TORSION EFFECTS IN ELASTIC COMPOSITES WITH HIGH ...

Author manuscript, published in "SIAM Journal on Mathematical Analysis 41, 6 (2010) 2514-2553"

TORSION EFFECTS IN ELASTIC COMPOSITES WITH HIGH CONTRAST MICHEL BELLIEUD

∗

Abstract. We establish a homogenization result and a corrector result for a vibration problem of elasticity. We assume that the data depend in a periodic way on a small parameter ε. We assume also that the Lam´ e coefficients take possibly high values in a periodical set of disconnected inclusions and take values of the order ε2 elsewhere. In the fibered case, torsional vibrations take place at an infinitesimal scale and give rise to non-local effects. Key words. homogenization, elasticity, non-local effects

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AMS subject classifications. 35B27, 35B40, 74B05, 74Q10

1. Introduction . In this paper, we analyze the behavior of solutions to initial boundary value problems describing vibrations of periodic elastic composites with rapidly varying elastic properties. More specifically, we analyze a two-phase medium whereby a set of ”stiff” unbounded fibers or bounded inclusions is embedded in a ”soft” matrix, i.e. what is often referred to as the ”high contrast case”. This task is set in the context of linearized elasticity. Problems of a high-contrast type have been studied extensively over the last decades. Nowadays, there are two main trends in asymptotic methods: the asymptotic expansions and the two-scale convergence. The first approach [14], [25], [26], [29], [30] gives often stronger results including all asymptotic information about the solution and error estimates of higher order with respect to small parameters. It also contains the formulation of strong rigorous theories, but requires sufficiently regular data and boundaries. Let us mention in particular the detailed paper [28] of G. Sandrakov, yielding full proofs of the convergence and the error estimates for various high contrast asymptotic and geometric regimes in hyperbolic elastic problems. Let us mention also a most recent work [5] on the application of the asymptotic approach to some scalar spectral problems with high contrasts in both ”stiffness” and ”density”, with rigorous convergence results and error bound obtained. The second approach [2], [4], [7], [9], [11], [13], [31] , employed in our paper, also yields the convergence to an asymptotic solution and a first order corrector result. It requires much less smoothness of the data but it does not allow to obtain any error estimates with respect to small parameters. Notice that the papers [13] and [14] apply the asymptotic expansions and the twoscale convergence respectively to the same problem: as a result, [13] ends with stronger results but for more regular boundaries. We are aiming at complementing this extensive material. From the point of view of what is already available on the subject in the litterature, the most challenging case is that of a set of disconnected parallel fibers with elastic moduli of order 1 embedded in a ”soft” matrix with moduli of order ε2 , where ε is the period of the medium in the plane transverse to the fibers. We will focuse on the vibratory case. However, we emphasize that our analysis goes through in the same way in the case of equilibrium equations. The results obtained in this way are relevant to Example II and to Example III of the paper [9] by the author with G. Bouchitt´e, where fibered structures with elastic moduli respectively of order 1 and of order ε12 embedded in a ∗ D´ epartement de Math´ ematiques, Universit´ e de Perpignan, 52 Av. Paul Alduy, 66860 Perpignan Cedex, France.([email protected]).

1

2

M. BELLIEUD

”soft” matrix were considered. We agree with the result obtained in Example III and we find that the result obtained in Example II is false. Indeed, the effective energy functional obtained in [9], Th. 2.4 turns out to be only a lower bound of the actual effective energy functional. We prove that the latter functional includes additional terms describing torsional stored energy (see Section 5). The study of the torsion effects is, essentially, the main new contribution our manuscript aims to target. We turn now to a more detailed introduction of the paper. For a given bounded smooth open subset Ω of R3 , we consider the vibration problem

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(1.1)

 ∂ 2 uε   − div(σ ε (uε )) = ρε f in Ω×(0, T ), (f ∈ L2 (0, T ; L2 (Ω, R3 ))), ρ  ε  ∂t2      σ (u ) = λ tr(e(u ))I + 2µ e(u ), e(u ) = 1 (∇u + ∇T u ), ε ε ε ε ε ε ε ε ε 2    uε ∈ C(0, T ; H01 (Ω, R3 )) ∩ C 1 (0, T ; L2 (Ω, R3 )),       uε (0) = a0 , ∂uε (0) = b0 , (a0 , b0 ) ∈ H 1 (Ω, R3 ) × L2 (Ω, R3 ). 0 ∂t

We assume that the Lam´e coefficients λε , µε take values of order 1 in an ε-periodic subset Bε of Ω consisting of parallel disjoint cylinders of Lebesgue measure of order 1 and take values of order ε2 in the surrounding matrix. Heuristically, the norm of the gradient of the solution uε of (1.1) is expected to take high values, of the order 1 ε , in the parts of the body where the coefficients are small. So, a gap between the mean displacement of the different constituent parts of the composite may take place, originating the non-local nature of the effective problem (see Remark 2.2 (i)). A commonly-used method consists in expressing the homogenized problem under the guise of a system of equations involving, besides the limit u0 of the sequence (uε ), the limit v of an auxiliary sequence (v ε ) (see (2.16)) designed to characterize the average displacement in the inclusions. It turns out (see Theorem 2.1) that torsional vibrations take place at a microscopic scale in the fibers constituting the composite material. They are described in terms of the limit θ of the sequence (θε ) defined by (2.16), which characterizes the effective rescaled angle of torsion of the fibers (see Remark 2.2 (iv)). The functions v and θ are defined on Ω×(0, T ) and take values respectively in R3 and R. The function u0 : Ω×(0, T )×(− 21 , 12 )3 → R3 is the two-scale limit of (uε ) (see [2], [23]). The effective displacement in the cylinders is governed by the coupled system of equations in Ω×(0, T )

(1.2)

    2 ∂2θ ∂2v ρ∂ θ   .e3 + m(u0 ).e3 ,  J ∂t2 − kJ ∂x2 = ρ1 (y G − y B ) ∧ f − ∂t2 3 2 2 2    ρ1∂ v −k|B| 3l + 2 ∂ v3 e3 = ρ1 f + g(u0 ) − ρ1 ∂ θ e3 ∧ (y G − y B ), ∂t2 l + 1 ∂x23 ∂t2

associated with the boundary and initial conditions given in (2.19), the constants k, Jρ , J, y G , y B , ρ1 being defined by (2.2), (2.9), (2.12). The first equation of (1.2), regarding θ, displays the torsional vibrations. The third component of the second equation shows extensional vibrations with regard to the longitudinal displacement v3 (see [20], p. 428-429). The coupling with the matrix is marked by the fields g(u0 ) and m(u0 ). They represent respectively the sum of the surface forces applied on each fiber by the surrounding medium and their total moment with respect to the center

TORSION EFFECTS IN ELASTIC COMPOSITES

3

of gravity of the geometric fiber. They are defined by (2.3), (2.4) in terms of the restriction to Ω×(0, T )×(Y \ B) of u0 , which characterizes the effective displacement in the matrix. The letters Y and B symbolize respectively the unit cell and the rescaled fiber. The effective displacement in the matrix is governed by the equation

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(1.3)

ρ

∂ 2 u0 − divy (σ 0y (u0 )) = ρf ∂t2

in Ω×(0, T )×(Y \ B),

coupled with the variables v, θ by the relation u0 = v + θe3 ∧ (y − y B ) in B, where ρ stands for the strong two-scale limit of the mass density (ρε ) and σ 0y is defined 2 by R (2.3). The weak limit in L of (uε ) satisfies the non-explicit equation u(x, t) = u0 (x, t, y)dy. We obtain corrector results (see (2.25) and Remark 2.2 (iv)). Y When the order of magnitude of the elasticity coefficients in the fibers is larger (namely when k := limε→0 µ1ε = +∞), the functions θ and v3 are equal to zero and the effective displacement in the fibers is governed by the system of equations of v1 , v2 given, in terms of the order of magnitude of the parameter κ := limε→0 ε2 µ1ε , by (2.20), (2.21) or (2.22). In the most interesting case 0 < κ < +∞, already investigated in the context of elliptic equations for fibers with a circular cross-section (see [9], Th. 2.5), this system involves the 4th derivative of v1 , v2 with respect to x3 , revealing bending effects (see [20], p. 430 ) similar to those studied in [10], [27]. Otherwise, the fibers display the behavior of a collection of unstretchable strings that do not twist if κ = 0 and k = +∞ and that of fixed bodies if κ = ∞. If Bε consists of totally disconnected particles, the particles behave asymptotically like rigid bodies regardless of the order of magnitude (≥ 1) of their stiffness. Their effective displacement is governed by the system of equations (3.6), where the field r, obtained as the limit of the sequence (r ε ) defined by (3.3), describes their effective rotation vector (in the fibered case, r = θe3 ). The displacement in the matrix is governed by the equation (1.3) coupled with v, r by the equation u0 = v+r ∧(y−y B ) in Ω×(0, T )×B. Grain-like inclusions have been also considered by G. P. Panasenko [26] and G. V. Sandrakov [28] by using the asymptotic approach. We can extend these results to the case of a multiphase medium comprising a finite collection Bε1 ,.., Bεm of non-intersecting ε-periodic families of grain-like inclusions or of fibers of various shapes and stiffness embedded in a ”soft” matrix, each family of fibers being for simplicity parallel to one of the coordinate axes. The effective displacement in Bεi is described in terms of a couple (v i , r i ) and governed by a system P hom i similar, up to a rotation of the coordinate axes, to one of the systems (1.2), (2.20), (2.21), (2.22), (3.6) depending on the shape and on the order of magnitude of the elastic moduli in the specified inclusions (see Section 4). The displacement in the matrix is governed by the equation (1.3), where B = B 1 ∪ ... ∪ B m . The coupling of P hom i with the matrix is marked by the equation u0 = v i +r i ∧(y −y B i ) in B i and by the presence of fields g i (u0 ) and mi (u0 ) in P hom i (see Section 4). Multiphase homogenized models have been also considered in [25], [26], [28], [29], [30]. The two-phase models of composites obtained theoretically by our process of homogenization turn out to be unsufficiently reinforced, in general, to resist to some specific body forces. More precisely, in the elliptic case, the boundedness in L2 (Ω; R3 ) of the solutions may fail to hold depending on f and, in the corresponding hyperbolic case, the effective equations may describe a motion of collapse. From a physical point of view, finding conditions ensuring the obtention of an effective elastic composite sufficiently reinforced to resist to body forces is an important task. We show (see Proposition 5.2) that the last mentioned boundedness is guaranteed for any choice

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M. BELLIEUD

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of the field of body forces f ∈ L2 (Ω; R3 ), if and only if a multiphase composite is considered whereby the set of inclusions comprises either one family of parallel fibers with elastic moduli of order ε12 , or three families of parallel fibers with elastic moduli of order 1 distributed in three independent directions. Hence, although two-phase media offer the convenient setting for the mathematical study of torsion effects, only multiphase media are likely to provide a physically satisfactory model of an elastic composite exhibiting torsion effects. The paper is organised as follows: the notations and the results relating to the fibered case are displayed in Section 2, those concerning grain-like inclusions are stated in Section 3. The case of multiphase media and of equilibrium equations are discussed respectively in Section 4 and in Section 5. Section 6 is devoted mainly to a priori estimates in the fibered case. The proofs of the main Theorem 2.1 (fibered case) and Theorem 3.1 (case of grain-like inclusions) and a sketch of the proof of Proposition 5.2 are presented respectively in Section 7, Section 8 and Section 9. 2. Fibered case. In the sequel, {e1 , e2 , e3 } stands for the canonical basis of R3 . Vectors and vector-valued functions are represented by symbols beginning by a boldface lower case letter (examples: u, f , g, div(σ),...). For any vector u ∈ R3 , P3 P3 we denote by ui or (u)i its components (that is u = i=1 ui ei = i=1 (u)i ei ). We do not use the repeated index convention for summation. We denote by (εijk ) P3 the orientation tensor and by u ∧ v = i,j,k=1 εijk uj vk ei the exterior product in 3 R . Matrices and matrix-valued functions are represented by symbols beginning by a boldface upper case letter with the following exceptions: ∇u (displacement gradient), e(u) (linearized strain tensor), σ(u) (linearized stress tensor). We denote by A : B = P3 i,j=1 Aij Bij the inner product of two matrices. We denote by C different constants whose precise values may vary. Fixing a non-empty connected open set D ⊂ R2 with a Lipschitz boundary, we set

(2.1)

 2    3 3 X 1 1 1 1 1 1 yi ei , , B := D × − , , Y := − , , y := ρD ⊂ − , 2 2 2 2 2 2 i=1 Z |B| :=

dy, y B := B

(2.2)

1 |B|

Z

Z ydy, J :=

B

|e3 ∧(y−yB )|2 dy,

B

Z Jαβ :=

(y − yB )α (y − yB )β dy. B

Denoting by S3 the set of all real symmetric matrices of order 3, we introduce the operators ey , σ 0y : H 1 (Y ; R3 ) → L2 (Y ; S3 ), g : H → R3 , m : H → R3 defined by   1 ∂wi ∂wj (ey (w))ij = + , σ 0y (w) := λ0 tr(ey (w))I + 2µ0 ey (w), 2 ∂yj ∂yi Z (2.3) g(w) := σ 0y (w).nB dH2 (y), ∂B∩Y Z m(w) := (y−y B ) ∧ (σ 0y (w).nB ) dH2 (y), ∂B∩Y

where nB stands for the outward pointing normal to ∂B, λ0 , µ0 are positive reals, and

TORSION EFFECTS IN ELASTIC COMPOSITES

(2.4)

5

 H := w ∈ H 1 (Y \ B; R3 ), div(σ 0y (w)) ∈ (H 1 (Y \ B; R3 ))0 ,

the symbol E 0 indicating the continuous dual of a Banach space E.
We denote by C]∞ (Y ) (resp. C] (Y )) the set of Y -periodic functions of C ∞ R3 (resp. C(R3 )), by C]∞ (Y \ B) the set of the restrictions of the elements of C]∞ (Y ) to Y \ B, by H]1 (Y ) (resp. H]1 (Y \ B)) the completion of C]∞ (Y ) (resp. C]∞ (Y \ B)) with respect  R  12 
12 R 2 2 2 2 to the norm w → Y (|w| + |∇w| )dy resp. w → Y \B (|w| + |∇w| )dy . Our proofs are based on the two-scale convergence method of G. Allaire [2] and G. Nguetseng [23] . A sequence (fε ) in L2 (0, T ; L2 (Ω)) is said to be two-scale convergent to f0 ∈ L2 (0, T ; L2 (Ω × Y )) with respect to x (notation: fε * * f0 ) if for each ϕ0 ∈ D(Ω×(0, T ), C]∞ (Y )), there holds

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Z (2.5)

fε (x, t)ϕ0

lim

ε→0

Ω×(0,T )



x dxdt = x, t, ε

Z f0 ϕ0 dxdtdy. Ω×(0,T )×Y

A sequence (ϕε ) ⊂ L2 (0, T ; L2 (Ω)) is said to be two-scale strongly convergent to ϕ0 ∈ L2 (0, T ; L2 (Ω × Y )) (notation ϕε −→ −→ ϕ0 ) if (2.6)

ϕε * * ϕ0

and

lim ||ϕε ||L2 (0,T ;L2 (Ω)) = ||ϕ0 ||L2 (0,T ;L2 (Ω×Y )) .

ε→0

The symbols * * and −→ −→ will be used also to denote the two-scale convergence and the strong two-scale convergence of sequences (fε ) in L2 (Ω) independent of t or functions of x only by formally regarding those as constant in t. We consider the vibration problem (1.1), where Ω := ω × 0, L, ω is a bounded regular domain of R2 and Bε is the ε-periodic set of parallel cylinders defined by (see fig. 1) (2.7)

Bε := Ω ∩ ε

[

({i} + B).

i∈Z3

We assume that the Lam´e coefficients satisfy

(2.8)

µε (x) = µ1ε 1Bε (x) + ε2 µ0 1Ω\Bε (x), λε (x) = λ1ε 1Bε (x) + ε2 λ0 1Ω\Bε (x), λ1ε , lim lε = l ∈ 0, +∞. µ1ε ≥ c > 0, lε := ε→0 µ1ε

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M. BELLIEUD

We set (2.9)

κ := lim ε2 µ1ε .

k := lim µ1ε , ε→0

ε→0

Under (2.8), the relative compactness of the sequence (uε ) of the solutions of (1.1) in the ?-weak topology of L∞ (0, T ; L2 (Ω; R3 )) is ensured by a0 = 0 if µ1ε >> 1, (2.10)

inf ρε > c > 0 Bε

or

0 ≤ ρε ≤ C < +∞ if {b0 6= 0} or {f 6= 0}, inf ρε > c > 0,

if κ = 0.

Ω\Bε

We suppose that

ρε −→ −→ ρ,

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(2.11)

for some ρ ∈ L2 (Ω × Y ). The effective mass, the positions of the principal axes, the positions of the geometric principal axes and the moments of inertia with respect to the last mentioned axes of the fibers are characterized respectively by the constants ρ1 , y G , y B , Jρ defined by (2.2) and Z

Z

ρ1 :=

ρdy,

ρ1 y G :=

B

(2.12) Jρ :=

Z

ρydy,

(y G = y B if ρ1 = 0),

B

ρ|e3 ∧ (y − y B )|2 dy.

B

We assume that (see Remark 2.2 (iii))  (2.13)

D=

(y1 , y2 ) ∈ R2 ,

q

y12 + y22 < R

 if

lim inf εµ1ε < +∞, ε→0

  for some R ∈ 0, 12 . For simplicity the main result is stated under the additional hypotheses (see Remark 2.2 (v))

(2.14)

ρε ≥ c > 0,

(2.15)

ρε ≤ C < +∞.

Representing by Int(s) the integer part of a real s, we set 1 uε (x, t)1Bε (x), |B|  h x i  1 − y B 1Bε (x), θε (x, t) := uε (x, t). e3 ∧ J ε  3 h hxi X hx i x  x xi i 1 i i i := ei , := − Int + . ε ε ε ε ε 2 i=1 v ε (x, t) :=

(2.16)

TORSION EFFECTS IN ELASTIC COMPOSITES

7

Under these assumptions, we show that (uε ,v ε ,θε ) converges, in the sense defined below, to (u0 , v, θ) (a geometrical interpretation of θ is given in Remark 2.2 (iv)) of ( hom (Pmatrix ), (2.17) (Pfhom ibers (k, κ)), where, setting r := θe3 and denoting by n the outward pointing normal to ∂Y ,

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 2 ∂ u0   in Ω×(0, T )×(Y \ B), ρ 2 − divy (σ 0y (u0 )) = ρf   ∂t     in Ω×(0, T )×B,   u0 = v + r ∧ (y − y B ) hom on Ω×(0, T ) × ∂Y, (2.18) (Pmatrix ) : σ 0y (u0 ).n(y) = −σ 0y (u0 ).n(−y)   2 1 3 1   u0 ∈ C([0, T ]; L (Ω, H] (Y ; R ))) ∩ C ([0, T ]; L2 (Ω × Y ; R3 )),      ∂u0  u (0)1 (0)1Y \B = b0 1Y \B , 0 Y \B = a0 1Y \B , ∂t and (Pfhom ibers (k, κ)) is given, in terms of the order of magnitude of the coefficients k, κ, by

(2.19)

 ∂2θ ∂2θ    Jρ 2 − kJ 2   ∂t ∂x3         ∂2v   = ρ1 (y G − y B ) ∧ f − 2 .e3 + m(u0 ).e3   ∂t      in Ω×(0, T ),     2 2  3l + 2 ∂ v3  ∂ v (Pfhom ibers (k, 0)) : e3 ρ1 2 − k|B| ∂t l + 1 ∂x23  (0 < k < +∞)    ∂2θ   ρ f + g(u ) − ρ e3 ∧ (y G − y B ) in Ω×(0, T ), =  0 1 1  ∂t2      v3 , θ ∈ C([0, T ]; L2 (ω; H01 (0, L))) ∩ C 1 ([0, T ]; L2 (Ω)),      v ∈ C 1 ([0, T ]; L2 (Ω; R3 )),        θ(0) = 0, ∂θ (0) = 0, v(0) = a0 , ∂v (0) = b0 , ∂t ∂t

 ∂ 2 vα    ρ1 2 = ρ1 fα + (g(u0 ))α α ∈ {1, 2} in Ω×(0, T ),  ∂t  (2.20) (Pfhom (+∞, 0)) : v ∈ C 1 ([0, T ], L2 (Ω; R3 )), ibers      vα (0) = 0, ∂vα (0) = (b0 )α , α ∈ {1, 2}, v3 = θ = 0, ∂t  2  ∂ 4 vβ ∂ 2 vα X 3l + 2    ρ1 2 + κ Jαβ   ∂t l+1 ∂x43   β=1   (Pfhom ibers (+∞, κ)) : = ρ1 fα + gα (u0 ), α ∈ {1, 2} in Ω × (0, T ), (2.21)  (0 < κ < +∞)  2 2 3 1 2 3   v ∈ C([0, T ]; L (ω, H0 (0, L; R )))∩C ([0, T ]; L (Ω; R )),     ∂vα   vα (0) = 0, (0) = (b0 )α , α ∈ {1, 2}, v3 = θ = 0, ∂t

8

(2.22)

M. BELLIEUD

(Pfhom ibers (+∞, +∞)) :

v = 0,

θ = 0.

We establish the corrector result (2.25) under the assumption (see Remark 2.2 (iv)) (2.23)

a0 = 0,

 x u0 x, t, −→ −→ u0 . ε

Theorem 2.1. Assume (2.1), (2.7), (2.8), (2.10), (2.11), (2.13), (2.14), (2.15), let (uε ) be the sequence of the solutions of (1.1) and let (v ε ), (θε ) be defined by (2.16). Then (uε ) two-scale converges to u0 with respect to x and (uε , v ε , θε ) converges starweakly in (L∞ (0, T ; L2 (Ω, R3 )))2 × L∞ (0, T ; L2 (Ω)) to (u, v, θ), where Z u :=

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u0 (., y)dy,

B

Y

(2.24)

1 θ= J

Z v = − u0 (., y)dy,

Z u0 (., y).(e3 ∧ (y − y B ))dy. B

The triple (u0 , v, θ) is the unique solution of (2.17). Moreover, (uε (τ )) two-scale converges to u0 (τ ) with respect to x, for each τ ∈ 0, T . Assume in addition (2.23), then (uε ) two-scale converges strongly to u0 and  x  (2.25) lim uε − u0 x, t, = 0. ε→0 ε L2 (Ω×(0,T );R3 ) Remark 2.2.

(i) If 0 < k < +∞, the variable θ satisfies the vibrating string equation q kJ 0 0 − (0) = 0, θ(x , 0, t) = θ(x , L, t) = 0, where c := = h, θ(0) = ∂θ ∂t Jρ ,     2 h := J1ρ ρ1 ((y G − y B ) ∧ f ) .e3 + m(u0 ).e3 − ρ1 (y G − y B ) ∧ ∂∂tv .e3 , hence is 2 given by ∂2θ ∂t2

(2.26)

∂2θ c2 ∂x 2 3

r Z t +∞   nπ   cnπ X 2 L (t − τ ) γn (x1 , x2 , τ )dτ sin x3 , θ(x, t) = sin cnπ L L L 0 n=1 r Z L  nπ  2 γn (x1 , x2 , t) = h(x1 , x2 , x3 , t) sin x3 dx3 . L L 0

The substitution of (2.26) in (2.18), (2.19) reveals the presence of memory terms in the limit problem. Memory effects induced by homogenization are studied also in [1], [3], [21], [32]. More generally, non-local effects are likely to come about in composites with high contrast [2], [4]-[7], [9]-[11], [13], [14], [17], [25], [28]-[31]. In the case of scalar linear elliptic equations, they can be interpreted in the context of Dirichlet forms [22]. This approach breaks down in the framework of linear elasticity, any non-negative lower-semicontinuous quadratic form on L2 (Ω; R3 ) being theoretically the limit of a suitable sequence of linear elasticity functionals on H 1 (Ω; R3 ) [12]. Passing from stationary to evolution equations, memory effects can add further to the possible non-local effects attendant on the elliptic case, even though the homogenization of the corresponding equilibrium equations leads to a classical local problem [10], Remark 3.2; [7], Remark 2.2 (v).

TORSION EFFECTS IN ELASTIC COMPOSITES

9

(ii) If the fibers have a vanishing measure (i.e. rε > 1 (see (6.49)). If (2.14) is not satisfied, some modifications of the data related to time in (2.17) are possibly required. 3. Case of grain-like inclusions . In this section, we assume that Ω and B are regular domains of R3 , and that (see fig. 2)  (3.1)

B ⊂ Y :=

1 1 − , 2 2

3 .

The relative compactness of the sequence of the solutions of (1.1) in the ?-weak topology of L∞ (0, T ; L2 (Ω; R3 )) is ensured by the assumptions (2.8), (2.10) and

10

M. BELLIEUD

(3.2)

inf ρε > c > 0

or

Bε

inf ρε > c > 0.

Ω\Bε

We introduce the inertia matrices J ρ , J and the sequence (r ε ) given by ρ := − Jij

Z ρ(y − y B )i (y − y B )j dy,

if

i 6= j,

ZB Jij := −

(y − y B )i (y− y B )j dy, B

(3.3) Jiiρ :=

XZ

ρ|(y−y B )j |2 dy,

if i 6= j, XZ Jii := |(y−y B )j|2 dy,

j6=i B

j6=i B

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  hxi −y B ∧uε 1Bε . r ε := J −1 ε Theorem 3.1. Assume (2.8), (2.10), (2.11), (2.14), (2.15), (3.1), (3.2), let (uε ) be the sequence of the solutions of (1.1) and let (v ε ), (r ε ) be defined by (2.16), (3.3). Then the sequence (uε ) two-scale converges to u0 with respect to x and the sequence (uε , v ε , r ε ) converges star-weakly in (L∞ (0, T ; L2 (Ω, R3 )))3 to the triple (u, v, r) given by (2.24) and Z

(3.4)

r=J

−1

 (y − y B ) ∧ u0 (., y)dy .

B

The triple (u0 , v, r) is the unique solution of the system ( (3.5)

hom (Pmatrix ), hom (Pinclusions ),

hom where (Pmatrix ) is given by (2.18) and

   2 ∂ v ∂2r   + ∧ (y − y ) ρ  1 G B  ∂t2 ∂t2      = ρ1 f + g(u0 ) in Ω×(0, T ),     2 2   ρ ∂ r ∂ v hom (3.6) (Pinclusions ) : J . ∂t2 + ρ1 (y G − y B ) ∧ ∂t2    = ρ1 (y G − y B ) ∧ f + m(u0 ) in Ω×(0, T ),     1 2 3    v, r ∈ C (0, T ; L (Ω; R )),    ∂v ∂r   v(0) = a0 , (0) = b0 , r(0) = (0) = 0. ∂t ∂t Moreover, (uε (τ )) two-scale converges to u0 (τ ) for each τ ∈ [0, T ]. Assume in addition (2.23), then (uε ) two-scale converges strongly to u0 and the corrector result (2.25) holds. Remark 3.2. (i) Grain-like inclusions are concerned as well with Remark 2.2 (i), (iv), (v). Regarding (ii), memory effects are obtained with particles of high mass density and diameter rε 0 is bounded in L2 (Ω; R2 ). 6. Preliminary results and a priori estimates.. The following section is devoted to the study, in the fibered case, of the asymptotic behavior of the sequence (uε ) of the solutions of (1.1) and of the sequences (v ε ) and (θε ) defined by (2.16) (cf. Proposition 6.4). It includes also a technical lemma (Lemma 6.1) concerning the twoscale convergence and a theorem (Theorem 6.2) gathering some classical theoretical results about hyperbolic equations that will be employed to establish the well-posed nature of Problem (2.17) and the corrector result (2.25). A fundamental property of the two-scale convergence (defined by (2.5)) is that any sequence bounded in L2 (0, T ; L2 (Ω)) admits a two-scale convergent subsequence. A sequence (ϕε ) ⊂ L2 (0, T ; L2 (Ω)) is said to be admissible if it two-scale converges to some ϕ0 ∈ L2 (0, T ; L2 (Ω × Y )) and if, for every two-scale convergent sequence (fε ), there holds Z Z (6.1) lim fε ϕε dxdt = f0 ϕ0 dxdtdy. ε→0

Ω×(0,T )

Ω×(0,T )×Y

16

M. BELLIEUD

It turns out that the set of all admissible sequences is equal to the set of all sequences (ϕε ) ⊂ L2 (0, T ; L2 (Ω)) satisfying (2.6) for some ϕ0 ∈ L2 (0, T ; L2 (Ω × Y )) (that is the set of all two-scale strongly convergent sequences). Indeed, the following implication is proved in [[2], Theorem 1.8] fε* *f0 Z lim

(6.2)

ε→0

ϕε −→ −→ϕ0 Z fε ϕε dxdt =

and

Ω×(0,T )

⇒ f0 ϕ0 dxdtdy.

Ω×(0,T )×Y

hal-00777686, version 1 - 17 Jan 2013

Conversely, if (ϕε ) is admissible, one sees by substituting ϕε for fε in (6.1) that (ϕε ) is two-scale strongly convergent. Lemma 6.1. (i) Let h0 ∈ L∞ (0, T ; L∞ (Ω, C] (Y ))) ∪ L∞ ] (Y, C(Ω×(0, T ))) and let
x 2 hε (x, t) := h0 x, t, ε . Then for every sequence (χε ) ⊂ L (0, T ; L2 (Ω)) the following implications hold: (6.3)

χε −→ −→ χ0

=⇒

χε hε −→ −→ χ0 h0 ,

(6.4)

χε * * χ0

=⇒

χε hε * * χ0 h0 .

(ii) If (fε ) is bounded in L∞ (0, T ; L2 (Ω)) and two-scale converges to f0 , then f0 ∈ 1,∞ L∞ (0, T ; L2 (Ω × Y )). If in addition (0, T ; L2 (Ω)), then f0 ∈   (fε ) is bounded in W

W 1,∞ (0, T ; L2 (Ω × Y )) and

∂fε ∂t

two-scale converges to

∂f0 ∂t .

Besides, if fε (0)* *a0 ,   ε 0 then a0 = f0 (0) and (fε (τ )) * * f0 (τ ), ∀τ ∈ [0, T ]. Moreover, if ∂f −→ −→ ∂f ∂t ∂t and fε (0)−→ −→a0 , then (fε (τ ))−→ −→f0 (τ ), ∀τ ∈ [0, T ]. Proof. (i) Assuming (χε )−→ −→χ0 , we fix a sequence (fε ) bounded in L2 (0, T ; 2 L (Ω)), a positive real η > 0 and a function ψ0 ∈ C(Ω×(0, T ), C] (Y )) such that (6.5)

fε * * f0 ,

||χ0 − ψ0 ||L2 (0,T ;L2 (Ω×Y )) < η.

Since h0 ψ0 ∈ L2 (0, T ; L2 (Ω, C]∞ (Y ))) ∪ L2] (Y, C(Ω×(0, T ))), the sequence (hε ψε ) (ψε (x, t) := ψ0 (x, t, xε )) is admissible with respect to the two-scale convergence (see [2], Lemma 5.2, Corollary 5.4). Thanks to (6.5) and to the strong two-scale convergence of (χε − ψε ) to χ0 − ψ0 we infer Z Z lim sup χε hε fε dxdt − χ0 h0 f0 dxdtdy ε→0 Ω×(0,T ) Ω×(0,T )×Y Z ≤ lim sup hε (χε − ψε ) fε dxdt ε→0 Ω×(0,T ) Z Z + lim sup hε ψε fε dxdt − χ0 h0 f0 dxdtdy ε→0 Ω×(0,T ) Ω×(0,T )×Y Z ≤ lim sup ||hε ||L∞ ||χε − ψε ||L2 ||fε ||L2 + h0 (ψ0 − χ0 )f0 dxdtdy Ω×(0,T )×Y ε→0 ≤ Cη,

TORSION EFFECTS IN ELASTIC COMPOSITES

17

hence χε hε −→ −→ χ0 h0 . Supposing now (χε ) * * χ0 , fixing a sequence Z (ϕε ) such that (ϕε )−→ −→ϕ0 , we deduce from (6.3) that (ϕε hε )−→ −→ϕ0 h0 , thus lim χε hε ϕε dxdt = ε→0 Ω×(0,T ) Z χ0 h0 ϕ0 dxdtdy. (ii) If (fε ) is bounded in L∞ (0, T ; L2 (Ω)) and two-scale converges Ω×(0,T )×Y
to f0 , fixing ϕ0 ∈ C(Ω×(0, T ), C] (Y )) and setting ϕε (x, t) := ϕ0 x, t, xε , noticing that Z

T

Z

||fε (., t)||L2 (Ω) ||ϕε (., t)||L2 (Ω) dt

fε ϕε dxdt ≤ 0

Ω×(0,T )

(6.6)

Z ≤C

T

||ϕε (., t)||L2 (Ω) dt,

hal-00777686, version 1 - 17 Jan 2013

0

and that limε→0 ||ϕε (., t)||L2 (Ω) = ||ϕ0 (., t)||L2 (Ω×Y ) , ∀ t ∈ (0, T ), by passing to the limit as ε → 0 in (6.6) in accordance with (6.2) and the Dominated Convergence Theorem we infer Z (6.7)

f0 ϕ0 dxdtdy Ω×(0,T )×Y

≤ C||ϕ0 ||L1 (0,T,L2 (Ω×Y )) ,

∀ ϕ0 ∈ C(Ω×(0, T ), C] (Y )),

∞ 2 ε hence f0 ∈ L∞ (0, T ; L2 (Ω×Y )). If in addition ( ∂f ∂t ) is bounded in L (0, T ; L (Ω)), ∂fε by the same argument ( ∂t ) two-scale converges up to a subsequence to some ξ0 ∈ L∞ (0, T ; L2 (Ω × Y )), thus

Z

Z

∂fε  x ψ0 x, t, dxdt ε→0 Ω×(0,T ) ∂t ε Z ∂ψ0  x = − lim fε x, t, dxdt ε→0 Ω×(0,T )∂t ε Z ∂ψ0 dxdtdy, ∀ ψ0 ∈ D(Ω×(0, T ); C]∞ (Y )). =− f0 ∂t Ω×(0,T )×Y

ξ0 ψ0 dxdtdy= lim Ω×(0,T )×Y

1,∞ 0 Hence ∂f (0, T ; L2 (Ω × Y )), and the convergence holds for the whole ∂t = ξ0 , f0 ∈ W sequence. If fε (0)* * a0 , fixing τ ∈ [0, T ] and an admissible sequence (ϕε ) ⊂ L2 (Ω) such that (ϕε ) −→ −→ ϕ0 ∈ L2 (Ω × Y ) and applying (6.4) with h0 (x, t, y) := 1[0,τ ] (t), we obtain ! Z Z Z T ∂fε 1[0,τ ] (t)dt + fε (0) ϕε dx lim fε (τ )ϕε dx= lim ε→0 Ω ε→0 Ω ∂t 0 Z Z ∂f0 (6.8) = (t)1[0,τ ] (t)ϕ0 dxdtdy+ a0 ϕ0 dxdy Ω×(0,T )×Y ∂t Ω×Y Z = (f0 (τ )−f0 (0)+a0 )ϕ0 dxdy, Ω×Y

hence fε (τ ) * * f0 (τ ) − f0 (0) + a0 , ∀τ ∈ [0, T ]. Fixing ϕ0 ∈ D(Ω×(0, T ); C]∞ (Y ))),
setting ϕε (x, t) := ϕ0 x, t, xε and applying the Dominated Convergence Theorem, we

18

M. BELLIEUD

infer Z

Z

Z

f0 ϕ0 dxdydt = lim

ε→0

Ω×(0,T )×Y

fε ϕε dxdt = 0

(f0 − f0 (0) + a0 )ϕ0 dxdtdy,

=

Z

 fε (τ )ϕε (τ )dx dt

lim

Ω×(0,T )

Z

T ε→0

Ω

∀ ϕ0 ∈ D(Ω×(0, T ); C]∞ (Y ))),

Ω×(0,T )×Y

hal-00777686, version 1 - 17 Jan 2013

0 ε −→ ∂f hence f0 (0) = a0 . If fε (0)−→ −→a0 and ∂f ∂t −→ ∂t , we deduce from the previous reasoning that f0 (0) = a0 and notice that (6.8) holds for any two-scale converging sequence (ϕε ).

The abstract results collected in the next theorem are proved in [19] (see Theorem 8.1 p. 287, Theorem 8.2 and Lemma 8.3 p. 298), [16] (see Formula (5.20) p. 667, and Theorem 1 p. 670), [18] (see Remark 1.3 p. 155). Henceforth, the derivatives in D0 (0, T ; H) are identified with the time derivatives in D0 (Ω × (0, T ) × Y ) and are 0 denoted both by ∂ζ ∂t or by ζ . Theorem 6.2. Let V and H be separable Hilbert spaces such that V ⊂ H = H 0 ⊂ V 0 , with continuous and dense imbeddings. Let ||.||V , |.|H , ((., .))V , (., .)H denote their respective norm and inner product. Let a : V × V → R be a con˜ = tinuous bilinear symmetric form on V . Let A ∈ L(V, V 0 ) be defined by a(ξ, ξ) 2 ˜ ˜ (Aξ, ξ)(V 0 ,V ) , ∀ (ξ, ξ) ∈ V . Assume that ∃(λ, α) ∈ R+ × R∗+ , a(ξ, ξ) + λ|ξ|2H ≥ α||ξ||2V , ∀ ξ ∈ V.

(6.9)

Let h ∈ L2 (0, T ; H), ξ0 ∈ V , ξ1 ∈ H. Then there exists a unique solution ξ of Aξ(t) + ξ 00 (t) = h(t),

(6.10)

ξ 0 ∈ L2 (0, T ; H),

where ξ 0 = (6.11)

∂ξ ∂t ,

ξ 00 =

∂2ξ ∂t2 .

1 2

What is more, ξ 0 ∈ L2 (0, T ; V ),

ξ 00 ∈ L2 (0, T ; V 0 ).

[(ξ 0 (τ ), ξ 0 (τ ))H + a(ξ(τ ), ξ(τ ))] , ∀ τ ∈ [0, T ], there holds Z

(6.12)

ξ 0 (0) = ξ1 ,

ξ(0) = ξ0 ,

ξ ∈ C([0, T ]; V ) ∩ C 1 ([0, T ]; H),

Besides, setting e(τ ) :=

ξ ∈ L2 (0, T ; V ),

e(τ ) = e(0) + 0

τ

(h, ξ 0 )H dt,

∀ τ ∈ [0, T ].

Moreover, Problem (6.10) is equivalent to Z

T

 ˜ ˜ H η 00 (t) dt + (ξ0 , ξ) ˜ H η 0 (0) a(ξ(t), ξ)η(t) + (ξ(t), ξ)

0

Z T ˜ H η(0) = (h, ξ) ˜ H η(t)dt, − (ξ1 , ξ)

(6.13)

0

∀ ξ˜ ∈ V,

∀ η ∈ D(] − ∞, T [);

ξ ∈ L2 (0, T ; V ), ξ 0 ∈ L2 (0, T ; H).

The next lemma concerns both the fibered case and the case of grain-like particles. The estimate (6.14) will be employed in the demonstration of Proposition 6.4 as a

19

TORSION EFFECTS IN ELASTIC COMPOSITES

means to prove the boundedness of the sequence (uε ) of the solutions of (1.1) and also in Section 7, in order to establish the corrector result (2.25). Lemma 6.3. Under the assumptions (2.7) and either (2.1) or (3.1), there holds if inf ρε > c > 0

or

Bε

inf ρε > c > 0,

then

Ω\Bε

∂w 2 dxdt |w| (τ )dx ≤ C ρε ∂t Ω Ω×(0,T ) Z Z 2 2 +C ε |e(w)| (τ )dx + C |w|2 (0)dx,

Z (6.14)

Z

2

Ω

Ω

∀ w ∈ C([0, T ]; H01 (Ω; R3 )) ∩ C 1 ([0, T ]; L2 (Ω; R3 )).

∀τ ∈ [0, T ],

hal-00777686, version 1 - 17 Jan 2013

Proof. For each w ∈ L2 (Ω) we define, setting w = 0 in Ω \ R3 ,

(6.15)

w bε (x) :=

!

Z −

X

Bεi

i∈Jε

Jε := {i ∈ Z3 , Yεi ∩ Ω 6= ∅}.

wds 1Yεi (x),

R By making suitable changes of variables in the Poincar´e-Wirtinger inequality Y |w
R R R b ε |2 dx ≤ − R−B wds |2 dx ≤ C Y |∇w|2 dx, ∀w ∈ H 1 (Y ), we infer that Ω |w − w C Ω ε2 |∇(w)|2 dx, ∀w ∈ H 1 (Ω; R3 ). Therefore, by Korn’s inequality in H01 (Ω; R3 ) we have Z (6.16) Ω

By (6.15) there holds from (6.16) Z

Z

2

b ε | dx ≤ C |w − w R Ω

b ε |2 dx ≤ C |w

|w|2 dx ≤ Cε2

(6.17)

ε2 |e(w)|2 dx, ∀ w ∈ H01 (Ω, R3 ).

Ω

Ω

Z

R Bε

|w|2 dx, ∀w ∈ L2 (Ω; R3 ), hence we infer

|e(w)|2 dx + C

Ω

Z

|w|2 dx, ∀ w ∈ H01 (Ω, R3 ).

Bε

If inf Bε ρε > c > 0, then Z Bε

(6.18)

2 ∂w (s)ds + w(0) dx ∂t Bε 0 Z Z ∂w 2 dxdt + C ≤C ρε |w|2 (0)dx, ∂t Ω×(0,T ) Ω

|w|2 (τ )dx =

Z

Z

τ

∀w ∈ C([0, T ]; H01 (Ω; R3 )) ∩ C 1 ([0, T ]; L2 (Ω; R3 )). Assertion (6.14) follows then from (6.17) and (6.18). Otherwise, if inf Ω\Bε ρε > c > 0, we repeat the same argument, substituting Y \ B for Y . The following proposition specifies, in the fibered case, the asymptotic behavior of several sequences associated to the sequence (uε ) of the solutions of (1.1). Proposition 6.4. There exists a unique solution uε of (1.1). Moreover,

20

M. BELLIEUD

∂uε ∈ L2 (0, T ; H01 (Ω; R3 )), ∂t

(6.19)

∂ 2 uε ∈ L2 (0, T ; H −1 (Ω; R3 )). ∂t2

Under (2.8), (2.10), there exists a constant C > 0 such that Z

2

Z

2

ε |e(uε )(τ )| dx + Ω\Bε

Ω

! ∂uε 2 2 2 2 + |uε | + |v ε | + |θε | (τ )dx ≤ C, ρε ∂t ∀τ ∈ [0, T ],

hal-00777686, version 1 - 17 Jan 2013

(6.20)

Z ∂vε3 2 C 2 |e(uε )(τ )| dx + ∂x3 (τ ) dx ≤ µ1ε , Ω ZBε v C ε3 2 (τ ) dx ≤ 2 , |vε1 (τ )|2 + |vε2 (τ )|2 + ε ε µ1ε Ω

Z

∀τ ∈ [0, T ], ∀τ ∈ [0, T ],

and fields u0 ∈ L∞ (0, T ; L2 (Ω × Y ; R3 )), u, v ∈ L∞ (0, T ; L2 (Ω; R3 )), θ ∈ L∞ (0, T ; L2 (Ω)), Ξm , Ξf ∈ L∞ (0, T ; L2 (Ω × Y ; S3 )), such that, up to a subsequence, * Ξm , e(uε )1Bε * * Ξf , εe(uε )1Ω\Bε * ∂vε3 ? ∂v3 ? ? ? * uε * u, v ε * v, θε * θ, star-weakly in L∞ (0, T ; L2 ). ∂x3 ∂x3 uε * * u0 ,

(6.21)

Besides the next relations are satisfied u0 ∈ L∞ (0, T ; L2 (Ω; H]1 (Y ; R3 ))),

v3 ∈ L∞ (0, T ; L2 (ω; H01 (0, L))), Z u0 = v + θe3 ∧(y − y B ) in Ω×(0, T ) × B, u = u0 (., y)dy , Y Z Z 1 u0 (., y).(e3 ∧ (y − y B ))dy, v = − u0 (., y)dy, θ = J B B (6.22)

Ξm = ey (u0 ), Ξf = 0 in Ω×(0, T ) × Y \B, Z 12 ∂v3 Ξf33 dy3 in Ω×(0, T ) × B, = ∂x3 − 12 Z  ∂θ 2 f = Ξ .(e3 ∧ (y − y B ))dy in Ω×(0, T ), ∂x3 J B 3 θ ∈ L∞ (0, T ; L2 (ω; H01 (0, L))),

the last two lines of (6.22) being obtained under the additional assumption (2.13). Moreover, (6.23)

θ = v3 = 0

if

k = +∞,

v=0

if

κ = +∞.

If κ ∈]0, +∞], there exists ζ0 ∈ L∞ (0, T ; L2 (ω × Y ; H01 (0, L))), Ξb ∈ L∞ (0, T ; L2 (Ω × Y ; S3 )), ξ ∈ L∞ (0, T ; L2 (ω; H01 (0, L))) such that up to a subsequence, (6.24)

uε3 1Bε * * ζ0 , ε

1 e(uε )1Bε * * Ξb , ε

TORSION EFFECTS IN ELASTIC COMPOSITES

21

and v1 , v2 ∈ L∞ (0, T ; L2 (ω; H02 (0, L))), ζ0 = ξ −

(6.25)

2 X ∂vα α=1

1 2

Z

− 21

∂x3

Ξb33 dy3 =

(y−y B )α

in Ω×(0, T )×B,

2 X ∂ξ ∂ 2 vα ∂ζ0 (y−y B )α = − ∂x3 ∂x3 α=1 ∂x23

in Ω×(0, T )×B.

Under the additional hypothesis (2.14), we have for any k ∈]0, +∞], u0 ∈ W 1,∞ (0, T ; L2 (Ω × Y ; R3 )), ∂uε ∂u0 * * , uε (τ ) * * u0 (τ ), ∀ τ ∈ [0, T ]. ∂t ∂t

hal-00777686, version 1 - 17 Jan 2013

(6.26)

˜ H := Proof. The problem (1.1) is equivalent to (6.13), where H := L2R(Ω; R3 ), (ξ, ξ) 1 3 0 −1 3 ˜ ˜ ˜ ρ ξ.ξdx, V := H0 (Ω; R ) (V = H (Ω; R )), a(ξ, ξ) := Ω σ ε (ξ) : e(ξ)dx, Ω ε (ξ 0 , ξ 1 , h) = (a0 , b0 , f ). By (2.14) and (2.15), H is a Hilbert space and the assumptions of Theorem 6.2 are satisfied. Therefore (1.1) has a unique solution and (6.19) follows from (6.11). By (6.12) we have, for all τ ∈ [0, T ],

R

! ∂uε 2 ρε + σ ε (uε ) : e(uε ) (τ )dx ∂t Ω Z Z   ∂uε 1 2 ρε |b0 | + σ ε (a0 ) : e(a0 ) dx + ρε f . (6.27) = dxdt. 2 Ω ∂t Ω×(0,τ )  R  R 2 2 By (2.10) there holds Ω ρε |b0 | +σ ε (a0 ) : e(a0 ) dx + Ω×(0,T ) ρε |f | dxdt ≤ C, hence 1 2

Z

Z Ω

(6.28)

! ∂uε 2 dx + σ ε (uε ) : e(uε ) (τ )dx ρε ∂t   sZ ∂uε 2 dxdt , ∀τ ∈ [0, T ]. ≤ C 1 + ρε ∂t Ω×(0,T )

u 2 R ε By integrating (6.28) with respect to τ over (0, T ), we deduce that Ω×(0,T ) ρε ∂∂t dxdt ≤ C and then, coming back to (6.28), that Z Z ∂uε 2 (τ )dx + (6.29) ρε σ ε (uε ) : e(uε )(τ )dx ≤ C, ∀τ ∈ [0, T ]. ∂t Ω Ω We infer from (1.1), (2.8), (2.16), (6.29) that ∂uε 2 (τ ) + ε2 |e(uε )|2 (τ )dx ≤ C, ρε ∂t Ω Z Z ∂vε3 2 C 2 |e(uε )| (τ )dx + ∂x3 (τ )dx ≤ µ1ε . Bε Ω

Z

(6.30)

22

M. BELLIEUD

By (6.17) and by the inequality (see [9], Formula (4.32)) Z  Z w 2  C 3 2 2 2 (6.31) |w1 | + |w2 | + dx ≤ 2 |e(w)| dx, ε ε Bε Bε

∀ w ∈ H01 (Ω; R3 ),

R deduced appropriate changes of variables in the Korn’s inequality B |w|2 dx R by making ≤ C B |e(w)|2 dx, ∀w ∈ {ξ ∈ H 1 (B; R3 ), ξ(y1 , y2 , − 12 ) = 0}, we have Z Z Z C |w|2 dx ≤ Cε2 |e(w)|2 dx, ∀ w ∈ H01 (Ω; R3 ). (6.32) |e(w)|2 dx + 2 ε Bε Ω Ω R R If κ > 0, then by (2.9), (6.29) and (6.32) there holds Ω |uε (τ )|2 dx ≤ C Ω σ ε (uε ) : e(uε )(τ )dx ≤ C. Otherwise, if κ = 0, then by (2.10), (6.14) and (6.29) we have Z Z ∂uε 2 dxdt + C σ ε (uε ) : e(uε )(τ )dx + C |a0 |2 dx |uε (τ )| (τ )dx≤ C ρε ∂t Ω Ω Ω Ω×(0,T ) ≤ C.

hal-00777686, version 1 - 17 Jan 2013

Z

2

Z

The estimate Z (6.33)

|uε (τ )|2 dx ≤ C,

∀τ ∈ [0, T ],

Ω

is proved. We deduce from (2.16) and (6.33) that Z (6.34) |v ε |2 (τ ) + |θε |2 (τ )dx ≤ C,

∀τ ∈ [0, T ].

Ω

By substituting uε (τ ) for w in (6.31), taking (2.16) and (6.30) into account we infer Z Z v C ε3 2 |vε1 (τ )|2 + |vε2 (τ )|2 + (τ ) dx≤ 2 |e(uε )|2 (τ )dx ε ε Ω Ω C ≤ 2 , ∀τ ∈ [0, T ], ε µ1ε which, joined with (6.30), (6.33), (6.34) completes the proof of (6.20). Taking Lemma 6.1 into account, we deduce that the convergences (6.21), (6.26) take place, up to a subsequence, for suitable u0 ∈ L∞ (0, T ; L2 (Ω × Y ; R3 )), (Ξm , Ξf ) ∈ (L∞ (0, T ; L2 (Ω × Y ; S)))2 , (u, v) ∈ (L∞ (0, T ; L2 (Ω; R3 )))2 , θ ∈ L∞ (0, T ; L2 (Ω)). In order to establish the identification relations (6.22), we test the convergences (6.21) with appropriate fields. Choosing first Ψ ∈ D(Ω×(0, T ); C]∞ (Y ; S)) and passing to the limit as ε → 0 in the equation Z  x εe(uε ) : Ψ x, t, dxdt = ε Ω×(0,T ) Z Z   x x −ε uε .divx Ψ x, t, dxdt − uε .divy Ψ x, t, dxdt, ε ε Ω×(0,T ) Ω×(0,T ) R R we find Ω×(0,T )×Y Ξm : Ψdxdtdy = − Ω×(0,T )×Y u0 .divy Ψdxdtdy and deduce by the arbitrary choice of Ψ that u0 ∈ L∞ (0, T ; L2 (Ω; H]1 (Y ))) and ey (u0 ) = Ξm . By (6.20), the sequence (εe(uε )1Bε ) converges strongly to 0 in L2 . Choosing Ψ ∈

TORSION EFFECTS IN ELASTIC COMPOSITES

23

R D(Ω×(0, T ); D] (B; S)) we deduce Ω×(0,T )×Y Ξm : Ψdxdtdy = 0. We infer that ey (u0 ) = 0 in Ω×(0, T )×B. Therefore, for a. e. (x, t) ∈ Ω×(0, T ), the restriction of u0 (x, t, .) to B is a rigid displacement. By the periodicity of u0 there holds u0 (x, t, y1 , y2 , − 21 ) = u0 (x, t, y1 , y2 , 12 ), hence

hal-00777686, version 1 - 17 Jan 2013

(6.35)

u0 = a + be3 ∧ (y − y B ), in Ω×(0, T )×B,

for a suitable (a, b) ∈ L∞ (0, T ; L2 (Ω; R3 )) × L∞ (0, T ; L2 (Ω)). Fixing ϕ ∈ D(Ω×(0, T ); ∗ R3 ), taking (2.2), (2.16), (6.35), the convergences v ε * v and uε * *u0 and Lemma 6.1 (i) into account, we get Z Z h x i 1 dxdt v.ϕdxdt= lim uε .ϕ(x, t)1B |B| ε→0 Ω×(0,T ) ε Ω×(0,T ) Z Z 1 = u0 .ϕ(x, t)1B (y)dxdtdy = (6.36) a.ϕdxdt, |B| Ω×(0,T )×B Ω×(0,T ) R ∗ *u0 with a hence a = v = −B u0 (., y)dy. By testing the convergences θε * θ, uε * function ϕ ∈ D(Ω×(0, T )) and with the sequence (ϕ ) given by ϕ (x, t) := ϕ(x, t)(e3 ∧ ε ε
 x 
x − y )1 ((ϕ ) is admissible by Lemma 6.1 (i)), thanks to (2.2), (2.16), B ε B ε ε (6.35) we find Z Z θε ϕdxdt θϕdxdt= lim ε→0 Ω×(0,T ) Ω×(0,T ) Z  h x i  1 = lim uε . e3 ∧ − y B ϕdxdt ε→0 J B ×(0,T ) ε ε Z 1 u0 .(e3 ∧ (y−y B ))ϕdxdtdy = J Ω×(0,T )×B Z 1 = (v+be3 ∧(y−y B )).(e3 ∧(y−y B ))ϕdxdtdy J Ω×(0,T )×B Z Z 1 2 b|e3 ∧(y−y B )| ϕdxdtdy = bϕdxdt, = J Ω×(0,T )×B Ω×(0,T ) R hence θ = b = J1 B u0 (., y).(e3 ∧(y−y B ))dy. By (1.1), (2.16) and (6.20), the sequence (vε3 ) is bounded in L∞ (0, T ; L2 (ω; H01 (0, L))), thus v3 ∈ L∞ (0, T ; L2 (ω; H01 (0, L))). ∂ϕ = 0 and ϕ = 0 in Ω×(0, T ) × Choosing ϕ ∈ D(Ω×(0, T ); C]∞ (Y )) such that ∂y 3
R x ε3 (Y \ B) and passing to the limit as ε → 0 in the equation Ω×(0,T ) ∂u ∂x3 ϕ x, t, ε dx =
R ∂ϕ − Ω×(0,T ) uε3 ∂x x, t, xε dxdt, we obtain 3 Z Z ∂ϕ f Ξ33 ϕdxdtdy= − (v + θe3 ∧ (y − y B ))3 dxdtdy ∂x 3 Ω×(0,T )×B Ω×(0,T )×B Z ∂ϕ =− v3 dxdtdy. ∂x 3 Ω×(0,T )×B R1 ∂v3 We infer from the arbitrary choice of ϕ that −2 1 Ξf33 (x, t, y1 , y2 , s)ds = ∂x a.e. in 3 2 Ω×(0, T )×D. The proof of (6.22) is achieved provided we establish that under (2.13), there holds (6.37)

θ ∈ L∞ (0, T ; L2 (ω; H01 (0, L))), Z ∂θ 2 = (−(y − y B )2 Ξf13 + (y − y B )1 Ξf23 )dy. ∂x3 J B

24

M. BELLIEUD

To that aim, we fix ϕ ∈ C ∞ (Ω×(0, T )), set ϕ = 0 on R3 × (0, T ) \ Ω×(0, T ) and define ! X Z − ϕ(s1 , s2 , x3 , t)ds1 ds2 1Pεi (x1 , x2 ), ϕε (x, t) := i∈Iε

(6.38)

Pεi := ε ({i} + P ) ,

Dεi

 P :=

1 1 − , 2 2

2 ,

Dεi := ε({i} + D), Iε := {i ∈ Z2 , Pεi ∩ ω 6= ∅},

(6.39)

M ε (x, t) :=  0  0   −( xε2 − (y B )2 )ϕε

0 0   ( xε1 − (y B )1 )ϕε

   −( xε2 − (y B )2 )ϕε ( xε1 − (y B )1 )ϕε  1Bε (x). 0

hal-00777686, version 1 - 17 Jan 2013

∂ϕε ε Denoting by n the outward pointing normal to ∂Bε , noticing that ∂ϕ ∂x1 = ∂x2 = 0 in Bε and that n3 = 0 on ∂Bε ∩ Ω, by integration by parts we get, for all τ ∈ [0, T ],

Z e(uε ) : M ε (τ )dx = Z  h x i   ∂ϕ h x i 2 1 ε − −uε1 −(y B )2 + uε2 −(y B )1 (τ )dx ε ε ∂x3 Bε ! Z  h x i  h x i 1 2 − (y B )2 n1 + − (y B )1 n2 uε3 ϕε (τ )dH2 (x). + − ε ε ∂Bε

Bε

(6.40)

If lim inf ε→0 εµ1ε < +∞, then under (2.13) the  set  D is  adisk
of center 0, hence by (2.1), (2.7) we have y B = 0, n1∂Bε ∩Ω = R1 xε1 e1 + xε2 e2 1∂Bε ∩Ω , therefore the term of the second line of (6.40) is equal to zero. Otherwise, if limε→0 εµ1ε = +∞, then by (6.44) the term of the second line of (6.40) is negligible. Taking (2.16) into account, we infer Z Z ∂ϕ (6.41) e(uε ) : M ε (τ )dx = −J θε ε (τ )dx + o(1). ∂x3 Bε Ω By (6.38), we have (6.42)

||ϕ−ϕε ||L∞ ≤ Cε ||∇ϕ||L∞ ,

2 ∂(ϕ − ϕε ) ∂x3 ∞ ≤ Cε ∇ ϕ L∞ .) L

By (6.42)   and (6.3) (applied with h0 = 1B , χ0 (x, y) := ϕ(x)(y − y B )α , χε (x) := χ0 (x, xε ), α ∈ {1, 2}), there holds 

 0 0 −(y − y B )2 ϕ 0 0 (y − y B )1 ϕ  1B (y). M ε −→ −→  −(y − y B )2 ϕ (y − y B )1 ϕ 0 By passing to the limit as ε → 0 in (6.41), in accordance with (6.2), (6.21), (6.42), we obtain Z   2 −(y − y B )2 Ξf13 + (y − y B )1 Ξf23 ϕdxdydt = Ω×(0,T )×B Z ∂ϕ −J θ (6.43) dxdt. ∂x 3 Ω×(0,T )

TORSION EFFECTS IN ELASTIC COMPOSITES

25

As (6.43) takes place for all ϕ ∈ C ∞ (Ω×(0, T )), we deduce (6.37). The proof of (6.22) is achieved. If k = +∞, we infer from (2.9), (6.20), (6.21) that v3 = 0 and Ξf = 0, then from (6.37) that θ = 0. If κ = +∞, then by (2.9), (6.20), (6.21) we have v = 0. Assertion (6.23) is proved. If κ > 0, Assertion (6.24) results from (2.9) and (6.20). The relations stated in (6.25) are obtained by fitting the argument developed in [9] (see Proposition 3.8 and the argumentation p.180). If (2.14) is verified, then by (6.20) the sequence (uε ) is bounded in W 1,∞ (0, T ; L2 (Ω; R3 )). Assertion (6.26) follows then from Lemma 6.1 (ii).

hal-00777686, version 1 - 17 Jan 2013

The estimate established in the next lemma is used in the proof of (6.37). Lemma 6.5. Assume (2.1), (2.7), (2.8). Let (uε ) be the sequence of the solutions of (1.1) and let n denote the outward pointing normal to Bε . Let ϕ ∈ C(Ω×(0, T )) and let ϕε be defined by (6.38). Then the following estimate holds Z  h x i  h x i   C 2 1 2 − (6.44) − (y B )2 n1 + − (y B )1 n2 uε3 ϕε (τ )dH ≤ . ε ε εµ 1ε ∂Bε Proof. By the inequality (proved below) 2 Z Z Z 2 1 w − − wdL dH ≤ C |∇w|2 dy, ∀ w ∈ H 1 (D), (6.45) D

∂D

we have Z

D

2

|w − w| dH2 ≤ C

∂D×]0,L[

Z

|∇w|2 dx, ∀ w ∈ H 1 (D × (0, L); R3 ),

D×(0,L)

R where w(x) :=−D w(s1 , s2 , x3 )ds1 ds2 . Since W := H 1 (D×(0, L); R3 )∩L2 (D; H01 (0, L; R3 )) contains no non-vanishing rigid displacement, we infer from Korn’s inequality (see [24], Theorem 2.5 p.19) that Z Z 2 (6.46) |w − w| dH2 ≤ C |e(w)|2 dx, ∀ w ∈ W. ∂D×]0,L[

D×(0,L)

Fixing i = (i1 , i2 ) ∈ Z2 , setting wα (y1 , y2 , y3 ) := uεα (ε(y1 −i1 ), ε(y2 −i2 ), y3 ), w3 (y) := 1 ε uε3 (ε(y1 − i1 ), ε(y2 − i2 ), y3 ), by making suitable changes of variables in (6.46) and by summation over i ∈ Iε , where Iε is defined by (6.38), taking (6.20) into account we deduce Z Z C C 2 2 (6.47) |uε3 − uε3 | dH2 ≤ |e(uε )| dx ≤ . ε Bε εµ1ε ∂Bε ∩Ω ∂g ε On the other hand, noticing that by (6.38) there holds ∂x = 0 in Bε for all α ∈ {1, 2} α 1 and g ∈ H (Ω), we infer from the Gauss-Green’s Theorem that Z  h x i  h x i   2 1 (6.48) − − (y B )2 n1 + − (y B )1 n2 uε3 ϕε (τ )dH2 (x) = 0. ε ε ∂Bε

Assertion (6.44) follows from (6.47) and (6.48). Proof of (6.45). RIf (6.45) is false, there exists Ra sequence (wn ) in H 1 (D) such that R 2 −D wn dL2 = 0, ∂D |wn | dH1 = 1, limn→+∞ D |∇wn |2 dy = 0. By the Poincar´e R R R w− − wdL2 2 dy ≤ C |∇w|2 dy, there holds wn → 0 in Wirtinger’s inequality D

D

D

26

M. BELLIEUD

1 by the continuity of the trace application from H 1 (D) to L2 (∂D), RH (D), 2hence, 1 |wn | dH → 0. This contradiction establishes (6.45). ∂D Justification of Remark 2.2 (v). Assume that ρε >> 1 on some ε-periodic subset Gε 
uε 1 ) of Ω that is 1Gε = 1G xε for some G ⊂ Y . Then by (6.20) the sequence ( ∂∂t Gε

two-scale converges to 0. Noticing that by (6.3) there holds (uε 1Gε ) * * u0 1G and u0 1 = 0, hence (uε (0)1Gε ) * * a0 1G , we deduce from Lemma 6.1 (ii) that ∂∂t G u0 (τ )1G = u0 (0)1G = a0 1G , ∀ τ ∈ [0, T ].

hal-00777686, version 1 - 17 Jan 2013

(6.49)

7. Proof of Theorem 2.1. Our proof, which combines the energy method of Tartar [33] with the two-scale convergence method of Allaire and Nguetseng [2], [23], relies on the appropriate choice of an admissible sequence of oscillating test fields (φε ). We will multiply (1.1) by (φε ) and, by passing to the limit as ε → 0 in accordance with the convergences (6.21) established in Proposition 6.4, we will obtain the variational problem satisfied by the triple (u0 , v, θ) given, according to the order of magnitude of k and κ, by (7.21), (7.38) or (7.44). Then, noticing that this variational problem is equivalent to (6.13) for a suitable choice of H, V, a, h, ξ0 , ξ1 , we will deduce from Theorem 6.2 the existence, the uniqueness and the regularity of its solution and the initial-boundary conditions. Consequently, the convergences established in (6.21) for subsequences of (uε ), (v ε ), (θε ), take place for the complete sequences. Then, we will prove that this variational problem is equivalent to (2.17). Finally, we will establish the corrector result (2.25). We set ( H :=

(w0 ,ψ, ϕ) ∈ L2 (Ω×Y;R3 )×L2 (Ω; R3 )×L2 (Ω), )

(7.1)

w0 = ψ+ϕe3 ∧(y−y B ) in Ω×B , ˜ ϕ)) ˜ 0 , ψ, ((w0 , ψ, ϕ), (w ˜ H :=

Z ˜ 0 dxdy. ρw0 .w Ω×Y

By (2.14), (2.15) there holds 0 < c ≤ ρ ≤ C < +∞, hence the application (., .)H is an R 1 inner product on H and the associated norm is equivalent to ( Ω×Y |w0 |2 dxdy) 2 = R R R 1 ( Ω×(Y \B) |w0 |2 dxdy+|B| Ω |ψ|2 dx+J Ω |ϕ|2 dx) 2 (see (2.2)). If ((w0n , ψ n , ϕn )) is a Cauchy sequence in H, then the sequences (w0n ), (ψ n ), (ϕn ) converge strongly in L2 and, up to a subsequence, almost everywhere respectively to some w0 , ψ, ϕ. Since w0n = ψ n +ϕn e3 ∧ (y − y B ) in Ω × B, ∀n ∈ N, there holds w0 = ψ+ϕe3 ∧ (y−y B ) in Ω × B, thus (w0 , ψ, ϕ) ∈ H. We infer that H is a Hilbert space. In order to define (φε ), we choose (w0 , ψ, ϕ) ∈ L2 (0, T ; H) satisfying (7.2)

(7.3)

w0 ∈ C ∞ ([0, T ]; D(Ω; C]∞ (Y ; R3 ))),

w0 (T ) =

∂w0 (T ) = 0, ∂t

set (7.4)

B ε := {y ∈ Y, dist(y, B) < ε},

B]ε :=

[ i∈Z3

{i}+B ε ,

Bεε = Ω ∩ εB]ε ,

27

TORSION EFFECTS IN ELASTIC COMPOSITES

(Bεε denotes the ε2-neighborhood of Bε in Ω), fix ηε ∈ C]∞ (Y ) such that (7.5) 0 ≤ ηε ≤ 1,

ηε = 1

in B,

ηε = 0 in

Y \ Bε,

|∇ηε |
0. Taking (7.22), (7.23) and (7.45) into account, we infer ˜ 2 (3) ≤ ||ξ|| ˜ 2 + Ca(3) ((ψ, ϕ), (ψ, ϕ)) ||ξ|| V V ˜ H + a(ξ, ˜ ξ) ˜ + a(3) ((ψ, ϕ), (ψ, ϕ))) ≤ C(|ξ| ˜ H (3) + a(3) (ξ, ˜ ξ)), ˜ ≤ C(|ξ| that is (6.9). We deduce from Theorem 6.2 that ξ = (u0 , v, θ) is the unique solution of (3) (7.44) and that ξ ∈ C([0, T ]; V (3) )∩C 1 ([0, T ]; H (3) ), ξ(0) = 0, ∂ξ ∂t (0) = ξ1 , yielding by P2 (7.25) and the inequality α=1 ||ψα ||L2 (ω;H02 (0,L)) ≤ C||(w0 , ψ, ϕ)||V (3) , ∀ (w0 , ψ, ϕ) ∈ V (3) , the initial-boundary conditions and regularity properties stated in (2.18), (2.21). Repeating the argument of the case 0 < k < +∞, we integrate (7.44) with respect to y over B, set ψ1 = ψ2 = 0, find (7.29), deduce (7.30), (7.31), (7.32), (7.33), subtract (7.33) from (7.44), get Z Z Z ∂2ψ ρ1 v. 2 dxdt + ρ1 b0 .ψ(0)dx − g(u0 ).ψdxdt ∂t Ω×(0,T ) Ω Ω×(0,T ) Z Z 2 X 3l + 2 ∂ 2 ψ α ∂ 2 vβ + κ Jαβ dxdt = ρ1 f .ψdxdt, 2 2 l+1 Ω×(0,T ) ∂x3 ∂x3 Ω×(0,T ) α,β=1

P2 2 ∂ 4 vβ then, making ψ1 , ψ2 vary in D(Ω×(0, T )), infer ρ1 ∂∂tv2α (x, t) + β=1 κ 3l+2 l+1 Jαβ ∂x43 = ρ1 fα + (g(u0 ))α , in Ω×(0, T ) for α ∈ {1, 2}, and deduce that (u0 , v, θ) satisfies (2.17), (2.21). Case κ = +∞. We set (7.46)

χε = 0,

ψ = 0,

ϕ = 0.

By (7.7), (7.12), we have I3ε = 0. By passing to the limit as ε → 0 in (7.8), we obtain the variational problem (7.29) and deduce that (u0 , v, θ) satisfies (2.17), (2.22).

36

M. BELLIEUD

Proof of the corrector result (2.25). We consider the fibered case, when 0 < k < +∞ (the other cases are similar). Setting (7.22), we introduce the continuous symmetric bilinear form on W 1,2 (0, T ; V, H) := {ζ ∈ L2 (0, T ; V ), ζ 0 ∈ L2 (0, T ; H)} defined by Z T     ˜ ∈ (W 1,2 (0, T ; V, H))2 . ˜ (7.47) a ∀ (ζ, ζ) ζ 0 , ζ˜0 +a ζ, ζ˜ dt, ˜(ζ, ζ) := H

0

We fix ξ˜ := (w0 , ψ, ϕ) ∈ W 1,2 (0, T ; V, H) satisfying (7.2) (not (7.3)) and set (7.7). There holds φε ∈ C([0, T ]; H01 (Ω; R3 )) ∩ C 1 ([0, T ]; L2 (Ω; R3 )) for small epsilons. By applying (6.14) to w = uε − φε and by integrating it over (0, T ), taking (2.23) into account, we infer Z Z 2 |uε − φε |2dxdt ≤ C (J1ε − 2J2ε + J3ε ) + C |φε (0)| dxdt, Ω×(0,T )

Ω×(0,T )

∂uε 2 + e(uε ) : σ ε (uε )dxdt, := ρε ∂t Ω×(0,T ) Z ∂uε ∂φε := ρε . + e(uε ) : σ ε (φε )dxdt, ∂t ∂t Ω×(0,T ) Z ∂φε 2 + e(φε ) : σ ε (φε )dxdt. := ρε ∂t Ω×(0,T ) Z

J1ε

hal-00777686, version 1 - 17 Jan 2013

(7.48) J2ε J3ε

In order to compute the limit of (J1ε ), we notice that by (2.23) and (6.27) we have a0 = 0 and ! Z Z T Z ∂uε 2 (7.49) J1ε = ρε |b0 | dxdt + 2 ρε f . dxds dt. ∂t Ω×(0,T ) 0 Ω×(0,t) R uε Since ρε 1Ω×(0,t) −→ −→ρ1(0,t) for all t ∈ (0, T ) and since, by (6.20), Ω×(0,t) ρε f . ∂∂t dxds ≤ C, we deduce from (6.26), (7.22) and from the Dominated Convergence Theorem that ! Z Z T Z ∂u0 2 dxdsdy dt lim J1ε = ρ |b0 | dxdydt + 2 ρf . ε→0 ∂t Ω×(0,T )×Y 0 Ω×(0,t)×Y  Z T Z t = (ξ1 , ξ1 )H + 2 (h, ξ 0 )H ds dt 0

0

Z =2 0

T

  Z t 0 e(0) + (h, ξ )H ds dt. 0

Applying the energy equation (6.12), taking (7.47) into account, we infer Z T Z T (7.50) lim J1ε = 2 e(t)dt = ((ξ 0 , ξ 0 )H + a(ξ, ξ)) dt = a ˜(ξ, ξ). ε→0

0

0

R By (6.26), (7.10), (7.12), (7.14), (7.16), (7.20), (7.22) we have limε→0 Ω×(0,T ) e(uε ) : RT R uε . ∂φε = R u0 . ∂ w 0 ˜ and limε→0 Ω×(0,T ) ρε∂∂t σ ε (φε )dxdt = 0 a(ξ, ξ)dt, ρ ∂∂t ∂t ∂t Ω×(0,T )×Y R T  0 0 dxdtdy = 0 ξ , ξ˜ dt, hence H

(7.51)

˜ lim J2ε = a ˜(ξ, ξ).

ε→0

37

TORSION EFFECTS IN ELASTIC COMPOSITES

The convergences deduced by substituting 1 for ρε and ρ in (7.10) hold true, hence Z ∂w0 2 ∂φε 2 dxdtdy lim ρε dxdt= ρ ε→0 Ω×(0,T ) ∂t ∂t Ω×(0,T )×Y Z T  dt. = ξ˜0 , ξ˜0 Z

(7.52)

H

0

We deduce from an explicit computation that εe(φε )1Ω\Bε −→ −→ ey (w0 )1Y \B ,  −l ∂ψ

−l ∂ψ3 2(l+1) ∂x3 1 ∂ϕ 2 ∂x3 (y − y B )1

 0 e(φε )1Bε −→ −→  1 ∂ϕ − 2 ∂x3 (y − y B )2

hal-00777686, version 1 - 17 Jan 2013

 ∂ϕ − 12 ∂x (y − y ) 2 B 3  1 ∂ϕ 2 ∂x3 (y − y B )1  1B (y),

0

3

2(l+1) ∂x3

∂ψ3 ∂x3

yielding, in accordance with (7.13), (7.15), (7.19), (7.22), (7.52) ˜ ξ). ˜ lim J3ε = a ˜(ξ,

(7.53)

ε→0

Joining (7.48), (7.50), (7.51), (7.53), and taking the strong two-scale convergence of
(u0 x, t, xε − φε ) to u0 − w0 into account (cf. (2.23)), we infer  2 x − uε lim sup u0 x, t, ε L2 ε→0 Z  2 x 2 ≤ C lim sup − φε + |uε − φε | dxdt u0 x, t, ε ε→0 Ω×(0,T ) Z T 2 ˜ 2 dt + C˜ ˜ ξ − ξ) ˜ + C (ξ − ξ)(0) ˜ ≤C |ξ − ξ| a (ξ − ξ, . H H

0

By the arbitrary choice of ξ˜ ∈ C ∞ ([0, T ]; W ) (W := {(w0 , ψ, ϕ) ∈ V, w0 ∈ D(Ω; C]∞ (Y ; R3 ))}), the density of C ∞([0,T ]; W ) in W 1,2 (0, T ; V, H)and the continuity of RT R 2 the application ζ → 0 |ζ|2H dt + a ˜(ζ, ζ) + Ω×Y |ζ(0)| dxdy on W 1,2 (0, T ; V, H), the corrector result (2.25) is proved. The convergence uε −→ −→u0 follows then from (2.23). Remark 7.1. If uε −→ −→u0 , then by Fatou’s Lemma Z

|u0 |2 dxdτ dy = lim

ε→0

Ω×(0,T )×Y

Z 0

T

Z

|uε (τ )|2 dxdτ ≥

Ω

Z 0

T

  Z lim inf |uε (τ )|2 dx dt. ε→0

Ω

On the other hand, as for all τ there holds uε (τ )* *u0 (τ ), we have (see [2], Theorem 0.2) Z lim inf ε→0

|uε (τ )|2 dx ≥

Ω

Z

|u0 (τ )|2 dxdy, ∀ τ ∈ [0, T ],

Ω×Y

thus lim inf ε→0 Ω |uε (τ )|2 dx = Ω×Y |u0 (τ )|2 dxdy, for a.e. τ ∈ [0, T ]. Hence for a.e. τ ∈ [0, T ], the sequence (uε (τ )) two-scale converges strongly, up to a subsequence, to u0 (τ ). R

R

38

M. BELLIEUD

8. Proof of Theorem 3.1. The first step consists in the study of the asymptotic behavior of some sequences associated with the sequence (uε ) of the solutions of (1.1). Repeating the argument of  the proof of Proposition 6.4, we obtain R  ∂ uε 2 2 2 2 ρε ∂t + ε |e(uε )| +µ1ε |e(uε )| 1Bε (τ )dx ≤ C, ∀τ ∈ [0, T ], and applying Ω R R (6.14) to w = uε , get Ω |uε |2 (τ )dx ≤ C and then Ω (|v ε |2 + |r ε |2 )(τ )dx ≤ C (see (2.16), (3.3)). We infer that, up to a subsequence, there holds uε * * u0 ,

(8.1)

?

uε * u,

∂uε ∂u0 * * , εe(uε )1Ω * * Ξm , εe(uε )1Bε * * 0, ∂t ∂t ? ? v ε * v, r ε * r star-weakly in L∞ (0, T ; L2 ).

hal-00777686, version 1 - 17 Jan 2013

We identify Ξm = ey (u0 ), deduce that u0 ∈ L∞ (0, T ; L2 (Ω; H]1 (Y ; R3 ))) and that ey (u0 ) = 0 in Ω×(0, T )×B, hence u0 = a + b ∧ (y − y B ) for a suitable (a, b) ∈ (L∞(0,T ; L2 (Ω; R3 )))2 . We find a = v (see (6.36)), then fixing γ ∈ D(Ω×(0, T ); R3 ), deduce from (3.3), (8.1) that Z

Z

r.γdxdt = lim ε→0 Ω×(0,T ) Z =

J −1

Bε ×(0,T )

h x i ε

  − y B ∧ uε .γdxdt

J −1 ((y − y B ) ∧ u0 ) .γdxdtdy

Ω×(0,T )×B

(8.2)

 (y − y B ) ∧ (b ∧ (y − y B ))dy .γdxdt Ω×(0,T ) B Z Z = (J −1 J b).γdxdt = b.γdxdt, Z

=

J −1

Z

Ω×(0,T )

Ω×(0,T )

thus b = r. The next step consists in the choice of a suitable sequence of test fields. We define  H (4) := (w0 , ψ, γ) ∈ L2 (Ω × Y ; R3 ) × (L2 (Ω; R3 ))2 , w0 (x, y) = ψ + γ ∧ (y − y B ) in Ω × B , (8.3) Z ˜ γ ˜ 0 , ψ, ˜ ))H (4) := ˜ 0 dxdy, ((w0 , ψ, γ), (w ρw0 .w Ω×Y

choose (w0 , ψ, γ) ∈ L2 (0, T ; H (4) ) satisfying (7.2), (7.3), and set

b (x, t) + γ b ε (x, t) ∧ (y − y B ) − w0 (x, t, y) , χε (x, t, y) := ψ ε

(8.4)



b and γ b ε are given by (6.15). We multiply (1.1) by φε := ηε xε χε x, t, xε + where ψ ε
w0 x, t, xε , where ηε is defined by (7.5) and get (7.8). We obtain (7.11), set (7.12), find (7.14), (7.16) and I3ε = 0 and, passing to the limit as ε → 0 in (7.8), get Z (8.5)

Ω×(0,T )×Y

∂ 2 w0 ρu0 . dxdtdy + ∂t2

Z + Ω×(0,T )×(Y \B)

Z

Z ∂w0 ρa0 . (0)dxdy − ρb0 .w0 (0)dxdy ∂t Ω×Y Ω×Y Z ey (u0 ) : σ 0y (w0 )dxdtdy = ρf .w0 dxdtdy. Ω×(0,T )×Y

39

TORSION EFFECTS IN ELASTIC COMPOSITES

We set n o V (4) := ξ˜ = (w0 , ψ, γ) ∈ H (4) , w0 ∈ L2 (Ω; H]1 (Y ; R3 )) , (4)

(8.6)

(4)

ξ := (u0 , v, r), ξ0 := (a0 , a0 , 0), ξ1 := (b0 , b0 , 0), h(4) := (f , f , 0), Z ˜ V (4) := (ξ, ξ) ˜ H (4) + ((ξ, ξ)) ∇y u0 .∇y w0 dxdy, Ω×Y \B Z a(4) ((u0 , v, r), (w0 , ψ, γ)) := ey (u0) : σ y (w0 )dxdy, a(4) := 0. Ω×(Y \B)

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2

(4)

0

Since there holds ξ ∈ L (0, T ; V ), ξ ∈ L2 (0, T ; H (4) ), the variational formulation ˜ 2 (4) ≤ C(|ξ| ˜ 2 (4) + (8.5) is equivalent to (6.10). By Korn’s inequality we have ||ξ|| V H ˜ ξ)), ˜ yielding (6.9). We deduce from Theorem 6.2 that (u0 ,v,r) is the unique a(4) (ξ, solution of (6.10) and satisfies the properties of continuity and the initial boundary conditions stated in (3.5). By integrating (8.5) with respect to y over B, thanks to (2.12), (3.3), we obtain Z Z ∂w0 ∂ 2 w0 dxdtdy + ρa0 . (0)dxdy ρu0 . 2 ∂t ∂t Ω×(Y \B) Ω×(0,T )×(Y \B) Z Z − ρb0 .w0 (0)dxdy + ey (u0 ) : σ 0y (w0 )dxdtdy Ω×(Y \B)

Ω×(0,T )×(Y \B) 2

Z

(ρ1 v + ρ1 r ∧ (y G − y B )) .

+ Ω×(0,T )

(8.7)

∂ ψ dxdt + ∂t2

Z ρ1 a0 . Ω

∂ψ (0)dx ∂t

∂2γ (J ρ .r + ρ1 ((y G − y B ) ∧ v)) . 2 dxdt ∂t Ω×(0,T ) Ω Z Z ∂γ + ρ1 ((y G − y B ) ∧ a0 ). (0)dx − ρ1 ((y G − y B ) ∧ b0 ).γ(0)dx ∂t Ω Z Z Ω = ρf .w0 dxdtdy + ρ1 f .ψdxdt Ω×(0,T )×(Y \B) Ω×(0,T ) Z + ρ1 ((y G − y B ) ∧ f ).γdxdt. Z

Z

−

ρ1 b0 .ψ(0)dx +

Ω×(0,T )

Choosing ψ = γ = 0 in (8.7) we deduce (7.30), (7.31) and find the equation obtained by replacing m(u0 ).e3 ϕ by m(u0 ).γ in (7.33). Subtracting it from (8.7), we get Z Z ∂2ψ ∂ψ ρ1 a0 . (0)dx (ρ1 v + ρ1 r ∧ (y G − y B )) . 2 dxdt + ∂t ∂t Ω×(0,T ) Ω Z Z ∂2γ − ρ1 b0 .ψ(0)dx + (J ρ r + ρ1 ((y G − y B ) ∧ v)) 2 dxdt ∂t Ω Ω×(0,T ) Z Z ∂γ + ρ1 ((y G − y B ) ∧ a0 ). (0)dx − ρ1 ((y G − y B ) ∧ b0 ).γ(0)dx ∂t Ω Z Ω Z (ρ1 ((y G −y B )∧ f )+m(u0 )).γdxdt, = (ρ1 f +g(u0 )).ψdxdt+ Ω×(0,T )

Ω×(0,T )

yielding the equations satisfied by (v, r) set forth in (3.5). The corrector result is obtained by fitting the argument of the fibered case. Remark 8.1. In the fibered case, by substituting θe3 for b in (8.2), we find that the sequence (r ε ) converges star-weakly in L∞ (0, T ; L2 (Ω : R3 ) to r := θe3 .

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M. BELLIEUD

9. Sketch of the proof of Proposition 5.2. a) (v) ⇒ (iii). If (a) (resp. (b)) is satisfied, the proof of the estimate (5.3) is similar to that of the estimate below formula (4.32) of [9] (resp. Formula (4.3) of [9]). (iii) ⇒ (iv). By multiplying (5.1) by uε and by integrating by parts, we infer from (5.3) that (uε ) is bounded in L2 (Ω; R3 ). (iv) ⇒ (v). Assume by contradiction that neither (a) nor (b) are satisfied, then the dimension of the subspace of R3 spanned by the directions of the fibers is lower than or equal to 2. We can assume without loss of generality that this subspace is spanned by (e2 , e3 ). Fix f := e1 . By (iv), (uε ) admits a two-scale converging subsequence which by Corollary 5.1 satisfies (5.2). Consider the constant field w0 (x, y) := e1 . It can be checked that w0 ∈ V , a(u0 , w0 ) = 0, (f , w0 )H 6= 0, hence (5.2) has no solution, a contradiction. (iii) ⇒ (i). We choose a smooth field w0 ∈ V and consider the sequence of test field (φε ) corresponding to that introduced in the proof of Theorem 2.1. Repeating the argument of the proof of (7.53), we get limε→0 Fε (φε ) = a(w0 , w0 ). Passing to the limit as ε → 0 in the inequality ||φε ||2L2 (Ω;R3 ) ≤ CFε (φε ), we infer ||w0 ||2L2 (Ω×Y ;R3 ) ≤ Ca(w0 , w0 ). Thanks to (4.2), (4.3) and to Korn’s inequality in H 1 (Y \ B; R3 ), we get |w0 |2V ≤ Ca(w0 , w0 ). (i) ⇒ (ii). This results from the Lax-Milgram Theorem. (ii) ⇒ (v). Similar to the proof of (iv) ⇒ (v). (vi) ⇒ (iv). Obvious. (iii) ⇒ (vi). If (iii) holds, then (uε ) is bounded in L2 (Ω; R3 ) (see the proof of (iii) ⇒ (iv)) and that (5.2) has a unique solution u0 (because (iii) ⇒ (ii)). Hence, by Corollary 5.1, (uε ) two-scale converges to u0 . b) Assume by contradiction that (5.2) has a solution u0 . Let P denote the subspace of R3 orthogonal to the space spanned by the directions of the fibers. Fix w ∈ D(Ω) such that w(x) ∈ P, ∀x ∈ Ω and (f , w)L2 (Ω;R3 ) > 0. Set w0 (x, y) := w(x). Then w0 ∈ V and (f, w0 )H > 0. On the other hand, since (v) b) is not satisfied and since w0 (x, y) ∈ P , we infer a(u0 , w0 ) = 0, which contradicts (5.2).

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