ONE-DIMENSIONAL VON KA´RMAŃ MODELS FOR ELASTIC RIBBONS LORENZOFREDDI,PETERHORNUNG,MARIAGIOVANNAMORA,ANDROBERTOPARONI Abstract. By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von Kármań plates, passing to the limit as the widthoftheplategoestozero. Theone-dimensionalmodelfoundstartingfromthe“linearized” vonKármańenergycorrespondstothatofalinearlyelasticbeamthatcantwistbutcandeform injustoneplane;whilethemodelfoundfromthevonKármańenergyisanon-linearmodelthat 7 comprises stretching, bendings, and twisting. The “constrained” von Kármań energy, instead, 1 leadstoanewSadowskytypeofmodel. 0 2 Keywords: Elasticribbons,vonKármańplates,Sadowskyfunctional,Gamma-convergence n MathematicsSubjectClassification: 49J45, 49S05,74B20, 74K20,74K99 a J 0 1 1. Introduction ] P Geometrically a ribbon is a body with three length scales: it is a parallelepiped whose length A ℓ is much larger than the width ε, which, in turn, is much larger than the thickness h. That is, . ℓ ≫ ε ≫ h. Since two characteristic dimensions are much smaller than the length, ribbons can h be efficiently modelled as a one-dimensional continuum, see [14]. In the literature, two types of t a one-dimensional models are found: rod models and “Sadowsky type” models. We shall mainly m discuss the latter, since we work within that framework; for rod type models we refer to [7] and [ the references therein. 1 So far, “Sadowsky type” models have been deduced starting from a plate model, that is, from v atwo-dimensionalmodelobtainedfromathree-dimensionalproblembyletting the thicknesshgo 1 to zero. Starting from a Kirchhoff plate model, a one-dimensional model for an isotropic elastic 2 ribbon was proposed by Sadowsky in 1930, [15, 20]. The model was formally justified in 1962 by 7 Wunderlich, [23, 26], by considering the Kirchhoff model for a plate of length ℓ and width ε, and 2 0 by letting ε go to zero. The justification given was only formal, since it was based on an ansatz . on the deformation. Wunderlich’s technique is quite ingenious, but it leads to a singular energy 1 density; we refer to [17] for a rigorous analysis of the so-called Wunderlich energy. A corrected 0 7 Sadowsky type of energy was derived in [9] and generalized in [1, 10]. 1 Athirdapproach,whichpartlyjustifiesthetwoapproachesmentionedabove,istoletthewidth : ε and the thickness h go to zero simultaneously. By appropriately tuning the rates at which ε v i and h convergeto zero, one obtains a hierarchyof one-dimensionalmodels: in [11, 12] severalrod X models have been deduced, and in a forthcoming paper we will show that also “Sadowsky type” r models can be obtained. a Beforedescribingthe contentsofthe presentpaper,we pointoutthatthe literatureonribbons is really blooming in several interesting directions, see, for instance, [2, 3, 4, 6, 8, 18, 21, 22]. OurstartingpointarethevonKármánplatemodels,whereasthepapersquotedabovehavethe Kirchhoffplatemodelasastartingpoint. ThevonKármánmodelforplateshasbeensuccessfully used in [5] to describe the plethora of morphological instabilities observed in a stretched and twisted ribbon. The von Kármán plate equations, formulated more than a hundred years ago [24], have been recentlyjustifiedbyFriesecke,James,andMu¨ller[13]. Theseauthorsconsiderathree-dimensional non-linear hyper-elastic material in a reference configuration Ω = S ×(−h,h) with a stored h ε 2 2 energy density W :R3×3 →[0,+∞) satisfying standard regularity and growthconditions. In [13] the set S is quite general, but in this introduction, in order to be consistent with the previous ε discussion,wetakeS =(−ℓ/2,ℓ/2)×(−ε/2,ε/2). Then,theenergyassociatedwithadeformation ε 1 2 L.FREDDI,P.HORNUNG,M.G.MORA,ANDR.PARONI y :Ω →R3 is given by h Eh(y)= W(∇y)dx. ˆ Ωh By scalingthe elasticenergy per unit volumeEh/h∼hβ, with β a positive realparameter,in [13] a hierarchy of plate models has been derived (by letting h go to zero) by means of Γ-convergence theory. The larger β is, the smaller the energy becomes. Therefore, heuristically, for large β the limit ofthe rescaledenergy should produce the linear plate equation. This is indeed corroborated in [13]. Still in the same paper it is shown that for β = 2α− 2 and for the regimes α > 3, α = 3, and 2 < α < 3 three different Γ-limits are obtained that correspond to von Kármán type of energies. Precisely, denoting by u: S →R2 and v : S →R the in-plane and the out-of-plane displace- ε ε ment fields, respectively, the three asymptotic energies are as follows (see [13, Theorem 2]): (LvK) for α>3 we have the “linearized” von Kármán theory, where u=0 and v minimizes the ε functional 1 ILvK(v):= Q (∇2v)dx, ε 24ˆ 2 Sε with Q :R2×2 →[0,+∞) the positive definite quadratic form of linearized elasticity, see 2 sym Remark 2.5 for a precise definition; (vK) for α = 3 we have the von Kármán theory, where the in-plane and the out-of-plane ε displacements u and v minimize the functional 1 1 1 IvK(u,v):= Q [∇u+(∇u)T +∇v⊗∇v] dx+ Q (∇2v)dx; ε 2ˆ 2 2 24ˆ 2 Sε (cid:16) (cid:17) Sε (CvK) for 2<α<3 we have the “constrained” von Kármán theory, in which the functional ε 1 ICvK(v):= Q (∇2v)dx ε 24ˆ 2 Sε has to be minimized under the non-linear constraint ∇u+(∇u)T +∇v⊗∇v =0, (1.1) or, equivalently, the functional ICvK has to be minimized under the constraint ε det(∇2v)=0 (which is, in turn, necessary and sufficient for the existence of a map u satisfying (1.1)). The existence of minimizers and the characterization of the Euler equations for constrained von Kármán plates have been studied in [16]. By letting h go to zero, the three-dimensional domain Ω = S ×(−h,h) is “squeezed” to h ε 2 2 become S . In this paper, we consider the von Kármán energies and we let ε go to zero, still by ε means of Γ-convergence, to find one-dimensional models for elastic ribbons in the von Kármán regimes. In this way, the two-dimensional domain S = (−ℓ/2,ℓ/2)×(−ε/2,ε/2) is “squeezed” ε to the segment I = (−ℓ/2,ℓ/2), which we parametrize with the coordinate x . In the limit, the 1 in-planedisplacementu:S →R2 generatestwodisplacements: anaxialdisplacementξ :I →R, ε 1 andanorthogonal“in-plane”displacementξ :I →R. Theout-of-planedisplacementv :S →R, 2 ε inturn,generatesan“out-of-plane”displacementw :I →Randthederivativeofvinthedirection orthogonal to the axis leads to a rotation ϑ:I →R. The limit energies that we find in the three regimes are the following: (LvK) the limit of the “linearized” von Kármán energy is 1 JLvK(w,ϑ):= Q (w′′,ϑ′)dx ; 24ˆ 1 1 I (vK) the limit of the von Kármán energy is 1 |w′|2 1 JvK(ξ,w,ϑ):= Q ξ′ + dx + Q (ξ′′)+Q (w′′,ϑ′) dx ; 2ˆ 0 1 2 1 24ˆ 0 2 1 1 I I (cid:16) (cid:17) (cid:0) (cid:1) ONE-DIMENSIONAL VON KA´RMAŃ MODELS FOR ELASTIC RIBBONS 3 (CvK) the limit of the “constrained” von Kármán energy is 1 JCvK(w,ϑ):= Q(w′′,ϑ′)dx . 24ˆ 1 I Here Q ,Q , and Q are energy densities whose precise definition can be found in Section 2; see 1 0 Remark 2.5 for the specialization of these energies in the isotropic case. Wenotethat,sinceQ isquadratic,thefunctional(LvK)correspondstotheenergyofalinearly 1 elastic “three-dimensional beam” in which the section S is unstretchable: the energy is simply ε due to the “out-of-plane” bending of the axis and to the torsion of the cross-section orthogonal to the axis. The limit functional (vK) is non-linear and penalizes stretching and both bendings of the axis, as well as the torsion of the cross-section. The functional (CvK) is sometimes called the energy of a beam with large deflections, see [25]. Despite the appearance, the energy functional (CvK) is very different from that of (LvK). Indeed, in contrastto Q , the energy density Q is not 1 quadratic. It incorporates into its definition the non-linear constraint (1.1) that appears into the two-dimensionalmodel(CvK) . The energydensity Qagreeswiththe correctedSadowskyenergy ε density found in [9] in the isotropic case, and with that found in [10] for the general anisotropic case. To the best of our knowledge, the model (CvK) is new. Weconcludethisintroductionbypointingoutthatthestatementsoftheresultsandtheprecise definitions are given in Section 2, while Section 3 is exclusively devoted to the the proofs of these results. 2. Narrow strips Let ℓ > 0, let I denote the interval (−ℓ/2,ℓ/2), and let S = I ×(−ε/2,ε/2) with ε > 0. For ε u ∈W1,2(S ;R2) and v ∈W2,2(S ) we consider the scaled von Kármán extensional and bending ε ε energies 11 1 1 1 Jext(u,v)= Q Eu+ ∇v⊗∇v dx, Jben(v)= Q (∇2v)dx, ε ε2ˆ 2 2 ε ε24ˆ 2 Sε (cid:16) (cid:17) Sε where Eu = 1(∇u+∇uT) is the symmetric part of the gradient of the in-plane displacement u, 2 while ∇2v denotes the Hessian matrix of the out-of-plane displacement v. The energy density Q :R2×2 →[0,+∞) is assumed to be a positive definite quadratic form. 2 sym To simplify our analysis we rewrite the energies over the domain S := S = I ×(−1/2,1/2). 1 Moreprecisely,weintroducethescaledversionsy :S →R2andw :S →Rofuandv,respectively, by setting y (x ,x ):=u (x ,εx ), y (x ,x ):=εu (x ,εx ), w(x ,x ):=v(x ,εx ), 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 and define the scaled differential operators ∂ y 1(∂ y +∂ y ) Eεy := 1 1 2ε 1 2 2 1 , 21ε(∂2y1+∂1y2) ε12∂2y2 ! 1 ∂2 w 1∂2 w ∇w:= ∂ w, ∂ w , ∇2w := 11 ε 12 , ε (cid:16) 1 ε 2 (cid:17) ε 1ε∂221w ε12∂222w! so that Eεy(x)=Eu(x ,εx ), ∇ w(x)=∇v(x ,εx ), ∇2w(x)=∇2v(x ,εx ). 1 2 ε 1 2 ε 1 2 By performing the changeof variables in the energy integralswe havethat Jext(u,v)=Jext(y,w) ε ε and Jben(v)=Jben(w), where ε ε 1 1 1 Jext(y,w):= Q Eεy+ ∇ w⊗∇ w dx, Jben(w):= Q (∇2w)dx. (2.1) ε 2ˆ 2 2 ε ε ε 24ˆ 2 ε S S (cid:16) (cid:17) 4 L.FREDDI,P.HORNUNG,M.G.MORA,ANDR.PARONI Sincewedonotimposeboundaryconditions,werequirethedisplacementstohavezeroaverage and, for the out-of plane component, also zero average gradient. That is, we shall work in the following spaces: for every open set Ω⊂Rα with α=1,2, we consider W1,2(Ω):= g ∈W1,2(Ω): g(x)dx=0 , h0i ˆ Ω n o W2,2(Ω):= g ∈W2,2(Ω): g(x)dx=0 and ∇g(x)dx=0 , h0i ˆ ˆ Ω Ω n o and similarly we define W1,2(Ω;R2). h0i Ourfirstresultisaboutcompactnessofsequenceswithboundedenergy;thelimitofthein-plane displacements will belong to the space of two-dimensional Bernoulli-Navier functions defined by BN (S;R2):={g ∈W1,2(S;R2) : (Eg) =(Eg) =0} h0i h0i 12 22 ={g ∈W1,2(S;R2) : ∃ξ ∈W1,2(I) and ξ ∈W1,2(I)∩W2,2(I) such that h0i 1 h0i 2 h0i g (x)=ξ (x )−x ξ′(x ), g (x)=ξ (x )}, 1 1 1 2 2 1 2 2 1 where the second characterizationcan be obtained by arguing as in [19, Section 4.1]. Lemma 2.1. Let (w )⊂W2,2(S) be a sequence such that ε h0i supJben(w )<∞. (2.2) ε ε ε Then, up to a subsequence, there exist a vertical displacement w ∈ W2,2(I) and a twist function h0i ϑ∈W1,2(I) such that h0i w ⇀w in W2,2(S), ∇ w ⇀(w′,ϑ) in W1,2(S;R2), (2.3) ε ε ε and w′′ ϑ′ ∇2w ⇀ in L2(S;R2×2) (2.4) ε ε ϑ′ γ sym (cid:18) (cid:19) for a suitable γ ∈L2(S). Moreover, if (y )⊂W1,2(S;R2) is a further sequence such that ε h0i supJext(y ,w )<∞, (2.5) ε ε ε ε then, up to a subsequence, there exists y ∈BN (S;R2) such that h0i y ⇀y in W1,2(S;R2). ε Also, Eεy ⇀E in L2(S;R2×2) ε sym for a suitable E ∈L2(S;R2×2) such that E =∂ y . sym 11 1 1 The rest of this section is devoted to state the Γ-convergence results starting from the simpler caseofthelinearizedtheory(LvK) ,andproceedingintheorderofincreasingdifficultytoconsider ε the standard and the constrained models (vK) and (CvK) , respectively. ε ε 2.1. The linearized von Kármán model. Inordertostateourfirstconvergenceresultweneed to introduce some definitions. Let Q :R×R→[0,+∞) be defined by 1 κ τ Q (κ,τ):=min Q (M) : M = . 1 γ∈R 2 τ γ n (cid:18) (cid:19)o Let JLvK :W2,2(I)×W1,2(I)→R be defined by h0i h0i 1 JLvK(w,ϑ):= Q (w′′,ϑ′)dx . 24ˆ 1 1 I Theorem 2.2. As ε→0, the functionals Jben Γ-converge to the functional JLvK in the following ε sense: ONE-DIMENSIONAL VON KA´RMAŃ MODELS FOR ELASTIC RIBBONS 5 (i) (liminf inequality) for every sequence (w ) ⊂ W2,2(S), w ∈ W2,2(I), and ϑ ∈ W1,2(I) ε h0i h0i h0i such that w ⇀w in W2,2(S), and ∇ w ⇀(w′,ϑ) in W1,2(S;R2), we have that ε ε ε liminfJben(w )≥JLvK(w,ϑ); ε ε ε→0 (ii) (recoverysequence)for everyw ∈W2,2(I) andϑ∈W1,2(I)thereexistsasequence(w )⊂ h0i h0i ε W2,2(S) such that w ⇀w in W2,2(S), ∇ w ⇀(w′,ϑ) in W1,2(S;R2), and h0i ε ε ε limsupJben(w )≤JLvK(w,ϑ). ε ε ε→0 2.2. The von Kármán model. The statement of our second convergence result needs some further definitions. Let Q :R→[0,+∞) be defined by 0 µ z Q (µ):=minQ (µ,z)= min Q (M) : M = 1 . 0 z∈R 1 (z1,z2)∈R2n 2 (cid:18)z1 z2(cid:19)o Let JvK :BN (S;R2)×W2,2(I)×W1,2(I)→R be defined by h0i h0i h0i 1 |w′|2 1 JvK(y,w,ϑ):= Q ∂ y + dx+ Q (w′′,ϑ′)dx . 2ˆ 0 1 1 2 24ˆ 1 1 S I (cid:16) (cid:17) Theorem 2.3. As ε → 0, the functionals JvK := Jext+Jben Γ-converge to the functional JvK ε ε ε in the following sense: (i) (liminf inequality) for every pair of sequences (y ) ⊂ W1,2(S;R2), (w ) ⊂ W2,2(S), y ∈ ε h0i ε h0i BN (S;R2), w ∈ W2,2(I), and ϑ ∈ W1,2(I) such that y ⇀ y in W1,2(S;R2), w ⇀ w h0i h0i h0i ε ε in W2,2(S), and ∇ w ⇀(w′,ϑ) in W1,2(S;R2), we have that ε ε liminfJvK(y ,w )≥JvK(y,w,ϑ); ε ε ε ε→0 (ii) (recoverysequence)for everyy ∈BN (S;R2), w∈W2,2(I)andϑ∈W1,2(I)thereexists h0i h0i h0i a pair of sequences (y )⊂W1,2(S;R2), (w )⊂W2,2(S) such that y ⇀y in W1,2(S;R2), ε h0i ε h0i ε w ⇀w in W2,2(S), ∇ w ⇀(w′,ϑ) in W1,2(S;R2), and ε ε ε limsupJvK(y ,w )≤JvK(y,w,ϑ). ε ε ε ε→0 2.3. The constrained von Kármán model. The constrained von Kármán energy of a displacement v ∈W2,2(S ) such that det∇2v =0 a.e. in S is Jben(v). We observe that the map w, h0i ε ε ε defined over the rescaled domain, belongs to the space W2,2 (S):= w ∈W2,2(S): det∇2w=0 a.e. in S . det,ε h0i ε We set JCvK :W2,2 (S)→R the f(cid:8)unctional JCvK(w)=Jben(w). (cid:9) ε det,ε ε ε Let Q:R×R→[0,+∞) be defined by κ τ Q(κ,τ):=min Q (M)+α+(detM)++α−(detM)− :M = , γ∈R 2 τ γ n (cid:18) (cid:19)o where α+ :=sup{α>0: Q (M)+αdetM ≥0 for every M ∈R2×2} 2 sym and α− :=sup{α>0: Q (M)−αdetM ≥0 for every M ∈R2×2}. 2 sym Let JCvK :W2,2(I)×W1,2(I)→R be defined by h0i h0i 1 JCvK(w,ϑ):= Q(w′′,ϑ′)dx . 24ˆ 1 I Theorem2.4. Asε→0,thefunctionalsJCvK Γ-convergetothefunctionalJCvK inthefollowing ε sense: 6 L.FREDDI,P.HORNUNG,M.G.MORA,ANDR.PARONI (i) (liminf inequality) for every sequence (w ) with w ∈ W2,2 (S), w ∈ W2,2(I), and ϑ ∈ ε ε det,ε h0i W1,2(I) such that w ⇀w in W2,2(S), and ∇ w ⇀(w′,ϑ) in W1,2(S;R2), we have that h0i ε ε ε liminfJCvK(w )≥JCvK(w,ϑ); ε ε ε→0 (ii) (recovery sequence) for every w ∈ W2,2(I) and ϑ ∈ W1,2(I) there exists a sequence (w ) h0i h0i ε with w ∈W2,2 (S) such that w ⇀w in W2,2(S), ∇ w ⇀(w′,ϑ) in W1,2(S;R2), and ε det,ε ε ε ε limsupJCvK(w )≤JCvK(w,ϑ). ε ε ε→0 Remark 2.5. The quadratic energy density Q can be computed from the non-linear energy 2 density W of the material, also mentioned in the introduction, by first computing the quadratic energy density Q , see [13], 3 ∂2W 3 ∂2W Q (F):= (I)(F,F)= (I)F F , F ∈R3×3, 3 ∂F2 ∂F ∂F ij kl ij kl i,j,k,l=1 X and then by minimizing over the third column and row: Q (A):=min{Q (F):F =A α,β =1,2}, A∈R2×2. 2 3 αβ αβ sym If the energy density W is isotropic, the quadratic energy density Q has the following repre- 3 sentation: F +FT Q (F)=2µ|F |2+λ(F ·I)2, F := ∈R3×3, 3 sym sym sym 2 where µ and λ are the so-called Lamé coefficients. A simple computation then leads to 2µλ Q (A)=2µ|A|2+ (A·I)2, A∈R2×2. 2 2µ+λ sym The energy densities Q ,Q , and Q, may be found to have the following representation 1 0 Q (κ,τ)=E κ2+4µτ2, 1 Y where E :=µ2µ+3λ is the Young modulus of the material, Y µ+λ Q (κ)=E κ2, 0 Y and (κ2+τ2)2 1 D if |κ|>|τ|, Q(κ,τ)= κ2 12  4Dτ2 if |κ|≤|τ|,  where D:= 3µ((2λµ++µλ)) is the bending stiffness. 3. Proofs Thissectionisdevotedtoprovethetheoremsstatedintheprevioussection. Foragivenfunction u∈L1(S), we shall denote by hui the integral mean value of u on S, that is, 1 hui:= u(x)dx. ℓ ˆ S We use the same notation to denote the average over I of functions defined on I. Proof of Lemma 2.1. Let (w ) ⊂ W2,2(S) be a sequence of vertical displacements of S satisfying ε h0i (2.2). This bound and the fact that Q is positive definite imply that 2 k∂121wεkL2(S)+kε−1∂122wεkL2(S)+kε−2∂222wεkL2(S) ≤C (3.1) for any ε. Since w (x)dx=0, ∇w (x)dx=0 ˆ ε ˆ ε S S ONE-DIMENSIONAL VON KA´RMAŃ MODELS FOR ELASTIC RIBBONS 7 for every ε > 0, by Poincaré-Wirtinger inequality the sequence (w ) is uniformly bounded in ε W2,2(S). Therefore, there exists w ∈ W2,2(S) such that wε ⇀ w weakly in W2,2(S), up to a h0i subsequence. By the previous bound, ∇(ε−1∂ w ) is a bounded sequence in L2(S;R2) and, by Poincaré- 2 ε Wirtinger inequality, also (ε−1∂ w ) is bounded in L2(S). It follows that w is independent of x 2 ε 2 and there exixts ϑ ∈ W1,2(S) such that ε−1∂ w ⇀ ϑ weakly in W1,2(S), up to a subsequence. h0i 2 ε Moreover,also ϑ is independent of x . 2 By (3.1), up to subsequences, we have that ∇2w converges to a matrix field A weakly in ε ε L2(S;R2×2). Byusingtheconvergencesestablishedabove,itfollowsthatA =w′′ andA =ϑ′. sym 11 12 The entry A , that cannot be identified in terms of w and ϑ, is denoted by γ in the statement. 22 This proves (2.4). We now prove the second part of the statement. The bound (2.5) implies that 1 Eεy + ∇ w ⊗∇ w ≤C (3.2) ε ε ε ε ε 2 L2 (cid:13) (cid:13) for any ε. Since (∇2εwε) is bound(cid:13)(cid:13)ed in L2, we have (cid:13)(cid:13) k∇εwε⊗∇εwεkL2 ≤Ck|∇εwε|2kL2 =Ck∇εwεk2L4 ≤C(k∂1wεk2L4 +kε−1∂2wεk2L4) ≤C(k∇∂ w k2 +k∇ε−1∂ w k2 )≤Ck∇2w k2 ≤C 1 ε L2 2 ε L2 ε ε L2 for any ε, and the third to last inequality follows by the imbedding W1,2(S) ⊂ Lq ∀q ∈ [2,+∞) and Poincaré-Wirtinger inequality. Together with (3.2), this implies that the sequence (Eεy ) is ε bounded in L2. By the definition of Eε and Korn-Poincaréinequality we have that kyεkW1,2 ≤CkEyεkL2 ≤CkEεyεkL2 ≤C. (3.3) Hence, up to subsequences, there exist E ∈L2(S;R2×2) and y ∈W1,2(S;R2) such that sym h0i Eεy ⇀E in L2(S;R2×2), ε sym y ⇀y in W1,2(S;R2). ε By the definition of Eε and (3.3) we have that (Ey ) ⇀0=(Ey) , (Ey ) ⇀0=(Ey) ; ε 12 12 ε 22 22 hence, y ∈ BN (S;R2). Finally, the observation that (Eεy ) = ∂ (y ) ⇀ ∂ y in L2(S) h0i ε 11 1 ε 1 1 1 concludes the proof. (cid:3) Proof of Theorem 2.2–(i). Let (w ) ⊂ W2,2(S) be such that w ⇀ w in W2,2(S), and ∇ w ⇀ ε h0i ε ε ε (w′,ϑ) in W1,2(S;R2), for some w ∈W2,2(I) andϑ∈W1,2(I). Without loss ofgenerality,we can h0i h0i assume that liminf Jben(w )<+∞. By Lemma 2.1 we infer that, up to subsequences, ε→0 ε ε w′′ ϑ′ ∇2w ⇀ =:M in L2(S;R2×2) ε ε ϑ′ γ γ sym (cid:18) (cid:19) for some γ ∈L2(S). By weak lower semicontinuity and the definition of Q we have 1 1 liminfJben(w )=liminf Q (∇2w )dx ε→0 ε ε ε→0 24ˆS 2 ε ε 1 1 ≥ Q (M )dx≥ Q (w′′,ϑ′)dx =JLvK(w,ϑ). 24ˆ 2 γ 24ˆ 1 1 S I (cid:3) Proof of Theorem 2.2–(ii). Let w∈W2,2(I) and ϑ∈W1,2(I). We set h0i h0i w′′ ϑ′ M := , γ ϑ′ γ (cid:18) (cid:19) where γ ∈L2(I) is such that Q (w′′,ϑ′)=Q (M ). 1 2 γ 8 L.FREDDI,P.HORNUNG,M.G.MORA,ANDR.PARONI The fact that γ belongs to L2(I) follows immediately by choosing M =w′′e ⊗e +ϑ′(e ⊗e + 0 1 1 1 2 e ⊗e ) as a competitor in the definition of Q and by using the positive definiteness of Q . 2 1 1 2 Letϑ ∈C∞(I) be suchthat ϑ (x )dx =0, ϑ →ϑin W1,2(I), andεϑ′′ →0 inL2(I). Let ε I ε 1 1 ε ε γ ∈C∞(I) be such that γ →γ´, εγ′ →0 and ε2γ′′ →0 in L2(I). Let ε ε ε ε ε2 w (x)=w(x )+εx ϑ (x )+ x2γ (x )−hx2γ i−x hx2γ′i . (3.4) ε 1 2 ε 1 2 2 ε 1 2 ε 1 2 ε It turns out that w ∈ W2,2(S) and, by the con(cid:0)vergences above, we have w (cid:1)→ w in W2,2(S), ε h0i ε ∇ w → (w′,ϑ) in W1,2(S;R2), and ∇2w → M in L2(S). Moreover, by strong continuity we ε ε ε ε γ have 1 1 limJben(w )= lim Q (∇2w )dx= Q (w′′,ϑ′)dx =JLvK(w,ϑ). ε→0 ε ε ε→024ˆS 2 ε ε 24Î 1 1 (cid:3) Proof of Theorem 2.3–(i). Let (y ) ⊂ W1,2(S;R2), (w ) ⊂ W2,2(S) be such that y ⇀ y in ε h0i ε h0i ε W1,2(S;R2), w ⇀w inW2,2(S),and∇ w ⇀(w′,ϑ)inW1,2(S;R2),forsomey ∈BN (S;R2), ε ε ε h0i w ∈ W2,2(I) and ϑ ∈ W1,2(I). As usual, we can assume that liminf JvK(y ,w ) < +∞ and h0i h0i ε→0 ε ε ε by Lemma 2.1 we deduce that, up to subsequences, w′′ ϑ′ Eεy ⇀E and ∇2w ⇀ =:M in L2(S;R2×2) ε ε ε ϑ′ γ γ sym (cid:18) (cid:19) forsomeE ∈L2(S;R2×2)withE =∂ y ,andγ ∈L2(S). Moreover,bythe convergencesabove, sym 11 1 1 1 1 Eεy + ∇ w ⊗∇ w ⇀E+ (w′,ϑ)⊗(w′,ϑ) in L2(S;R2×2). ε 2 ε ε ε ε 2 sym Then, by lower semicontinuity, we have 1 1 liminfJvK(y ,w ) ≥ liminf J (y ,w )+liminf Jlin(w ) ε→0 ε ε ε ε→0 2 ε ε ε ε→0 24 ε ε 1 1 1 = liminf Q Eεy + ∇ w ⊗∇ w dx+liminf Q (∇2w )dx ε→0 2ˆS 2 ε 2 ε ε ε ε ε→0 24ˆS 2 ε ε (cid:16) (cid:17) 1 1 1 ≥ Q E+ (w′,ϑ)⊗(w′,ϑ) dx+ Q (M )dx 2ˆ 2 2 24ˆ 2 γ S S (cid:16) (cid:17) 1 1 1 ≥ Q ∂ y + |w′|2 dx+ Q (w′′,ϑ′)dx 2ˆ 0 1 1 2 24ˆ 1 1 S I (cid:16) (cid:17) = JvK(y,w,ϑ). (cid:3) Proof of Theorem 2.3–(ii). Let y ∈ BN (S;R2), w ∈ W2,2(I), and ϑ ∈ W1,2(I). As before, h0i h0i h0i there exists γ ∈L2(I) such that the matrix w′′ ϑ′ M := γ ϑ′ γ (cid:18) (cid:19) satisfies Q (w′′,ϑ′)=Q (M ). 1 2 γ There exist ξ ∈ W1,2(I) and ξ ∈ W1,2(I)∩W2,2(I) such that y (x) = ξ (x )−x ξ′(x ) and 1 h0i 2 h0i 1 1 1 2 2 1 y (x)=ξ (x ). Moreover,there exists z ∈L2(S;R2) such that the matrix 2 2 1 ∂ y + 1|w′|2 z ξ′(x )−x ξ′′(x )+ 1|w′(x )|2 z M := 1 1 2 1 = 1 1 2 2 1 2 1 1 z z z z z 1 2 1 2 (cid:18) (cid:19) (cid:18) (cid:19) satisfies 1 Q ∂ y + |w′|2 =Q (M ). 0 1 1 2 z 2 (cid:16) (cid:17) ONE-DIMENSIONAL VON KA´RMAŃ MODELS FOR ELASTIC RIBBONS 9 Itis easilyseenthatz andz dependlinearlyon∂ y +1|w′|2. Since ∂ y (x ,x )+1|w′(x )|2 = 1 2 1 1 2 1 1 1 2 2 1 ξ′(x )−x ξ′′(x )+ 1|w′(x )|2, there exist ζ ∈L2(I) and η ∈L2(I) such that 1 1 2 2 1 2 1 α α z (x ,x )=ζ (x )+x η (x ), α=1,2. α 1 2 α 1 2 α 1 Let w be as in the proof of Theorem 2.2–(ii) (see (3.4)), and let ζε, ηε ∈C∞(I) be such that ε α α ζε →ζ and ηε →η in L2(I) and εζε′ →0 and εηε′ →0 in L2(I). Let us define α α α α α α (y ) (x ,x ) := ξ (x )−x ξ′(x )+ε x2ηε(x )−hx2ηεi , ε 1 1 2 1 1 2 2 1 2 1 1 2 1 x1 x1 (y ) (x ,x ) := ξ (x )+ε 2ζε((cid:0)s)−w′(s)ϑ(s) ds−(cid:1) h 2ζε(s)−w′(s)ϑ(s) dsi ε 2 1 2 2 1 ˆ 1 ˆ 1 0 0 ε2 (cid:16) (cid:0) ε2 (cid:1) (cid:0) (cid:1) (cid:17) + x 2ζε(x )−ϑ2(x ) + x2ηε(x )−hx2ηεi . 2 2 2 1 1 2 2 2 1 2 2 Then it is easy to check that(cid:0) (cid:1) (cid:0) (cid:1) 1 Eεy + ∇ w ⊗∇ w →M in L2(S;R2×2). ε 2 ε ε ε ε z sym Thus, by strong continuity, 1 1 limJvK(y ,w ) = lim Jext(y ,w )+ Jben(w ) ε→0 ε ε ε ε→0 2 ε ε ε 24 ε ε (cid:16)1 1 (cid:17) 1 = lim Q Eεy + ∇ w ⊗∇ w dx+ Q (∇2w )dx ε→0 2ˆS 2 ε 2 ε ε ε ε 24ˆS 2 ε ε (cid:16) (cid:16) (cid:17) (cid:17) 1 1 = Q (M )dx+ Q (M )dx 2ˆ 2 z 24ˆ 2 γ S S 1 1 1 = Q ∂ y + |w′|2 dx+ Q (w′′,ϑ′)dx 2ˆ 0 1 1 2 24ˆ 1 1 S I (cid:16) (cid:17) = JvK(y,w,ϑ). (cid:3) The proof of the Γ-convergence theorem 2.4 is based on a relaxation result for a quadratic integralfunctionalwithaconstraintonthedeterminant,thathasbeenprovedin[10,Proposition9] and recalled here for reader’s convenience. LetB be a bounded open subsetof Rn. Let Q:B×R2×2 →[0,+∞)be measurable inthe first sym variable and quadratic in the second. Define the functional F :L2 B;R2×2 →[0,+∞] sym by (cid:0) (cid:1) Q(x,M(x))dx if detM =0 a.e. in B, F(M):= ˆB  +∞ otherwise. Theorem 3.1 ([10]). The weak-L2 lower semicontinuous envelope of F is the functional F :L2 B;R2×2 →[0,+∞) sym given by (cid:0) (cid:1) F(M)= Q(x,M(x))+α+(x)(detM(x))++α−(x)(detM(x))− dx, ˆ B (cid:0) (cid:1) where for every x∈B α+(x):=sup{α>0: Q(x,M)+αdetM ≥0 for every M ∈R2×2} sym and α−(x):=sup{α>0: Q(x,M)−αdetM ≥0 for every M ∈R2×2}. sym 10 L.FREDDI,P.HORNUNG,M.G.MORA,ANDR.PARONI Proof of Theorem 2.4–(i). Let (w ) be such that w ∈ W2,2 (S), w ⇀ w in W2,2(S), and ε ε det,ε ε ∇ w ⇀ (w′,ϑ) in W1,2(S;R2), for some w ∈ W2,2(I) and ϑ ∈ W1,2(I). Under the assump- ε ε h0i h0i tion that liminf JvK(w )<+∞, by Lemma 2.1 we deduce that, up to subsequences, ε→0 ε ε w′′ ϑ′ ∇2w ⇀ in L2(S;R2×2) ε ε ϑ′ γ sym (cid:18) (cid:19) for some γ ∈L2(S). Since det∇2w =0, an application of Theorem 3.1 with Q(x,M):=Q (M) ε ε 2 and B =S shows that 1 1 w′′ ϑ′ 1 liminfJCvK(w )=liminf F(∇2w )≥ F ≥ Q(w′′,ϑ′)dx . ε→0 ε ε ε→0 24 ε ε 24 (cid:18)ϑ′ γ(cid:19) 24Î 1 Therefore, we conclude that liminfJCvK(w )≥J(w,ϑ). ε ε ε→0 (cid:3) Proof of Theorem 2.4–(ii). Let w∈W2,2(I) and ϑ∈W1,2(I). We set h0i h0i w′′ ϑ′ M := , ϑ′ γ (cid:18) (cid:19) where γ ∈L2(I) is such that Q(w′′,ϑ′)=Q (M)+α+(detM)++α−(detM)−. 2 As before, the fact that γ belongs to L2(I) follows immediately by choosing M = w′′e ⊗e + 0 1 1 ϑ′(e ⊗e +e ⊗e ) as a competitor in the definition of Q and by using the positive definiteness 1 2 2 1 of Q . 2 By Theorem 3.1 with Q(x,M) := Q (M) and B = I, there exist Mj ∈ L2(I;R2×2) with 2 sym detMj =0 and such that Mj ⇀M weakly in L2(I;R2×2) and F(Mj)→F(M), as j →∞. We sym canalsoassume that Mj ∈C∞(I¯;R2×2). The proofof this fact relies ona constructiondescribed sym in [9, Theorem 2.2–(ii)]. We give here full details for convenience of the reader. Suppose that (M ) be a sequence of matrices with the same properties of (Mj) apart from the regularity, and n denote by λ ∈ L2(I) the trace of M . Since M is symmetric with detM = 0, there exists n n n n β =β (x )∈(−π/2,π/2] such that n n 1 cosβ −sinβ λ 0 cosβ sinβ M = n n n n n , n sinβ cosβ 0 0 −sinβ cosβ n n n n (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) and β is uniquely determined if λ 6= 0. When λ (x ) = 0, we set β (x ) = 0. We may n n n 1 n 1 assume without loss of generality that λ ∈ L∞(I), possibly after truncating λ in modulus n n by n, while M still enjoys the same properties as before. We can find λ ∈ C∞(I¯) and n n,k β ∈ C∞(I¯;(−π/2,π/2)) such that, as k → ∞, λ → λ and β → β in Lp(I) for every n,k n,k n n,k n p<+∞. Set cosβ −sinβ λ 0 cosβ sinβ M := n,k n,k n,k n,k n,k . n,k sinβ cosβ 0 0 −sinβ cosβ n,k n,k n,k n,k (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) Then, detM =0 for every n,k and M →M in L2(I;R2×2), as k →∞. n,k n,k n sym Thus, by a diagonal argument, we may assume that there exist λj ∈ C∞(I¯) and βj ∈ C∞(I¯) such that |βj|<π/2 on I¯, and with cosβj −sinβj λj 0 cosβj sinβj Mj := sinβj cosβj 0 0 −sinβj cosβj (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) cos2βj sinβjcosβj = λj sinβjcosβj sin2βj (cid:18) (cid:19) wehavethatMj ∈C∞(I¯;R2×2), detMj =0 foreveryj, Mj ⇀M inL2(I;R2×2), andF(M )→ sym sym j F(M), as j →∞.

One-dimensional von K\'arm\'an models for elastic ribbons

Similar One-dimensional von K\'arm\'an models for elastic ribbons