Quasiconvex envelopes of energies for nematic elastomers in the small strain regime and applications

Cesana, Pierluigi; DeSimone, Antonio

doi:10.1016/j.jmps.2011.01.007

Cited by 33 publications

(31 citation statements)

References 23 publications

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“…This has been shown by detailed x-ray experiments [8] and by numerical study [41,43], and is a result of the nonconvex energy of nematic LCEs [44]. The characteristic stress-strain response of Sm-A elastomers [12] also exhibits microstructure if the sample is clamped during stretching [45].…”

Section: Discussionmentioning

confidence: 85%

Negative Poisson's ratio and semisoft elasticity of smectic-Cliquid-crystal elastomers

Brown

Adams

2012

Phys. Rev. E

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Models of smectic-C liquid-crystal elastomers predict that they can display soft elasticity, in which the shape of the elastomer changes at no energy cost. The amplitude of the soft mode and the accompanying shears are dependent on the orientation of the layer normal and the director with respect to the stretch axis. We demonstrate that in some geometries the director is forced to rotate perpendicular to the stretch axis, causing lateral expansion of the sample-a negative Poisson's ratio. Current models do not include the effect of imperfections that must be present in the physical sample. We investigate the effect of a simple model of these imperfections on the soft modes in monodomain smectic-C elastomers in a variety of geometries. When stretching parallel to the layer normal (with imposed strain) the elastomer has a negative stiffness once the director starts to rotate. We show that this is a result of the negative Poisson's ratio in this geometry through a simple scalar model.

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Section: Discussionmentioning

confidence: 85%

Negative Poisson's ratio and semisoft elasticity of smectic-Cliquid-crystal elastomers

Brown

Adams

2012

Phys. Rev. E

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“…In the fourth example, we want to investigate the creation of so-called stripe domains in a mechanically stretched monodomain sample 8 . To this end, we consider a specimen of size 100 × 100 [nm 2 ] with homogeneously oriented order parameter and a modified set of material parameters (λ = 49 N/m 2 , µ = 1 N/m 2 ).…”

Section: Formation Of Stripe Domains Under Mechanical Tensionmentioning

confidence: 99%

“…The starting point of the simulation is based on an equilibrated monodomain microstructure with maximum eigenvectors pointing in vertical direction. Horizontal mechanical tension is imposed through the application of horizontal 8 For recent investigations related to the creation of microstructures and wrinkles in nematic sheets we refer to [44]. Onset of stripe domain pattern; c) and d) Formation of + 1 2 disclination traveling from right to left boundary; e) Microstructure after the disclination has traversed the sample.…”

Section: Formation Of Stripe Domains Under Mechanical Tensionmentioning

confidence: 99%

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An electro‐elastic phase‐field model for nematic liquid crystal elastomers based on Landau‐de‐Gennes theory

Keip

Nadgir

2017

GAMM-Mitteilungen

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The present contribution proposes a continuum-mechanical phase-field approach to model the electro-elastic behavior of nematic liquid crystal elastomers. The nemato-electro-elastic model is embedded into the fundamental Landau-de-Gennes theory for isotropic-nematic phase transition and employs a tensorial order parameter as the phase field. Nematic deformations are incorporated through a constitutive ansatz in terms of the order parameter. The phase-field formulation is implemented into the finite element method, so that the microstructure evolution of nematic elastomers under different boundary conditions can be studied. We show that the model is capable to capture characteristic features of nematic elastomers including mechanically and electrically induced microstructure evolution as well as the occurrence of stripe domains.

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“…In particular, it lends itself more easily to the exploration of model extensions such as, for example, accounting for the presence of electric or magnetic fields (see [8]) and including curvature elasticity terms typical of liquid crystals by simply adding new terms in the governing energy. Furthermore, the linear theory is simpler in many respects and, importantly, the resulting energy landscapes have an easier geometric structure (see also [7] and [9]). Therefore, rigorous mathematical results such as, for example, the explicit construction of the relaxed energies are more complete, leading to a great insight into the energetically optimal states of the material.…”

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confidence: 99%

Nematic Elastomers: Gamma-Limits for Large Bodies and Small Particles

Cesana¹

2011

SIAM J. Math. Anal.

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Abstract. We compute the large-body and the small-particle Gamma-limit of a family of energies for nematic elastomers. We work under the assumption of small deformations (linearized kinematics) and consider both compressible and incompressible materials. In the large-body asymptotics, even if we describe the local orientation of the liquid crystal molecules according to the model of perfect order (Frank theory), we prove that we obtain a fully biaxial nematic texture (that of the de Gennes theory) as a by-product of the relaxation phenomenon connected to Gamma-convergence. In the case of small particles, we show that formation of new microstructure is not possible, and we describe the map of minimizers of the Gamma-limit as the phase diagram of the mechanical model. 1. Introduction. Studying the microstructure of complex materials is one of the most interesting problems in modern applied mathematics and statistical mechanics. A paradigmatic case is represented by nematic liquid crystal elastomers (LCEs), a class of materials which associate a liquid crystalline texture composed of rigid rod-like molecules (nematic mesogens) with an elastic medium. To sketch the internal organization of this material, we recall that the backbone of the elastomer is constituted by long polymeric chains which are cross-linked to a substrate. Then rod-like molecules of a nematic liquid crystal are linked to the chains. As a result, the topology of the mesogens is fixed and a mechanical deformation can reorient locally the nematic molecules and modify the optical properties of the elastomer. Their interesting properties stem from the interaction between liquid crystalline order and the elastic response of the chains. A very relevant phenomenon observed in nematic elastomers is the large spontaneous deformation accompanying a temperature-induced phase transformation from the isotropic to the nematic state. This deformation can reach 400% with respect to the reference configuration. Moreover, LCEs can deform and bend under UV-light excitation or in the presence of electric or magnetic fields. These properties make them extremely interesting for applications in bioengineering and robotics (e.g., artificial muscles and crystalline).This paper is part of a series of articles concerning the analysis of functionals for LCEs in the scenario of the linearized elasticity. Our main reference on the mathematical theory of elasticity is [10]. We refer to [15], [28], and [29] for a physical and mathematical introduction to liquid crystals and liquid crystal elastomers. The general approach to modeling the nematic mesogens is to define an order tensor [15], [28]. 2354Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. Γ-LIMITS FOR LARGE BODIES AND SMALL PARTICLES 2355In brief, the information on the orientation and the degree of order of the nematic molecules is encoded in the eigenvectors and in the eigenvalues of the symmetric matrix Q, a state variable which can be defined in three different ways according to the three...

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Quasiconvex envelopes of energies for nematic elastomers in the small strain regime and applications

Cited by 33 publications

References 23 publications

Negative Poisson's ratio and semisoft elasticity of smectic-Cliquid-crystal elastomers

Negative Poisson's ratio and semisoft elasticity of smectic-Cliquid-crystal elastomers

An electro‐elastic phase‐field model for nematic liquid crystal elastomers based on Landau‐de‐Gennes theory

Nematic Elastomers: Gamma-Limits for Large Bodies and Small Particles

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