International audienceIn a previous work, it was shown how the linearized strain tensor field e := (∇u^T +∇u)/2 ∈ L^2(Ω) can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain Ω ⊂ R3 , instead of the displacement vector field u ∈ H^1 (Ω ) in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition u = 0 on a portion Γ_0 of the boundary of Ω can be recast, again as boundary conditions on Γ_0, but this time expressed only in terms of the new unknown e∈L^2(Ω)

Cristinel Mardare

Philippe Ciarlet

Comptes Rendus Mathematique

Ciarlet, Philippe G.

Mardare, Cristinel

Hal-Diderot

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 329–334Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical Problems in MechanicsExpression of Dirichlet boundary conditions in terms of the strain tensorin linearized elasticityExpression de conditions aux limites de Dirichlet en fonction du tenseur linéarisé desdéformationsPhilippe Ciarlet a, Cristinel Mardare ba Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kongb Université Pierre-et-Marie-Curie, Laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, Francea r t i c l e i n f o a b s t r a c tArticle history:Received and accepted 18 April 2013Available online 20 May 2013Presented by Philippe G. CiarletIn a previous work, it was shown how the linearized strain tensor field e := 12 (∇uT +∇u) ∈L2(Ω) can be considered as the sole unknown in the Neumann problem of linearizedelasticity posed over a domain Ω ⊂R3, instead of the displacement vector field u ∈ H 1(Ω)in the usual approach. The purpose of this Note is to show that the same approach appliesas well to the Dirichlet–Neumann problem. To this end, we show how the boundarycondition u = 0 on a portion Γ0 of the boundary of Ω can be recast, again as boundaryconditions on Γ0, but this time expressed only in terms of the new unknown e ∈ L2(Ω).© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans un travail antérieur, on a montré comment le champ e := 12 (∇uT + ∇u) ∈ L2(Ω) destenseurs linéarisés des déformations peut être considéré comme la seule inconnue dansle problème de Neumann pour l’élasticité linéarisée posé sur un domaine Ω ⊂ R3, au lieudu champ u ∈ H 1(Ω) des déplacements dans l’approche habituelle. L’objet de cette Noteest de montrer que la même approche s’applique aussi bien au problème de Dirichlet–Neumann. À cette fin, nous montrons comment la condition aux limites u = 0 sur uneportion Γ0 de la frontière de Ω peut être ré-écrite, à nouveau sous forme de conditions auxlimites sur Γ0, mais exprimées cette fois uniquement en fonction de la nouvelle inconnuee ∈ L2(Ω).© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. PreliminariesGreek indices, resp. Latin indices, range over the set {1,2}, resp. {1,2,3}. The summation convention with respect torepeated indices is used in conjunction with these rules. The notations |a|, a ∧ b, a ⊗ b, and a · b respectively denote theEuclidean norm, the exterior product, the dyadic product, and the inner product of vectors a, b ∈R3.The notation Sm , resp. Am , designates the space of all symmetric, resp. antisymmetric, tensors of order m. The innerproduct of two m × m tensors e and τ is denoted and defined by e : τ = tr(eT τ ). Given a normed vector space X , thenotation L2sym(X × X) designates the space of all continuous symmetric bilinear forms defined on the product X × X .E-mail addresses: mapgc@cityu.edu.hk (P. Ciarlet), mardare@ann.jussieu.fr (C. Mardare).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.2013.04.015330 P. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 329–334Let Ω ⊂ R3 be a connected, bounded, open set whose boundary ∂Ω is of class C4. This means that there exist a finitenumber N of open sets ωk ⊂ R2 and of mappings θk ∈ C4(ωk;R3), k = 1,2, . . . , N , such that ∂Ω = ⋃Nk=1 θk(ωk). It alsoimplies that there exists ε > 0 such that the mappings Θk ∈ C3(U k;R3), defined by:Θk(y, y3) := θk(y) + y3ak3(y) for all (y, y3) ∈ Uk := ωk × (−ε, ε),where ak3 denotes the unit inner normal vector field along the portion θk(ωk) of the boundary of Ω , are C3-diffeomorphismsonto their image (cf. [2, Theorem 4.1-1]). Thus the mappings {Θk; 1  k  N} form an atlas of local charts for the openset Ωε := {x ∈ Ω;dist(x, ∂Ω) < ε} ⊂ R3, while the mappings {θk;1  k  N} form an atlas of local charts for the surfaceΓ = ∂Ω ⊂ R3. When no confusion should arise, we will drop the explicit dependence on k for notational brevity.A generic point in ω is denoted y = (yα) and a generic point in U = ω × (−ε, ε) is denoted (y, y3). Partial derivativeswith respect to yi are denoted ∂i . The vectors aα(y) := ∂αθ(y) form a basis in the tangent space at θ(y) to the surfaceΓ := ∂Ω ⊂ R3 and the vectors g i(y, y3) := ∂iΘ(y, y3) form a basis in the tangent space at Θ(y, y3) to the open setΘ(U ) ⊂ Ωε ⊂ R3. Note that:gα(y, y3) = aα(y) + y3∂αa3(y) and g3(y, y3) = a3(y).By exchanging if necessary the coordinates y1 and y2, we may always assume that:a3(y) = a1(y) ∧ a2(y)|a1(y) ∧ a2(y)| .The vectors aα(y) in the tangent space at θ(y) to Γ and g i(y, y3) in the tangent space at Θ(y, y3) are defined by:aα(y) · aβ(y) = δαβ and g i(y, y3) · g j(y, y3) = δij,the area element on Γ is dΓ := √ady, where a := |a1 ∧ a2|, and the Christoffel symbols Cσαβ and Γ ki j , respectively inducedby the immersions θ and Θ , are defined by:Cσαβ := ∂αβθ · aσ and Γ ki j := ∂i jΘ · gk.A point in Ω will be specified either by its Cartesian coordinates x = (xi) with respect to a given orthonormal basis êiin R3, or, when x ∈ Ωε ⊂ Ω , by its curvilinear coordinates (y, y3) corresponding to a local chart Θ ; thus x = Θ(y, y3) insuch a local chart.Vector fields, resp. tensor fields, on Ω will be expanded at each x = Θ(y, y3) ∈ Ωε over the contravariant bases g i(y, y3),resp. (g i ⊗ g j)(y, y3). Covariant derivatives with respect to the local chart Θ are defined as usual, and denoted ui‖ j , ui‖ jk ,ei j‖k , etc.Let Γ0 be a connected and relatively open subset of the boundary Γ of Ω . Since Γ is a manifold of class C4, so is Γ0.It follows that functions, vector fields, and tensor fields, of class Cm , m = 0,1,2, can be defined on Γ0. The Lebesgue andSobolev spaces on Γ0 and their norms are then defined as in, e.g., Aubin [1].We also let Cmc (Γ0) denote the space of all functions f :Γ0 → R of class Cm with compact support contained in Γ0. Thenthe Sobolev space Hm0 (Γ0) is defined as the completion of the space Cmc (Γ0) with respect to the norm ‖ · ‖Hm(Γ0) . Its dualspace is denoted H−m(Γ0).Spaces of vector fields, resp. symmetric tensor fields, with values in R3, resp. in S3, are defined by using a given Cartesianbasis {êi,1  i  3} in R3, resp. the basis { 12 (êi ⊗ ê j + ê j ⊗ êi),1  i, j  3} in S3. They will be denoted by bold letters andby capital Roman letters, respectively.Complete proofs and complements will be found in [5].2. Linearized change of metric and curvature tensors on ∂Ω associated with a linearized strain tensor in C1(Ω)Given any displacement field u ∈ C2(Ω), the restriction ζ := u|Γ 0 ∈ C2(Γ 0) is a displacement field of the surfaceΓ 0 ⊂ R3. The linearized change of metric and change of curvature tensor fields induced by ζ are then respectively de-fined in each local chart by:γ (ζ ) = γαβ(ζ )aα ⊗ aβ, where γαβ(ζ ) := 12(∂αζ · aβ + ∂βζ · aα),ρ(ζ ) = ραβ(ζ )aα ⊗ aβ, where ραβ(ζ ) :=(∂αβζ − Cσαβ∂σ ζ) · a3, (1)where for convenience the same notation ζ denotes either the vector field ζ :Γ 0 → R3 or the vector field ζ := ζ ◦θ :ω →R3in a local chart θ :ω → Γ 0 of Γ 0.Let TxΓ0 ⊂ R3 denote the tangent space at each point x of the surface Γ0. Given any matrix field e ∈ C1(Ω), let thetensor fields:P. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 329–334 331γ (e) : x ∈ Γ0 →(γ (e))(x) ∈ L2sym(TxΓ0 × TxΓ0),ρ(e) : x ∈ Γ0 →(ρ(e))(x) ∈ L2sym(TxΓ0 × TxΓ0),be defined in a local chart θ :ω → Γ0 by:γ (e) = γ αβ(e)aα ⊗ aβ, where γ αβ(e) := eαβ |ω×{0},ρ(e) = ραβ(e)aα ⊗ aβ, where ραβ(e) :=(eα3‖β + eβ3‖α − eαβ‖3 + Γ 3αβe33)∣∣ω×{0}, (2)and the functions ei j are defined by e(x) = (ei j g i ⊗ g j)(y, y3) for all x = Θ(y, y3).The following theorem shows that the tensors fields γ (ζ ) and ρ(ζ ), which are defined in terms of the trace on Γ0 ofthe displacement field u ∈ C2(Ω), can be in fact expressed in terms of the traces on Γ0 of the linearized strain tensor field:e = ∇su := 12(∇uT + ∇u) ∈C1(Ω).Theorem 1. Let u ∈ C2(Ω), let e = ∇su, and let ζ := u|Γ 0 . Then:γ (ζ ) = γ (e) and ρ(ζ ) = ρ(e) on Γ 0,where the tensor fields appearing in these equalities are defined in (1) and (2).Sketch of proof. Proving the above equalities amounts to proving the equalities:γαβ(e) = γαβ(ζ ) and ραβ(e) = ραβ(ζ ) in ω,in any local chart θ :ω → Γ 0 of the surface Γ 0. These equalities follow from direct computations. Let:Im ∇s :={∇su; u ∈ C2(Ω)} ⊂ C1(Ω),and let the linear operators:γ  : e ∈ Im ∇s → γ (e) ∈C1(Γ 0) and ρ : e ∈ Im ∇s → ρ(e) ∈C0(Γ 0)be defined by the relations (2). The next theorem shows that these operators are continuous with respect to appropriate“weak” norms.Theorem 2.(a) There exists a constant C such that:∥∥γ (e)∥∥H−1(Γ0) +∥∥ρ(e)∥∥H−2(Γ0)  C infr∈R(Ω)∥∥(u + r)|Γ0∥∥L2(Γ0) for all e = ∇su, u ∈ C2(Ω),whereR(Ω) := {r :Ω →R3; there exist a ∈R3 and B ∈ A3 such that r(x) = a + Bx, x ∈ Ω}.(b) There exists a constant C such that:∥∥γ (e)∥∥H−1(Γ0) +∥∥ρ(e)∥∥H−2(Γ0)  C‖e‖L2(Ω) for all e = ∇su, u ∈ C2(Ω).Sketch of proof. Let e ∈ Im∇s and let u ∈ C2(Ω) be any vector field that satisfies e = ∇su. Since the spaces C1c (Γ0) andC2c (Γ0) are respectively dense in the spaces H10(Γ0) and H20(Γ0), we have:∥∥γ (e)∥∥H−1(Γ0) = supτ∈C1c (Γ0)| ∫Γ0γ (e) : τ dΓ |‖τ‖H1(Γ0)and∥∥ρ(e)∥∥H−2(Γ0) = supτ∈C2c (Γ0)| ∫Γ0ρ(e) : τ dΓ |‖τ‖H2(Γ0).The basic idea of the proof then relies on a careful re-writing of the numerators found in the above expressions. Com-bined with a partition of unity associated with a covering Γ0 ⊂ ⋃Nk=1 θk(ωk), this re-writing shows that there exist constantsC1 and C2 such that:332 P. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 329–334∣∣∣∣∫Γ0γ (e) : τ dΓ∣∣∣∣  C1‖ζ‖L2(Γ0)‖τ‖H1(Γ0) for all τ ∈C1c (Γ0),and ∣∣∣∣∫Γ0ρ(e) : τ dΓ∣∣∣∣  C2‖ζ‖L2(Γ0)‖τ‖H2(Γ0) for all τ ∈C2c (Γ0),where ζ := u|Γ0 , so that:∥∥γ (e)∥∥H−1(Γ0)  C1‖ζ‖L2(Γ0) = C1‖u|Γ0‖L2(Γ0) (3)and∥∥ρ(e)∥∥H−2(Γ0)  C2‖ζ‖L2(Γ0) = C2‖u|Γ0‖L2(Γ0). (4)Since inequalities (3) and (4) hold for all vector fields u ∈ C2(Ω) that satisfy e = ∇su, and since ∇sr = 0 for all r ∈ R(Ω),there exists a constant C3 such that:∥∥γ (e)∥∥H−1(Γ0) +∥∥ρ(e)∥∥H−2(Γ0)  C3 infr∈R(Ω)∥∥(u + r)|Γ0∥∥L2(Γ0).Furthermore, the continuity of the trace operator from H 1(Ω) into L2(Γ0) shows that there exists a constant C4 suchthat:∥∥γ (e)∥∥H−1(Γ0) +∥∥ρ(e)∥∥H−2(Γ0)  C4 infr∈R(Ω)‖u + r‖H 1(Ω).Finally, the classical Korn’s inequality shows that there exists a constant C5 such that:∥∥γ (e)∥∥H−1(Γ0) +∥∥ρ(e)∥∥H−2(Γ0)  C5‖∇su‖L2(Ω) = C5‖e‖L2(Ω). 3. Linearized change of metric and curvature tensors on ∂Ω associated with a linearized strain tensor in L2(Ω)The classical Korn’s inequality shows that the closure of the space Im ∇s in the space L2(Ω) is the space:Im ∇s ={∇su; u ∈ H 1(Ω)} ⊂ L2(Ω).The next theorem shows that the definition of the tensor fields γ (e) and ρ(e) given in Section 2 for fields e ∈ Im∇s ⊂C1(Ω) can be extended in a natural way to linearized strain vector fields e that belong to the closed subspace Im∇sof L2(Ω).Theorem 3. Let the linear operators:γ  : e ∈ Im ∇s → γ (e) ∈C1(Γ 0) and ρ : e ∈ Im ∇s → ρ(e) ∈C0(Γ 0)be defined by (2).(a) There exist continuous linear operators:γ  : e ∈ Im ∇s → γ (e) ∈H−1(Γ0) and ρ : e ∈ Im ∇s → ρ(e) ∈H−2(Γ0)such that:γ (e) = γ (e) and ρ(e) = ρ(e) for all e ∈ Im ∇s,and there exists a constant C0 such that:∥∥γ (e)∥∥H−1(Γ0) +∥∥ρ(e)∥∥H−2(Γ0)  C0‖e‖L2(Ω) for all e = ∇su, u ∈ H1(Ω).(b) There exists a constant C1 such that:∥∥γ (e)∥∥H−1(Γ0) +∥∥ρ(e)∥∥H−2(Γ0)  C1 infr∈R(Ω)∥∥(u + r)|Γ0∥∥L2(Γ0) for all e = ∇su, u ∈ H 1(Ω).Proof. It suffices to combine Theorem 2 with the classical theorem about the extension of densely defined continuous linearoperators with values in a Banach space. P. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 329–334 3334. A Korn inequality on a surface in Sobolev spaces with negative exponentsIn the proof of Theorem 5 in the next section, we will need a “weak” variant of the Korn inequality on a surface (The-orem 4), the difference with the classical Korn inequality on a surface (see, e.g., [2, Theorem 4.3-5]) being that it is nowexpressed in terms of negative Sobolev norms. If ζ is only in the space L2(Γ0), the corresponding linearized change ofmetric, and change of curvature, tensor fields γ (ζ ) ∈ H−1(Γ0), and ρ(ζ ) ∈ H−2(Γ0), are defined in a local chart θ by theformulas (1).Theorem 4. Let the sets Ω and Γ0 satisfy the assumptions of Section 1 and let the space R(Γ0) be defined by:R(Γ0) :={r :Γ0 →R3; there exist a ∈R3 and B ∈ A3 such that r(x) = a + Bx, x ∈ Γ0}.Then there exists a constant C such that:infr∈R(Γ0)‖ζ + r‖L2(Γ0)  C(∥∥γ (ζ )∥∥H−1(Γ0) +∥∥ρ(ζ )∥∥H−2(Γ0))for all ζ ∈ L2(Γ0).Sketch of proof. (i) It suffices first to prove that there exists a constant C such that:‖ζ‖L2(Γ0)  C(‖ζ‖H−1(Γ0) +∥∥γ (ζ )∥∥H−1(Γ0) +∥∥ρ(ζ )∥∥H−2(Γ0))(5)for all ζ ∈ L2(Γ0), then to establish the following weak form of the infinitesimal rigid displacement lemma on a surface: If adisplacement field ζ ∈ L2(Γ0) satisfies γ (ζ ) = 0 in H−1(Γ0) and ρ(ζ ) = 0 in H−2(Γ0), then there exist a vector a ∈ R3 andan antisymmetric matrix B ∈ A3 such that ζ (x) = a + Bx for dΓ -almost all x ∈ Γ0.(ii) Proof of inequality (5). It suffices to prove (5) only for ζ ∈ C2(Γ 0). To this end, a crucial use is made of the relation:2∂αβζσ = ∂α(∂βζσ + ∂σ ζβ) + ∂β(∂αζσ + ∂σ ζα) − ∂σ (∂βζα + ∂αζβ),and of a crucial inequality due to Nec̆as [7] (see also Theorem 6.14-1 in [3]), which shows that there exists a constant Cindependent of ζ such that:‖ζ‖L2(ω)  C(‖ζ‖H−1(ω) +∑α‖∂αζ‖H−1(ω))for all ζ ∈ L2(ω).(iii) Proving the infinitesimal rigid displacement lemma hinges on a careful adaptation of an argument due to P.G. Ciarletand S. Mardare [6, Lemma 2] to vector fields ζ that are only in L2(Γ0). 5. An intrinsic formulation of the boundary conditionsUsing Theorems 3 and 4, we now show how a homogeneous Dirichlet boundary condition imposed on the displacement fieldappearing in the displacement–traction problem of linearized elasticity can be replaced by a homogeneous boundary condition imposedon the linearized strain tensor field:Theorem 5. Let the sets Ω and Γ0 satisfy the assumptions of Section 1. Given a vector field u ∈ H 1(Ω), let:e = ∇su ∈ L2(Ω).(a) If u = 0 on Γ0 , then γ (e) = 0 in H−1(Γ0) and ρ(e) = 0 in H−2(Γ0), where the tensor fields γ (e) and ρ(e) are those definedin Theorem 3.(b) If γ (e) = 0 in H−1(Γ0) and ρ(e) = 0 in H−2(Γ0), then there exists a unique infinitesimal rigid displacement r ∈ R(Ω) suchthat (u + r) = 0 on Γ0 .Sketch of proof. (a) Assume that u = 0 on Γ0. The second inequality of Theorem 3 shows that:∥∥γ (e)∥∥H−1(Γ0) +∥∥ρ(e)∥∥H−2(Γ0)  C1‖u|Γ0‖L2(Γ0).Since u vanishes on Γ0, it then follows that γ (e) = 0 in H−1(Γ0) and ρ(e) = 0 in H−2(Γ0).(b) Assume that γ (e) = 0 in H−1(Γ0) and that ρ(e) = 0 in H−2(Γ0). Since the space C2(Ω) is dense in H 1(Ω), thereexists a sequence (un) in C2(Ω) such that un → u in H 1(Ω) as n → ∞. Therefore,∇sun → ∇su = e in L2(Ω) as n → ∞.By Theorem 3, this implies that:γ (∇sun) → γ (e) = 0 in H−1(Γ0) and ρ(∇sun) → ρ(e) = 0 in H−2(Γ0) as n → ∞.334 P. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 329–334Let ζ := u|Γ0 and ζn := un|Γ0 . Since un ∈ C2(Ω), Theorems 1 and 3 together show that:γ (∇sun) = γ (∇sun) = γ (ζn) and ρ(∇sun) = ρ(∇sun) = ρ(ζn).Hence the previous convergences become:γ (ζn) → 0 in H−1(Γ0) and ρ(ζn) → 0 in H−2(Γ0) as n → ∞.Combined with the Korn inequality established in Theorem 4, these convergences imply that:infr∈R(Γ0)‖ζn + r‖L2(Γ0) → 0 as n → ∞.There thus exists a sequence (rn) in the space R(Γ0) such that:ζn + rn → 0 in L2(Γ0) as n → ∞.The space R(Γ0) being finite-dimensional, the sequence (rn) possesses a convergent subsequence. Let r denote thelimit of this subsequence; we then have r ∈ R(Γ0) and ζ + r = 0 in L2(Γ0). Hence the trace on Γ0 of the vector field(u + r) ∈ H 1(Ω) vanishes.If r̃ ∈ R(Ω) is such that the trace on Γ0 of the vector field (u + r) ∈ H 1(Ω) also vanishes, then (r̃ − r)|Γ0 = 0. Thisimplies that (r̃ − r) = 0 in Ω . We refer to the extended article [5] for applications of the results presented in this Note. There it will be shown inparticular how the Dirichlet–Neumann boundary value problem of three-dimensional linearized elasticity can be completely recastas a boundary value problem with the tensor field e = ∇su as the sole unknown. Such a result thus complements the approachof [4], which was restricted to the pure Neumann problem of the three-dimensional linearized elasticity.AcknowledgementsThis work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region,China [Project No. 9041529 – CityU 100710].References[1] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer, Berlin, 2010.[2] P.G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005.[3] P.G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Providence, 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, C. Mardare, Intrinsic formulation of the displacement–traction problem in linearized elasticity, in preparation.[6] P.G. Ciarlet, S. Mardare, On Korn’s inequalities in curvilinear coordinates, Math. Models Methods Appl. Sci. 11 (2001) 1379–1391.[7] J. Nec̆as, Equations aux Dérivées Partielles, Presses de l’université de Montréal, 1966.

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Ciarlet, Philippe

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Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

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