Examples of pure bending actuation under the metric constraint

ACTUATION OF HETEROGENEOUSLY PATTERNED THIN SHEETS

4.5 Examples of pure bending actuation under the metric constraint

Indeed, suppose the first condition does not hold. Then(u_s^h(t)²−σ₁)² ≥ ¹₄min{σ²₁,(hσi−

σ1)²}> 0 on the interval [−1,−1+c3/2], which gives I_s^h(u^h_s) ≥

Z −1+c₃/2

−1

1

h((u^h_s)²−σ₁)²dt ≥ c₃

8hmin{σ²₁,(hσi −σ₁)²}. (4.96) Taking h > 0 sufficiently small, we eventually arrive at a contradiction to (4.94).

The second condition in (4.95) holds by an identical argument.

Now, by the Sobolev embedding theorem u^h_s ∈ W^1,2((−1,1),R) has a continuous representative. This continuity and the observation that (4.95) holds leads to the non-zero lower bound on the energy. Indeed, we have the estimate

I_s^h(u^h_s) ≥ 2√ c₁

Z 1

−1

|(u^h_s)²−σ(s,t)||(u_s^h)⁰|dt. (4.97) Hence, we define

a:=max

t ∈[−1,1] : u^h_s(t)² = 1

2(σ1+hσi) , b:= min

t ∈ (a,1] : u^h_s(t)²= 1

2(σ₂+hσi) .

(4.98)

By the continuity ofu^h_s and the observation (4.95), these quantities (as asserted) do, in fact, exist. Moreover,

Z 1

−1

|(u^h_s)²−σ(s,t)||(u^h_s)⁰|dt

≥ Z b

a

|(u_s^h)²−σ(s,t)||(u^h_s)⁰|dt ≥ 1 2min

1,2 {|hσi −σi|}

Z b a

|(u^h_s)⁰|dt

≥ 1 2min

1,2 {|hσi −σi|}

Z b a

(u^h_s)⁰dt

= 1 2min

1,2 {|hσi −σi|}|u_s^h(b)−u^h_s(a)|

= 1 2√

2min

1,2 {|hσi −σi|}|(σ2+hσi)^1/2−(σ1+hσi)^1/2|

(4.99) by the fundamental theorem of calculus. Since this lower bound is positive and independent ofsandh, combining (4.97) and (4.99) completes the proof.

Theorem 4.5.1 (Smooth Surfaces) Letr¯ ∈ (0,1) orr¯ > 1. Let n₀ andn₀ satisfy (4.15). If y ∈C³(ω,¯ R³) andn0 ∈ C²(ω,¯ S²) such that (∇y)˜ ^T∇y˜ = `˜n₀ everywhere onω, then forh> 0sufficiently small, there exists ay^h ∈C¹(Ωh,R³)such that

y^h(x,˜ 0) = y(x),˜ x˜ ∈ω I^h

n^h₀(y^h)=O(h³). (4.100) Notice that for this theorem we assume y andn0areC³andC² respectively. Such smoothness is not always necessary. To highlight this, we introduce a large class of y,n which automatically satisfy the two-dimensional metric constraint (4.17).

These surfaces are given as the graph of a function, combined with an appropriate contraction (here we consider cooling, so ¯r > 1). We call these “lifted surfaces”.

They are defined by

y(x)˜ =r¯^−1/6(x1e1+x2e2)+ϕ(r¯^−1/6x˜)e3, (4.101) where the functionϕis from the following set

φ∈W^2,∞(r¯^−1/6ω,R): k∇φk˜ L^∞ < λr_¯ :=r¯−1, suppφ⊂ r¯^−1/6ωm . (4.102) Here, we set ωm := {x˜ ∈ ω: dist(x, ∂ω)˜ > m > 0} (recall that ω ⊂ R² is the midplane of the sheet, a bounded Lipschitz domain). The corresponding director field of a lifted surface is

n0(x)˜ = 1 λ^1/2_r¯

* . . . ,

∂1ϕ(r¯^−1/6x)˜

∂2ϕ(r¯^−1/6x)˜ (λr_¯− |∇ϕ(˜ r¯^−1/6x)˜ |²)^1/2

+ / / / -

. (4.103)

We emphasize that any such choice of y,n₀satisfies (4.17). This fact can be proved by rewriting (4.17) in an equivalent form, which is in fact more practical from the perspective of design, and we discuss this in Section 4.7, which has a focus towards applications.. The first main result is then that lifted surfaces have entropic energy ofO(h³) (and therefore they are good candidates for designable actuation).

Corollary 4.5.2 (Lifted Surfaces) Letr¯ > 1and m > 0. Given a midplane defor- mationyas in (4.101) with ϕtaken from the set (4.102), define the director fieldn0

as in (4.103). Letn^h₀be close ton0in the sense of (4.15).

Then, for every h > 0 sufficiently small, there exists a yδ_h ∈ C³(ω,¯ R³) and an extension y^h ∈C¹(Ωh,R³) such that

y^h(x,˜ 0) = yδ_h(x),˜ x˜ ∈ω, kyδ_h −yk_W^1,∞_(ω) =O(h), I^h

n^h₀(y^h) =O(h³). (4.104)

The key reason why the lifted surface configurations satisfy the O(h³) scaling is that they satisfy the metric constraint, they are sufficiently smooth and (for our proof technique) they can be approximated by even smoother configurations which satisfy the metric constraint (see Remark 4.5.3(ii)). Thus, we can generalize the proof of Theorem 4.5.1 to obtain this result.

Remark 4.5.3 (i) The surfaces of revolution in Aharoni et al. [2] and the designs exploring Gaussian curvature in Mostajeran [72] satisfy the conditions of Corollary 4.5.2. Thus, these designs and their predicted actuation are pure bending configurations in that they have entropic energy of O(h³) (which justifies that they are good candidates to be realized in actuation).

(ii) To arrive at the results presented in this section, we employ techniques of Conti and Dolzmann [30, 31] to construct incompressible three dimensional defor- mationsy^h ∈C¹(Ωh,R³). These techniques rely on the ability to approximate Sobolev functions by sufficiently smooth functions (see the details of Section 2 in Plucinsky et al. [81]). In this direction, an important feature of lifted surfaces is that given any y as in (4.101) with ϕ as in (4.102), there exists a smoothyδ_happroximatingyin theW^2,2(ω,R³)norm which additionally satis- fies∇y˜ δ_h ∈ D_r¯onω(see Theorem 4.7.1 for the context for which the spaceD_r¯ arises). The spaceD_r¯can be thought of as the appropriate generalization to ne- matic anisotropy of the space of matrices representing isometries. Specifically, in the isotropic caser¯= 1,D_r¯reduces toD₁= {F˜ ∈R^3×2: ˜F^TF˜ = I2×2}. The corresponding function space

W_iso^2,2(ω,R³) :={y ∈W^2,2(ω,R³): (∇y)˜ ^T∇y˜ = I2×2 a.e.} (4.105) has been studied extensively in the literature as this is the space of all bending deformations for isotropic sheets (as detailed rigorously by Friesecke et al.

[46]). For instance, Pakzad [77] showed that smooth isometric immersions are dense in W_iso^2,2 as long as the initially flat sheet ω is a convex regular domain. This was later generalized by Hornung [52] for flat sheets which belong to a much larger class of bounded and Lipschitz domains. For nematic elastomers, an appealing analogue to these results would be a similar density result for the space

W_r¯^2,2(ω,R³) := {y ∈W^2,2(ω,R³): ˜∇y ∈ D_r¯ a.e.}. (4.106)

For instance, this space arises in compactness at the bending scale for the combined entropic and Frank energy studied in section 4.6. It does not appear that a result of this type has been considered so far. Our result for non- smooth midplane deformations satisfying ∇y˜ ∈ D_r¯ a.e. is only stated for lifted surfaces, as these are the examples we can explicitly construct and approximate.

Proof of Theorems 4.3.3&4.5.1.

Each of the idealized two dimensional actuations detailed in this work (i.e., noniso- metric origami, sufficiently smooth surface, and lifted surfaces) satisfy the effective metric constraint: (∇y)˜ ^T∇y˜ = `˜n<sub>0</sub> a.e. onω. For extending these to three dimen- sional deformations with low energy, the basic idea is to construct an incompressible extension y<sup>h</sup> which satisfies y<sup>h</sup>(x) ≈ y(x)˜ + x<sub>3</sub>b(x), where˜ b: ω → R<sup>3</sup> is chosen so that (∇y|b)˜ <sup>T</sup>(∇y˜ |b) = `n₀ a.e. onω. The energy of the constructions depends on the regularity of the idealized two dimensional fields y,band n0. (The energy is O(h²) for nonisometric origami and O(h³) for lifted surfaces and sufficiently smooth surfaces).

The importance of regularity considerations is reflected in the construction of y^h. Indeed, if the fields are not regular enough, as is the case with nonisometric origami and some lifted surfaces, we develop δ-dependent approximations yδ and bδ that satisfy(∇y˜ δ|bδ)^T(∇y˜ δ|bδ) ≈`n₀for smallδ. In extending these approximations to a three dimensional incompressible deformation (by parameterizingδ by h), thelow energy argument emerges as a balance between deviations from the metric constraint and abendingpenalty related to localized deformation induced by nearly satisfying the metric constraint.

First, we construct three dimensional incompressible deformations starting from sufficiently smooth two dimensional deformations. These constructions cover all cases of idealized actuation considered in this work. Then, we use these construc- tions to prove theO(h³)energy statement for lifted surfaces and sufficiently smooth surfaces. As part of the proof, we develop two dimensional approximations to the lifted surface ansatz as needed. Finally, we follow analogous steps to prove the O(h²)energy statement for nonisometric origami.

Incompressible extensions

We begin with extensions of the deformations of a planar domain to three dimen- sional incompressible deformations of a thin domain based on the techniques of

Conti and Dolzmann[30, 31].

Lemma 4.5.4 Letα ∈ {−1,0,1}. Suppose for anyδ > 0sufficiently small we have yδ ≡ yδ(α) ∈C³(ω,¯ R³)andbδ ≡ bδ(α) ∈C²(ω,¯ R³)satisfying

det(∇y˜ δ|bδ) = 1 onω, k∇y˜ _δk_L^∞ +kb_δk_L^∞_(ω) ≤ M,

k∇˜∇y˜ _δk_L^∞ +k∇˜bδk_L^∞ ≤ Mδ^min{−α,0}, k∇˜⁽³⁾yδk_L^∞+ k∇˜∇b˜ δk_L^∞ ≤ Mδ^−α−1

(4.107)

for some uniform constantM > 0. Then there exists anm ≡m(M, α) ≥ 1such that for anyh > 0sufficiently small, there exists a uniqueξ^h ≡ ξ^h(α) ∈C¹(Ω_h,R)and an extensiony^h ≡ y^h(α) ∈C¹(Ωh,R³) satisfying

y^h = y_δh+ξ^hbδ_h, with δh =mh

and det∇y^h= 1 onΩ_h. (4.108)

In addition,ξ^hsatisfies the pointwise estimates

|ξ^h−x3| ≤Ch^min{−α,0}|x3|²,

|∂₃ξ^h−1| ≤Ch^min{−α,0}|x₃|,

|∇ξ˜ ^h| ≤ Ch^−α−1|x3|².

(4.109)

everywhere onΩ_h. Here, eachC ≡C(M)and does not depend onh.

We prove this lemma at the end of the section.

Remark 4.5.5 (i) The α ∈ {−1,0,1} dependent hypotheses (4.107) is related to the (sufficiently) smooth approximations to midplane fields y and b which satisfy(∇y|b)˜ ^T(∇y˜ |b) =`n₀ a.e. onωfor idealized actuation. These approxi- mations depend on the regularity of the midplane fieldy,bandn₀. If the fields are smooth enough, then no approximation is required, and this is reflected in the hypotheses withα = −1. Lifted surfaces need not be smooth (i.e., we can havey ∈W^2,∞(ω,R³)\C³(ω,¯ R³)). Consequently, approximations in this case (i.e., yδ(α)) correspond to α = 0. Finally, nonisometric origami actuations are strictly Lipschitz continuous, and as such, the approximations correspond toα= 1.

(ii) We will show below that the three dimensional extensionsy defined in (4.108) have low energy for appropriate choices ofyδandbδ. Moreover, the estimates (4.109) precisely quantify the approximation ξ^h ≈ x3, and these are crucial for the energy argument.

(iii) We can choosem=1forα= {−1,0}. Forα =1, we generally have to choose msuch thatm ≥ max{C(M),1}whereC(M)is a constant that depends on M but is independent ofh.

TheO(h³)energy argument for lifted surfaces and sufficiently smooth surfaces We begin with the case of sufficiently smooth surfaces and programs which satisfy the metric constraint. In this case, we do not have to approximate the midplane fields associated to idealized actuation, and so the approach is straightforward.

Definition of three dimensional deformation for smooth surfaces:Let ¯r > 0. We sup- pose thatn₀∈C²(ω,¯ S²)andy ∈C³(ω,¯ R³)such that(∇y)˜ ^T∇y˜ =`˜n<sub>0</sub> onω. Follow- ing Proposition A.1.9, there exists ab ∈C<sup>2</sup>(ω,¯ R<sup>3</sup>) such that (∇y˜ |b)<sup>T</sup>(∇y|b)˜ = `n₀

and det(∇y˜ |b) =1. The smoothness is due to the regularity ofn₀andy by explicit differentiation of the parameterization in (A.24). Nowyandbsatisfy the hypotheses of Lemma 4.5.4 withα=−1 since these fields areδ-independent. Hence, forh> 0 sufficiently small there exists a ξ^h ∈C¹(Ωh,R) and an extension y^h ∈ C¹(Ωh,R³) with the properties:











y^h := y+ξ^hb, det∇y^h =1 onΩ_h,

|ξ^h−x3| ≤C|x3|², |∂3ξ^h−1| ≤C|x3|, |∇˜ξ^h| ≤ C|x3|² onΩh. (4.110) Proof of Theorem 4.5.1. We first note they^h(x,˜ 0) = y(x)˜ for ˜x ∈ωsinceξ^h(x,˜ 0) = 0 by the first estimate forξ^hin (4.110). So it remains to prove only theO(h³)scaling of the energyI_nh

0(y^h).

We compute explicitly

∇y^h= (∇y|˜ b)+x3(∇˜b|0)+(ξ^h− x3)(∇˜b|0)

+(∂₃ξ^h−1)b⊗e₃+b⊗∇ξ˜ ^h. (4.111) Hence, by the estimates onξ^hin (4.110), we conclude

∇y^h = (∇y|˜ b)+O(x3). (4.112)

By hypothesis,(∇y˜ |b)^T(∇y|b)˜ =`n₀, and so we find that

W^e((∇y|˜ b),n,n0) =WnH((`n<sup>f</sup>)<sup>−1/2</sup>(∇y|˜ b)(`⁰_n0)^1/2) =0 onω (4.113) following Proposition A.1.7 and the identity (2.3). Here, we have set

n:= (∇y˜ |b)n₀

|(∇y˜ |b)n₀| onω. (4.114)

Since the energy density (4.113) vanishes, we deduce from Proposition A.1.4 that (`<sub>n</sub><sup>f</sup>)<sup>−1/2</sup>(∇y|b)(`˜ ⁰_n0)^1/2=: R ∈SO(3) onω. (4.115) Now, we letn^h:= (∇y^h)n^h₀/|(∇y^h)n^h₀|onΩ_h, and observe that

(`⁰_nh

0

)^1/2 = (`⁰_n0)^1/2+O(h), (4.116) where the equality follows from the scaling of the non-ideal terms in (4.15). Addi- tionally given (4.112), we conclude

(`^f

n^h)^−1/2= (`_n^f)^−1/2+O(x₃)+O(h). (4.117) Hence, combining the estimates (4.112), (4.116) and (4.117), we find

W^e(∇y^h,n^h,n^h₀) =WnH((`<sub>n</sub><sup>f</sup>)<sup>−1/2</sup>(∇y|b)(`˜ ⁰_n0)^1/2+O(x₃)+O(h))

=WnH(R^T(`<sub>n</sub><sup>f</sup>)<sup>−1/2</sup>(∇y|b)(`˜ ⁰_n0)^1/2+O(x₃)+O(h))

=WnH(I_3×3+O(x₃)+O(h))= O(h²) onω.

(4.118)

For the last equality, we used the definition of R in (4.115) and for the inequality, we used the estimate in Proposition A.1.3. Since this inequality holds on all ofω,

I^h

n^h₀(y^h) = Z h/2

−h/2

Z

ωW^e(∇y^h,n^h,n₀^h)dx= O(h³). (4.119)

This completes the proof.

We now apply Lemma 4.5.4 to the case of lifted surfaces.

Definition of three dimensional deformations for lifted surfaces: Let ¯r > 1. We suppose{ϕ,y,n₀}are as in the lifted surface ansatz (i.e.,ysatsifying (4.101) andn₀ satisfying (4.103) forϕas in (4.102) for somem> 0) andn₀^his as in (4.15). Forδ >0 sufficiently small, there existδ−dependent functions{ϕδ,yδ,n0,δ,bδ}approximating this ansatz as detailed in Propositions 4.5.6-4.5.8 at the end of this section. The approximations satisfy (∇y˜ δ|bδ)^T(∇y˜ δ|bδ) ≈ `n₀, and they are sufficiently smooth

so that we can apply Lemma 4.5.4 withα=0 when we setδh≡ δ = h(see Remark 4.5.5(iii)). Thus forh > 0 sufficiently small, there exists aξ^h ∈ C¹(Ωh,R) and an extension y^h ∈C¹(Ωh,R³)with the properties:











y^h:= y_δh +ξ^hbδ_h, det∇y^h =1 onΩ_h,

|ξ^h− x₃| ≤C|x₃|², |∂₃ξ^h−1| ≤C|x₃|, |∇ξ˜ ^h| ≤ Ch⁻¹|x₃|² onΩ_h

(4.120) forC = C(∇y)˜ > 0 independent of h. With this construction, we prove Corollary 4.5.2:

Proof of Corollary 4.5.2. We first note the y^h(x,˜ 0) = y_δh(x)˜ for ˜x ∈ ω since ξ^h(x,˜ 0) =0 by the first estimate forξ^hin (4.120). Moreover,ky_δh−yk_W1,∞ =O(h) is shown in Proposition 4.5.7. So it remains to prove only theO(h³) scaling of the energyI_nh

0(y^h).

For this, we note that given the estimates and properties established in Proposition 4.5.6-4.5.8, the fact that we are smoothing on a length scale δh ≡ δ = h and the estimates forξ^hin (4.120), the proof here follows exactly the same line of arguments as in the theorem above by replacing{y,n0,n,b,R}with{yδ_h,n0,δ_h,nδ_h,bδ_h,Rδ_h}.

It remains to construct theδ-dependent smoothings{ϕ_δ,n0,δ,y_δ,bδ}asserted in the definition of three dimensional deformations for lifted surfaces.

Construction of ϕδ. Consider anyϕas in (4.102) form > 0. We extendϕ to all of R²yieldingϕ∈W^2,∞(R²,R³)(the extension is not relabeled), and we set

ϕδ :=ηδ ∗ϕ on ¯r^−1/6ω (4.121) for a standard mollifierηδ supported on a ball of radiusδ/2. For this mollification, we have:

Proposition 4.5.6 Forδ >0sufficiently small,ϕδin (4.121) belongs toC₀^∞(r¯^−1/6ω,R) and satisfies the estimates

kϕ−ϕδk_W1,∞ = O(δ), k∇˜ϕδk_L^∞ < λr,

k∇˜⁽ⁿ⁾ϕδk_L^∞ =O(δ²⁻ⁿ), for any integern ≥ 2. (4.122) Proof. ϕδis smooth by mollification. It vanishes on the boundary of ¯r^−1/6ωforδ >0 sufficiently small since by (4.102), sptϕ ⊂ r¯^−1/6ωm := r¯^−1/6{x˜ ∈ ω: dist(x, ω)˜ >

m}and sinceηδis supported on a ball of radiusδ/2. From standard manipulation of the mollification (4.121), the estimate on theW^1,∞norm follows from the Lipschitz continuity ofϕand ˜∇ϕ, the estimate on ˜∇ϕδfollows from that fact thatk∇˜ϕk_L^∞ < λr, and the estimates on the higher derivatives follow from the fact that ˜∇∇˜ϕ∈ L^∞. Construction ofn_0,δ and y_δ. We replace ϕ in the lifted surface ansatz (4.101) and (4.103) with ϕδ from the proposition above and define n0,δ as in (4.103) andyδ as in (4.101) with this replacement. We make the following observations:

Proposition 4.5.7 Letδ > 0sufficiently small. Letn0,δ andyδbe as defined above for ϕδ as in (4.121), ϕ as in (4.102), n₀ as in (4.103) and y as in (4.101). Then n0,δ ∈C^∞(ω,¯ S²)andyδ ∈C^∞(ω,¯ R³)and they satisfy

(∇y˜ δ)^T(∇y˜ δ)= `˜n0,δ onω

kn0,δ−n₀k_L^∞ =O(δ), kyδ−yk_W1,∞ =O(δ), k∇y˜ _δk_L^∞ + k∇˜∇y˜ _δk_L^∞ + k∇˜n0,δk_L^∞ ≤C, k∇˜⁽³⁾yδk_L^∞ +k∇˜∇n˜ 0,δk_L^∞ ≤ Cδ⁻¹

(4.123)

forCindependent ofδ.

Proof. These properties are a consequence of the properties on ϕδ established in Proposition 4.5.6. In particular, smoothness follows since ϕδ is a mollification;

the metric constraint holds by the equivalence (4.205) since k∇˜ϕδk_L^∞ < λr; the estimates on the approximationsn0,δ−n0andy_δ−yfollow from theW^1,∞estimate ofϕ_δ−ϕusing the explicit definition of each field; and theδ-dependent derivative estimates follow from the δ-dependent derivative estimates of ϕδ again using the

explicit definition of each field.

Construction ofbδ. We construct the out-of-plane vectorbδ: ω →R³to ensure the metric constraint is satisfied at the midplane:

Proposition 4.5.8 Let δ > 0 sufficiently small. Let n^δ₀ and yδ as in Proposition 4.5.7. There exists abδ ∈C^∞(ω,¯ R³)such that

(∇y˜ δ|bδ)^T(∇y˜ δ|bδ)= `_n^δ

0, det(∇y˜ δ|bδ) =1,

kbδk_L^∞ +k∇b˜ δk_L^∞ ≤ C, k∇˜∇b˜ δk_L^∞ ≤ Cδ⁻¹ (4.124) forCindependent ofδ.

Proof. Since by Proposition 4.5.7, we have (∇yδ) ∇yδ = `n0,δ everywhere on ω, we apply Proposition A.1.9 pointwise everywhere on ω. Thus, we define the vector bδ: ω → R³ as in (A.24) with ˜∇y_δ replacing ˜F and n0,δ replacing n0 in these relations. Hence, (4.124) holds on ω. Smoothness follows since n0,δ, y_δ and the parameterization (A.24) are each themselves smooth. The estimates on the derivatives of b_δ follow from the estimates on the derivative of y_δ and n₀^δ in Proposition 4.5.7 by explicit differentiation of the parameterization in (A.24).

TheO(h²) energy argument for nonisometric origami

We now apply Lemma 4.5.4 to the case of nonisometric origami. This requires the existence of aδ-smoothing of yin the sense of Definition 4.3.2.

Definition of three dimensional deformations for nonisometric orgiami: Let ¯r > 0.

We supposeω ⊂ R²andn0satisfy Definition 4.3.1(i), ysatsifies Definition 4.3.1(ii) and n^h₀ satisfies (4.15). In addition, we assume there exists a δ-smoothing y^δ ∈ C³(ω,¯ R³)as in Definition 4.3.2.

In Proposition 4.5.9 below, we prove that the existence of a δ-smoothing yδ also guarantees the existence of a vector field bδ that complements yδ. (By this, we mean that it satisfies (∇y˜ δ|bδ)^T(∇y˜ δ|bδ) ≈`n₀, and it is sufficiently smooth so that we can apply Lemma 4.5.4 with α = 1.) Thus by Lemma 4.5.4, there exists a m =m(∇y)˜ ≥ 1 such that forh> 0 sufficiently small there exists aξ^h ∈C¹(Ωh,R) and an extensiony^h ∈C¹(Ωh,R³)with the properties:











y^h := y_δh+ξ^hbδ_h with δh= mh, det∇y^h= 1 onΩh,

|ξ^h−x₃| ≤ Ch⁻¹|x₃|², |∂₃ξ^h−1| ≤Ch⁻¹|x₃|, |∇ξ˜ ^h| ≤Ch⁻²|x₃|² onΩ_h (4.125) forC = C(∇y)˜ > 0 independent of h. With this construction, we prove Theorem 4.3.3.

Proof of Theorem 4.3.3. We first remark that y^h(x,˜ 0) = y_δh(x)˜ for every ˜x ∈ ω sinceξ^h(x,˜ 0)= 0 following the first estimate forξ^hin (4.125). Further since y_δh is aδh-smoothing of y(recall definition 4.3.2), we find ky_δh −yk_W1,2 = O(h). Thus, it remains only to show that the energy scales asO(h²) for this deformation.

To this end, we first compute∇y^hexplicitly. We find that

∇y^h= (∇y˜ δ_h|0)+ξ^h(∇b˜ δ_h|0)+(∂₁ξ^hbδ_h|∂₂ξ^hbδ_h|∂₃ξ^hbδ_h), (4.126)

and note that from Proposition 4.5.9, ˜∇bδ_h = 0 on the setω \ω˜δ_h where |ω˜δ_h| = O(δh) = O(h). It follows thatξ^h = x3on this set. Indeed, since det∇y^h = 1, we find that onω\ω˜δ_h,

1=det((∇y˜ _δh|0)+(∂₁ξ^hbδ_h|∂₂ξ^hbδ_h|∂₃ξ^hbδ_h)) =∂₃ξ^hdet(∇y˜ _δh|bδ_h). (4.127) Also from Proposition 4.5.9, det(∇y˜ δ_h|bδ_h) = 1. Thus, ∂₃ξ^h = 1 on ω \ω˜δ_h. Consequently,ξ^h= x₃on this set since we have the conditionξ^h(x,˜ 0) =0.

To recap, we find that

∇y^h = (∇y˜ δ_h|bδ_h) onω\ω˜δ_h. (4.128) On the exceptional set ˜ωδ_h, we find that

|∇y^h|=

(∇y˜ δ_h|bδ_h)+(∂₃ξ^h−1)bδ_h ⊗e₃

+ x3(∇b˜ δ_h|0)+(ξ^h− x3)(∇b˜ δ_h|0)+bδ_h ⊗∇ξ˜ ^h

≤ |(∇y˜ _δh|b_δh)|+|∂₃ξ^h−1||b_δh|

+ |x₃||∇b˜ δ_h|+|ξ^h− x₃||∇b˜ δ_h|+|bδ_h||∇ξ˜ ^h|

≤ C

1+h⁻¹|x₃|+h⁻²|x₃|²

≤ C,

(4.129)

where each C = C(∇˜y,m(∇˜y)) > 0 is independent of h. These estimates follow from the estimates (4.21), (4.139) in Proposition 4.5.9, and (4.125).

Now, we recall from Proposition 4.5.9 that(∇y˜ δ_h|bδ_h)^T(∇y˜ δ_h|bδ_h) =`n₀onω\ω˜δ_h. Thus,

W^e((∇y˜ _δh|bδ_h),nδ_h,n0)

=WnH((`_n^f_δ

h)^−1/2(∇y˜ δ_h|bδ_h)(`⁰_n0)^1/2) =0 onω\ω˜δ_h

(4.130) following Proposition A.1.7 and the identity (2.3). Here, we have set

n_δh := (∇y˜ δ_h|bδ_h)n₀

|(∇y˜ δ_h|bδ_h)n₀| onω. (4.131) Since the energy density (4.130) vanishes, we deduce from Proposition A.1.4 that

(`_n^f_δ

h)^−1/2(∇y˜ δ_h|bδ_h)(`⁰_n0)^1/2=: R^h ∈SO(3) onω\ω˜δ_h. (4.132) We have yet to account for the non-ideal terms on this set as n₀^h in (4.15) is the appropriate argument for the energy density, not n₀. To do this, we exploit the observation in (4.132). Indeed, we set

n^h:= (∇y^h)n₀^h

|(∇y^h)n₀^h| onΩ_h (4.133)

and observe

(`⁰

n^h₀)^1/2= (`<sup>0</sup><sub>n0</sub>)<sup>1/2</sup>+O(h), (`_n^f_h)^−1/2= (`n^f_δ

h)^−1/2+O(h) onω\ω˜δ_h

(4.134) following (4.128) and the scaling of the non-ideal term in (4.15). Hence onω\ω˜δ_h, we find

W^e(∇y^h,n^h,n₀^h)=WnH((`n^f_δ

h)^−1/2(∇y˜ _δh|bδ_h)(`⁰_n0)^1/2+O(h))

=WnH((R^h)^T((`_n^f_δ

h)^−1/2(∇y˜ _δh|b_δh)(`⁰_n0)^1/2+O(h)))

=WnH(I_3×3+O(h)) =O(h²).

(4.135)

For the equalities, we used (4.128), (4.134), the frame invariance of WnH, and (4.132). For the inequality, we used the estimate in Proposition A.1.3.

Now, on the exceptional set ˜ωδ_h, we have

W^e(∇y^h,n^h,n^h₀) ≤ c(|∇y^h|²+1) ≤C (4.136) given the estimate in Proposition A.1.2 and (4.129). Thus, on the set ˜ωδ_h, the energy is|O(1)|compared to hbut this set is small for nonisometric origami, i.e.,

|ω˜δ_h| = O(δh) = O(h)given δh = mh in (4.125). Hence, combining the estimates (4.135) and (4.136), we conclude

I^h

n^h₀(y^h) = Z h/2

−h/2

Z

ω˜_δhW^e(∇y^h,n^h,n₀^h)dx +

Z h/2

−h/2

Z

ω\ω˜δh

W^e(∇y^h,n^h,n₀^h)dx

≤Ch|ω˜δ_h|+O(h³) =O(h²).

(4.137)

This completes the proof.

Proposition 4.5.9 Let r¯ > 0. Let ω and n₀ satisfy Definition 4.3.1(i) and y ∈ W^1,∞(ω,R³)satisfy Definition 4.3.1(ii). If there exists a δ-smoothing yδ of y as in definition 4.3.2, then forδ >0sufficiently small, there exists abδ ∈C²(ω,¯ R³)such that

(∇y˜ _δ|bδ)^T(∇y˜ _δ|bδ)= `n₀ and ∇˜bδ = 0 onω\ω˜δ with|ω˜δ|=O(δ),

det(∇y˜ δ|bδ)= 1 everywhere onω.

(4.138)

Moreover,bδ satisfies

kbδk_L^∞ ≤C, k∇b˜ δk_L^∞ ≤ Cδ⁻¹, k∇˜∇b˜ δk_L^∞ ≤ Cδ⁻² (4.139)

everywhere onωfor someC > 0which can depend on yandn₀but is independent ofδ.

Proof. From Proposition A.1.9, if ˜F ∈R^3×2andn₀∈S²such that ˜F^TF˜ = `˜n<sub>0</sub>, then there exists ab≡ b(F,˜ n<sub>0</sub>) ∈R<sup>3</sup>such that(F˜|b)<sup>T</sup>(F˜|b) =`n₀and det(F˜|b) =1. The parameterization is explicit, i.e., (A.24). Hence, we set

bδ := b(∇y˜ δ,n0,δ) onω (4.140) for theδ-smoothingyδand the directorn0,δ ∈C^∞(ω,¯ S²)given below in Proposition 4.5.10. The parameterization b(F,˜ n0) is smooth in its arguments when |F˜e˜1 × F˜e˜2| is bounded away from zero. Consequently, (4.139) holds by the chain rule given the properties of the δ-smoothing y_δ and that n_0,δ satisfies (4.141). Further det(∇y˜ _δ|b_δ) =1 everywhere onω as the parameterization ensures this (even when the metric constraint is not satisfied).

It remains to verify the first two properties in (4.138). To this end, note for δ sufficiently small we have thatyδ = yexcept on a set of measureO(δ)(by hypothesis of aδ-smoothing) and thatn0,δ = n0except on (perhaps a different) set of measure O(δ) (Proposition 4.5.10 below). Therefore, we conclude that there is a set ˜ωδ of measureO(δ) such that y_δ = y andn0,δ = n0 onω\ω˜_δ. Moreover, ˜∇y = const., n₀ = const. and (∇y)˜ ^T∇y˜ = `˜n₀ in any connected region in ω \ω_δ. Hence, we conclude the first two properties in (4.138) given (4.140) for b as in Proposition

A.1.9.

To construct bδ, we utilized a smoothing approximation for the piecewise constant direction designn0: ω →S²akin to a construction of DeSimone (Assertion 1 [36]).

Precisely:

Proposition 4.5.10 Letr¯ >0. Letωandn₀satisfy (4.19). For anyδ >0sufficiently small, there exists ann0,δ ∈C^∞(ω,¯ S²)which satisfies

n0,δ = n0 onω\ωδ with |ωδ|=O(δ),

k∇n˜ 0,δk_L^∞ ≤Cδ⁻¹ and k∇˜∇n˜ 0,δk_L^∞ ≤Cδ⁻². (4.141) HereC≡C(n₀) > 0is independent ofδ.

Proof. Given thatω= ∪_α=1,...,Nωαfor connected polygonal regionsωαandn0: ω → S²satisfiesn0= n0αon eachωα, there exists aν ∈S²such thatB(ν)∩range{n0} =∅

for some >0. We letΠ_ν: S \ {ν} →R denote the stereographic projection with projection pointν. This map is bijective (i.e., there exists a Π⁻¹_ν : R² → S²\ {ν}).

Thus, we extendn0 to all ofR² by setting n0 = n01 forR²\ω (we do not relabel) and we define

n0,δ(x)˜ = (Π⁻¹_ν ◦ (η_δ∗(Π_ν◦ n0)))(x),˜ x˜ ∈ω. (4.142) Hereη_δ ∈C^∞(R²,R) is the standard mollifier onR²supported on a ball of radius δ/2.

We claim that this map has all the properties stated in the proposition. Indeed,Π_ν◦n0

maps to a compact subset ofR²given thatν is at least away from anyn_0α. Thus, kΠ_ν ◦n₀k_L^∞ ≤ CforC ≡C(n₀) > 0. Consequently,η_δ∗(Π_ν ◦ n₀) ∈C^∞(R²,R²) with

k∇˜(ηδ ∗(Πν◦ n0))k_L^∞ ≤ Cδ⁻¹

k∇˜∇(η˜ δ∗ (Πν ◦ n₀))kL^∞ ≤ Cδ⁻² (4.143) given thatηδis the mollifier as above. HereC ≡C(n₀) > 0 is independent ofδ >0.

NowΠ_ν⁻¹is smooth. Thus, n_0,δ ∈C^∞(ω,¯ S²) and by the chain rule, we deduce the estimates in (4.141).

For the equality condition in (4.141), we set ωδ = {x˜ ∈ ω: dist(x, ∂ω˜ α) ≤ δ/2 for someα ∈ {1, . . . ,N})}. Clearly this set has measure O(δ) for δ > 0 sufficiently small. Moreover, we observe that

ηδ∗(Πν◦ n₀) =IIν◦n₀ onω\ωδ (4.144) sincen₀ = const. on Bδ/2(x)˜ for any ˜x ∈ ω\ωδ. Given this and the definition of n0,δ in (4.142), we deduce the equality in (4.141). This completes the proof.

Proof of Lemma 4.5.4. (On incompressibility)

Here we prove Lemma 4.5.4, which develops (and catalogues properties associ- ated with) explicit constructions of incompressible deformations for thin elastomer sheets. Proof of Lemma 4.5.4. We set δh = mh for m ≥ 1 to be determined in Proposition 4.5.11 below. We consider the functionv^h≡ v^h(α)given by

v^h(x,˜ x3) := y_δh(x)˜ +x3bδ_h(x)˜ (4.145) and assumex₃ ∈(−h/2,h/2). Since∇v^h= (∇y˜ _δh|b_δh)+x₃(∇b˜ _δh|0)and det(∇y˜ _δh|b_δh)= 1, we letS^h≡ S^h(α) := (∇y˜ δ_h|bδ_h)⁻¹(∇b˜ δ_h|0)and find

det∇v^h= det((∇y˜ _δh|bδ_h)⁻¹∇v^h) =det(I+x3S^h)

= 1+x₃Tr(S^h)+x²₃Tr(cofS^h)+ x³₃det(S^h). (4.146)

For the estimates below,C ≡ C(M). We note that k(∇y˜ δ_h|bδ_h)⁻¹k_L^∞_(ω) ≤ C since the determinant is unity, and therefore |S^h| ≤Cδ^min{−α,0}_h by hypothesis and

|det∇v^h−1| ≤C

|x3|δ^min{−α,0}_h +|x3|²δ2 min{−α,0}

h +|x3|³δ3 min{−α,0}

h

≤ C|x₃|δ^min{−α,0}_h for α ∈ {−1,0,1}, m ≥ 1. (4.147) In addition, for β =1,2, sincek∂βS^hk_L^∞_(ω) ≤ C(δ2 min{−α,0}

h +δ^−α−1_h ) ≤Cδ^−α−1_h for α ∈ {−1,0,1} and since |∂βTr(S^h)| ≤ |∂βS^h|, |∂βTr(cofS^h)| ≤ 2|S^h||∂βS^h| and

|∂βdet(S^h)| ≤ |S^h|²|∂βS^h|, we conclude that

|∂_βdet∇v^h| ≤ C

|x₃|δ^−α−1_h +|x₃|²δ^min{−α,0}_h δ^−α−1_h +|x₃|³δ2 min{−α,0}

h δ^−α−1_h

≤ C|x3|δ^−α−1_h for β ∈ {1,2}, α ∈ {−1,0,1}, m ≥1.

(4.148)

Now since v^h is not incompressible, we modify it through a non-linear change in coordinates. We letΞ^h(x,˜ x₃) = (x, ξ˜ ^h(x,˜ x₃))forξ^h∈C¹(Ωh,R)to be determined, and we define y^h ≡ y^h(α) := v^h ◦ Ξ^h. Hence, by the column linearity of the determinant, we find that

det∇y^h= det(∇v^h◦Ξ^h)∂₃ξ^h. (4.149) Thus, satisfying the determinant constraint on∇y^hamounts to satisfying the ordinary differential equation

∂₃ξ^h = 1

det(∇v^h◦Ξ^h) on Ω_h (4.150)

for some ξ^h. There is anm = m(α,M) ≥ 1 such that for h > 0 sufficiently small, there is a solution to (4.150), i.e.,ξ^h ≡ ξ^h(α) for aξ^h ∈ C¹(Ωh,R) with the initial conditionξ^h(x,˜ 0)= 0, see Proposition 4.5.11.

It remains to prove the estimates in (4.109). By Proposition 4.5.11, the map ξ^h satisfies pointwise

|ξ^h| ≤ 2|x₃|, |∂₃ξ^h| ≤ 2 (4.151) everywhere onΩ_h. Thus, given (4.150),(4.147) and the estimates above, we deduce

|∂₃ξ^h−1| ≤ |∂₃ξ^h||det(∇v^h◦Ξ^h)−1|

≤Ch^min{−α,0}|ξ^h| ≤ Ch^min{−α,0}|x3| (4.152)