Characterization of fine-scale features

MECHANICAL RESPONSE AND INSTABILITIES OF THIN SHEETS

3.5 Characterization of fine-scale features

Now, let ˜F ∈ M with d = δ(F) ≥ r and q = λM(F) ∈ [d ,r d ]. This corresponds to the pointC in Figure 3.3a. Let

K˜ = G˜ ∈R^3×2: λM(G)˜ =r^1/4d^1/2, δ(G)˜ = d (3.128) that corresponds to the point D in Figure 3.3(a). Therefore, by Proposition 3.4.6, we have ˜F ∈K˜⁽¹⁾. Therefore, arguing as before,W_2D^rc(F˜) ≤ W^mem(F˜).

Finally, let ˜F ∈ L with q = λM(F)˜ ≤ r^1/3 and d = δ(F˜) ≤ min{q²,r^1/6}. This corresponds to the pointPin Figure 3.3a. Let

K˜ = G˜ ∈R^3×2: λM(G)˜ =r^1/3, δ(G)˜ =d (3.129) and

K = G˜ ∈R^3×2 :λM(G)˜ =r^1/3, δ(G)˜ =r^1/6 (3.130) that correspond to the pointsQandZ in Figure 3.3(a), respectively. From Proposi- tion 3.4.6, ˜F ∈K˜⁽¹⁾. Further, again by Proposition 3.4.6, ˜K ⊂ K⁽¹⁾. In other words, F˜ ∈ K⁽²⁾. We can again argue as above to show thatW_2D^rc(F)˜ ≤ 0 =W^mem(F)˜ as

required.

Let νx_˜ be anyW^1,2 gradient Young measure generated by such a sequence. Since {yj} is a minimizing sequence for I2D, it is also a minimizing sequence for the relaxationR

ωW_2D^qc(∇y)d˜ x. Further, since˜ W_2D^qc is non-negative and bounded as in (3.80), it follows from Theorem 1.3 of Kinderlehrer and Pedregal [55] that

f(∇y_j) *hνx˜, fi inL¹(A) for any f ∈









g ∈C(R^3×2): sup

F∈˜ R^3×2

|g(F)˜ |

|F˜|²+1 < +∞









, (3.133)

and for every measurable A ⊂ ω whenever the sequence {f(∇y_j)} converges. As an immediate consequence, we obtain the identities

hνx_˜,idi= F˜₀, hνx_˜,W_2D^qci=W_2D^qc(F˜₀) a.e. x ∈ω. (3.134) Now, sinceW_2Dis a normal integrand, the fundamental theorem of Young measures gives an inequality, lim infj→∞I_2D(yj) ≥ R

ωhνx_˜,W_2Didx˜ (cf. Definition 6.27 and Theorem 8.6 in Fonseca and Leoni [43]). Thus,

|ω|W_2D^qc(F˜₀) = lim

j→∞I_2D(yj) ≥ Z

ωhνx_˜,W_2Didx˜

≥ Z

ωhνx˜,W_2D^qcidx˜ = |ω|W_2D^qc(F˜0),

(3.135)

where we use the fact thatW_2D ≥ W_2D^qc. It follows |ω|W_2D^qc(F˜₀) = R

ωhνx_˜,W_2Didx.˜ Again using the fact thatW_2D ≥W_2D^qc and (3.134), we conclude

hνx_˜,W_2Di=W_2D^qc(F˜₀) a.e. ˜x ∈ω. (3.136) By the localizing properties ofW^1,2gradient Young measures (cf. Theorem 2.3 of Kinderlehrer and Pedregal [54]), we conclude that the fine-scale features which arise from minimizing sequences ofW2D are described by the homogenousW^1,2gradient Young measures admitting the identities

hν,idi=F˜₀, hν,W_2Di=W_2D^qc(F˜₀). (3.137) The support of such Young measures is highly restricted:

Theorem 3.5.1 Letr > 1,F˜₀ ∈R^3×2and let MF˜₀ := (

ν ∈ M^qc, hν,idi= F˜₀, hν,W_2Di= W_2D^qc(F˜₀))

(3.138) be the set of homogenousW^1,2gradient Young measures that satisfy (3.137). This set is non-empty and the following is true:

(i) (The regionM.) Suppose (λM(F₀), δ(F₀)) ∈ M. Set δ := δ(F₀) and λM := λM(F˜0). Let the singular value decomposition ofF˜0be given byF˜0:=Q0D˜0R˜0

forQ0∈ SO(3),R˜0 ∈O(2)andD˜0:= diag(λ¯M,δ/¯ λ¯M) ∈R^3×2. Then anyνF˜₀ ∈ MF˜₀ satisfies

suppνF˜₀ ⊂ K_M := (

F˜ ∈R^3×2: ˜F =Q₀QD˜δ¯R,˜

s.t. Q ∈SO(3), R˜ ∈O(2), det(R)Qe˜ 3= det(R˜0)e3

), (3.139) wheree₃ ∈S² is orthogonal to the plane of the reference configuration of the membrane andD˜δ¯ := δ¯^1/2diag(r^1/4,r^−1/4) ∈R^3×2.

(ii) (The regionW.) Suppose(λM(F˜0), δ(F˜0)) ∈ W. Setλ¯M = λM(F˜0). Further, leteM ∈S²ande˜M ∈S¹be the unique pair (up to a change in sign) of vectors which satisfyF˜₀e˜M = λ¯MeM.

Then anyν_F˜

0 ∈ MF˜₀ satisfies suppνF˜₀ ⊂ K_W := (

F˜ ∈R^3×2 :(λM, δ)(F)˜ = (λ¯M,λ¯^1/2_M ), s.t. F˜e˜M = λ¯MeM

). (3.140)

(iii) (The regionL.) Suppose(λM(F˜₀), δ(F˜₀)) ∈ L.

Then anyνF˜₀ ∈ MF˜₀ satisfies suppν_F˜

0 ⊂ K_L := (

F˜ ∈R^3×2 :(λM, δ)(F)˜ ∈ L s.t. λM(F)/δ(˜ F)˜ =r^−1/6)

. (3.141)

(iv) (The regionS.) Suppose(λM(F˜₀), δ(F˜₀)) ∈ S. Then anyνF˜₀ ∈ MF˜₀is a Dirac mass, i.e., suppνF˜₀ = {F˜0}.

This result has striking physical implications related to the instabilities ofmicrostruc- tureandwrinkling:

Remark 3.5.2 (i) (The regionM.) First consider the particular case whenQ₀ = I. Consider any F˜ ∈ suppνF˜₀ and its characterization in (3.139). Since det(R)Qe˜ 3 = det(R˜0)e3, it follows QD˜δ¯R˜v˜ · e3 = 0 for each v˜ ∈ R². In other words, QD˜δ¯R˜ maps R² to R². Thus, all the oscillations are in the plane. Further, for such matrices F,˜ W2D(F˜) = W3D(F˜|c) = W_iso^e (F˜|b,n) for b = (0,0,δ¯⁻¹)^T and n·e3 = 0. The first of these identities follows from

the fact that (λM(F), δ(˜ F˜)) ∈ S (see Lemma 5.3 and 6.2 in Cesana et al.

[25]), while the second follows from the fact that the largest principal value of (F˜|c)isλM(F˜). Importantly, the director is always in the plane. In summary, the director oscillates in the plane and oscillations create no out-of-plane deformation. The caseQ0 , I3×3 is similar except the plane is oriented by the rotation Q₀. Thus, the fine-scale features in M are limited to in-plane oscillations of the director. In other words, they are related to the formation of material microstructure and not structural wrinkling.

(ii) (The region W.) First consider the case when eM = (e˜M,0)^T. Using an argument as before, for anyF˜ ∈suppνF˜₀,W_2D(F)˜ =W_3D(F˜|c) =W^e(F|b,˜ n) forb·eM =0, |b| = λ¯^−1/2_M andn= eM. In other words, the directornis fixed with an in-plane directioneM. Further, noticeF˜in the{eM,(e˜^⊥_M,0)^T,e₃}frame is necessarily of the formF˜ = QD˜M for D˜M := diag(λ¯M,λ¯^−1/2_M ) ∈ R^3×2and for someQ ∈ SO(3) that satisfiesQeM = eM. In other words, the membrane is uniformly stretched by a factorλ¯Min theeMdirection, uniformly contracted transverse to the stretch by a factorλ¯^−1/2_M , and with the fine features related to rotations about the fixed axis of stretcheM. That is, these oscillations represent wrinkling or undulations perpendicular toeM (i.e., the direction of maximum stretch). The general caseeM , (e˜M,0)^T is similar except a uniform rotation orients (e˜M,0)^T to eM. In summary, the maximum stretch and director are fixed for this region, and undulations occur transverse to this direction. Thus, the fine-scale features inW are related to the structural wrinkling and not material microstructure.

(iii) (The regionL.) The regionLinvolves only the spontaneously deformed states.

However, these can be arranged in a manner such that the effective deformation is due to soft elasticity and/or crumpling.

(iv) (The region S.) There are no fine-scale features associated to S since the support ofνF˜₀is a Dirac mass.

In summary, this theorem guarantees that effective deformation in regionMneces- sarily corresponds to material microstructure only, effective deformation in region W necessarily corresponds to structural wrinkling only, and effective deformation in regionSis without instability.

Proof of Theorem 3.5.1.

We now turn to the proof of the Theorem 3.5.1, which provides a characterization of the fine-scale features in the membrane. It relies on the following lemmas.

For definiteness and completeness, we introduce the notion of a gradient Young measure, which characterizes the statistics of the fine-scale oscillations in the gradi- ents weakly converging sequences (cf. Müller [74]). We define a homogenousW^1,2 gradient Young measure to be a probability measure that satisfies Jensen’s inequality for every quasiconvex function f : R^3×2 → R whose norm can be bounded by a quadratic function. LetMdenote the space of signed Radon measures onR^3×2with the finite mass paring

hµ, fi=Z

R^3×2

f(F˜)dµ(F).˜ (3.142) Then the space of homogenousW^1,2gradient Young measures is given by

M^qc :=ν ∈ M : ||ν||= 1,hν, fi ≥ f(hν,idi)

∀ f :R^3×2→Rquasiconvex with|f(F)˜ | ≤C(|F˜|²+1) . (3.143) The proof of the theorem relies on the following lemmas:

Lemma 3.5.3 LetF˜₀∈R^3×2andδ¯satisfy the hypotheses in Theorem 3.5.1 Part (i).

Then anyνF˜₀ ∈ MF˜₀ satisfies

suppνF˜₀ ⊂ {F˜ ∈R^3×2: (λM, δ)(F)˜ = (r^1/4δ¯^1/2,δ)¯ }. (3.144) Lemma 3.5.4 LetF˜0 ∈R^3×2 andλ¯M satisfy the hypotheses of Theorem 3.5.1 Part (ii). Then anyν_F˜

0 ∈ M_F˜

0 satisfies

suppνF˜₀ ⊂ {F˜ ∈R^3×2: (λM, δ)(F˜)= (λ¯M,λ¯^1/2_M )}. (3.145) Proof of Lemma 3.5.3. Recall from the previous section that we may writeW_2D^qc = ψ◦(λM, δ)andW2D = ϕ◦(λM, δ)whereψ (ϕ) :R →R (R∪ {+∞}), respectively for R = {(s,t) ∈ R² : s² ≥ t,t ≥ 0}. Recall also that ψ is a convex, and it is non-decreasing in each argument. Also, ψ ≤ ϕ. Finally, (λM, δ) : R^3×2 → R are quasiconvex functions bounded quadratically. Therefore, for every homogenous W^1,2gradient Young measure withhν,idi= F˜₀,

W_2D^qc(F˜0) =ψ◦(λM, δ)(hν,idi)

≤ ψ(hν, λMi,hν, δi)

≤ hν, ψ◦(λM, δ)i

≤ hν, ϕ◦(λM, δ)i= hν,W_2Di.

(3.146)

Here, the first inequality follows from the Jensen’s inequality satisfied by homoge- nousW^1,2 gradient Young measures since(λM, δ) are quasiconvex with the appro- priate growth and ψ is non-decreasing in each argument. The second inequality follows from the convexity ofψ, and the third follows sinceψ ≤ ϕ.

Now, for anyνF˜₀ ∈ MF˜₀, each inequality in (3.146) is an equality. This restricts the support ofνF˜₀. To deduce this restriction, suppose that the point(λM(F˜₀), δ(F˜₀)) ∈ M corresponds to point C in Figure 3.3(a).

Consider the first inequality. By quasiconvexity and growth conditions,hνF˜₀, λMi ≥ λM(F˜0)andhνF˜₀, δi ≥ δ(F˜0). In the(λM, δ)space in Figure 3.3(a), these inequalities imply the point(hνF˜₀, λMi,hνF˜₀, δi) cannot be to the left or below point C. Further, every point to the right and above the point C has higherψ (cf. Figure 3.2) except the line between and including the points C and D. Hence,

(hνF˜₀, λMi,hνF˜₀, δi) ∈C D. (3.147)

Next, consider the last inequality. Since ϕ = ψ only on S ∪ {(s,t) ∈ L : t/s = r^−1/6} =:S⁰(see Figure 3.3(b)), we conclude

suppνF˜₀ ⊂ (

F˜ ∈R^3×2 :(λM, δ)(F)˜ ∈ S⁰)

. (3.148)

It remains to consider the middle inequality in (3.146). We do this in Proposition 3.5.5 below. If the middle inequality is an equality, we show in the proposition the support ofνF˜₀ satisfies (3.144). This completes the proof.

Proposition 3.5.5 LetF˜0andδ¯be as in the Theorem 3.5.1 Part (i). If νF˜₀ satisfies (3.147),(3.148) and

ψ(hνF˜₀, λMi,hνF˜₀, δi) =hνF˜₀, ψ◦(λM, δ)i, (3.149) then the support ofνF˜₀ satisfies (3.144) in Lemma 3.5.3.

Proof. SetA⁺ = {F˜ : (λM, δ)(F)˜ ∈ S ∩ {δ >δ¯}}and θ⁺ =Z

A⁺

dνF˜₀(F˜). (3.150)

Ifθ = 1, then by the polyconvexity of δ, hνF˜₀, δi> δcontradicting (3.147). Now consider the case 1> θ⁺ >0. Set

λ⁺_M := 1 θ⁺

Z

A⁺

λM(F)˜ dνF˜₀(F˜), λ⁻_M := 1 1−θ⁺

Z

R^3×2\A⁺

λM(F)˜ dνF˜₀(F˜), δ⁺ := 1

θ⁺ Z

A⁺

δ(F)d˜ νF˜₀(F˜), δ⁻ := 1 1−θ⁺

Z

R^3×2\A⁺

δ(F)d˜ νF˜₀(F).˜ (3.151) Clearly,δ⁺ > δ¯and

θ⁺λ⁺_M+(1−θ⁺)λ⁻_M =hνF˜₀, λMi,

θ⁺δ⁺+(1−θ⁺)δ⁻ = hνF˜₀, δi. (3.152) From the equality in (3.152),δ⁻ < δ. Further, notice from the convexity of¯ ψ that

ψ(λ⁺_M, δ⁺) ≤ 1 θ⁺

Z

A⁺

ψ

λM(F˜), δ(F)˜

dνF˜₀(F˜), ψ(λ⁻_M, δ⁻) ≤ 1

1−θ⁺ Z

R^3×2\A⁺

ψ

λM(F), δ˜ (F˜)

dνF˜₀(F˜).

(3.153)

Now, in the (λM, δ) space shown in Figure 3.3(a), the definitions above imply that the point(λ⁺_M, δ⁺)is a point above the line CD while(λ⁻_M, δ⁻)is below the line CD such that the line joining these points intersects CD. It is easy to verify by explicitly computing the derivative along such lines (or by inspecting Figure 3.2), that ψ is strictly convex in such segments. Therefore,

ψ(hνF˜₀, λMi,hνF˜₀, δi) =ψ

θ⁺λ⁺_M +(1−θ⁺)λ⁻_M, θ⁺δ⁺+(1−θ⁺)δ⁻

< θ⁺ψ(λ⁺_M, δ⁺)+(1−θ⁺)ψ(λ⁻_M, δ⁻)

≤ hνF˜₀, ψ◦(λM, δ)i.

(3.154)

The last inequality follows from (3.153). However, this contradicts the assumption (3.149).

Therefore,θ⁺ = 0, and

suppνF˜₀ ⊂ {(λM, δ)(F˜) ∈ S⁰: δ(F˜) ≤ δ¯}, (3.155) which is the compliment ofA⁺ in the set given in (3.148).

Finally, given (3.155) and since ¯δ = hνF˜₀, δi(see 3.147), it follows that suppνF˜₀ ⊂ {(λM, δ)(F)˜ ∈ S⁰: δ(F)˜ = δ¯}. But this is just a single point in the (λM, δ) space, and it’s given by (3.144). Thus, we conclude the proposition.

The proof of Lemma 3.5.4 is very similar and omitted.

Proof of Theorem 3.5.1. Existence of a ν_F˜

0 ∈ M_F˜

0 follows from the construction in the previous section.

Part (i). For any ˜F0with (λM(F˜0), δ(F˜0)) ∈ M and for anyνF˜₀ ∈ MF˜₀, the support of νF˜₀ satisfies (3.144) by Lemma 3.5.3. Note that (λM, δ)(F)˜ = (r^1/4δ¯^1/2,δ)¯ is equivalent to stating that the principal values of ˜F are r^1/4δ¯^1/2 and r^−1/4δ¯^1/2. Therefore, by the singular value decomposition theorem, it follows that

suppνF˜₀ ⊂ {F˜ ∈R^3×2: ˜F =QD˜δ¯R,˜ Q ∈ SO(3),R˜ ∈O(2)}=:K_supp (3.156) for ˜Dδ¯ given in the statement of the theorem for Part (i).

Now, for any ˜D ∈ R^3×2,Q ∈ SO(3),R˜ ∈ O(2), it is an easy calculation to find that adj(QD˜R)˜ = det(R)Q˜ adj ˜D. Further for ˜D = diag(λ₁, λ₂), adj ˜D = λ₁λ₂e₃. Further, the adjugate is a minor and therefore hνF˜₀,adji = adj(hνF˜₀,idi) = adj ˜F₀. Recalling the support (3.156) ofνF˜₀, we conclude

det(R˜₀)Q₀e₃ = 1

δ¯hνF˜₀,adji= 1 δ¯

Z

R^3×2

adj ˜F dνF˜₀(F˜)

= Z

K_{su p p}

det(R(˜ F))Q(˜ F)e˜ ₃

dνF˜₀(F˜).

(3.157)

Note that det(R˜₀)Q₀e₃ ∈ S², and det(R(˜ F˜))Q(F˜)e₃ ∈ S² for each ˜F ∈ K_supp. In other words, the equation above states that an average of a distribution on S² yields an element ofS². However, since each element ofS² is an extreme point, it means that the distribution is concentrated at a single point onS². That is, if we let Q^∗(F)˜ =Q^T₀Q(F˜), then det(R˜₀)e₃= det(R(˜ F˜))Q^∗(F˜)e₃. The result follows.

Part (ii). For any ˜F₀ with(λM(F˜₀), δ(F˜₀)) ∈ W and for anyνF˜₀ ∈ MF˜₀, it follows from the definition ofeM,e˜M that

Z

R^3×2

F˜e˜MdνF˜₀(F˜)= λ¯MeM. (3.158) So,

Z

R^3×2

|F˜e˜M|dνF˜₀(F˜) ≥ Z

R^3×2

F˜e˜MdνF˜₀(F˜)

= |F˜0e˜M|= λ¯M. (3.159) However, from Lemma 3.5.4, we see that max_e∈S1|Fe|˜ = λM(F˜) = λ¯M for each F˜ ∈ suppνF˜₀. Therefore, |F˜e˜M| ≤ λ¯M for each ˜F ∈ suppνF˜₀. We conclude that

|F˜e˜M| = λ¯M for each ˜F ∈ suppνF˜₀. Setting ˜Fe˜M = λ¯Me(F˜) for e(F)˜ ∈ S² and

substituting in (3.158), we conclude that e(F) = eM for each ˜F ∈ suppνF˜₀. The result follows.

Part (iii). For any ˜F₀ with (λM(F˜₀), δ(F˜₀)) ∈ L, the result follows from the fact thatW_2D is non-negative andW_2D(F˜)= 0 if and only if ˜F ∈ K_L.

Part (iv). Finally, let ˜F₀ ∈R^3×2such that (λM(F˜₀), δ(F˜₀)) ∈ S, νF˜₀ ∈ MF˜₀. Recall W_2D = ϕ◦(λM, δ) andϕis strictly convex inS. Thus,

suppν_F˜

0 ⊂ {F˜ ∈R^3×2: (λM, δ)(F˜)= (λM, δ)(F˜0)}. (3.160) This is actually equivalent to the set (3.162) given in Proposition 3.5.6 below since λm = δ/λM. The result follows from the proposition.

Proposition 3.5.6 Let F˜₀ ∈ R^3×2 such that the singular values satisfy the strict inequality

λM(F˜0) > λm(F˜0) ≥ 0. (3.161) Supposeνis a probability measure on the space ofR^3×2matrices such thathν,idi= F˜₀and

suppν ⊂ {F˜ ∈R^3×2: (λM, λm)(F)˜ = (λM, λm)(F˜0)}. (3.162) Thenν(up to a set of measure zero) is a Dirac mass at F˜₀.

Proof. To begin, set (λ¯M,λ¯m) = (λM(F˜0), λm(F˜0)). We let {g₀₁,g₀₂} ⊂ R³ and {f˜₀₁, f˜₀₂} ⊂ R²be sets of orthonormal vectors such that

F˜₀= λ¯Mg₀₁⊗ f˜₀₁+λ¯mg₀₂⊗ f˜₀₂. (3.163) Letϕf˜₀₁(F˜):= |F˜f˜01|². This is a convex function. Therefore, by Jensen’s inequality and givenhν,idi= F˜0with ˜F0satisfying (3.163),

hν, ϕf˜₀₁i ≥ ϕf˜₀₁(F˜₀) = λ¯²_M. (3.164) Conversely, applying a similar change of variables (3.163) to the ˜F ∈suppν, we see

hν, ϕf˜₀₁i=Z

|(λ¯Mg₁⊗ f˜₁+ λ¯mg₂⊗ f˜₂)f˜₀₁|²

(F)d˜ ν(F˜)

=Z

λ¯²_Mcos(θ(F))˜ ²+λ¯²_msin(θ(F˜))²

dν(F˜)











= λ¯²_M if ν({F˜ ∈R^3×2: sin(θ(F˜)) ,0}) =0

< λ¯²_M otherwise,

(3.165)

since by assumption ¯λM > λ¯m. Here, cosθ denotes the direction cosine between f˜1 and ˜f01. Combining this observation with (3.164), we deduce (up to a set of measure zero), sin(θ(F))˜ = 0. This implies (up to a change in sign) ˜f1 = f˜01 in measure. Since ˜f1 and ˜f2 are orthogonal, it follows that (up to a change in sign)

f˜2 = f˜02in measure.

We repeat this argument substituting ϕf˜₀₁ with the convex function ϕ_g01(F)˜ =

|F˜^Tg₀₁|². It follows that (up to a change in sign)g₁= g₀₁andg₂= g₀₂in measure.

The fact thathν,idi = F˜0 ensures the eigenvectors are fixed and not oscillating in sign with some non-zero measure. The conclusion follows.