Alain Damlamian

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 350 (2012) 861–865Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical Problems in MechanicsSome unilateral Korn inequalities with application to a contact problemwith inclusionsQuelques inégalités de Korn unilatérales et leur application à un problème de contactavec inclusionsAlain DamlamianUniversité Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-94010, Créteil, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 12 July 2012Accepted 4 September 2012Available online 12 October 2012Presented by Philippe G. CiarletIn the first part of this note, some unilateral inequalities of the Korn type are established.These inequalities seem to be new.In the second part, these inequalities are used in an essential way to prove the existenceof a solution (which is not necessarily unique) for a unilateral contact problem involving amatrix material with inclusions of various shapes (the conditions depend on the shape ofeach inclusion in a remarkable way).© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans une première partie sont démontrées quelques inégalités de Korn unilatérales quisemblent nouvelles.Ces inégalités sont alors utilisées de façon essentielle dans la démontration de l’existenced’une solution pour un problème de contact unilatéral dans une matrice avec des inclusionsde diverses formes (les conditions dépendent de façon remarquable de la forme de chaqueinclusion).© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeDans la première partie de cette note, on démontre quelques inégalités de Korn unilatérales qui semblent nouvelles (Pro-positions 2.8 et 2.9), ne faisant intervenir sur le bord de l’ouvert considéré que la partie positive de la composante normalede la déformation (et, uniquement pour des inclusions de forme sphérique ou cylindrique, la composante tangentielle de ladéformation).Ces inégalités sont utilisées dans la seconde partie pour traiter un problème de contact unilatéral dans une matrice avecdes inclusions de diverses formes. Il s’agit alors d’une inequation variationnelle de type Signorini sans frottement ou avecfrottement de type Tresca (Problem P ′). Celle-ci s’exprime de manière équivalente sous forme de la minimisation d’unefonctionnelle convexe sur un convexe fermé (Problem P).E-mail address: damlamian@u-pec.fr.1631-073X/$ – see front matter © 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2012.09.006862 A. Damlamian / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 861–865Le résultat d’existence est donné dans la Proposition 3.2. Les hypothèses spécifiques portent sur le second membreuniquement dans les inclusions de forme sphérique ou cylindrique (hypothèses (H j)). L’unicité n’est pas nécessairementgarantie dans les inclusions à cause de l’absence de stricte convexité de la fonctionnelle qui est minimisée.1. IntroductionContact problems in the framework of three-dimensional linear elasticity have been studied by many authors. The firsttreatment of the subject appeared in the papers of Fichera [3–5]. We also refer to the book of G. Duvaut and J.-L. Lions[2] and references therein. More recently J. Nečas and his co-workers have studied the numerical approximation of suchproblems [7]. See the book of I. Hlaváček, J. Haslinger, J. Nečas and J. Lovíšek [8] for references on the subject.In this note, we consider a Signorini problem with possible Tresca friction for inclusions surrounded by a matrix. Theproblem here differs from the previous references insofar as the contact conditions are posed on the full boundaries of eachinclusion. The conditions for the existence of the solution(s) are related to those of [3–5], but explicitly involve the momentsof the right-hand side with respect to spherical and cylindrical inclusions only (hypotheses (H j)).The result is based on a series of unilateral Korn inequalities which seem to be new and which are adapted to thevarious shapes of inclusions.1.1. Notations• The normal component of a vector field v on the boundary of a domain is denoted vν , while the tangential componentv − vνν is denoted vτ (here ν is the outward unit normal to the boundary of the domain);• the strain tensor of a vector field v is denoted by e(v);• the kernel of the strain operator e in a connected domain is the six-dimensional space of rigid motions denoted R:R .= {x → va,b(x) = a ∧ x + b; a and b ∈ R3}. (1)2. Korn inequalities2.1. Classical Korn inequalitiesDefinition 2.1. A bounded domain O is a Korn domain whenever it satisfies the second Korn inequality, i.e., if there exists aconstant C such that∀v ∈ H1(O ), |v|H1(O )  C(|v|L2(O ) + ∣∣e(v)∣∣L2(O )). (2)It is known (see Gobert [6]) that this holds for every connected bounded domain with Lipschitz boundary, but is alsotrue for more general domains since a finite union of Korn domains is a Korn domain.Definition 2.2. A bounded and connected domain O is a Korn–Wirtinger domain whenever it satisfies the following Korn-type inequality (similar to the Poincaré–Wirtinger inequality for scalar fields):there exists a constant C such that for every v ∈ H1(O ) there is an element rv ∈R with|v − rv |H1(O )  C∣∣e(v)∣∣L2(O ). (3)Examples of Korn–Wirtinger domains are given by the following proposition which is proved by a standard contradictionargument.Proposition 2.3. Let O be a connected Korn domain for which the embedding from H1(O ) into L2(O ) is compact. Then O is aKorn–Wirtinger domain.The following is a straightforward application of inequality (3):Corollary 2.4. Let O be a Korn–Wirtinger domain and Γ be a closed subset of ∂ O with non-zero boundary measure. Then the firstKorn inequality is satisfied for vector fields which vanish on Γ , i.e. there exists a constant C such that∀v ∈ H1(O ;Γ ), |v|H1(O )  C∣∣e(v)∣∣L2(O ). (4)A. Damlamian / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 861–865 8632.2. Two types of nonlinear Korn inequalitiesWe distinguish two classes of connected bounded domains according to a geometric property (of their shapes).Definition 2.5. A locked domain is a bounded connected domain with Lipschitz boundary for which the only rigid motionwhich is tangent on its boundary is zero.Remark 2.6. The only non-locked bounded Lipschitz domains (in R3) are the spherical balls and the bounded right-sectionsof circular cylindric bars, for which the tangent rigid motions are obvious.We start with a simple remark concerning the rigid motions in a bounded domain with a Lipschitz boundary (since arigid motion is divergence-free).Lemma 2.7. Let O be a bounded connected and Lipschitz domain. Then,r → ‖r‖ .= |rτ |L1(∂ O ) +∣∣(rν)+∣∣L1(∂ O ) is a norm on R. (5)If O is a locked domain thenr → ‖r‖l .=∣∣(rν)+∣∣L1(∂ O ) is a norm on R. (6)Furthermore, if O is a spherical ball centered at the origin, there exists a constant C such that for all r in R, there exists a vector b(r) inR3 with∣∣r − b(r) ∧ Id∣∣R  C∣∣(rν)+∣∣L1(∂ O ). (7)Similarly, if O is a right-section of a circular cylinder with axis going through the origin and with unit vector dO , there exists a constantC such that for all r in R, there is a scalar (r) in R with∣∣r − (r)dO ∧ Id∣∣R  C∣∣(rν)+∣∣L1(∂ O ). (8)The previous lemma yields two new types of Korn inequalities. It is of interest that they only involve the positive part v+νof the normal component of the trace of v on the boundary so that they are, in some sense, unilateral. Obviously, applyingthem to −v would involve v−ν instead.Proposition 2.8 (Unilateral Korn inequality for locked domains). Let O be a locked domain. Then there exists a constant C1 such that∀v ∈ H1(O ), |v|H1(O )  C1(∣∣e(v)∣∣L2(O ) +∣∣(vν)+∣∣L1(∂ O )). (9)Proposition 2.9 (Unilateral Korn inequalities for non-locked domains). Let O be a connected non-locked domain with Lipschitz bound-ary. Then there exists a constant C2 such that∀v ∈ H1(O ), |v|H1(O )  C2(∣∣e(v)∣∣L2(O ) +∣∣(vν)+∣∣L1(∂ O ) +∣∣(vτ )∣∣L1(∂ O )). (10)Furthermore, if O is a ball centered at the origin, there exists a constant C3 and a map b : H1(O ) → R3 such that for every v in H1(O )∣∣v − b(v) ∧ Id∣∣H1(O )  C3(∣∣e(v)∣∣L2(O ) +∣∣(vν)+∣∣L1(∂ O )). (11)If O is a right-section of a circular right-cylinder with axis going through the origin and unit vector b, there is a constant C3 andmap  : H1(O ) → R such that∣∣v − (v)b ∧ Id∣∣H1(O )  C3(∣∣e(v)∣∣L2(O ) +∣∣(vν)+∣∣L1(∂ O )). (12)3. A linear elasticity contact problem in a domain with inclusions3.1. The problem PLet Ω be a bounded domain in R3 with Lipschitz boundary, and ΓD be a non-empty open subset of the boundary ∂Ω(ΓD is the set where a homogeneous Dirichlet condition will be prescribed).Inside Ω are given a finite number m of open subsets Ω1, . . . ,Ωm which are relatively compact in Ω . Their respectiveboundaries denoted S j are assumed to be Lipschitz. Let Ω0 be the subset Ω \ ⋃ j=1,m Ω j (see Fig. 1). Finally, we denote864 A. Damlamian / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 861–865Fig. 1. The various subdomains of Ω .by Ω∗ the open set Ω \ ⋃ j=1,...,m S j = ⋃ j=0,...,m Ω j . The unit normal vector to S j is denoted by ν and is normalized asinwards for Ω0.Consider a symmetric bilinear form:a(e(u), e(v)) .=m∑j=0∫Ω j3∑α,β,γ ,δ=1aαβγ δ(x)e(u)γ δ(x)e(v)αβ(x)dx,where the tensor field a = (aαβγ δ) has the usual properties of symmetry, boundedness and coercivity (with constant α > 0)when operating on symmetric tensors of order two, i.e. (with the summation convention):aαβγ δ = aβαγ δ = aαβδγ = aγ δαβ,maxαβγ δ∣∣aεαβγ δ∣∣L∞(Ω) < ∞, αηαβηαβ  aεαβγ δηαβηγ δ.Let K be the convex set defined, for given non-negative g j on S j , byv ∈ K .= {v = (v0, . . . , vm) ∣∣ v ∈ H1(Ω0) × · · · × H1(Ωm),v0|ΓD ≡ 0, v jν − v0ν  g j on S j}.The vector fields v are the admissible deformation fields with respect to the reference configuration Ω . By standard tracetheorems, the jump [v]S j of v across each surface S j exists in H1/2(S j) (it is v j|S j − v0|S j ).The tensor field σαβ(v).= ∑3γ ,δ=1 aαβγ δe(v)γ δ is the stress tensor associated to the deformation v and is defined in Ω∗(not to be confused with the surface measures dσ !).With the choice of the unit normals ν , the functions g j ’s are the original gaps (in the reference configuration), and thecorresponding inequalities represent the non-penetration conditions. In case there is contact in the reference configuration,these functions are just 0. More generally, we will assume that g j ∈ L1(S j) for j = 0, . . . ,m (which implies that the originalgaps have finite volumes).Clearly, K is a closed convex subset of the space H1(Ω0) × · · · × H1(Ωm).Consider also a family of convex functions Ψ j , 1  j  m, where Ψ j is continuous on H1/2(S j) and is such that thereexist a non-negative number M j withΨ j(w)  M j|wτ |L1(S j) for |wτ |L1(S j) large. (13)In the case of Tresca friction, the functions Ψ j are actually explicitly given byΨ j(v).=∫S jG j(x)∣∣v jτ (x) − v0τ (x)∣∣ dσ(x), (14)with G j bounded below by M j for j = 1, . . . ,m. This friction term plays a role in the case of non-locked inclusions.Problem P . Given f = ( f 0, . . . , f m) in L2(Ω) find a minimizer over K of the functionalE(v) .= 12a(e(v), e(v)) +m∑j=0Ψ j([v]S j ) −∫∗f v dx. (15)ΩA. Damlamian / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 861–865 865From standard arguments, Problem P is equivalent to the variational inequalityProblem P ′ . Find u ∈K such that for every v ∈K,a(e(u), e(v − u)) +m∑j=0(Ψ j([v]S j ) − Ψ j([u]S j ))∫Ω∗f (v − u)dx. (16)We refer to Kikuchi–Oden [9, Section 10.3] for a detailed study of the Tresca conditions.3.2. Statement of the main resultLet the number δ j be defined for j = 1, . . . ,m as follows:δ j ={0 if Ω j is a locked domain,1 otherwise.(17)For non-locked domains, we make an extra assumption on some moments of the right-hand side f j .Definition 3.1. For a spherical domain Ω j , M j denotes the moment of f j with respect to the center of Ω j .For a cylindrical domain Ω j , M j denotes the moment of f j with respect to the axis of Ω j .The extra hypothesis is(H j):{For δ j = 1 the moment M j is assumed to be “small”compared to M j .(18)Proposition 3.2 (Existence and uniqueness of the solution of P). Let f be in L2(Ω), g j in L1(S j) for j = 1, . . . ,m. Assume that for δ j =1 (non-locked domains), the function Ψ j is coercive in the sense that the constant M j of (13) is strictly positive and that hypothesis(H j) holds. Then, there is a solution u = (u0, . . . , um) in K for Problem P ′ . Uniqueness holds only for u0 and for u j − projRu j ,j = 1, . . . ,m (equivalently for e(u j)).Remark 3.3. Proposition 3.2 is still true with M j = 0 provided the corresponding M j = 0 (only for δ j = 1). This covers thecase of frictionless contact.On the other hand, no condition is required on M j if the corresponding Ψ j has superlinear growth.In a joint forthcoming paper with Doina Cioranescu and Julia Orlik [1], the periodic homogenization of this problem isstudied.References[1] D. Cioranescu, A. Damlamian, J. Orlik, Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptot. Anal., inpress.[2] G. Duvaut, J.-L. Lions, Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques, vol. 21, Dunod, Paris, 1972.[3] G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti Acc. Naz. Lincei (s. VIII) VII(1964).[4] G. Fichera, Elastostatics problems with unilateral constraints, Séminaire Jean Leray 3 (1966–1967) 64–68, http://www.numdam.org/item?id=SJL_1966-1967___3_64_0.[5] G. Fichera, Unilateral constraints in elasticity, in: Actes Congrès Int. Math. (Nice 1970), vol. 3, Gauthier–Villars, Paris, 1970, pp. 79–84.[6] J. Gobert, Une inégalité fondamentale de la théorie de l’élasticité, Bull. Soc. Roy. Sci. Liège 31 (1962) 182–191.[7] I. Hlaváček, J. Haslinger, J. Nečas, Numerical methods for unilateral problems in solid mechanics, in: Handbook of Numerical Analysis, vol. IV, North-Holland, Amsterdam, 1996, pp. 313–485.[8] I. Hlaváček, J. Haslinger, J. Nečas, J. Lovíšek, Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences, vol. 66, Springer-Verlag,New York, ISBN 0-387-96597-1, 1988, x+275 pp. Translated from Slovak by J. Jarník.[9] N. Kikuchi, J.T. Oden, Contact Problems in Elasticity, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1988.

Some unilateral Korn inequalities with application to a contact problem with inclusions

Damlamian, Alain

Numérisation de Documents Anciens Mathématiques

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Some unilateral Korn inequalities with application to a contact problem with inclusions

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