REDUCED MEMBRANE MODEL FOR LIQUID CRYSTAL POLYMER NETWORKS ASYMPTOTICS AND COMPUTATION LUCAS BOUCK RICARDO H. NOCHETTO AND SHUO YANG

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REDUCED MEMBRANE MODEL FOR LIQUID CRYSTAL

POLYMER NETWORKS: ASYMPTOTICS AND COMPUTATION

LUCAS BOUCK, RICARDO H. NOCHETTO, AND SHUO YANG

Abstract. We examine a reduced membrane model of liquid crystal poly-

mer networks (LCNs) via asymptotics and computation. This model requires

solving a minimization problem for a non-convex stretching energy. We show

a formal asymptotic derivation of the 2Dmembrane model from 3Drubber

elasticity. We construct approximate solutions with point defects. We design

a ﬁnite element method with regularization, and propose a nonlinear gradient

ﬂow with Newton inner iteration to solve the non-convex discrete minimization

problem. We present numerical simulations of practical interests to illustrate

the ability of the model and our method to capture rich physical phenomena.

1. Introduction

Liquid crystal polymer networks (LCNs) are materials that combine elastomeric

polymer networks with mesogens (compounds that display liquid crystal proper-

ties). The long rod-like molecules of liquid crystals are densely crosslinked with the

elastomeric polymer network. This contrasts with liquid crystal elastomers (LCEs),

whose crosslinks are less dense. The orientation of the liquid crystal (LC) molecules

can be represented by a director. The orientation of the director inﬂuences defor-

mation of materials when actuated. Common modes of actuation are heating [1, 2]

and light [3, 4]. In this study, we focus on such actuated deformations of LCNs.

LCNs are one of many possible materials that enable spontaneous mechanical

motion under a stimulus. This has been referred to as “the material is the machine”

[5]. Due to this feature, engineers create soft robots using LCNs/LCEs materials,

such as thermo-responsive micro-robots with autonomous locomotion in unstruc-

tured environments [6], soft materials that “swim” away from light [3], and LCN

actuators that can lift an object tens of times its weight [2]. They oﬀer abundant

application prospects, for instance in the design of biomedical devices [7, 8].

Since the deformation of LCNs depend on the orientation of the nematic direc-

tor, the director can be blueprinted or programmed so that the materials achieve

desired shapes [1, 9, 10, 11]. Some methods to program the orientation of the

liquid crystals include mechanical alignment [3], photoalignment [2], and additive

manufacturing [12], which is a subset of 4D printing [13]. Even if the director is

(Lucas Bouck) Department of Mathematics, Carnegie Mellon University, Pittsburgh,

Pennsylvania 15213, USA.

(Ricardo H. Nochetto) Department of Mathematics and Institute for Physical Science

and Technology, University of Maryland, College Park, Maryland 20742, USA.

(Shuo Yang) Beijing Institute of Mathematical Sciences and Applications, Beijing

101408, China.

E-mail addresses:lbouck@andrew.cmu.edu, rhn@umd.edu, shuoyang@bimsa.cn.

Date: March 22, 2024.

arXiv:2210.02710v2 [math.NA] 21 Mar 2024

2 MEMBRANE MODEL FOR LIQUID CRYSTAL POLYMER NETWORKS

constant throughout the material, interesting shapes may occur due to nonuniform

actuation. An example of nonuniform actuation from light is the LCE swimmer in

[3]. Two reviews of experimental work on LCEs/LCNs can be found in [9, 14].

For 3D bodies, one of the most accepted elastic energies for modeling the in-

teraction of the material deformation with the LCs is known as the trace for-

mula [15, 16, 17], although other types of elastic energies have been proposed [18].

Depending on the density of crosslinks, the director ﬁeld may be either totally

free [19, 20] or subject to a Frank elasticity term [21, 22, 23, 24]. This kind of

blueprinted conﬁguration describes the situation where the LCs are unconstrained

or constrained only on a low-level by rubbery polymers. On the other hand, the

LCs may also be frozen into the material via a direct algebraic constraint [25, 26],

an approach that we also follow; see our Eq. (1). The director’s deviation from

(1) may be penalized with a nonideal energy contribution [23, 27]. Although the

LCs may also be subject to a Frank energy term, a key modeling diﬀerence between

LCE/LCNs and nematic LCs is that the former are constrained by the rubber. It is

known that higher degree defects are unstable [28] for the one constant Frank model

of nematic LCs. However in LCNs, higher degree defects do not split apart due

to the constraining nature of the polymer network. We refer to [4] for blueprinted

defects with degrees up to order 10 in LCNs.

The energy scaling with respect to thickness in models of thin 3Delastic bodies

dictates 2D models of LCNs/LCEs. If the energy is scaled linearly with the thick-

ness, the resulting model is a membrane model: the energy is a function of the

ﬁrst fundamental form of the deformed surface and encodes stretching. Works that

studied membrane models include [19, 25, 29]. For LCNs, the ﬁrst fundamental

form of zero stretching energy states satisfy a pointwise metric condition. Exten-

sive work dedicated to examining conﬁgurations that satisfy this metric condition

include [30, 31, 32, 33, 34, 10, 35, 36, 37]. For a review of these techniques, we refer

to [11]. The second common scaling is a cubic scaling in the thickness, and results

in a plate model driven by bending. The metric condition giving zero stretching

energy becomes a constraint in the bending model. Some existing bending models

include theory derived via formal asymptotics [25], a von Karman plate model de-

rived in [38] using asymptotics, a rigorous Gamma convergence theory for a model

of bilayer materials composed of LCEs and a classical isotropic elastic plate [24],

or a plate model where the LC dramatically changes its orientation through the

thickness [39]. Moreover, reduced 1Dmodels for LCNs/LCEs have been explored

as well; we refer to [40] for a rod model and to [41, 42] for ribbon models.

The computation of LCEs/LCNs has received considerable attention in recent

years. Publications include computations of various membrane models [34], a mem-

brane model with regularization [26], a bending model of LCE bilayer structure [24],

a relevant 2D model for LCEs [22], 3D models [43, 44], and LCE rods [40]. Paper

[22] proves well-posedness of a mixed method for a 2D model with Frank-Oseen

regularization.

The goal of this paper is to predict actuated equilibrium shapes of thin LCN

membranes using a ﬁnite element method (FEM). We discretize a membrane energy

of LCNs using piecewise linear ﬁnite elements and add a numerical regularization

that mimics a higher order bending energy. To solve the discrete minimization

problem, we design a nonlinear gradient ﬂow with an embedded Newton method.

Our FEM is able to predict conﬁgurations of LCNs of practical signiﬁcance, whose

MEMBRANE MODEL FOR LIQUID CRYSTAL POLYMER NETWORKS 3

solutions are hard or impossible to derive by hand. We present salient examples

in Section 5.2 of LCNs with preferred discontinuous metric. We complement the

numerical study with the derivation of the LCN model via Kirchhoﬀ-Love asymp-

totics, and the development of a new formal asymptotic method to approximate

shapes of membranes that arise from higher order defects, which we discuss ﬁrst.

Our companion paper [45] provides a numerical analysis of the ﬁnite element

method (FEM) described in this article. We refer to Section 1.4 for a list of our

main contributions and outline.

1.1. 3D elastic energy: Neo-classical energy. We are concerned with thin

ﬁlms of LCNs. Slender materials are usually modeled as 3Dhyper-elastic bodies

B:= Ω ×(−t/2, t/2), with Ω ⊂R2being a bounded Lipschitz domain and t > 0

being a small thickness parameter. We denote by u:B → R3the 3D deformation

and by F:=∇u∈R3×3the deformation gradient of the LCNs material.

We denote by m:B → S2the blueprinted nematic director ﬁeld on the reference

conﬁguration and by n:B → S2the director ﬁeld on the deformed conﬁguration.

The former is dictated by construction of the LCNs material, whereas the latter

obeys an equation that depends on the density of crosslinks between the mesogens

and polymer network. LCNs (also called liquid crystal glasses) have moderate to

dense crosslinks whereas liquid crystals elastomers (LCEs) have low crosslinks [9].

In this paper, we focus on LCNs and leave a numerical study of LCEs for future

research. Mathematically, the strong coupling in LCNs is expressed in terms of the

following kinematic constraint between mand n[25]:

(1) n:=Fm

|Fm|.

In contrast to LCEs [16, 24], nis not a free variable but rather a frozen director

ﬁeld for LCNs [26]. For LCEs the energy density may be minimized over nﬁrst

and next over F, like in [19, 43], or a Frank elastic energy for nmay be introduced

(c.f. [21, 24, 22]). Moreover, we note that a director ﬁeld description may not be

the only choice for modeling LC components. One can also formulate a model with

Q-tensor descriptions like in [46].

The LC eﬀect on the material is governed by the so-called step-length tensors

(2) Lm:= (s0+ 1)−1/3(I3+s0m⊗m)

in the reference conﬁguration, and

(3) Ln:= (s+ 1)−1/3(I3+sn⊗n)

in the deformed conﬁguration. Both of these are uniaxial tensor ﬁelds that exhibit

the typical head-to-tail symmetry commonly observed in LCs. Our deﬁnition of

these step length tensors follows the notation of [25, 47]. The tensor Lmcan be

related to the more commonly encountered step length tensor ℓ0

⊥I3+(ℓ0

∥−ℓ0

⊥)m⊗m

[48, Eq. (13)] by setting ℓ0

⊥= (s0+ 1)−1/3and ℓ0

∥= (s0+ 1)2/3. These step length

tensors measure the anisotropy contributed by nematogenic molecular units to ne-

matic elastomers/networks, which are isotropic solids with a ﬂuid-like anisotropic

ordering. Moreover, s0, s ∈L∞(Ω) are nematic order parameters that refer to the

reference conﬁguration and deformed conﬁguration respectively. They are typically

constant and depend on temperature, but may also vary in Ω if the liquid crystal

4 MEMBRANE MODEL FOR LIQUID CRYSTAL POLYMER NETWORKS

polymers are actuated non-uniformly. These parameters have a physical range

−1< s0, s ≤C < ∞.

Consequently, both Lmand Lnare SPD tensor ﬁelds, which reduce to the identity

matrix, i.e. Lm=Ln=I3∈R3×3, provided s=s0= 0 (no actuation).

The neo-classical energy density for incompressible nematic elastomers/networks

has been proposed by Bladon, Warner and Terentjev in [15, 16, 17] and reads

(4) W3D(x,F):= trFTL−1

nFLm−3;

this energy depends explicitly on the space variable x:= (x′, x3):= (x1, x2, x3)∈ B

due to the dependence of m,n, s, s0on x. The energy (4) can be rewritten as the

neo-Hookean energy density

(5) W3D(x,F) = L−1/2

nFL1/2

m2−3,

and reduces to the classical neo-Hookean energy density for rubber-like materials

W3D(F) = |F|2−3 provided s=s0= 0. Moreover, the material is assumed to be

incompressible, i.e,

(6) det F= 1.

The 3D energy density W3Dis non-degenerate, namely

(7) W3D(x,F)≥distL−1/2

nFL1/2

m, SO(3)2≥0

for all F∈R3×3such that det F= 1. We refer to [23, Appendix A] and [45] for a

proof of this fundamental property.

The 3D elastic energy is given in terms of the energy densities (4) or (5) by

(8) E3D[u] = Zt/2

−t/2ZΩ

W3D(x,∇u)dx′dx3,

where x′∈Ω, x3∈(−t/2, t/2), and det ∇u= 1.

1.2. Model reduction. We assume the 3Dblueprinted director ﬁeld m= ( e

m,0) :

B → S2is planar and it depends only on x′; hence, with a slight abuse of notation,

we identify mand e

mand let m:Ω→S1be the 2Dblueprinted director ﬁeld. We

denote by m⊥:Ω→S1a director ﬁeld perpendicular to meverywhere in Ω, and

by y:Ω→R3the deformation of 2D midplane Ω.

The 2Dmembrane model requires solving the following minimization problem:

ﬁnd y∗∈H1(Ω; R3) such that

(9) y∗= argminy∈H1(Ω;R3)Estr[y], Estr[y]:=ZΩ

Wstr(x′,∇y)dx′,

where Wstr is a stretching energy density that is only a function of x′∈Ω and of

the ﬁrst fundamental form I[y]:=∇yT∇y. It is given by

(10) Wstr(x′,∇y):=λ"1

J[y]+1

s+ 1 tr I[y] + s0Cm[y] + sJ[y]

Cm[y]#−3,

and, since s, s0>−1, the actuation parameter λ:Ω→R+is well-deﬁned by

(11) λ=λs,s0:=3

rs+ 1

s0+ 1.

MEMBRANE MODEL FOR LIQUID CRYSTAL POLYMER NETWORKS 5

If the material is heated, then λ < 1, whereas if it is cooled, then λ > 1. Moreover,

J[y], Cm[y] are among the following abbreviations:

(12) J[y] = det I[y], Cm[y] = m·I[y]m, Cm⊥[y] = m⊥·I[y]m⊥.

When the second argument of Wstr is F∈R3×2, we then use the following nota-

tional abbreviations:

(13)

I(F):=FTF, J(F):= det I(F),

Cm(F):=m·I(F)m, Cm⊥(F):=m⊥·I(F)m⊥.

We emphasize that since s, s0,mdepend on x′∈Ω then Wstr also has an explicit

dependence on x′.

We note that (10) is consistent with the stretching energy in [25] after addition-

ally assuming an inextensibility constraint J[y] = 1 and up to the multiplicative

parameter λand the constant −3.

The energy Estr in (9) is not weakly lower semicontinuous in H1(Ω; R3), which

we show in [45]. As a result, Estr may not have minimizers in H1(Ω; R3), but

may admit minimizing sequences [49]. In fact, one can adapt [45, Example 2.8]

to show that the energy density (10) is not quasiconvex in the sense of [50, Def.

1.5], the correct notion of convexity for a vector-valued problem. The lack of weak

lower semi-continuity is responsible for the main diﬃculties to prove convergence

of our discretization as well as to design eﬃcient iterative solvers for the discrete

minimization problem. We discuss convergence of discrete minimizers in [45], and

present a nonlinear iterative scheme with inner Newton solver in Section 4.3.

Throughout this work, we do not impose any boundary condition so that the ma-

terial under consideration has free boundaries. If necessary, one can take Dirichlet

boundary conditions into account with a simple modiﬁcation on the method.

An important property of the stretching energy is that Wstr(x′,∇y) = 0 if and

only if I[y] = gpointwise, where g∈R2×2is the target metric

(14) g=λ2m⊗m+λ−1m⊥⊗m⊥;

we show this property in Section 2.2. In the physics literature, maps ythat satisfy

the metric constraint I[y] = gare known as spontaneous distortions [16, 30]. The

physics community has developed techniques to ﬁnd such deformations in special

situations. Some examples are radially symmetric director ﬁelds [35], cylindrical

shapes [36], and nonisometric origami [10, 23, 32, 33]. We refer to [11] for a re-

view of the techniques to predict shapes based on the metric. The purpose of this

work is to provide a diﬀerent approach via energy minimization and approxima-

tion. Rather than constructing yanalytically such that I[y] = g, we numerically

approximate minimizers to the stretching energy. We will validate our numerical

method in Section 5 by successfully reproducing the intricate shapes resulting from

higher order defects observed in experimental studies [4], as well as exact noniso-

metric origami solutions [32]. An advantage of employing energy minimization and

numerical approximation is the ability to tackle more general scenarios that lack

exact analytical solutions. We extensively explore incompatible metrics in Section

5.2, which present signiﬁcant challenges when attempting to solve I[y] = gexactly

or study solutions analytically. Our computations inspired lab experiments [51].

1.3. Discretizations. In this work, we propose a FEM discretization to (9). We

consider the space Vhof continuous piecewise linear ﬁnite elements over a shape

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REDUCEDMEMBRANEMODELFORLIQUIDCRYSTALPOLYMERNETWORKS:ASYMPTOTICSANDCOMPUTATIONLUCASBOUCK,RICARDOH.NOCHETTO,ANDSHUOYANGAbstract.Weexamineareducedmembranemodelofliquidcrystalpoly-mernetworks(LCNs)viaasymptoticsandcomputation.Thismodelrequiressolvingaminimizationproblemforanon-convexstretchingenergy.Wes...

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REDUCED MEMBRANE MODEL FOR LIQUID CRYSTAL POLYMER NETWORKS ASYMPTOTICS AND COMPUTATION LUCAS BOUCK RICARDO H. NOCHETTO AND SHUO YANG

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