Preliminaries: Mechanical response and instabilities Monodomain samples and the isotropic reference configurationMonodomain samples and the isotropic reference configuration

THEORY FOR THE ELASTICITY OF NEMATIC ELASTOMERS

2.2 Preliminaries: Mechanical response and instabilities Monodomain samples and the isotropic reference configurationMonodomain samples and the isotropic reference configuration

where the final equality holds when detF = 1 as cofF ∈ R^3×3denotes the cofactor matrix ofF.

Hence, the free energy of a specimen of nematic elastomer occupying a region Ω when it is undeformed and under the deformationy: Ω→R³and with undeformed and deformed director fieldsn₀,n: Ω →S²is written as

E_n0(y,n) := Z

Ω

(

WD(∇y,n,n0)+ κ

µ|(∇n)(cof∇y)^T|² )

dx, (2.14) where we have normalized the energy by the shear modulus via µ/2 as the energy densityWD: R^3×3: R³: R³ →R∪ {+∞}is defined by

WD(F,n,n₀):= (µ/2)⁻¹

W^e(F,n,n₀)+Wⁿⁱ(F,n,n₀)

. (2.15)

The parameterε = pκ/µis likely quite small in nematic elastomers. Specifically, in liquid crystal fluids, the moduli κi (which bound κ) have been measured in detail, and these moduli are likely similar for nematic elastomers (see, for instance, the discussion in Chapter 3 [105]). Further, the shear modulus µof the rubbery network, which is distinct to elastomers, is much larger. Substituting the typical values for these parameters, we findε∼10−100nm. Thus, entropic elasticity will often dominate Frank elasticity in these elastomers. This observation is a key point in many of the results developed herein.

2.2 Preliminaries: Mechanical response and instabilities

wheren₀ ∈ S denotes the undeformed uniform director of the monodomain sam- ple5. This is the isotropic reference configuration. Indeed, we can relate the deformation and deformed director from the monodomain sample to a deformation and deformed director as mapped from the isotropic reference configuration via

y(x) := ym (`^∗_n0)^1/2x

and n(x):= nm (`^∗_n0)^1/2x, x ∈Ω. (2.17) In doing this, we find that

E^m_n

0(ym,nm) = Z

Ω_m

WD(∇ym,nm,n0)+ κ

µ|(∇nm)(cof∇ym)^T|²

dxm

=Z

Ω

WD_iso^e (∇y,n)+WDⁿⁱ((∇y)(`^∗_n0)^−1/2,n,n₀) + κ

µ|(∇n)(cof∇y)^T|² dx.

(2.18)

Here, xm := (`<sup>∗</sup><sub>n0</sub>)<sup>1/2</sup>x for x ∈ Ω, L(·) := (µ/2)<sup>−1</sup>(·), and the identity uses the fact that det(`^∗_n0) =1. Further,W_iso^e : R^3×3×R³→R∪ {+∞}is given by

W_iso^e (F,n) := µ 2











Tr(F^T(`^∗_n)⁻¹F)−3, ifn∈S²,detF =1

+∞ otherwise, (2.19)

whereF ∈R^3×3now denotes the deformation gradient from the isotropic reference Ω, and the anisotropy is all encoded in the deformed configuration through the step-length tensor6

`^∗_n:=r^1/3(I_3×3+(r −1)n⊗n). (2.20) This energy density is frame indifferent and isotropic (i.e., W_iso^e (RFQ,Rn) = W_iso^e (F,n)for allR,Q ∈SO(3); see Proposition A.1.1).

Now, the Finkelmann samples exhibiting soft elasticity and fine-scale material mi- crostructure, as in the sample highlighted in Figure 1.4, are cross-linked in the high temperature isotropic phase. In this case, the parameterαis likely quite small, i.e., α 1 (cf. [17, 18, 105]). In Chapter 3, we are interested in the competition between instabilities inherent to thin sheets and the fine-scale material microstructure asso- ciated to (some) nematic elastomers. At a high level, the reason for soft elasticity and fine-scale material microstructure is the existence of soft modes of deformation

5Note,(`<sup>∗</sup><sub>n0</sub>)<sup>1/2</sup>is equal to(`⁰_n0)^1/2forr₀=r. We introduce the notation(·)^∗so as to distinguish this step-length tensor from a step-length tensor defined later for actuation.

6Again, this step-length tensor is simply`_n^f evaluated forr_f =r.

inherent to the ideal entropic elastic energy (i.e., the deformations depicted in Fig- ure 2.1). This behavior would be suppressed for largeα, but it is well-established (cf. [17, 27, 101]) that for small α, semi-soft behavior and fine-scale material microstructure are still pervasive in these nematic elastomers. Thus, we study the caseα=0, for which the free energy from an isotropic reference (2.18) reduces to

E_iso(y,n) :=Z

Ω

WD_iso^e (∇y,n)dx+ κ

µ|(∇n)(cof∇y)^T|²

dx, (2.21) as this is likely not a far departure from reality for samples cross-linked in the isotropic phase. (We refer the interested reader to Conti and Dolzmann [29] for a recent numerical approach to deriving effective theories for the mechanical behavior of nematic elastomers in the caseα >0.)

Notice that this energy (i.e., (2.21)) now has no dependence on the initial reference director of the undeformed monodomain sample (hence, the terminology: isotropic reference configuration). In Chapter 3, we introduce effective and two dimensional theories for nematic elastomers which are systematically derived from this free energy. That is, the deformations in these theories are all mappings from the isotropic reference configuration, rather than the physical undeformed monodomain sample. Nevertheless, for a monodomain sheet in which the initial director n0 is in the plane of the sheet, one can properly account for the distinction between the isotropic reference state and monodomain reference state via the change of variables (2.17), and thus derive the corresponding theories as energies from the undeformed monodomain state. For clarity, we develop this in Appendix A.2.

As a final remark regarding this energy, we note that Verwey et al. [101] argued that stripe domains with alternating directors can emerge as minimizers of this energy functional, with transition zones proportional to the length-scalep

κ/µ(recall that this length-scale is on the order of 1−100nm). Thus, Frank elasticity need not inhibit very fine material microstructure in nematic elastomers.

A macroscopic three dimensional description via relaxation

In this section, we recall the results of DeSimone and Dolzmann [37] concerning the macroscopic behavior of nematic elastomers. Since the Frank energy is small, we can neglect it while studying the behavior of specimens that are large compared top

κ/µ. Thus, we can define a purely mechanical energy density by minimizing

M(cof F)

M(F) r^1/3 r^1/6

M S

L

(a) (b)

(c) '3D 3D

W_3D^qc t

s t

r^1/3 s r^1/6

r^1/3 r^1/6

'3D 3D

W_3D^qc t

M S

L

Figure 2.4: Macroscopic three-dimensional energy of nematic elastomers following DeSimone and Dolzmann [37]. (a) Contour plots of the functionϕ_3D that describes the entropic elastic energyW3D, (b) contour plots of the functionψ3D that describes the relaxed elastic energyW_3D^qc (i.e., one that implicitly accounts for microstructure and (c) Identification of the regions L, MandS.

out the effect of the directorninW_iso^e . This is given by

W_3D(F) := inf

n∈S²

W_iso^e (F,n) = 









ϕ3D(λM(F), λM(cofF)) if detF =1

+∞ otherwise, (2.22)

where

ϕ3D(s,t) := µ

2 r^1/3 s² r + t²

s² + 1 t²

!

−3

!

, (2.23)

λM(F) is the maximum singular value of F (i.e., the largest principal stretch or the square-root of the maximum eigenvalue ofF^TF andF F^T) andλM(cofF)is the maximum singular value of cofF ∈R^3×3(it is easy to show that the this is also equal to the product of the largest two principal values of F). A contour plot of ϕ_3D is given in Figure 2.4(a).

This energy density is not quasiconvex. Thus, fine-scale microstructure can drive energy minimization in the variational formulation of the elastic energy with this strain energy density, and this leads to a possible non-existence of minimizers. We account for this by replacing W_3D with its relaxation. Mathematically, this is the quasiconvex envelope ofW_3D (see Dacorogna [33]),

W_3D^qc(F)= inf (?

Ω

W3D(F+∇φ)dx: φ ∈W₀^1,∞(Ω,R³) )

, (2.24)

whereW₀^1,∞(Ω,R³)is the space of Lipschitz continuous functionsφ: Ω →R³which vanish on the boundary ofΩ, and>

Ω = _|Ω|¹ R

Ωaverages the energy density overΩ.

DeSimone and Dolzmann [37] computed the analytical expression forW_3D^qc forW3D

in (2.22),

W_3D^qc(F)= 









ψ_3D(λM(F), λM(cofF)) if detF =1

+∞ otherwise,

(2.25)

where

ψ3D(s,t) := µ 2



















0 if (s,t) ∈ L

r^1/3 _2t

r^1/2 + _t¹2

−3 if (s,t) ∈ M r^1/3_s2

r + ^t_s²2 + _t¹2

−3 if (s,t) ∈S, L := {(s,t) ∈R⁺×R⁺: t ≤ s², t ≤r^1/6, t ≥ s^1/2}, M :={(s,t) ∈R⁺×R⁺: t ≤ s², t ≥r^−1/2s², t ≥r^1/6}, S := {(s,t) ∈R⁺ ×R⁺: t ≤ r^−1/2s², t ≥ s^1/2}.

(2.26)

A contour plot of W_3D is given in Figure 2.4(b), and the regions L (ofliquid-like behavior), M (related to stressed microstructure) and S(of normalsolid behavior) are identified in in Figure 2.4(c). Specifically, note that the relaxationW_3D^qc deviates from the energy densityW3Din regionsLandMin Figure 2.4(c). Importantly, these are the regions where macroscopic deformation can be accommodated by fine-scale oscillations in the director field of a nematic elastomer, resulting in the relaxation having lower energy in region M and zero energy in region L. These features were used by Conti et al. [27, 28] to explain soft elasticity and the complex deformation states in the clamped-stretch experiments of Kundler and Finkelmann [58] (Figure 1.4) assuming purely planar deformations.