | DOI
[31] , and , Liquid crystal elastomer strips as soft crawlers. J. Mech. Phy. Solids 84 (2015) 254–272. | MR | Zbl | DOI
[32] and , Elastic energies for nematic elastomers. Eur. Phys. J. E 29 (2009) 191–204. | DOI
[33] , Liquid crystals with variable degree of orientation. Arch. Ratl. Mech. Anal. 113 (1991) 97–120. | MR | Zbl | DOI
[34] , and , On the theory of liquid crystals. Discuss. Faraday Soc. 25 (1958) 19–28. | DOI
[35] , , , , , , , and , Reversible heat-induced microwrinkling of PEDOT: PSS nanofilm surface over a monodomain liquid crystal elastomer. Mol. Cryst. Liquid Crys. 572 (2013) 40–49. | DOI
[36] , , , and , A nonlinear beam model of photomotile structures. Proc. Natl. Acad. Sci. 117 (2020) 9762. | MR | DOI
[37] , , , , , and , Blueprinting photothermal shape-morphing of liquid crystal elastomers. Adv. Mater. 32 (2020) 2000609. | DOI
[38] , and , Automated solution of differential equations by the finite element method: The FEniCS book, vol. 84. Springer Science & Business Media (2012). | Zbl | MR | DOI
[39] , and , An extension of the Landau-Ginzburg-de Gennes theory for liquid crystals. Liquid Cryst. 2 (1987) 769–796. | DOI
[40] , , and , Patterning nonisometric origami in nematic elastomer sheets. Soft Matter 14 (2018) 3127–3134. | DOI
[41] , and , Programming complex shapes in thin nematic elastomer and glass sheets. Phys. Rev. E 94 (2016) 010701. | DOI
[42] , , , , and , Hierarchical wrinkling patterns. Soft Matter 6 (2010) 5751–5756. | DOI
[43] , Variational theories for liquid crystals, vol. 8. CRC Press (1995). | MR | Zbl
[44] and , Liquid Crystal Elastomers. Oxford University Press (2003). | DOI
[45] and , Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers. Nat. Mater. 14 (2015) 1087–1098. | DOI
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