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

arXiv:math/0407459v1 [math.AP] 27 Jul 2004 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. 975-980) Juan CASADO-D´ IAZa, Manuel LUNA-LAYNEZa, Franc¸ois MURATb aDepartamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Espa˜ na bLaboratoire Jacques-Louis Lions, Universit´ e Pierre et Marie Curie, boˆıte courrier 187, 75252 Paris cedex 05, France Abstract. We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter εtends 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 εrε) of the second one; the Neumann boundary condition is assumed on the remainder of the boundary. We show that the result depends on rε, and that there are 3 critical sizes, namely rε=ε3,rε=ε, and rε=ε1/3, and in total 7 different regimes. We also prove a corrector result for each behavior of rε.c XXXX Acad´emie des sciences/´ Editions scientifiques et m´edicales Elsevier SAS Comportement asymptotique d’une poutre ´ elastique fix´ ee sur une petite partie de l’une de ses extr´ emit´ es R´ esum´ e. Nous ´etudions le comportement asymptotique de la solution d’un probl`eme d’´elasticit´e lin´eaire anisotrope et h´et´erog`ene dans un cylindre dont le diam`etre εtend vers z´ero. Le cylindre est fix´e (condition de Dirichlet homog`ene) sur la totalit´e de l’une de ses extr´emit´es, mais seulement sur une petite partie (de taille εrε) de l’autre base; sur le reste de la fronti`ere on a la condition de Neumann. Nous montrons que le r´esultat depend de rε, et qu’il existe 3 tailles critiques, `a savoir rε=ε3,rε=εet rε=ε1/3, et au total 7 comportements diff´erents. Nous donnons un r´esultat de correcteur pour tous les comportements de rε.c XXXX Acad´emie des sciences/´ Editions scientifiques et m´edicales Elsevier SAS Version franc¸aise abr´ eg´ ee Dans cette Note, nous ´etudions le comportement asymptotique de la solution d’un probleme d’´elasticit´e lin´eaire anisotrope et h´et´erog`ene pos´e dans un cylindre Ωε= (0,1) ×εS ⊂R3dont le diam`etre εtend vers z´ero et dont l’axe est le premier axe de coordonn´ees Ox1. A l’une de ses extr´emit´es (x1= 1) le cylindre est fix´e (condition de Dirichlet homog`ene) sur la totalit´e de la base Γε 1={1} × εS, tandis qu’`a l’autre extr´emit´e (x1= 0), il est seulement fix´e sur une petite partie Γε 0={0} × εrεS0de la base, partie qui J. Casado-D´ıaz et al. Figure 1. est bien plus petite que εcar rεtend vers z´ero. Sur le reste de la fronti`ere de Ωε, on a la condition de Neumann (voir Figure 1). Pour Atenseur d’ordre 4 coercif `a coefficients continus dans Ω(Ω = (0,1) ×S), et f∈L2(Ω)3,h∈L2(Ω)3×3 sdonn´es, on d´efinit Aε,Fεet Hεpar (1) et (2) et on d´efinit Uεcomme la solution du probl`eme d’´elasticit´e (3). Le but de cette Note est de d´ecrire le comportement asymptotique de Uεet de donner un r´esultat de correcteur pour e(Uε)quand εet rεtendent vers z´ero. Ce r´esultat est d´ecrit dans le Th´eor`eme enonc´e dans la version anglaise ci-dessous, qui fait apparaitre 3 tailles critiques, `a savoir rε≈ε3,rε≈εet rε≈ε1/3, qui s´eparent 4 zones correspondant `a rε≪ε3,ε3≪rε≪ε,ε≪rε≪ε1/3et ε1/3≪rε⩽C, donc au total 7 cas diff´erents. Le r´esultat de convergence est donn´e par (5), o`u uest d´efini par la solution de (4), tandis que (6) est un r´esultat de correcteur pour e(Uε). Quand rε≪ε3, l’ensemble Γε 0={0} × εrεS0est si petit que la condition de Dirichlet impos´ee `a Uεpour x1= 0 disparaˆıt compl`etement `a la limite. Quand ε3≪rε≪ε, l’ensemble Γε 0est suffisamment grand pour qu’`a la limite on ait ζα(0) = 0 pour α∈ {2,3}. Quand ε≪rε≪ε1/3, on a en outre ζ1(0) = 0. Finalement, quand ε1/3≪rε, l’ensemble Γε 0est si grand que toutes les conditions de Dirichlet possibles sont satisfaites `a la limite, c’est `a dire ζα(0) = ζ1(0) = dζα dy1(0) = c(0) = 0 (voir la version anglaise pour les d´efinitions des espaces fonctionnels qui interviennent dans le probl`eme limite (4) et pour les d´efinitions de ces fonctions). Sauf quand rεest de taille critique (c’est `a dire quand rε≈ελavec λ= 3,1ou 1/3), la forme bilin´eaire Bqui intervient dans (4) et la fonction Pεqui intervient dans (6) sont nulles. Mais pour ces trois tailles critiques, Best une forme bilin´eaire coercive (en ζα(0) pour λ= 3, en ζ1(0) pour λ= 1, et en c(0) et dζα dy1(0) pour λ= 1/3, voir (7) de la version anglaise), de sorte que pour ces tailles critiques rε≈ελ,Best une p´enalisation, par rapport au cas rε≪ελ, des nouvelles conditions de Dirichlet qui apparaˆıtront pour rε≫ελ, ce qui r´etablit une certaine continuit´e dans la transition. La pr´esente Note est la g´en´eralisation au cas de l’´elasticit´e lin´eaire de l’´etude men´ee dans [2], [3] pour le probl`eme de diffusion analogue quand le param`etre tεqui intervient dans ces travaux est tε= 0. Les d´emonstrations d´etaill´ees seront donn´ees dans [4]. 1. Position of the problem and notation In this Note we study the asymptoticbehavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem posed in a thin cylinder Ωεwhose diameter εtends to zero and whose axis is the first axis of coordinates Ox1. On one of its extremities (x1= 1) the cylinder is fixed on its whole basis Γε 1whereas on the second one (x1= 0) it is fixed only on a small part Γε 0of it, of diameter εrεmuch smaller than ε. The Neumann boundary condition is assumed on the remainder of the boundary of Ωε. Mathematically the problem can be formulated as follows. For ε > 0, we consider rεa positive parameter which tends to zero with ε. Let S0and Sbe two bounded smooth domains in R2, with 0∈S. We define Ω = (0,1) ×S,Ωε= (0,1) ×εS and Γε= Γε 0∪Γε 1, where Γε 0={0} × εrεS0,Γε 1={1} × εS. Observe that the size of Γε 0is much smaller than the size of the basis {0} × εS since rεtends to zero. The thin cylinder Ωεis represented in Figure 1 in the case where S0and Sare both balls of R2. 2 An elastic beam fixed on a small part of one of its extremities The elements of R3are decomposed as x= (x1, x′),x1∈R,x′= (x2, x3)∈R2. We denote by {e1,e2,e3}the canonical basis of R3and by L(R3×3 s)the space of linear maps of R3×3 sinto itself (or in other terms of fourth order tensors), where R3×3 sis the space of the 3×3symmetric matrices. We adopt Einstein’s convention of repeated indices. Greek indices (αand β) take only the values 2and 3, while latin indices (iand j) take the values 1,2and 3. We consider A∈C0(Ω; L(R3×3 s)) such that there exists m > 0with A(y)ξξ ⩾m|ξ|2,∀ξ∈R3×3 s,∀y∈Ω, and we define Aε∈C0(Ωε;L(R3×3 s)) by Aε(x) = A(x1,x′ ε),∀x∈Ωε.(1) We also consider f∈L2(Ω)3and h∈L2(Ω)3×3 s, and we define Fε∈L2(Ωε)3and Hε∈L2(Ωε)3×3 sby Fε(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 domain Ωεwe consider the elasticity problem    Uε∈H1 Γε(Ωε)3, ZΩε Aεe(Uε) : e(Uε)dx =ZΩε FεUεdx +ZΩε Hε:e(Uε)dx, ∀Uε∈H1 Γε(Ωε)3,(3) where H1 Γε(Ωε) = U∈H1(Ωε) : U= 0 on Γε; observe that in the above formulation, as well as in the remainder of the present Note, complex numbers never appear, and that Uε(and later u,v,w) denotes the test function associated to the solution Uε, and not its complex conjugate. Observe also that the solution Uεof (3) satisfies a non homogeneous Neumann boundary condition on the part ∂Ωε\Γεwhere it is not fixed, since integrating by parts RΩεHε:e(Uε)dx (when hand thereforeHεis sufficiently smooth) produces both body forces and surface forces. Similarly to the body forces Fεwe could have introduced explicit surface forces Gεon ∂Ωε\Γε, but we have preferred not to include them for the sake of simplicity. It is well known that problem (3) has an unique solution (see, e.g., [5]). The aim of the present Note, which announces our paper [4], is to describe the asymptotic behavior of the solution Uεand to give a corrector result for e(Uε)as εtends to zero. The result depends on the behavior of rεand exhibits 3 critical sizes, namely ε3,ε, and ε1/3, so that there are 7 different regimes: rε≪ε3,rε≈ε3,ε3≪rε≪ε,rε≈ε, ε≪rε≪ε1/3,rε≈ε1/3, and ε1/3≪rε⩽C, where rε≪ελstands for rε/ελ→0(and equivalently ελ≪rεfor rε/ελ→+∞), while rε≈ελstands for rε/ελ→ρ, for some ρwith 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 by y=yε(x)with y1=x1, y′=x′ ε, which transforms the variable domain Ωεinto the fixed domain Ω. This is the usual change of variables used to study equations in thin cylinders (see, e.g., [6], [9], [10], [11]). When rε= 1 and S0=S(but the same proof works for rε=Cindependent of εsuch that CS0⊂S), it was used successfully in [6], [9], [10] to pass to the limit in (3). But when rεtends to zero with ε, this first change of variables does not provide the information we need about the behavior of Uεin the part of Ωεclose to Γε 0. Thus we introduce a second change of variables given by z=zε(x)with z=x εrε, which transforms the variabledomain Ωεinto a variable domain Zε, the limit of which is the half space Z= (0,+∞)×R2. Observe that the Dirichlet boundary condition is now imposed on the fixed part {0} × S0 3 J. Casado-D´ıaz et al. of the boundary of Zε. This change of variables provides a suitable rescaling near x1= 0. It was used successfully in [2], [3] to study the diffusion problem similar to (3) in the geometry that we consider in the present Note (and even in a more complicated one, where Ωεis made of union of two cylinders {(−tε,0) ×εrεS0}∪{(0,1) ×εS}and where the Dirichlet condition is imposed on x1=−tεand x1= 1; the geometry considered in the present Note corresponds to tε= 0; a problem of conduction in a notched beam of the same type was solved in [1] by using the same change of variables). We denote by D1,2(Z)the Deny’s space D1,2(Z) = p:p∈L6(Z),∇p∈L2(Z)3. We will also use the functional spaces (already used in [7], [8], [9], [10])        BNb(Ω) = nu:∃ζα∈H2(0,1), ζα(1) = dζα dy1 (1) = 0, uα(y) = ζα(y1),∀α∈ {2,3}, ∃ζ1∈H1(0,1), ζ1(1) = 0, u1(y) = ζ1(y)−dζα dy1 (y1)yαo,      Rb(Ω) = nv:v1∈L2(0,1; H1(S)),ZS v1(y1, y′)dy′= 0 a.e. y1∈(0,1), ∃c∈H1(0,1), c(1) = 0, v2(y) = c(y1)y3, v3(y) = −c(y1)y2o,        RD⊥ 2(Ω) = nw:w1= 0, wα∈L2(0,1; H1(S)),ZS wα(y1, y′)dy′= 0,∀α∈ {2,3}, ZS (y3w2(y1, y′)−y2w3(y1, y′)) dy′= 0 a.e. y1∈(0,1)o. For a given (u, v, w)∈BNb(Ω)×Rb(Ω)×RD⊥ 2(Ω), we denote by E(u, v, w)the second order symmetric tensor with values in R3×3 sdefined by E11(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 comments The asymptotic behavior of the solution of (3) depends on the size of rεwith respect to ε. Seven regimes appear in the following Theorem which describes the asymptotic behavior of Uεand provides a corrector result for Uεand e(Uε). THEOREM Let Uεbe the solution of (3). There exist a closed linear subspace Eof BNb(Ω)×Rb(Ω)× RD⊥ 2(Ω), a function Pε∈L2(Ωε)3×3 s, and a bilinear continuous nonnegative form Bon E × E such that, defining (u, v, w)as the solution of the variational problem            (u, v, w)∈ E, ZΩ AE(u, v, w) : E(u, v, w)dy +h B(u, v, w),(u, v, w)i= =ZΩ fudy +ZΩ h:E(u, v, w)dy, ∀(u, v, w)∈ E, (4) then, when εtends to zero, we have 1 |Ωε|ZΩε |Uε 1(x)−u1(x1,x′ ε)|2+ 3 X α=2 |εUε α(x)−uα(x1)|2!dx −→ 0,(5) 1 |Ωε|ZΩε |e(Uε)(x)−E(u, v, w)(x1,x′ ε)−Pε(x εrε)|2dx −→ 0.(6) 4 An elastic beam fixed on a small part of one of its extremities The definitions of E,Pεand Bdo not depend on the forces fand hwhich define Fεand Hε, but only on the set S0, on the fourth order tensor A, and on the behaviorof rεwhen εtends to zero, and more specifically of its behavior with respect to ε3,ε, and ε1/3, such that there are 7 regimes, which are described now. •If rε≪ε3, then E=BNb(Ω) ×Rb(Ω) ×RD⊥ 2(Ω), Pε= 0,B= 0. •If rε≈ε3with rε/ε3→ρ, then E=BNb(Ω) ×Rb(Ω) ×RD⊥ 2(Ω), and defining ϕα,α∈ {2,3}, as the solution of    ϕα∈D1,2(Z)3, ϕα=eαon {0} × S0, ZZ A(0)e(ϕα) : e(ϕ)dz = 0,∀ϕ∈D1,2(Z)3, ϕ = 0 on {0} × S0, then one has Pε(z) = −1 ε2rεζα(0) e(ϕα)(z),a.e. z∈Z, h B(u, v, w),(u, v, w)i=ρZZ A(0) (ζα(0) e(ϕα)) : ζβ(0) e(ϕβ)dz, ∀(u, v, w),(u, v, w)∈ E. •If ε3≪rε≪ε, then E=(u, v, w)∈BNb(Ω) ×Rb(Ω) ×RD⊥ 2(Ω) : ζα(0) = 0,∀α∈ {2,3}, Pε= 0,B= 0. •If rε≈εwith rε/ε →ρ, then E=(u, v, w)∈BNb(Ω) ×Rb(Ω) ×RD⊥ 2(Ω) : ζα(0) = 0,∀α∈ {2,3}, and defining ϕ1as the solution of    ϕ1∈D1,2(Z)3, ϕ1=e1on {0} × S0, ZZ A(0)e(ϕ1) : e(ϕ)dz = 0,∀ϕ∈D1,2(Z)3, ϕ = 0 on {0} × S0, and then setting ˆϕ1=ϕ1+aαϕα, where (a2, a3)is defined by    (a2, a3)∈R2, ZZ A(0) e(ϕ1) + aαe(ϕα):aβe(ϕβ)dz = 0,∀(a2, a3)∈R2, then one has Pε(z) = −1 εrεζ1(0) e( ˆϕ1)(z),a.e. z∈Z, h B(u, v, w),(u, v, w)i=ρZZ A(0) ζ1(0) e( ˆϕ1):ζ1(0) e( ˆϕ1)dz, ∀(u, v, w),(u, v, w)∈ E. •If ε≪rε≪ε1/3, then E=(u, v, w)∈BNb(Ω)×Rb(Ω)×RD⊥ 2(Ω) : ζα(0)=ζ1(0)=0,∀α∈{2,3}, Pε=0,B=0. •If rε≈ε1/3with (rε)3/ε →ρ, then E=(u, v, w)∈BNb(Ω) ×Rb(Ω) ×RD⊥ 2(Ω) : ζα(0) = ζ1(0) = 0,∀α∈ {2,3}, 5 J. Casado-D´ıaz et al. and defining ψ1as the solution of    ψ1∈D1,2(Z)3, ψ1=z3e2−z2e3on {0} × S0, ZZ A(0)e(ψ1) : e(ψ)dz = 0,∀ψ∈D1,2(Z)3, ψ = 0 on {0} × S0, and ψα,α∈ {2,3}, as the solution of    ψα∈D1,2(Z)3, ψα=z1eα−zαe1on {0} × S0, ZZ A(0)e(ψα) : e(ψ)dz = 0,∀ψ∈D1,2(Z)3, ψ = 0 on {0} × S0, and then setting ˆ ψi=ψi+bi kϕk, where (bi 1, bi 2, bi 3),i∈ {1,2,3}, is defined by    (bi 1, bi 2, bi 3)∈R3, ZZ A(0) e(ψi) + bi ke(ϕk):bi le(ϕl)dz = 0,∀(bi 1, bi 2, bi 3)∈R3, then one has Pε(z) = −1 εc(0) e(ˆ ψ1)(z) + dζα dy1 (0) e(ˆ ψα)(z),a.e. z∈Z,        h B(u, v, w),(u, v, w)i=ρZZ A(0) c(0) e(ˆ ψ1) + dζα dy1 (0) e(ˆ ψα): :c(0) e(ˆ ψ1) + dζα dy1 (0) e(ˆ ψα)dz, ∀(u, v, w),(u, v, w)∈ E. •If ε1/3≪rε⩽C, then E=n(u, v, w)∈BNb(Ω)×Rb(Ω)×RD⊥ 2(Ω) : ζα(0)= ζ1(0)= dζα dy1 (0)= c(0)= 0,∀α∈{2,3}o, Pε= 0,B= 0. Let us make some comments about the statement of this Theorem. Assertion (6) is a corrector result, since using Uε=Uεas test function in (3) and Korn’s inequality, one can prove that 1 |Ωε|ZΩε |e(Uε)|2dx ⩽C1 |Ω|ZΩ |f|2dx +1 |Ω|ZΩ |h|2dx, and since one can also prove that 1 |Ωε|ZΩε |E(u, v, w)(x1,x′ ε)|2dx +1 |Ωε|ZΩε |Pε(x εrε)|2dx ≈1. IfoneexaminesthedefinitionofE, one realizesthat thenumberofDirichletboundaryconditionsimposed in the definition of Eincreases with the size of rε. Indeed, in view of the definitions of BNb(Ω),Rb(Ω) and RD⊥ 2(Ω), the sole functionswhich have a trace for x1= 0 are ζα,ζ1,dζα dy1and c, where α∈ {2,3}(the other functions, namely v1and wα, have no trace for x1= 0). When rε≪ε3, the set Γε 0={0} × εrεS0is too small and the homogeneous Dirichlet boundary condition imposed for x1= 0 to the solution Uεof (3) completely disappears at the limit. When ε3≪rε≪ε, the set Γε 0is sufficiently large to impose at the limit that ζα(0) = 0 for α∈ {2,3}. When ε≪rε≪ε1/3, one further has ζ1(0) = 0. Finally when 6 An elastic beam fixed on a small part of one of its extremities ε1/3≪rε, the set Γε 0is so large that all the possible Dirichlet boundary conditions are imposed at x1= 0, namely ζα(0) = ζ1(0) = dζα dy1(0) = c(0) = 0. Except in the three regimes where the size of rεis critical (i.e. when rε≈ελ, with λ= 3,1, or 1/3), one always has Pε= 0 and B= 0. For these three critical sizes, one can show that Bis a coercive bilinear form, in the sense that there exists some n > 0such that                h B(u, v, w),(u, v, w)i⩾nρ X α=2,3 |ζα(0)|2,when rε≈ε3, h B(u, v, w),(u, v, w)i⩾nρ|ζ1(0)|2,when rε≈ε, h B(u, v, w),(u, v, w)i⩾nρ |c(0)|2+X α=2,3 dζα dy1 (0) 2!,when rε≈ε1/3. (7) This implies that for every critical size rε≈ελ, the new Dirichlet boundary conditions which appear for rε≫ελ(with respect to those imposed for rε≪ελ) are penalized by the value of ρ. This introduces some type of continuity in the transition of the Dirichlet condition between the two regimes which are separated by a critical size ελ. For these three critical sizes, the functions ϕα,ϕ1,ˆϕ1,ψi, and ˆ ψiare in some sense generalized capacitary potentials of {0} × S0in Z, and the bilinear form Bis in some sense an asymptotic trace of some type of capacity of Γε 0in Ωεfor the weighted energy 1 |Ωε|RΩεA(x)e(ϕ) : e(ϕ)dx. The present work is the natural generalization to the elastic case of [2], [3], where diffusion problems were posed in the union of two cylinders {(−tε,0) ×εrεS0}∪{(0,1) ×εS}, with both tεand rεtending to zero (the present geometry corresponds to tε= 0). When tε= 0, the diffusion problem was in comparison more simple, since only one critical size, namely rε≈ε, appeared in the analysis, separating a pure Neumann boundary condition [corresponding to the analogue of usatisfying u∈H1(0,1),u(1) = 0] for rε≪εand a pure Dirichlet boundary condition [corresponding to the analogue of usatisfying u∈ H1(0,1),u(0) = u(1) = 0] for rε≫ε. These works were related to [1], where a notched beam for diffusion problems was considered. The present work is also related to [7], [8], where a multidomain made of an elastic vertical beam of length 1 and of radius rεand of an horizontal plate of radius 1 and of height ε was considered. The detailed proofs of the results of the present Note will be given in a forthcoming paper [4]. Acknowledgements. This work has been partly supported by the project BFM2002-00672 of the DGI of Spain and by the HMS2000 Training and Research Network. It was begun during a visit of the third author to the Departamento de Ecuaciones Diferenciales y An´alisis Num´erico de la Universidad de Sevilla. References [1] Casado-D´ıaz J., Murat F., The diffusion equation in a notched beam, to appear. [2] Casado-D´ıaz J., Luna-Laynez M., Murat F., Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths, C. R. Acad. Sci. Paris, Ser. I, 338 (2004), 133-138. [3] Casado-D´ıaz J., Luna-Laynez M., Murat F., Diffusion problems in a domain made of two thin cylinders of different diameters and lengths, to appear. [4] Casado-D´ıaz J., Luna-Laynez M., Murat F., Elasticity problems in a beam fixed on only a small part of one of its extremities, to appear. [5] Ciarlet P.G., Mathematical elasticity, Vol. I: Three-dimensional elasticity, North-Holland, 1988. 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