Françoise Krasucki

Giuseppe Geymonat

Philippe G. Ciarlet

Ciarlet, Philippe G.

Geymonat, Giuseppe

Krasucki, Françoise

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 405–409Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical Problems in MechanicsNonlinear Donati compatibility conditions for the nonlinearKirchhoff–von Kármán–Love plate theoryConditions de compatibilité non linéaires de Donati pour la théorienon linéaire des plaques de Kirchhoff–von Kármán–LovePhilippe G. Ciarlet a, Giuseppe Geymonat b, Françoise Krasucki ca Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kongb Laboratoire de mécanique des solides, UMR 7649-0176, École polytechnique, 91128 Palaiseau cedex, Francec I3M, UMR–CNRS 5149, université de Montpellier-2, place Eugène-Bataillon, 34095 Montpellier cedex 5, Francea r t i c l e i n f o a b s t r a c tArticle history:Received and accepted 24 May 2013Available online 12 June 2013Presented by Philippe G. CiarletLet ω be a simply-connected domain in R2 and let (Eαβ) and (Fαβ) be two symmetric 2×2matrix fields with components in L2(ω). In this Note, we identify nonlinear compatibilityconditions “of Donati type” that the components Eαβ and Fαβ must satisfy in order thatthere exists a vector field (η1, η2, w) ∈ H10(ω) × H10(ω) × H20(ω) such that:12(∂αηβ + ∂βηα + ∂α w∂β w) = Eαβ and ∂αβ w = Fαβ in ω.The left-hand sides of these relations are the components of tensors found in theKirchhoff–von Kármán–Love theory of nonlinearly elastic plates.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoit ω un domaine simplement connexe de R2 et soient (Eαβ) et (Fαβ) deux champsde matrices 2 × 2 symétriques dont les composantes sont dans L2(ω). Dans cette Note,on identifie et justifie des conditions non linéaires de compatibilité « de type Donati »que doivent satisfaire les composantes Eαβ et Fαβ afin qu’il existe un champ de vecteurs(η1, η2, w) ∈ H10(ω) × H10(ω) × H20(ω) tel que :12(∂αηβ + ∂βηα + ∂α w∂β w) = Eαβ et ∂αβ w = Fαβ dans ω.Les membres de gauche de ces relations sont les composantes de tenseurs trouvés dans lathéorie de Kirchhoff–von Kármán–Love des plaques non linéairement élastiques.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.E-mail addresses: mapgc@cityu.edu.hk (P.G. Ciarlet), giuseppe.geymonat@lms.polytechnique.fr (G. Geymonat), krasucki@math.univ-montp2.fr(F. Krasucki).1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2013.05.012406 P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 405–4091. PreliminariesGreek indices vary in {1,2} and the convention summation with respect to repeated indices is used. A domain in R2is a bounded, open, and connected subset ω of R2 with a Lipschitz-continuous boundary ∂ω, the set ω being locally onthe same side of ∂ω. Partial derivatives of the first, second, and third, order of functions of y = (yα) ∈ ω are denoted∂α := ∂/∂ yα , ∂αβ := ∂2/∂ yα∂ yβ , and ∂αβσ := ∂3/∂ yα∂ yβ∂ yσ ; the same notations are used for partial derivatives in thesense of distributions.Vector fields and matrix fields, and spaces of vector fields, defined over ω are denoted by boldface letters. The usualSobolev spaces over ω are denoted Hm(ω), m ∈ Z, and Hm0 (ω), m  1; the norm in Hm(ω), m ∈ Z, is denoted ‖ · ‖m,ω; inparticular then, ‖ ·‖0,ω is the norm of H0(ω) = L2(ω). The notation L2(ω) designates the space of all 2×2 symmetric matrixfields with components in L2(ω). If S = (Sαβ) is a 2 × 2 matrix field with smooth enough components defined over ω, itsdivergence div S is the vector field defined by (div S)α = ∂β Sαβ .If X and Y are two (real) vector spaces and A is a linear operator from X to Y ,Im A := {y ∈ Y ; y = Ax for at least one x ∈ X} and Ker A := {x ∈ X; Ax = 0}.The notation X ′ 〈·,·〉X designates the duality between a normed vector space X and its dual X ′ .In the classical Kirchhoff–von Kármán–Love theory of nonlinearly elastic plates (see, e.g., Chapters 4 and 5 in [2]), theunknown displacement field of the middle surface ω of the plate minimizes an energy whose integrand contains a positive-definite quadratic function of the change of metric and change of curvature tensors, respectively defined byEαβ := 12(∂αηβ + ∂βηα + ∂α w∂β w) and Fαβ := ∂αβ w, (1)for an arbitrary displacement field (η1, η2, w) ∈ H10(ω) × H10(ω) × H20(ω) (we consider here plates that are clamped overtheir entire lateral face).In the intrinsic approach to the same theory, the matrix fields (Eαβ) ∈ L2(ω) and (Fαβ) ∈ L2(ω) are considered as the soleunknowns. There thus arises the question as to whether there exist suitable compatibility conditions that the components Eαβand Fαβ of these matrix fields should satisfy in order that there exists a vector field (η1, η2, w) ∈ H10(ω) × H10(ω) × H20(ω)satisfying (1). As shown in [6], if the domain ω is simply-connected, the nonlinear Saint-Venant compatibility conditions:∂στ Eαβ + ∂αβ Eστ − ∂ασ Eβτ − ∂βτ Eασ + Fαβ Fστ − Fασ Fβτ = 0 in H−2(ω),∂σ Fαβ − ∂β Fασ = 0 in H−1(ω),constitute one possible answer to this question. The objective of this Note is to give (cf. Theorem 4.2) a different answerto the same question, this time in the form of variational equations, which as such constitute examples of nonlinear Do-nati compatibility conditions (a general presentation of Saint-Venant and Donati compatibility conditions as they arise inthree-dimensional linearized elasticity is found in Chapter 6 in [3]).Complete proofs and an application to the intrinsic nonlinear plate theory will be found in [5].2. An existence theorem for an Airy-functionThe following result is a “weak” version (already used in [6]) of a classical result for smooth functions. Its proof is basedon the two-dimensional version of the weak Poincaré lemma due to [4] and then given a substantially simpler proof in [10];cf. also Theorem 6.17-4 in [3].Theorem 2.1. Let ω be a simply-connected domain in R2 , and let there be given a matrix field (Fαβ) ∈ L2(ω) that satisfies∂σ Fαβ − ∂β Fασ = 0 in H−1(ω).Then there exists a function ϕ ∈ H2(ω), unique up to the addition of a polynomial of degree  1, such that∂αβϕ = Fαβ in L2(ω).Theorem 2.1 can be immediately recast as an existence result of an ad hoc Airy function (denoted ϕ in the next theorem)under low regularity assumptions. As such, it complements Theorem 2 of [7], where the existence of an Airy function wasestablished, again in the space H2(ω), but for non-simply-connected domains, under the assumption that the tensor fieldnoted S in the next theorem satisfies in addition the usual global equilibrium equations.Theorem 2.2. Let ω be a simply-connected domain in R2 , and let there be given a matrix field S = (Sαβ) ∈ L2(ω) that satisfiesdiv S = 0 in H−1(ω).P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 405–409 407Then there exists a function ϕ ∈ H2(ω), unique up to the addition of a polynomial of degree  1, such that∂11ϕ = S22, ∂12ϕ = −S12, ∂22ϕ = S11 in L2(ω). (2)A function ϕ satisfying the relations (2) is called an Airy function associated with the matrix field S .3. Linear Donati compatibility conditions for linearly elastic platesFor convenience, we consider separately the existence of the “horizontal” components ηα , and that of the “vertical”component w of the unknown vector field.Theorem 3.1. Let ω be a domain in R2 and let there be given a matrix field (eαβ) ∈ L2(ω) that satisfies∫ωeαβ sαβ dy = 0 for all (sαβ) ∈ L2(ω) such that ∂β sαβ = 0 in H−1(ω). (3)Then there exists a vector field (ηα) ∈ H10(ω) × H10(ω) such that12(∂αηβ + ∂βηα) = eαβ in L2(ω),and such a vector field (ηα) is uniquely determined.Proof. See, e.g., [1], or [8] and [9]. Theorem 3.2. Let ω be a domain in R2 and let there be given a matrix field (Fαβ) ∈ L2(ω) that satisfies∫ωFαβ Tαβ dy = 0 for all (Tαβ) ∈ L2 such that ∂αβ Tαβ = 0 in H−2(ω).Then there exists a function w ∈ H20(ω) such that∂αβ w = Fαβ in L2(ω), (4)and this function is uniquely determined.Sketch of proof. Let the continuous linear operator H : H20(ω) → L2(ω) be defined byH w =(∂11 w ∂12 w∂21 w ∂22 w)∈ L2(ω) for each w ∈ H20(ω).Then one shows that Im H is a closed subspace of L2(ω), and that the dual operator of H is div div : L2(ω) → H−2(ω). Theconclusion then follows from Banach closed range theorem. Relations (3) and (4) constitute the linear Donati compatibility conditions corresponding to the Kirchhoff–Love theory oflinearly elastic plates. Note that they hold regardless of whether the domain ω is simply-connected.4. Nonlinear Donati compatibility conditions for nonlinearly elastic platesThe Green’s formula (5) established in the next theorem is crucial to our subsequent analysis.Theorem 4.1. For all functions w ∈ H20(ω) and ϕ ∈ H2(ω),∫ω(∂11 w∂22 w − ∂12 w∂12 w)ϕ dy =∫ω{−12(∂1 w)2∂22ϕ − 12(∂2 w)2∂11ϕ + ∂1 w∂2 w∂12ϕ}dy. (5)Sketch of proof. Both sides of (5) being continuous functions of (w,ϕ) ∈ H20(ω) × H2(ω), it is enough to establish (5) forall (w,ϕ) ∈D(ω) × H2(ω). To this end, one uses the integration by parts formulas in Sobolev spaces. The next theorem constitutes the main result of this Note.408 P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 405–409Theorem 4.2. Let ω be a simply-connected domain in R2 . Given a matrix field S ∈ L2(ω) that satisfies div S = 0 in H−1(ω), thereexists a unique function ϕ ∈ H2(ω) such that (cf. Theorem 2.2):∂11ϕ = S22, ∂12ϕ = −S12, ∂22ϕ = S11 in L2(ω) and∫ωϕ dy =∫ω∂αϕ dy = 0.LetΦ : {S ∈ L2(ω); div S = 0 in H−1(ω)} →{ψ ∈ H2(ω);∫ωψ dy =∫ω∂αψ dy = 0}denote the mapping defined in this fashion, i.e., by Φ(S) := ϕ .Let there be given two matrix fields (Eαβ) ∈ L2(ω) and F = (Fαβ) ∈ L2(ω) that satisfy∫ωFαβ Tαβ dy = 0 for all T = (Tαβ) ∈ L2(ω) such that div div T = 0 in H−2(ω), (6)∫ω{Eαβ Sαβ + (det F )Φ(S)}dy = 0 for all S = (Sαβ) ∈ L2(ω) such that div S = 0 in H−1(ω). (7)Then there exists a vector field (η1, η2, w) ∈ H10(ω) × H10(ω) × H20(ω) such that12(∂αηβ + ∂βηα + ∂α w∂β w) = Eαβ in L2(ω),∂αβ w = Fαβ in L2(ω),and such a vector field (η1, η2, w) is uniquely determined.Proof. Relation (6) shows that there exists a uniquely determined function w ∈ H20(ω) such that (cf. Theorem 3.2)Fαβ = ∂αβ w in L2(ω). (8)Given any matrix field S = (Sαβ) ∈ L2(ω) such that div S = 0 in H−1(ω), there exists a uniquely determined functionϕ ∈ H2(ω) such that (cf. Theorem 2.2):S11 = ∂22ϕ, S12 = −∂12ϕ, S22 = ∂11ϕ in L2(ω).Therefore, for any such matrix field S ,∫ωEαβ Sαβ dy =∫ω{E11∂22ϕ + E22∂11ϕ − 2E12∂12ϕ}dy,and (cf. (8))∫ω(det F )Φ(S)dy =∫ω(F11 F22 − (F12)2)ϕ dy=∫ω{(∂11 w∂22 w − ∂12 w∂12 w)ϕ dy}.The left-hand side of relation (6) can thus be rewritten as (cf. (5)):∫ω{Eαβ Sαβ + (det F )Φ(S)}dy=∫ω{(E11 − 12(∂1 w)2)∂22ϕ − 2(E12 − 12∂1 w∂2 w)∂12ϕ +(E22 − 12(∂2 w)2)∂11ϕ}dy=∫ {(E11 − 12(∂1 w)2)S11 + 2(E22 − 12∂1 w∂2 w)S12 +(E22 − 12(∂2 w)2)S22}dy.ωP.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 405–409 409Since this last relation holds for all S = (Sαβ) ∈ L2(ω) such that div S = 0 in H−1(ω), there exists a uniquely determinedvector field (ηα) ∈ H10(ω) × H10(ω) such that (cf. Theorem 3.1):Eαβ − 12∂α w∂β w = 12(∂αηβ + ∂βηα) in L2(ω).This completes the proof. Relations (6) and (7) constitute the nonlinear Donati compatibility conditions corresponding to the Kirchhoff–von Kármán–Love theory of nonlinearly elastic plates. Note that, when properly extended, they can also cover the case where the domain ωis not simply-connected; cf. [5].Finally, note that, as expected, the linearization of the nonlinear Donati compatibility conditions (7) reduce to the linear ones(cf. (3)).AcknowledgementThis work was partially supported by a grant from the Research Grants Council of the Hong Kong Special AdministrativeRegion, China [Project No. 9041738-CityU 100612].References[1] C. Amrouche, P.G. Ciarlet, L. Gratie, S. Kesavan, On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl. 86 (2006)116–132.[2] P.G. Ciarlet, Mathematical Elasticity, Volume II: Theory of Plates, North-Holland, Amsterdam, 1997.[3] P.G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, 2013.[4] P.G. Ciarlet, P. Ciarlet Jr., Another approach to linearized elasticity and a new proof of Korn’s inequality, Math. Models Methods Appl. Sci. 15 (2005)259–271.[5] P.G. Ciarlet, G. Geymonat, F. Krasucki, Nonlinear Donati compatibility conditions and the intrinsic approach for nonlinearly elastic plates, in preparation.[6] P.G. Ciarlet, S. Mardare, Nonlinear Saint-Venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates, Math. Models MethodsAppl. Sci. (2013), http://dx.doi.org/10.1142/S0218202513500322, in press.[7] G. Geymonat, F. Krasucki, On the existence of the Airy function in Lipschitz domains. Application to the traces of H2, C. R. Acad. Sci. Paris, Ser. I 330(2000) 355–360.[8] G. Geymonat, F. Krasucki, Some remarks on the compatibility conditions in elasticity, Accad. Naz. Sci. XL 123 (2005) 175–182.[9] G. Geymonat, F. Krasucki, Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains, Commun. Pure Appl.Anal. 8 (2009) 295–309.[10] S. Kesavan, On Poincaré’s and J.-L. Lions’ lemmas, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 27–30.

Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff–von Kármán–Love plate theory

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Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff–von Kármán–Love plate theory

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