A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry

DOI:10.1051/cocv/2010002 www.esaim-cocv.org

A NOTE ON CONVERGENCE OF LOW ENERGY CRITICAL POINTS OF NONLINEAR ELASTICITY FUNCTIONALS,

FOR THIN SHELLS OF ARBITRARY GEOMETRY

Marta Lewicka

Abstract. We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thicknessh and around the mid-surface S of arbitrary geometry, converge as h → 0 to the critical points of the von K´arm´an functional onS, recently proposed in [Lewicka et al.,Ann. Scuola Norm.

Sup. Pisa Cl. Sci.(to appear)]. This result extends the statement in [M¨uller and Pakzad,Comm. Part.

Differ. Equ. 33(2008) 1018–1032], derived for the case of plates whenS ⊂R². The convergence holds provided the elastic energies of the 3d deformations scale likeh⁴ and the external body forces scale likeh³.

Mathematics Subject Classification.74K20, 74B20.

Received December 10, 2008. Revised December 3rd, 2009.

Published online March 24, 2010.

1. Introduction and statement of the main results

Since the beginning of research in nonlinear elasticity, a major topic has been the derivation of lower dimen- sional theories, appropriately approximating the three dimensional theory on structures which are thin in one or more directions (such as beams, rods, plates or shells). Recently, the application of variational methods, notably the Γ-convergence [3], lead to many significant and rigorous results in this setting [5–8]. Roughly speaking, a Γ-limit approach guarantees the convergence of minimizers of a sequence of functionals, to the minimizers of the limit. However, it does not usually imply convergence of the possibly non-minimizing critical points (the equilibria) and hence other tools must be applied to study this problem.

In this note, following works [12,14,15] in which beams, rods and plates were analyzed, we study critical points of the 3d nonlinear elasticity functional on a thin shell of arbitrary geometry, in the von K´arm´an scaling regime. A Γ-convergence result in this framework was recently derived in [10] (see also [9]), providing the natural from the minimization point of view generalization of the von K´arm´an functional [5] to shells. In analogy with the analysis done in [15] for plates, we now proceed to prove convergence of the weak solutions to the static equilibrium equations of nonlinear elasticity (1.13) (the Euler-Lagrange equations associated to the elasticity functional), which in the present setting are the critical points of the 3d energy (1.1), to the critical points of the functional (1.7) obtained in [10]. In case of plates, the Euler-Lagrange equations of (1.7) reduce

Keywords and phrases. Shell theories, nonlinear elasticity, Gamma convergence, calculus of variations.

1 Marta Lewicka, University of Minnesota, Department of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA.

lewicka@math.umn.edu

Article published by EDP Sciences c EDP Sciences, SMAI 2010

to the von K´arm´an system of 4th order PDEs written in terms of the minimizing out-of-plate displacement and the Airy stress potential [11,17].

We now introduce the basic framework for our results. We consider a 2-dimensional surface S embedded in R³, which is compact, connected, oriented, of classC^1,1, and with boundary ∂S being the union of finitely many (possibly none) Lipschitz curves. A family {S^h}_h>0 of shells of small thickness h around S is given through:

S^h={z=x+tn(x); x∈S, −h/2< t < h/2}, 0< h < h₀.

Byn(x) we denote the unit normal to S and by T_xS the tangent space. With a slight abuse of notation, we shall treat T_xS as a 2d subspace of R³, so that both 2d and 3d operators can be applied on vectors τ ∈T_xS.

By Π(x) =∇n(x) we denote the shape operator onS (the negative second fundamental form). Recall that Π is symmetric and Π(x)τ∈T_xS for allτ∈T_xS. Again, we shall view Π as a linear operator fromT_xS toR³, or a linear operator fromT_xS toT_xS, or otherwise as a symmetric bilinear form, or as its matrix representation, whichever is more convenient.

The projection onto Salongnis denoted byπ, so that:

π(z) =x ∀z=x+tn(x)∈S^h.

We assume thath < h₀, withh₀sufficiently small to haveπdefined on eachS^h.

To a deformationu∈W^1,2(S^h,R³) we associate its elastic energy (scaled per unit thickness):

I^h(u) = 1 h _Sh

W(∇u). (1.1)

The stored energy densityW :R^3×3−→[0,∞] is assumed to beC²in a neighborhood ofSO(3), and to satisfy the following normalization, frame indifference and nondegeneracy conditions:

∀F ∈R^3×3 ∀R∈SO(3) W(R) = 0, W(RF) =W(F),

W(F)≥cdist²(F, SO(3)) (1.2)

(with a uniform constant c > 0). Our objective is to describe the limiting behavior, as h → 0, of critical pointsu^hto the following total energy functionals:

J^h(u) =I^h(u)−1 h _Sh

f^hu, (1.3)

subject to external forcesf^h, where we assume that:

f^h(x+tn) =h√

e^hf(x) det(Id +tΠ)⁻¹, f ∈L²(S,R³) and

S

f = 0.

Above,e^his a given sequence of positive numbers obeying a prescribed scaling law. It can be shown [5,10] that iff^hscale likeh^α, then the minimizersu^h of (1.3) satisfyI^h(u^h)∼h^β withβ=αif 0≤α≤2 andβ= 2α−2 ifα >2. Throughout this paper we shall assume that β≥4, or more generally:

h→0lime^h/h⁴=κ <+∞, (1.4) which for S ⊂R² corresponds to the von K´arm´an and the purely linear theories of plates, derived rigorously in [5].

In our recent paper [10], the Γ-limit of 1/e^hJ^hhas been identified in the scaling range corresponding to (1.4), and for arbitrary surfaces S. It turns out that the elastic energy scaling I^h(u^h) ≤ Ce^h implies that on S

the deformationsu^h_|S must be close to some rigid motion ¯Qx+c, and that the first order term in the expansion of ¯Q^T(u^h_|S −c)−id with respect to h, is an element V of the class V of infinitesimal isometries on S [16].

The spaceV consists of vector fields V ∈W^2,2(S,R³) for whom there exists a matrix fieldA∈W^1,2(S,R^3×3) so that:

∂_τV(x) =A(x)τ and A(x)^T =−A(x) ∀a.e. x∈S ∀τ∈T_xS. (1.5) Equivalently, the change of metric on S induced by the deformation id +hV is at most of orderh², for each V ∈ V.

When in (1.4)κ= 0, the limiting total energy is given by:

J(V,Q) =¯ 1

24 _SQ2(x,(∇(An)−AΠ)_tan) dx−

S

f·QV¯ dx, ∀V ∈ V, Q¯∈SO(3). (1.6) The first term above measures the first order change in the second fundamental form Π of S, produced byV. The subscript “tan” refers to the tangential minor of a given matrix field; given F ∈ L²(S,R^3×3) we put:

F_tan(x) = [(F(x)τ)η]_τη∈TxS.

The quadratic formsQ₂(x,·) are given as follows:

Q2(x, F_tan) = min{Q3( ˜F); ( ˜F−F)_tan= 0}, Q3(F) =D²W(Id)(F, F).

The formQ3is defined for allF ∈R^3×3, whileQ2(x,·) for a givenx∈S, is defined on tangential minorsF_tan of such matrices (as explained above). Both forms depend only on the symmetric parts of their arguments and are positive definite on the space of symmetric matrices [4]. In the weak formulation of the Euler-Lagrange equations of (1.6) one naturally encounters the linear operatorsL3 andL2(x,·), defined on and valued in the matrix spacesR^3×3andR^2×2(identified here with the space of bilinear forms onT_xS), respectively, given by:

∀F ∈R^3×3 Q3(F) =L3F :F and Q2(x, F_tan) =L2(x, F_tan) :F_tan.

Recall that, for two square matricesF₁andF₂, of the same dimension, their inner product isF₁:F₂= tr(F₁^TF₂).

Forκ >0, the Γ-limit (which is the generalization of the von K´arm´an functional [5] to shells), contains also a stretching term, measuring the total second order change in the metric ofS:

J^vK(V, B_tan,Q) =¯ κ 2 _SQ2

x, B_tan−1 2(A²)_tan

+ 1

24 _SQ2(x,(∇(An)−AΠ)_tan)−

S

f·QV.¯ (1.7) It involves a symmetric matrix field B_tan on S belonging to the finite strain space B. Given a vector field w∈W^1,2(S,R³), by sym∇wwe mean the bilinear form onT_xS, given by: ((sym∇w(x))τ)η =¹₂[(∂_τV(x))η+ (∂_ηV(x))τ] for allτ, η∈T_xS. Then we define [10]:

B= cl_L2(S)

sym∇w^h; w^h∈W^1,2(S,R³) .

The two terms in (1.7) correspond, in appearing order, to the stretching and bending energies of a sequence of deformationsv^h= id +V +²w^h ofS (where=√

e^h/h) which is induced by:

(i) a first order displacementV ∈ V;

(ii) the second order displacementsw^h satisfying lim_h→0sym∇w^h=B_tan.

In view of the fundamental theorem of calculus of variations, the crucial property of (1.7) is the one-to-one correspondence between the minimizing sequences u^h of the total energies J^h(u^h), and their approximations (modulo rigid motions ¯Qx+c) given byv^h as above with (V, B_tan,Q) minimizing the Γ-limit¯ J^vK of 1/e^hJ^h (or (V,Q) minimizing¯ J whenκ= 0).

The purpose of this paper is to show that under the following extra assumption of [15]:

∀F ∈R^3×3 |DW(F)| ≤C(|F|+ 1) (1.8)

also the equilibria (possibly non-minimizing) of (1.1) converge to the equilibria of (1.7) or (1.3).

Theorem 1.1. Assume(1.2)and(1.8). Letu^h∈W^1,2(S^h,R³)be a sequence of deformations, satisfying:

(a) the following equilibrium equations hold, for eachh:

∀φ^h∈W^1,2(S^h,R³)

S^h

DW(∇u^h) :∇φ^h=

S^h

f^hφ^h; (1.9)

(b) I^h(u^h)≤Ce^h, wheree^h is the scaling with(1.4).

Then there exist a sequence Q^h ∈SO(3), converging (up to a subsequence) to some Q¯ ∈SO(3), and c^h∈R³ such that for the normalized rescaled deformations:

y^h(x+tn) = (Q^h)^Tu^h(x+h/h₀tn)−c^h defined on the common domain S^h0, we have:

(i) y^h converge in W^1,2(S^h0)toπ.

(ii) The scaled average displacements:

V^h(x) = h

√e^h

h0/2

−h0/2

y^h(x+tn)−xdt (1.10)

converge (up to a subsequence) in W^1,2(S)to someV ∈ V.

(iii) h/√

e^h sym∇V^h converge (up to a subsequence) in L²(S)to some B_tan∈ B.

(iv) The triple (V, B_tan,Q)¯ satisfies the Euler-Lagrange equations of the functional J^vK. That is, for all V˜ ∈ V with A˜=∇V˜ given as in the formula(1.5), and allB˜_tan∈ B, there holds:

SL2

x, B_tan−1 2(A²)_tan

: ˜B_tan= 0, (1.11)

−κ

SL₂

x, B_tan−1 2(A²)_tan

: (AA)˜ _tan + 1

12 _SL₂(x,(∇(An)−AΠ)_tan) : (∇( ˜An)−AΠ)˜ _tan=

S

f·Q¯V .˜ (1.12) Whenκ= 0then the couple (V,Q)¯ satisfies(1.12)for all V˜ ∈ V, which is the Euler-Lagrange equations of the functional (1.6).

We prove Theorem1.1in Section2. In Section3we derive the third Euler-Lagrange equation (after the first two (1.11) and (1.12)), corresponding to variation in ¯Q∈SO(3). We first notice that the limiting ¯Qnecessarily satisfies the constraint of the average torque τ( ¯Q) = _Sf ×Qx¯ dx being 0. The main difficulty arises now from the fact that the variations must be taken inside SO(3) in a way that this constraint remains satisfied.

Assuming that such variations exist, we establish the limit equation under the nondegeneracy condition that Q^h approach ¯Qalong a directionU ∈T_Q¯SO(3) for which∂_Uτ( ¯Q)= 0.

Remark 1.2. Integrating by parts we see that (1.9) is the weak formulation of the following fundamental balance law [1]:

div [DW(∇u^h)] +f^h= 0 inS^h, DW(∇u^h)n= 0 on∂S^h, (1.13) where the operator div above is understood as acting on rows of the matrix fieldDW(∇u^h).

The definition of an equilibrium of the 3d energy J^h may be understood in two different manners, corre- sponding to passing with the scaling of a variationφto 0 outside or inside the integral sign. Namely, for a fixedh >0, we may either require (1.9) or require that:

∀φ^h∈W^1,2(S^h,R³) lim

→0

1

J^h(u^h+φ^h)−J^h(u^h)

= 0. (1.14)

Whether (1.14) and (1.9) are equivalent, even for local minimizers (without assuming extra regularity,e.g. their Lipschitz continuity) is an open problem of nonlinear elasticity, listed by Ball as Problem 5 in [1]. However, condition (1.8) readily implies that:

→0lim S^h

1

[W(∇u^h+∇φ^h)−W(∇u^h)] =

S^h

DW(∇u^h) :∇φ^h,

because of the pointwise convergence of the integrands and of their boundedness by anL¹function independent of:

1

|W(∇u^h+∇φ^h)−W(∇u^h)| ≤ ¹

0 |DW(∇u^h+s∇φ^h)| · |∇φ^h| ds≤C

|∇u^h|+|∇φ^h|+ 1

|∇φ^h|. Hence in presence of (1.8), the equilibrium conditions (1.14) and (1.9) are equivalent.

Remark 1.3. Notice that, in view of (1.2) resulting in DW(F) = 0 for all F ∈ SO(3), condition (1.8) is equivalent to:

∀F ∈R^3×3 |DW(F)| ≤Cdist(F, SO(3)).

Using the last assumption in (1.2), the above implies that: |DW(F)| ≤ CW(F)^1/2 for all F ∈ R^3×3. Hence, roughly speaking,W has a quadratic growth and we see that (1.8) is actually very restrictive. Independently, Mora and Scardia [13] have recently proved a complementary result where the requirement (1.8) is relaxed, while the equilibrium condition of (1.3) is understood in a different manner, related to Ball’s inner variations and the Cauchy stress balance law [1].

2. Convergence of weak solutions to the Euler-Lagrange equations (equilibria) of the 3 d energies

We first gather the relevant information from [10]:

Lemma 2.1 [10]. Let u^h∈W^1,2(S^h,R³) be a sequence of deformations of shellsS^h. Assume (1.4)and let the scaled energiesI^h(u^h)/e^h be uniformly bounded. Then there exists a sequence of matrix fieldsR^h∈W^1,2(S,R³) with R^h(x)∈SO(3) for a.e. x∈S, such that:

∇u^h−R^hπ_L²_(S^h₎≤Ch^1/2√

e^h and ∇R^h_L²_(S)≤Ch⁻¹√ e^h

and there exists a sequence of matrices Q^h∈SO(3) such that:

(i) (Q^h)^TR^h−Id_W^1,2_(S)≤C√ e^h/h.

(ii) h/√

e^h((Q^h)^TR^h −Id) converges (up to a subsequence) to a skew-symmetric matrix field A, weakly in W^1,2(S).

Moreover, there exists a sequencec^h∈R³ such that for the normalized rescaled deformations:

y^h(x+tn) = (Q^h)^Tu^h(x+h/h₀tn)−c^h defined on the common domain S^h0, the following holds.

(iii) y^h converge in W^1,2(S^h0)toπ.

(iv) The scaled average displacements V^h, defined in (1.10)converge (up to a subsequence) in W^1,2(S)to someV ∈ V, whose gradient is given by A, as in(1.5).

(v) h/√

e^h sym∇V^h converge (up to a subsequence) in L²(S)to some B_tan∈ B.

The statements in Theorem 1.1(i), (ii), (iii) are contained in the lemma above. It therefore suffices to use the extra assumptions (1.9) and (1.8) to recover equations (1.11) and (1.12) ash→0.

We start by rewriting the equilibrium equation (1.9) in a more convenient form. Clearly, every variation φ^h∈W^1,2(S^h,R³) can be by a change of variables expressed as:

φ^h(x+tn) =ψ(x+th₀/hn), (2.1) for the correspondingψ∈W^1,2(S^h0,R³). Then, (1.9) becomes:

h²√ e^h

S

f(x)

h0/2

−h0/2

ψ(x+tn) dt dx

=h

S

h0/2

−h0/2det(Id +th/h₀Π)DW(∇u^h(x+th/h₀n)) :∇φ^h(x+th/h₀n) dtdx. (2.2) Notice also that:

∇φ^h(x+th/h₀n) =∇ψ(x+tn)P(x+tn), (2.3) where the matrix fieldP ∈L^∞(S^h0,R^3×3) has the following non-zero entries:

P(x+tn)_tan= (Id +th/h₀Π(x))⁻¹(Id +tΠ(x)), n^TP(x+tn)n=h₀/h.

In view of Lemma2.1, define the matrix fieldsE^h, G^h∈L²(S^h0,R^3×3):

E^h= 1

√e^hDW(Id +√

e^hG^h), G^h(x+tn) = 1

√e^h

(R^h)^T∇u^h(x+th/h₀n)−Id .

With this notation, recalling the frame invariance ofW in (1.2) we get, for everyF ∈R^3×3:

√1

e^hDW(∇u^h(x+th/h₀n)) :F = 1

√e^hDW(R^h(Id +√

e^hG^h)) :F

= 1

√e^hDW(Id +√

e^hG^h) : (R^h)F =R^hE^h:F.

In particular, (2.2) becomes, after exchangingψto (Q^h)^Tψ, using (2.3) and dividing both sides by√ e^h: h²

Sf(x)

h0/2

−h0/2Q^hψdtdx

=h

S

h0/2

−h0/2det(Id +th/h₀Π)

(Q^h)^TR^h(x)E^h(x+tn) :∇φ^h(x+th/h₀n) dtdx

=h

S

h0/2

−h0/2det(Id +th/h₀Π)

(Q^h)^TR^hE^h _{T S} :

(∇tanψ)(Id +th/h₀Π)⁻¹(Id +tΠ) dtdx

+h₀

S

h0/2

−h0/2

det(Id +th/h₀Π) ((Q^h)^TR^hE^hn)∂_nψ(x+tn) dtdx, (2.4) where ∇tan denotes gradient in the tangent directions of T_xS. The subscript T S stands for taking the 3×2 minor of the matrix under consideration, for example: ∇tanψ= [∇ψ]_{T S}. Also, with a slight abuse of notation, Id stands for the identity map onR³ orT_xS, whichever is appropriate.

Lemma 2.2. The sequence G^h converges (up to a subsequence), weakly inL²(S^h0,R^3×3)to aL²(S^h0)matrix fieldG, whose symmetrized tangential minor has the form:

symG(x+tn)_tan=√κ

B_tan−1 2(A²)_tan

+ t

h₀(∇(An)−AΠ)_tan. (2.5) Moreover, if (1.8)holds, then:

(i) E^h converges (up to a subsequence) weakly in L²(S^h0,R^3×3)to the matrix field E=L₃G.

(ii) The sequence(Q^h)^TR^h(x)E^h(x+tn)converges (up to a subsequence) to E, weakly inL²(S^h0,R^3×3).

Proof. The convergence ofG^hand the formula (2.5) follow from Lemma 3.6 and Lemma 4.1 in [10]. Convergence in (i) is a consequence of Proposition 2.3 in [15], where the crucial role was played by the following equivalent form of the assumption (1.8):

∀F ∈R^3×3 |DW(Id +F)| ≤C|F|.

Finally, (ii) is an immediate consequence of (i) in view of Lemma 2.1 (i) and the boundedness of (Q^h)^TR^h

in L^∞(S^h0).

Lemma 2.3. The matrix field E∈L²(S^h0,R^3×3), defined in Lemma 2.2(i) satisfies the following properties, a.e. inS^h0:

(i) En= 0.

(ii) E^T =E, that is: E is symmetric.

(iii) E_tan(x+tn) =L2(x, G_tan(x+tn)).

Proof. To prove (i), one needs to passh→0 in (2.4) and use Lemma2.2(ii) to obtain:

S

h0/2

−h0/2

E(x+tn)n

∂_nψ(x+tn) dtdx= 0. (2.6)

Now, any vector fieldφ∈L²(S^h0,R³) has the formφ=∂_nψ, whereψ(x+tn) =_−h^t

0/2φ(x+sn) ds. Therefore (i) follows from (2.6).

By frame indifference (1.2) and the fact that W is minimized at Id, it follows that DW(F) = 0 for all F ∈SO(3). It implies that for allH ∈so(3) there holds L₃H = 0, and soE :H =L₃G:H =L₃H :G= 0, proving (ii). Hereso(3) stands for the space of 3×3 skew-symmetric matrices.

The assertion (iii) follows fromE=L3Gand the reasoning exactly as in the proof of Proposition 3.2 [15].

A more precise information, with respect to that in Lemma 2.3(ii) is given by:

Lemma 2.4. There holds:

(i) skewE^h_L¹_(S^h0)≤C√ e^h. (ii) lim

h→0

1

h skewE^h_Lp(S^h0)= 0,for some exponentp∈(1,2).

Proof. By frame indifference (1.2) one has: 0 = DW(F) : HF =DW(F)F^T : H, for all F ∈ R^3×3 and all H ∈so(3) (sinceHF is a tangent vector toSO(3)F atF). We further obtain thatDW(F)F^T is a symmetric matrix. Apply this statement pointwise to the matrix fieldF = Id +√

e^hG^h: 0 = 1

√e^h

DW(Id +√

e^hG^h) (Id +√

e^h(G^h)^T)−(Id +√

e^hG^h)DW^T(Id +√ e^hG^h)

=E^h−(E^h)^T +√ e^h

E^h(G^h)^T −G^h(E^h)^T . Hence the claim in (i) is proved, as by Lemma 2.2:

sym (E^h(G^h)^T)_L1(S^h0)≤CE^h_L2(S^h0)G^h_L2(S^h0)≤C.

Now, (ii) follows from (i) in view of the boundedness ofE^h in L²(S^h0), (1.4), and through an interpolation inequality:

1

hskewE^h_Lp(S^h0)≤ 1

h skewE^hθ_L1skewE^h1−θ_L2 ≤C/h√

e^hθ=C√

e^h/h²_θ h^2θ−1,

where 1/p=θ+ (1−θ)/2 andθ∈(0,1). Clearly, the above converges to 0, whenθ >1/2.

Introduce now the two matrix fields ¯E, Eˆ∈L²(S,R³) given by the 0th and 1st moments ofE:

E(x) =¯

h0/2

−h0/2

E(x+tn) dt, E(x) =ˆ

h0/2

−h0/2

tE(x+tn) dt.

It easily follows by Lemma2.3(iii), Lemma2.2and the fact thatL₂(x,·) depends only on the symmetric part of its argument, that:

E¯_tan(x) =

h0/2

−h0/2L₂(x, G_tan(x+tn)) dt=√κL₂

x, B_tan−1 2(A²)_tan

, (2.7)

Eˆ_tan(x) =

h0/2

−h0/2L₂(x, tG_tan(x+tn)) dt= h₀

12L₂(x,(∇(An)−AΠ)_tan). (2.8) We will now use the fundamental balance (2.4) and the above formulas to recover the Euler-Lagrange equa- tions (1.11), (1.12) in the limit ash→0.

Proof of the first Euler-Lagrange equation (1.11). Use the variation of the form: ψ(x+tn) =φ(x) in (2.4), divide both sides byhand pass to the limit to obtain:

0 = lim

h→0 S

h0/2

−h0/2det(Id +th/h₀Π)

(Q^h)^TR^hE^h

T S:

∇_tanφ(x)(Id +th/h₀Π)⁻¹ dtdx

=

S

h0/2

−h0/2

E_{T S}:∇tanψ(x) dtdx=√κ

SL2

x, B_tan−1 2(A²)_tan

: [∇φ(x)]_tan dx

=√ κ

SL2

x, B_tan−1 2(A²)_tan

: [sym∇φ(x)]_tan dx (2.9)

where we have used Lemma2.2(i), Lemma2.3and (2.7). Therefore, by density of{sym∇φ}in the spaceB, (1.11)

follows immediately.

Proof of the second Euler-Lagrange equation (1.12). Let ˜V ∈ V and denote by ˜A the skew-symmetric matrix field representing∇V˜, as in (1.5).

1. We now apply (2.4) to a variation of the form: ψ(x+tn) = tAn(x). For simplicity, write˜ η = ˜An ∈ W^1,2(S,R³). Upon dividing (2.4) byhand passing to the limit, we obtain:

0 = lim

h→0

S

h0/2

−h0/2det(Id +th/h₀Π)

(Q^h)^TR^htE^h _{T S} :

∇tanη(x)(Id +th/h₀Π)⁻¹ dtdx

+h₀ h _S

h0/2

−h0/2

((Q^h)^TR^hE^hn)η(x) dtdx

+

S

h0/2

−h0/2

(ttrace Π +t²h/h₀det Π)((Q^h)^TR^hE^hn)η(x) dtdx

, (2.10)

where we used the identity:

det(Id +th/h₀Π) = 1 +th/h₀trace Π +t²h²/h²₀det Π.

The first term in (2.10), in view of Lemma2.2 (ii), Lemma2.3and (2.8), converges to:

S

h0/2

−h0/2

tE_{T S} :∇tanη(x) dtdx=

S

Eˆ_tan: (∇η(x))_tan dtdx

= h₀

12 _SL₂(x,(∇(An)−AΠ)_tan) : (∇η(x))_tan dx.

In turn, the third term in (2.10) converges to 0. This is because (Q^h)^TR^hE^hn converge weakly inL²(S^h0,R³) toEn= 0, by Lemma2.2(ii) and Lemma2.3(i). Summarizing, (2.10) yields:

h→0lim 1 h _S

h0/2

−h0/2

((Q^h)^TR^hE^hn) ˜Andtdx=−1

12 _SL₂(x,(∇(An)−AΠ)_tan) : (∇( ˜An))_tan dx. (2.11) 2. Now, apply (2.4) to the variationψ(x+tn) = ˜V(x), and pass to the limit after dividing both sides of (2.4) byh²:

Sf(x)·Q¯V˜(x) dx= lim

h→0 Sf(x)·Q^hV˜(x) dx

= lim

h→0

S

h0/2

−h0/2

1

h((Q^h)^TR^h−Id)E^h

T S

:

A(x)˜ _{T S}(Id +th/h₀adj Π) dtdx

+

S

h0/2

−h0/2

1 hE^h_{T S}:

A(x)˜ _{T S}(Id +th/h₀adj Π) dtdx

:= lim

h→0[I_h+II_h], (2.12)

where we used the definition of the adjoint matrix:

det(Id +th/h₀Π) (Id +th/h₀Π)⁻¹= adj (Id +th/h₀Π) = Id +th/h₀adj Π.

Notice that, by Lemma2.1(ii) and (1.4), the matrix field:

1/h((Q^h)^TR^h−Id) = (√

e^h/h²)h/√

e^h((Q^h)^TR^h−Id)

converges to√κA, weakly inW^1,2(S) and hence strongly inL²(S). Hence, by the weak convergence ofE^h toE and the uniform convergence of (Id +th/h₀adj Π) to Id, the first term of (2.12) converges to:

h→0limI_h=√κ

S

h0/2

−h0/2

(AE)_{T S} : ˜A_{T S}=√κ

S

(AE)¯ _{T S} : ˜A_{T S}=√κ

S

(AE) : ˜¯ A

=−√ κ

S

E¯: (AA) =˜ −√ κ

S

E¯_tan: (AA)˜ _tan

=−κ

SL2

x, B_tan−1 2(A²)_tan

: (AA)˜ _tan dx, (2.13)

where we also have used Lemma2.3and (2.7).

3. Towards finding the limit of II_h in (2.12), consider first the contribution of the tangential minors. By Lemma2.4 (ii) and since ˜A∈L^p(S^h0) for allp≥1, one observes that:

h→0lim 1 h _S

h0/2

−h0/2 skewE_tan^h : ˜A_tan = 0. (2.14) Hence:

h→0lim S

h0/2

−h0/2

1 hE_tan^h :

A(x)˜ _tan(Id +th/h₀adj Π) dtdx

= 1 h₀ lim

h→0 S

h0/2

−h0/2

tE_tan^h :

A˜_tan adj Π

= 1 h₀ lim

h→0 S

Eˆ_tan :

A˜_tan adj Π

=−1 h₀ lim

h→0 S

Eˆ_tan : ( ˜A_tanΠ)^T =−1

12 _SL₂(x,(∇(An)−AΠ)_tan) : ( ˜AΠ)_tan dx, (2.15) where we have used (2.8) and Lemma2.3(ii), combined with the following formula, which can be easily checked for ˜A_tan∈so(2):

A˜_tan adj Π =−( ˜A_tanΠ)^T. Further, by (2.11):

h→0lim S

h0/2

−h0/2

1 h

(E^h)^Tn ( ˜A)^Tn

dtdx=−lim

h→0

1 h _S

h0/2

−h0/2

((Q^h)^TR^hE^hn)( ˜An)

+ lim

h→0 S

h0/2

−h0/2

1

h((Q^h)^TR^h−Id)(E^hn)

( ˜An) + 2 lim

h→0 S

h0/2

−h0/2

1

h( skewE^h)n

( ˜An)

= 1

12 _SL2(x,(∇(An)−AΠ)_tan) : (∇( ˜An))_tan dx. (2.16) Indeed, 1/h((Q^h)^TR^h−Id) converges toκAweakly inL⁴(S) while ˜An∈L⁴(S) and ¯E^hnconverges to 0 weakly inL²(S). Therefore the second term in (2.16) converges to 0. The last limiting term there vanishes as well, by Lemma2.4 (ii) as in (2.14).

Finally, we have:

h→0lim 1 h_{0 S}

h0/2

−h0/2

n^TtE^h

tan(adj Π)

( ˜A)^Tn

tan dtdx= 0, (2.17)

because ( ˆE^h)^Tnconverges to 0 weakly inL²(S) by Lemmas2.2(i) and2.3.

Adding now (2.15), (2.16) and (2.17) we obtain:

h→0limII_h= 1

12 _SL2(x,(∇(An)−AΠ)_tan) : (∇( ˜An)−AΠ)˜ _tan dx. (2.18)

Together with (2.12) and (2.13), the formula (2.18) implies (1.12).

3. The limiting rotations Q ¯

In this section we will derive the third Euler-Lagrange equation (after the first two (1.11) and (1.12)), corresponding to variation in ¯Q∈SO(3), and under certain nondegeneracy condition. We first notice that the limiting ¯Qnecessarily satisfies the constraint of the average torque:

τ( ¯Q) =

S

f×Qx¯ dx= 0. (3.1)

The main difficulty arises now from the fact that the variations must be taken insideSO(3) in a way that this constraint remains satisfied. Assuming that such variations exist, we establish the limit equation under the additional condition thatQ^happroach ¯Qalong a directionU ∈T_Q¯SO(3) for which∂_Uτ( ¯Q)= 0.

In what follows, the crucial role is played by the function g(Q) = _Sf ·Qx dx defined on SO(3). Let K∈R^3×3 be such that: g(Q) =K:Q, for allQ∈SO(3).

Lemma 3.1. Assume the hypothesis of Theorem 1.1. Then the limitQ¯∈SO(3) ofQ^h must satisfy:

S

f·QF x¯ dx= 0 ∀F ∈so(3), (3.2) or equivalently (3.1). Another equivalent formulation of(3.2)is: skew( ¯Q^TK) = 0.

Proof. First, for any given H ∈ so(3), consider the variation φ^h = Hu^h in the equilibrium equation (1.9).

Recalling that DW(∇u^h)(∇u^h)^T is symmetric (see the proof of Lem.2.4) we obtain:

S^h

f^h·Hu^h=

S^h

DW(∇u^h) :H∇u^h=

S^h

DW(∇u^h)(∇u^h)^T

:H = 0. (3.3)

Similarly, takingφ^h=¹(exp(H)u^h−u^h) in (1.14), by frame indifference ofW we get:

S^h

f^h·Hu^h= lim

→0

1 _Sh

f^h·(exp(H)u^h−u^h) =hlim

→0

1

J^h(exp(H)u^h)−J^h(u^h) = 0.

Now, for any sequence of skew-symmetric matricesF^h we have:

S

f·Q^hF^hV^h= 1 he^h _Sh

f^h·Q^hF^h

(Q^h)^Tu^h−c^h−id

= 1 he^h _Sh

f^h·

Q^hF^h(Q^h)^T

u^h− 1 he^h _Sh

f^h dz·Q^hF^hc^h− 1 he^h _Sh

f^h·Q^hF^hz dz

=− h

√e_{h S}f·Q^hF^hxdx, (3.4)

where the first two terms in the second line above vanish by takingH =Q^hF^h(Q^h)^T ∈so(3) in (3.3), and by the normalization off^h. Passing to the limit withh→0 in (3.4), whereF^h=F, we see that: −_Sf·QF V¯ = lim_h→0h/√

e^h_Sf·Q^hF xdx. This implies (3.2).

Clearly, (3.2) is also equivalent to 0 =K: ¯QF = ¯Q^TK:F for allF ∈so(3), which means exactly that ¯Q^TK is a symmetric matrix.

To prove the other equivalent formulation of (3.2), notice that:

Sf·QF x¯ =

S

Q¯^Tf·F x=−c_F·

S

Q¯^Tf×x=−c_F

Sf×Qx,¯

where c_F ∈R³ is such that F x=c_F×xfor all x∈R³. Since there is a one to one correspondence between

vectorsc_F and skew matricesF, the proof is achieved.

Define now the set of the rotation equilibria:

M={Q¯∈SO(3); skew ( ¯Q^TK) = 0}.

Our goal is to derive the third Euler-Lagrange equation, with respect to the variations of ¯QinM. For ¯Q∈ M, letF ∈so(3) be such that:

QF¯ = lim

n→∞

Q¯_n−Q¯ Q¯_n−Q¯, for some ¯Q_n ∈ Mconverging to ¯Q. Clearly, the above implies that:

skew (FQ¯^TK) = 0. (3.5)

Lemma 3.2. Under the hypothesis of Theorem 1.1, assume moreover that:

h→0lim

Q^h−Q¯

Q^h−Q¯ = ¯QH, with skew(HQ¯^TK)= 0.

Then for every F∈so(3)satisfying (3.5)there holds:

S

f·QF V¯ dx= 0.

Proof. We will find a sequenceF^h∈so(3), converging toF and such that, for allh:

S

f ·Q^hF^hxdx= 0. (3.6)

In view of (3.4) this will prove the lemma. Existence of such approximating sequenceF^h is guaranteed by the assumed nondegeneracy condition: skew (HQ¯^TK)= 0.

Firstly, notice that forQ^h∈ M one can takeF^h=F. Otherwise, define:

F^h=F− (Q^h)^TK:F

|skew ((Q^h)^TK)|² skew ((Q^h)^TK).

Then:

S

f·Q^hF^hxdx=K:Q^hF^h= (Q^h)^TK:F^h

= (Q^h)^TK:F − (Q^h)^TK:F

|skew ((Q^h)^TK)|² (Q^h)^TK: skew

(Q^h)^TK

= 0,