## Abstract

Continuum models describing ideal nematic solids are widely used in theoretical studies of liquid crystal elastomers. However, experiments on nematic elastomers show a type of anisotropic response that is not predicted by the ideal models. Therefore, their description requires an additional term coupling elastic and nematic responses, to account for aeolotropic effects. In order to better understand the observed elastic response of liquid crystal elastomers, we analyse theoretically and computationally different stretch and shear deformations. We then compare the elastic moduli in the infinitesimal elastic strain limit obtained from the molecular dynamics simulations with the ones derived theoretically, and show that they are better explained by including nematic order effects within the continuum framework.

### 1. Introduction

Liquid crystalline solids are responsive multifunctional materials that combine the flexibility of polymeric networks with the nematic order of liquid crystals [1,2]. Owing to their molecular architecture, they can exhibit dramatic spontaneous deformations and phase transitions, which are reversible and repeatable under heat, light, solvents and electric or magnetic fields [3–12]. However, their physical behaviour under combined mechanical loading and external stimuli still needs to be fully elucidated.

For ideal monodomain nematic elastomers, with the mesogens uniaxially aligned throughout the material, a continuum model is given by the so-called neoclassical strain-energy density introduced in [13–16]. This is a phenomenological strain-energy function that extends the neo-Hookean model for rubber [17], where the parameters are derived from macroscopic shape changes at small strain or through statistical averaging at microscopic scale [18,19]. Since elastic stresses dominate over Frank elasticity induced by the distortion of mesogens alignment [20–22], Frank energy [23,24] is generally neglected. Extensions to polydomains where every domain has the same strain-energy density as a monodomain are provided in [25,26]. These descriptions have been generalized by employing other hyperelastic models, such as Mooney–Rivlin, Gent and Ogden, which better capture the nonlinear elastic behaviour at large strains [27–29] (see [30,31] for molecular interpretations of the Mooney–Rivlin and Gent strain energies in rubber elasticity). Further generalizations can be found in [32,33]. Numerical studies of liquid crystal elastomers (LCEs) are presented, for example, in [33–35], where the finite-element method is used, and in [36,37], where molecular Monte Carlo simulations are employed.

Usually, when the elastic properties of a material are investigated, uniaxial deformations, which are easier to reproduce experimentally, are examined first [38–41]. For a purely elastic isotropic material, the shear modulus is then inferred from a universal relation between elastic moduli from the classical theory. To study the elastic responses of nematic monodomains, uniaxial deformations were assumed in [38] (see also [42]), where it was found that, if only the elastic energy was considered, then the stretch moduli in the direction parallel to the director and in a perpendicular direction were equal. However, experiments clearly show an aeolotropic effect; namely, the stretch moduli depend on the direction in which they are measured. To capture this experimentally observed response of the material, the nematic energy [43] was then also taken into account. Experimental results for monodomains where the tensile load formed different angles with the initial nematic director were reported in [39,44]. In [45], measurements of five independent elastic constants derived from three uniaxial tests, with the director parallel, perpendicular or at an angle of 45° relative to the loading direction, respectively, were obtained for a nematic monodomain treated as a classical transversely isotropic material. However, for many complex materials, shear deformations can reveal important additional mechanical effects, which may not be observed or inferred from uniaxial tests. In particular, to assess LCE materials, shear deformations with the direction of shear either parallel or perpendicular to the nematic director need to be considered independently of uniaxial stretches [46–52]. For example, on the one hand, it was found experimentally in [38] that, if ${E}_{\parallel}$ and *E*_{⊥} are the stretch moduli in a direction parallel or perpendicular to the nematic director, respectively, then ${E}_{\parallel}/{E}_{\perp}>1$ at low temperature, ${E}_{\parallel}/{E}_{\perp}<1$ at high temperature and ${E}_{\parallel}/{E}_{\perp}=1$ at the transition point. On the other hand, if ${\mu}_{\parallel}$ and *μ*_{⊥} denote the shear moduli in a direction parallel or perpendicular to the nematic director, respectively, then it was reported in [47,48,51] that ${\mu}_{\parallel}/{\mu}_{\perp}\approx 1$ in the isotropic phase and ${\mu}_{\parallel}/{\mu}_{\perp}<1$ at temperatures below that for the nematic–isotropic phase transition. Therefore, from a symmetry point of view, monodomain LCEs are transversely isotropic materials with five independent elastic constants and the distinguished direction given by the nematic director. However, despite the constitutive symmetry about the direction given by the nematic field, the mechanical responses of LCEs differ from the known elastic behaviours in traditional transversely isotropic elastic materials where, typically, ${E}_{\parallel}>{E}_{\perp}$ and${\mu}_{\parallel}>{\mu}_{\perp}$ [53–56].

The aim of this study is to develop an explicit approach for the derivation of elastic moduli that captures the aeolotropy of liquid crystalline elastomers. This approach represents an extension of the general theoretical framework by which similar elastic moduli were obtained for hyperelastic materials [57]. In the case of nematic solids, these moduli include information about both the elasticity of the polymeric network and the mechanical responses of the liquid crystal molecules. In §2, we recall the neoclassical model for ideal nematic elastomers, with the isotropic phase at high temperature as the reference configuration [29,58–61], instead of the nematic phase at cross-linking [14–16,32,33,62,63]. Phenomenologically, this choice is motivated by the multiplicative decomposition of the effective deformation into an elastic distortion, followed by a natural stress-free shape change [64–67]. This multiplicative decomposition is similar to those found in the constitutive theories of thermoelasticity, elastoplasticity and growth [68,69] (see also [70,71]), but it is also different in the sense that the elastic deformation is directly applied to the reference state. The elastic stresses can then be used to study the final deformation where the stress-free geometrical change also plays a role [65]. In §3, we calculate the two *stretch moduli* under small elastic uniaxial tension and finite natural deformation when the nematic director is either parallel or perpendicular to the tensile direction, respectively. In §4, we further consider three shear deformations where the elastic component is a simple shear, while the nematic director is either parallel to the shear direction, perpendicular to the shear direction or perpendicular to the shear plane, respectively. When the elastic shear strain is small and the natural deformation is finite, we obtain effective *shear moduli* with the relative ratio equal to the natural anisotropy parameter of the nematic material. Note that we use the uniaxial and simple shear deformation, respectively, to describe the elastic contribution to the deformation rather than the overall deformation, which also contains a natural deformation component. To derive the elastic moduli, we then take the limit of small elastic strain, while the natural deformation remains finite. This enables us to rigorously adapt the elasticity theory to nematic elastomers (for a review on elastic moduli, see [57]). To account for the physical aeolotropy of real nematic solids, in §5, we extend the continuum model by incorporating a nematic energy, and show how the stretch and shear moduli corresponding to the ideal case are modified by the additional information. The parameters entering the LCE model characterize either the change of the microstructure due to nematic effects or the behaviour of the material in large natural deformations. Nevertheless, in the limit of small elastic deformations and for fixed nematic parameters, the system behaves indeed like a transverse isotropic material with five independent elastic constants. In physics, *aeolotropy* refers to materials exhibiting different properties depending on the direction in which they are measured, or simply defined by Lord Kelvin as ‘That which is different in different directions’ [72, p. 122]. While traditional anisotropic elastic materials also exhibit aeolotropy, we refer to nematic solids as *aeolotropic materials*, and reserve the characterization of *isotropic* or *anisotropic* for the elastic part of the energy. In §6, we present a molecular dynamics simulation of a nematic elastomer, and analyse its response under similar stretch and shear deformations as for the continuum model to illustrate the aeolotropic mechanical responses. In §§3–5, physical quantities are treated symbolically, and units of measure only appear in §6, where datasets are used to illustrate the theory. The final section containsconcluding remarks.

### 2. Prerequisites

The strain-energy density describing an ideal monodomain nematic liquid crystalline (NLC) solid takes the general form [64–67,73]

**F**represents the deformation gradient from the isotropic state,

**n**is a unit vector, known as the

*director*, for the orientation of the nematic field and

*W*(

**A**) denotes 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 following relation:

*a*> 0 represents a temperature-dependent stretch parameter,

*ν*is the optothermal analogue to the Poisson ratio [74] and relates responses in directions parallel or perpendicular to the director

**n**, ⊗ denotes the tensor product of two vectors and

**I**= diag(1, 1, 1) is the identity tensor. It is assumed here that

*a*and

*ν*are spatially independent. The ratio

*r*=

*a*

^{1/3}/

*a*

^{−ν/3}=

*a*

^{(ν+1)/3}represents the anisotropy parameter, which, in an ideal nematic solid, is the same in all directions. In the nematic phase, both the cases with

*r*> 1 (prolate molecules) and

*r*< 1 (oblate molecules) are possible, while when

*r*= 1 the energy function reduces to that of an isotropic hyperelastic material. Nematic elastomers have

*ν*= 1/2, while for nematic glasses

*ν*∈ (1/2, 2) [10,75]. Natural strains in NLC glasses are typically of up to 4%, whereas for NLC elastomers these may be up to 400%. The nematic director

**n**is an observable (spatial) quantity. Denoting by

**n**

_{0}the reference orientation of the local director corresponding to the cross-linking state,

**n**may differ from

**n**

_{0}both by a rotation and by a change in

*r*. In nematic elastomers, which are weakly cross-linked, the director can rotate freely, and the material exhibits isotropic mechanical effects. In nematic glasses, which are densely cross-linked, the director

**n**cannot rotate relative to the elastic matrix, but changes through convection due to elastic strain and satisfies [21,22,58,74,76]

For a hyperelastic material described by the strain-energy density *W* = *W*(**A**), the Cauchy stress tensor is equal to

*p*denotes the Lagrange multiplier for the incompressibility constraint $det\mathbf{\text{A}}=1$,

*β*

_{1}= 2∂

*W*/∂

*I*

_{1}and

*β*

_{−1}= −2∂

*W*/∂

*I*

_{2}are material parameters,

**B**=

**A**

**A**

^{T}is the left Cauchy–Green elastic deformation tensor and

*I*

_{1},

*I*

_{2}are its first two principal invariants (

*I*

_{3}= 1 owing to incompressibility). The corresponding first Piola–Kirchhoff stress tensor is equal to

In the next sections, we obtain two *stretch moduli* under small elastic uniaxial tension and three nonlinear *shear moduli* under elastic simple shear deformations, respectively, which combine elastic and nematic effects. In our calculations, the nematic director is ‘frozen’, but the case where the director is ‘free’ can be treated similarly, provided that the elastic deformation is small. In particular, when the director is free to rotate and a tensile force is applied perpendicular to the director, experimental results show that there is a range of strains, up to 10% (e.g. [78,79]), before the director rotates in response to the applied force. However, the local nematic order might be altered. We further note that, in finite elasticity, the strain-energy density *W*(**A**) and the stress relationship (2.5) characterize an isotropic material. Yet, the response of a nematic solid also depends on the director orientation. For instance, we show that the ideal model exhibits an anisotropic response under stretch if the natural deformation is finite. There are, in fact, two possible contributions to aeolotropy. First, in the ideal case, the elastic energy of the system is isotropic but the full energy depends on the orientation of the nematic field as well [59]. Second, there is a component of the energy that depends directly on the nematic order through the so-called **Q**-tensor [23]. In this case, we need to extend the discussion by including this nematic energy density as developed later in the paper.

### 3. Stretch moduli

The stretch modulus of a homogeneous isotropic elastic material is obtained under uniaxial tension with the gradient tensor in a Cartesian system of coordinates taking the form [57]

*λ*> 1 is the stretch ratio in the direction of the applied tensile force. Assuming that the only non-zero component of the associated first Piola–Kirchhoff stress, given by (2.6), is in the tensile direction, i.e.

*P*

_{22}> 0, it follows that

*P*

_{22}is the tensile first Piola–Kirchhoff stress given by (3.2),

*λ*− 1 is the corresponding tensile strain (for different definitions of an elastic strain, see [57]) and

*μ*= lim

_{λ→1}(

*β*

_{1}−

*β*

_{−1}) is the shear modulus at small strain.

To derive stretch moduli for the nematic material, we apply the tensile force in a direction that is either parallel or perpendicular to the reference nematic director [45]. In each case, we assume an overall deformation where the elastic component is a uniaxial tension with the deformation tensor given by (3.1). Then, we calculate the stretch moduli by taking the ratio between the first Piola–Kirchhoff tensile stress and the associated strain in the limit of *small elastic tensile strain*, while the natural deformation remains finite.

#### (a) Nematic director parallel or perpendicular to the tensile direction

When the tensile force is acting in the second Cartesian direction and the reference director is parallel to the tensile force, the nematic director in the current configuration and the associated spontaneous deformation tensor take the following form, respectively (figure 1*a*):

*a*),

**A**is of the form given by (3.1), for the deformation gradient

**F**and the associated first Piola–Kirchhoff stress, the principal components in the direction of the tensile load are, respectively,

*P*

_{22}is the first Piola–Kirchhoff stress given by (3.2). These are illustrated in figures 1

*b*and 2

*b*, respectively, for

*μ*= 1 and

*ν*= 1/2. We define the following stretch moduli for the nematic material under the above two deformations, respectively:

*E*= 3

*μ*is Young’s modulus defined by (3.3). Therefore, the stretch moduli given by (3.8) and (3.9) satisfy the following relation:

*r*> 1, then

*E*

^{(1)}<

*E*

^{(2)}; if

*r*< 1, then

*E*

^{(1)}>

*E*

^{(2)}; if

*r*= 1, then

*E*

^{(1)}=

*E*

^{(2)}=

*E*.

### 4. Shear moduli

To obtain the shear modulus of a homogeneous isotropic elastic material, a standard deformation is the simple shear, with the following gradient tensor in a Cartesian system ofcoordinates [57]:

*k*> 0 is the shear parameter. The non-zero components of the associated Cauchy stress tensor, given by (2.5), are

*small elastic shearing strains*, while the natural deformation remains finite.

#### (a) Nematic director parallel to the shear direction

When the reference nematic director ${\mathbf{\text{n}}}_{0}^{(1)}=[1,0,0{]}^{T}$ is parallel to the direction of applied shear force, in the current configuration, we have (figure 3*a*)

*P*

_{12}is the shear component of the elastic first Piola–Kirchhoff stress given by (4.3). These are represented in figure 3

*b*, for

*μ*= 1 and

*ν*= 1/2. We now define the shear modulus for the nematic material at small shear as follows:

#### (b) Nematic director perpendicular to the shear direction

When the reference nematic director ${\mathbf{\text{n}}}_{0}^{(2)}=[0,1,0{]}^{T}$ is perpendicular to the direction of shear, we obtain in the current configuration (figure 4*a*)

*k*, the corresponding shear strain and first Piola–Kirchhoff shear stress of the nematic material take the form

*P*

_{12}and

*P*

_{22}of the elastic first Piola–Kirchhoff stress given by (4.3). These are illustrated in figure 4

*b*, for

*μ*= 1 and

*ν*= 1/2. In this case, the associated shear modulus at small shear is equal to

#### (c) Nematic director perpendicular to the shear plane

When the reference nematic director ${\mathbf{\text{n}}}_{0}^{(3)}=[0,0,1{]}^{T}$ is perpendicular to the shear plane, we have (figure 5*a*)

*P*

_{12}is the shear component of the elastic first Piola–Kirchhoff stress given by (4.3). These are represented in figure 5

*b*, for

*μ*= 1 and

*ν*= 1/2. We now define the shear modulus for the nematic material at small shear as follows:

*r*> 1, then

*μ*

^{(1)}<

*μ*

^{(2)}<

*μ*

^{(3)}; if

*r*< 1, then

*μ*

^{(1)}>

*μ*

^{(2)}>

*μ*

^{(3)}; if

*r*= 1, then

*μ*

^{(1)}=

*μ*

^{(2)}=

*μ*

^{(3)}=

*μ*.

### 5. Contribution of the nematic free energy

The results of §§3 and 4 imply that, for an ideal nematic material, the effective shear and stretch moduli respect the same inequalities (e.g. if *r* > 1, then we have both *E*^{(1)} < *E*^{(2)} and *μ*^{(1)} < *μ*^{(2)} < *μ*^{(3)}, and so on). However, numerous experimental results have demonstrated that there are significant differences between the behaviour of real nematic solids and that of ideal nematic materials analysed in the previous sections [38,42,45]. In particular, it was found that the ideal behaviour of the stretch and shear moduli ratio does not match experimental results. We therefore extend the strain-energy function by taking into account the nematic free-energy density, in addition to the isotropic strain-energy density *W*^{(nc)} = *W*^{(nc)}(**F**, **n**), given by (2.1), as

*W*

_{n}is equal to [38,43] (see also [19], ch. 2)

**Q**the order parameter tensor. This macroscopic tensor parameter is used to describe the orientational order in nematic liquid crystals [23] (see also [80]). For incompressible nematic elastomers subjected to uniaxial stretches, the contribution given by (5.2) to the total strain-energy density described by (5.1) was originally analysed in [38]. Following a similar approach, we restrict our attention to the one-term strain-energy density of the form (2.1) (see also [58]),

*μ*> 0 is the elastic shear modulus at small strain. This strain-energy density can be expressed equivalently as follows [14]:

**Q**= diag( − (

*Q*−

*b*)/2, − (

*Q*+

*b*)/2,

*Q*), where

*Q*and

*b*are scalar values (see also [19], §2.2), we consider the nematic energy described by (5.2) and approximate it by [38]

**G**

^{2}, the first-order approximation of the Taylor expansions about the backbone order parameters in the initial state (

*Q*,

*b*) = (

*Q*

_{0}, 0) are [38]

*l*

_{⊥,Q}= ∂

*l*

_{2}/∂

*Q*= ∂

*l*

_{3}/∂

*Q*denote the first derivatives of

*l*

_{1}and

*l*

_{2}(and also

*l*

_{3}) with respect to

*Q*at

*b*= 0, respectively, and

*l*

_{2,b}= ∂

*l*

_{2}/∂

*b*and

*l*

_{3,b}= ∂

*l*

_{3}/∂

*b*are the first derivatives of

*l*

_{2}and

*l*

_{3}, respectively, with respect to

*b*at

*Q*=

*Q*

_{0}.

For a nematic solid with the strain-energy density given by (5.6), we derive the stretch and shear moduli in a direction parallel or perpendicular to the nematic director, as follows.

#### (a) Extension parallel to the director

When the nematic director is uniformly aligned in the first direction, we take $\overline{\mathbf{\text{F}}}=\mathrm{diag}({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})$, where *λ*_{1} = *λ* > 1 and ${\lambda}_{2}={\lambda}_{3}=1/\sqrt{\lambda}$. Assuming infinitesimal extension, we have *λ* = 1 + ε and *λ*^{−1} = 1 − ε, where ε denotes the infinitesimal strain. At *b* = 0, the elastic strain-energy density is approximated by

*δQ*has a minimum of

*E*= 3

*μ*, the stretch modulus in a direction parallel to the nematic director is then

#### (b) Extension perpendicular to the director

When the director is aligned in the first direction and $\overline{\mathbf{\text{F}}}=\mathrm{diag}({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})$, where ${\lambda}_{1}={\lambda}_{3}=1/\sqrt{\lambda}$ and *λ*_{2} = *λ* > 1, assuming *λ* = 1 + ε and *λ*^{−1} = 1 − ε, with ε the infinitesimal strain, the elastic strain-energy density given by (5.4) is approximated by

*δQ*and

*b*gives

*E*= 3

*μ*, the stretch modulus in a direction perpendicular to the nematic director is

#### (c) Shear parallel to the director

We recall that a simple shear deformation of a hyperelastic material is equivalent to a biaxial stretch (‘pure shear’ [81]) in the principal directions [82]. We assume that the director is aligned uniformly in the first (or second direction), and $\overline{\mathbf{\text{F}}}=\mathbf{\text{R}}\hspace{0.17em}\mathrm{diag}({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})$, where *λ*_{1} = *λ* > 1, *λ*_{2} = 1/*λ* and *λ*_{3} = 1, and **R** is the rotation by an angle of *π*/4 in the plane formed by the first two directions. Taking *λ* = 1 + ε and *λ*^{−1} = 1 − ε, with ε the infinitesimal strain, the elastic strain-energy density is approximated as follows:

*δQ*and

*b*,

#### (d) Shear perpendicular to the director

Next, assuming that the director is aligned in the first direction, we set $\overline{\mathbf{\text{F}}}=\mathrm{diag}({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})$, where *λ*_{1} = 1, *λ*_{2} = *λ* > 1 and *λ*_{3} = 1/*λ*. Taking *λ* = 1 + ε and *λ*^{−1} = 1 − ε, with ε the infinitesimal strain, the elastic strain-energy density is approximated by

*b*has a minimum value of

#### (e) Deviation from isotropy

The stretch and shear moduli ${E}_{\parallel}$, *E*_{⊥}, ${\mu}_{\parallel}$ and *μ*_{⊥}, defined by (5.11), (5.15), (5.19) and (5.23), respectively, contain information about the nematic order, in addition to the small strain elasticity of the polymer matrix, described by Young’s modulus *E* and shear modulus *μ*. Proceeding as in §§3 and 4, with ${\stackrel{~}{W}}^{(\text{lce})}$ instead of *W*^{(nc)}, by replacing *E* with ${E}_{\parallel}$ in (3.8) and with *E*_{⊥} in (3.9), *μ* with ${\mu}_{\parallel}$ in (4.7) and (4.10), and with *μ*_{⊥} in (4.13), we obtain

*E*

^{(2)}/

*E*

^{(1)}<

*r*

^{2}; if ${E}_{\parallel}/{E}_{\perp}<1$, then

*E*

^{(2)}/

*E*

^{(1)}>

*r*

^{2}; if ${\mu}_{\parallel}/{\mu}_{\perp}>1$, then

*μ*

^{(3)}/

*μ*

^{(2)}<

*r*; if ${\mu}_{\parallel}/{\mu}_{\perp}<1$, then

*μ*

^{(3)}/

*μ*

^{(2)}>

*r*. In particular, when

*r*> 1, if ${E}_{\parallel}/{E}_{\perp}>{r}^{2}>1$ and ${\mu}_{\parallel}/{\mu}_{\perp}<1$, then

*E*

^{(1)}>

*E*

^{(2)}and

*μ*

^{(1)}<

*μ*

^{(2)}<

*μ*

^{(3)}, which is qualitatively different from the behaviour of ideal nematic solids or any standard anisotropic hyperelastic material. In the next section, we show how to access these moduli through molecular dynamics simulations.

### 6. Molecular dynamics simulation

We performed molecular dynamics (MD) simulations to synthesize LCEs, form nematic LCEs and characterize their response under stretch and shear deformations. Given its accuracy and efficiency for modelling mesogen–polymer systems, we used a hybrid force field, including Gay–Berne coarse-graining potentials for mesogen–mesogen interaction and Lennard-Jones (LJ) potentials for united atoms of hydrocarbon groups, CH_{x}, in polymer chains. Our computer simulations [83] can serve as a virtual experiment to observe the macroscopic mechanical behaviour, which can then be compared directly with the continuum theory, and to calculate the physical quantities arising from atomistic movement and gain a mechanistic understanding.

In the MD simulations of LCEs, the potential energy of the whole system includes contributions from bond stretch, angle bending, dihedral rotation, non-bonded LJ interaction between united atoms (*a*-*a*), anisotropic non-bonded Gay–Berne interaction between mesogens (*m*-*m*) and the extended Gay–Berne interaction between united atoms and mesogens (*a*-*m*):

*a*

*b*

*c*

*d*

*e*

*f*

*E*

_{bond}represents a harmonic bond style, with

*l*

_{i}the

*i*th bond length,

*l*

^{(1)}the equilibrium bond length of CH

_{x}–CH

_{y}and

*l*

^{(2)}the equilibrium bond length of CH

_{x}–mesogen;

*E*

_{angle}denotes a harmonic angle style, with

*θ*

_{i}the

*i*th angle,

*θ*

^{(1)}the equilibrium angle of the non-branched X–

*CH*

_{2}–

*X*and

*θ*

^{(2)}the equilibrium angle of branched X–CH–X;

*E*

_{dihedral}is a multiple-term harmonic dihedral style, with

*ϕ*

_{i}the

*i*th torsion angle, ${C}_{n}^{(1)}$ the non-branched X–CH

_{2}–CH

_{2}–X and ${C}_{n}^{(2)}$ the branched X–CH

_{2}–CH–X;

*E*

_{(a–a)}represents the LJ potential between non-bonded united atoms CH

_{x}, with

*r*

_{ij}the distance between two united atoms and

*a*

_{i},

*c*

_{i}the factorized energy parameters for CH

_{x};

*E*

_{m−m}is the Gay–Berne potential between non-bonded mesogens, with

*U*

_{r}the shifted distance-dependent interaction,

*η*,

*χ*the orientation-dependent energy,

**A**

_{i}the transformation matrix for mesogen

*i*,

**r**

_{ij}the centre-to-centre vector between the

*i*th and

*j*th mesogens and all the rest of the parameters specified as constants in table 1; and

*E*

_{a−m}denotes the extended Gay–Berne potential between a non-bonded mesogen and CH

_{x}following the standard mixing rule [86]. The cut-off distance is 9.8 Å for the LJ potential and 16.8 Å for the Gay–Berne potential. For the detailed explanation and explicit form of the Gay–Berne potential, readers should refer to [87,88]. First, 64 molecules of side-chain liquid crystal polymers were created, where every molecule has a backbone of 100 hydrocarbon monomers and 50 side chains attaching to the backbone in a syndiotactic way. Among the 50 side chains for each molecule, 20% were randomly selected to be attached with cross-linking sites, and the rest were attached with mesogens, as shown in figure 6

*a*. Thus, every LC molecule has a different configuration. The 64 molecules were mixed by heating at 800 K for 50 ns. Then the system was quenched down to 500 K during 10 ns. Cross-linking of the first step was established while equilibrating the system at 500 K. A weakly cross-linked isotropic LCE was constructed, as shown in figure 6

*b*. The system was found to have its isotropic–nematic phase-transition temperature below 490 K, consistent with other MD studies of similar LCE systems [89]. To form nematic LCEs, the whole system was quenched down to 450 K with an external field ${U}_{i}^{\text{efield}}=-1.0\cdot {P}_{2}(\mathrm{cos}{\theta}_{i})$, where

*P*

_{2}(

*x*) = (3

*x*

^{2}− 1)/2 is the second Legendre polynomial and

*θ*

_{i}is the angle between the long axis of the

*i*th mesogen and the external field. The external field has been experimentally and computationally proved to accelerate the formation of the nematic phase [90,91]. The quenching stage from 500 K to 450 K lasted 10 ns and the equilibrium stage at 450 K lasted 20 ns, for both of which the external field was applied along the

*z*-direction. The external field was removed while the LCE system was equilibrated at 450 K for another 17 ns and the nematic phase was found to be stable. The nematic order

*S*

_{2}= 〈

*P*

_{2}(cos

*θ*

_{i})〉 is shown in figure 6

*b*during the quenching and equilibrium state. Subsequently, the second stage of cross-linking was performed within the nematic LCEs. The nematic system was then quenched down to 300 K and the nematic order was found to be well maintained. Throughout the whole process of constructing nematic LCEs, NPT calculations were performed using a time step of 1 fs in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) with periodic boundary conditions along three dimensions. At the isotropic–nematic transition, a spontaneous deformation was observed in the MD simulations shown in figure 6

*b*, owing to the alignment of ellipsoidal mesogens along the

*Z*-direction. The nematic LCE at 300 K has three dimensions: ${l}_{x}^{\mathrm{nc}}=99.6537\hspace{0.17em}\text{\xc5}$, ${l}_{y}^{\mathrm{nc}}=88.1515\hspace{0.17em}\text{\xc5}$ and ${l}_{z}^{\mathrm{nc}}=185.7362\hspace{0.17em}\text{\xc5}$. During the isotropic–nematic phase transition, the volume of LCEs should demonstrate a negligible change [92]. However, the MD simulations of LCEs using Gay–Berne potentials are known to show thermal expansion effects [93]. To eliminate these non-physical effects, the isotropic LCE at 500 K was quickly quenched down to 300 K and the three dimensions for the isotropic phase were measured as ${l}_{x}^{\text{iso}}=120.3505\hspace{0.17em}\text{\xc5}$, ${l}_{y}^{\text{iso}}=118.7118\hspace{0.17em}\text{\xc5}$ and ${l}_{z}^{\text{iso}}=117.2290\hspace{0.17em}\text{\xc5}$. The respective stretch ratios were then calculated as

*λ*

_{x}= 0.8280,

*λ*

_{y}= 0.7426 and

*λ*

_{z}= 1.5844. Ideally, the biaxial Gay–Berne potential for mesogen–mesogen interaction should yield the same stretch ratio along the

*X*- and

*Y*-directions. Here, however, the small size of the LCE system in the MD simulations gives rise to slightly different

*λ*

_{x}and

*λ*

_{y}. To eliminate the size effect on the spontaneous deformation, these stretch ratios were averaged to

*λ*

_{x−y}= 0.7853. Then the anisotropy ratio can be estimated as

*r*≈

*λ*

_{z}/

*λ*

_{x−y}= 2.0176, Poisson’s ratio is

*ν*≈ 0.5 and

*a*=

*r*

^{2}≈ 4.0706.

parameters | value (units) |
---|---|

k_{b}, bond energy constant |
520.0156 (kcal/mol/Å^{2}) |

l^{(1)}, equilibrium CH_{x}–CH_{y} bond length |
1.540 (Å) |

l^{(2)}, equilibrium CH_{x}–mesogen bond length |
7.075 (Å) |

k_{a}, angle energy constant |
124.2009 (kcal/mol) |

θ^{(1)}, equilibrium non-branched angle |
114.0014 (°) |

θ^{(2)}, equilibrium branched angle |
112.0018 (°) |

${C}_{0}^{(1)}$, non-branched torsion energy constant | 2.0066 (kcal/mol) |

${C}_{0}^{(2)}$, non-branched torsion energy constant | 4.0111 (kcal/mol) |

${C}_{0}^{(3)}$, non-branched torsion energy constant | 0.2709 (kcal/mol) |

${C}_{0}^{(4)}$, non-branched torsion energy constant | −6.2885 (kcal/mol) |

${C}_{1}^{(1)}$, branched torsion energy constant | 0.7413 (kcal/mol) |

${C}_{1}^{(2)}$, branched torsion energy constant | 1.8264 (kcal/mol) |

${C}_{1}^{(3)}$, branched torsion energy constant | 0.5329 (kcal/mol) |

${C}_{1}^{(4)}$, branched torsion energy constant | −3.4521 (kcal/mol) |

a_{i}, energy parameter for CH_{3} |
$2534.0341\hspace{0.17em}(\sqrt{\text{kcal/mol}}\cdot {\text{\xc5}}^{6})$ |

c_{i}, energy parameter for CH_{3} |
$47.2921\hspace{0.17em}(\sqrt{\text{kcal/mol}}\cdot {\text{\xc5}}^{3})$ |

a_{i}, energy parameter for CH_{2} |
$2251.9322\hspace{0.17em}(\sqrt{\text{kcal/mol}}\cdot {\text{\xc5}}^{6})$ |

c_{i}, energy parameter for CH_{2} |
$37.0955\hspace{0.17em}(\sqrt{\text{kcal/mol}}\cdot {\text{\xc5}}^{3})$ |

a_{i}, energy parameter for CH_{1} |
$1467.0522\hspace{0.17em}(\sqrt{\text{kcal/mol}}\cdot {\text{\xc5}}^{6})$ |

c_{i}, energy parameter for CH_{1} |
$21.2860\hspace{0.17em}(\sqrt{\text{kcal/mol}}\cdot {\text{\xc5}}^{3})$ |

ϵ, well depth for U_{r} function in (6.1a–f) |
0.8079 (kcal/mol) |

σ, minimum effective particle radii in (6.1a–f) |
5 (Å) |

ϵ_{i}, relative well depth for end-to-end |
5 |

ϵ_{i}, relative well depth for side-to-side |
1 |

v, exponent for η function in (6.1a–f) |
1 |

ξ, exponent for χ function in (6.1a–f) |
2 |

κ, length/breadth ratio of mesogen |
3 |

mass, united atoms CH_{x} |
12.0 + x (g/mol) |

mass, mesogen | 226.0 (g/mol) |

After obtaining the nematic LCE at 300 K, we subjected the system to either stretch or shear deformations as described in §§3 and 4, respectively. For each deformation, we can calculate the evolution of the first Piola–Kirchhoff stresses with respect to the strains. We first calculate the Cauchy stress tensor in LAMMPS based on ${T}_{ij}={\Sigma}_{k}^{N}{m}_{k}{v}_{ki}{v}_{kj}/V+{\Sigma}_{k}^{{N}^{\prime}}{r}_{ki}{f}_{kj}/V$, where *m*_{k} represents the mass of the *k*th atom, *v*_{ki} denotes the velocity of the *k*th atom in the *i*th dimension, *N* is the total number of atoms, *V* is the total volume of the system, *N*′ is the number of atoms pairs and *r*_{ki} and *f*_{kj} are the positions and forces of atom *k* along the *i*th and *j*th dimensions, respectively [94]. Next, given the deformation imposed, we construct the elastic deformation tensor **A** and, subsequently, the corresponding first Piola–Kirchhoff stress tensor **P**, following (2.6). Then, imposing the spontaneous deformation tensor **G**, or simply following (3.6), (3.7), (4.6), (4.9) or (4.12), we obtain the final ${\hat{P}}_{ij}^{(\text{nc})}$. The results are shown in figures 7 and 8, respectively, where all the cases demonstrate some nonlinearity in the stress-deformation relations. From the two stretch deformations, we derive the effective stretch moduli *E*^{(1)} = 2253.4 MPa and *E*^{(2)} = 775.475 MPa. From the three shear deformations, we find the effective shear moduli *μ*^{(1)} = 132.78 MPa, *μ*^{(2)} = 258.26 MPa and *μ*^{(3)} = 889.97 MPa. Thus,

### 7. Conclusion

We studied theoretically and computationally the mechanical behaviour of nematic LCEs under different stretch and shear deformations. Theoretically, we first examined ideal nematic elastomers characterized by a homogeneous isotropic elastic strain-energy density, then also phenomenological models incorporating an additional nematic energy. We showed that these cases are qualitatively different, and that the generalized model does not necessarily order stretch moduli in the same way as the shear moduli. We also performed molecular dynamics simulations to analyse numerically the responses of simulated systems under a similar set of deformations, and found different mechanical responses in different directions. Therefore, the trifecta of experiments, computations and theory leads us to conclude that the contribution of the nematic free energy cannot be ignored, even in small deformations, and that LCEs are best understood as aeolotropic materials. When Frank effects also play an important role, they need to be taken into account as well. However, the deviation from isotropy is well captured by including the nematic energy, and this constitutes an important step in the constitutive modelling of liquid crystalline solids.

### Data accessibility

All data for this research are openly available at https://doi.org/10.5281/zenodo.5156815.

### Authors' contributions

L.A.M. and J.G. conceived of and designed the study. L.A.M. and A.G. carried out the theoretical analysis, and drafted and revised the manuscript. J.G. and H.W. carried out the computational simulations and the numerical analysis, and helped to draft and revise the manuscript. All the authors gave final approval for publication and agree to be held accountable for the work performed herein.

### Competing interests

The authors declare that they have no conflict of interest.

### Funding

We are grateful for the support by the Engineering and Physical Sciences Research Council of Great Britain under research grant nos. EP/R020205/1 to A.G. and EP/S028870/1 to L.A.M.

## Acknowledgements

The support and resources from the Center for High Performance Computing at the University of Utah are gratefully acknowledged.

## Appendix A. Stresses in nematic solids

In this appendix, we formulate the stress tensors of a nematic material described by the strain-energy function given by (2.1) in terms of the stresses in the base polymeric network. These relations were originally presented in [65] and are provided here for convenience.

If the nematic director is ‘free’ to rotate relative to the elastic matrix, then **F** and **n** are independent variables, and the Cauchy stress tensor for the nematic material with the strain-energy function described by (2.1) is calculated as follows:

**T**is the Cauchy stress tensor defined by (2.5), $J=det\mathbf{\text{F}}$ and the scalar

*p*

^{(nc)}represents the Lagrange multiplier for the internal constraint

*J*= 1. Then

**P**the first Piola–Kirchhoff stress given by (2.6).

If the nematic director is ‘frozen’, the Cauchy stress tensor for the nematic material takes the form

**T**is the Cauchy stress defined by (2.5), $J=det\mathbf{\text{F}}$,

*p*

^{(nc)}is the Lagrange multiplier for the volume constraint

*J*= 1 and

*q*is the Lagrange multiplier for the constraint (2.4). Then

### References

- 1.
de Gennes PG . 1975 Physique moléculaire—réflexions sur un type de polymères nématiques.**C. R. Acad. Sci. B**, 101-103. Google Scholar**281** - 2.
Finkelmann H, Kock HJ, Rehage G . 1981 Investigations on liquid crystalline polysiloxanes 3. Liquid crystalline elastomers—a new type of liquid crystalline material.**Makromol. Chem. Rapid Commun.**, 317-322. (doi:10.1002/marc.1981.030020413) Crossref, Google Scholar**2** - 3.
de Haan LT, Schenning AP, Broer DJ . 2014 Programmed morphing of liquid crystal networks.**Polymer**, 5885-5896. (doi:10.1016/j.polymer.2014.08.023) Crossref, ISI, Google Scholar**55** - 4.
Jiang ZC, Xiao YY, Zhao Y . 2019 Shining light on liquid crystal polymer networks: preparing, reconfiguring, and driving soft actuators.**Adv. Opt. Mater.**, 1900262. (doi:10.1002/adom.201900262) Crossref, ISI, Google Scholar**7** - 5.
Kuenstler AS, Hayward RC . 2019 Light-induced shape morphing of thin films.**Curr. Opin. Colloid Interface Sci.**, 70-86. (doi:10.1016/j.cocis.2019.01.009) Crossref, ISI, Google Scholar**40** - 6.
McCracken JM, Donovan BR, White TJ . 2020 Materials as machines.**Adv. Mater.**, 1906564. (doi:10.1002/adma.201906564) Crossref, ISI, Google Scholar**32** - 7.
Pang X, Lv J-a, Zhu C, Qin L, Yu Y . 2019 Photodeformable azobenzene-containing liquid crystal polymers and soft actuators.**Adv. Mater.**, 1904224. (doi:10.1002/adma.201904224) Crossref, ISI, Google Scholar**31** - 8.
Ula SW, Traugutt NA, Volpe RH, Patel RP, Yu K, Yakacki CM . 2018 Liquid crystal elastomers: an introduction and review of emerging technologies.**Liquid Crystals Rev.**, 78-107. (doi:10.1080/21680396.2018.1530155) Crossref, ISI, Google Scholar**6** - 9.
Wan G, Jin C, Trase I, Zhao S, Chen Z . 2018 Helical structures mimicking chiral seedpod opening and tendril coiling.**Sensors**, 2973. (doi:10.3390/s18092973) Crossref, ISI, Google Scholar**18** - 10.
Warner M . 2020 Topographic mechanics and applications of liquid crystalline solids.**Annu. Rev. Condens. Matter Phys.**, 125-145. (doi:10.1146/annurev-conmatphys-031119-050738) Crossref, ISI, Google Scholar**11** - 11.
White TJ, Broer DJ . 2015 Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers.**Nat. Mater.**, 1087-1098. (doi:10.1038/nmat4433) Crossref, PubMed, ISI, Google Scholar**14** - 12.
Xia Y, Honglawan A, Yang S . 2019 Tailoring surface patterns to direct the assembly of liquid crystalline materials.**Liquid Crystals Rev.**, 30-59. (doi:10.1080/21680396.2019.1598295) Crossref, ISI, Google Scholar**7** - 13.
Bladon P, Terentjev EM, Warner M . 1993 Transitions and instabilities in liquid-crystal elastomers.**Phys. Rev. E**, R3838-R3840. Crossref, ISI, Google Scholar**47** - 14.
Bladon P, Terentjev EM, Warner M . 1994 Deformation-induced orientational transitions in liquid crystal elastomers.**J. Phys. II**, 75-91. (doi:10.1051/jp2:1994100) Crossref, Google Scholar**4** - 15.
Warner M, Gelling KP, Vilgis TA . 1988 Theory of nematic networks.**J. Chem. Phys.**, 4008-4013. (doi:10.1063/1.453852) Crossref, ISI, Google Scholar**88** - 16.
Warner M, Wang XJ . 1991 Elasticity and phase behavior of nematic elastomers.**Macromolecules**, 4932-4941. (doi:10.1021/ma00017a033) Crossref, ISI, Google Scholar**24** - 17.
Treloar LRG . 2005**The physics of rubber elasticity**, 3rd edn. Oxford, UK: Oxford University Press. Google Scholar - 18.
Warner M, Terentjev EM . 1996 Nematic elastomers—a new state of matter?**Prog. Polym. Sci.**, 853-891. Crossref, ISI, Google Scholar**21** - 19.
Warner M, Terentjev EM . 2007**Liquid crystal elastomers**. Oxford, UK: Oxford University Press. Google Scholar - 20.
Bai R, Bhattacharya K . 2020 Photomechanical coupling in photoactive nematic elastomers.**J. Mech. Phys. Solids**, 104115. (doi:10.1016/j.jmps.2020.104115) Crossref, ISI, Google Scholar**144** - 21.
Modes CD, Bhattacharya K, Warner M . 2010 Disclination-mediated thermo-optical response in nematic glass sheets.**Phys. Rev. E**, 060701(R). (doi:10.1103/PhysRevE.81.060701) Crossref, ISI, Google Scholar**81** - 22.
Modes CD, Bhattacharya K, Warner M . 2011 Gaussian curvature from flat elastica sheets.**Proc. R. Soc. A**, 1121-1140. (doi:10.1098/rspa.2010.0352) Link, Google Scholar**467** - 23.
de Gennes PG, Prost J . 1993**The physics of liquid crystals**, 2nd edn. Oxford, UK: Clarendon Press. Google Scholar - 24.
Frank FC . 1958 I. Liquid crystals. On the theory of liquid crystals.**Discuss. Faraday Soc.**, 19-28. Crossref, Google Scholar**25** - 25.
Biggins JS, Warner M, Bhattacharya K . 2009 Supersoft elasticity in polydomain nematic elastomers.**Phys. Rev. Lett.**, 037802. (doi:10.1103/PhysRevLett.103.037802) Crossref, PubMed, ISI, Google Scholar**103** - 26.
Biggins JS, Warner M, Bhattacharya K . 2012 Elasticity of polydomain liquid crystal elastomers.**J. Mech. Phys. Solids**, 573-590. (doi:10.1016/j.jmps.2012.01.008) Crossref, ISI, Google Scholar**60** - 27.
Agostiniani V, Dal Maso G, DeSimone A . 2015 Attainment results for nematic elastomers.**Proc. R. Soc. Edinb. A**, 669-701. (doi:10.1017/S0308210515000128) Crossref, ISI, Google Scholar**145** - 28.
Agostiniani V, DeSimone A . 2012 Ogden-type energies for nematic elastomers.**Int. J. Non-Linear Mech.**, 402-412. (doi:10.1016/j.ijnonlinmec.2011.10.001) Crossref, ISI, Google Scholar**47** - 29.
DeSimone A, Teresi L . 2009 Elastic energies for nematic elastomers.**Eur. Phys. J. E**, 191-204. (doi:10.1140/epje/i2009-10467-9) Crossref, PubMed, ISI, Google Scholar**29** - 30.
Fried E . 2002 An elementary molecular-statistical basis for the Mooney and Rivlin-Saunders theories of rubber elasticity.**J. Mech. Phys. Solids**, 571-582. (doi:10.1016/S0022-5096(01)00086-2) Crossref, ISI, Google Scholar**50** - 31.
Horgan CO, Saccomandi G . 2002 A molecular-statistical basis for the Gent constitutive model of rubber elasticity.**J. Elast.**, 167-176. (doi:10.1023/A:1026029111723) Crossref, ISI, Google Scholar**68** - 32.
Anderson DR, Carlson DE, Fried E . 1999 A continuum-mechanical theory for nematic elastomers.**J. Elast.**, 33-58. (doi:10.1023/A:1007647913363) Crossref, ISI, Google Scholar**56** - 33.
Zhang Y, Xuan C, Jiang Y, Huo Y . 2019 Continuum mechanical modeling of liquid crystal elastomers as dissipative ordered solids.**J. Mech. Phys. Solids**, 285-303. (doi:10.1016/j.jmps.2019.02.018) Crossref, ISI, Google Scholar**126** - 34.
Conti S, DeSimone A, Dolzmann G . 2002 Soft elastic response of stretched sheets of nematic elastomers: a numerical study.**J. Mech. Phys. Solids**, 1431-1451. (doi:10.1016/S0022-5096(01)00120-X) Crossref, ISI, Google Scholar**50** - 35.
Conti S, DeSimone A, Dolzmann G . 2002 Semi-soft elasticity and director reorientation in stretched sheets of nematic elastomers.**Phys. Rev. E**, 61710. (doi:10.1103/PhysRevE.66.061710) Crossref, Google Scholar**60** - 36.
Skačej G, Zannoni C . 2012 Molecular simulations elucidate electric field actuation in swollen liquid crystal elastomers.**Proc. Natl Acad. Sci. USA**, 10 193-10 198. (doi:10.1073/pnas.1121235109) Crossref, ISI, Google Scholar**109** - 37.
Skačej G, Zannoni C . 2014 Molecular simulations shed light on supersoft elasticity in polydomain liquid crystal elastomers.**Macromolecules**, 8824-8832. (doi:10.1021/ma501836j) Crossref, ISI, Google Scholar**47** - 38.
Finkelmann H, Greve A, Warner M . 2001 The elastic anisotropy of nematic elastomers.**Eur. Phys. J. E**, 281-293. (doi:10.1007/s101890170060) Crossref, ISI, Google Scholar**5** - 39.
Mistry D, Gleeson HF . 2019 Mechanical deformations of a liquid crystal elastomer at director angles between 0° and 90°: deducing an empirical model encompassing anisotropic nonlinearity.**J. Polymer Sci.**, 1367-1377. (doi:10.1002/polb.24879) ISI, Google Scholar**57** - 40.
Mistry D, Nikkhou M, Raistrick T, Hussain M, Jull EIL, Baker DL, Gleeson HF . 2020 Isotropic liquid crystal elastomers as exceptional photoelastic strain sensors.**Macromolecules**, 3709-3718. (doi:10.1021/acs.macromol.9b02456) Crossref, ISI, Google Scholar**53** - 41.
Urayama K, Mashita R, Kobayashi I, Takigawa T . 2007 Stretching-induced director rotation in thin films of liquid crystal elastomers with homeotropic alignment.**Macromolecules**, 7665-7670. (doi:10.1021/ma071104y) Crossref, ISI, Google Scholar**40** - 42.
Pereira GG, Warner M . 2001 Mechanical and order rigidity of nematic elastomers.**Eur. Phys. J. E**, 295-307. (doi:10.1007/s101890170061) Crossref, ISI, Google Scholar**5** - 43.
Gramsbergen EF, Longa L, de Jeu WH . 1986 Landau theory of the nematic-isotropic phase transition.**Phys. Rep.**, 195-257. (doi:10.1016/0370-1573(86)90007-4) Crossref, ISI, Google Scholar**135** - 44.
Okamoto S, Sakurai S, Urayama K . 2021 Effect of stretching angle on the stress plateau behavior of main-chain liquid crystal elastomers.**Soft Matter**, 3128-3136. (doi:10.1039/d0sm02244f) Crossref, PubMed, ISI, Google Scholar**17** - 45.
Oh SW, Guo T, Kuenstler AS, Hayward R, Palffy-Muhoray P, Zheng X . 2020 Measuring the five elastic constants of a nematic liquid crystal elastomer.**Liq. Cryst.**, 511-520. (doi:10.1080/02678292.2020.1790680) Crossref, ISI, Google Scholar**48** - 46.
Brand HR, Pleiner H, Martinoty P . 2006 Selected macroscopic properties of liquid crystalline elastomers.**Soft Matter**, 182-189. (doi:10.1039/B512693M) Crossref, PubMed, ISI, Google Scholar**2** - 47.
Clarke SM, Tajbakhsh AR, Terentjev EM, Warner M . 2001 Anomalous viscoelastic response of nematic elastomers.**Phys. Rev. Lett.**, 4044-4047. (doi:10.1103/PhysRevLett.86.4044) Crossref, PubMed, ISI, Google Scholar**86** - 48.
Martinoty P, Stein P, Finkelmann H, Pleiner H, Brand HR . 2004 Mechanical properties of monodomain side chain nematic elastomers.**Eur. Phys. J. E**, 311-321. (doi:10.1140/epje/i2003-10154-y) Crossref, PubMed, ISI, Google Scholar**14** - 49.
Rogez D, Brandt H, Finkelmann H, Martinoty P . 2006 Shear mechanical properties of main chain liquid crystalline elastomers.**Macromol. Chem. Phys.**, 735-745. (doi:10.1002/macp.200500573) Crossref, ISI, Google Scholar**207** - 50.
Rogez D, Francius G, Finkelmann H, Martinoty P . 2006 Shear mechanical anisotropy of side chain liquid-crystal elastomers: influence of sample preparation.**Eur. Phys. J. E**, 369-378. (doi:10.1140/epje/i2005-10132-5) Crossref, PubMed, ISI, Google Scholar**20** - 51.
Rogez D, Krause S, Martinoty P . 2018 Main-chain liquid-crystal elastomers versus side-chain liquid-crystal elastomers: similarities and differences in their mechanical properties.**Soft Matter**, 6449-6462. (doi:10.1039/C8SM00936H) Crossref, PubMed, ISI, Google Scholar**14** - 52.
Rogez D, Martinoty P . 2011 Mechanical properties of monodomain nematic side-chain liquid-crystalline elastomers with homeotropic and in-plane orientation of the director.**Eur. Phys. J. E**, 69. (doi:10.1140/epje/i2011-11069-8) Crossref, PubMed, ISI, Google Scholar**34** - 53.
Horgan CO, Murphy JG . 2017 Fiber orientation effects in simple shearing of fibrous soft tissues.**J. Biomech.**, 131-135. Crossref, PubMed, ISI, Google Scholar**64** - 54.
Horgan CO, Murphy JG . 2021 Incompressible transversely isotropic hyperelastic materials and their linearized counterparts.**J. Elast.**, 187-194. (doi:10.1007/s10659-020-09803-7) Crossref, ISI, Google Scholar**143** - 55.
Murphy JG . 2013 Transversely isotropic biological soft tissue must be modeled using both anisotropic invariants.**Eur. J. Mech. A/Solids**, 90-96. (doi:10.1016/j.euromechsol.2013.04.003) Crossref, ISI, Google Scholar**42** - 56.
Murphy JG . 2014 Evolution of anisotropy in soft tissue.**Proc. R. Soc. A**, 20130548. (doi:10.1098/rspa.2013.0548) Link, Google Scholar**470** - 57.
Mihai LA, Goriely A . 2017 How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity.**Proc. R. Soc. A**, 20170607. (doi:10.1098/rspa.2017.0607) Link, Google Scholar**473** - 58.
Cirak F, Long Q, Bhattacharya K, Warner M . 2014 Computational analysis of liquid crystalline elastomer membranes: changing Gaussian curvature without stretch energy.**Int. J. Solids Struct.**, 144-153. (doi:10.1016/j.ijsolstr.2013.09.019) Crossref, ISI, Google Scholar**51** - 59.
DeSimone A . 1999 Energetics of fine domain structures.**Ferroelectrics**, 275-284. (doi:10.1080/00150199908014827) Crossref, ISI, Google Scholar**222** - 60.
DeSimone A, Dolzmann G . 2000 Material instabilities in nematic elastomers.**Physica D**, 175-191. (doi:S0167-2789(99)00153-0) Crossref, ISI, Google Scholar**136** - 61.
DeSimone A, Dolzmann G . 2002 Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies.**Arch. Rational Mech. Anal.**, 181-204. (doi:10.1007/s002050100174) Crossref, ISI, Google Scholar**161** - 62.
Verwey GC, Warner M, Terentjev EM . 1996 Elastic instability and stripe domains in liquid crystalline elastomers.**J. Phys.**, 1273-1290. (doi:10.1051/jp2:1996130) Google Scholar**6** - 63.
Warner M, Bladon P, Terentjev E . 1994 ‘Soft elasticity’—deformation without resistance in liquid crystal elastomers.**J. Phys. II**, 93-102. (doi:10.1051/jp2:1994116) Crossref, Google Scholar**4** - 64.
Goriely A, Mihai LA . 2021 Liquid crystal elastomers wrinkling.**Nonlinearity**, 5599-5629. (doi:10.1088/1361-6544/ac09c1) Crossref, ISI, Google Scholar**34** - 65.
Mihai LA, Goriely A . 2020 A plate theory for nematic liquid crystalline solids.**J. Mech. Phys. Solids**, 104101. (doi:10.1016/j.jmps.2020.104101) Crossref, ISI, Google Scholar**144** - 66.
Mihai LA, Goriely A . 2020 A pseudo-anelastic model for stress softening in liquid crystal elastomers.**Proc. R. Soc. A**, 20200558. (doi:10.1098/rspa.2020.0558) Link, Google Scholar**476** - 67.
Mihai LA, Goriely A . 2021 Instabilities in liquid crystal elastomers.**Mater. Res. Soc. (MRS) Bull.**, 1-11. (doi:10.1557/s43577-021-00115-2) Google Scholar**46** - 68.
Goriely A . 2017**The mathematics and mechanics of biological growth**. New York, NY: Springer. Crossref, Google Scholar - 69.
Lubarda VA . 2004 Constitutive theories based on the multiplicative decomposition of deformation gradient: thermoelasticity, elastoplasticity and biomechanics.**Appl. Mech. Rev.**, 95-108. (doi:10.1115/1.1591000) Crossref, Google Scholar**57** - 70.
Goodbrake C, Goriely A, Yavari A . 2021 The mathematical foundations of anelasticity: existence of smooth global intermediate configurations.**Proc. R. Soc. A**, 20200462. (doi:10.1098/rspa.2020.0462) Link, Google Scholar**477** - 71.
Sadik S, Yavari A . 2017 On the origins of the idea of the multiplicative decomposition of the deformation gradient.**Math. Mech. Solids**, 771-772. (doi:10.1177/1081286515612280) Crossref, ISI, Google Scholar**22** - 72.
Thomson W . 1904**Baltimore lectures on molecular dynamics and the wave theory of light**. Cambridge, UK: Cambridge University Press. Google Scholar - 73.
Mihai LA, Goriely A . 2020 Likely striping in stochastic nematic elastomers.**Math. Mech. Solids**, 1-22. (doi:10.1177/1081286520914958) Crossref, ISI, Google Scholar**25** - 74.
Modes CD, Warner M . 2011 Blueprinting nematic glass: systematically constructing and combining active points of curvature for emergent morphology.**Phys. Rev. E**, 021711. (doi:10.1103/PhysRevE.84.021711) Crossref, ISI, Google Scholar**84** - 75.
van Oosten CL, Harris KD, Bastiaansen CWM, Broer DJ . 2007 Glassy photomechanical liquid-crystal network actuators for microscale devices.**Eur. Phys. J. E**, 329-333. (doi:10.1140/epje/i2007-10196-1) Crossref, PubMed, ISI, Google Scholar**23** - 76.
Mostajeran C . 2015 Curvature generation in nematic surfaces.**Phys. Rev. E**, 062405. (doi:10.1103/PhysRevE.91.062405) Crossref, ISI, Google Scholar**91** - 77.
Warner M, Modes CD, Corbett D . 2010 Curvature in nematic elastica responding to light and heat.**Proc. R. Soc. A**, 2975-2989. (doi:10.1098/rspa.2010.0135) Link, Google Scholar**466** - 78.
Kundler I, Finkelmann H . 1995 Strain-induced director reorientation in nematic liquid single crystal elastomers.**Macromol. Rapid Commun.**, 679-686. (doi:10.1002/marc.1995.030160908) Crossref, ISI, Google Scholar**16** - 79.
Mitchell GR, Davis FJ, Guo W . 1993 Strain-induced transitions in liquid-crystal elastomers.**Phys. Rev. Lett.**, 2947. (doi:10.1103/PhysRevLett.71.2947) Crossref, PubMed, ISI, Google Scholar**71** - 80.
Ball JM, Majumdar A . 2010 Nematic liquid crystals: from Maier-Saupe to a continuum theory.**Mol. Crystals Liquid Crystals**, 1-11. (doi:10.1080/15421401003795555) Crossref, ISI, Google Scholar**525** - 81.
Rivlin RS, Saunders DW . 1951 Large elastic deformations of isotropic materials. VII. Experiments on the deformation of rubber.**Phil. Trans. R. Soc. Lond. A**, 251-288. Link, ISI, Google Scholar**243** - 82.
- 83.
Wang H, Guilleminot J, Soize C . 2019 Modeling uncertainties in molecular dynamics simulations using a stochastic reduced-order basis.**Comput. Methods Appl. Mech. Eng.**, 37-55. (doi:10.1016/j.cma.2019.05.020) Crossref, ISI, Google Scholar**354** - 84.
Ilnytskyi JM, Neher D . 2007 Structure and internal dynamics of a side chain liquid crystalline polymer in various phases by molecular dynamics simulations: a step towards coarse graining.**J. Chem. Phys.**, 174905. (doi:10.1063/1.2712438) Crossref, PubMed, ISI, Google Scholar**126** - 85.
Vlugt TJH, Krishna R, Smit B . 1999 Molecular simulations of adsorption isotherms for linear and branched alkanes and their mixtures in silicalite.**J. Phys. Chem. B**, 1102-1118. (doi:10.1021/jp982736c) Crossref, ISI, Google Scholar**103** - 86.
Wilson MR . 1997 Molecular dynamics simulations of flexible liquid crystal molecules using a Gay-Berne/Lennard-Jones model.**J. Chem. Phys.**, 8654-8663. (doi:10.1063/1.475017) Crossref, ISI, Google Scholar**107** - 87.
Brown WM, Petersen MK, Plimpton SJ, Grest GS . 2009 Liquid crystal nanodroplets in solution.**J. Chem. Phys.**, 044901. (doi:10.1063/1.3058435) Crossref, PubMed, ISI, Google Scholar**130** - 88.
Everaers R, Ejtehadi M . 2003 Interaction potentials for soft and hard ellipsoids.**Phys. Rev. E**, 041710. (doi:10.1103/PhysRevE.67.041710) Crossref, ISI, Google Scholar**67** - 89.
Ilnytskyi JM, Saphiannikova M, Allen MP . 2012 Modelling elasticity and memory effects in liquid crystalline elastomers by molecular dynamics simulations.**Soft Matter**, 11 123-11 134. (doi:10.1039/C2SM26499D) Crossref, ISI, Google Scholar**8** - 90.
Boamfǎ MI, Viertler K, Wewerka A, Stelzer F, Christianen PCM, Maan JC . 2003 Magnetic-field-induced changes of the isotropic-nematic phase transition in side-chain polymer liquid crystals.**Phys. Rev. E**, 050701. (doi:10.1103/PhysRevE.67.050701) Crossref, ISI, Google Scholar**67** - 91.
Stimson LM, Wilson MR . 2005 Molecular dynamics simulations of side chain liquid crystal polymer molecules in isotropic and liquid-crystalline melts.**J. Chem. Phys.**, 034908. (doi:10.1063/1.1948376) Crossref, PubMed, ISI, Google Scholar**123** - 92.
Pritchard RH, Lava P, Debruyne D, Terentjev EM . 2013 Precise determination of the Poisson ratio in soft materials with 2D digital image correlation.**Soft Matter**, 6037-6045. (doi:10.1039/C3SM50901J) Crossref, ISI, Google Scholar**9** - 93.
Wilson MR . 2005 Progress in computer simulations of liquid crystals.**Int. Rev. Phys. Chem.**, 421-455. (doi:10.1080/01442350500361244) Crossref, ISI, Google Scholar**24** - 94.
Thompson AP, Plimpton SJ, Mattson W . 2009 General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions.**J. Chem. Phys.**, 154107. (doi:10.1063/1.3245303) Crossref, PubMed, ISI, Google Scholar**131**