On the Justification of the Nonlinear Inextensional ... - Semantic Scholar
Arch. Rational Mech. Anal. 167 (2003) 179–209 Digital Object Identifier (DOI) 10.1007/s00205-002-0238-1
On the Justification of the Nonlinear Inextensional Plate Model Olivier Pantz Communicated by S. Müller
Abstract We consider a cylindrical three-dimensional body, made of a Saint Venant-Kirchhoff material, and we let its thickness go to zero. For a specific order of magnitude for the applied loads and under appropriate restrictions on the set of admissible deformations, we show that the almost-minimizers of the total energy converge toward deformations that minimize the nonlinear bending energy obtained by Fox, Raoult and Simo using formal asymptotic expansions. Our result is obtained by -convergence arguments.
1. Introduction Plates are three-dimensional bodies that are characterized by a reference placement possessing one dimension, the thickness, which is small in comparison with the other two dimensions. In order to make use of this feature, it is natural to perform an asymptotic analysis by letting the thickness go to zero. In this way, two-dimensional models can be derived from the genuinely three-dimensional model. For a specific order of magnitude for the applied loads, a nonlinear inextensional bending model was obtained by Fox, Raoult & Simo [12] by formal asymptotic-expansion methods. In this model, the equilibrium state is a minimizer of a bending energy, in the sense that the internal energy only depends on the second fundamental form of the midsurface, which is to say, on the curvature. Furthermore, admissible deformations are inextensional. In other words, they are isometries and therefore their first fundamental form is the identity. Moreover, Coutand [8] proved the existence of a solution for this model and obtained similar results with Ciarlet [6, 7] for shells. Nevertheless, no rigorous convergence result of the three-dimensional nonlinear model towards the nonlinear inextensional bending model seem to be known. The purpose of this article is to give a justification of this plate model by using -convergence arguments. This method led to rigorous convergence results
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in the case of body forces of order 1. Le Dret & Raoult [18] and Ben Belgacem [4] thus obtained nonlinear membrane models via this approach. We must also mention the pioneering work of Acerbi, Butazzo & Percivale [1] for strings and the work of Anzellotti, Balbo & Percivale [3]. Frieseke, James & Müller have recently obtained similar results to the one we present here; see [13, 14]. The main difference is that they do not use the same estimate of the deformation in terms of average strains. Indeed, they obtained a better estimate than the one (due to John [16], see Section 2) we use in this paper. This enabled them to avoid the restriction on admissible deformations we made. Moreover, we consider plates made of Saint Venant-Kirchhoff material whereas their analysis applies to more general hyper-elastic bodies. Those three authors and Mora extended those results to the case of elastic shells [15]. As in [12], we consider here the case of body forces of the order of ε2 . Under appropriate hypotheses on the set of admissible deformations, we show that the almost-minimizers of the genuine three-dimensional elasticity converge toward minimizers of the nonlinear inextensional bending energy of Fox, Raoult and Simo. This article is set out as follows. In Section 2, we describe the nonlinear inextensional bending model, the three-dimensional model of Saint Venant-Kirchhoff, and recall some basic results about -convergence theory. We also recall an estimate of the deformations in terms of their elastic strain due to F. John. To be able to make use of this estimate, we restrict the set of admissible deformations. The formulation of our problem thus obtained is presented in Section 2.5. Our main result is stated in Section 2.6. Section 3 is devoted to giving a similar estimate to that of F. John in the case of a plate. We show that the -limit is larger than the nonlinear bending energy in Section 4 and prove the converse inequality in Section 5. The conclusions of this paper were announced in Pantz [21].
2. Preliminaries In what follows, Greek indices take the values 1 or 2, while Latin indices take the values 1, 2 or 3. We adopt the standard summation convention on repeated indices. The space Rn is Endowed with the Euclidean norm |.|. If F is a matrix, we denote by F and |F | respectively its Euclidean and inducted norm, that is F 2 = tr(F T F ) and |F | = sup |F · x|. |x|=1
The letter C will denote a generic constant whose value may change from line to line and we will write x in place of x1 , x2 . 2.1. The nonlinear inextensional bending plate model In this section, we describe the model obtained by Fox, Raoult & Simo [12] using a formal asymptotic expansion. The plate is described as an open set ω of R2 with Lipschitz boundary. It is assumed to be clamped on a nonempty Lipschitz open subset ωφ ⊂ ω and submitted to surface forces fσ . Let VF (ω) be the set of
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isometric deformations of a plate with normal to the surface belonging to class H 1 , satisfying the clamping condition, VF (ω) = ψ ∈ W 1,∞ (ω; R3 ) : Dψ T Dψ = I2 ;
n(ψ) = ∂1 ψ ∧ ∂2 ψ ∈ H 1 (ω); ψ(x) = x on ωφ .
Remark 1. The clamping condition assumed here is not the usual one. Usually, both conditions ψ(x) = x and Dψ · ν = ν are imposed on a subset of the boundary of the plate, where ν is the normal to ∂ω. However, our approach can be easily adapted to this other setting. For all ψ ∈ VF (ω), we denote by n(ψ) the normal to the deformed surface n(ψ) = ∂1 ψ ∧ ∂2 ψ and by bαβ (ψ) its second fundamental form bαβ (ψ) = −∂α ψ · ∂β n(ψ). Remark 2. In [8], Coutand defined the space of admissible deformations as the set of H 2 (ω) isometric deformations. The equality between the two spaces has been pointed out by Friesecke, James & Müller in [13]. First, by a density argument, any two maps ϕ and n in W 1,2 (ω; R3 ) satisfy ∂1 (∂2 ϕ ∧ n) − ∂2 (∂1 ϕ ∧ n) = ∂2 ϕ ∧ ∂1 n − ∂1 ϕ ∧ ∂2 n in the sense of distributions. Secondly, if ϕ is an isometry and n the normal to the surface, then ∂2 ϕ ∧ n = ∂1 ϕ and −∂1 ϕ ∧ n = ∂2 ϕ. Thus ϕ 2 and that ∂ ϕ = −(∂ ϕ · ∂ n)n. belongs to L2 . This implies that ϕ belongs to Hloc ij i j 2 Hence, ϕ ∈ H (ω) as claimed. For simplicity, except in case of ambiguity, we will drop the dependence of n and bαβ on ψ. In the bending model of Fox, Raoult & Simo [12], an equilibrium state is a minimizer on the space VF (ω) of the energy functional JF (ψ) = ω σ (ψ) =
1 αβγ δ bαβ bγ δ 3C
dx − σ (ψ),
fσ (x) · ψ(x) dx, ω
and C
αβγ δ
=
2µλ αβ γ δ δ δ + µ(δ αγ δ βδ + δ αδ δ βγ ), λ + 2µ
where λ and µ are the Lamé moduli of the material, which we assume to be such that µ > 0 and λ 0. Since the total stored energy only depends on the second fundamental form of the surface, the model is said to be of pure bending type. The only interesting case is when VF is not reduced to the identity.
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2.2. A few facts about -convergence A countable sequence Gε of functions from a metric space X into R is said to -converge toward G0 for the topology of X if the following two conditions are satisfied for all x ∈ X: ∀ xε → x, lim inf Gε (xε ) G0 (x), ∃ yε → x, Gε (yε ) → G0 (x). The notion of -convergence presents two major interests, a compactness property and the preservation of minimizers. More precisely, any sequence Gε : X → R admits a -convergent subsequence, and if quasi-minimizers of Gε stay in a compact set of X, then their limit points are minimizers of G0 . In other words, if xε is a sequence of elements in X such that |Gε (xε ) − inf Gε (x)| → 0, x
and if xε stay in a compact set of X, then the limit points of the sequence xε are minimizers for G0 ; see De Giorgi & Franzoni [10], Attouch [2], Dal Maso [9]. 2.3. The Saint Venant-Kirchhoff model A Saint Venant-Kirchhoff material is completely determined by its Lamé moduli µ and λ. The stored-energy function takes the form W (F ) =
λ µ tr((F T F − I )2 ) + (tr(F T F − I ))2 , 4 8
or W (F ) = 21 Cij kl Eij Ekl , with E = 21 (F T F − I ), and Cij kl = λδ ij δ kl + µ(δ ik δ j l + δ il δ j k ). The function W satisfies the following growth and coercivity conditions (see Ciarlet [5]): ∃C > 0, ∀F ∈ M3,3 , |W (F )| C(1 + F 4 ), ∃α > 0, ∃β 0, ∀F ∈ M3,3 , |W (F )| αF 4 − β.
(1)
Let us first remark that a Saint Venant-Kirchhoff material offers no resistance to compression. Such a feature is preserved by passing to the -limit. Let us give an example. Let r : R → R be defined by r(t) = −1 on ]3m, 3m + 1], r(t) = +1 on ]3m + 1, 3m + 3] for all integers m.
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1
−3
−2
−1
1
2
3
−1 Fig. 1. Graph of the function r.
Let ψ n :] − 1, 1[2 ×] − 1/n, 1/n[→ R3 be defined by x1 ψ n (x1 , x2 , x3 ) = r(n · t) dt, x2 , x3 . −1
For all n, we have W (Dψ n ) dx = 0.
(2)
ε
The sequence ψ n weakly converges, in a sense made clear later, towards ψ :] − 1, 1[2 → R3 defined by ψ(x1 , x2 ) = (x1 /2, x2 ). Unfortunately, (2) implies that the -limit energy applied on ψ is zero, whereas ψ is a homogeneous compression. Thus is it necessary to modify the Saint Venant-Kirchhoff model, to get a reasonable inextensional model for plates. There are several ways to achieve this. In the following, we choose to restrict the set of admissible deformations. 2.4. Estimate of the distance between a deformation and the set of rigid-body motions For all n, we define the set A(n) of affine deformations as A(n) = {ϕ : Rn → Rn ; ϕ(x) = g · x + d with g ∈ M n and d ∈ Rn }, endowed with the norm 1/2 |ϕ|A = |g|2 + |d|2 . The subset of n-dimensional rigid motions R(n) is defined as R(n) = {ϕ : Rn → Rn ; ϕ(x) = g · x + d with g ∈ O(n) and d ∈ Rn }.
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Let be a open connected subset of Rn . For all maps ϕ : → Rn , we set Uϕ (x) = (Dϕ(x)T Dϕ(x))1/2 , Eϕ (x) = Uϕ (x) − I. It is well known that if ϕ is regular, ϕ is a rigid motion if and only if Eϕ = 0. In 1972, John [16] studied the stability of this equivalence for “small” Eϕ : If Eϕ is near zero, is ϕ near a rigid motion? John gave a positive answer to this question if is a cube, provided that ϕ belongs to a specific class of functions. For all δ > 0, we define I,δ = {ϕ ∈ C 1 (; R3 ) : Eϕ (x) < δ for all x ∈ }. Theorem 1 (F. John). For all p > 1, there exist δ = δ(p, n) > 0 and C = C(p, n) such that if is a cube of Rn , and ϕ ∈ I,δ , then there exists a matrix g ∈ O(n) such that Dϕ − gLp () C(p, n)Eϕ Lp () Corollary 1. Let =] − 1, 1[3 . There exists C > 0 such that if ϕ ∈ I,δ(2,3) , then there exists a rigid motion γ = g · x + d such that Dϕ − g2L2 () C ϕ − γ 2L6 () C
W (Dϕ) dx, W (Dϕ) dx. 1
Proof. For all ϕ ∈ W 1,4 (; R3 ), let λi be the eigenvalues of Uϕ = (Dϕ T Dϕ) 2 , Eϕ (x)2 = i (λi − 1)2 and Dϕ T Dϕ − I 2 =
(λ2i − 1)2 = (λi − 1)2 (λi + 1)2 . i
i
As each λi is positive, we have Eϕ (x) Dϕ T Dϕ − I for almost all x ∈ . Furthermore, Dϕ T Dϕ − I 2 4/µW (Dϕ(x)) almost everywhere. Thus, by Theorem 1, for all ϕ ∈ I,δ(2,3) , Dϕ − g2L2 () C
Eϕ (x)2 dx C
W (Dϕ) dx.
Let d = ϕ dx. By the Poincaré-Wirtinger inequality and the Sobolev embeddings, we obtain the second estimate.
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2.5. The three-dimensional problems We consider a sequence of three-dimensional plates made of the same Saint Venant-Kirchhoff material and parametrize the sequence by the thickness 2ε. For all ε > 0, the reference placement is defined by ε = ω×] − ε, ε[, where ω, the midsurface of the plate, is a Lipschitz open connected bounded subset of R2 . We assume that the plates are subjected to dead body forces densities fε ∈ L1 (ε ; R3 ) and surface traction densities gε ∈ L1 (Sε ; R3 ) on Sε = ω × {−ε, ε}. φ Furthermore the plates are assumed to be clamped on ε = ∂(ω×] − ε × ε[) ∩ ∂(ωφ ×] − ε × ε[), where ωφ is a nonempty Lipschitz open subset of ω. Let δ be given by Corollary 1. For all open sets of R3 , we define Vε () = {ψ ε ∈ C 1 (ε ; R3 ) : Eψ ε (x) < δ for all x ∈ ε }. In order to be able to use John’s estimates, we restrict the set of admissible deformations by introducing Vεφ = {ψ ∈ Vε (); ψ ε (x) = x on εφ }. We define the energies Jε : L4 (ε ; R3 ) → R by φ ε ε ε ε ε W (Dψ ) dx − ε (ψ ) for ψ ∈ Vε , Jε (ψ ) = φ +∞ for ψ ε ∈ / Vε ,
where ε (ψ ) =
ψ · fε dx +
ε
(3)
ψ ε · gε dσ.
ε
ε
Sε
To look for equilibria, it is natural to try and minimize the energy Jε . Such minimizers do not however always exist. Nevertheless, there always exists a sequence φ ε of almost-minimizers of the energies such that Jε (φ ε )
inf
ψ ε ∈L4 (ε ;R3 )
Jε (ψ ε ) + ε 3 h(ε),
(4)
where h is a positive function such that h(ε) → 0 when ε → 0. Our aim is to study the behavior of φ ε as the thickness goes to zero by using -convergence arguments. For this purpose, it is useful to rescale the problem. We set = 1 , and define a rescaling operator π(ε) : → ε by π(ε)( x , x3 ) = ( x , εx3 ). For each sequence ψ ε : ε → R3 , we set ψ(ε) = ψ ε ◦ π(ε), and for each ϕ : → R3 , J (ε)(ϕ) = Jε (ϕ ◦ π(ε)−1 ). We are interested in the cases of total energy J (ε) and virtual work of the external loads of the order of the cube of the thickness. In effect, Fox, Raoult and Simo obtained their inextensional bending model for such an order of magnitude of the energy. For simplicity, we assume that f (ε) = fε ◦ π(ε) = ε 2 f
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and g(ε) = gε ◦ π(ε) = ε 3 g. The rescaled energies are defined by I (ε) =
J (ε) , ε3
and the rescaled virtual work of a deformation ϕ(ε) is (ε)(ϕ(ε)) (ϕ(ε)) = = f · ϕ(ε) dx + g · ϕ(ε) dσ. ε3 ω×{−1;1} Let I (εn ) be a -convergent subsequence of I (ε) for the strong topology of L4 () and I0 its limit:
I (εn ) − → I0 for the strong topology of L4 (). Our aim is to compute the -limit I0 of I (εn ). The uniqueness of the -limit will make the extraction of the subsequence superfluous a posteriori. In fact the whole sequence I (ε) -converges toward I0 . For notational brevity, we will simply use the notation ε instead of εn . It is easy to show that the almost-minimizers φ ε stay in a compact set of L4 (; R3 ); thus their limit points are minimizers of the -limit I0 of I (ε), which characterize the behavior of thin plates. Remark 3. The energies I (ε) are defined on the space L4 (; R3 ) instead of W 1,4 (; R3 ), which is the natural energy space. This allows us to work with the metric space L4 (; R3 ) endowed with its strong topology. This is a classical trick used in the applications of -convergence. As we will see later, the sequence of quasi-minimizers of the energies is strongly compact in W 1,4 . Consequently, we can in fact work in W 1,4 as well. Remark 4. Some of the results in this paper remain valid for larger sets of admissible deformations. For this purpose, we use in [20] other estimates of the deformations in terms of their elastic strain similar to the one obtained by Kohn in [17]. Remark 5. In [12], the model derived from three-dimensional genuine elasticity by formal asymptotic expansion also takes into account loading terms due to resultant couples. This could be done in our analysis with minor changes. 2.6. Main results We recall that VF (ω) is the set of isometric deformations of a plate with normal to the midsurface belonging to H 1 (ω; S 1 ), clamped on ωφ , and that JF is the nonlinear bending energy (see Section 2.1) of a plate submitted to dead surface forces x) = fσ (
1
−1
f ( x , x3 ) dx3 + g( x , +1) + g( x , −1).
Our main results are as follows.
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Theorem 2. Let φ ε be a sequence of almost-minimizers of the energy (see (4)) and φ(ε) be the associate rescaled deformations. Then, the sequence φ(ε) is compact in W 1,4 (; R3 ). If φ 0 is one of its limit points, then there exists φ ∈ VF (ω) such that φ 0 ( x , x3 ) = φ( x ). Moreover, φ is a minimizer of the nonlinear bending energy JF . To prove this theorem, we compute the -limit of the sequence of functional φ I (ε). Let us introduce the functional JF defined on L4 (; R3 ) by
x , x3 ) = ϕ( x ), JF (ϕ) if there exists ϕ ∈ VF (ω) such that ϕ 0 ( φ JF (ϕ 0 ) = +∞ otherwise. φ
Theorem 3. The sequence I (ε) -converges towards I0 = JF for the strong topology of L4 (). Thus, Theorem 2 is an obvious consequence of the properties of -convergence recalled in Section 2.2 and the following compactness result. Proposition 1. Let ϕ(ε) be a sequence of L4 (; R3 ) such that I (ε)(ϕ(ε)) is bounded. Then, the sequence ϕ(ε) is compact in W 1,4 (). The proof of Theorem 3 will be given in Sections 4 and 5, where we respectively φ φ prove that I0 JF and I0 JF . Moreover, the proof of Proposition 1 is given in Section 4.3.
3. Estimating the deformation of a plate in terms of the energy The purpose of this section is to obtain an estimate similar to the one given in Corollary 1 for square plates Rε =] − 1, 1[2 ×] − ε, ε[. More precisely, we prove Theorem 4. There exists C > 0 such that for all ε, 1/4 > ε > 0, for all functions ϕ ε ∈ Vε (Rε ), there exists a rigid motion γ ε ∈ R(3), such that ϕ ε − γ ε 2L2 (R ) Cε−2 W (Dϕ ε ) dx. ε
Rε
To this end, we begin to prove a similar estimate in the case of beams Bε = ] − 1; 1[×] − ε, ε[2 . 3.1. Case of beams This section is devoted to the proof of following theorem. Theorem 5. There exists C > 0 such that for all 1/4 > ε > 0, for all ϕ ε ∈ Vε (Bε ), there exists a rigid motion γ ε such that ϕ ε − γ ε 2L2 (B ) Cε−2 W (Dϕ ε ) dx. ε
Bε
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Olivier Pantz ε/2
ε xi
xi+1 yi
δ
Qi Fig. 2. Covering of Bε by ∪Qi .
Let ε, 1/4 > ε > 0, and N ∈ N be such that 2 2 − 2 N < − 1. ε ε Let δx =
(5)
2 − 2ε . N
Obviously, we have 6ε < δx ε. 7
(6)
Let us consider the family of points (xi )i=0,... ,N and the family of cubes (Qi )i=0,... ,N defined by xi = ε + δ x i − 1 and Qi =]xi − ε, xi + ε[×] − ε, ε[2 . The beam Bε is included in the union of the cubes Qi . The dependence of the integer N, the families (xi )i=0,... ,N and (Qi )i=0,... ,N on the parameter ε is understood.
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Lemma 1. There exists a constant C such that for all ε > 0 and ϕ ε ∈ Vε (]−ε, ε[3 ), there exists a rigid motion γ ε ∈ R(3), such that ε ε 2 2 ϕ − γ L2 (]−ε,ε[3 ;R3 ) Cε W (Dϕ ε ) dx. ]−ε,ε[3
Proof. Let ε > 0, ϕ ε ∈ Vε (] − ε, ε[3 ). We define ϕ(ε) :] − 1, 1[3 → R3 by ϕ(ε)(x) =
ϕ ε (εx) . ε
By Corollary 1, there exists a rigid motion γ (ε) ∈ R(3) such that ϕ(ε) − γ (ε)2L2 (]−1,1[3 ;R3 ) C W (Dϕ(ε)) dx. ]−1,1[3
Let γ ε ∈ R(3) be the rigid motion defined by γ ε (εx) = εγ (ε)(x). We have ϕ ε − γ ε 2L2 ( ]−ε,ε[3 ;R3 ) = ε 5 ϕ(ε) − γ (ε)2L2 ( ]−1,1[3 ;R3 ) Cε 5 W (Dϕ(ε))L1 ( ]−1,1[3 ) = Cε 2 W (Dϕ ε )L1 ( ]−ε,ε[3 ) , hence Lemma 1 follows. Let us consider ϕ ε ∈ Vε (Bε ). Clearly, ϕ ε |Qi belongs to Vε (Qi ) for all i. Therefore, by Lemma 1, there exists a family of rigid motions γiε , γiε (x) = giε · x + diε such that ϕ ε − γiε 2L2 (Q ;R3 ) Cε2 W (Dϕ ε )L1 (Qi ) for all i = 1, . . . , N.
(7)
i
We first prove that Lemma 2. There exists a constant C such that for all ε, 1/4 > ε > 0, for all functions ϕ ε ∈ Vε (Rε ) and for all integers i, j ∈ {0, . . . , N}, |γiε − γjε |2A Cε−4 W (Dϕ ε )L1 (Bε ) . Proof. In a first step, we estimate the distance in the norm of A between two ε . successive rigid motions γkε and γk+1 For all integers k, k < N , we have by Lemma 1, ε ε − γkε L2 (Qk+1 ∩Qk ) γk+1 − ϕ ε L2 (Qk+1 ) + γkε − ϕ ε L2 (Qk ) γk+1 1/2 1/2 Cε W (Dϕ ε )L1 (Q ) + W (Dϕ ε )L1 (Q ) . k+1
k
(8) Let us consider yk = (see Fig. 2)
xk+1 +xk 2
the middle point of xk and xk+1 . By (5), we have
k ⊂ Qk+1 ∩ Qk Q
(9)
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Olivier Pantz
k =]yk − ε/2, yk + ε/2[×] − ε, ε[2 . Obviously, where Q ε ε ε 2 γk+1 − γkε 2L2 (Q ) γk+1 − γk L2 (Q
k+1 ∩Qk )
k
.
Furthermore, by a simple changing of scale, we have ε 3 2 γk+1 − γkε 2L2 (Q ) = ε (gk+1 − gk )εx + dk+1 − dk + (gk+1 − gk ) · yk L2 (Q) k
= ε 3 (gk+1 − gk )εx2L2 (Q) +ε 3 dk+1 − dk + (gk+1 − gk ) · yk 2L2 (Q) =] − 1/2, 1/2[×] − 1, 1[2 . where Q Moreover, for all matrices g ∈ M3,3 , g · x2L2 (Q) |g|. xL2 (Q) Setting C1 = xL2 (Q) , we obtain ε 5 2 3 2 γk+1 − γkε 2L2 (Q ) ε C1 |gk+1 − gk | + 4ε |dk+1 − dk + (gk+1 − gk ) · yk | . k
Due to the convexity of the map x → |x|2 , for all real α, 0 < α < 1 and for all vectors a and b, 1−α 2 |a + b|2 (1 − α)|a|2 − |b| . α Applying this inequality to a = dk+1 − dk , b = (gk+1 − gk ) · yk and α = 2 −1 , we get 1 + C18ε ε − γkε 2L2 (Q γk+1 ) k
C1 ε 5 |(gk+1 − gk )|2 + α|dk+1 − dk |2 . 2
Finally, by (8), we find that there exists a constant C such that 1/2 1/2 ε |γk+1 − γkε |A Cε−3/2 W (Dϕ ε )L1 (Q ) + W (Dϕ ε )L1 (Q ) . k+1
k
(10)
For all i and j , we now estimate the distance between a rigid motion γiε and any other rigid motion γjε in terms of W (Dϕ ε )L1 (Bε ) . For all elements x ∈ Qk , we have |x − xk | < 2ε, so that x ∈ Qk and x ∈ Qk ⇒ |kδx − k δx | < 4ε ⇒ |k − k |δx < 4ε ε ⇒ |k − k | < 4ε, 2 hence, Card({k : k ∈ {0, ..., N }; x ∈ Qk }) 7.
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In other words, our covering is such that any given point belongs to at most seven cubes. It follows that N
W (Dϕ ε )L1 (Qk ) = W (Dϕ ε ) Card({k : k ∈ {0, ..., N}; x ∈ Qk }) dx Bε
k=0
7W (Dϕ ε )L1 (Bε ) . From the above inequality and (10), we deduce that |γi − γj |A
j −1
|γk+1 − γk |A
k=i N
Cε −3/2
1/2
W (Dϕ ε )L1 (Q
k)
k=0
Cε
−3/2
Cε
−2
N
1/2 W (Dϕ )L1 (Qk ) ε
(N + 1)1/2
k=0 1/2
W (Dϕ ε )L1 (B ) . ε
Theorem 5 easily results from Lemma 2: Proof of Theorem 5. We can choose as an approximation γ ∈ R(3) for ϕ ε any one of the γi defined above, with i ∈ {0, . . . , N}. We first estimate ϕ ε − γi L2 (Bε ) in terms of ϕ ε − γk L2 (Bε ) and |γi − γk |A , which are controlled by Lemmas 1 and 2. ϕ
ε
− γiε 2L2 (B ) ε
N
ϕ ε − γiε 2L2 (Q
k)
k=0
2
N
ϕ ε − γkε 2L2 (Q ) + γkε − γiε 2L2 (Q
k)
k
.
k=0
A straightforward computation yields the inequality γkε − γiε 2L2 (Q ) 4ε3 (2 + ε 2 )|gk − gi |2 + 2|dk − di |2 . k
Therefore, ϕ ε − γiε 2L2 (B ) C
N
ε
ϕ ε − γkε 2L2 (Q ) + ε 3 |γkε − γiε |2A . k
k=0
Applying estimate (7) and Lemma 2, we obtain ϕ ε − γiε 2L2 (B ) C
N
ε 2 W (Dϕ ε )L1 (Qk ) + ε −1 W (Dϕ ε )L1 (Bε )
ε
k=0 −2
Cε
W (Dϕ ε L1 (Bε ) .
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3.2. The case of square plates We now are in a position to give the proof of Theorem 4, which uses Lemma 2 and Theorem 5 proved in the previous section. As in the case of a beam, let us consider ε > 0, ε < 1/4 and N ∈ N such that 2 2 − 2 N < − 1. ε ε We set δx =
2 − 2ε ; N
xi = ε + iδx − 1;
xi,j = (xi , xj )
and Qi,j =]xi − ε, xi + ε[×]xj − ε, xj + ε[×] − ε, ε[. The plate Rε is included in the union of the cubes Qi,j . Let ϕ ε ∈ Vε (Rε ). By ε such that Lemma 1, there exists a family of rigid motions γi,j ε 2 ϕ ε − γi,j L2 (Q
i,j )
Cε2 W (Dϕ ε )L1 (Qi,j ) .
Proof of Theorem 4. We divide the square plate Rε into rows Li = N j =0 Qi,j N and columns Tj = i=0 Qi,j . Each of these rows and columns can be considered as a beam. Since any given point can belong to at most seven columns or rows, it follows that N
W (Dϕ ε )L1 (Ti ) 7W (Dϕ ε )L1 (Rε ) .
(11)
i=0
Therefore, there exists n ∈ {0, . . . , N} such that W (Dϕ ε )L1 (Tn )
7 W (Dϕ ε )L1 (Rε ) CεW (Dϕ ε )L1 (Rε ) . N +1
(12)
We can apply Lemma 2 to the beam Tn . In particular, for each i ∈ {0, . . . , N}, we have ε ε 2 |γi,n − γ0,n |A Cε−4 W (Dϕ ε )L1 (Tn ) , so that by (12), ε ε 2 |γi,n − γ0,n |A Cε−3 W (Dϕ ε )L1 (Rε ) .
(13)
Let us now apply Theorem 5 to each beam Li (as seen in the proof of the theorem, ε ). We get we can choose γ ε = γi,n N
ε 2 ϕ ε − γi,n L2 (L ) Cε−2
N
i
i=0
i=0
W (Dϕ ε )L1 (Li ) Cε−2 W (Dϕ ε )L1 (Rε ) .
On the Justification of the Nonlinear Inextensional Plate Model
193
So that ε 2 L2 (R ε ) Cε−2 W (Dϕ ε )L1 (Rε ) + 2 ϕ ε − γ0,n
N
ε ε 2 γ0,n − γi,n L2 (L ) . (14) i
i=0
In order to estimate the second part of the right-hand side, we observe that ε ε 2 ε ε 2 γ0,n − γi,n L2 (L ) 4ε2 |γ0,n − γi,n |A . i
Combining this last estimate with (13), estimate (14) yields ε 2 L2 (R ε ) Cε−2 W (Dϕ ε L1 (R ε ) , ϕ ε − γ0,n
concluding the proof of Theorem 4. 4. Computation of a bound from below for the -limit 4.1. Deformations with zero-limit energy are rigid motions Let us introduce the set of rigid motions of a plate, Rσ = {γ : R2 → R3 : γ (x) = c · x + b; c ∈ M3,2 ; cT c = I2 ; b ∈ R3 }, which is canonically isomorphic to R = {γ : R3 → R3 : γ = γ ◦ p2 : γ ∈ Rσ }, where p2 is the projection of R3 onto the first two components. Lemma 3. There exists C > 0 such that for all sequences ϕ(ε) → ϕ 0 strongly in L4 (R1 ), there exists γ ∈ R such that 1/2 ε Rε W (Dϕ ) dx 0 ϕ − γ L2 (R1 ) C lim inf . ε→0 ε3 Proof. Without loss of generality, we may assume that ε Rε W (Dϕ ) dx < ∞. lim inf ε→0 ε3 Otherwise, there is nothing to prove. By Theorem 4, there exists γ ε ∈ R(3), such that 1/2 ε ϕ ε − γ ε L2 (Rε ) Rε W (Dϕ ) dx C . ε3 ε 1/2 Let us recall that γ (ε) = γ ε ◦ π(ε) and ϕ(ε) = ϕ ε ◦ π(ε) where π(ε)( x , x3 ) = ( x , εx3 ). Thus, rewriting the above estimate in terms of ϕ(ε) and γ (ε), we obtain 1/2 ε Rε W (Dϕ ) dx ϕ(ε) − γ (ε)L2 (R1 ) C . (15) ε3
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There exists a sequence εn → 0 such that εn ε Rεn W (Dϕ ) dx Rε W (Dϕ ) dx = lim inf . lim n→+∞ ε→0 εn3 ε3 According to the definition of R(3), x , εx3 ) + dεn , γ (εn )(x) = gεn ( where gεn ∈ O(3) and dε ∈ R3 . As O(3) is compact, and dεn stay in a bounded set of R3 , we can extract a subsequence from εn , still denoted εn , such that gεn converges towards an element g of O(3) and dεn towards an element d of R3 . Therefore, γεn → g · ( x , 0) + d strongly in L2 (R1 ). The limit point γ = g · ( x , 0) + d is an element of R . We conclude by passing to the limit when n goes to +∞ in (15), with ε = εn . In the special case where ω is a square, we deduce from the above result that Corollary 2. Without external loads, i.e., L = 0, any deformation ϕ 0 ∈ L4 (R1 ; R3 ) such I0 (ϕ 0 ) = 0 is a rigid motion, i.e., belongs to R . Remark 6. The purpose of this corollary is just to illustrate the meaning of Lemma 3. It may be extended to the more general case of any Lipschitz open set ω. 4.2. Deformations with finite energy are isometries The aim of this section is to prove that any deformation with finite -limit energy is an isometry. Proposition 2. Let ϕ 0 be such that I0 (ϕ 0 ) < ∞. Then – there exists ϕ ∈ W 1,4 (ω) such that ϕ 0 ( x , x3 ) = ϕ( x ); – ϕ is an isometric deformation, that is, Dϕ T Dϕ = I2 . We divide the proof of this proposition into two parts. Proof of the first part of Proposition 2. Let ϕ 0 be such a deformation. There exists a sequence ϕ(ε) → ϕ 0 strongly in L4 (; R3 ) such that lim I (ε)(ϕ(ε)) < +∞. Due to the coercivity condition (1) satisfied by W , (∂1 ϕ(ε)|∂2 ϕ(ε)|ε −1 ∂3 ϕ(ε))4 dx α −1 ε W (Dϕ ε ) dx + β ε −1 ε ε ε W (Dϕ ) dx − ε (ϕ ) + ε −1 ε (ϕ ε ) + β ε
ε I (ε)(ϕ(ε)) + ε 2 (ϕ(ε)) + β ε2 I (ε)(ϕ(ε)) + ε 2 Cϕ(ε)W 1,4 () + β. 2
On the Justification of the Nonlinear Inextensional Plate Model
Thus, as soon as ε < 1, Dϕ(ε)4 dx ε2 I (ε)(ϕ(ε)) + ε 2 Cϕ(ε)W 1,4 () + β.
195
(16)
Let us recall that there exists a constant C, depending only on and φ , such that for all ψ ∈ W 1,4 (; R3 ), ψW 1,4 (;R3 ) C DψL4 () + ψW 1,4 ( φ ;R3 ) . As ϕ(ε)( x , x3 ) = ( x , εx3 ) on φ , the Poincaré inequality above, together with estimate (16), implies that ϕ(ε) is bounded in W 1,4 (; R3 ) so that ϕ 0 is an element of W 1,4 (; R3 ). Moreover, as |∂3 ϕ(ε)| ε(∂1 ϕ(ε)|∂2 ϕ(ε)|ε −1 ∂3 ϕ(ε)), inequality (16) implies that ∂3 ϕ(ε) → 0 strongly in L4 and ∂3 ϕ 0 = 0, which implies that there exists ϕ ∈ W 1,4 (ω) such that ϕ 0 = ϕ. We have already proved that Lemma 4. Let ϕ(ε) be a sequence of rescaled deformations, such that the energies I (ε)(ϕ(ε)) remains bounded. Then ϕ(ε) is bounded in W 1,4 (; R3 ). In order to prove the second part of Proposition 2, we need a technical lemma. As the open set ω is Lipschitz, there exists an linear extension operator Rω from W 1,4 (ω; R3 ) to W 1,4 (R2 ; R3 ). For all δ > 0, we introduce the application Sω (δ) from W 1,4 (ω; R3 ) to W 1,4 (ω×] − 1; 1[2 ; R3 ) defined as follows: for all x ∈] − 1; 1[2 , y ∈ ω and v ∈ W 1,4 (ω; R3 ), we set Sω (δ) · v(y, x) =
Rω (v)(δx + y) − Rω (v)(y) . δ
We claim that Lemma 5. For all v ∈ W 1,4 (ω; R3 ), Sω (δ) · v −−→ Dy v · x strongly in L4 (ω×] − 1; 1[2 ; R3 ). δ→0
Proof. First of all, let us consider a function u ∈ C0∞ (R2 ; R3 ). Clearly, we have u(δx + y) − u(y) − Dy u · x −−→ 0. δ δ→0 Moreover, u(δx+y)−u(y) − Dy u · x is bounded by an L4 function independently δ of δ. Hence, by the dominated-convergence theorem, u(δx + y) − u(y) − Dy u.x −→ 0. (17) 4 2 2 −δ→0 δ L (R ×R )
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Next, let v ∈ W 1,4 (ω; R3 ). For all η > 0, there exists uη ∈ C0∞ (R2 ; R3 ) such that η Rω (v) − uη W 1,4 (R2 ;R3 ) √ . 3 2 We have to prove that for δ small enough, Dy v · x − Sω (δ) · v
L4 (ω×]−1;1[2 ;R3 )
(18)
η.
Setting uη (δx + y) − uη (y) A = Dy uη · x − 4 L δ uη (δx + y) − uη (y) and B = Sω (δ) · v − 4, L δ Dy v · x − Sω (δ) · v
L4 (ω×]−1;1[2 ;R3 )
(19) (20)
Dy v · x − Dy uη · x L4 + A + B. (21)
It follows that we just have to estimate each term of the right-hand side. First of all, by (18), √ η Dy v · x − Dy uη · x 4 2Dy v − Dy uη L4 (ω;M3,3 ) . L (ω×]−1;1[2 ;R3 ) 3 (22) Furthermore, (17) implies that we can chose δ small enough so that uη (δx + y) − uη (y) η A = Dy uη · x − . δ 3 L4 (ω×]−1;1[2 ) Finally, B=
1
0
D(Rω (v) − uη )(tδx + y) · x dt
L4 (ω×]−1;1[2 )
ω×]−1;1[2
1
1/4
|D(Rω (v) − uη )(tδx + y) · x| dt dx dy 4
0
R2 ×]−1;1[2 0
1
1/4 |D(Rω (v) − uη )(tδx + y) · x| dt dx dy 4
with the change of variable(x, y) → (x, z), z = tδx + y, 1/4 1 4 |D(Rω (v) − uη )(z) · x| dt dx dz =
√
R2 ×]−1;1[2 0
R2 ×]−1;1[2
1/4
|D(Rω (v) − uη )(z) · x|4 dx dz
2D(Rω (v) − uη )(z)L4 (R2 ;M3,3 )
This estimate completes the proof.
η . 3
(23)
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We are now in a position to continue the proof of Proposition 2. Proof of the second part of Proposition 2. For all ρ > 0, we introduce √ ωρ = {x ∈ ω : dist(x, R2 − ω) > 2ρ}.
(24)
Let ϕ 0 such that I0 (ϕ 0 ) < ∞. There exists M < +∞ and ϕ(ε) → ϕ 0 strongly in L4 () such that I (ε)(ϕ(ε)) < M. Thus, by Lemma 4, the sequence ϕ(ε) is bounded in W 1,4 () and ϕ(ε) ϕ 0 weakly in W 1,4 (; R3 ). Hence, by the Rellich-Kondrachov theorem, ϕ(ε) → ϕ 0
in C(; R3 ).
(25)
Let ρ > 0 and δ < ρ. We define ϕ ε,δ : ωρ ×] − 1, 1[2 ×] − ε/δ, ε/δ[ → R3 ϕ ε (δ x˜ + y, δx3 ) − ϕ ε (y, 0) . (y, x, ˜ x3 ) → δ The convergence (25) of ϕ(ε) in C(; R3 ) and the definition of Sω (δ) imply that for all y ∈ ωρ , (ϕ ε,δ (y)) ◦ π(ε) → (Sω (δ) · ϕ 0 (y)) ◦ p2 as ε → 0 in C(R1 ; R3 ), hence strongly in L4 (R1 ; R3 ). Lemma 3 applied to the sequence ϕ ε,δ (y) ◦ π(ε) implies that there exists γδ (y) ∈ R , such that 1/2 ε,δ Rε/δ W (Dϕ (y)) dx . Sω (δ) · ϕ(y)) − γδ (y)L2 (R1 ) C lim inf ε→0 (ε/δ)3 After an integration of the square of this inequality, we obtain, using Fatou’s lemma, |Sω (δ) · ϕ(y)(x) − γδ (y) · x|2 dx dy ωρ
x∈]−1,1[2
C lim inf y∈ωρ
Rε/δ
W (Dϕ ε,δ (y)) 3 δ dx dy. ε3
Let us now look for a bound for the right-hand side of the above inequality: W (Dϕ ε,δ ) 3 W (Dϕ ε (δx + (y, 0))) 3 δ dx dy = δ dx dy ε3 ε3 ωρ ×Rε/δ
ωρ ×Rε/δ
= ωρ ×]−1,1[2
W (Dϕ ε (δ x˜ + y, δx3 )) 3 δ dx3 dx˜ dy. ε3 x3 =−ε/δ ε/δ
Let us perform a simple change of variable using the map ωρ ×] − 1; 1[2 ×] − 1, 1[ → R 2 (y; δ)×] − 1, 1[2 ×] − 1, 1[ y∈ωρ
(y, x , x3 ) → (z, x , x3 ) = (δ x + y, x , x3 ).
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As
Olivier Pantz
R 2 (y; δ) ⊂ ω, we get
y∈ωρ
y∈ωρ
Rε/δ
W (Dϕ ε,δ ) 3 δ dx dy ε3 ε/δ
W (Dϕ ε (z, δx3 )) 3 δ dx3 dx˜ dz 2 x =−ε/δ ε3 z∈ω x∈]−1,1[ ˜ 3 ε W (Dϕ ε (z, x3 )) 2 δ dx3 dz = 4δ 2 I (ε)(ϕ(ε)). =4 ε3 z∈ω x3 =−ε
Putting all these estimates together, we have thus shown that |Sω (δ) · ϕ(y)(x) − γδ (y).x|2 dx dy ωρ
x∈]−1,1[2
Cδ 2 lim inf I (ε)(ϕ(ε)) Cδ 2 M. By Lemma 5, Sω (δ) · ϕ → Dy ϕ · x strongly in L2 (ωρ ×] − 1, 1[2 ) when δ goes to zero. Thus, γδ converges strongly in L2 (ωρ ×] − 1, 1[2 ; R3 ) toward Dy ϕ.x. Let cδ (y) ∈ L2 (ωρ ; M3,2 ) and bδ ∈ L2 (ωρ ; R3 ) such that for almost all y, γδ (y).x = cδ (y) · x + bδ (y). We have just proved that up to a subsequence cδ · x + bδ → Dy ϕ · x strongly in L2 (ωρ ; R3 ) for almost all x ∈] − 1, 1[2 . Since the dependence in x is affine, it follows that this convergence holds true for all x ∈] − 1, 1[2 . Therefore, bδ → 0 strongly in L2 (ωρ ; R3 ) and cδ → Dy ϕ strongly in L2 (ωρ ; M3,2 ). In particular, cδT cδ → Dy ϕ T Dy ϕ strongly in L1 (ωρ ; M2,2 ). Since for any δ, cδ ∈ R , we have cδT cδ = I2 . Therefore, for almost all y ∈ ωρ , Dy ϕ T Dy ϕ = I2 , i.e., ϕ is an isometry. To conclude the proof, it suffices to note that ω= ωρ . ρ
On the Justification of the Nonlinear Inextensional Plate Model
199
4.3. Compactness We are now able to prove the compactness result stated in Proposition 1. We have already proved the compactness of bounded-energy sequences in L4 (). Hence, it only remains to prove the following proposition. Proposition 3. Let ϕ(ε) → ϕ 0 strongly in L4 (), such that there exists M < +∞, with I (ε)(ϕ(ε)) < M for all ε. Then, ϕ(ε) converges toward ϕ 0 strongly in W 1,4 (; R3 ). Proof. Let ϕ(ε) be such a sequence. We already know that ϕ(ε) ϕ 0 weakly in W 1,4 (; R3 ). For all (f1 |f2 |f3 ) ∈ M3,3 , (|f1 |2 − 1)2 + (|f2 |2 − 1)2 Thus,
4 W ((f1 |f2 |f3 )). µ
(|∂1 ϕ(ε)|2 − 1)2 + (|∂2 ϕ(ε)|2 − 1)2 dx 4 W ((∂1 ϕ(ε)|∂2 ϕ(ε)|ε −1 ∂3 ϕ(ε))) dx µ 4 = ε −1 W (Dϕ ε ) dx µ ε 4 2 = ε (I (ε)(ϕ(ε) − (ϕ(ε))) µ 4 ε 2 (M + Cϕ(ε)W 1,4 (;R3 ) ). µ
As ϕ(ε)W 1,4 (;R3 ) is bounded, it follows that |∂1 ϕ(ε)|2 → 1 strongly in L2 (), |∂2 ϕ(ε)|2 → 1 strongly in L2 (). Let us recall that ∂3 ϕ(ε) → 0 strongly in L4 . Thus, |Dϕ(ε)|2 dx → 2||.
On the other hand, by Proposition 2, 0 2 |Dϕ | dx = 2 dx = 2||.
We have thus shown that ϕ(ε)W 1,4 (;R3 ) → ϕ 0 W 1,4 (;R3 ) . Since the unit ball of W 1,4 is uniformly convex, weak convergence together with the convergence of the norms implies strong convergence.
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Olivier Pantz
We define the operator D ε = (∂1ε |∂2ε |∂3ε ) on D () by D ε ψ = D(ψ ◦ π(ε)−1 ) ◦ π(ε) for all ψ, that is, for all u ∈ D(), < ∂iε ψ, u >= − < ψ, ∂i (u ◦ π(ε)) ◦ π(ε)−1 > . In particularly, D ε ϕ(ε) = Dϕ ε ◦ π(ε). Proposition 4. Let ϕ(ε) → ϕ 0 strongly in L4 (; R3 ) such that there exists a constant M < ∞, I (ε)(ϕ(ε)) < M. Then ∂αε ϕ(ε) → ∂α ϕ 0 strongly in L4 (; R3 ), ∂3ε ϕ(ε) → n strongly in L2 (; R3 ), where n = ∂1 ϕ 0 ∧ ∂2 ϕ 0 . Proof. Let us remark that ∂αε ϕ(ε) = ∂α ϕ(ε). Thus, the first part of the proposition is just a part of Corollary 3. Let U (ε) be the positive-definite symmetric matrix such that U (ε)2 = ε D ϕ(ε)T D ε ϕ(ε). For almost all x ∈ , there exists a single matrix C(ε)(x) ∈ O(3), such that D ε ϕ(ε) = C(ε) · U (ε). φ
According to the definition of Vε , det(Dx ϕ(ε)) > 0 for almost all x ∈ . We deduce that det(Dxε ϕ(ε)) > 0 and det(C(ε)(x)) > 0 almost everywhere, thus C(ε)(x) ∈ SO(3). Let us remark that, D ε ϕ(ε) − C(ε)2 = U (ε) − I 2 CW (D ε ϕ(ε)). Thus,
D ε ϕ(ε) − C(ε)2L2 (;M
3,3 )
C
W (D ε ϕ(ε)) dx
C = ε
W (Dϕε ) dx −−→ 0. ε
(26)
ε→0
As ∂αε ϕ(ε) → ∂α ϕ 0 strongly in L4 (; R3 ) hence in L2 (; R3 ), we have Cα (ε) → ∂α ϕ 0 strongly in L2 (; R3 ). Without loss of generality, we may assume that Cα (ε) converges almost everywhere. As C(ε)(x) ∈ SO(3), C3 (ε)(x) = C1 (ε)(x) ∧ C2 (ε)(x). Thus, C3 (ε)(x) converges toward ∂1 ϕ 0 (x)∧∂2 ϕ 0 (x) for almost all x ∈ . As |C3 (ε)| is bounded by 1, by the dominated-convergence theorem, C3 (ε) → ∂1 ϕ 0 ∧ ∂2 ϕ 0 = n strongly in L2 (; R3 ). As the limit is independent of the chosen subsequence, the whole sequence converges. The conclusion follows from (26).
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201
4.4. The -limit is bounded from below by the bending energy We are now in a position to prove that φ
I0 JF ,
(27)
which will follow directly from Propositions 5 and 6 proved in this section. Let ϕ(ε) ∈ W 1,4 (; R3 ). The rescaled Green-Lagrange tensor of ϕ ε is given by E(ϕ(ε)) = 21 ((D ε ϕ(ε))T D ε ϕ(ε) − I3 ). For simplicity, we will denote E(ϕ(ε)) by E(ε). In the following, we will need a technical lemma Lemma 6. Let U be an open set of Rn−1 ; Let ψ ∈ Lp (U ×] − 1, 1[) such that ∂nn ψ = 0. Then there exists a ∈ Lp (U ), b ∈ Lp (U ) such that ψ = a + x3 b. Furthermore, ∂n ψ ∈ Lp (U ×] − 1, 1[) and for almost all ( x , xn ) ∈ U ×] − 1, 1[, ∂n ψ( x , xn ) = b( x ). We omit the proof of this lemma which does not raise any particular difficulty. Lemma 7. Let ϕ(ε) → ϕ 0 strongly in L4 (; R3 ), such that I (ε)(ϕ(ε)) is bound1 (ω). Moreover, there exists a subsequence still ed. Then ϕ 0 ∈ W 1,∞ , and n ∈ Hloc noted ε, and there exists wαβ ∈ L2 (ω; R3 ) such that Eαβ (ε) x3 wαβ ( x ) + (∂α n · ∂β ϕ 0 + ∂β n · ∂α ϕ 0 ) weakly in L2 (; R3 ). ε 2 Proof. We have 1 I (ε)(ϕ(ε)) = 2 2ε
Cij kl Eij (ε)Ekl (ε) dx − (ϕ(ε)). E (ε)
The coercivity of the elasticity tensor implies that for all i and j , ijε is bounded 2 in L2 (). Hence, we may assume that E(ε) ε weakly converges in L (; M3,3 ). √ Let ρ > 0 and ε > 0 such that 2ε < ρ. Let us introduce Rε,i,j =]ε(i − 1), ε(i + 1)[×]ε(j − 1), ε(j + 1)[ and S = {(i, j ) ∈ N × N : Rε,i,j ⊂ ω}. We have ωρ ⊂ Rε,s , s∈S
where ωρ is defined by (24). Figure 3 represents in bold the elements Rε,s such that s ∈ S. The element Rε,i,j is shown in gray.
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Olivier Pantz j
ε
i ω
ωρ
Fig. 3.
By Theorem 1, there exists a function gε ∈ L2 ( each square Rε,s ,
s∈S
Rε,s ; O(3)) such that on
Dϕε − gε L2 (Rε,s ×]−ε,ε[;M3,3 ) C
Rε,s ×]−ε,ε[
W (Dϕε ).
After the standard change of variable, we obtain εD
ε
ϕ(ε) − g(ε)2L2 (R ×]−1,1[;M ) ε,s 3,3
C
Rε,s ×]−ε,ε[
W (Dϕε ).
By summation on those estimates, we have εD ε ϕ(ε) − g(ε)2L2 (ω
ρ ×]−1,1[;M3,3 )
C
W (Dϕε ) ω×]−ε,ε[
and
D ε ϕ(ε) − g(ε)L2 (ωρ ×]−1,1[;M3,3 ) ε
2 C(I (ϕ(ε)) + ϕ(ε)W 1,4 (;R3 ) ). (28)
On the Justification of the Nonlinear Inextensional Plate Model D ε ϕ(ε)−g(ε) is bounded in L2 , there exists a subsequence, still ε ε D ϕ(ε)−g(ε) , that weakly converges in L2 (ωρ ×] − 1, 1[; M3,3 ). Let be ε
As
limit. We have
∂3
∂αε ϕ(ε) − gα (ε) ε
203
denoted its weak
∂3 ∂αε ϕ(ε) ε = ∂α ∂3ε ϕ(ε). =
Furthermore, ∂3ε ϕ(ε) → n strongly in L2 (ωρ ×] − 1, 1[; R3 ). Thus, ∂3 α = ∂α n. As ∂3 n = 0, ∂33 α = 0. Thus, by Lemma 6, ∂α n ∈ L2 (ωρ ) and there exists vα ∈ L2 (ωρ ) such that = vα + x3 ∂3 α = vα + x3 ∂α n. We have proved that ∂αε ϕ(ε) − gα (ε) x ) + x3 ∂α n weakly in L2 (ωρ ×] − 1, 1[; R3 ). vα ( ε
(29)
Let us rewrite E(ε) as 1 ε (D ϕ(ε)T D ε ϕ(ε) − I ) 2 (D ε ϕ(ε) − g(ε))T D ε ϕ(ε) − g(ε) = g(ε) + D ε ϕ(ε)T . 2 2
E(ε) =
By (29) we deduce that Eαβ (ε) x3 vα · ∂β ϕ 0 + vβ · ∂α ϕ 0 + (∂α n · ∂β ϕ 0 + ∂β n · ∂α ϕ 0 ) ε 2 in D (ωρ ×] − 1, 1[). Indeed, gα → ∂α ϕ 0 strongly in L2 (ω) and g3 → n strongly in L2 (ωρ ). Moreover, ∂3
Eαβ (ε) ε
1 (∂α n · ∂β ϕ 0 + ∂β n · ∂α ϕ 0 ) in D (ωρ ×] − 1, 1[). 2
(30)
Since the limit in (30) is independent of δ, the convergence holds true in D (). Lemma 6 completes the proof. Lemma 8. For all symmetric matrices F ∈ M3,3 , C
αβγ δ
Fαβ Fγ δ Cij kl Fij Fkl .
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Olivier Pantz
Proof. We have 2 2 2 Cij kl Fij Fkl = Cαβγ δ Fαβ Fγ δ + (λ + 2µ)F33 + 2λF33 (F11 + F22 ) + 4µ(F13 + F23 ).
The function t → (λ + 2µ)t 2 + 2λt (F11 + F22 ) reaches its minimum for t = 11 +F22 ) − λ(Fλ+2µ . Thus we have λ2 (F11 + F22 )2 λ + 2µ λ2 = Cαβγ δ Fαβ Fγ δ − δ αβ δ γ δ Fαβ Fαβ λ + 2µ λ2 = Cαβγ δ − δ αβ δ γ δ Fαβ Fαβ λ + 2µ
Cij kl Fij Fkl Cαβγ δ Fαβ Fγ δ −
=C
αβγ δ
Fαβ Fγ δ .
Proposition 5. If ϕ 0 ∈ L4 (; R3 ) such that I0 (ϕ 0 ) < ∞, then there exists ϕ ∈ H 1 (ω), n(ϕ) ∈ H 1 (ω), ϕ 0 ( x , x3 ) = ϕ( x ) and 1 αβγ δ I0 (ϕ 0 ) C (∂α n.∂β ϕ)(∂γ n.∂δ ϕ) − σ (ϕ). 3 ω Proof. Let ϕ 0 ∈ L4 (; R3 ) be such that I0 (ϕ 0 ) < +∞. By Proposition 2, ϕ 0 ( x , x3 ) = ϕ( x ). There exists ϕ(ε) → ϕ 0 strongly in L4 (; R3 ) such that lim I (ε)(ϕ(ε)) = I0 (ϕ 0 ). As
I (ε)(ϕ(ε)) =
1 ij kl Eij Ekl C dx − (ϕ(ε)), 2 ε ε
by Lemma 8, I (ε)(ϕ(ε))
1 αβγ δ Eαβ Eγ δ C dx − (ϕ(ε)). 2 ε ε
Passing to the limit, when ε goes to zero, since the right-hand side is sequentially weakly lower semicontinuous for the topology L2 () with respect to Eαβ (ε), we have by Lemma 7,
x3 1 αβγ δ C wαβ + (∂α n · ∂β ϕ + ∂β n · ∂α ϕ) · · · 2 ω −1 2 x3 · · · wγ δ + (∂γ n · ∂δ ϕ + ∂δ n · ∂γ ϕ) dx − (ϕ 0 ) 2 1 1 αβγ δ 2 C x3 (∂α n · ∂β ϕ)(∂γ n · ∂δ ϕ) dx3 dx1 dx2 − (ϕ 0 ) ω −1 2 1 αβγ δ = C (∂α n · ∂β ϕ)(∂γ n · ∂δ ϕ) dx − σ (ϕ). (31) 3 ω
I0 (ϕ ) 0
1
On the Justification of the Nonlinear Inextensional Plate Model
205
1 (ω). It remains to prove that n ∈ H 1 (ω).At this stage, we just know that n ∈ Hloc 2 Since ∂α ϕ ∈ L (ω), it is easy to show that the derivative of the following product makes sense: 0 = ∂α (n · ∂β ϕ) = n · ∂αβ ϕ + ∂α n · ∂β ϕ.
Thus, ∂α n · ∂β ϕ = ∂β n · ∂α ϕ.
(32)
For all symmetric matrices F ∈ M2,2 , we have C
αβδγ
Fαβ Fδγ 2µF .
(33)
Combining (31), (32) and (33), we conclude that ∂α n · ∂β ϕ ∈ L2 (ω). Finally, for almost all x, |∂α n|2 = δ βγ (∂α n · ∂β ϕ)(∂α n · ∂γ ϕ). As the left-hand side belongs to L1 (ω), it follows that n ∈ H 1 (ω). To complete the proof of (27), we just have to prove that any deformation with finite -limit energy satisfies the clamping condition. Proposition 6. Let ϕ 0 ∈ L4 () such that I0 (ϕ 0 ) < ∞. Then there exists ϕ ∈ W 1,∞ (ω; R3 ) such that ϕ 0 ( x , x3 ) = ϕ( x ), and ϕ( x ) = ( x , 0)
on ωφ .
Proof. There exists ϕ(ε) → ϕ 0 strongly in W 1,4 (; R3 ) such that I0 (ϕ 0 ) = lim I (ε)(ϕ(ε)). As, for all ε, ϕ(ε)( x , 1) = ( x , ε)
on ωφ × {1}.
Thus, passing to the limit, as ϕ(ε) converges in C(; R3 ), ϕ 0 ( x , 1) = ( x , 0)
on ωφ × {1}.
We already know that there exists ϕ ∈ W 1,∞ (ω) such that ϕ 0 ( x , x3 ) = ϕ( x ) for almost all x. Thus, ϕ( x ) = ( x , 0) on ωφ . 5. Computation of the -limit To prove Theorem 3, it remains to establish the converse inequality of (27) for regular deformations, that is, φ
I0 (ϕ 0 ) JF (ϕ). x , x3 ) = ϕ( x ). In Proof of Theorem 3. Let ϕ ∈ VF and ϕ 0 ∈ H 2 () such that ϕ 0 ( the following, we will construct a family of sequences ϕψ (ε) of rescaled deformations, depending on a parameter ψ ∈ C0∞ (ω − ωφ ; R3 ) such that ϕψ (ε) converges
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toward ϕ 0 strongly in H 2 () as ε goes to zero and inf lim I (ε)(ϕ(ε)) JF (ϕ).
(34)
ψ ε→0
To this end, we proceed in two steps. In a first step, we approximate the deformation ϕ and the normal to the midsurface n = ∂1 ϕ ∧ ∂2 ϕ by regular maps. Then, using this approximation, we build sequences ϕψ (ε) satisfying (34). The first step is based on a truncation result for Sobolev maps [11, 19, 22]. This argument has been adapted by Frieseke, James & Müller (see [13]) to preserve the clamping conditions. They show (Proposition A.3) that for all λ > 0, there exist ϕλ ∈ W 2,∞ (ω; R3 ) and nλ ∈ C 1 (ω; R3 ), ϕλ = (ϕ − id)λ + id; nλ = (n − e3 )λ + e3 such that ϕλ and nλ fulfill the clamping condition, that is, ϕλ = id
and
nλ = e3
on ωφ .
Furthermore, the maps ϕλ and nλ agree with ϕ and n respectively on a set ω − ωλ , where ωλ is given by ωλ = {x ∈ ω : ϕλ (x) = ϕ(x) or nλ (x) = n(x)}, and such that |ωλ | CRλ /λ2 . In addition, Rλ =
{x∈ω:|ϕ−id |+|Dϕ−id |+|D 2 ϕ|λ/2}
(35)
|ϕ − id |2 + |Dϕ − id |2 + |D 2 ϕ|2 dx
··· +
{x∈ω:|n−e3 |+|Dn|λ/2}
|n − e3 |2 + |Dn|2 dx −−−−→ 0. λ→+∞
Moreover, the derivatives of ϕλ and nλ are controlled by λ: Dnλ L∞ (ω) ,
D 2 ϕλ L∞ (ω) Cλ.
(36)
From (35) and (36), it is easy to deduce that nλ converges toward n strongly in H 1 (ω) and ϕλ toward ϕ strongly in H 2 (ω) as λ goes to infinity. For sufficiently large λ, we have 1/2
f λ (x) = dist((Dϕλ , nλ ); SO(3)) CRλ
(37)
for all x ∈ ω. Indeed, f λ is Lipschitz with Lipschitz constant lower or equal to Cλ. As ω is Lipschitz, there exists a constant A such that for r small enough, |B(x0 , r) ∩ ω| Ar 2 for all x0 ∈ ω. Thus, |B(x0 , r) ∩ (ω − ωλ )| Ar 2 − |B(x0 , r)| Ar 2 − |ωλ | Ar 2 − CRλ /λ2 .
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λ Choosing r 2 = 2CR , we deduce that B(x0 , r) ∩ (ω − ωλ ) is not empty, and from Aλ2 the Lipschitz property of fλ that for all x0 ∈ ω, λ f (x0 ) Crλ = CRλ 1/2 .
Let ψ ∈ C0∞ (ω − ωφ ; R3 ). For all ε, we define ϕψε by x , x3 ) = ϕλε ( x ) + x3 nλε ( x ) + x32 ψ( x )/2. ϕψε ( The deformation ϕψε satisfies the clamping condition. Moreover, for ε small enough, Eϕψε δ,
(38) φ
where δ is given by Corollary 1. Hence, ϕψε belongs to Vε . To show this, we note that Dϕψε = (Dϕλε , nλε ) + x3 (Dnϕε , ψ) + x32 /2(Dψ, 0) and |Dϕψε − (Dϕλε ; nλε )| ε(Dnλε L∞ + ψL∞ ) + ε 2 DψL∞ Cελε + (ε + ε 2 )ψC 1 . From (37), choosing λε = ε−1 , we obtain (38). Consider ϕψ (ε) = ϕψε ◦π(ε). The sequence ϕψ (ε) converges toward ϕ 0 strongφ
ly in H 2 (ω). Moreover, as ϕψε belongs to Vε , −3 W (Dϕψε ) dx − (ϕψ (ε)). I (ε)(ϕψ (ε)) = ε ω×]−ε,ε[
Let us now compute the limit of the right-hand side as ε goes to zero. First of all, (ϕψ (ε)) → (ϕ 0 ) as ε → 0. To compute the limit of the internal energy, we split the integral on ω into two parts. We first show that the contribution due to the bad set ωλε vanishes as ε goes to zero: −3 ε W (Dϕψε ) dx Cε−2 |ωλε |Dϕψε S ϕε ×]−ε,ε[
Cε−2 |ωλε | C(λε ε)−2 Rλε = CRλε −−→ 0. ε→0
On the good set ω − ωλε , ∂α ϕψε .∂β ϕψε − δαβ = ∂α ϕ.∂β n + (∂β ϕ.∂α n)x3 + o(x3 ) = −2bαβ x3 + o(x3 ). After integration with respect to x3 , we obtain 1 ε −3 ε W (Dϕψ ) dx = Cαβγ δ bαβ − 2Cαβ33 bαβ ψ · n · · · 3 ω−ωλε ω×]−ε,ε[−ωλε + 2Cα3β3 (ψ · ∂α ϕ)(ψ · ∂β ϕ) + C3333 (ψ · n)2 d x + o(1).
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Letting ε go to zero, as |ωλε | → 0, as ϕλε and nλε converge toward ϕ and n strongly in H 2 (ω) and H 1 (ω) respectively, we have W (Dϕψε ) dx lim I (ε)(ϕψ (ε)) = ε −3 1 = 3
ω×]−ε,ε[−ωλε
Cαβγ δ bαβ − 2Cαβ33 bαβ ψ · n ω−ωλε α3β3
+ 2C
(ψ · ∂α ϕ)(ψ · ∂β ϕ) + C3333 (ψ · n)2 d x − (ϕ 0 ).
It remains to compute the infimum of the limit energy over the set of functions ψ ∈ C0∞ (ω − ωφ ; R3 ). For all α and β, ∂αβ ϕ = 0 on ωφ . As ωφ is a Lipschitz open set, there exists a sequence ψn ∈ C0∞ (ω − ωφ ; R3 ) which converges toλ ward λ+2µ δ αβ ∂αβ ϕ strongly in L2 (ω; R3 ). A straightforward computation yields the expected inequality, that is, φ
I0 (ϕ 0 ) lim lim I (ε)(ϕψ (ε)) = JF (ϕ). n→∞ ε
Remark 7. The truncation argument used in the first part of the proof is due to Friesecke, James & Müller (see [13]). This argument would be unnecessary if we could prove the density of VF ∩ C 2 in VF . Unfortunately, it is not clear whether such a density result holds true or not. Acknowledgements. Olivier Pantz is grateful to Tudor S. Ratiu for giving him the opportunity of working one year at the Swiss Federal Institute of Technology (EPFL), where this paper was written. He also wishes to thank Hervé Le Dret for reading the manuscript and for making many corrections and suggestions, and S. Müller for pointing out to him the truncation argument.
References 1. Acerbi, E., Buttazzo, G., Percivale, D.: A variational definition of the strain energy for an elastic string. J. Elasticity 25(2), 137–148 (1991) 2. Attouch, H.: Variational convergence for functions and operators. Pitman (Advanced Publishing Program), Boston, Mass., 1984 3. Anzellotti, G., Baldo, S., Percivale, D.: Dimension reduction in variational problems, asymptotic development in -convergence and thin structures in elasticity. Asymptotic Anal. 9, 61–100 (1994) 4. Ben Belgacem, H.: Une méthode de -convergence pour un modèle de membrane non linéaire. C. R. Acad. Sci. Paris Sér. I Math. 324, 845–849 (1997) 5. Ciarlet, P.G.: Élasticité tridimensionnelle. Masson, Paris, 1986 6. Ciarlet, P.G., Coutand, D.: An existence theorem for nonlinearly elastic “flexural” shells. J. Elasticity 50, 261–277 (1998) 7. Ciarlet, P.G., Coutand, D.: Un théorème d’existence pour une coque non linéairement élastique “en flexion”. C. R. Acad. Sci. Paris Sér. I Math. 326, 903–907 (1998) 8. Coutand, D.: Existence d’un minimiseur pour le modèle “proprement invariant” de plaque “en flexion” non linéairement élastique. C. R. Acad. Sci. Paris Sér. I Math. 324, 245–248 (1997) 9. Dal Maso, G.: An introduction to -convergence. Birkhäuser Boston Inc., Boston, MA, 1993
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10. De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58(8), 842–850 (1975) 11. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Studies Advanced Mathematics, CRC Press: Boca Raton, 1992 12. Fox, D.D., Raoult, A., Simo, J.C.: A justification of nonlinear properly invariant plate theories. Arch. Rational Mech. Anal. 124, 157–199 (1993) 13. Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Max-Planck-Institut, Leipzig. Preprint 5/2002 to appear in Comm. Pure Appl. Math. 14. Friescke, G., James, R.D., Müller, S.: Rigorous derivation of nonlinear plate theory and geometric rigidity. C. R. Math. Acad. Sci. Paris 334, 173–178 (2002) 15. Friesecke, G., James, R.D., Mora, M.G., Müller, S.: Derivation of nonlinear bending theory for shells from three dimensional nonlinear elasticity by Gamma-convergence Max-Planck-Institut, Leipzig. Preprint 10/2002 to appear in CRAS. 16. John, F.: Bounds for deformations in terms of average strains. In Inequalities III, pages 129–144, Academic Press, New York, 1972 17. Kohn, R.V.: New integral estimates for deformations in terms of their nonlinear strains. Arch. Rational Mech. Anal. 78, 131–172 (1982) 18. Le Dret, H., Raoult, A.: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74(9), 549–578 (1995) 19. Liu, F.-C.: A Lusin property of Sobolev functions. Indiana U. Math. J. 26, 645–651 (1977) 20. Pantz, O.: Quelques problèmes de modélisation en élasticité non linéaire. Thèse Université Pierre et Marie Curie, 2001 21. Pantz, O.: Une justification partielle du modèle de plaque en flexion par -convergence. C. R. Acad. Sci. Paris Série I 332, 587–592 (2001) 22. Ziemer, W.: Weakly Differentiable Functions. Springer Verlag, New York, 1989 Centre de Mathématiques Appliquées Ecole Polytechnique 91128 PALAISEAU Cedex France (Accepted September 18, 2002) Published online January 15, 2003 – © Springer-Verlag (2003)
On the Justification of the Nonlinear Inextensional Plate Model Olivier Pantz Communicated by S. Müller
Abstract We consider a cylindrical three-dimensional body, made of a Saint Venant-Kirchhoff material, and we let its thickness go to zero. For a specific order of magnitude for the applied loads and under appropriate restrictions on the set of admissible deformations, we show that the almost-minimizers of the total energy converge toward deformations that minimize the nonlinear bending energy obtained by Fox, Raoult and Simo using formal asymptotic expansions. Our result is obtained by -convergence arguments.
1. Introduction Plates are three-dimensional bodies that are characterized by a reference placement possessing one dimension, the thickness, which is small in comparison with the other two dimensions. In order to make use of this feature, it is natural to perform an asymptotic analysis by letting the thickness go to zero. In this way, two-dimensional models can be derived from the genuinely three-dimensional model. For a specific order of magnitude for the applied loads, a nonlinear inextensional bending model was obtained by Fox, Raoult & Simo [12] by formal asymptotic-expansion methods. In this model, the equilibrium state is a minimizer of a bending energy, in the sense that the internal energy only depends on the second fundamental form of the midsurface, which is to say, on the curvature. Furthermore, admissible deformations are inextensional. In other words, they are isometries and therefore their first fundamental form is the identity. Moreover, Coutand [8] proved the existence of a solution for this model and obtained similar results with Ciarlet [6, 7] for shells. Nevertheless, no rigorous convergence result of the three-dimensional nonlinear model towards the nonlinear inextensional bending model seem to be known. The purpose of this article is to give a justification of this plate model by using -convergence arguments. This method led to rigorous convergence results
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in the case of body forces of order 1. Le Dret & Raoult [18] and Ben Belgacem [4] thus obtained nonlinear membrane models via this approach. We must also mention the pioneering work of Acerbi, Butazzo & Percivale [1] for strings and the work of Anzellotti, Balbo & Percivale [3]. Frieseke, James & Müller have recently obtained similar results to the one we present here; see [13, 14]. The main difference is that they do not use the same estimate of the deformation in terms of average strains. Indeed, they obtained a better estimate than the one (due to John [16], see Section 2) we use in this paper. This enabled them to avoid the restriction on admissible deformations we made. Moreover, we consider plates made of Saint Venant-Kirchhoff material whereas their analysis applies to more general hyper-elastic bodies. Those three authors and Mora extended those results to the case of elastic shells [15]. As in [12], we consider here the case of body forces of the order of ε2 . Under appropriate hypotheses on the set of admissible deformations, we show that the almost-minimizers of the genuine three-dimensional elasticity converge toward minimizers of the nonlinear inextensional bending energy of Fox, Raoult and Simo. This article is set out as follows. In Section 2, we describe the nonlinear inextensional bending model, the three-dimensional model of Saint Venant-Kirchhoff, and recall some basic results about -convergence theory. We also recall an estimate of the deformations in terms of their elastic strain due to F. John. To be able to make use of this estimate, we restrict the set of admissible deformations. The formulation of our problem thus obtained is presented in Section 2.5. Our main result is stated in Section 2.6. Section 3 is devoted to giving a similar estimate to that of F. John in the case of a plate. We show that the -limit is larger than the nonlinear bending energy in Section 4 and prove the converse inequality in Section 5. The conclusions of this paper were announced in Pantz [21].
2. Preliminaries In what follows, Greek indices take the values 1 or 2, while Latin indices take the values 1, 2 or 3. We adopt the standard summation convention on repeated indices. The space Rn is Endowed with the Euclidean norm |.|. If F is a matrix, we denote by F and |F | respectively its Euclidean and inducted norm, that is F 2 = tr(F T F ) and |F | = sup |F · x|. |x|=1
The letter C will denote a generic constant whose value may change from line to line and we will write x in place of x1 , x2 . 2.1. The nonlinear inextensional bending plate model In this section, we describe the model obtained by Fox, Raoult & Simo [12] using a formal asymptotic expansion. The plate is described as an open set ω of R2 with Lipschitz boundary. It is assumed to be clamped on a nonempty Lipschitz open subset ωφ ⊂ ω and submitted to surface forces fσ . Let VF (ω) be the set of
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isometric deformations of a plate with normal to the surface belonging to class H 1 , satisfying the clamping condition, VF (ω) = ψ ∈ W 1,∞ (ω; R3 ) : Dψ T Dψ = I2 ;
n(ψ) = ∂1 ψ ∧ ∂2 ψ ∈ H 1 (ω); ψ(x) = x on ωφ .
Remark 1. The clamping condition assumed here is not the usual one. Usually, both conditions ψ(x) = x and Dψ · ν = ν are imposed on a subset of the boundary of the plate, where ν is the normal to ∂ω. However, our approach can be easily adapted to this other setting. For all ψ ∈ VF (ω), we denote by n(ψ) the normal to the deformed surface n(ψ) = ∂1 ψ ∧ ∂2 ψ and by bαβ (ψ) its second fundamental form bαβ (ψ) = −∂α ψ · ∂β n(ψ). Remark 2. In [8], Coutand defined the space of admissible deformations as the set of H 2 (ω) isometric deformations. The equality between the two spaces has been pointed out by Friesecke, James & Müller in [13]. First, by a density argument, any two maps ϕ and n in W 1,2 (ω; R3 ) satisfy ∂1 (∂2 ϕ ∧ n) − ∂2 (∂1 ϕ ∧ n) = ∂2 ϕ ∧ ∂1 n − ∂1 ϕ ∧ ∂2 n in the sense of distributions. Secondly, if ϕ is an isometry and n the normal to the surface, then ∂2 ϕ ∧ n = ∂1 ϕ and −∂1 ϕ ∧ n = ∂2 ϕ. Thus ϕ 2 and that ∂ ϕ = −(∂ ϕ · ∂ n)n. belongs to L2 . This implies that ϕ belongs to Hloc ij i j 2 Hence, ϕ ∈ H (ω) as claimed. For simplicity, except in case of ambiguity, we will drop the dependence of n and bαβ on ψ. In the bending model of Fox, Raoult & Simo [12], an equilibrium state is a minimizer on the space VF (ω) of the energy functional JF (ψ) = ω σ (ψ) =
1 αβγ δ bαβ bγ δ 3C
dx − σ (ψ),
fσ (x) · ψ(x) dx, ω
and C
αβγ δ
=
2µλ αβ γ δ δ δ + µ(δ αγ δ βδ + δ αδ δ βγ ), λ + 2µ
where λ and µ are the Lamé moduli of the material, which we assume to be such that µ > 0 and λ 0. Since the total stored energy only depends on the second fundamental form of the surface, the model is said to be of pure bending type. The only interesting case is when VF is not reduced to the identity.
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2.2. A few facts about -convergence A countable sequence Gε of functions from a metric space X into R is said to -converge toward G0 for the topology of X if the following two conditions are satisfied for all x ∈ X: ∀ xε → x, lim inf Gε (xε ) G0 (x), ∃ yε → x, Gε (yε ) → G0 (x). The notion of -convergence presents two major interests, a compactness property and the preservation of minimizers. More precisely, any sequence Gε : X → R admits a -convergent subsequence, and if quasi-minimizers of Gε stay in a compact set of X, then their limit points are minimizers of G0 . In other words, if xε is a sequence of elements in X such that |Gε (xε ) − inf Gε (x)| → 0, x
and if xε stay in a compact set of X, then the limit points of the sequence xε are minimizers for G0 ; see De Giorgi & Franzoni [10], Attouch [2], Dal Maso [9]. 2.3. The Saint Venant-Kirchhoff model A Saint Venant-Kirchhoff material is completely determined by its Lamé moduli µ and λ. The stored-energy function takes the form W (F ) =
λ µ tr((F T F − I )2 ) + (tr(F T F − I ))2 , 4 8
or W (F ) = 21 Cij kl Eij Ekl , with E = 21 (F T F − I ), and Cij kl = λδ ij δ kl + µ(δ ik δ j l + δ il δ j k ). The function W satisfies the following growth and coercivity conditions (see Ciarlet [5]): ∃C > 0, ∀F ∈ M3,3 , |W (F )| C(1 + F 4 ), ∃α > 0, ∃β 0, ∀F ∈ M3,3 , |W (F )| αF 4 − β.
(1)
Let us first remark that a Saint Venant-Kirchhoff material offers no resistance to compression. Such a feature is preserved by passing to the -limit. Let us give an example. Let r : R → R be defined by r(t) = −1 on ]3m, 3m + 1], r(t) = +1 on ]3m + 1, 3m + 3] for all integers m.
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1
−3
−2
−1
1
2
3
−1 Fig. 1. Graph of the function r.
Let ψ n :] − 1, 1[2 ×] − 1/n, 1/n[→ R3 be defined by x1 ψ n (x1 , x2 , x3 ) = r(n · t) dt, x2 , x3 . −1
For all n, we have W (Dψ n ) dx = 0.
(2)
ε
The sequence ψ n weakly converges, in a sense made clear later, towards ψ :] − 1, 1[2 → R3 defined by ψ(x1 , x2 ) = (x1 /2, x2 ). Unfortunately, (2) implies that the -limit energy applied on ψ is zero, whereas ψ is a homogeneous compression. Thus is it necessary to modify the Saint Venant-Kirchhoff model, to get a reasonable inextensional model for plates. There are several ways to achieve this. In the following, we choose to restrict the set of admissible deformations. 2.4. Estimate of the distance between a deformation and the set of rigid-body motions For all n, we define the set A(n) of affine deformations as A(n) = {ϕ : Rn → Rn ; ϕ(x) = g · x + d with g ∈ M n and d ∈ Rn }, endowed with the norm 1/2 |ϕ|A = |g|2 + |d|2 . The subset of n-dimensional rigid motions R(n) is defined as R(n) = {ϕ : Rn → Rn ; ϕ(x) = g · x + d with g ∈ O(n) and d ∈ Rn }.
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Let be a open connected subset of Rn . For all maps ϕ : → Rn , we set Uϕ (x) = (Dϕ(x)T Dϕ(x))1/2 , Eϕ (x) = Uϕ (x) − I. It is well known that if ϕ is regular, ϕ is a rigid motion if and only if Eϕ = 0. In 1972, John [16] studied the stability of this equivalence for “small” Eϕ : If Eϕ is near zero, is ϕ near a rigid motion? John gave a positive answer to this question if is a cube, provided that ϕ belongs to a specific class of functions. For all δ > 0, we define I,δ = {ϕ ∈ C 1 (; R3 ) : Eϕ (x) < δ for all x ∈ }. Theorem 1 (F. John). For all p > 1, there exist δ = δ(p, n) > 0 and C = C(p, n) such that if is a cube of Rn , and ϕ ∈ I,δ , then there exists a matrix g ∈ O(n) such that Dϕ − gLp () C(p, n)Eϕ Lp () Corollary 1. Let =] − 1, 1[3 . There exists C > 0 such that if ϕ ∈ I,δ(2,3) , then there exists a rigid motion γ = g · x + d such that Dϕ − g2L2 () C ϕ − γ 2L6 () C
W (Dϕ) dx, W (Dϕ) dx. 1
Proof. For all ϕ ∈ W 1,4 (; R3 ), let λi be the eigenvalues of Uϕ = (Dϕ T Dϕ) 2 , Eϕ (x)2 = i (λi − 1)2 and Dϕ T Dϕ − I 2 =
(λ2i − 1)2 = (λi − 1)2 (λi + 1)2 . i
i
As each λi is positive, we have Eϕ (x) Dϕ T Dϕ − I for almost all x ∈ . Furthermore, Dϕ T Dϕ − I 2 4/µW (Dϕ(x)) almost everywhere. Thus, by Theorem 1, for all ϕ ∈ I,δ(2,3) , Dϕ − g2L2 () C
Eϕ (x)2 dx C
W (Dϕ) dx.
Let d = ϕ dx. By the Poincaré-Wirtinger inequality and the Sobolev embeddings, we obtain the second estimate.
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2.5. The three-dimensional problems We consider a sequence of three-dimensional plates made of the same Saint Venant-Kirchhoff material and parametrize the sequence by the thickness 2ε. For all ε > 0, the reference placement is defined by ε = ω×] − ε, ε[, where ω, the midsurface of the plate, is a Lipschitz open connected bounded subset of R2 . We assume that the plates are subjected to dead body forces densities fε ∈ L1 (ε ; R3 ) and surface traction densities gε ∈ L1 (Sε ; R3 ) on Sε = ω × {−ε, ε}. φ Furthermore the plates are assumed to be clamped on ε = ∂(ω×] − ε × ε[) ∩ ∂(ωφ ×] − ε × ε[), where ωφ is a nonempty Lipschitz open subset of ω. Let δ be given by Corollary 1. For all open sets of R3 , we define Vε () = {ψ ε ∈ C 1 (ε ; R3 ) : Eψ ε (x) < δ for all x ∈ ε }. In order to be able to use John’s estimates, we restrict the set of admissible deformations by introducing Vεφ = {ψ ∈ Vε (); ψ ε (x) = x on εφ }. We define the energies Jε : L4 (ε ; R3 ) → R by φ ε ε ε ε ε W (Dψ ) dx − ε (ψ ) for ψ ∈ Vε , Jε (ψ ) = φ +∞ for ψ ε ∈ / Vε ,
where ε (ψ ) =
ψ · fε dx +
ε
(3)
ψ ε · gε dσ.
ε
ε
Sε
To look for equilibria, it is natural to try and minimize the energy Jε . Such minimizers do not however always exist. Nevertheless, there always exists a sequence φ ε of almost-minimizers of the energies such that Jε (φ ε )
inf
ψ ε ∈L4 (ε ;R3 )
Jε (ψ ε ) + ε 3 h(ε),
(4)
where h is a positive function such that h(ε) → 0 when ε → 0. Our aim is to study the behavior of φ ε as the thickness goes to zero by using -convergence arguments. For this purpose, it is useful to rescale the problem. We set = 1 , and define a rescaling operator π(ε) : → ε by π(ε)( x , x3 ) = ( x , εx3 ). For each sequence ψ ε : ε → R3 , we set ψ(ε) = ψ ε ◦ π(ε), and for each ϕ : → R3 , J (ε)(ϕ) = Jε (ϕ ◦ π(ε)−1 ). We are interested in the cases of total energy J (ε) and virtual work of the external loads of the order of the cube of the thickness. In effect, Fox, Raoult and Simo obtained their inextensional bending model for such an order of magnitude of the energy. For simplicity, we assume that f (ε) = fε ◦ π(ε) = ε 2 f
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and g(ε) = gε ◦ π(ε) = ε 3 g. The rescaled energies are defined by I (ε) =
J (ε) , ε3
and the rescaled virtual work of a deformation ϕ(ε) is (ε)(ϕ(ε)) (ϕ(ε)) = = f · ϕ(ε) dx + g · ϕ(ε) dσ. ε3 ω×{−1;1} Let I (εn ) be a -convergent subsequence of I (ε) for the strong topology of L4 () and I0 its limit:
I (εn ) − → I0 for the strong topology of L4 (). Our aim is to compute the -limit I0 of I (εn ). The uniqueness of the -limit will make the extraction of the subsequence superfluous a posteriori. In fact the whole sequence I (ε) -converges toward I0 . For notational brevity, we will simply use the notation ε instead of εn . It is easy to show that the almost-minimizers φ ε stay in a compact set of L4 (; R3 ); thus their limit points are minimizers of the -limit I0 of I (ε), which characterize the behavior of thin plates. Remark 3. The energies I (ε) are defined on the space L4 (; R3 ) instead of W 1,4 (; R3 ), which is the natural energy space. This allows us to work with the metric space L4 (; R3 ) endowed with its strong topology. This is a classical trick used in the applications of -convergence. As we will see later, the sequence of quasi-minimizers of the energies is strongly compact in W 1,4 . Consequently, we can in fact work in W 1,4 as well. Remark 4. Some of the results in this paper remain valid for larger sets of admissible deformations. For this purpose, we use in [20] other estimates of the deformations in terms of their elastic strain similar to the one obtained by Kohn in [17]. Remark 5. In [12], the model derived from three-dimensional genuine elasticity by formal asymptotic expansion also takes into account loading terms due to resultant couples. This could be done in our analysis with minor changes. 2.6. Main results We recall that VF (ω) is the set of isometric deformations of a plate with normal to the midsurface belonging to H 1 (ω; S 1 ), clamped on ωφ , and that JF is the nonlinear bending energy (see Section 2.1) of a plate submitted to dead surface forces x) = fσ (
1
−1
f ( x , x3 ) dx3 + g( x , +1) + g( x , −1).
Our main results are as follows.
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Theorem 2. Let φ ε be a sequence of almost-minimizers of the energy (see (4)) and φ(ε) be the associate rescaled deformations. Then, the sequence φ(ε) is compact in W 1,4 (; R3 ). If φ 0 is one of its limit points, then there exists φ ∈ VF (ω) such that φ 0 ( x , x3 ) = φ( x ). Moreover, φ is a minimizer of the nonlinear bending energy JF . To prove this theorem, we compute the -limit of the sequence of functional φ I (ε). Let us introduce the functional JF defined on L4 (; R3 ) by
x , x3 ) = ϕ( x ), JF (ϕ) if there exists ϕ ∈ VF (ω) such that ϕ 0 ( φ JF (ϕ 0 ) = +∞ otherwise. φ
Theorem 3. The sequence I (ε) -converges towards I0 = JF for the strong topology of L4 (). Thus, Theorem 2 is an obvious consequence of the properties of -convergence recalled in Section 2.2 and the following compactness result. Proposition 1. Let ϕ(ε) be a sequence of L4 (; R3 ) such that I (ε)(ϕ(ε)) is bounded. Then, the sequence ϕ(ε) is compact in W 1,4 (). The proof of Theorem 3 will be given in Sections 4 and 5, where we respectively φ φ prove that I0 JF and I0 JF . Moreover, the proof of Proposition 1 is given in Section 4.3.
3. Estimating the deformation of a plate in terms of the energy The purpose of this section is to obtain an estimate similar to the one given in Corollary 1 for square plates Rε =] − 1, 1[2 ×] − ε, ε[. More precisely, we prove Theorem 4. There exists C > 0 such that for all ε, 1/4 > ε > 0, for all functions ϕ ε ∈ Vε (Rε ), there exists a rigid motion γ ε ∈ R(3), such that ϕ ε − γ ε 2L2 (R ) Cε−2 W (Dϕ ε ) dx. ε
Rε
To this end, we begin to prove a similar estimate in the case of beams Bε = ] − 1; 1[×] − ε, ε[2 . 3.1. Case of beams This section is devoted to the proof of following theorem. Theorem 5. There exists C > 0 such that for all 1/4 > ε > 0, for all ϕ ε ∈ Vε (Bε ), there exists a rigid motion γ ε such that ϕ ε − γ ε 2L2 (B ) Cε−2 W (Dϕ ε ) dx. ε
Bε
188
Olivier Pantz ε/2
ε xi
xi+1 yi
δ
Qi Fig. 2. Covering of Bε by ∪Qi .
Let ε, 1/4 > ε > 0, and N ∈ N be such that 2 2 − 2 N < − 1. ε ε Let δx =
(5)
2 − 2ε . N
Obviously, we have 6ε < δx ε. 7
(6)
Let us consider the family of points (xi )i=0,... ,N and the family of cubes (Qi )i=0,... ,N defined by xi = ε + δ x i − 1 and Qi =]xi − ε, xi + ε[×] − ε, ε[2 . The beam Bε is included in the union of the cubes Qi . The dependence of the integer N, the families (xi )i=0,... ,N and (Qi )i=0,... ,N on the parameter ε is understood.
On the Justification of the Nonlinear Inextensional Plate Model
189
Lemma 1. There exists a constant C such that for all ε > 0 and ϕ ε ∈ Vε (]−ε, ε[3 ), there exists a rigid motion γ ε ∈ R(3), such that ε ε 2 2 ϕ − γ L2 (]−ε,ε[3 ;R3 ) Cε W (Dϕ ε ) dx. ]−ε,ε[3
Proof. Let ε > 0, ϕ ε ∈ Vε (] − ε, ε[3 ). We define ϕ(ε) :] − 1, 1[3 → R3 by ϕ(ε)(x) =
ϕ ε (εx) . ε
By Corollary 1, there exists a rigid motion γ (ε) ∈ R(3) such that ϕ(ε) − γ (ε)2L2 (]−1,1[3 ;R3 ) C W (Dϕ(ε)) dx. ]−1,1[3
Let γ ε ∈ R(3) be the rigid motion defined by γ ε (εx) = εγ (ε)(x). We have ϕ ε − γ ε 2L2 ( ]−ε,ε[3 ;R3 ) = ε 5 ϕ(ε) − γ (ε)2L2 ( ]−1,1[3 ;R3 ) Cε 5 W (Dϕ(ε))L1 ( ]−1,1[3 ) = Cε 2 W (Dϕ ε )L1 ( ]−ε,ε[3 ) , hence Lemma 1 follows. Let us consider ϕ ε ∈ Vε (Bε ). Clearly, ϕ ε |Qi belongs to Vε (Qi ) for all i. Therefore, by Lemma 1, there exists a family of rigid motions γiε , γiε (x) = giε · x + diε such that ϕ ε − γiε 2L2 (Q ;R3 ) Cε2 W (Dϕ ε )L1 (Qi ) for all i = 1, . . . , N.
(7)
i
We first prove that Lemma 2. There exists a constant C such that for all ε, 1/4 > ε > 0, for all functions ϕ ε ∈ Vε (Rε ) and for all integers i, j ∈ {0, . . . , N}, |γiε − γjε |2A Cε−4 W (Dϕ ε )L1 (Bε ) . Proof. In a first step, we estimate the distance in the norm of A between two ε . successive rigid motions γkε and γk+1 For all integers k, k < N , we have by Lemma 1, ε ε − γkε L2 (Qk+1 ∩Qk ) γk+1 − ϕ ε L2 (Qk+1 ) + γkε − ϕ ε L2 (Qk ) γk+1 1/2 1/2 Cε W (Dϕ ε )L1 (Q ) + W (Dϕ ε )L1 (Q ) . k+1
k
(8) Let us consider yk = (see Fig. 2)
xk+1 +xk 2
the middle point of xk and xk+1 . By (5), we have
k ⊂ Qk+1 ∩ Qk Q
(9)
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Olivier Pantz
k =]yk − ε/2, yk + ε/2[×] − ε, ε[2 . Obviously, where Q ε ε ε 2 γk+1 − γkε 2L2 (Q ) γk+1 − γk L2 (Q
k+1 ∩Qk )
k
.
Furthermore, by a simple changing of scale, we have ε 3 2 γk+1 − γkε 2L2 (Q ) = ε (gk+1 − gk )εx + dk+1 − dk + (gk+1 − gk ) · yk L2 (Q) k
= ε 3 (gk+1 − gk )εx2L2 (Q) +ε 3 dk+1 − dk + (gk+1 − gk ) · yk 2L2 (Q) =] − 1/2, 1/2[×] − 1, 1[2 . where Q Moreover, for all matrices g ∈ M3,3 , g · x2L2 (Q) |g|. xL2 (Q) Setting C1 = xL2 (Q) , we obtain ε 5 2 3 2 γk+1 − γkε 2L2 (Q ) ε C1 |gk+1 − gk | + 4ε |dk+1 − dk + (gk+1 − gk ) · yk | . k
Due to the convexity of the map x → |x|2 , for all real α, 0 < α < 1 and for all vectors a and b, 1−α 2 |a + b|2 (1 − α)|a|2 − |b| . α Applying this inequality to a = dk+1 − dk , b = (gk+1 − gk ) · yk and α = 2 −1 , we get 1 + C18ε ε − γkε 2L2 (Q γk+1 ) k
C1 ε 5 |(gk+1 − gk )|2 + α|dk+1 − dk |2 . 2
Finally, by (8), we find that there exists a constant C such that 1/2 1/2 ε |γk+1 − γkε |A Cε−3/2 W (Dϕ ε )L1 (Q ) + W (Dϕ ε )L1 (Q ) . k+1
k
(10)
For all i and j , we now estimate the distance between a rigid motion γiε and any other rigid motion γjε in terms of W (Dϕ ε )L1 (Bε ) . For all elements x ∈ Qk , we have |x − xk | < 2ε, so that x ∈ Qk and x ∈ Qk ⇒ |kδx − k δx | < 4ε ⇒ |k − k |δx < 4ε ε ⇒ |k − k | < 4ε, 2 hence, Card({k : k ∈ {0, ..., N }; x ∈ Qk }) 7.
On the Justification of the Nonlinear Inextensional Plate Model
191
In other words, our covering is such that any given point belongs to at most seven cubes. It follows that N
W (Dϕ ε )L1 (Qk ) = W (Dϕ ε ) Card({k : k ∈ {0, ..., N}; x ∈ Qk }) dx Bε
k=0
7W (Dϕ ε )L1 (Bε ) . From the above inequality and (10), we deduce that |γi − γj |A
j −1
|γk+1 − γk |A
k=i N
Cε −3/2
1/2
W (Dϕ ε )L1 (Q
k)
k=0
Cε
−3/2
Cε
−2
N
1/2 W (Dϕ )L1 (Qk ) ε
(N + 1)1/2
k=0 1/2
W (Dϕ ε )L1 (B ) . ε
Theorem 5 easily results from Lemma 2: Proof of Theorem 5. We can choose as an approximation γ ∈ R(3) for ϕ ε any one of the γi defined above, with i ∈ {0, . . . , N}. We first estimate ϕ ε − γi L2 (Bε ) in terms of ϕ ε − γk L2 (Bε ) and |γi − γk |A , which are controlled by Lemmas 1 and 2. ϕ
ε
− γiε 2L2 (B ) ε
N
ϕ ε − γiε 2L2 (Q
k)
k=0
2
N
ϕ ε − γkε 2L2 (Q ) + γkε − γiε 2L2 (Q
k)
k
.
k=0
A straightforward computation yields the inequality γkε − γiε 2L2 (Q ) 4ε3 (2 + ε 2 )|gk − gi |2 + 2|dk − di |2 . k
Therefore, ϕ ε − γiε 2L2 (B ) C
N
ε
ϕ ε − γkε 2L2 (Q ) + ε 3 |γkε − γiε |2A . k
k=0
Applying estimate (7) and Lemma 2, we obtain ϕ ε − γiε 2L2 (B ) C
N
ε 2 W (Dϕ ε )L1 (Qk ) + ε −1 W (Dϕ ε )L1 (Bε )
ε
k=0 −2
Cε
W (Dϕ ε L1 (Bε ) .
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Olivier Pantz
3.2. The case of square plates We now are in a position to give the proof of Theorem 4, which uses Lemma 2 and Theorem 5 proved in the previous section. As in the case of a beam, let us consider ε > 0, ε < 1/4 and N ∈ N such that 2 2 − 2 N < − 1. ε ε We set δx =
2 − 2ε ; N
xi = ε + iδx − 1;
xi,j = (xi , xj )
and Qi,j =]xi − ε, xi + ε[×]xj − ε, xj + ε[×] − ε, ε[. The plate Rε is included in the union of the cubes Qi,j . Let ϕ ε ∈ Vε (Rε ). By ε such that Lemma 1, there exists a family of rigid motions γi,j ε 2 ϕ ε − γi,j L2 (Q
i,j )
Cε2 W (Dϕ ε )L1 (Qi,j ) .
Proof of Theorem 4. We divide the square plate Rε into rows Li = N j =0 Qi,j N and columns Tj = i=0 Qi,j . Each of these rows and columns can be considered as a beam. Since any given point can belong to at most seven columns or rows, it follows that N
W (Dϕ ε )L1 (Ti ) 7W (Dϕ ε )L1 (Rε ) .
(11)
i=0
Therefore, there exists n ∈ {0, . . . , N} such that W (Dϕ ε )L1 (Tn )
7 W (Dϕ ε )L1 (Rε ) CεW (Dϕ ε )L1 (Rε ) . N +1
(12)
We can apply Lemma 2 to the beam Tn . In particular, for each i ∈ {0, . . . , N}, we have ε ε 2 |γi,n − γ0,n |A Cε−4 W (Dϕ ε )L1 (Tn ) , so that by (12), ε ε 2 |γi,n − γ0,n |A Cε−3 W (Dϕ ε )L1 (Rε ) .
(13)
Let us now apply Theorem 5 to each beam Li (as seen in the proof of the theorem, ε ). We get we can choose γ ε = γi,n N
ε 2 ϕ ε − γi,n L2 (L ) Cε−2
N
i
i=0
i=0
W (Dϕ ε )L1 (Li ) Cε−2 W (Dϕ ε )L1 (Rε ) .
On the Justification of the Nonlinear Inextensional Plate Model
193
So that ε 2 L2 (R ε ) Cε−2 W (Dϕ ε )L1 (Rε ) + 2 ϕ ε − γ0,n
N
ε ε 2 γ0,n − γi,n L2 (L ) . (14) i
i=0
In order to estimate the second part of the right-hand side, we observe that ε ε 2 ε ε 2 γ0,n − γi,n L2 (L ) 4ε2 |γ0,n − γi,n |A . i
Combining this last estimate with (13), estimate (14) yields ε 2 L2 (R ε ) Cε−2 W (Dϕ ε L1 (R ε ) , ϕ ε − γ0,n
concluding the proof of Theorem 4. 4. Computation of a bound from below for the -limit 4.1. Deformations with zero-limit energy are rigid motions Let us introduce the set of rigid motions of a plate, Rσ = {γ : R2 → R3 : γ (x) = c · x + b; c ∈ M3,2 ; cT c = I2 ; b ∈ R3 }, which is canonically isomorphic to R = {γ : R3 → R3 : γ = γ ◦ p2 : γ ∈ Rσ }, where p2 is the projection of R3 onto the first two components. Lemma 3. There exists C > 0 such that for all sequences ϕ(ε) → ϕ 0 strongly in L4 (R1 ), there exists γ ∈ R such that 1/2 ε Rε W (Dϕ ) dx 0 ϕ − γ L2 (R1 ) C lim inf . ε→0 ε3 Proof. Without loss of generality, we may assume that ε Rε W (Dϕ ) dx < ∞. lim inf ε→0 ε3 Otherwise, there is nothing to prove. By Theorem 4, there exists γ ε ∈ R(3), such that 1/2 ε ϕ ε − γ ε L2 (Rε ) Rε W (Dϕ ) dx C . ε3 ε 1/2 Let us recall that γ (ε) = γ ε ◦ π(ε) and ϕ(ε) = ϕ ε ◦ π(ε) where π(ε)( x , x3 ) = ( x , εx3 ). Thus, rewriting the above estimate in terms of ϕ(ε) and γ (ε), we obtain 1/2 ε Rε W (Dϕ ) dx ϕ(ε) − γ (ε)L2 (R1 ) C . (15) ε3
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Olivier Pantz
There exists a sequence εn → 0 such that εn ε Rεn W (Dϕ ) dx Rε W (Dϕ ) dx = lim inf . lim n→+∞ ε→0 εn3 ε3 According to the definition of R(3), x , εx3 ) + dεn , γ (εn )(x) = gεn ( where gεn ∈ O(3) and dε ∈ R3 . As O(3) is compact, and dεn stay in a bounded set of R3 , we can extract a subsequence from εn , still denoted εn , such that gεn converges towards an element g of O(3) and dεn towards an element d of R3 . Therefore, γεn → g · ( x , 0) + d strongly in L2 (R1 ). The limit point γ = g · ( x , 0) + d is an element of R . We conclude by passing to the limit when n goes to +∞ in (15), with ε = εn . In the special case where ω is a square, we deduce from the above result that Corollary 2. Without external loads, i.e., L = 0, any deformation ϕ 0 ∈ L4 (R1 ; R3 ) such I0 (ϕ 0 ) = 0 is a rigid motion, i.e., belongs to R . Remark 6. The purpose of this corollary is just to illustrate the meaning of Lemma 3. It may be extended to the more general case of any Lipschitz open set ω. 4.2. Deformations with finite energy are isometries The aim of this section is to prove that any deformation with finite -limit energy is an isometry. Proposition 2. Let ϕ 0 be such that I0 (ϕ 0 ) < ∞. Then – there exists ϕ ∈ W 1,4 (ω) such that ϕ 0 ( x , x3 ) = ϕ( x ); – ϕ is an isometric deformation, that is, Dϕ T Dϕ = I2 . We divide the proof of this proposition into two parts. Proof of the first part of Proposition 2. Let ϕ 0 be such a deformation. There exists a sequence ϕ(ε) → ϕ 0 strongly in L4 (; R3 ) such that lim I (ε)(ϕ(ε)) < +∞. Due to the coercivity condition (1) satisfied by W , (∂1 ϕ(ε)|∂2 ϕ(ε)|ε −1 ∂3 ϕ(ε))4 dx α −1 ε W (Dϕ ε ) dx + β ε −1 ε ε ε W (Dϕ ) dx − ε (ϕ ) + ε −1 ε (ϕ ε ) + β ε
ε I (ε)(ϕ(ε)) + ε 2 (ϕ(ε)) + β ε2 I (ε)(ϕ(ε)) + ε 2 Cϕ(ε)W 1,4 () + β. 2
On the Justification of the Nonlinear Inextensional Plate Model
Thus, as soon as ε < 1, Dϕ(ε)4 dx ε2 I (ε)(ϕ(ε)) + ε 2 Cϕ(ε)W 1,4 () + β.
195
(16)
Let us recall that there exists a constant C, depending only on and φ , such that for all ψ ∈ W 1,4 (; R3 ), ψW 1,4 (;R3 ) C DψL4 () + ψW 1,4 ( φ ;R3 ) . As ϕ(ε)( x , x3 ) = ( x , εx3 ) on φ , the Poincaré inequality above, together with estimate (16), implies that ϕ(ε) is bounded in W 1,4 (; R3 ) so that ϕ 0 is an element of W 1,4 (; R3 ). Moreover, as |∂3 ϕ(ε)| ε(∂1 ϕ(ε)|∂2 ϕ(ε)|ε −1 ∂3 ϕ(ε)), inequality (16) implies that ∂3 ϕ(ε) → 0 strongly in L4 and ∂3 ϕ 0 = 0, which implies that there exists ϕ ∈ W 1,4 (ω) such that ϕ 0 = ϕ. We have already proved that Lemma 4. Let ϕ(ε) be a sequence of rescaled deformations, such that the energies I (ε)(ϕ(ε)) remains bounded. Then ϕ(ε) is bounded in W 1,4 (; R3 ). In order to prove the second part of Proposition 2, we need a technical lemma. As the open set ω is Lipschitz, there exists an linear extension operator Rω from W 1,4 (ω; R3 ) to W 1,4 (R2 ; R3 ). For all δ > 0, we introduce the application Sω (δ) from W 1,4 (ω; R3 ) to W 1,4 (ω×] − 1; 1[2 ; R3 ) defined as follows: for all x ∈] − 1; 1[2 , y ∈ ω and v ∈ W 1,4 (ω; R3 ), we set Sω (δ) · v(y, x) =
Rω (v)(δx + y) − Rω (v)(y) . δ
We claim that Lemma 5. For all v ∈ W 1,4 (ω; R3 ), Sω (δ) · v −−→ Dy v · x strongly in L4 (ω×] − 1; 1[2 ; R3 ). δ→0
Proof. First of all, let us consider a function u ∈ C0∞ (R2 ; R3 ). Clearly, we have u(δx + y) − u(y) − Dy u · x −−→ 0. δ δ→0 Moreover, u(δx+y)−u(y) − Dy u · x is bounded by an L4 function independently δ of δ. Hence, by the dominated-convergence theorem, u(δx + y) − u(y) − Dy u.x −→ 0. (17) 4 2 2 −δ→0 δ L (R ×R )
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Next, let v ∈ W 1,4 (ω; R3 ). For all η > 0, there exists uη ∈ C0∞ (R2 ; R3 ) such that η Rω (v) − uη W 1,4 (R2 ;R3 ) √ . 3 2 We have to prove that for δ small enough, Dy v · x − Sω (δ) · v
L4 (ω×]−1;1[2 ;R3 )
(18)
η.
Setting uη (δx + y) − uη (y) A = Dy uη · x − 4 L δ uη (δx + y) − uη (y) and B = Sω (δ) · v − 4, L δ Dy v · x − Sω (δ) · v
L4 (ω×]−1;1[2 ;R3 )
(19) (20)
Dy v · x − Dy uη · x L4 + A + B. (21)
It follows that we just have to estimate each term of the right-hand side. First of all, by (18), √ η Dy v · x − Dy uη · x 4 2Dy v − Dy uη L4 (ω;M3,3 ) . L (ω×]−1;1[2 ;R3 ) 3 (22) Furthermore, (17) implies that we can chose δ small enough so that uη (δx + y) − uη (y) η A = Dy uη · x − . δ 3 L4 (ω×]−1;1[2 ) Finally, B=
1
0
D(Rω (v) − uη )(tδx + y) · x dt
L4 (ω×]−1;1[2 )
ω×]−1;1[2
1
1/4
|D(Rω (v) − uη )(tδx + y) · x| dt dx dy 4
0
R2 ×]−1;1[2 0
1
1/4 |D(Rω (v) − uη )(tδx + y) · x| dt dx dy 4
with the change of variable(x, y) → (x, z), z = tδx + y, 1/4 1 4 |D(Rω (v) − uη )(z) · x| dt dx dz =
√
R2 ×]−1;1[2 0
R2 ×]−1;1[2
1/4
|D(Rω (v) − uη )(z) · x|4 dx dz
2D(Rω (v) − uη )(z)L4 (R2 ;M3,3 )
This estimate completes the proof.
η . 3
(23)
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197
We are now in a position to continue the proof of Proposition 2. Proof of the second part of Proposition 2. For all ρ > 0, we introduce √ ωρ = {x ∈ ω : dist(x, R2 − ω) > 2ρ}.
(24)
Let ϕ 0 such that I0 (ϕ 0 ) < ∞. There exists M < +∞ and ϕ(ε) → ϕ 0 strongly in L4 () such that I (ε)(ϕ(ε)) < M. Thus, by Lemma 4, the sequence ϕ(ε) is bounded in W 1,4 () and ϕ(ε) ϕ 0 weakly in W 1,4 (; R3 ). Hence, by the Rellich-Kondrachov theorem, ϕ(ε) → ϕ 0
in C(; R3 ).
(25)
Let ρ > 0 and δ < ρ. We define ϕ ε,δ : ωρ ×] − 1, 1[2 ×] − ε/δ, ε/δ[ → R3 ϕ ε (δ x˜ + y, δx3 ) − ϕ ε (y, 0) . (y, x, ˜ x3 ) → δ The convergence (25) of ϕ(ε) in C(; R3 ) and the definition of Sω (δ) imply that for all y ∈ ωρ , (ϕ ε,δ (y)) ◦ π(ε) → (Sω (δ) · ϕ 0 (y)) ◦ p2 as ε → 0 in C(R1 ; R3 ), hence strongly in L4 (R1 ; R3 ). Lemma 3 applied to the sequence ϕ ε,δ (y) ◦ π(ε) implies that there exists γδ (y) ∈ R , such that 1/2 ε,δ Rε/δ W (Dϕ (y)) dx . Sω (δ) · ϕ(y)) − γδ (y)L2 (R1 ) C lim inf ε→0 (ε/δ)3 After an integration of the square of this inequality, we obtain, using Fatou’s lemma, |Sω (δ) · ϕ(y)(x) − γδ (y) · x|2 dx dy ωρ
x∈]−1,1[2
C lim inf y∈ωρ
Rε/δ
W (Dϕ ε,δ (y)) 3 δ dx dy. ε3
Let us now look for a bound for the right-hand side of the above inequality: W (Dϕ ε,δ ) 3 W (Dϕ ε (δx + (y, 0))) 3 δ dx dy = δ dx dy ε3 ε3 ωρ ×Rε/δ
ωρ ×Rε/δ
= ωρ ×]−1,1[2
W (Dϕ ε (δ x˜ + y, δx3 )) 3 δ dx3 dx˜ dy. ε3 x3 =−ε/δ ε/δ
Let us perform a simple change of variable using the map ωρ ×] − 1; 1[2 ×] − 1, 1[ → R 2 (y; δ)×] − 1, 1[2 ×] − 1, 1[ y∈ωρ
(y, x , x3 ) → (z, x , x3 ) = (δ x + y, x , x3 ).
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Olivier Pantz
R 2 (y; δ) ⊂ ω, we get
y∈ωρ
y∈ωρ
Rε/δ
W (Dϕ ε,δ ) 3 δ dx dy ε3 ε/δ
W (Dϕ ε (z, δx3 )) 3 δ dx3 dx˜ dz 2 x =−ε/δ ε3 z∈ω x∈]−1,1[ ˜ 3 ε W (Dϕ ε (z, x3 )) 2 δ dx3 dz = 4δ 2 I (ε)(ϕ(ε)). =4 ε3 z∈ω x3 =−ε
Putting all these estimates together, we have thus shown that |Sω (δ) · ϕ(y)(x) − γδ (y).x|2 dx dy ωρ
x∈]−1,1[2
Cδ 2 lim inf I (ε)(ϕ(ε)) Cδ 2 M. By Lemma 5, Sω (δ) · ϕ → Dy ϕ · x strongly in L2 (ωρ ×] − 1, 1[2 ) when δ goes to zero. Thus, γδ converges strongly in L2 (ωρ ×] − 1, 1[2 ; R3 ) toward Dy ϕ.x. Let cδ (y) ∈ L2 (ωρ ; M3,2 ) and bδ ∈ L2 (ωρ ; R3 ) such that for almost all y, γδ (y).x = cδ (y) · x + bδ (y). We have just proved that up to a subsequence cδ · x + bδ → Dy ϕ · x strongly in L2 (ωρ ; R3 ) for almost all x ∈] − 1, 1[2 . Since the dependence in x is affine, it follows that this convergence holds true for all x ∈] − 1, 1[2 . Therefore, bδ → 0 strongly in L2 (ωρ ; R3 ) and cδ → Dy ϕ strongly in L2 (ωρ ; M3,2 ). In particular, cδT cδ → Dy ϕ T Dy ϕ strongly in L1 (ωρ ; M2,2 ). Since for any δ, cδ ∈ R , we have cδT cδ = I2 . Therefore, for almost all y ∈ ωρ , Dy ϕ T Dy ϕ = I2 , i.e., ϕ is an isometry. To conclude the proof, it suffices to note that ω= ωρ . ρ
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4.3. Compactness We are now able to prove the compactness result stated in Proposition 1. We have already proved the compactness of bounded-energy sequences in L4 (). Hence, it only remains to prove the following proposition. Proposition 3. Let ϕ(ε) → ϕ 0 strongly in L4 (), such that there exists M < +∞, with I (ε)(ϕ(ε)) < M for all ε. Then, ϕ(ε) converges toward ϕ 0 strongly in W 1,4 (; R3 ). Proof. Let ϕ(ε) be such a sequence. We already know that ϕ(ε) ϕ 0 weakly in W 1,4 (; R3 ). For all (f1 |f2 |f3 ) ∈ M3,3 , (|f1 |2 − 1)2 + (|f2 |2 − 1)2 Thus,
4 W ((f1 |f2 |f3 )). µ
(|∂1 ϕ(ε)|2 − 1)2 + (|∂2 ϕ(ε)|2 − 1)2 dx 4 W ((∂1 ϕ(ε)|∂2 ϕ(ε)|ε −1 ∂3 ϕ(ε))) dx µ 4 = ε −1 W (Dϕ ε ) dx µ ε 4 2 = ε (I (ε)(ϕ(ε) − (ϕ(ε))) µ 4 ε 2 (M + Cϕ(ε)W 1,4 (;R3 ) ). µ
As ϕ(ε)W 1,4 (;R3 ) is bounded, it follows that |∂1 ϕ(ε)|2 → 1 strongly in L2 (), |∂2 ϕ(ε)|2 → 1 strongly in L2 (). Let us recall that ∂3 ϕ(ε) → 0 strongly in L4 . Thus, |Dϕ(ε)|2 dx → 2||.
On the other hand, by Proposition 2, 0 2 |Dϕ | dx = 2 dx = 2||.
We have thus shown that ϕ(ε)W 1,4 (;R3 ) → ϕ 0 W 1,4 (;R3 ) . Since the unit ball of W 1,4 is uniformly convex, weak convergence together with the convergence of the norms implies strong convergence.
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We define the operator D ε = (∂1ε |∂2ε |∂3ε ) on D () by D ε ψ = D(ψ ◦ π(ε)−1 ) ◦ π(ε) for all ψ, that is, for all u ∈ D(), < ∂iε ψ, u >= − < ψ, ∂i (u ◦ π(ε)) ◦ π(ε)−1 > . In particularly, D ε ϕ(ε) = Dϕ ε ◦ π(ε). Proposition 4. Let ϕ(ε) → ϕ 0 strongly in L4 (; R3 ) such that there exists a constant M < ∞, I (ε)(ϕ(ε)) < M. Then ∂αε ϕ(ε) → ∂α ϕ 0 strongly in L4 (; R3 ), ∂3ε ϕ(ε) → n strongly in L2 (; R3 ), where n = ∂1 ϕ 0 ∧ ∂2 ϕ 0 . Proof. Let us remark that ∂αε ϕ(ε) = ∂α ϕ(ε). Thus, the first part of the proposition is just a part of Corollary 3. Let U (ε) be the positive-definite symmetric matrix such that U (ε)2 = ε D ϕ(ε)T D ε ϕ(ε). For almost all x ∈ , there exists a single matrix C(ε)(x) ∈ O(3), such that D ε ϕ(ε) = C(ε) · U (ε). φ
According to the definition of Vε , det(Dx ϕ(ε)) > 0 for almost all x ∈ . We deduce that det(Dxε ϕ(ε)) > 0 and det(C(ε)(x)) > 0 almost everywhere, thus C(ε)(x) ∈ SO(3). Let us remark that, D ε ϕ(ε) − C(ε)2 = U (ε) − I 2 CW (D ε ϕ(ε)). Thus,
D ε ϕ(ε) − C(ε)2L2 (;M
3,3 )
C
W (D ε ϕ(ε)) dx
C = ε
W (Dϕε ) dx −−→ 0. ε
(26)
ε→0
As ∂αε ϕ(ε) → ∂α ϕ 0 strongly in L4 (; R3 ) hence in L2 (; R3 ), we have Cα (ε) → ∂α ϕ 0 strongly in L2 (; R3 ). Without loss of generality, we may assume that Cα (ε) converges almost everywhere. As C(ε)(x) ∈ SO(3), C3 (ε)(x) = C1 (ε)(x) ∧ C2 (ε)(x). Thus, C3 (ε)(x) converges toward ∂1 ϕ 0 (x)∧∂2 ϕ 0 (x) for almost all x ∈ . As |C3 (ε)| is bounded by 1, by the dominated-convergence theorem, C3 (ε) → ∂1 ϕ 0 ∧ ∂2 ϕ 0 = n strongly in L2 (; R3 ). As the limit is independent of the chosen subsequence, the whole sequence converges. The conclusion follows from (26).
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4.4. The -limit is bounded from below by the bending energy We are now in a position to prove that φ
I0 JF ,
(27)
which will follow directly from Propositions 5 and 6 proved in this section. Let ϕ(ε) ∈ W 1,4 (; R3 ). The rescaled Green-Lagrange tensor of ϕ ε is given by E(ϕ(ε)) = 21 ((D ε ϕ(ε))T D ε ϕ(ε) − I3 ). For simplicity, we will denote E(ϕ(ε)) by E(ε). In the following, we will need a technical lemma Lemma 6. Let U be an open set of Rn−1 ; Let ψ ∈ Lp (U ×] − 1, 1[) such that ∂nn ψ = 0. Then there exists a ∈ Lp (U ), b ∈ Lp (U ) such that ψ = a + x3 b. Furthermore, ∂n ψ ∈ Lp (U ×] − 1, 1[) and for almost all ( x , xn ) ∈ U ×] − 1, 1[, ∂n ψ( x , xn ) = b( x ). We omit the proof of this lemma which does not raise any particular difficulty. Lemma 7. Let ϕ(ε) → ϕ 0 strongly in L4 (; R3 ), such that I (ε)(ϕ(ε)) is bound1 (ω). Moreover, there exists a subsequence still ed. Then ϕ 0 ∈ W 1,∞ , and n ∈ Hloc noted ε, and there exists wαβ ∈ L2 (ω; R3 ) such that Eαβ (ε) x3 wαβ ( x ) + (∂α n · ∂β ϕ 0 + ∂β n · ∂α ϕ 0 ) weakly in L2 (; R3 ). ε 2 Proof. We have 1 I (ε)(ϕ(ε)) = 2 2ε
Cij kl Eij (ε)Ekl (ε) dx − (ϕ(ε)). E (ε)
The coercivity of the elasticity tensor implies that for all i and j , ijε is bounded 2 in L2 (). Hence, we may assume that E(ε) ε weakly converges in L (; M3,3 ). √ Let ρ > 0 and ε > 0 such that 2ε < ρ. Let us introduce Rε,i,j =]ε(i − 1), ε(i + 1)[×]ε(j − 1), ε(j + 1)[ and S = {(i, j ) ∈ N × N : Rε,i,j ⊂ ω}. We have ωρ ⊂ Rε,s , s∈S
where ωρ is defined by (24). Figure 3 represents in bold the elements Rε,s such that s ∈ S. The element Rε,i,j is shown in gray.
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Olivier Pantz j
ε
i ω
ωρ
Fig. 3.
By Theorem 1, there exists a function gε ∈ L2 ( each square Rε,s ,
s∈S
Rε,s ; O(3)) such that on
Dϕε − gε L2 (Rε,s ×]−ε,ε[;M3,3 ) C
Rε,s ×]−ε,ε[
W (Dϕε ).
After the standard change of variable, we obtain εD
ε
ϕ(ε) − g(ε)2L2 (R ×]−1,1[;M ) ε,s 3,3
C
Rε,s ×]−ε,ε[
W (Dϕε ).
By summation on those estimates, we have εD ε ϕ(ε) − g(ε)2L2 (ω
ρ ×]−1,1[;M3,3 )
C
W (Dϕε ) ω×]−ε,ε[
and
D ε ϕ(ε) − g(ε)L2 (ωρ ×]−1,1[;M3,3 ) ε
2 C(I (ϕ(ε)) + ϕ(ε)W 1,4 (;R3 ) ). (28)
On the Justification of the Nonlinear Inextensional Plate Model D ε ϕ(ε)−g(ε) is bounded in L2 , there exists a subsequence, still ε ε D ϕ(ε)−g(ε) , that weakly converges in L2 (ωρ ×] − 1, 1[; M3,3 ). Let be ε
As
limit. We have
∂3
∂αε ϕ(ε) − gα (ε) ε
203
denoted its weak
∂3 ∂αε ϕ(ε) ε = ∂α ∂3ε ϕ(ε). =
Furthermore, ∂3ε ϕ(ε) → n strongly in L2 (ωρ ×] − 1, 1[; R3 ). Thus, ∂3 α = ∂α n. As ∂3 n = 0, ∂33 α = 0. Thus, by Lemma 6, ∂α n ∈ L2 (ωρ ) and there exists vα ∈ L2 (ωρ ) such that = vα + x3 ∂3 α = vα + x3 ∂α n. We have proved that ∂αε ϕ(ε) − gα (ε) x ) + x3 ∂α n weakly in L2 (ωρ ×] − 1, 1[; R3 ). vα ( ε
(29)
Let us rewrite E(ε) as 1 ε (D ϕ(ε)T D ε ϕ(ε) − I ) 2 (D ε ϕ(ε) − g(ε))T D ε ϕ(ε) − g(ε) = g(ε) + D ε ϕ(ε)T . 2 2
E(ε) =
By (29) we deduce that Eαβ (ε) x3 vα · ∂β ϕ 0 + vβ · ∂α ϕ 0 + (∂α n · ∂β ϕ 0 + ∂β n · ∂α ϕ 0 ) ε 2 in D (ωρ ×] − 1, 1[). Indeed, gα → ∂α ϕ 0 strongly in L2 (ω) and g3 → n strongly in L2 (ωρ ). Moreover, ∂3
Eαβ (ε) ε
1 (∂α n · ∂β ϕ 0 + ∂β n · ∂α ϕ 0 ) in D (ωρ ×] − 1, 1[). 2
(30)
Since the limit in (30) is independent of δ, the convergence holds true in D (). Lemma 6 completes the proof. Lemma 8. For all symmetric matrices F ∈ M3,3 , C
αβγ δ
Fαβ Fγ δ Cij kl Fij Fkl .
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Proof. We have 2 2 2 Cij kl Fij Fkl = Cαβγ δ Fαβ Fγ δ + (λ + 2µ)F33 + 2λF33 (F11 + F22 ) + 4µ(F13 + F23 ).
The function t → (λ + 2µ)t 2 + 2λt (F11 + F22 ) reaches its minimum for t = 11 +F22 ) − λ(Fλ+2µ . Thus we have λ2 (F11 + F22 )2 λ + 2µ λ2 = Cαβγ δ Fαβ Fγ δ − δ αβ δ γ δ Fαβ Fαβ λ + 2µ λ2 = Cαβγ δ − δ αβ δ γ δ Fαβ Fαβ λ + 2µ
Cij kl Fij Fkl Cαβγ δ Fαβ Fγ δ −
=C
αβγ δ
Fαβ Fγ δ .
Proposition 5. If ϕ 0 ∈ L4 (; R3 ) such that I0 (ϕ 0 ) < ∞, then there exists ϕ ∈ H 1 (ω), n(ϕ) ∈ H 1 (ω), ϕ 0 ( x , x3 ) = ϕ( x ) and 1 αβγ δ I0 (ϕ 0 ) C (∂α n.∂β ϕ)(∂γ n.∂δ ϕ) − σ (ϕ). 3 ω Proof. Let ϕ 0 ∈ L4 (; R3 ) be such that I0 (ϕ 0 ) < +∞. By Proposition 2, ϕ 0 ( x , x3 ) = ϕ( x ). There exists ϕ(ε) → ϕ 0 strongly in L4 (; R3 ) such that lim I (ε)(ϕ(ε)) = I0 (ϕ 0 ). As
I (ε)(ϕ(ε)) =
1 ij kl Eij Ekl C dx − (ϕ(ε)), 2 ε ε
by Lemma 8, I (ε)(ϕ(ε))
1 αβγ δ Eαβ Eγ δ C dx − (ϕ(ε)). 2 ε ε
Passing to the limit, when ε goes to zero, since the right-hand side is sequentially weakly lower semicontinuous for the topology L2 () with respect to Eαβ (ε), we have by Lemma 7,
x3 1 αβγ δ C wαβ + (∂α n · ∂β ϕ + ∂β n · ∂α ϕ) · · · 2 ω −1 2 x3 · · · wγ δ + (∂γ n · ∂δ ϕ + ∂δ n · ∂γ ϕ) dx − (ϕ 0 ) 2 1 1 αβγ δ 2 C x3 (∂α n · ∂β ϕ)(∂γ n · ∂δ ϕ) dx3 dx1 dx2 − (ϕ 0 ) ω −1 2 1 αβγ δ = C (∂α n · ∂β ϕ)(∂γ n · ∂δ ϕ) dx − σ (ϕ). (31) 3 ω
I0 (ϕ ) 0
1
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1 (ω). It remains to prove that n ∈ H 1 (ω).At this stage, we just know that n ∈ Hloc 2 Since ∂α ϕ ∈ L (ω), it is easy to show that the derivative of the following product makes sense: 0 = ∂α (n · ∂β ϕ) = n · ∂αβ ϕ + ∂α n · ∂β ϕ.
Thus, ∂α n · ∂β ϕ = ∂β n · ∂α ϕ.
(32)
For all symmetric matrices F ∈ M2,2 , we have C
αβδγ
Fαβ Fδγ 2µF .
(33)
Combining (31), (32) and (33), we conclude that ∂α n · ∂β ϕ ∈ L2 (ω). Finally, for almost all x, |∂α n|2 = δ βγ (∂α n · ∂β ϕ)(∂α n · ∂γ ϕ). As the left-hand side belongs to L1 (ω), it follows that n ∈ H 1 (ω). To complete the proof of (27), we just have to prove that any deformation with finite -limit energy satisfies the clamping condition. Proposition 6. Let ϕ 0 ∈ L4 () such that I0 (ϕ 0 ) < ∞. Then there exists ϕ ∈ W 1,∞ (ω; R3 ) such that ϕ 0 ( x , x3 ) = ϕ( x ), and ϕ( x ) = ( x , 0)
on ωφ .
Proof. There exists ϕ(ε) → ϕ 0 strongly in W 1,4 (; R3 ) such that I0 (ϕ 0 ) = lim I (ε)(ϕ(ε)). As, for all ε, ϕ(ε)( x , 1) = ( x , ε)
on ωφ × {1}.
Thus, passing to the limit, as ϕ(ε) converges in C(; R3 ), ϕ 0 ( x , 1) = ( x , 0)
on ωφ × {1}.
We already know that there exists ϕ ∈ W 1,∞ (ω) such that ϕ 0 ( x , x3 ) = ϕ( x ) for almost all x. Thus, ϕ( x ) = ( x , 0) on ωφ . 5. Computation of the -limit To prove Theorem 3, it remains to establish the converse inequality of (27) for regular deformations, that is, φ
I0 (ϕ 0 ) JF (ϕ). x , x3 ) = ϕ( x ). In Proof of Theorem 3. Let ϕ ∈ VF and ϕ 0 ∈ H 2 () such that ϕ 0 ( the following, we will construct a family of sequences ϕψ (ε) of rescaled deformations, depending on a parameter ψ ∈ C0∞ (ω − ωφ ; R3 ) such that ϕψ (ε) converges
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Olivier Pantz
toward ϕ 0 strongly in H 2 () as ε goes to zero and inf lim I (ε)(ϕ(ε)) JF (ϕ).
(34)
ψ ε→0
To this end, we proceed in two steps. In a first step, we approximate the deformation ϕ and the normal to the midsurface n = ∂1 ϕ ∧ ∂2 ϕ by regular maps. Then, using this approximation, we build sequences ϕψ (ε) satisfying (34). The first step is based on a truncation result for Sobolev maps [11, 19, 22]. This argument has been adapted by Frieseke, James & Müller (see [13]) to preserve the clamping conditions. They show (Proposition A.3) that for all λ > 0, there exist ϕλ ∈ W 2,∞ (ω; R3 ) and nλ ∈ C 1 (ω; R3 ), ϕλ = (ϕ − id)λ + id; nλ = (n − e3 )λ + e3 such that ϕλ and nλ fulfill the clamping condition, that is, ϕλ = id
and
nλ = e3
on ωφ .
Furthermore, the maps ϕλ and nλ agree with ϕ and n respectively on a set ω − ωλ , where ωλ is given by ωλ = {x ∈ ω : ϕλ (x) = ϕ(x) or nλ (x) = n(x)}, and such that |ωλ | CRλ /λ2 . In addition, Rλ =
{x∈ω:|ϕ−id |+|Dϕ−id |+|D 2 ϕ|λ/2}
(35)
|ϕ − id |2 + |Dϕ − id |2 + |D 2 ϕ|2 dx
··· +
{x∈ω:|n−e3 |+|Dn|λ/2}
|n − e3 |2 + |Dn|2 dx −−−−→ 0. λ→+∞
Moreover, the derivatives of ϕλ and nλ are controlled by λ: Dnλ L∞ (ω) ,
D 2 ϕλ L∞ (ω) Cλ.
(36)
From (35) and (36), it is easy to deduce that nλ converges toward n strongly in H 1 (ω) and ϕλ toward ϕ strongly in H 2 (ω) as λ goes to infinity. For sufficiently large λ, we have 1/2
f λ (x) = dist((Dϕλ , nλ ); SO(3)) CRλ
(37)
for all x ∈ ω. Indeed, f λ is Lipschitz with Lipschitz constant lower or equal to Cλ. As ω is Lipschitz, there exists a constant A such that for r small enough, |B(x0 , r) ∩ ω| Ar 2 for all x0 ∈ ω. Thus, |B(x0 , r) ∩ (ω − ωλ )| Ar 2 − |B(x0 , r)| Ar 2 − |ωλ | Ar 2 − CRλ /λ2 .
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λ Choosing r 2 = 2CR , we deduce that B(x0 , r) ∩ (ω − ωλ ) is not empty, and from Aλ2 the Lipschitz property of fλ that for all x0 ∈ ω, λ f (x0 ) Crλ = CRλ 1/2 .
Let ψ ∈ C0∞ (ω − ωφ ; R3 ). For all ε, we define ϕψε by x , x3 ) = ϕλε ( x ) + x3 nλε ( x ) + x32 ψ( x )/2. ϕψε ( The deformation ϕψε satisfies the clamping condition. Moreover, for ε small enough, Eϕψε δ,
(38) φ
where δ is given by Corollary 1. Hence, ϕψε belongs to Vε . To show this, we note that Dϕψε = (Dϕλε , nλε ) + x3 (Dnϕε , ψ) + x32 /2(Dψ, 0) and |Dϕψε − (Dϕλε ; nλε )| ε(Dnλε L∞ + ψL∞ ) + ε 2 DψL∞ Cελε + (ε + ε 2 )ψC 1 . From (37), choosing λε = ε−1 , we obtain (38). Consider ϕψ (ε) = ϕψε ◦π(ε). The sequence ϕψ (ε) converges toward ϕ 0 strongφ
ly in H 2 (ω). Moreover, as ϕψε belongs to Vε , −3 W (Dϕψε ) dx − (ϕψ (ε)). I (ε)(ϕψ (ε)) = ε ω×]−ε,ε[
Let us now compute the limit of the right-hand side as ε goes to zero. First of all, (ϕψ (ε)) → (ϕ 0 ) as ε → 0. To compute the limit of the internal energy, we split the integral on ω into two parts. We first show that the contribution due to the bad set ωλε vanishes as ε goes to zero: −3 ε W (Dϕψε ) dx Cε−2 |ωλε |Dϕψε S ϕε ×]−ε,ε[
Cε−2 |ωλε | C(λε ε)−2 Rλε = CRλε −−→ 0. ε→0
On the good set ω − ωλε , ∂α ϕψε .∂β ϕψε − δαβ = ∂α ϕ.∂β n + (∂β ϕ.∂α n)x3 + o(x3 ) = −2bαβ x3 + o(x3 ). After integration with respect to x3 , we obtain 1 ε −3 ε W (Dϕψ ) dx = Cαβγ δ bαβ − 2Cαβ33 bαβ ψ · n · · · 3 ω−ωλε ω×]−ε,ε[−ωλε + 2Cα3β3 (ψ · ∂α ϕ)(ψ · ∂β ϕ) + C3333 (ψ · n)2 d x + o(1).
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Letting ε go to zero, as |ωλε | → 0, as ϕλε and nλε converge toward ϕ and n strongly in H 2 (ω) and H 1 (ω) respectively, we have W (Dϕψε ) dx lim I (ε)(ϕψ (ε)) = ε −3 1 = 3
ω×]−ε,ε[−ωλε
Cαβγ δ bαβ − 2Cαβ33 bαβ ψ · n ω−ωλε α3β3
+ 2C
(ψ · ∂α ϕ)(ψ · ∂β ϕ) + C3333 (ψ · n)2 d x − (ϕ 0 ).
It remains to compute the infimum of the limit energy over the set of functions ψ ∈ C0∞ (ω − ωφ ; R3 ). For all α and β, ∂αβ ϕ = 0 on ωφ . As ωφ is a Lipschitz open set, there exists a sequence ψn ∈ C0∞ (ω − ωφ ; R3 ) which converges toλ ward λ+2µ δ αβ ∂αβ ϕ strongly in L2 (ω; R3 ). A straightforward computation yields the expected inequality, that is, φ
I0 (ϕ 0 ) lim lim I (ε)(ϕψ (ε)) = JF (ϕ). n→∞ ε
Remark 7. The truncation argument used in the first part of the proof is due to Friesecke, James & Müller (see [13]). This argument would be unnecessary if we could prove the density of VF ∩ C 2 in VF . Unfortunately, it is not clear whether such a density result holds true or not. Acknowledgements. Olivier Pantz is grateful to Tudor S. Ratiu for giving him the opportunity of working one year at the Swiss Federal Institute of Technology (EPFL), where this paper was written. He also wishes to thank Hervé Le Dret for reading the manuscript and for making many corrections and suggestions, and S. Müller for pointing out to him the truncation argument.
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