A plate theory for nematic liquid crystalline solids
Introduction
Liquid crystalline (LC) solids are complex materials that combine the elasticity of polymeric solids with the self-organisation of liquid crystal structures (Finkelmann, Kock, Rehage, 1981, de Gennes, 1975). Due to their molecular architecture, consisting of cross-linked networks of polymeric chains containing liquid crystal mesogens, their deformations are typically large and nonlinear, and can arise spontaneously and reversibly under certain external stimuli (heat, light, solvents, electric or magnetic field) (Agrawal, Yun, Pesek, Chapman, Verduzco, 2014, Brömmel, Zou, Finkelmann, Hoffmann, 2013, Corbett, Warner, 2008, Davidson, Kotikian, Li, Aizenberg, Lewis, 2019, Finkelmann, Nishikawa, Pereita, Warner, 2001, Kaiser, Winkler, Krause, Finkelmann, Schmidt, 2009, Küpfer, Finkelmann, 1991, Küpfer, Finkelmann, 1994, Torras, Zinoviev, Marshall, Terentjev, Esteve, 2011, Urayama, Honda, Takigawa, 2005, Urayama, Honda, Takigawa, 2006, Wang, Tian, He, Cai, 2017, Ware, McConney, Wie, Tondiglia, White, 2015, Wei, Yu, 2012, Winkler, Kaiser, Krause, Finkelmann, Schmidt, 2010, Zentel, 1986, Zhao, Liu, 2020). These qualities suggest many avenues for technological applications, but more research efforts are needed before they can be exploited on an industrial scale (Camacho-Lopez, Finkelmann, Palffy-Muhoray, Shelley, 2004, DeSimone, Gidoni, Noselli, 2015, Ford, Ambulo, Kent, Markvicka, Pan, Malen, Ware, Majidi, 2019, Gelebart, Mulder, Varga, Konya, Vantomme, Meijer, RLB, Broer, 2017, Haghiashtiani, Habtour, Park, Gardea, McAlpine, 2018, Mori, Cukelj, Prévôt, Ustunel, Story, Gao, Diabre, McDonough, Johnson Freeman, Hegmann, Clements, 2020, van Oosten, Harris, CWM, Broer, 2007, Prévôt, Andro, SLM, Ustunel, Zhu, Nikolov, Rafferty, Brannum, Kinsel, LTJ, Freeman, McDonough, Clements, Hegmann, 2018, Tian, Wang, Chen, Shao, Gao, Cai, 2018, Tottori, Zhang, Qiu, Krawczyk, Franco-Obregón, Nelson, 2012, Ula, Traugutt, Volpe, Patel, Yu, Yakacki, 2018, Wan, Jin, Trase, Zhao, Chen, 2018, Wang, He, Wang, Cai, 2019, Wie, Shankar, White, 2016).
For thin nematic bodies, large deformations have been studied extensively, both theoretically and in laboratory. Rectangular geometries were assumed in Aharoni et al. (2014), Cirak et al. (2014), circular discs were considered in Ahn et al. (2015), Sk et al. (2016), Konya et al. (2016), Kowalski et al. (2017), Modes et al. (2010), Modes et al. (2011), Modes and Warner (2011), Mostajeran (2015), Mostajeran et al. (2016), Pismen (2014), Warner and Mostajeran (2018), thin ribbons were treated theoretically in Agostiniani, DeSimone, 2017, Agostiniani, DeSimone, 2017, Agostiniani et al. (2017), Tomassetti and Varano (2017) and experimentally in Sawa, Urayama, Takigawa, Gimenez-Pinto, Mbanga, Ye, Selinger, Selingerb, 2013, Sawa, Ye, Urayama, Takigawa, Gimenez-Pinto, RLB, Selingerb, 2011, Uruyama (2013). Different molecular structures and compositions, and the complex morphing behaviour that can be achieved in liquid crystal polymer networks were reviewed in de Haan et al. (2014), Kuenstler and Hayward (2019), Modes and Warner (2016), Pang et al. (2019), Warner (2020), White and Broer (2015).
At the constitutive level, for ideal monodomain nematic solids, where the mesogens are aligned throughout the material, a general formulation is usually provided by the phenomenological neoclassical strain-energy function proposed in Bladon et al. (1994), Warner et al. (1988), Warner and Wang (1991). This model is based on the molecular network theory of rubber elasticity (Treloar, 2005), where the parameters are directly measurable experimentally or derived from macroscopic shape changes (Warner, Terentjev, 1996, Warner, Terentjev, 2007). For nematic polydomains, where the mesogens are separated into many domains, such that in every domain, they that are aligned along a local director, in Biggins, Warner, 2009, Biggins, Warner, 2012, it is assumed that each domain has the same strain-energy density as a monodomain. Extensions to strain-energy functions for nematic elastomer plates with out-of-plane heterogeneities were proposed in Agostiniani, DeSimone, 2017, Agostiniani, DeSimone, 2017, Agostiniani et al. (2017). General continuum mechanical theories for nematic elastomers are provided in Anderson et al. (1999), Zhang et al. (2019).
In this study, we derive a reduced two-dimensional (2D) model describing the equilibrium of thin nematic solids made of a liquid crystalline material subject to optothermal stimuli and mechanical loads. We first define the strain-energy function for a solid nematic material where the director may or may not rotate freely relative to the elastic matrix (Section 2), then formulate the corresponding stress tensors similar to those from the finite elasticity theory (Section 3). Here, we take as reference configuration the isotropic phase at high temperature (Cirak, Long, Bhattacharya, Warner, 2014, DeSimone, 1999, DeSimone, Dolzmann, 2000, DeSimone, Dolzmann, 2002, DeSimone, Teresi, 2009) (inspired by the classical work of Flory (1961) on polymer elasticity), rather than the nematic phase in which the cross-linking was produced (Anderson, Carlson, Fried, 1999, Bladon, Terentjev, Warner, 1994, Verwey, Warner, Terentjev, 1996, Warner, Bladon, Terentjev, 1994, Warner, Gelling, Vilgis, 1988, Warner, Wang, 1991, Zhang, Xuan, Jiang, Huo, 2019). The relation between the trace formula of Bladon et al. (1994) (using as reference orientation the one corresponding to the cross-linking state) and the neo-Hookean-based strain-energy density defined in DeSimone (1999) (with a “virtual” isotropic state as reference configuration) is explained in DeSimone and Teresi (2009). Similar nematic strain-energy densities based on other classical hyperelastic models (e.g., Money-Rivlin, Ogden) are also discussed in DeSimone and Teresi (2009). In adopting the isotropic phase as reference configuration, we follow Cirak et al. (2014), where strain-energy functions with either free or frozen nematic director are defined, and the director has an initial direction which may be spatially varying. Our choice is phenomenologically motivated by the multiplicative decomposition of the deformation gradient from the reference configuration to the current configuration into an elastic distortion followed by a natural (stress free) shape change. This multiplicative decomposition is similar to those found in the constitutive theories of thermoelasticity, elastoplasticity, and growth (Goriely, 2017, Lubarda, 2004), but it is fundamentally different as the stress free geometrical change of liquid crystalline solids is superposed on the elastic deformation, which is applied directly to the reference state. Such difference is important since, although the elastic configuration obtained by this deformation may not be observed in practice, it may still be possible for the nematic body to assume such a configuration under suitable external stimuli. The elastic stresses can then be used to analyse the final deformation where the particular geometry also plays a role. However, in liquid crystalline materials, asymmetric Cauchy stresses generally occur, unlike in purely elastic materials (Warner and Terentjev, 2007, p. 80). We employ the method of asymptotic expansions, with the thickness of the body as the small parameter, and show that the leading term of the expansion is the solution of a system of equations of the Föppl-von Kármán-type (Föppl, 1907, von Kármán, 1910) (Section 5). A similar model for the elastic growth of thin biological plates was developed in Dervaux et al. (2009) (see also Dervaux and Ben Amar (2008)). For an initial application of the plate theory, we consider ‘spontaneous’ nonlinear deformations of annular circular discs (Section 6). We conclude with a summary of these results and further remarks (Section 7).
Section snippets
An ideal nematic solid
For an ideal nematic liquid crystalline (NLC) solid, the neoclassical strain-energy density function takes the generic form where F represents the deformation gradient from the isotropic state, n denotes the unit vector (or ‘director’) for the orientation of the nematic field, and W(A) is the strain-energy density of the isotropic polymer network, depending only on the (local) elastic deformation tensor A. The tensors F and A satisfy the relation (see Fig. 1) where G is the
Stress tensors
We restrict our attention to the case when W(A) in (1) describes an incompressible neo-Hookean material (Treloar, 1944), i.e., where “tr” denotes the trace operator, and μ > 0 represents the constant shear modulus at small strain.
Of particular significance are the left and right Cauchy-Green tensors defined, respectively, by
Using these deformation tensors, the elastic Almansi strain tensor is equal to Ogden (1997, pp. 90-91) and the elastic
The 3D equilibrium equations
We consider a solid nematic body characterised by the strain-energy function defined by (12), and occupying a compact domain such that the interior of the body is an open, bounded, connected set and its boundary is Lipschitz continuous (in particular, we assume that a unit normal vector N exists almost everywhere on ∂Ω). The elastic energy stored by the body is equal to Ogden (1997, p. 205) where enforces the condition that
The nematic plate model
Our goal is to devise a 2D model for a nematic solid which is sufficiently thin so that it can be approximated by a plate equation. In our derivation of the constitutive equations for the nematic plate model, we rely on the following conditions assumed a priori:
(P1) Surface normals to the plane of the plate remain perpendicular to the plate after deformation;
(P2) Changes in the thickness of the plate during deformation are negligible;
(P3) The stress field in the deformed plate is parallel to
Applications: Nematic rings
For illustration, we apply the plate equations to some classical problems of temperature-driven shape changes based on the disc geometry. We focus our attention on nematic circular annular discs, or rings, with the ‘frozen’ director uniformly distributed throughout the thickness and circularly symmetric around the centre. These rings can deform homogeneously through the thickness and inhomogeneously in the plane (Agrawal, Luchette, Palffy-Muhoray, Biswal, Chapman, Verduzco, 2012, Ahn, Liang,
Conclusion
We have developed here a Föppl-von Kármán-type model to describe the elastic behaviour of NLC plates, subject to combined optothermal stimulation and mechanical loading. We achieve this by exploiting the multiplicative decomposition of the deformation gradient into an elastic component and a ‘spontaneous’ deformation tensor. To illustrate the application of this model, we provide analytical solutions to combined natural and forced shape changes of circular rings with ‘frozen’ nematic director
CRediT authorship contribution statement
L. Angela Mihai: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing - original draft, Writing - review & editing. Alain Goriely: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing - original draft, Writing - review
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
We thank Fehmi Cirak (University of Cambridge) for a discussion on constitutive modelling of liquid crystalline solids. The support by the Engineering and Physical Sciences Research Council of Great Britain under research grants EP/R020205/1 to Alain Goriely and EP/S028870/1 to L. Angela Mihai is gratefully acknowledged.
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