A MATHEMATICAL THEORY FOR NEMATIC ELASTOMERS WITH

A MATHEMATICAL THEORY FOR NEMATIC ELASTOMERS WITH NON-UNIFORM PROLATE SPHEROIDS MARIA-CARME CALDERER, CHUN LIU, AND BAISHENG YAN Abstract. We address some problems arising in the mathematical modeling of nematic elastomers with variable step length tensors and directors with non-uniform length (which represents, in turn, the nonuniform prolate spheroid shapes). Using a similar idea of the Ericksen theory for liquid crystals with variable degree of orientation, we introduce a non-unit vector, called the shape parameter, to describe the step length and the director simultaneously. In addition to the usually used uniaxial elastic energy, we provide a model for the energy contribution due to change of the shape parameters associated with the elastic deformation of the elastomer. This energy is a generalization of the Oseen-Frank free energy for nematic orientational distortions of liquid crystals, but it incorporates coherently the elastic deformation of the network. The total free energy we obtain for the elastomer is suitable for the existence of energy minimizers and other mathematical studies of the elastomers and provides a concrete example that is consistent with both the continuum mechanics theory and the molecular-statistical theory.

1. Introduction Nematic elastomers are unusual materials that simultaneously combine the elastic properties of rubbers with the anisotropy of liquid crystals. They consist of networks of elastic solid chains (the backbones) formed by the cross-linking of nematic crystalline molecules (the mesogens) as the elements of their main-chains and/or pendant side-groups. Because of this structure, any stress on the polymer network influences the nematic order of the nematic, and, conversely, any change in the orientational order will affects the mechanical-shape of the elastomer. The interplay between elastic and orientational changes is responsible for many fascinating properties of such materials that are different from the classical elastic solids and liquid crystals. These materials have been actively studied for both fundamental research and industrial applications. The specific properties of nematic elastomers have been predicted by PG. de Gennes (see, e.g., [7]) long before the first example of such materials has been produced by K¨ ufper and Finkelmann (see [16]). We refer the 1

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MARIA-CARME CALDERER, CHUN LIU, AND BAISHENG YAN

reader to the monograph of Warner & Terentjev [21] for the basic molecularstatistical study and for more references on liquid crystalline elastomers (see also [12, 13, 16, 19, 20]), and to the work of Andersen, Carlson & Fried [2] for the continuum mechanics model of nematic elastomers (seel also [14]). The mathematical study on such materials has been initiated by DeSimone & Dolzmann [8] (see also [5, 9]). The nematic elastomer can be formed in two steps: First, in the melt the material is an isotropic elastic rubber with no defined orientation. Under some preferred procedures (applied fields or temperature change) the nematic chooses an initial shape of a spheroid with a fixed direction n0 and asphericity q0 (that is, the aspect ratio of lengths parallel and perpendicular to the spheroid axis n0 ). Then, the network undergoes a deformation y from the reference domain to the current domain where the nematic takes a shape of a spheroid at a deformed point z with director n ˜ (z) and asphericity q˜(z). We assume the elastic deformation is incompressible and the spheroids are all prolate in the uniaxial direction n ˜ (z). This is to assume the asphericity q(z) ≥ 1 at all points z in the current configuration. These spheroids are usually described by the so-called step-length tensors given by the matrix L(q, n) = q −1/3 [I + (q − 1)n ⊗ n]. In this paper we propose a study of the nematic elastomers following the idea of Ericksen’s theory on nematic liquid crystals with variable degree of ˜ orientation (see [11]). We describe the nematic spheroid by a vector d = d(z) at each point z of the current domain, which we call the shape parameter, through the matrix (1.1)

L(d) = (1 + |d|2 )−1/3 (I + d ⊗ d).

d . If d = 0 Therefore if d 6= 0 then L(d) = L(q, n) with q = 1 + |d|2 and n = |d| 2 then L(0) = L(1, n) = I for all n ∈ S , and the elastomer is in the isotropic state even at the cross-linking; in this case, we consider the material has a ˜ = 0. The choice of the non-unit length vector defect at the point where d(z) ˜ d is our first step towards the complete picture for the coupling between the mechnical deformation and the orientational alignment. Using the shape parameter, a new form of the total free energy of the nematic elastomer consists of three contributions: the nematic elastic energy due to the anisotropy of the nematic, the asphericity energy that memorizes the initial shape of the spheroid, and the Oseen-Frank penalty energy due to the coupling of non-uniform nematic orders and elastic deformations. The last part of the energy has been usually ignored in the study of nematic elastomers. We stress that our main contribution here is to provide a mathematically workable and physically sound theory for the total energy of elastomers including the reasonable Oseen-Frank penalty energy for the cross-linking,

A MATHEMATICAL THEORY FOR NEMATIC ELASTOMERS

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and like Ericksen’s theory for liquid crystals (see [11, 15, 17]), our model permits the defects in the elastomer network. We want to point out the subtlety of the orientation director d. As this microscopic (continuum) variable is obtained through macroscopic average of the microscopic orientation vectors [7], it stands as a variable in the Eulerian coordinate (observer’s coordinate). As is obtained in the context, it already takes into account the influence of the mechanical deformation. We choose to neglect this effect and leave it to our future work. 2. The total free-energy density function The total free energy of the nematic elastomer consists of three contributions: the nematic elastic energy, the asphericity energy, and finally the Oseen-Frank penalty energy due to the non-uniform distribution of the nematic order. 2.1. The nematic elastic energy. The ideal nematic elastic free energy density is a simple extension of the isotropic Gaussian elasticity and is given by the so-called trace formula: µ tr(L0 F T L−1 F ) − 3 , Eel = Eel (d, F ) = 2 where L0 = L(d0 ) and L = L(d) are the step-length tensors (matrices) given by initial vector d0 and current vector d through the formula (1.1). Throughout this paper we assume d0 6= 0 is a constant vector. Note that we can write µ µ 1/2 (2.1) Eel (d, F ) = (|L−1/2 F L0 |2 − 3) = (|B(d)F A0 |2 − 3), 2 2 1/2

where A0 = L0 (2.2) In fact, (2.3)

and −1/2

B(d) = L

2 1/6

= (1 + |d| )

"

# d⊗d p . I−p 1 + |d|2 (1 + 1 + |d|2 )

µ 1 T 2 2 1/3 2 Eel (d, F ) = |A0 F d| − 3 , (1 + |d| ) |F A0 | − 2 1 + |d|2

and hence for all d ∈ R3 and matrices F ∈ M 3×3 we have (2.4)

1 c0 |F |2 − c1 ≤ Eel (d, F ) ≤ |F |2 (1 + |d|2 )1/3 , 2/3 2 c0 (1 + |d| )

where c0 , c1 > 0 are constants depending on µ and d0 . Moreover, if det F = 1, then Eel (d, F ) ≥ 0, and Eel (d, F ) = 0 if and only if B(d)F A0 ∈ SO(3); that is, (2.5)

F ∈ U(d) ≡ B(d)−1 SO(3)A−1 0 .

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Now suppose y(x) is the elastic deformation which maps the reference domain Ω to the current domain D and at each point z ∈ D the shape ˜ of the nematic elastomer is given. Throughout the paper we parameter d(z) assume the reference domain Ω is a bounded open domain with Lipschitz boundary Ω in R3 . ˜ follows the anisotropic uniWe propose the interaction of y(x) and d(z) ˜ axial elasticity and y prefers the axis defined by the vector d(x) = d(y(x)), which we choose to be a state variable defining a shape parameter in the reference domain Ω. The nematic elastic energy is then defined by Z Eel (d(x), ∇y(x)) dx. (2.6) Eel = Eel (d, y) = Ω

2.2. The asphericity energy. We postulate that after deformation the nematic elastomer penalizes the change of asphericity except for the isotropic sphere case and remembers only the initial asphericity constant. we assume the asphericity energy is given by Z ˜ Eãsph = Φ(|d(z)|) dz, D

where Φ(s) is a continuous function satisfying, for some constants c0 > 0, c1 > 0 and 1 ≤ p < 6, (2.7) (2.8)

max{0, c0 |s|2 − 1} ≤ Φ(s) ≤ c1 (|s|p + 1), Φ(s) = 0 ⇐⇒ s = |d0 | or s = 0.

2.3. The modified Oseen-Frank curvature energy. For a nematic elastomer in the current configuration D, we use a modified Oseen-Frank curvature energy to penalize the spatial change of both the directors n ˜ (z) and the asphericity constants q˜(z) wherever they are well-defined. Since our ˜ to represent the both quantities, we propose an theory uses the vector d(z) ˜ 6= 0. Oseen-Frank energy that penalizes the gradient of d˜ wherever d(z) In general, in the absence of applied fields and surface energy terms, an Oseen-Frank curvature energy density function for a unit vector field n ˜ (z) is given by Z D

W (˜ n(z), ∇˜ n(z)) dz,

where W (n, P ) is a function of n ∈ S 2 and P ∈ M 3×3 with the following invariance property: (2.9) (2.10)

W (−n, −P ) = W (n, P ),

W (Rn, RP RT ) = W (n, P )

∀ R ∈ SO(3).

˜ is We assume the modified Oseen-Frank energy for shape parameter d(z) given by Z ˜ ˜ K(d(z), ∇d(z)) dz EÕF = D


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with the density function K(d, P ) given by an Oseen-Frank density function W (n, P ) in terms of ( d/|d| if d 6= 0, (2.11) K(d, P ) = W (ω(d), P ); ω(d) = n0 if d = 0, where n0 = d0 /|d0 |. Note that the assumption K(0, P ) = W (n0 , P ) reflects a postulation that the elastomer may have some memory of its initial director when the current shape is isotropic. The commonly used Oseen-Frank free energy density function is given by (see, e.g., [15]) (2.12) (2.13)

W0 (n, P ) = k1 (tr P )2 + k2 (n · ax(P ))2 + k3 |n × ax(P )|2 +k4 [tr(P 2 ) − (tr P )2 ],

where ax(P ) ∈ R3 denotes the axial vector of matrix P ∈ M 3×3 satisfying (2.14)

(P − P T )v = ax(P ) × v,

∀ v ∈ R3 .

This special form of Oseen-Frank energy defines a special modified OseenFrank energy (2.15)

K0 (d, P ) = W0 (ω(d), P ).

Note that since tr(∇m) = div m and ax(∇m) = curl m for all smooth vector fields m : D → R3 , the constants k1 , k2 and k3 in the energy density function W0 (m, ∇m) represent the corresponding splay, twist and bend curvature elastic constants, respectively. Using the identity (2.16)

|P |2 = (tr P )2 + (n · ax(P ))2 + |n × ax(P )|2 + N (P )

for all |n| = 1, we see that, if k2 = k3 then W0 (n, P ) and thus K0 (d, P ) depend only on P. Furthermore, if (2.17) we have (2.18)

κ = min{k1 , k2 , k3 } > 0, κ|P |2 ≤ K0 (d, P ) + (κ − k4 )N (P ).

If k1 = k2 = k3 = k4 = κ > 0, then one obtains the so-called one-constant Oseen-Frank energy formula: (2.19)

K0 (d, P ) = κ|P |2 .

2.4. The total free energy density function. For the nematic elastomer network we define the total free energy by (2.20) Etotal = Eel + Eãsph + EÕF .

To derive the total energy density function in the reference configuration, we assume for the moment that the domain D in the current configuration is the deformed domain of a domain Ω in the reference configuration by a bi-Lipschitz map y : Ω → D. Let y −1 : D → Ω be the inverse of y.

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For any given function ϕ˜ : D → Rd , we define a function ϕ : Ω → Rd by letting ϕ(x) = ϕ(y(x)) ˜ ∀ x ∈ Ω, d and vice versa any function ϕ : Ω → R defines a function ϕ˜ : D → Rd through ϕ(z) ˜ = ϕ(y −1 (z)). Note that we have ∇ϕ(x) = ∇ϕ(y(x))∇y(x) ˜

and hence

∇ϕ(z) ˜ = ∇ϕ(x)(∇y(x))−1 . We change all the energy integrals on current domain D to the integrals on the reference configuration Ω by the coordinate change z = y(x). Since det ∇y(x) = 1, we have Z K(d(x), ∇d(x)(∇y(x))−1 ) dx EÕF = EOF (d, y) = Ω

and

Eãsph = Easph (d, y) =

Z

Φ(|d(x)|) dx.

Ω

Therefore, we have the total energy is given by Z ψ(d, ∇y, ∇d) dx, (2.21) Etotal = Etotal (d, y) = Ω

where the total free energy density function ψ = ψ(d, F, G) is given by (2.22)

ψ(d, F, G) = Eel (d, F ) + Φ(|d|) + K(d, GF −1 ).

The state variables (d, F, G) for this free energy density function ψ are (2.23)

d ∈ R3 ,

F, G ∈ M 3×3

with det F = 1.

Remark 1. (i) We derived the formula (2.21) under the assumption that y : Ω → D is a bi-Lipschitz map; so once we have d(x) in the reference domain ˜ Ω we would have d(z) = d(y −1 (z)) in the current domain D. However, our final formula (2.21) or (2.22) for the total energy Etotal (d, y) makes no requirement that the deformation y be one-to-one from Ω onto D = y(Ω); ˜ therefore, we may not be able to recover the current shape parameter d(z). (ii) Mathematically, the injectivity of the deformations in nonlinear elasticity has been addressed in Ball [3] and Ciarlet & Neˇcas [4], either by a pure displacement (Dirichlet) boundary with one-to-one boundary data [3] or by imposing the condition below [4], mainly suggested by the area formula: Z det ∇y(x) dx ≤ |y(Ω)|. Ω

(iii) However, there is still a way to recover the current shape parameter ˜ d(z) by defining the admissible class for the total energy as a subset of the joint class C of (d, y) with ˜ C = {(d, y) | y ∈ W 1,∞ (Ω; R3 ), d(x) = d(y(x)) ∃d˜ ∈ W 1,2 (y(Ω); R3 )};


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this is equivalent to requiring d be constant on each level set of y. Remark 2. Our density function ψ(d, F, G) defined by (2.22), when restricted to the unit vector d ∈ S 2 , satisfies all the frame-indifference and material symmetry properties required in Andersen et al [2] for their density functions (cf. (4.38), (4.49) and (4.52) in [2]) based on continuum mechanics theory: (2.24) (2.25) (2.26)

ψ(Rd, RF, RG) = ψ(d, F, G) ∀ R ∈ SO(3),

ψ(d, F Q, GQ) = ψ(d, F, G), ∀ Q ∈ SO(3) with Qd0 = d0 , ψ(−d, F, −G) = ψ(d, F, G).

Therefore, our energy density functions are consistent with the continuum mechanics principles. Remark 3. By a linear change of coordinate x = A0 x ˆ with x ∈ Ω and ˆ ˆ x ˆ ∈ Ω and the change of any function f (x) by f (ˆ x) = f (x), we deduce Z ˆ yˆ) = ˆ d, ˆ ∇ˆ ˆ dˆ (2.27) Etotal (d, y) = Eˆtotal (d, ψ( y , ∇d) x, ˆ Ω

ˆ d, ˆ Fˆ , G) ˆ is the new energy density given by where ψˆ = ψ( µ ˆ Fˆ |2 − 3) + Φ(|d|) ˆ + K(d, ˆG ˆ Fˆ −1 ) (2.28) ψˆ = (|B(d) 2 with the same functions B, Φ and K as above. Therefore, one can assume the initial shape to be an isotropic sphere (i.e., A0 = I); however, the information on d0 can still be encoded in the asphericity energy function Φ(|d|). Therefore, in the rest of the paper, we shall assume A0 = I and use only the special Oseen-Frank energy K0 (d, P ) defined by (2.15). 3. Variational properties of the total energy 3.1. Notation and preliminaries. We first introduce some notation, some of which has already used above. Let M m×n denote the space of m × n matrices with inner product and norm defined by P : Q = tr(P T Q),

|P | = (P : P )1/2 ,

where P T ∈ M n×m denotes the transpose of P. The components of matrix P ∈ M m×n are written as Pij with 1 ≤ i ≤ m and 1 ≤ j ≤ n. In this way, given vector a ∈ Rm and b ∈ Rn , we use a ⊗ b to denote the (rank-one) matrix in M m×n with components given by (a ⊗ b)ij = ai bj .

For P ∈ M m×n , E ∈ M n×d , denote by P E ∈ M m×d the product matrix defined by n X Pik Ekj . (P E)ij = k=1

8


One can easily prove that for all P ∈ M m×n and E ∈ M n×d , |P E| ≤ |P ||E|.

(3.1)

A vector v ∈ Rn will be considered as a matrix v ∈ M 1×n if it is written as a row vector or as a matrix v ∈ M n×1 if it is written as a column vector, depending on the formula used. For our purpose in this paper, it suffices to consider the cases with 1 ≤ m, n, d ≤ 3. Given a square matrix P ∈ M 3×3 we denote by det P the determinant of P and define adj P to be the (unique) matrix satisfying ∀ P ∈ M 3×3 .

P (adj P ) = (adj P )P = (det P )I

In particular, if P is invertible (i.e., det P 6= 0), we have (det P )P −1 = adj P.

(3.2)

Let Ω be a bounded domain with Lipschitz boundary in R3 . For a smooth function u : Ω → Rd we denote by ∇u(x) the Jacobi matrix of u as a d × 3 matrix defined by ∂ui , i = 1, · · · , d; j = 1, 2, 3. ∂xj In this notation, the Jacobi matrix ∇ϕ of a scalar function ϕ : Ω → R will be considered as a row vector. The spaces Lp (Ω; M m×n ) and W 1,p (Ω; Rd ) (1 ≤ p ≤ ∞) will be defined by requiring each component being the usual Lebesgue Lp and Sobolev W 1,p function, respectively. In particular, W 1,∞ (Ω; Rd ) will become the space of all Lipschitz continuous functions from Ω to Rd . The norms of Lp (Ω; M m×n ) and W 1,p (Ω; Rd ) will be the standard extension of the usual Banach space norms. If P : Ω → M d×3 is a smooth d × 3 matrix-valued function, then div P ∈ Rd (a d-column) with component (∇u)ij =

(div P )i =

3 X ∂Pij j=1

∂xj

.

Lemma 3.1. For smooth functions g ∈ C 1 (Ω; R3 ) a direct calculation shows that div[adj(∇g(x))]T = 0, ∀ x ∈ Ω.

3.2. The energy density function. We shall assume A0 = I and use only the special Oseen-Frank energy K0 (d, P ) defined by (2.15). In view of (3.2), we can be write (3.3)

ψ(d, F, G) = Eel (d, F ) + Φ(|d|) + K0 (d, G adj F ).

Let W0 (n, P ) = WOF (n, P ) + k4 N (P ) with (3.4) (3.5)

WOF (n, P ) = k1 (tr P )2 + k2 (n · ax(P ))2 + k3 |n × ax(P )|2 , N (P ) = tr(P 2 ) − (tr P )2 .


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Lemma 3.2. For all P ∈ M 3×3 , N (P ) = −2 tr(adj P ). Since the map ax : M 3×3 → R3 defined by (2.14) above is linear, we have the following Lemma 3.3. Suppose k1 , k2 , k3 are all positive. Then the function WOF (n, P ) is convex in P for any given n ∈ R3 . Hence, for any P, Q ∈ M 3×3 , (3.6)

WOF (n, P + Q) ≥ WOF (n, P ) +

∂WOF (n, P ) : Q. ∂P

Note that the term N (P ) = tr(P 2 ) − (tr P )2 is a null-Lagrangian; that is, Z Z N (∇g0 (x)) dx N (∇g(x)) dx = (3.7) Ω

Ω

W 1,2 (Ω; R3 )

holds for all g, g0 ∈ with g = g0 on ∂Ω. This equality will be generalized later; see (3.18) below. 3.3. Compensated compactness and lower semicontinuity of energies. The total energy Etotal contains terms like ∇d(x) adj ∇y(x). Therefore, to handle these terms, we need the following important result. (Some part of the result should be well-known to the experts, but we include it here for the completeness of the paper.) Theorem 3.4. Let f ∈ W 1,p (Ω; R3 ) and g ∈ W 1,q (Ω; R3 ) with p, q satisfying 1 2 2 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, + ≤ 1. q p Then the following conclusions hold: (i) For all ζ ∈ C01 (Ω) Z Z (3.8) (ζ∇g) adj ∇f dx = − (g ⊗ ∇ζ) adj ∇f dx. Ω

Ω

(ii) There exists a bilinear map L : M 3×3 × M 3×3 → M 3×3 such that, for all ζ ∈ C01 (Ω), Z Z ζ adj ∇f dx = − L(∇f, f ⊗ ∇ζ) dx; (3.9) Ω

Ω

in fact, 1 [adj(P + Q) − adj P − adj Q] . 2 (iii) If f1 ∈ W 1,p (Ω; R3 ) and g1 ∈ W 1,q (Ω; R3 ) satisfy f = f1 and g = g1 on the boundary ∂Ω in the sense of trace, then Z Z ∇g1 adj ∇f1 dx. ∇g adj ∇f dx = (3.10) L(P, Q) =

Ω

Ω

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Proof. (i) Note that ∇(ζg) = ζ∇g + g ⊗ ∇ζ,

(3.11)

hence (3.8) follows from Lemma 3.1. (ii) By considering Ω± = {x ∈ Ω| ± ζ(x)√> 0} and suitable approximations, we may assume ζ ≥ 0 in Ω. Let ϕ = ζ. Since each element of adj P is 2 × 2 subdeterminant of P , it follows that Z Z adj(ϕ∇f + f ⊗ ∇ϕ) dx. adj ∇(ϕf ) dx = (3.12) 0= Ω

Ω

Using adj(a ⊗ b) = 0 and the bilinearity of L we have adj(ϕ∇f + f ⊗ ∇ϕ) = adj(ϕ∇f ) + 2L(ϕ∇f, f ⊗ ∇ϕ) = ϕ2 adj ∇f + L(∇f, f ⊗ ∇(ϕ2 )) = ζ adj ∇f + L(∇f, f ⊗ ∇ζ).

Hence (3.9) follows by (3.12). (iii) Finally to show (3.10), we note that the (i, j)-element of ∇g adj ∇f is exactly the Jacobi determinant of a map u : Ω → R3 : (∇g adj ∇f )ij = det ∇u, where u = (u1 , u2 , u3 ) is defined by uj = g i ,

uk = f k (k 6= j).

Therefore Z

Ω

(∇g adj ∇f )ij dx =

Z

Ω

det ∇u dx

is determined by the boundary values of this u (in the sense of trace). This proves (iii). Theorem 3.5. Let fν ∈ W 1,∞ (Ω; R3 ) and gν ∈ W 1,2 (Ω; R3 ) satisfy fν ⇀ f¯ weakly * in W 1,∞ ,

gν ⇀ g¯

weakly in W 1,2 .

Then it follows that (3.13) (3.14)

adj ∇fν ⇀ adj ∇f¯ weakly * in L∞ (Ω; M 3×3 ), ∇gν adj ∇fν ⇀ ∇¯ g adj ∇f¯ weakly in L2 (Ω; M 3×3 ),

and therefore Z

Ω

|∇¯ g adj ∇f¯|2 dx ≤ lim inf ν→∞

Z

Ω

|∇gν adj ∇fν |2 dx.

We now prove the weak lower semicontinuity for the energies involved in the total energy Etotal .


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Theorem 3.6. Let yν ∈ W 1,∞ (Ω; R3 ) and dν ∈ W 1,2 (Ω; R3 ) satisfy yν ⇀ y¯ weakly * in W 1,∞ , dν ⇀ d¯ weakly in W 1,2 . Let Φ(|d|), WOF (n, P ) and ω(d) be the functions defined above. Then one has Z Z ¯ dx ≤ lim inf (3.15) Φ(|dν |) dx, Φ(|d|) ν→∞ Ω Ω Z Z ¯ ¯ (3.16) WOF (ω(d), ∇d adj ∇¯ WOF (ω(dν ), ∇dν adj ∇yν ). y ) ≤ lim inf ν→∞

Ω

Ω

Proof. (3.15) is easy; so we prove (3.16). Let P¯ = ∇d¯adj ∇¯ y and Qν = 2 3×3 ¯ ∇dν adj ∇yν − ∇d adj ∇¯ y . Then Qν ⇀ 0 weakly in L (Ω; M ). Let nν = ¯ By (3.6), ω(dν ) and n ¯ = ω(d). ∂WOF (3.17) WOF (nν , P¯ + Qν ) ≥ WOF (nν , P¯ ) + (nν , P¯ ) : Qν . ∂P Note that there is a constant C0 such that for any |n| = 1 and P ∈ M 3×3 2 ∂WOF ∂ WOF ≤ C0 . ≤ C |P |, (n, P ) (n, P ) 0 ∂P ∂P 2

¯ Without loss of generality, one assumes dν (x) → d(x) as ν → ∞ for almost ′ ¯ every x ∈ Ω. Let Ω = {x ∈ Ω|d(x) 6= 0}. Then nν (x) → n ¯ ν (x) for a.e. ′ ¯ ¯ x ∈ Ω and WOF (ω(d(x)), ∇d(x) adj ∇¯ y (x)) = 0 a.e. on Ω \ Ω′ . Hence it follows that ∂WOF ∂WOF (nν , P¯ ) → (¯ n, P¯ ) strongly in L2 (Ω′ ; M 3×3 ). ∂P ∂P Therefore Z ∂WOF (nν , P¯ ) : Qν dx = 0, lim ν→∞ Ω′ ∂P from which the theorem follows by integrating (3.17). 3.4. The null-Lagrangian term. Let N (P ) = tr(P 2 ) − (tr P )2 be the null-Lagrangian in the Oseen-Frank energy density function W0 (n, P ). We have the following result, which generalizes (3.7). Theorem 3.7. Let f1 , f2 ∈ W 1,∞ (Ω; R3 ) and g1 , g2 ∈ W 1,2 (Ω; R3 ) with det ∇fi (x) = 1

a.e. x ∈ Ω,

(i = 1, 2)

satisfy f1 = f2 and g1 = g2 on the boundary ∂Ω in the sense of trace. Then Z Z N (∇g2 adj ∇f2 ) dx. N (∇g1 adj ∇f1 ) dx = (3.18) Ω

Ω

Proof. From Lemma 3.2 and using

adj(P Q) = adj Q adj P, we have that for all G, F ∈ (3.19)

M 3×3

adj(adj P ) = (det P )P,

with det F = 1

N (G adj F ) = −2 tr(F adj G).

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Therefore the theorem follows from (3.10) of Theorem 3.4.

Corollary 3.8. Assume κ = min{k1 , k2 , k3 } > 0. Then Z Z K0 (d, ∇d(∇y)−1 ) dx + C(d1 , y1 ) (3.20) κ |∇d(∇y)−1 |2 dx ≤ Ω 1,∞ W (Ω; R3 )

Ω W 1,2 (Ω; R3 ),

for all d ∈ y ∈ with det ∇y(x) = 1 a.e. and d|∂Ω = d1 , y|∂Ω = y1 , where d1 ∈ W 1,2 (Ω; R3 ) and y1 ∈ W 1,∞ (Ω; R3 ) with det ∇y1 (x) = 1 a.e. are given functions, and C(d1 , y1 ) is a constant only depending on d1 , y1 . Proof. This follows from the previous theorem and inequality (2.18).

4. Admissible classes and the existence of minimizers We study the minimization problem for the total nematic elastomer energy in different classes of admissible deformations and shape parameters. Assume the total energy is given by Z ψ(d, ∇y, ∇d) dx Etotal (d, y) = Ω

with ψ defined by (2.22) with K(d, P ) = K0 (d, P ), or equivalently by (3.3). 4.1. Admissible classes. The natural admissible class for shape parameter d is the Sobolev space D = W 1,2 (Ω; R3 ).

The natural class for deformation y is the volume-preserving Lipschitz maps. However since we can not have a priori bounds on the Lipschitz constant or the W 1,∞ -norm of y, we have to assume such a bound in advance. So we define the admissible class of deformations to be the volume-preserving Lipschitz maps with a given uniform Lipschitz constant Λ > 0. Let (4.1)

YΛ = {y ∈ W 1,∞ (Ω; R3 ) | det ∇y(x) = 1 a.e., k∇ykL∞ (Ω) ≤ Λ},

and, for simplicity, Y = Y∞ . 4.2. Minimization with boundary anchoring of both deformation and shape. Let d¯ ∈ D, y¯ ∈ YΛ be given. We define the admissible Dirichlet class for shape and deformation with the boundary anchoring by ¯ (4.2) D ¯ = {d ∈ D | d|∂Ω = d}, d

(4.3)

YΛ,¯y = {y ∈ YΛ | y|Ω = y¯}.

For simplicity, we denote A1 = Dd¯×YΛ,¯y . The strong and weak convergences in these Dirichlet classes will be those induced by the same convergence in the Banach space W 1,2 × W 1,∞ . We easily see that they are sequentially compact in the weak topology. We can prove the following existence result.


13

Theorem 4.1. Assume Etotal is defined by (2.21) and (2.22) with all constants α, µ, k1 , k2 and k3 being positive. Let Λ < ∞. Then there exists a minimizer (d∗ , y ∗ ) ∈ A1 such that Etotal (d∗ , y ∗ ) =

min Etotal (d, y).

(d,y)∈A1

Proof. Let (dν , yν ) ∈ A1 be a minimizing sequence; that is, Eν = Etotal (dν , yν ) → E0 =

inf

(d,y)∈A1

Etotal (d, y).

1. Without loss of generality, we assume yν ⇀ y ∗ weakly star in W 1,∞ (Ω; R3 ). Then y ∗ ∈ YΛ,¯y . 2. We derive a bound for {dν }. By (3.20), we have {∇dν adj ∇yν } is bounded in L2 (Ω; M 3×3 ). Note that, by (3.1), for E, F ∈ M 3×3 with det F = 1 and |F | ≤ Λ, (4.4)

|E| ≤ |F ||EF −1 | ≤ Λ|E adj F |.

This gives a uniform L2 -bound of {∇dν }. Since dν ∈ Dd¯, this implies {dν } is bounded in W 1,2 (Ω; R3 ). Without loss of generality we assume that dν ⇀ d∗ weakly in W 1,2 (Ω; R3 ) and d∗ ∈ Dd¯. 3. From (2.1) we see that the term ψ1 (d, F ) = Eel (d, F ) + Φ(|d|) in the total free energy density function ψ is a convex function in F and hence, by a theorem in [1], the corresponding energy is weakly lower semicontinuous in the class A1 . Therefore, by Theorem 3.6, the whole energy Etotal is weakly lower semicontinuous in the class A1 . 4. Finally we have Etotal (d∗ , y ∗ ) ≤ lim inf Etotal (dν , yν ) = E0 . ν→∞

This shows that (d∗ , y ∗ ) ∈ A1 is a minimizer.

5. Effective elastic response to shape fluctuations In many situations, we need to minimize the energy with the director or shape fluctuating freely without anchoring to the boundary. In such cases, we do not have control on the null-Lagrangian term N (P ) in K0 (d, P ) for general choice of Frank elasticity constants ki . In this section, we will use the one-constant formula for K0 (d, P ); therefore, only consider the energy functional Z h i µ (|B(d)∇y|2 − 3) + Φ(|d|) + κ|∇d(∇y)−1 |2 dx (5.1) I(d, y) = Ω 2

with the same functions B(d) and Φ(|d|) as defined above and constants µ, κ > 0.

14


5.1. Asphericity control. We define, for given constants 0 ≤ θ1 ≤ θ1 ≤ ∞, the class of asphericity-controlled shape parameters D1 = Dθ1 ,θ2 = {d ∈ W 1,2 (Ω; R3 ) | θ1 ≤ |d(x)| ≤ θ2 a.e.}. Note that, in this notation, D = D0,∞ and D1,1 = W 1,2 (Ω; S 2 ). Due to the property of the asphericity energy density function Φ(|d|), we shall see below that the values of θ1 , θ2 affect the elastic response of the elastomer. 5.2. Effective elastic response energy. We introduce an effective elastic response energy due to the shape fluctuation in D1 by (5.2)

J (y) = Jθ1 ,θ2 (y) = inf I(d, y) = min I(d, y). d∈D1

d∈D1

Remark 4. (i) The fact that the minimum in (5.2) is attained can be proved easily by the direct method in a similar way as in Theorem 4.1 above. (ii) Suppose κ = 0 and θ1 = θ2 = |d0 |. DeSimone & Dolzmann [8, 9] (see also [5]) introduced the similar energy J(y) by minimizing I(d, y) over d ∈ L∞ (Ω; |d0 |S 2 ). The functional J(y) they obtained is local and in the form of Z W (∇y(x)) dx

J(y) =

Ω

for a (nonquasiconvex) function W (F ). Then they computed the quasiconvexification of W (F ). (iii) The situation we study here with Frank elasticity constant κ > 0 is quite different, even still with θ1 = θ2 = |d0 | = 1; our response energy J (y) is non-local but weak * lower semicontinuous on y ∈ Y (see Theorem 5.2 below). 5.3. The effective energy-well. We define the energy well for the effective elastic response energy to be Kθ1 ,θ2 = {y ∈ Y | Jθ1 ,θ2 (y) = 0}. Theorem 5.1. (i) Kθ1 ,θ2 = ∅ if 0 < θ1 ≤ θ2 < |d0 | or if |d0 | < θ1 ≤ θ2 . (ii) Let θ1 = 0 and 0 ≤ θ2 < |d0 |. Then y ∈ K0,θ2 if and only if y(x) = F x + b for some F ∈ SO(3) and b ∈ R3 . (iii) Let 0 < θ1 ≤ |d0 | ≤ θ2 . Then y ∈ Kθ1 ,θ2 if and only if y(x) = F x + b for some F ∈ K and b ∈ R3 , where  −1/6  q0 0 0   −1/6 2 (5.3) K = SO(3)  0 q0 0  SO(3), q0 = 1 + |d0 | . 0

0

1/3

q0

(iv) Let θ1 = 0 and θ2 ≥ |d0 |. Then y ∈ K0,θ2 if and only if y(x) = F x + b for some F ∈ K ∪ SO(3) and b ∈ R3 .


15

Proof. Note that J (y) = I(d, y) = 0 if and only if (5.4)

∇d(x) = 0,

|d(x)| ∈ {0, |d0 |},

B(d(x))∇y(x) ∈ SO(3)

for almost every x ∈ Ω. Thus d must be a constant satisfying |d| ∈ {0, |d0 |}. However the condition d ∈ D1 = Dθ1 ,θ2 also restricts such a constant. The result follows by examining the different situations. For example, we only give a proof of the case (iv). In this case, all these constants are in D1 . If d = 0 then ∇y(x) ∈ SO(3) and Liouville’s theorem implies y(x) = F x+b for some F ∈ SO(3). If d 6= 0, we assume d = Rd0 for a constant R ∈ SO(3). Note that B(Rd0 ) = RB(d0 )RT . Hence RB(d0 )RT ∇y(x) ∈ SO(3). This shows ∇˜ y (x) ∈ SO(3) where y˜(x) = RB(d0 )RT y(x). By Liouville’s theorem again, y˜(x) = F x + b for some F ∈ SO(3). Hence y(x) = F˜ x + ˜b where F˜ = RB(d0 )−1 RT F ∈ K as simple algebra shows. R Remark 5. (i) For the functional J(y) = Ω W (∇y(x)) dx, where W (F ) is the function used in DeSimone and Dolzmann [9] (see also (6.4) below), the energy-well would be defined by the set W = {y ∈ W 1,∞ (Ω; R3 ) | W (∇y(x)) = 0}. However, by a result of Dacorogna and Tanteri (see Example 1.6 and Theorem 10.1 in [6]), this energy-well W contains many non-linear deformations which are very singular and thus physically unacceptable. (ii) With the Frank elasticity constant κ > 0, even when θ1 = θ2 = |d0 |, the energy-well here turns out to contain only linear deformations (but may be empty). Also, our model in the regime (i) of the theorem provides a different model for the so-called semi-soft energy (see [21]) for the nematic elastomers, which is mainly due to the adding of the asphericity energy Φ(|d|). 5.4. Lower semicontinuity of the response energy. The following theorem indicates that, unlike the (un-quasiconvexified) response function introduced in [9] for the case κ = 0 and θ1 = θ2 = |d0 | = 1, our response function J (y) is weak * lower semicontinuous in Y. Therefore it has minimizers over any class YΛ with finite Λ < ∞. Theorem 5.2. The energy J : Y → R+ is continuous in the strong topology and lower semicontinuous in the weak * topology of Y; namely, (5.5)

J (¯ y ) ≤ lim inf J (yν )

(5.6)

J (¯ y ) = lim J (yν )

ν→∞

ν→∞

∀ yν ⇀ y¯ weakly * in Y, ∀ yν → y¯ strongly in Y.

Moreover, if yν → y¯ strongly in Y and let dν ∈ D1 be any minimizer of J (yν ); that is, J (yν ) = I(dν , yν ). Then {dν } has a subsequence {dνj } strongly converging in D to a minimizer d¯ of J (¯ y ).

16


Proof. Given y¯, yν ∈ Y, there exist minimizers d∗ , dν ∈ D1 such that J (¯ y ) = I(d∗ , y¯) ≤ I(d, y¯),

(5.7)

J (yν ) = I(dν , yν ) ≤ I(d, yν )

(5.8)

for all d ∈ D1 . In particular, J (yν ) ≤ I(d∗ , yν ). 1. Assume yν ⇀ y¯ weakly * in Y, and assume

lim inf J (yν ) = lim J (yν ) < ∞. ν→∞

ν→∞

Then one can easily show that {dν } is bounded in W 1,2 (Ω; R3 ) and thus can ¯ The definition of J and lower semicontinuity of energy I on assume dν ⇀ d. D × Y imply ¯ y¯) ≤ lim inf I(dν , yν ) = lim inf J (yν ). J (¯ y ) ≤ I(d, ν→∞

ν→∞

This proves (5.5). 2. Assume yν → y¯ strongly in Y. Since I(d∗ , y) is strongly continuous on y ∈ Y, it easily follows that y ). lim sup J (yν ) ≤ lim sup I(d∗ , yν ) = I(d∗ , y¯) = J (¯ ν→∞

ν→∞

Hence (5.6) follows by (5.5). (Note that (5.6) also follows from Step 3 below.) ¯ 3. Again, assume yν → y¯ strongly in Y. As in Step 1, we assume dν ⇀ d. 1,2 3 ¯ We claim dν → d strongly in W (Ω; R ). If this is proved, then it is easily seen that d¯ is a minimizer of J (¯ y ) by taking the limit in I(dν , yν ) ≤ I(d∗ , yν ). To prove dν → d¯ strongly in W 1,2 (Ω; R3 ), we note that −1 ¯ ≤ |∇yν ||∇dν (∇yν )−1 − ∇d(∇y ¯ |∇dν − ∇d| ν) | ¯ ≤ C[|∇dν adj ∇yν − ∇d¯adj ∇¯ y | + |∇d(adj ∇yν − adj ∇¯ y )|]. Hence it suffices to show ∇dν adj yν → ∇d¯adj ∇¯ y strongly in L2 (Ω; M 3×3 ). We already know the weak convergence, so it suffices to show Z Z 2 |∇d¯adj ∇¯ y |2 dx. (5.9) lim sup |∇dν adj ∇yν | dx ≤ ν→∞

Ω

Ω

This can be shown by the energy minimization property of (dν , yν ). In fact, ¯ yν ) it follows that from I(dν , yν ) ≤ I(d, Z Z −1 2 −1 2 ¯ κ |∇dν (∇yν ) | ≤ κ |∇d(∇y ν) | Ω Ω Z Z µ 2 2 ¯ ¯ − Φ(|dν |)] dx; + [|B(d)∇yν | − |B(dν )∇yν | ] + [Φ(|d|) 2 Ω Ω hence (5.9) follows by the Lebesgue dominated convergence theorem.

Corollary 5.3. Let Y1 be either the class YΛ or a Dirichlet class YΛ,¯y with Λ < ∞. Then there exists a y∗ ∈ Y1 such that J (y∗) = min J (y) = y∈Y1

min

(d,y)∈D1 ×Y1

I(d, y).


17

6. Boundary response in the constant asphericity case Similar to the quasiconvexification, we define the boundary response energy to be y ) = min J (y), JΛ# (¯

(6.1)

y∈YΛ,¯ y

and define J # (¯ y ) = inf JΛ# (¯ y ) = lim JΛ# (¯ y ). Λ>0

Λ→∞

JΛ#

(Note that also depends on the class D1 and thus θ1 , θ2 .) The previous corollary implies that JΛ# (¯ y ) = 0 if and only if there exists a linear function Ax + b in the energy well Kθ1 ,θ2 such that y¯|∂Ω = Ax + b. Therefore, the only boundary conditions corresponding to a soft elasticity deformation are those given by the affine maps in the energy-well. We say a boundary function y¯ gives a soft-boundary mode if y ) = 0. J # (¯ y ) = lim JΛ# (¯ Λ→∞

In this section, we assume |d| = |d0 |; that is, in the admissible class Dθ1 ,θ2 we assume θ1 = θ2 = |d0 |. In this case d = |d0 |n with n ∈ S 2 . We consider the energy defined by (5.1) and scale it by some constant to obtain the new energy functional Z (6.2) E(n, y) = [(|∇y|2 − α|∇y T n|2 − β) + γ|∇n(∇y)−1 |2 ] dx, Ω

where α, β, γ are the corresponding rescaled constants. As before, we define, for Λ < ∞, (6.3)

EΛ# (¯ y) =

min

n∈W 1,2 (Ω;S 2 )

y∈YΛ,¯ y

E(n, y);

E # (¯ y ) = lim EΛ# (¯ y ). Λ→∞

We now define the function used in [9] as mentioned before to be (6.4)

W (F ) = min (|F |2 − α|F T n|2 − β) ∀ det F = 1. n∈S 2

Then W (F ) ≥ 0 and W (F ) = 0 if and only if F ∈ K, where K is defined before by (5.3). Let W qc = W rc = W pc be the quasiconvexification of W computed in [9]. Then there are three regimes for the deformations: (6.5) (6.6) (6.7)

R1 = {F | det F = 1, W qc (F ) = W (F )},

R2 = {F | det F = 1, 0 < W qc (F ) < W (F )},

R3 = {F | det F = 1, 0 = W qc (F ) < W (F )}.

Note that the energy-well K ⊂ R1 . Theorem 6.1. Let F ∈ R1 . Then E # (F x + b) = W (F )|Ω|.

18


Proof. Given F ∈ R1 and let Λ > |F | be any number. Hence F x + b ∈ YΛ,F x+b . By the corollary above, there exist y ∗ ∈ YΛ,F x+b and n∗ ∈ W 1,2 (Ω; S 2 ) ≡ D0 such that EΛ# (F x + b) = E(n∗ , y ∗ ). Then it follows that Z qc W (∇y ∗ (x)) dx ≤ E(n∗ , y ∗ ) W (F )|Ω| ≤ Ω Z W (F )dx = W (F )|Ω|, ≤ E(nF , F x + b) = Ω

where nF ∈ S 2 is such that W (F ) = |F |2 − α|F T nF |2 − β by the definition of W (F ). Certainly this implies EΛ# (F x + b) = W (F )|Ω| for all Λ > |F | since W qc (F ) = W (F ). Hence the theorem is proved. Remark 6. (i) This theorem says that in the solid regime R1 the addition of Frank-type energy penalty does not affect the boundary response to both the director and deformation fluctuations. (ii) From the proof, it follows that E # (F x+b) ≥ W qc (F )|Ω| for all F with det F = 1. Hence any nontrivial soft-boundary modes must come from the regime R3 , where, in the case of no Frank energy, the stripe microstructures can render the average deformation with zero cost of energy [5, 13, 20]. Acknowledgement. This work was finished when the authors were visiting the Institute of Mathematics and its Applications of University of Minnesota for the special program “Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities.” We would like to thank the hospitality of the Institute. The authors also want to thank Professors Georg Dolzmann, Eugene Terentjev and Aaron Yip for many helpful discussions. MCC is supported by NSF grant DMS-0128832. CL is partially supported by NSF grant DMS-0405850. BY is partially supported by NSF through the IMA’s New Direction Visiting Professorship. References [1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (1984), 125–145. [2] D. Anderson, D. Carlson and E. Fried, A continuum-mechanical theory for nematic elastomers, J. Elasticity, 56 (1999), 33–58. [3] J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proceedings Royal Soc. Edinburgh, 88A (1981), 315–328. [4] P-G. Ciarlet and J. Neˇcas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal., 97 (1987), 171–188. [5] S. Conti, A. DeSimone and G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: a numerical study, Journal of the Mech. and Physics of Solids, 50 (2002), 1431–1451. [6] B. Dacorogna and C. Tanteri, Implicit partial differential equations and the constraints of nonlinear elasticity, J. Math. Pures Appl., 81 (2002), 311–341. [7] P.G. de Gennes and J Prost, “The Physics of Liquid Crystals,” Clarendon, Oxford, 1993.


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[8] A. DeSimone and G. Dolzmann, Material instabilities in nematic elastomers, Physica D, 136 (2000), 175–191. [9] A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Arch. Rational Mech. Anal., 161 (2002), 175–191. [10] A. DeSimone and R. James, A constrained theory of magnetoelasticity, Journal of the Mech. and Physics of Solids, 50 (2002), 181–204. [11] J. Ericksen, Liquid-crystals with variable degree of orientation, Arch. Rational Mech. Anal., 113 (1991), 97–120. [12] H. Finkelmann, H. Kock and G. Rehage, Liquid crystalline elastomers –a new type of liquid of liquid crystalline materials, Makromol. Chem., Rapid Commun., 2 (1981), 317–322. [13] H. Finkelmann, I. Kundler, E. Terentjev and M. Warner, Critical stripe domain instability of nematic elastomers, J. Physique II, 7 (1997), 1059–1069. [14] E. Fried and R. Todres, Disclinated states in nematic elastomers, Journal of Mechanics and Physics of Solids, 50 (2002), 2691–1716. [15] R. Hart, D. Kinderlehrer and F.H. Lin, Existence and partial regularity of static liquid crystal configurations, Commun. Math. Phys., 105 (1986), 547–570. [16] J. K¨ upfer and H. Finkelmann, Nematic liquid single-crystal elastomers, Makromol. Chem. Rapid Commun., 12 (1991), 717–726. [17] F.-H. Lin, On nematic liquid crystals with variable degree of orientation, Comm. Pure Appl. Math., 44(3) (1991), 453–486. [18] C. B. Morrey, Jr., Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25–53. [19] J. Selinger, H. Jeon and B. Ratna, Isotropic-nematic transition in liquid-crystalline elastomers, Physical Review Letters, 89(22) (2002), (225701), 1-4. [20] M. Warner, P. Blendon, and E. Terentjev, Soft elasticity –deformation without resistance in liquid crystal elastomers, J. Physique II, 4 (1994), 93–102. [21] M. Warner and E. Tenrentjev, “Liquid Crystal Elastomers,” Clarendon, Oxford, 2003. School of Mathematics, University of Minnesota, Minneapolis, MN 55455 E-mail address: [email protected] Department of Mathematics, The Penn State University, University Park, PA 16802 E-mail address: [email protected] Department of Mathematics, Michigan State University, East Lansing, MI 48824 E-mail address: [email protected]