Calc. Var. (2013) 48:185–209 DOI 10.1007/s00526-012-0549-5

Calculus of Variations

The von Kármán theory for incompressible elastic shells Hui Li · Milena Chermisi

Received: 4 August 2011 / Accepted: 20 July 2012 / Published online: 21 August 2012 © Springer-Verlag 2012

Abstract We rigorously derive the von Kármán shell theory for incompressible materials, starting from the 3D nonlinear elasticity. In case of thin plates, the Euler-Lagrange equations of the limiting energy functional reduce to the incompressible version of the classical von Kármán equations, obtained formally in the limit of Poisson’s ratio ν → 1/2. More generally, the midsurface of the shell to which our analysis applies, is only assumed to have the following approximation property: C 3 first order infinitesimal isometries are dense in the space of all W 2,2 infinitesimal isometries. The class of surfaces with this property includes: subsets of R2 , convex surfaces, developable surfaces and rotationally invariant surfaces. Our analysis relies on the methods and extends the results of Conti and Dolzmann (Calc Var PDE 34:531–551, 2009, Lewicka et al. (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX:253–295, 2010, Friesecke et al. (Comm. Pure. Appl. Math. 55, no. 2, 1461–1506, 2002). Mathematics Subject Classification (2000) 74K20, 74K25

1 Introduction In this paper we rigorously derive the von Kármán shell theory for incompressible materials, starting from the 3D nonlinear elasticity. In the particular case of thin isotropic plates, the Euler-Lagrange equations of the limiting energy functional reduce to the following system:

Communicated by J. Ball. H. Li (B) Department of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, MN 55455, USA e-mail: [email protected] M. Chermisi Department of Mathematical Sciences, New Jersey Institute of Technology, 323 Dr. M.L.King, Jr. Blvd., Newark, NJ 07102, USA e-mail: [email protected]

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μ 2  v = [v, ], 3

2  = −

3μ [v, v], 2

(1)

which is the incompressible version of the classical von Kármán equations, obtained formally in the limit of Poisson’s ratio ν → 21 . The system (1) may be combined with the natural (free) or clamped boundary conditions. Given a 2d surface S in R3 , a thin shell is defined as:   h h , S h = z = x + tn(x); x ∈ S, − < t < 2 2 where n(x) is the unit normal to S at the point x, and h > 0 is the shell’s thickness, assumed to be small compared to the other two dimensions. When S is a subset of R2 , S h is called a plate. According to the static (variational) theory of elasticity, the deformation of S h in response to an external force f h : S h −→ R3 , is described by a minimizer of the total energy functional J h :  1 f h · u h , ∀u h ∈ W 1,2 (S h , R3 ). J h (u h ) = I h (u h ) − h Sh

The elastic energy I h (u h ) above has the following expression:   1   W ∇u h , I h uh = h Sh

where the incompressible energy density W : R3×3 −→ [0, ∞] obeys:  Wc (F), if detF = 1, W (F) = ∞, otherwise, reflecting that the incompressible elastic bodies only perform deformations with determinants of their gradients equal to 1. The effective energy density Wc is assumed to satisfy the appropriate physically relevant conditions (6); since these conditions generally imply that I h (u h ) is non-convex in its argument u h , while the second term in J h (u h ) is linear, the main variational analysis concerns the functional I h . It consists of computing the -limit Iβ of the sequence of scaled energy functionals h −β I h , where the exponent β is determined by the scaling α of the external forces f h . Namely, if f h ∼ h α , the elastic energy I h (u h ) at minimizers u h of J h scales like h β , where β = α if 0 ≤ α ≤ 2, and β = 2α − 2 if α ≥ 2. The limiting energy Iβ plays hence the role of the 2D counterpart of the 3D energy functionals I h . Recalling [9,10], a sequence of functionals Fn : X −→ [−∞, +∞] defined on a metric space X , -converges to the limit functional F : X −→ [−∞, +∞] whenever: (i) (ii)

(Lower bound) For any converging sequence x n → x in X , one has F(x) ≤ lim inf n→∞ Fn (xn ). (Recovery sequence) For any x ∈ X , there exists a sequence x n converging to x, such that limn→∞ Fn (xn ) = F(x).

The fundamental property of -convergence is the following. If xn is a sequence of approximate minimizers of Fn in X :   lim Fn (xn ) − inf Fn = 0, n→∞

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and if xn → x, then x is a minimizer of F. In turn, any recovery sequence associated to a minimizer of F is an approximate minimizing sequence for Fn . In the context of shells (plates), the sequence of minimizing deformations for J h can be hence recovered from minimizers of the -limit of h −β J h whose crucial term is provided by Iβ . Along this line, many results have been obtained in recent years. Particularly, the incompressibility assumption imposes a further constraint on the recovery sequence and causes the subsequent analytical difficulties. In [4,30,31], Conti and Dolzmann and also Trabelsi investigated the incompressible case for plates with β = 0, and derived the appropriate membrane theory. In [5,6], Conti and Dolzmann analyzed the scaling β = 2, still in the case of plates, and established the resulting incompressible version of the Kirchhoff theory. We presently study the case of incompressible shells and energy scaling β = 4, deriving hence the generalized incompressible von Kármán theory and extending the results in [21,5,6] (see also [8,20] in the context of prestrained films). In order to put our results in proper perspective, we now briefly review the theories obtained in the compressible case (i.e. when W = Wc ). For plates, -convergence as above was first established by LeDret and Raoult in [16] for β = 0 (see also [1] for the derivation starting from the linearized isotropic elastic energy), and then in a sequence of seminal papers by Friesecke et al. [11,12] for all β ≥ 2. In the case when 0 < β < 53 , the related result was obtained by Conti and Maggi in [7], while the regime 53 ≤ β < 2 remains open and is conjectured to be relevant to crumpling of elastic sheets. For shells, LeDret and Raoult studied the case β = 0 in [17], thus obtaining the membrane model, where the limit energy I0 depends only on the stretching and shearing produced by the deformation on the surface S. Another study belongs to Friesecke et al. [13], pertaining to the case β = 2. This scaling corresponds to a flexural shell model, where the only admissible deformations are those preserving the metric on S. The energy I2 depends then on the change of curvature due to the deformation. Further, Lewicka et al. studied the case of β ≥ 4 in [21]. For β = 4, the -limit I4 obtained therein is a generalization of the von Kármán theory for plates. The crucial role in describing the functional I4 is played by the space of first order infinitesimal isometries V on S. This space consists of displacements V ∈ W 2,2 (S, R3 ) with skew-symmetric covariant gradient:   (2) (∂τ V (x)) η + ∂η V (x) τ = 0 ∀a.e. x ∈ S ∀τ, η ∈ Tx S. The equivalent defining property of V ∈ V is that the family of deformations u  : S → R3 given by u  = id + V preserves the metric on S up to first order in :   |∂τ u  |2 − |τ |2 = O  2 ∀a.e. x ∈ S ∀τ ∈ Tx S. (3) The functional I4 consists then of two terms, measuring the second order change in the metric and the first order change in curvature, due to the admissible deformations of the form id + hV + h 2 w, with V ∈ V . For β > 4 this theory reduces to the linearized flexural shell model. The case of shells, whose thickness varies and is given by functions g1h , g2h rather than the constant h, has been discussed in [22,26]. Convergence of the critical points or equilibria of the energy functionals above has been studied in [19,28,29]. For an overview, we refer to [18,25]. In this paper we study the case of incompressible shells and the energy scaling β = 4. Our results apply to any sufficiently smooth midsurface S satisfying the following approximation property: ¯ R3 ) ∩ V is dense in V with respect to the W 2,2 (S, R3 ) The space V0 = C 3 ( S, (H ) topology.

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Surfaces enjoying condition (H) include: - flat surfaces S ⊂ R2 ; this is an easy consequence of the structure of first order isometries V for plates, where up to rigid motions V = (0, 0, v)T , - strictly convex C 5,α surfaces, as showed in [23] (for any α ∈ (0, 1)), - developable C 4,1 surfaces without flat parts, as showed in [15], - rotationally invariant C 3 surfaces, as showed in Appendix B. We organize our paper as follows. In Sect. 2, we introduce the notations and the main result, while in Sects. 3 and 4 we present the proofs. The proofs follow the analysis as in Theorem 1.1 [6], where the main point is the discussion of the ODE (15) obtained from the constraint of incompressibility, in the context of constructing the recovery sequence u h . The main difficulty in the present case of shells (and energy scaling β = 4) lies in the derivation of the crucial bounds (17). We overcome the technical difficulties by working with the determinant of the strain (∇u h )T ∇u h , rather than directly computing the determinant of the deformation gradient ∇u h , and by applying a combination of calculations in [21] plus an iteration argument discussed in detail in Appendix A. In Sect. 5, we derive equations (1) and the appropriate free boundary conditions (30).

2 Basic setting and the main results Consider a 2D compact, connected, oriented surface S of class C 4 , embedded in R3 , whose boundary ∂ S is the union of finitely many (possibly none) Lipschitz continuous curves. Let n(x) be the unit normal of S, Tx S the tangent space of S at x, and (x) = ∇n(x) the corresponding shape operator. Consider further a family of thin shells S h around S of the following form:   h h S h = z = x + tn(x); x ∈ S, − < t < . (4) 2 2 We will assume that h < h 0 << 1, so that: 1 1 < det(Id + t (x)) < 2, < |Id + t (x)| < 2, ∀x ∈ S ∀t ∈ (−h 0 , h 0 ), 2 2

(5)

and so that the projection π : S h −→ S along n(x) is well defined: π(z) = x if z = x + tn(x) ∈ S h . We shall use the following notation. For a 3 × 3 matrix field M : S −→ R3×3 on S, Mtan stands for its tangential minor, that is: Mtan (τ, η) = τ T Mη, ∀τ, η ∈ Tx S. By symM, we mean a bilinear form on Tx S given by:  1 T τ Mη + η T Mτ , (symM)(τ, η) = ∀ τ, η ∈ Tx S. 2 Also, when calculations in coordinates are necessary, we will perform these in some orthonormal basis {τ1 (x), τ2 (x), n(x)}, where τ1 (x), τ2 (x) ∈ Tx S. For a deformation u h ∈ W 1,2 (S h , R3 ), its scaled elastic energy is given by:   1   I h uh = W ∇u h , h Sh

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where the stored-energy density W : R3×3 −→ [0, ∞] obeys:  Wc (F), if detF = 1, W (F) = ∞, otherwise. The effective energy density Wc is assumed to be C 2 in a neighborhood of the special orthogonal group S O(3), and to satisfy the assumptions of frame indifference, normalization and growth [12]: ⎤ (i) Wc (R F) = Wc (F), ∀F ∈ R3×3 ∀R ∈ S O(3). ⎦ (6) (ii) Wc (Id) = 0. (iii) Wc (F) ≥ c dist2 (F, S O(3)), with some constant c > 0 independent of F. We shall now introduce the von Kármán energy functional I for incompressible shells. Define first two quadratic forms: Q3 (F) = ∇ 2 Wc (Id)(F, F),

acting on matrices F ∈ R3×3 and: Q2 (x, Ftan ) = min {Q3 (Ftan + d ⊗ n + n ⊗ d) ; Tr(Ftan + d ⊗ n + n ⊗ d) = 0} , d∈R3

acting on tangential minors Ftan ∈ R2×2 . Here, Tr stands for the trace of a given matrix. The crucial role in defining I : V × B → R is played by the two important spaces: space of first order infinitesimal isometries V and the space of finite strains B. The space V , given in (2), consists of vector fields V ∈ W 2,2 (S, R3 ), for which there exists a matrix field A ∈ W 1,2 (S, R3×3 ) with: ∂τ V (x) = A(x)τ and A(x)T = −A(x), ∀a.e. x ∈ S, ∀τ ∈ Tx S. Thus, under the displacement V ∈ V , there is no first order change in the Riemannian metric of S, see (3). Recall that S enjoys the approximation property (H) when   ¯ R3 ∩ V . V0 = C 3 S, is dense in V with respect to the W 2,2 (S, R3 ) topology. The finite strain space B (see [21]) consists of all symmetric matrix fields which are the L 2 limits of symmetric gradients of W 1,2 vector fields on S:   B = L 2 − lim h→0 sym∇w h ; w h ∈ W 1,2 S, R3 . Note that the smooth finite strain space:   ¯ R3 B0 = sym∇w; w ∈ C ∞ S, is dense in B with respect to the L 2 (S, R3 ) topology. For each V ∈ V and Btan ∈ B, the functional I calculated on V and Btan , returns the value:    1 1  2 A tan I (V, Btan ) = Q2 x, Btan − 2 2 S (7)    1 Q2 x, (∇(An) − A )tan . + 24 S

Our main result is contained in the theorem below. In what follows, by C we denote any constant which is independent of h (but it may depend on S, Wc , or other involved quantities).

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Theorem 1 Assume that S additionally enjoys the approximation property (H). Then: (a) (Compactness and lower bound) For any sequence of deformations u h ∈ W 1,2 (S h , R3 ) with:   I h u h ≤ Ch 4 , (8) there exists a sequence Q h ∈ S O(3) and ch ∈ R3 such that for a subsequence (which we will not relabel) of the normalized scaled deformations:    T h h h h y (x + tn(x)) = Q u x + t n(x) − ch : S h 0 −→ R3 h0 there hold: (i) y h converges in W 1,2 (S h 0 ) to the projection π. (ii) There exists an infinitesimal isometry V ∈ V which is the limit of the scaled average displacements: 

V h yh



h0

1 = h

2  h − 20

  h y h (x + tn(x)) − x + t n dt h0

(9)

in W 1,2 (S, R3 ). 1 (iii) sym∇V h [y h ] converges weakly in L 2 (S) to some Btan ∈ B. h 1 (iv) lim inf 4 I h (u h ) ≥ I (V, Btan ). h→0 h (b) (Recovery sequence) Conversely, for every V ∈ V and Btan ∈ B, there exists a sequence u h ∈ C 1 (S h , R3 ), such that:   h h h (i) y (x + tn(x)) = u x + t n(x) → π in W 1,2 (S h 0 , R3 ). h0 (ii) The scaled average displacements V h [y h ] defined in (9) converges in W 1,2 (S, R3 ) to V . 1 (iii) sym∇V h [y h ] → Btan in L 2 (S). h (iv) u h realizes the limiting energy, that is: lim

h→0

1 h  h I u = I (V, Btan ) . h4

3 Proof of compactness and lower bound The compactness and the lower bound follow from a combination of the corresponding results for the compressible von Kármán shell theory in [21] and the technique developed for the incompressible Kirchhoff model in [6]. Define, for all k > 0: Wk (F) = Wc (F) +

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Incompressible von Kármán shell theory

and:

191

  1   Ikh u h = Wk ∇u h dz. h Sh

As in [6], consider the quadratic forms associated with Wk : Qk3 (F) = ∇ 2 Wk (Id)(F, F) = Q3 (F) + k(TrF)2 ,

and the corresponding reduced forms:   Qk2 (x, Ftan ) = min Qk3 Ftan + d ⊗ n + n ⊗ d , ∀Ftan ∈ R2×2 . d∈R3

Applying the same argument as in the proof of Lemma 2.1 in [6], we obtain for each Ftan ∈ R2×2 , x ∈ S, k > 0: C

Q2 (x, Ftan ) − √ Ftan 2 ≤ Qk2 (x, Ftan ) ≤ Q2 (x, Ftan ) ,

(10)

k

where the constant C is independent of k. Proof (Proof of Theorem 1, part (a):) For a fixed k > 0, the approximate density Wk satisfies all the assumptions (6). Hence, Ikh (u h ) ≤ Ch 4 implies in virtue of Theorem 2.1 in [21] that (i), (ii) and (iii) hold. Indeed, by the definition of Wk , we have Wk (F) ≤ W (F), which yields Ikh (u h ) ≤ I h (u h ) ≤ Ch 4 . Further, still by [21]:      1   1 1  2 1 A tan + Qk2 x, Btan − Qk2 x, (∇(An) − A )tan . lim inf 4 Ikh u h ≥ h→0 h 2 2 24 S

S

By (10) we obtain:  2  1  2 1  2  C   A tan − √  A B − tan tan  2 2 k   1  2 k A tan , ≤ Q2 x, Btan − 2 2   C  Q2 x, (∇(An) − A )tan − √ (∇(An) − A )tan  k   k ≤ Q2 x, (∇(An) − A )tan , 

Q2 x, Btan −

which imply for each k > 0:     1 1 lim inf 4 I h u h ≥ lim inf 4 Ikh u h h→0 h h→0 h 2     1  2 1  2  1 C   dx A tan − √  A B Q2 x, Btan − − ≥ tan tan  2 2 2 k S  2   C  1 + Q2 x, (∇(An) − A )tan − √ (∇(An) − A )tan  dx 24 k S    2   2  2 C  1   A tan  Btan − = I (V, Btan ) − √   2 + (∇(An) − A )tan L 2 (S) . 2 k  L (S) Taking k → ∞, we obtain the result.

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4 Construction of the recovery sequence Since by assumption (H) the space V0 is dense in V , while B0 is always dense in B, part (b) in Theorem 1 will follow via the diagonal argument and the application of the next two Lemmas. Lemma 1 For any V ∈ V0 , Btan = sym∇w ∈ B0 , and c0 , c1 ∈ C ∞ (S, R3 ) with   1  2 Tr Btan − A tan + c0 ⊗ n + n ⊗ c0 = 0, 2   Tr sym (∇(An) − A )tan + c1 ⊗ n + n ⊗ c1 = 0, there exists a sequence of incompressible deformations u h ∈ C 1 (S h , R3 ), such that: (i), (ii) and (iii) of Theorem 1 (b) hold, together with:     1 1 1  2 A tan + c0 ⊗ n + n ⊗ c0 Q3 Btan − lim 4 I h u h = h→0 h 2 2 S (11)    1 Q3 sym (∇(An) − A )tan + c1 ⊗ n + n ⊗ c1 . + 24 S

Proof 1. Following [21], we define the rescaled compressible deformations ych ∈ W 1,2 (S 2h 0 , R3 ) in agreement with the Kirchhoff-Love ansatz: h h2 h3 ych (x +tn(x)) = x + hV (x) + h 2 w(x)+t n(x)+t A(x)n(x)−t (∇w(x))T n(x) h0 h0 h0     h3 1 2 T 2 2c0 + A(x) n(x) − +t n(x) A(x) n(x) n(x) h0 2  h3  +t 2 2 2c1 + (A(x) (x) − ∇(A(x)n(x)))T n(x) . 2h 0 Let u ch ∈ W 1,2 (S 2h , R3 ) be given by u ch (x + tn(x)) = ych (x + t hh0 n(x)). By a straightforward calculation:    h h h h ∂τ yc (x + tn(x)) = ∇u c x + t n(x) Id + t (x) (Id + t (x))−1 τ, h0 h0   h h ∂n ych (x + tn(x)) = ∇u ch x + t n(x) n(x), h0 h0 where the first equality holds for each τ ∈ Tx S. Thus:  −1  h  h h h τ ∇u c x + t n τ = ∇ y (x + tn)(Id + t ) Id + t h0 h0  h h2 = Id + h A + h 2 ∇w + t + t ∇(An) h0 h0     3 3 h h 1 T 2  T 2 n A n n −t ∇ (∇w) n + t ∇ 2c0 + A n − h0 h0 2 −1  h h3  +t 2 2 ∇ 2c1 + (A − ∇(An))T n Id + t τ, h0 2h 0

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 h  ∇u ch x + t n n = n + h An − h 2 (∇w)T n h0    1 T 2  h3  n A n n +t 2c1 + (A − ∇(An))T n . +h 2 2c0 + A2 n − 2 h0 2. We shall now calculate det∇u ch by looking at the strain (∇u ch )T ∇u ch − Id. We borrow the calculations from [21] and for each τ1 , τ2 , τ ∈ Tx S we obtain:   T    h h T h h τ1 ∇u c x + t n ∇u c x + t n − Id τ2T h0 h0     1 t = 2h 2 τ1T Btan − A2 + sym (∇(An) − A ) τ2T + O h 3 , 2 h0   T    h h T h h n ∇u c x + t n ∇u c x + t n − Id n h0 h0 (12)  3 h2 T 2 T = 4h n c0 + 4t n c1 + O h , h0   T    h h T h h n ∇u c x + t n ∇u c x + t n − Id τ h0 h0     t = h 2 c0T + c1T τ + O h 3 . h0 Hence, using the formula det(Id + F) = 1 + TrF + Tr(cofF) + det F, we see that: 

T   h h h det x + t n ∇u c x + t n(x) h0 h0   T    h h h h = det ∇u c x + t n ∇u c x + t n − Id + Id h0 h0     1  2 A tan + 2nT c0 = 1 + 2h 2 Tr Btan − 2  h2  Tr (sym (∇(An) − A )) + 2nT c1 + 2t h0   + h 4 E 1 Btan , A2 , n · c0 , τ1 · c0 , τ2 · c0   + th 4 E 2 Btan , A2 , n · c0 , τ1 · c0 , τ2 · c0 , ∇(An), A , n · c1 , τ1 · c1 , τ2 · c1   + t 2 h 4 E 3 Btan , A2 , n · c0 , τ1 · c0 , τ2 · c0 , ∇(An), A , n · c1 , τ1 · c1 , τ2 · c1 

∇u ch

+ o(h 4 )



= 1 + 2h Tr Btan 2

1  2 A tan + c0 ⊗ n + n ⊗ c0 − 2



h2 Tr (sym (∇(An) − A ) + c1 ⊗ n + n ⊗ c1 ) h0   + h 4 E 1 + th 4 E 2 + t 2 h 4 E 3 + o h 4   = 1 + h 4 E 1 + th 4 E 2 + t 2 h 4 E 3 + o h 4 . + 2t

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where E 1 , E 2 , E 3 are polynomials in their arguments. The above thus implies:  det∇u ch

     T 1/2 h h h h h x + t n = det ∇u c x + t n ∇u c x + t n h0 h0 h0  3 =1+O h .

Therefore, for each τ ∈ Tx S we deduce that:            det∇u h x + t h n − 1 + ∂τ det∇u h x + t h n  ≤ Ch 3 . c c    h0 h0  3. We shall seek the scaled recovery sequence in the form y h (x + tn) = ych (x + φ h (x, t)n), or equivalently:     h h h0 h h u (x + sn) = u c x + φ x, s n . (13) h0 h for some change of variable function φ h : S × (−h 0 , h 0 ) −→ R. For each τ ∈ Tx S, we have:         h h0 h0 h ∂τ u(x + sn) = ∇u ch x + φ h x, s n Id + ∇ φ h x, s n h0 h h0 h ·(Id + s )−1 τ. Meanwhile:         h0 h0 h0 h h h ∂τ φ x, s n(x) = φ x, s (x)τ + ∂τ φ x, s n. h h h Therefore: ∂τ u(x + sn) ⎡     h h 0 = ∇u ch x + φ h x, s n ⎣ h0 h

  ⎤ Id + hh0 φ h x, hh0 s    ⎦ h h x, h 0 s , h ∂ φ h x, h 0 s ∂ φ τ τ h0 1 h h0 2 h

· (Id + s )−1 τ     h h0 = ∇u ch x + φ h x, s n h0 h  ⎤ ⎡    h h Id + h 0 φ x, hh0 s (Id + s )−1 ⎦ τ.     · ⎣ h h x, h 0 s , h ∂ φ h x, h 0 s (Id + s )−1 h 0 ∂τ1 φ h h 0 τ2 h For the normal derivative: ∂n u (x + sn) = ∂t φ h

 h

     h0 h h h0 h x, s ∇u c x + φ x, s n(x) n. h h0 h

Consequently:     h h0 ∇u h (x + sn) = ∇u ch x + φ h x, s n(x) M1 , h0 h

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where:    ⎤  0 Id + hh0 φ h x, hh0 s (Id + s )−1      ⎦ . M 1 = ⎣ h h x, h 0 s , h ∂ φ h x, h 0 s −1 ∂ φ h x, h 0 s ∂ φ (Id + s ) τ τ t 1 2 h0 h h0 h h ⎡

Changing the variable s = t hh0 , we obtain:  ∇u

h

   h h h h x + t n = ∇u c x + φ (x, t)n(x) M2 , h0 h0

(14)

where: ⎡

⎤  −1 h Id + t h 0 0 ⎢ ⎥ M 2 = ⎣  −1 ⎦. h h h h h Id + t h 0 ∂t φ h (x, t) h 0 ∂τ1 φ (x, t), h 0 ∂τ2 φ (x, t)  Id +

h h h 0 φ (x, t)

Therefore:     h h det∇u h x + t n = det∇u ch x + φ h (x, t)n(x) h0 h0   −1  h h h · det Id + φ (x, t) Id + t ∂t φ h (x, t), h0 h0 and we see that the incompressibility of u h will follow, if φ h obey the following ODE: 

∂t φ h (x, t) = f (x, φ h (x, t), t), φ h (x, 0) = 0.

(15)

Here:   h yn f (x, y, t)−1 = det∇u ch x + h0   −1  h h · det Id + . y Id + t h0 h0

(16)

By the theory of families of solutions of parameter dependent ordinary differential equations, for each x ∈ S, there exists a unique solution φ h (x, t) ∈ C 1 (− h20 , h20 ) for each x ∈ S. Moreover, for each x ∈ S, t ∈ (− h20 , h20 ), we have the following bounds:             h ∂t φ (x, t) − 1 ≤ Ch 3 , φ h (x, t) − t  ≤ Ch 3 , ∂τ φ h (x, t) ≤ Ch 3 .

(17)

The detailed analysis of (15) and the proof of (17) can be found in Appendix A. 4. With bounds in (17), we obtain (i), (ii) and (iii) of Theorem 1 (b) by direct calculation. To deduce (11), we shall now find:   T    1 t t h h lim (18) ∇u x + h n ∇u x + h n − Id . h→0 2h 2 h0 h0

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We will first prove that the expression below is of the order O(h 3 ). By (14), we have:   T  h h ∇u h x + t n ∇u h x + t n h0 h0   T  h h − ∇u ch x + t n ∇u ch x + t n h0 h0 T    h h = M2T ∇u ch x + φ h (x, t)n ∇u ch x + φ h (x, t)n M2 h0 h0    T   h h h h ∇u c x + t n − ∇u c x + t n h0 h0 (19) T     h h h h T h h = M2 ∇u c x + φ (x, t)n ∇u c x + φ (x, t)n h0 h0 T    h h − ∇u ch x + t n ∇u ch x + t n M2 h0 h0 T    h h T h h + (M2 − Id) ∇u c x + t n ∇u c x + t n M2 h0 h0    T   h h ∇u ch x + t n + ∇u ch x + t n (M2 − Id). h0 h0 For the first term above, since M2 is bounded when h is small, we only need to consider: T    h h h h h h M3 =∇u c x + φ (x, t)n ∇u c x + φ (x, t)n h0 h0   T  h h − ∇u ch x + t n ∇u ch x + t n . h0 h0 According to (12) and the estimates (17) for φ h , we have for each τ1 , τ2 ∈ Tx S:      h2  h φ (x, t) − t τ1T sym (∇(An) − A ) τ2T + O h 3 = O h 3 , τ1T M3 τ2 = 2 h0  2  h n T M3 n = 4 φ h (x, t) − t nT c1 + O(h 3 ) = O(h 3 ), h0      h2  h φ (x, t) − t c1T + O h 3 = O h 3 . n T M3 τ1 = h0 Consequently: |M2T M3 M2 | ≤ Ch 3 .

(20)

Further, by (12): T    h h ∇u ch x + t n ∇u ch x + t n = O(1). h0 h0 Recall that M2 = O(1) as well, and consider M2 − Id, for which we have the following: ⎤ ⎡  −1 h h h − Id 0 Id + φ (x, t) Id + t h 0 ⎦. M2 − Id = ⎣  h 0  h h (x, t) ∂ φ h (x, t) (Id + t )−1 ∂ φ h (x, t) − 1 ∂ φ τ1 τ2 t h0

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Incompressible von Kármán shell theory

197

Since: 

 −1 h h h Id + t φ (x, t) − Id h0 h0 −1   h h  h φ (x, t) − t Id + t , = h0 h0

Id +

in view of (17), we conclude that: |M2 − Id| ≤ Ch 3 .

(21)

Denote:  K (x + tn) = ∇u h

h

h x +t n h0

T

 ∇u

h

 h x + t n − Id. h0

By (19), (20), (21), we obtain: T      h h K h (x + tn) = ∇u ch x + t n ∇u ch x + t n − Id + O h 3 . h0 h0

(22)

5. Repeating the calculations in the proof of Theorem 2.2 in [21] and taking into account (22), we get: 1 h K (x + tn) 2h 2   T     3 1 h h h h ∇u c x + t n ∇u c x + t n − Id + O h = lim h→0 2h 2 h0 h0

lim

h→0

1  2 A tan + c0 ⊗ n + n ⊗ c0 = Btan − 2   t + sym(∇(An) − A )tan + c1 ⊗ n + n ⊗ c1 . h0

(23)

Next, we examine the scaled energy functional I h (u h ). We have:   1   I h uh = W ∇u h dz h Sh



h0 /2

 W ∇u h

= S −h 0 /2



h x +t n h0





 h det Id + t dtdx. h0

(24)

By polar decomposition theorem and frame indifference of Wc , we see: ⎞ ⎛  T      h h h Wc ∇u h x + t n = Wc ⎝ ∇u h x + t n ∇u h x + t n ⎠ h0 h0 h0    1 = Wc Id + K h + O |K h |2 , 2

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which implies, as in [21]:       1 h 1 h 1 1  h 2 h 2 |K ∇u x + t Id + h W n(x) = W K + O | c c h4 h0 h4 2h 2 h2    1 h 1 1 h 2 = Q3 K + 2 O |K | 2 2h 2 h    1 h 1  h 2 2  + o(1)  2 K + 2 O |K |  . 2h h Thus for the limiting energy functional, by (23) and (24), we have:   1 lim 4 I h u h h→0 h   h0 /2   1 h 1 1  h 2 |K Q3 K + O | = lim h→0 2 2h 2 h2 S −h 0 /2

  2   1 h 1  + o(1)  2 K h + 2 O |K h |2  det Id + t dxdt 2h h h0 1 = lim h→0 2 1 = 2 =

1 2







h0 /2

Q3 S −h 0 /2



h0 /2

Q3 S −h 0 /2

 1 h K dtdx 2h 2

 t K 1 (x) + K 2 (x) dxdt h0



Q3 (K 1 (x))dx + S

1 24

 Q3 (K 2 (x))dx, S

where: 1  2 A tan + c0 ⊗ n + n ⊗ c0 , 2 K 2 (x) = (∇(An) − A )tan + c1 ⊗ n + n ⊗ c1 . K 1 (x) = Btan −

This ends the proof of Lemma 1. Lemma 2 For each V ∈ V0 , Btan = sym∇w ∈ B0 , there exists a sequence u h ∈ C 1 (S h , R3 ), such that (i), (ii), (iii) and (iv) of Theorem 1 part (b) hold. Proof By the definition of Q2 and positive definiteness of Q3 , there exist L 2 vector fields c0 , c1 : S −→ R3 such that:     1  2 1  2 A tan = Q3 Btan − A tan + c0 ⊗ n + n ⊗ c0 , Q2 x, Btan − 2 2   1  2 A tan + c0 ⊗ n + n ⊗ c0 = 0; with Tr Btan − 2 Q2 (x, sym(∇(An) − A )) = Q3 (sym (∇(An) − A ) + c1 ⊗ n + n ⊗ c1 ) ,

with Tr (sym (∇(An) − A ) + c1 ⊗ n + n ⊗ c1 ) = 0.

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Incompressible von Kármán shell theory

199

¯ for From trace zero condition and the regularity of V and Btan , we obtain ci · n ∈ C ∞ ( S) n ∞ 2 3 ¯ i = 0, 1. Let {ci } be a sequence of functions in C ( S), converging to ci in L (S, R ), with cin · n = ci · n for i = 0, 1. Notice that c1n , c2n satisfies the trace zero condition as well and that:    1  2 lim A tan + c0n ⊗ n + n ⊗ c0n Q3 Btan − n→∞ 2 S    1  2 = Q3 Btan − A tan + c0 ⊗ n + n ⊗ c0 , 2 S    lim Q3 sym (∇(An) − A )tan + c1n ⊗ n + n ⊗ c1n n→∞

S

 =





Q3 sym (∇(An) − A )tan + c1 ⊗ n + n ⊗ c1 . S

Applying Lemma 1 to V, Btan , c1n , c2n and taking a diagonal sequence yields the required recovery sequence u h .

5 The von Kármán equations for incompressible plates In this section, we will derive the Euler-Lagrange equations of the functional I in (7) for plates, that is when: S =  ⊂ R2 , and under assumption that the effective energy density Wc is isotropic: ∀F ∈ R3×3 ∀R ∈ S O(3) Wc (F R) = Wc (F),

(25)

In this situation, the finite strain space B takes the form:   B = sym∇w; w ∈ W 1,2 , R2 , while the space V becomes: # " V = (v1 , v2 , v3 )T ; v3 ∈ W 2,2 (), (v1 , v2 )T (x) = A1 x + b, A1 ∈ so(2), b ∈ R2 . ¯ is dense Here, so(2) denotes the space of 2 × 2 skew-symmetric matrices. Note that C ∞ () in W 2,2 (), thus the assumption (H) holds.  T Taking Btan = sym∇ w˜ and V (x1 , x2 ) = ax2 + b1 , −ax1 + b2 , v3 (x1 , x2 ) , we calculate: ⎞ ⎛ 0 a −∂1 v3 A = ⎝ −a 0 −∂2 v3 ⎠ , ∂1 v3 ∂2 v3 0 and hence, in view of Q2 not depending on x and = 0, we obtain:       1 1 1 2 1 ˜ + ∇v3 ⊗ ∇v3 + a Id + I (V, Btan ) = Q2 sym∇ w Q2 ∇ 2 v3 . 2 2 2 24 



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The limiting functional can be therefore expressed in terms of the in-plane displacement w = w˜ − 21 a 2 id and the out-plane displacement v = v3 :       1 1 1 I (w, v) = Q2 sym∇w + ∇v ⊗ ∇v + Q2 ∇ 2 v . (26) 2 2 24 



Indeed, based on the results in [11], and in view of the similar argument to the one in the proof of Theorem 1, one can show that for appropriately chosen rotations Q h and translations ch , the scaled average displacements V h [y h ] converge (up to subsequence) in W 1,2 (, R3 ) to the vector field of the form (0, 0, v)T , while the scaled in-plane displacements h −1 (V h [y h ])tan converge (up to a subsequence) weakly in W 1,2 (, R2 ) to the in-plane displacement field w ∈ W 1,2 (, R2 ). Now, for Wc isotropic as in (25), the quadratic form Q3 has the expression (see e.g.[12]): Q3 (F) = 2μ|symF|2 + λ|TrF|2 ,

(27)

where λ and μ are first and second Lamé constants respectively, which in addition satisfy: μ ≥ 0, 3λ + μ ≥ 0. Taking into account the incompressibility constraint, we shall rewrite (27) in terms of the second Lamé constant μ and Poisson ratio ν, in the virtue of the relation λ = 2μν/(1 − 2ν), as: Q3 (F) = 2μ|symF|2 +

2μν |TrF|2 . 1 − 2ν

We further obtain: 



Q2 (Ftan ) = 2μ |symFtan |2 + |TrFtan |2 ,

(28)

and following exactly the same calculation as in e.g. [20], we finally derive the Euler-Lagrange equations of the limiting energy I in (26): ⎧μ ⎨ 2 v = [v, ], 3 (29) ⎩ 2  = − 3μ [v, v]. 2 The Airy stress potential  ∈ W 2,2 (, R) serves then for recovering the in-plane displacement w by means of:    1 1 cof∇ 2  = 2μ sym∇w + ∇v ⊗ ∇v + divw + |∇v|2 Id , 2 2 and the Airy’s bracket [·, ·] is defined as [v, ] = ∇ 2 v : (cof∇ 2 ). The natural (free) boundary conditions associated to (26) are: ⎧ ⎨  = ∂nb  = 0, 2∇ 2 v : (nb ⊗ nb ) + ∇ 2 v : (τ ⊗ τ ) = 0, on (30)  ∂, ⎩  2 ∂τ ∇ v : (nb ⊗ τ ) + div 2∇ 2 v + cof∇ 2 v · nb = 0. where nb ∈ R2 denotes the normal and τ the tangent vector to ∂. Remark By means of change of variables  = 2μ1 , the system (29) is equivalent to: 3 2 v = 6[v, 1 ], 2 1 = − [v, v], 4

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Incompressible von Kármán shell theory

201

which eliminates the parameter μ entirely. However, it is preferred to keep the form (29), because it can be seen as the incompressible limit of the classical (compressible) von Kármán systems: S B2 v = [v, ], 2  = − [v, v]. 2 Here, S = 2μ(1 + ν) is Young’s modulus and B =

S 12(1−ν 2 )

is the bending stiffness. Thus, as

ν → 21 , the material becomes incompressible and the system above indeed converges to (29). Concerning von Kárnán theory, we thus establish the commutativity of the asymptotic limits corresponding to incompressibility on the one hand and thinness (i.e. decreasing thickness) on the other hand. For more discussion of commuting properties in elasticity, one may refer to [2,14]. Acknowledgements This research was partially supported by Professor Marta Lewicka’s NSF Grants DMS0707275 and DMS-0846996 and by her Polish MN Grant N N201 547438. We want to thank Professor Marta Lewicka for her advice and guidance during the work. We are also grateful to B. Davidovich for a helpful discussion about incompressibility. This project started out from discussions with Georg Dolzmann at University of Regensburg while MC was a postdoctoral fellow there. MC wishes to thank the University of Regensburg for its financial support during her stay.

Appendix A. Existence of φ h and the bounds (17) Lemma 3 When h is sufficiently small, then for each x ∈ S the ODE (15) has a unique solution φ h (x, ·) : (− h20 , h20 ) −→ R and     h0 h0  h  . φ (x, t) < h 0 , ∀x ∈ S ∀t ∈ − , 2 2

(31)

Furthermore, φ h is C 1 regular in x and its tangential gradient ∇tan φ h (x, t) is bounded:     h0 h0   h ∀x ∈ S ∀τ ∈ Tx S. ∂τ φ (x, t) < C|τ |, ∀t ∈ − , 2 2

(32)

Proof 1. Recall f : S × (−h 0 , h 0 ) × (− h20 , h20 ) −→ R, defined in (16), has the following expression:  f (x, y, t) =



det∇u ch

   −1 −1 h h h Id + t . x + y n det Id + y h0 h0 h0

First, we notice that f, ∇x f and ∇ y f are Lipschitz in y. Note that: 

h Id + y h0



h Id + t h0

−1



h = Id + y h0 = Id +



 2  3 h 2h 2 Id − t + t 2 + O h h0 h0

  h2 h (y − t) − t 2 (y − t) 2 + O h 3 , h0 h0

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which yields:   −1 h h det Id + y Id + t h0 h0 = 1+

  h2 h2 h (y − t) Tr − t 2 (y −t) Tr( 2 )+ 2 (y −t)2 E( ) + O h 3 , h0 h0 h0

where E is a polynomial. Hence: f (x, y, t)−1     1 1 1 = 1 + h 4 E 1 + yh 4 E 2 + y 2 h 4 E 3 + o h 4 · 2 2 2    3  (33) h2 h2 h 2 2 · 1+ (y −t) Tr − t 2 (y − t) Tr( )+ 2 (y −t) E( )+ O h h0 h0 h0 =1+

  h2   h2 h (y − t) Tr − t 2 (y − t) Tr 2 + 2 (y − t)2 E( ) + O h 3 , h0 h0 h0

which implies that: f (x, y, t) 

  h h2 h2 = 1 + (y −t)Tr −t 2 (y − t)Tr( 2 ) + 2 (y −t)2 E( )+ O h 3 h0 h0 h0

−1 .

(34)

It follows that, for sufficiently small h:

  h0 h0 1 < f (x, y, t) < 2, ∀x ∈ S, ∀y ∈ (−h 0 , h 0 ), ∀t ∈ − , . 2 2 2

(35)

2. Towards proving (32), we now expand the derivatives ∇x f (x, y, t) and ∇ y f (x, y, t) of f . For each τ ∈ Tx S, we have: ∇x f (x, y, t)τ

  − t)2 ∂τ (E( )) + O h 3 =−   .   2 h h2 h2 2 2 3 1 + h 0 (y − t)Tr − t h 2 (y − t)Tr( ) + h 2 (y − t) E( ) + O h h h 0 (y

2

− t)∂τ (Tr ) − t hh 2 (y − t)∂τ (Tr( 2 )) + 0

0

h2 (y h 20

0

(36) Meanwhile: ∇ y f (x, y, t) h h2 h 0 Tr − t h 2 Tr

= −

0

1+

h h 0 (y

   2 2 + 2 hh 2 (y − t)E( ) + O h 3

2 − t)Tr − t hh 2 (y 0

0

− t)Tr



2



+

h2 (y h 20

− t)2 E( ) + O

 .   2 3 h (37)

Choose h so small that: ∇x f (x, y, t)∞ + ∇ y f (x, y, t)∞ ≤ 1.

123

(38)

Incompressible von Kármán shell theory

203

Consider the Banach space B = C (S × (− h20 , h20 ); (−h 0 , h 0 )) equipped with the L ∞ norm and define the operator T : B −→ B: t f (x, u(x, s), s)ds, ∀u ∈ B.

(T u)(x, t) = 0

By (35), −h 0 < (T u)(x, t) < h 0 and so indeed T u ∈ B. By (38), for each u, v ∈ B:   t   t   f (x, v(x, s), s)ds  |(T u − T v)(x, t)| =  f (x, u(x, s), s)ds −   0

0

|t| | f (x, u(x, s), s) − f (x, v(x, s), s)| ds

≤ 0

≤

1 1 h 0 u − v∞ ≤ u − v∞ . 2 2

In view of the Banach fixed point theorem, there exists a unique φ h ∈ B, such that T φ h = φ h , which is equivalent to φ h solving Eq. (15). Also (31) is a direct consequence of the definition of the space B. 3. Our next goal is to show that φ h is C 1 regular in x. For a fixed τ ∈ Tx S, define: t ∇x f (x, u(x, s), s)τ + ∇ y f (x, u(x, s), s)v(x, s)ds.

S(u, v)(x, t) = 0

Let u 0 (x, t) = 0 and v0 (x, t) = ∂τ u 0 (x, t) = 0, and define sequences (u k ), (vk ) iteratively as follows: u k+1 = T u k , vk+1 = S(u k , vk ), so that lim u k = φ h . We will now show that: k→∞

lim vk = ∂τ φ h .

(39)

k→∞

Note first that: t ∂τ (T u)(x, t) =

∇x f (x, u(x, s), s)τ + ∇ y f (x, u(x, s), s)∂τ u(x, s)ds 0

= S(u, ∂τ u).

(40)

In fact, an inductive argument shows that for each k ∈ N, vk (x, t) = ∂τ u k (x, t), because: (i). For k = 0, this identity holds by the definition of v0 . (ii). If vk = ∂τ u k then, by (40): vk+1 = S(u k , vk ) = S(u k , ∂τ u k ) = ∂τ (T u k ) = ∂τ u k+1 .

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Now, for each v ∈ C (S × (−h 0 /2, h 0 /2)), define: t     h (Rv)(x, t) = S φ , v (x, t) = ∇x f x, φ h (x, s), s τ 

0

 +∇ y f x, φ (x, s), s v(x, s) ds. h

For any two v1 , v2 , we have:  t        1 |(Rv1 − Rv2 )(x, t)| =  ∇ y f x, φ h (x, s), s (v1 (x, t) − v2 (x, t)) ≤ v1 − v2 ∞ .   2 0

Thus R is a contraction, and hence it has a fixed point z. Further notice that:   vk+1 − Rvk = S (u k , vk ) − S φ h , vk t ∇x f (x, u k (x, s), s) τ + ∇ y f (x, u k (x, s), s)vk (x, s)ds

= 0

t −

    ∇x f x, φ h (x, s), s τ + ∇ y f x, φ h (x, s), s vk (x, s)dx

0

≤ L 1 u k − φ h ∞ + L 2 u k − φ h ∞ vk ∞ . where L 1 and L 2 are the Lipschitz constants of ∇x f (x, y, t) and ∇ y f (x, y, t) with respect to y. Since u k − φ h ∞ → 0 as k → ∞, we may use Ostrowski’s theorem on approximate iteration [32], and conclude that vk → z uniformly in S × (− h20 , h20 ). Therefore, there must be z = ∂τ φ h (x, t) and (39) follows. 4. We devote the following to proving (32). This will be a consequence of (39) and the boundedness of the sequence vk . Indeed, by (38):  t      1 |v1 (x, t)| =  ∇x f (x, u 0 (x, s), s) τ + ∇ y f (x, u 0 (x, s), s) v0 (x, s)ds  ≤ h 0 ,   2 0  t      |v2 (x, t)| =  ∇x f (x, u(x, s), s) τ + ∇ y f (x, u(x, s), s) v1 (x, s)ds    0

|t| ≤

1 1 1 + h 0 ds ≤ h 0 + 2 2

0

Inductively, we obtain vk ∞ ≤



1 h0 2

'k

h0 i i=1 ( 2 ) .

vk ∞ ≤

2 . Thus:

1 , 2 − h0

and so (32) follows. Note that the conclusion φ h (x, t) ∈ (−h 0 , h 0 ) is needed to ensure that the recovery sequence y h is well-defined.

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Incompressible von Kármán shell theory

205

Lemma 4 In the framework of Lemma 3, the bounds (17) hold. Proof By (34), we see that: ∂t φ h (x, t) − 1 = − +

    h  h h2  φ − t Tr + t 2 φ h − t Tr 2 h0 h0

(41)

2     h2  h φ −t (Tr )2 − E( ) + O h 3 , 2 h0

which implies: |∂t φ h (x, t) − 1| ≤ Ch,   |t|    t         h    h  ∂t φ (x, s) − 1 ds  ≤ ∂t φ h (x, s) − 1 ds ≤ Ch. φ (x, t) − t  =    0

(42) (43)

0

Using the estimate (43) in (41), we obtain that the quantity in (42) is bounded by Ch 2 , which implies the same quadratic bound in (43). Repeating the same procedure, we further improve the bounds to cubic ones:      h   h  (44) ∂t φ (x, t) − 1 ≤ Ch 3 , φ (x, t) − t  ≤ Ch 3 . Regarding the tangential derivative ∂τ φ h , observe that: ∂τ φ h (x, t) t ∂τ

=



1+

0

    h  h h2  φ − s Tr − s 2 φ h − s Tr 2 h0 h0

2   −1  h2  h ds − s E( ) + O φ h3 h 20      2 2  t − hh0 ∂τ φ h Tr + s hh 2 ∂τ φ h Tr 2 − 2 hh 2 φ h − s ∂τ φ h E( ) + O h 3 0 0 =  ds    2    2 2  2  h h 0 1 + h 0 φ h − s Tr − s h 2 φ h − s Tr( 2 ) + hh 2 φ h − s E( ) + O h 3 +

t + 0

h h0



s

− φh

 1+

h h0

0



2 ∂τ (Tr ) + s hh 2 0





 2 φ h −s Tr − s hh 2 0



∂τ (Tr



0

2   2  )− hh 2 φ h − s ∂τ (E( ))+ O h 3 0 ds.     2    h2  2 h 2 h 3 φ −s Tr + h 2 φ −s E( )+ O h

φ h −s

2



0

  Using (44) and a bootstrap argument again, we see that indeed: ∂τ φ h (x, t) ≤ Ch 3 .

Appendix B. Rotationally invariant surfaces In this section, we will show that a rotationally invariant C 3 surface S without flat parts (perpendicular to its rotational axis), whose closure has no intersection with its own rotational axis satisfies (H). 1. Consider the following parametrization of S: r : (s0 , s1 ) × [0, 2π] −→ R3 , r (s, θ ) = g(s)γ (θ ) + se3 ,

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H. Li, M. Chermisi

where g ∈ C 3 ([s0 , s1 ], R), e3 = (0, 0, 1) and γ (θ ) = (cos θ, sin θ, 0). We may write any W 2,2 deformation V of S in the form: V (s, θ ) = a(s, θ )γ (θ ) + b(s, θ )γ  (θ ) + c(s, θ )e3 , where a, b, c ∈ W 2,2 ((s0 , s1 ) × (0, 2π)) are periodic in θ . By Sobolev embedding a, b, c ∈ C 0,α ([s0 , s1 ] × [0, 2π]), 0 < α < 1. For each s ∈ (s0 , s1 ), we express a, b, c in the form of Fourier series. Take a(s, θ ) for example: +∞

a(s, θ ) =

() * 1 ak,0 (s) cos kθ + ak,1 (s) sin kθ , a0,0 (s) + 2

(45)

k=1

where: 1 ak,0 (s) = π ak,1 (s) =

1 π

2π a(s, θ ) cos kθ dθ, for k = 0, 1, 2, . . . 0

2π a(s, θ ) sin kθ dθ, for k = 1, 2, . . . . 0

For the future purpose, we shall also study the Fourier series of ∂θ a(s, θ ) and of ∂s a(s, θ ). 1 The regularity of a implies that for a.e. s ∈ (s0 , s1 ), the function a(s, ·) ∈ C 1, 2 ([0, 2π]). Integrating by parts, we get: 2π

2π ∂θ a(s, θ ) sin kθ dθ = −k

0

a(s, θ ) cos kθ dθ = −kπak,0 (s), ∀k ≥ 1 0

2π

(46)

2π ∂θ a(s, θ ) cos kθ dθ = k

0

a(s, θ ) sin kθ dθ = kπak,1 (s), ∀k ≥ 0. 0

and hence, for a.e. s ∈ (s0 , s1 ) the Fourier series of ∂θ a(s, θ ): ∂θ a(s, θ ) =

∞ ( * ) kak,1 (s) cos kθ − kak,0 (s) sin kθ ,

(47)

k=1

indeed converges to ∂θ a(s, θ ) in L 2 (0, 2π), as ∂θ a(s, θ ) ∈ L 2 (0, 2π) for a.e. s ∈ (s0 , s1 ). We claim that the Fourier series of ∂s a(s, θ ) converges for each s ∈ (s0 , s1 ): ∞

∂s a(s, θ ) =

() * 1    a0,0 (s) + (s) cos kθ + ak,1 (s) sin kθ . ak,0 2

(48)

k=1

Indeed, ak,0 , ak,1 ∈ W 2,2 (s0 , s1 ) and:  (s) = ak,0

1 π

1  ak,1 (s) = π

123

2π ∂s a(s, θ ) cos kθ dθ, 0

2π 0

 ak,0 (s) =

1 π

2π ∂s ∂s a(s, θ ) cos kθ dθ, ∀k ≥ 0 0

1  ∂s a(s, θ ) sin kθ dθ, ak,1 (s) = π

2π

(49) ∂s ∂s a(s, θ ) sin kθ dθ, ∀k ≥ 1.

0

Incompressible von Kármán shell theory

207

In fact, for a.e. s ∈ (s0 , s1 ) we have: ∞

∂s ∂s a(s, θ ) =

() * 1    ak,0 (s) cos kθ + ak,1 (s) sin kθ , a0,0 (s) + 2

(50)

k=0

The analysis for b(s, θ ), c(s, θ ) is identical. 2. Notice that since V ∈ V , then for a.e. (s, θ ) ∈ (s0 , s1 ) × (0, 2π) we have: ⎧ ⎨ ∂s V · ∂s r = 0, ∂θ V · ∂θ r = 0, ⎩ ∂θ V · ∂s r + ∂s V · ∂θ r = 0, which by direct calculation is equivalent to: ⎧  ⎨ g (s)∂s a(s, θ ) + ∂s c(s, θ ) = 0, ∂θ b(s, θ ) + a(s, θ ) = 0, ⎩  g (s)(∂θ a(s, θ ) − b(s, θ )) + g(s)∂s b(s, θ ) + ∂θ c(s, θ ) = 0.

(51)

Substituting (45), (47) (48) and the related expressions for b(s, θ ) and c(s, θ ) into (51) and equating the coefficients we obtain:   g  (s)a0,0 (s) + c0,0 (s) = 0,

(52a)

a0,0 (s) = 0,

(52b)

 (s) = 0, −g  (s)b0,0 (s) + g(s)b0,0

(52c)

while for all k ≥ 1:   (s) + ck,0 (s) = 0, g  (s)ak,0

(53a)

  g  (s)ak,1 (s) + ck,1 (s) = 0,

(53b)

kbk,1 (s) + ak,0 (s) = 0,

(53c)

−kbk,0 (s) + ak,1 (s) = 0,

(53d)

 g  (s)(−kak,0 (s) − bk,1 (s)) + g(s)bk,1 (s) − kck,0 (s) = 0,

(53e)



g (s)(kak,1 (s) − bk,0 (s)) +

 g(s)bk,0 (s) + kck,1 (s)

= 0,

(53f)

1

with ak,0 , ak,1 , bk,0 , bk,1 , ck,0 , ck,1 ∈ C 1, 2 ([s0 , s1 ]). By (52a) c0,0 ≡ c, and by (52c) 1 ˜ where c˜ is a constant. By (53e), (53f) bk,0 , bk,1 ∈ C 2, 2 and by (53c) and b0,0 = cg, 1 1 (53d) ak,0 , ak,1 ∈ C 2, 2 . By (53a) and (53b) ck,0 , ck,1 ∈ C 2, 2 . In view of g ∈ C 3 ([s0 , s1 ]), repeating such procedure, we obtain:   ak,0 , ak,1 , bk,0 , bk,1 , ck,0 , ck,1 ∈ C 3 [s0 , s1 ] , ∀k ≥ 1. 3. Define: Vn (s, θ ) = an (s, θ )γ (θ ) + bn (s, θ )γ  (θ ) + cn (s, θ )e3 ,

123

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H. Li, M. Chermisi

where: () * 1 a0,0 + ak,0 (s) cos kθ + ak,1 (s) sin kθ , 2 n

an (s, θ ) =

k=1 n

() * 1 bn (s, θ ) = b0,0 + bk,0 (s) cos kθ + bk,1 (s) sin kθ , 2 cn (s, θ ) =

1 c0,0 + 2

k=1 n (

* ) ck,0 (s) cos kθ + bk,1 (s) sin kθ .

k=1

Thanks to the previous analysis Vn ∈ C 3 ([s0 , s1 ] × [0, 2π]) and since an , bn , cn satisfy (51), Vn ∈ V0 is a C 3 infinitesimal isometry on S. We now only need to prove that lim Vn = V in W 2,2 ((s0 , s1 ) × (0, 2π)), which is equivalent to: lim an = a in W 2,2 , n→∞

n→∞

plus the same statements for b and c. This follows from the expansions below and (50), valid for a.e. s ∈ (s0 , s1 ): ∂θ ∂θ a(s, θ ) =

∞ ( )

   * −k 2 ak,0 (s) cos kθ + −k 2 ak,1 (s) sin kθ ,

k=1 ∞ ( )  *  kak,1 (s) cos kθ − kak,0 ∂θ ∂s a(s, θ ) = (s) sin kθ , k=1

in view of the following straightforward result: Lemma 5 Let An (s, θ ) be a sequence of C 3 functions on [s0 , s1 ] × [0, 2π] and A(s, θ ) ∈ L 2 ((s0 , s1 ) × (0, 2π)). For a.e. s ∈ (s0 , s1 ), A(s, ·) ∈ L 2 (0, 2π) thus has its Fourier series in θ , given by: ∞

A(s, θ ) =

() * 1 Ak,0 (s) cos kθ + Ak,1 sin kθ . A0,0 (s) + 2 k=1

If An (s, θ ) is the n-th partial sum of the Fourier series above, then: lim An (s, θ ) = A(s, θ ), in L 2 ((s0 , s1 ) × (0, 2π)) .

n→∞

References 1. Bourquin, F., Ciarlet, P.G., Geymonat, G., Raoult, A.: Gamma-convergence for the asymptotic theory of plates. C. R. Acad. Sci. Paris Ser. I 315, 1017–1024 (1992) 2. Chapelle, D., Mardare, C., Münch, A.: Asymptotic considerations shedding light on incompressible shell models. J Elast 76, 199–246 (2004) 3. Ciarlet PG.: Mathematical Elasticity. North-Holland, Amsterdam (2000) 4. Conti, S., Dolzmann, G.: Derivation of Elastic Theories for Thin Sheets and the Constraint of Incompressibility. Analysis, Modeling and Simulation of Multiscale Problems. pp 225–247. Springer, Berlin (2006) 5. Conti, S., Dolzmann, G.: Derivation of a plate theory for incompressible materials. C.R. Math. Acad. Sci. Paris 344(8), 541–544 (2007) 6. Conti, S., Dolzmann, G.: Gamma-convergence for incompressible elastic plates. Calc. Var. PDE 34, 531–551 (2009) 7. Conti, S., Maggi, F.: Confining thin sheets and folding paper. Arch. Ration. Mech. Anal 187(1), 1–48 (2008)

123

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209

8. Dervaux, J., Ben Amar, M.: Morphogenesis of growing soft tissues. Phys. Rev. Lett 101, 068101– 068104 (2008) 9. Dal Maso G.: An Introduction to Gamma-Convergence. Birkhäuser Boston Inc., Boston (1993) 10. De Giorgi, E., Franzoni, T.: (1975) Su un tipo di convergenza variazionale. (Italian) Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur (8). 58(6), 842–850 11. Friesecke, G., James, R., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure. Appl. Math 55(2), 1461–1506 (2002) 12. Friesecke, G., James, R., Müller, S.: A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Ration. Mech. Anal 180(2), 183–236 (2006) 13. Friesecke, G., James, R., Mora, M.G., Müller, S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris 336(8), 697– 702 (2003) 14. Geymonat, G., Krasucki, F., Marigo, J.J.: Sur la commutativité des passages à la limite en thoréie asymptotique des poutres composites. C. R. Acad. Sci. Paris Ser. II t. 305, 225–228 (1987) 15. Hornung, P., Lewicka M., Pakzad, M.: The matching property for isometries on developable surfaces and elasticity of thin shells. J. Elast. (10 May 2012). doi:10.1007/s10659-012-9391-4. 16. LeDret, H., Raoult, A.: The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl 73, 549–578 (1995) 17. LeDret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci 6, 59–84 (1996) 18. Lewicka, M.: Reduced theories in nonlinear elasticity. Nonlinear Conservation Laws and Applications. IMA vol. 153, pp 393–403, Springer (2011) 19. Lewicka, M.: A note on the convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry. ESAIM. Control Optim. Calc. Var 17, 493–505 (2011) 20. Lewicka, M., Mahadevan, L., Pakzad, M.R.: The Föppl–von Kármán equations for plates with incompatible strains. Proc. R. Soc. A 467, 402–426 (2011) 21. Lewicka, M., Mora, M.G., Pakzad, M.: Shell theories arising as low energy Gamma-limit of 3D nonlinear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci IX 5, 253–295 (2010) 22. Lewicka, M., Mora, M.G., Pakzad, M.: A nonlinear theory for shells with slowly varying thickness. C.R. Acad. Sci. Paris Sér. I 347, 211–216 (2009) 23. Lewicka, M., Mora, M.G., Pakzad, M.: The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells. Arch. Ration. Mech. Anal 200(3), 1023–1050 (2011) 24. Lewicka, M., Müller, S.: The uniform Korn–Poincaré inequality in thin domains. Ann. Inst. Henri Poincaré (C) 28(3), 443–469 (2011) 25. Lewicka, M., Pakzad, M.: The infinite hierarchy of elastic shell models: some recent results and a conjecture. Fields Inst. Commun. (accepted). http://www.math.pitt.edu/~lewicka/publications.html 26. Li, H.: A note on the von Kármán theory for elastic shells with variable thickness. Acta Math. Appl. Sin. (accepted). http://math.umn.edu/~lixxx609/ 27. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge University Press, Cambridge (1927) 28. Mora, M.G., Scardia, L.: Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density. J. Differ. Equ. 252, 35–55 (2012) 29. Müller, S., Pakzad, M.R.: Convergence of equilibria of thin elastic plates: the von Kármán case. Comm. Partial. Differ. Equ 33, 1018–1032 (2008) 30. Trabelsi, K.: Incompressible nonlinearly elastic thin membranes. C. R. Acad. Sci. Paris Ser. I 340, 75–80 (2005) 31. Trabelsi, K.: Modeling of a membrane for nonlinearly elastic incompressible materials via Gammaconvergence. Anal. Appl. (Singap.) 4(1), 31–60 (2006) 32. Walter, W.: Ordinary Differential Equations. Springer, New York (1998)

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