We study the asymptotic behavior of the solution of an anisotropic,
heterogeneous, linearized elasticity problem in a cylinder whose diameter
$\epsilon$ tends to zero. The cylinder is assumed to be fixed (homogeneous
Dirichlet boundary condition) on the whole of one of its extremities, but only
on a small part (of size $\epsilon r^\epsilon$) of the second one; the Neumann
boundary condition is assumed on the remainder of the boundary. We show that
the result depends on $r^\epsilon$, and that there are 3 critical sizes, namely
$r^\epsilon=\epsilon^3$, $r^\epsilon=\epsilon$, and
$r^\epsilon=\epsilon^{1/3}$, and in total 7 different regimes. We also prove a
corrector result for each behavior of $r^\epsilon$.Comment: Preliminary version of a Note to be published in a slightly
  abbreviated form in C. R. Acad. Sci. Paris, Ser. I, 338 (2004), pp. 975-98

Casado-Diaz, Juan

Luna-Laynez, Manuel

Murat, Francois

English

arXiv

Juan Casado-Díaz

Manuel Luna-Laynez

François Murat

Comptes Rendus Mathématique

tcylinderdans unde sesn delinéairetet.C. R. Acad. Sci. Paris, Ser. I 338 (2004) 975–980Mathematical Problems in MechanicsAsymptotic behavior of an elastic beam fixed on a small parof one of its extremitiesJuan Casado-Díaza, Manuel Luna-Layneza, François Muratba Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spainb Laboratoire Jacques-Louis Lions, Université Pierre etMarie Curie, boîte courrier 187, 75252 Paris cedex 05, FranceReceived 16 February 2004; accepted 23 February 2004Available online 7 May 2004Presented by Philippe G. CiarletAbstractWe study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in awhose diameterε tends to zero. The cylinder is assumed to be fixed (homogeneous Dirichlet boundarycondition) on the wholeof one of its extremities, but only on a small part (of sizeεrε) of the second one; the Neumann boundary condition is imposed onthe remainder of the boundary. We show that the result depends onrε , a d that there are 3 critical sizes, namelyrε = ε3, rε = ε,andrε = ε1/3, and in total 7 different regimes. We also prove a corrector result for each behavior ofrε . To cite this article:J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméComportement asymptotique d’une poutre élastique fixée sur une petite partie de l’une de ses extrémités.Nousétudions le comportement asymptotique de la solution d’un problème d’élasticité linéaire anisotrope et hétérogènecylindre dont le diamètreε tend vers zéro. Le cylindre est fixé (condition de Dirichlet homogène) sur la totalité de l’uneextrémités, mais seulement sur une petite partie (de tailleεrε) de l’autre base ; sur le reste de la frontière on a la conditioNeumann. Nous montrons que le résultat depend derε , t qu’il existe 3 tailles critiques, à savoirrε = ε3, rε = ε et rε = ε1/3,et au total 7 comportements différents. Nous donnons un résultat de correcteur pour tous les comportements derε. Pour citercet article : J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeDans cette Note, nous étudions le comportement asymptotique de la solution d’un probleme d’élasticitéanisotrope et hétérogène posé dans un cylindreΩε = (0,1) × εS ⊂ R3 dont le diamètreε tend vers zéro et donl’axe est le premier axe de coordonnéesOx1. À l’une de ses extrémités (x1 = 1) le cylindre est fixé (condition dDirichlet homogène) sur la totalité de la baseΓ ε1 = {1}×εS, tandis qu’à l’autre extrémité (x1 = 0), il est seulemenfixé sur une petite partieΓ ε0 = {0} × εrεS0 de la base, partie qui est bien plus petite queε car rε tend vers zéroE-mail addresses:jcasadod@us.es (J. Casado-Díaz), mllaynez@us.es (M.Luna-Laynez), murat@ann.jussieu.fr (F. Murat).1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.02.020976 J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 975–980tsuranglaiseset deearizedxiseatedain-nyfaceSur le reste de la frontière deΩε, on a la condition de Neumann. PourA tenseur d’ordre 4 coercif à coefficiencontinus dansΩ (Ω = (0,1) × S), etf ∈ L2(Ω)3, h ∈ L2(Ω)3×3s donnés, on définitAε, F ε et Hε par (1) et (2)et on définitUε comme la solution du problème d’élasticité (3).Le but de cette Note est de décrire le comportement asymptotique deUε et de donner un résultat de correctepoure(Uε) quandε et rε tendent vers zéro. Ce résultat est décrit dans le théorème enoncé dans la versionci-dessous, qui fait apparaitre 3 tailles critiques, à savoirrε ≈ ε3, rε ≈ ε et rε ≈ ε1/3, qui séparent 4 zonecorrespondant àrε  ε3, ε3  rε  ε, ε  rε  ε1/3 et ε1/3  rε  C, donc au total 7 cas différents. Lrésultat de convergence est donné par (5), oùu est défini par la solution de (4), tandis que (6) est un résultacorrecteur poure(Uε).1. Position of the problem and notationIn this Note we study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linelasticity problem posed in a thin cylinderΩε whose diameterε tends to zero and whose axis is the first aof coordinatesOx1. On one of its extremities (x1 = 1) the cylinder is fixed on its whole basisΓ ε1 whereas on thesecond one (x1 = 0) it is fixed only on a small partΓ ε0 of it, of diameterεrε much smaller thanε. The Neumannboundary condition is imposed on the remainder of the boundary ofΩε. Mathematically the problem can bformulated as follows.For ε > 0, we consider ε a positive parameter which tends to zero withε. Let S0 and S be two boundedsmooth domains inR2, with 0 ∈ S. We defineΩ = (0,1) × S, Ωε = (0,1) × εS and Γ ε = Γ ε0 ∪ Γ ε1 , whereΓ ε0 = {0} × εrεS0, Γ ε1 = {1} × εS. Observe that the size ofΓ ε0 is much smaller than the size of the basis{0} × εSsincerε tends to zero.The elements ofR3 are decomposed asx = (x1, x ′), x1 ∈ R, x ′ = (x2, x3) ∈ R2. We denote by{e1,e2,e3}the canonical basis ofR3 and byL(R3×3s ) the space of linear maps ofR3×3s into itself (or in other terms of fourthorder tensors), whereR3×3s is the space of the 3×3 symmetric matrices. We adopt Einstein’s convention of repeindices. Greek indices (α andβ) take only the values 2 and 3, while latin indices (i andj ) take the values 1, 2 and 3.We considerA ∈ C0(Ω;L(R3×3s )) such that there existsm > 0 with A(y)ξξ  m|ξ |2, for all ξ ∈ R3×3s and forall y ∈ Ω, and we defineAε ∈ C0(Ωε;L(R3×3s )) byAε(x) = A(x1,x ′ε), ∀x ∈ Ωε. (1)We also considerf ∈ L2(Ω)3 andh ∈ L2(Ω)3×3s , and we defineFε ∈ L2(Ωε)3 andHε ∈ L2(Ωε)3×3s byFε(x) = f1(x1,x ′ε)e1 + εfα(x1,x ′ε)eα, Hε(x) = h(x1,x ′ε), a.e.x ∈ Ωε. (2)In the thin domainΩε we consider the elasticity problemUε ∈ H 1Γ ε(Ωε)3,∫ΩεAεe(Uε) : e(Uε)dx = ∫ΩεF ε Uε dx +∫ΩεHε : e(Uε)dx, ∀Uε ∈ H 1Γ ε (Ωε)3, (3)whereH 1Γ ε (Ωε) = {U ∈ H 1(Ωε): U = 0 onΓ ε}; observe that in the above formulation, as well as in the remder of the present Note, complex numbers never appear, and thatUε ( nd laterū, v̄, w) denotes the test functioassociated to the solutionUε , and not its complex conjugate. Observe also that the solutionUε of (3) satisfies anonhomogeneous Neumann boundary condition on the part∂Ωε \ Γ ε where it is not fixed, since integrating bparts∫Ωε Hε : e(Uε)dx (whenh and thereforeHε is sufficiently smooth) produces both body forces and surforces. Similarly to the body forcesFε we could have introduced explicit surface forcesGε on ∂Ωε \ Γ ε, but wehave preferred not to include them for the sake of simplicity.J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 975–980 977whichrenders--im-also [1]).icarIt is well known that problem (3) has an unique solution (see, e.g., [5]). The aim of the present Note,announces our paper [4], is to describe the asymptotic behavior of the solutionUε a d to give a corrector result foe(Uε) asε tends to zero. The result depends on the behavior ofrε and exhibits 3 critical sizes, namelyε3, ε, andε1/3, so that there are 7 different regimes:rε  ε3, rε ≈ ε3, ε3  rε  ε, rε ≈ ε, ε  rε  ε1/3, rε ≈ ε1/3, andε1/3  rε  C, whererε  ελ stands forrε/ελ → 0 (and equivalentlyελ  rε for rε/ελ → +∞), while rε ≈ ελstands forrε/ελ → ρ, for someρ with 0 < ρ < +∞.To express the results and to make the proofs, we will use two changes of variables.The first change of variables is given byy = yε(x) with y1 = x1, y ′ = x ′/ε, which transforms the variabldomainΩε into the fixed domainΩ . This is the usual change of variables used to study equations in thin cyli(see, e.g., [6,9–12]). Whenrε = 1 andS0 = S (but the same proof works forrε = c independent ofε such thatcS0 ⊂ S), it was used successfully in [6,9–11] to pass to the limit in (3). Whenrε tends to zero withε, this firstchange of variables allows us to describe the behavior ofUε in the part ofΩε far from Γ ε0 (this behavior is thesame as in the case whererε = c), but it does not provide the information we need about the behavior ofUε in thepart ofΩε close toΓ ε0 .Thus we introduce a second change of variables given byz = zε(x) with z = x/εrε, which transforms the variable domainΩε into a variable domainZε, the limit of which is the half spaceZ = (0,+∞)×R2. Observe that theDirichlet boundary condition is now imposed on the fixed part{0}×S0 of the boundary ofZε. This change of variables provides a suitable rescaling nearx1 = 0. It was used successfully in [2,3] to study the diffusion problem silar to (3) in the geometry that we consider in the present Note, and even in a more complicated one (seeWe denote byD1,2(Z) the Deny’s spaceD1,2(Z) = {p: p ∈ L6(Z), ∇p ∈ L2(Z)3}. We will also use thefunctional spaces (already used in [7,8,10,11])BNb(Ω) ={u: ∃ζα ∈ H 2(0,1), ζα(1) = dζαdy1(1) = 0, uα(y) = ζα(y1), ∀α ∈ {2,3},∃ζ1 ∈ H 1(0,1), ζ1(1) = 0, u1(y) = ζ1(y) − dζαdy1(y1)yα},Rb(Ω) ={v: v1 ∈ L2(0,1;H 1(S)),∫Sv1(y1, y′)dy ′ = 0 a.e.y1 ∈ (0,1),∃c ∈ H 1(0,1), c(1) = 0, v2(y) = c(y1)y3, v3(y) = −c(y1)y2},RD⊥2 (Ω) ={w: w1 = 0, wα ∈ L2(0,1;H 1(S)), ∫Swα(y1, y′)dy ′ = 0, ∀α ∈ {2,3},∫S(y3w2(y1, y′) − y2w3(y1, y ′))dy ′ = 0 a.e.y1 ∈ (0,1)}.For a given(u, v,w) ∈ BNb(Ω) × Rb(Ω) × RD⊥2 (Ω), we denote byE(u,v,w) the second order symmetrtensor with values inR3×3s defined byE11(u, v,w) = e11(u), E1β(u, v,w) = e1β(v), Eαβ(u, v,w) = eαβ(w),∀α,β ∈ {2,3}.2. The result and some commentsThe asymptotic behavior of the solution of (3) depends on the size ofrε with respect toε. Seven regimes appein the following theorem which describes the asymptotic behavior ofUε and provides a corrector result forUεande(Uε).Theorem 2.1.Let Uε be the solution of(3). There exist a closed linear subspaceE of BNb(Ω) × Rb(Ω) ×RD⊥2 (Ω), a functionPε ∈ L2(Ωε)3×3s , and a bilinear continuous nonnegative formB onE ×E such that, defining(u, v,w) as the solution of the variational problem978 J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 975–980its(u, v,w) ∈ E,∫ΩAE(u, v,w) : E(ū, v̄, w)dy +B((u, v,w), (ū, v̄, w)) = ∫Ωf ūdy +∫Ωh : E(ū, v̄, w)dy,∀(ū, v̄, w) ∈ E,(4)then, whenε tends to zero, we have1|Ωε|∫Ωε(∣∣∣∣Uε1(x) − u1(x1,x ′ε)∣∣∣∣2+3∑α=2∣∣εUεα(x) − uα(x1)∣∣2)dx −→ 0, (5)1|Ωε|∫Ωε∣∣∣∣e(Uε)(x) − E(u,v,w)(x1,x ′ε)− Pε(xεrε)∣∣∣∣2dx −→ 0. (6)The definitions ofE, P ε andB do not depend on the forcesf andh which defineFε andHε, but only on thesetS0, on the fourth order tensorA, and on the behavior ofrε whenε tends to zero, and more specifically ofbehavior with respect toε3, ε, andε1/3, such that there are7 regimes, which are described now.• If rε  ε3, thenE = BNb(Ω) × Rb(Ω) × RD⊥2 (Ω), P ε = 0, B = 0.• If rε ≈ ε3 with rε/ε3 → ρ, thenE = BNb(Ω) × Rb(Ω) × RD⊥2 (Ω), and definingϕi, i ∈ {1,2,3}, as thesolution ofϕi ∈ D1,2(Z)3, ϕi = ei on {0} × S0,∫ZA(0)e(ϕi) : e(ϕ̄)dz = 0, ∀ϕ̄ ∈ D1,2(Z)3, ϕ̄ = 0 on {0} × S0, (7)then one hasPε(z) = − 1ε2rεζα(0) e(ϕα)(z), a.e.z ∈ Z,B((u, v,w), (ū, v̄, w)) = ρ ∫ZA(0)(ζα(0) e(ϕα)) : (ζ̄β(0) e(ϕβ))dz, ∀(u, v,w), (ū, v̄, w) ∈ E .• If ε3  rε  ε, thenE = {(u, v,w) ∈ BNb(Ω) × Rb(Ω) × RD⊥2 (Ω): ζα(0) = 0, ∀α ∈ {2,3}}, P ε = 0, B = 0.• If rε ≈ ε with rε/ε → ρ, thenE = {(u, v,w) ∈ BNb(Ω) × Rb(Ω) × RD⊥2 (Ω): ζα(0) = 0, ∀α ∈ {2,3}},and settingϕ̂1 = ϕ1 + aα ϕα , whereϕi is defined by(7) and where(a2, a3) is defined by(a2, a3) ∈ R2,∫ZA(0)(e(ϕ1) + aα e(ϕα)) : (āβ e(ϕβ))dz = 0, ∀(ā2, ā3) ∈ R2,then one hasPε(z) = − 1εrεζ1(0)e(ϕ̂1)(z), a.e.z ∈ Z,B((u, v,w), (ū, v̄, w)) = ρ ∫ZA(0)(ζ1(0)e(ϕ̂1)) : (ζ̄1(0)e(ϕ̂1))dz, ∀(u, v,w), (ū, v̄, w) ∈ E .J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 975–980 979dsle• If ε  rε  ε1/3, thenE = {(u, v,w) ∈ BNb(Ω) × Rb(Ω) × RD⊥2 (Ω): ζα(0) = ζ1(0) = 0, ∀α ∈ {2,3}}, P ε = 0, B = 0.• If rε ≈ ε1/3 with (rε)3/ε → ρ, thenE = {(u, v,w) ∈ BNb(Ω) × Rb(Ω) × RD⊥2 (Ω): ζα(0) = ζ1(0) = 0, ∀α ∈ {2,3}},and definingψ1 as the solution ofψ1 ∈ D1,2(Z)3, ψ1 = z3e2 − z2e3 on {0} × S0,∫ZA(0)e(ψ1) : e(ψ̄)dz = 0, ∀ψ̄ ∈ D1,2(Z)3, ψ̄ = 0 on {0} × S0,andψα, α ∈ {2,3}, as the solution ofψα ∈ D1,2(Z)3, ψα = z1eα − zαe1 on {0} × S0,∫ZA(0)e(ψα) : e(ψ̄)dz = 0, ∀ψ̄ ∈ D1,2(Z)3, ψ̄ = 0 on {0} × S0,and then settinĝψi = ψi + bik ϕk, whereϕk is defined by(7) and where(bi1, bi2, bi3), i ∈ {1,2,3}, is defined by(bi1, bi2, bi3) ∈ R3, ∫ZA(0)(e(ψi) + bike(ϕk)) : (b̄il e(ϕl))dz = 0, ∀(b̄i1, b̄i2, b̄i3) ∈ R3,then one hasPε(z) = −1ε(c(0) e(ψ̂1)(z) + dζαdy1(0)e(ψ̂α)(z)), a.e.z ∈ Z,∀(u, v,w), (ū, v̄, w) ∈ E, B((u, v,w), (ū, v̄, w))= ρ∫ZA(0)(c(0)e(ψ̂1) + dζαdy1(0)e(ψ̂α)) : (c̄(0)e(ψ̂1) + dζ̄αdy1(0)e(ψ̂α))dz.• If ε1/3  rε  C, thenE ={(u, v,w) ∈ BNb(Ω) × Rb(Ω) × RD⊥2 (Ω): ζα(0) = ζ1(0) =dζαdy1(0) = c(0) = 0, ∀α ∈ {2,3}},P ε = 0, B = 0.Let us make some comments about the statement of this theorem.Assertion (6) is a corrector result, since one can prove that1|Ωε |∫Ωε|e(Uε)|2 dx + 1|Ωε |∫Ωε|E(u,v,w) ×(x1,x ′ε)|2 dx + 1|Ωε |∫Ωε |Pε( xεrε )|2 dx ≈ 1.If one examines the definition ofE , one realizes that the number of Dirichlet boundary conditions imposein the definition ofE increases with the size ofrε. Indeed, in view of the definitions ofBNb(Ω), Rb(Ω) andRD⊥2 (Ω), the sole functions which have a trace forx1 = 0 areζα, ζ1, dζαdy1 , andc, whereα ∈ {2,3} (the otherfunctions, namelyv1 andwα , have no trace forx1 = 0). Whenrε  ε3, the setΓ ε0 = {0} × εrεS0 is too small andthe homogeneous Dirichletboundary condition imposed forx1 = 0 to the solutionUε of (3) completely disappearat the limit. Whenε3  rε  ε, the setΓ ε0 is sufficiently large to impose at the limit thatζα(0) = 0 for α ∈ {2,3}.Whenε  rε  ε1/3, one further hasζ1(0) = 0. Finally whenε1/3  rε, the setΓ ε0 is so large that all the possibDirichlet boundary conditions are imposed atx1 = 0, namelyζα(0) = ζ1(0) = dζαdy (0) = c(0) = 0.1980 J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 975–980seofle posedntncenyatedl,11],S2000acionesntths,paration.n335tic Anal. 10indresExcept in the three regimes where the size ofrε is critical (i.e., whenrε ≈ ελ, with λ = 3, 1, or 1/3), one alwayshasPε = 0 andB = 0. For these three critical sizes, one can show thatB is a coercive bilinear form, in the senthat there exists somen > 0 such thatB((u, v,w), (u, v,w)) is greater thanρ(|ζ2(0)|2 + |ζ3(0)|2) whenrε ≈ ε3,thannρ|ζ1(0)|2 whenrε ≈ ε, and thannρ(|c(0)|2 + | dζ2dy1 (0)|2 + |dζ3dy1(0)|2) whenrε ≈ ε1/3. This implies that forevery critical sizerε ≈ ελ, the new Dirichlet boundary conditions which appear forrε  ελ (with respect to thoseimposed forrε  ελ) are penalized by the value ofρ. This introduces some type of continuity in the transitionthe Dirichlet condition between the two regimes which are separated by a critical sizeελ. For these three criticasizes, the functionsϕα, ϕ1, ϕ̂1, ψi , andψ̂i are in some sense generalized capacitary potentials of{0} × S0 in Z,and the bilinear formB is in some sense an asymptotic trace of some type of capacity ofΓ ε0 in Ωε for the weightedenergy 1|Ωε |∫ΩεA(x)e(ϕ) : e(ϕ)dx.The present work is the natural generalization to the elastic case of [2,3], where diffusion problems werin the union of two cylinders{(−tε,0) × εrεS0} ∪ {(0,1) × εS}, with both tε andrε tending to zero (the presegeometry corresponds totε = 0). When tε = 0, the diffusion problem was in comparison more simple, sionly one critical size, namelyrε ≈ ε, appeared in the analysis, separating the Neumann boundary conditio(corresponding to the analogue ofu satisfyingu ∈ H 1(0,1), u(1) = 0) for rε  ε, and the Dirichlet boundarcondition (corresponding to the analogue ofu satisfyingu ∈ H 1(0,1), u(0) = u(1) = 0) for rε  ε. These workswere related to [1], where a notched beam for diffusion problems was considered. The present work is also relto [7,8], where a multidomain made of an elastic vertical beam of length 1 and of radiusrε and of an horizontaplate of radius 1 and of heightε was considered. Finally, the present work is also the generalization of [10which were concerned with the caserε = c, to the present case whererε tends to zero.The detailed proofs of the results of the present Note will be given in a forthcoming paper [4].AcknowledgementsThis work has been partly supported by the project BFM2002-00672 of the DGI of Spain and by the HMTraining and Research Network. It was begun during a visit of the third author to the Departamento de EcuDiferenciales y Análisis Numérico de la Universidad de Sevilla.References[1] J. Casado-Díaz, F. Murat, The diffusionequation in a notched beam, in preparation.[2] J. Casado-Díaz, M. Luna-Laynez, F. Murat,Asymptotic behavior of diffusion problems in a domain made of two cylinders of differediameters and lengths, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 133–138.[3] J. Casado-Díaz, M. Luna-Laynez, F. Murat, Diffusion problems in adomain made of two thin cylinders of different diameters and lengin preparation.[4] J. Casado-Díaz, M. Luna-Laynez, F. Murat, Elasticity problems in a beam fixed on only a small part of one of its extremities, in pre[5] P.G. Ciarlet, Mathematical Elasticity, vol. I: Three-Dimensional Elasticity, North-Holland, 1988.[6] A. Cimetière, G. Geymonat, H. Le Dret, A. Raoult, Z. Tutek, Asymptotic theory and analysis for displacemnts and stress distribution inonlinear straight slender rods, J. Elasticity 19 (1988) 111–161.[7] A. Gaudiello, R. Monneau, J. Mossino, F. Murat, A. Sili, On the junction of elastic plates and beams, C. R. Acad. Sci. Paris, Ser. I(2002) 717–722.[8] A. Gaudiello, R. Monneau, J. Mossino, F. Murat, A. Sili, Junction of elastic plates and beams, in press.[9] H. Le Dret, Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero, Asympto(1995) 367–402.[10] F. Murat, A. Sili, Comportement asymptotique des solutions du système de l’élasticité linéarisée anisotrope hétérogène dans des cylminces, C. R. Acad. Sci. Paris, Ser. I 328 (1999) 179–184.[11] F. Murat, A. Sili, Anisotropic, heterogeneous, linearized elasticity in thin cylinders, in preparation.[12] L. Trabucho, J.M. Viaño, Mathematical Modelling of Rods, in: Handbook of Numerical Analysis, vol. IV, North-Holland, 1996.

Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities

Preliminary version of a Note to be published in a slightly abbreviated form in C. R. Acad. Sci. Paris, Ser. I, 338 (2004), pp. 975-980We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter $\varepsilon$ tends to zero. The cylinder is assumed to be fixed (homogeneous Dirichlet boundary condition) on the whole of one of its extremities, but only on a small part (of size $\varepsilon r^\varepsilon$) of the second one; the Neumann boundary condition is assumed on the remainder of the boundary. We show that the result depends on $r^\varepsilon$, and that there are 3 critical sizes, namely $r^\varepsilon=\varepsilon^3$, $r^\varepsilon=\varepsilon$, and $r^\varepsilon=\varepsilon^{1/3}$, and in total 7 different regimes. We also prove a corrector result for each behavior of $r^\varepsilon$

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Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities

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