Homogenization via unfolding in periodic elasticity with contact on ...

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Asymptotic Analysis 00 (20xx) 1–32 DOI 10.3233/ASY-2012-1141 IOS Press

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Homogenization via unfolding in periodic elasticity with contact on closed and open cracks

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Doina Cioranescu a , Alain Damlamian b,∗ and Julia Orlik c

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Laboratoire J.-L. Lions–CNRS, Université Pierre et Marie Curie, Paris, France E-mail: [email protected] b Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, France E-mail: [email protected] c Fraunhofer-Zentrum Kaiserslautern, Kaiserslautern, Germany E-mail: [email protected]

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1. Introduction

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Keywords: ???

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Abstract. We consider the elasticity problem in a heterogeneous domain with an ε-periodic micro-structure, ε 1, including multiple micro-contacts between the structural components. These components can be a simply connected matrix domain with open cracks or inclusions completely surrounded by cracks, which do not touch the outer boundary. The contacts are described by the Signorini and Tresca-friction contact conditions. The Signorini condition is described mathematically by a closed convex cone, while the friction condition is a nonlinear convex functional over the interface jump of the solution on the oscillating interface. The difficulties appear when the inclusions are completely surrounded by cracks and can have rigid displacements. In this case, in order to obtain preliminary estimates for the solution in the ε-domain, the Korn inequality should be modified, first in the fixed context and then for the ε-dependent periodic case. Additionally, for all states of the contact (inclusions can freely move, or are locked/stuck to the interface with the matrix, or the frictional traction is achieved on the inclusion-matrix interface and the inclusions can slide in the tangential to the interface direction) we obtain estimates for the solution in the ε-domain, uniform with respect to ε. An asymptotic analysis (as ε → 0) for the nonlinear functionals over the growing interface is carried out, based on the application of the periodic unfolding method for sequences of jumps of the solution on the oscillating interface. This allows to obtain the homogenized limit as well as a corrector result.

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Contact problems for inner oscillating interface were considered in [14,17,18] and in the books [16] and [13]. In particular, [14] and [16] (Chapter 3, Section 5) deal with frictionless Signorini problems on oscillating interfaces, while in [13] (Chapter 10), contact problems with friction are considered. In [16] small cracks are imposed and the homogenization is done in a formal way. In [14] one considers the case *

Corresponding author: Alain Damlamian, Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, Centre Multidisciplinaire de Créteil, 94010 Créteil Cedex, France. E-mail: [email protected].

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0921-7134/12/$27.50 © 2012 – IOS Press and the authors. All rights reserved


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• perfect normal contact and tangential friction; • frictionless unilateral non-penetration. But, since the proof is essentially the same, we give it for a combination of both condition, even though the mechanical relevance of such a problem is not clear. The full problem of non-penetration together with a friction condition only where there is actual contact is not variational therefore much more complex and beyond the scope of this contribution. It is actually a dynamic problem, and is usually time-discretized, and it is this situation we consider here (see [9] and [10] for details). One key ingredient in the proofs is a couple of unilateral Korn inequalities for Lipschitz domains which seem new (see Proposition 4.5). The paper is organized as follows. Section 2 gives the geometric setting for the ε-periodic problem, including the unit cell. Section 3 presents the standard forms of the Korn inequality which will be used in the paper. In Section 4, we give a few inequalities related to the unfolding operator on the interface surfaces, then establish a uniform Korn inequality for the perforated matrix domain. Then, two unilateral Korn inequalities are proved with their applications to the oscillating inclusions in the domain. Section 5 concerns the problem for fixed ε-periodicity. Existence of the solution is proved and its uniqueness is analyzed. Finally, in Section 6, under more precise hypotheses, the homogenization of the problem is proved and the corrector result is established.

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of particles (which can rotate) diluted in the matrix material. The problem is handled via the two-scale homogenization.1 The papers [17,18] provide tools for weak convergence of some nonlinear Robin-type conditions on the inner oscillating interface under appropriate assumptions on the nonlinearity. The physically interesting scaling of frictional functional on the inner oscillating interface limε→0 S ε Ψ (ε−1 x, uε (x)) ds is considered also, but under some strong restrictions on the growth and the mean value of the function Ψ . In [12], the correct scaling is considered and the linear Robin-type interface conditions are handled by the two-scale convergence on the oscillating interface. In this paper, we consider a problem which combines unilateral contacts (which gives a condition on the jump of the normal components of the strain) with a Tresca-type friction condition (which is on the jump of the tangential strain – cf. (5.7) for their explicit form). We make extensive use of the periodic unfolding technique [4,8], which has two advantages: first, it transforms sequences of functions defined on varying domains and interfaces in sequences defined on fixed domains and interfaces; second, it significantly simplifies the proofs. We consider an example in which the stiffness of the matrix and of inclusions surrounded by open cracks are of the same order. An interesting result is that if the initial gap between the inclusions and the matrix is of the order of ε, the rigid part of the displacements of the particles strongly converges to the homogenized displacements. The non-penetration and friction contact conditions are given by periodic nonlinear functions of a weak convergent unfolded argument. Several possibilities are known to prove their weak or two-scale convergence. In [11] and [18], the function Ψ = Ψ (y, uε ) in the nonlinear Robin-type condition on the oscillating interface is linearized via approximations by Taylor series and reduced on this way to the linear case [12]. We use the convexity of these nonlinear functions and prove their convergence by lower semi-continuity with respect to weak convergences. One can apply the method in the two simpler physically relevant cases:

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D. Cioranescu et al. / Homogenization via unfolding in periodic elasticity with contact on closed and open cracks

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However, in [14], there is a flaw in the main estimate which implicitly assumes no jump in the tangential strain; hardly frictionless!

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D. Cioranescu et al. / Homogenization via unfolding in periodic elasticity with contact on closed and open cracks 1

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1.1. Notations

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• The normal component of a vector field v on the boundary of a domain is denoted vν , while the tangential component v − vν ν is denoted vτ (where ν is the outward unit normal to the boundary); • the strain tensor of a vector field v is denoted by e(v); its values are symmetric 3 × 3 real matrices, the set of which is denoted M3S (R); • the kernel of e in a connected domain is the finite dimensional space of rigid motions denoted R; • C indicates a generic constant which does not depend upon the scaling factor; • we use the notations of [4] and [3] for the unfolding method.

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The problem is set in the natural space R3 , although the results can be obtained in any dimension N 2. In the following, Y ⊂ R3 is the unit cell (or a set having the paving property with respect to the group of periods), and Y ∗ is the “cracked” Y , i.e. Y without all the cracks within Y (see Fig. 1). The set Y ∗ has one or several single connected components which intersect the boundary ∂Y , it is assumed that their union with the corresponding part of neighboring cells is still connected. We denote them by Y 0 and call this set the “matrix”. We have in mind either a cracked composite structure or a meshlike fiber set such as textiles. The other connected components of Y ∗ , denoted Y 1 , . . . , Y m , are finitely many, they correspond to . the inclusions completely surrounded by “closed” cracks: S j = ∂Y j inside Y . The other cracks, whose union is denoted S 0 , are “open cracks” which are included in the boundary of Y 0 inside Y . They are open in the sense that Y 0 lies on both sides of the connected components of S 0 . They can go up to the boundary of Y and even in such a way that, when extended by periodicity, such a crack produces a connected crack across several neighboring cells. Note that

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2. Geometric set up and first hypotheses

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We make the assumption that the set Y satisfies the hypotheses of the paper on the unfolding for holes [3] (concerning the set which is denoted in [3] by Y ∗ ). The results from [3] will be our essential tools in the present paper. As in [3], we require that Y 0 satisfy the Poincaré–Wirtinger inequality (for the space H 1 ), and that the union of Y 0 and its (3) direct neighbors be connected.

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Fig. 1. The unit cell Y . (Colors are visible in the online version of the article; http://dx.doi.org/10.3233/ASY-2012-1141.)

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Fig. 2. Definition of [z]Y and {z}Y . (Colors are visible in the online version of the article; http://dx.doi.org/10.3233/ ASY-2012-1141.)

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Fig. 3. Bounded domain with periodically distributed open and closed cracks. (Colors are visible in the online version of the article; http://dx.doi.org/10.3233/ASY-2012-1141.)

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R

z = [z]Y + {z}Y ,

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where, as in the one-dimensional case, [z]Y for z ∈ R3 , denotes the unique integer combination of periods such that {z} = z − [z]Y belongs to Y (see Fig. 2). Let Ω be a bounded domain in R3 with Lipschitz boundary, and ΓD be a non-empty open subset of the boundary ∂Ω (ΓD is the set where a Dirichlet condition will be prescribed). ε denote the set of the points x ∈ Ω whose ε-cell, ε[ x ]Y + εY , does not intersect the Neumann Let Ω ε . ε , the subset of Ω containing the parts from cells boundary ΓN = ∂Ω \ ΓD . Denote also by Λε = Ω \ Ω intersecting the boundary ΓN . For j = 1, . . . , m, introduce the set

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Recall that in the periodic setting, (almost) every point z ∈ R3 , can be written as

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x j j Ωε = x | x ∈ Ωε such that ε ∈Y . ε Y

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D. Cioranescu et al. / Homogenization via unfolding in periodic elasticity with contact on closed and open cracks 1 2 3 4

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The boundary ∂Ωεj is the set of “closed cracks” associated with S j ,

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j . j j ∈S . ∂Ωε = Sε = x x ∈ Ωε such that ε ε Y

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and . Ωε0 = Ω

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Ωεj ∪ Sε0 .

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For j = 0, set

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The union of all the cracks is denoted Sε∗ = Sεj .

Sε∗ ,

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Finally, set

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for j = 0, . . . , m.

We use the same notation Tε for the unfolding operator applied to functions defined on Ω or subsets of Ω (such as the Ωεj ’s and the Sεj ’s) with all its properties from [4]. The properties of its restrictions to functions defined on Ωε0 or on the Sεj follow from [3], with a slight variation on the open cracks Sε0 . In the following, for any bounded set O and ϕ ∈ L1 (O), MO (ϕ) denotes the mean value of ϕ over O, i.e.,

1 ϕ dy. MO (ϕ) = |O| O

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. v j = v|Ωεj

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Note that from these definitions, it is clear that there are no cracks in the layer Λε . For a function v defined on Ωε∗ , for simplicity, we denote its restriction to Ωεj by v j :

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3. Korn inequalities

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As mentioned in the Introduction, we will need some Korn inequalities in order to obtain a priori estimates for the solutions of the problem to be considered (see (5.5) and (5.6) below). Recall that the kernel of the strain operator e in an open set is formed by the vector fields which are a rigid motion (in each connected component of the set). We denote by R the finite vector space of rigid motions: . R = x → va,b (x) = a ∧ x + b; a and b ∈ R3 .

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Definition 3.1. A bounded connected domain O is a Korn domain whenever it satisfies the second Korn inequality, i.e., if there exists a constant C such that

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v H 1 (O)

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It is known (see [15]) that this holds for every connected bounded Lipschitz domain, but it is true for more general domains. Analogously, the following Korn inequality (which, because of its similarity with the Poincaré– Wirtinger inequality, we call the Korn-W inequality) holds for connected bounded Lipschitz domains, but does hold in more general connected domains:

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v − rv H 1 (O) C e(v)L2 (O) .

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In particular, for v ∈ W (O),

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v H 1 (O) C e(v)L2 (O) .

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. . As a consequence, P = Id − projW 1 (O) is the orthogonal projection from H 1 (O) to R and v 2 =

e(v) 2L2 (O) + P(v) 2R defines an equivalent norm on H 1 (O). Of course, the operator P depends upon O, but, for simplicity, we will not indicate this dependence (as it is always obvious for which domain it applies and it needlessly complicates the notations). Consider a bounded Lipschitz domain O, hence satisfying both the Korn-W and the Poincaré– Wirtinger inequalities together with a Dirichlet part ΓD (with positive measure) of its boundary ∂O. Then the following result holds:

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Proposition 3.5. Let O and ΓD be as above. There is a constant C such that

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Remark 3.4. If the Korn-W inequality holds, it follows that

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Definition 3.3. For a bounded domain O which satisfies the Korn-W inequality, we denote by W 1 (O) the orthogonal subspace of R in H 1 (O). Therefore H 1 (O) is the orthogonal sum of W 1 (O) and R.

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One could expect that the Korn-W and the Poincaré–Wirtinger inequalities are related. This seems to be an open question.

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Definition 3.2. The bounded domain O satisfies the Korn-W inequality if there exists a constant C such that for every v ∈ H 1 (O) there is a rigid motion rv ∈ R with

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C v L2 (O) + e(v)L2 (O) .

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(3.2)

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D. Cioranescu et al. / Homogenization via unfolding in periodic elasticity with contact on closed and open cracks 1 2

Proof. By (3.2), there is a constant C with v − rv H 1 (O) C e(v) L2 (O) . By the trace theorem,

v − rv L2 (∂Ω ) C e(v) L2 (O) . In particular, by restricting to ΓD ,

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Regarding the set Y , we make the assumption that it also satisfies the Korn-W inequality. This holds whenever Y 0 is connected and Lipschitz, even with Lipschitz open cracks that intersect ∂Y (since it is then a finite union of overlapping Lipschitz domains). Consequently, by the same type of argument as that in the proof of Proposition 3.5, if we consider the subspace V (ω) of H 1 (Y 0 ) consisting of the functions vanishing on the given open subset ω of Y 0 , the following holds: (3.3)

v H 1 (Y 0 ) C e(v)L2 (Y 0 ) .

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4.1. Some extra formulas from unfolding

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For ϕ ∈ H 1 (Ωεj ), recall that ∇y (Tε (ϕ)) = εTε (∇ϕ) on Ω × Y j . In a similar way, ey (Tε (ϕ)) = εTε (e(ϕ)). For ϕ ∈ Lp (Sεj ), Tε (ϕ) is similarly defined for y in S j . Furthermore,

1 ϕ dσ(x) = Tε (ϕ)(x, y) dx dσ(y) (4.1) ε|Y | Ω×S j Sεj

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ϕ Lp (Sεj ) =

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and

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Recall some formulas related to the unfolding operator (see for more details [3]) defined for any function φ Lebesgue-measurable on Ωεj by ⎧ x ⎨ ε × Y j , φ ε + εy a.e. for (x, y) ∈ Ω Tε (φ)(x, y) = ε Y ⎩ 0, otherwise.

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4. Some inequalities related to unfolding and the geometric domain

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Since R is finite dimensional, this implies

rv L2 (∂Ω ) C e(v)L2 (O) , whence the result.

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1 Tε (ϕ) p . L (Ω×S j ) (ε|Y |)1/p

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We will need a reverse Hölder inequality for functions such as images via unfolding which are piecewise constant on ε-cells.

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ε × Y j and piece-wise constant with respect to x on subsets all of Let ψ be a function defined on Ω 3 measure ε |Y |. Then, for 1 < p ∞, 1/p + 1/p = 1,

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ψ Lp (Ω ε ;L1 (Y j )) |Y |−1/p ε−3/p ψ L1 (Ω ε ×Y j ) .

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4.2. A Korn inequality for the perforated matrix domain Ωε

Let η be a fixed positive number, sufficiently small in order that the set . Yη = y ∈ Y 0 | dist y, Y ∩ ∂Y 0 > η , 0

be connected. Clearly, Yη0 is Lipschitz. Therefore

• there is an extension operator P from H 1 (Yη0 ) to H 1 (Y ) which is the identity on constant functions and a constant C such that P (v) 2 C v L2 (Y 0 ) and ∇P (v) 2 C ∇v L2 (Y 0 ) L (Y ) L (Y ) η η (see Cioranescu-Saint Jean Paulin extension from [7]), • Yη0 is a Korn-domain. So there is an equivalent Hilbert norm on H 1 (Yη0 ) such that the projection on the subspace of rigid motions R in the sense of this norm, is the same as the projection in the L2 . norm (choose as equivalent norm v 2 = v 2L2 (Y 0 ) + e(v) 2L2 (Y 0 ) ). We denote the projection of v η η on R as rv . By the previous argument (all the norms being equivalent on R),

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• by the Korn-W inequality there is a constant C such that

v − rv H 1 (Yη0 ) C e(v)L2 (Y 0 ) .

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rv R C rv L2 (Yη0 ) C v L2 (Yη0 ) ,

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Note that the reverse inequalities are sharp (in case that ψ is supported only in one cell).

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ψ L1 (Ω ε ×S j ) .

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ψ L1 (Ω ε ×S j ) |Ω|1/p ψ Lp (Ω ε ;L1 (S j )) ,

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Similarly,

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. Now, we define another extension operator Q from P . To do so, for v ∈ H 1 (Yη0 ), set Q(v) = P (v − rv ) + rv . It is easily seen that Q is the identity on constant functions and that Q(v) 1 C v H 1 (Y 0 ) , Q(v) 2 C v L2 (Y 0 ) , L (Y ) H (Y ) η η (4.5) e Q(v) 2 C e(v) 2 0 . L (Y ) L (Y ) η

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from which, by the Poincaré–Wirtinger inequality (which holds in Yη ), (4.6)

Since Q(v) − v vanishes on Yη0 , by (3.3) H 1 (Y 0 )

C e Q(v) − v L2 (Y 0 ) C e(v)L2 (Y 0 ) .

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Proposition 4.1. With the choice of η indicated above, there exists an extension operator Q from H 1 (Yη0 ) to H 1 (Y 0 ) such that (4.5), (4.6) and (4.7) hold. As a consequence, there exists a constant C independent of ε such that ∀u ∈ H 1 Ωε0 with u|ΓD = 0,

u H 1 (Ωε0 ) C e(u)L2 (Ω 0 ) . (4.8)

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∇Q(v) 2 C ∇v L2 (Y 0 ) . L (Y ) η

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Hence, by applying the second inequality of (4.5) to v − MYη0 (v) (keeping in mind that MYη0 (v) belongs to R) gives ∇Q(v) 2 C v − MY 0 (v) 2 0 + ∇v L2 (Y 0 ) , L (Y ) L (Y ) η η

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Furthermore, since u vanishes on ΓD so does Qε (u). By the Korn inequality (3.2) in Ω it follows that Qε (u) 1 C e(u) 2 0 , H (Ω ) L (Ω )

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4.3. Two unilateral Korn inequalities for the inclusions Ωεj

Regarding the inclusions Y their shapes).

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we need to distinguish two classes according to a geometric property (of

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from which (4.8) follows. 2

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Proof. The properties of Q are already proven. Let Ωε0 η denote the subset of (Ωε0 ) corresponding to Yη0 . By rescaling (4.5), (4.6) and (4.7), then adding on all the small periodic cells of Ωε0 η , one defines an “extension” operator Qε from H 1 (Ωε0 η ) to H 1 (Ω) such that for u ∈ H 1 (Ωε0 ) Qε (u) 2 C u L2 (Ω 0 ) , L (Ω ) ε Qε (u) − u 2 0 εC e(u) 2 0 , L (Ωε ) L (Ωε ) ∇ Qε (u) − u 2 0 C e(u) 2 0 , L (Ωε ) L (Ωε ) e Qε (u) 2 C e(u) 2 0 . L (Ω ) L (Ω )

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Definition 4.2. A locked domain is a bounded domain with Lipschitz boundary for which the only rigid motion which is tangent on its boundary is zero.

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Remark 4.3. Via the exponential map, one can see that a locked domain is a bounded Lipschitz domain whose group of isometries is discrete.

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In the next inequalities, it is of interest that only the positive part of vν (the normal component of v, see Notations in Section 1) appears. Obviously, applying them to −v would involve vν− alone. We start by a simple remark concerning the rigid motions in a bounded domain with a Lipschitz boundary.

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If furthermore, O is a locked domain, then . r −→ r l = (rν )+ L1 (∂O) is a norm on R.

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Proof. Every r ∈ R is divergence-free. Hence (rν )

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dσ = 0, so that

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Proposition 4.5 (Unilateral Korn inequalities). If O is a bounded connected and Lipschitz domain, there exists a constant C1 such that ∀v ∈ H 1 (O),

v H 1 (O) C1 e(v)L2 (O) + (vν )+ L1 (∂O) + vτ L1 (∂O) .

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(4.10)

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Proof. We give the proof for the second inequality, the first is obtained in a similar way. The proof is done by contradiction. If inequality (4.10) is false, there exists a sequence vn in H 1 (O) such that

vn H 1 (O) ≡ 1,

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If O is a locked domain, then there exists a constant C2 such that

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The previous lemma yields two new Korn-type inequalities which are essential in order to get uniform estimates for our problem. These inequalities seem new.

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13

Therefore · is a norm provided it vanishes only for r = 0. If r = 0, then r|∂O is identically zero. It is well-known that such a rigid motion has to be 0. This follows from the equi-projectivity of r itself (for each point x of O the projection of r(x + tei ) on a basis vector ei is independent of t, and for some t, x + tei belongs to ∂O). Similarly, if r l = 0, then r|∂O is tangent to ∂O. If O is a locked domain, this implies r = 0. 2

36 37

12

R EC

25

7

11

R

24

∂O rν

O

23

L1 (∂O)

= 2(rν )+ L1 (∂O) .

C

22

6

10

D

18

F

. r −→ r = rτ L1 (∂O) + (rν )+ L1 (∂O) is a norm on R.

O

12

5

9

Lemma 4.4. Let O be a bounded connected and Lipschitz domain. Then,

O

11

4

8

PR

9

2 3

8

10

1

e(vn )

L2 (O)

→ 0,

(vν )+

41 42 43 44

L1 (∂O)

→ 0.

45 46




11

By the compactness of the embedding of H 1 (O) into L2 (O), there is a v in H 1 (O) such that

1

2 3

2

1

vn v

in H (O),

vn → v

e(vn ) → 0 in L (O),

2

2

in L (O),

3

4

8

12 13 14 15

δj =

18

21 22 23 24 25 26 27 28 29 30 31

u L2 (Ωεj )) + ε ∇u L2 (Ωεj ) Cεe(u)L2 (Ω j ) ε + + C Tε (uν ) L2 (Ω ;L1 (S j )) + δj Tε (uτ )L2 (Ω ;L1 (S j )) .

H 1 (Y j )

L

39 40 41 42 43

2 2 2 Cε e(u) T ε 1 j (Ω ;H (Y )) L (Ω×Y j ) 2 + 2 + C Tε (u)ν L2 (Ω ;L1 (∂Y j )) + Tε (u)τ L2 (Ω ;L1 (∂Y j )) .

46

But from (4.1) and (4.2), the following equalities hold: ∇y Tε (u) 2 Tε (u) 2 = |Y | u L2 (Ωεj ) , = ε |Y | ∇u L2 (Ωεj ) , L (Ω×Y j ) L (Ω×Y j ) Tε e(u) 2 = |Y |e(u)L2 (Ω j ) , L (Ω×Y j ) ε

which, used in (4.12), give the result.

22 23 24

(4.11)

25 26 27 28 29 30 31

33 34 35 36

(4.12)

37 38

44 45

21

32

U

38

20

2 2 + 2 C e Tε (u) L2 (Y j ) + Tε (u)ν L1 (∂Y j ) + Tε (u)τ L1 (∂Y j ) .

N

36 37

19

Now using the facts that e(Tε (u)) = εTε (e(u)) and Y j is a Korn domain, we integrate over Ω to get Tε (u)2 2

15

18

O

35

14

17

Proof. We consider the case δj = 1, the proof being similar for δj = 0. Apply (4.9) or (4.10) to v = Tε (u) to obtain Tε (u)2

13

16

Proposition 4.6. For every j = 1, . . . , m, there exists a constant C such that for every u in H 1 (Ωεj ),

C

34

12

By scaling, we get the following result:

32 33

11

D

20

TE

19

2

0 if Y j is a locked domain, 1 otherwise.

7

10

We now turn back to functions defined on the domain Ωεj . From now on, the number δj will be defined for j = 1, . . . , m as follows:

16 17

F

which contradicts the fact that vn H 1 (O) ≡ 1.

6

9

O

11

vn H 1 (O)

O

9 10

5

8

C vn L2 (O) + e(vn )L2 (O) → 0,

PR

7

together with (vν → 0 in L (O). Consequently, e(v) = 0 so that v is a rigid motion, and by the continuity of the trace operator, (vν )+ = 0. Since O is a locked domain, this implies that v = 0 and

vn L2 (O) → 0. But, recalling that O is a Korn domain, by (3.1),

R EC

6

1

R

5

4

)+

2

39 40 41 42 43 44 45 46


12 1



5. The contact problem for fixed ε

1

2 3 4 5

2

For every ε we assume that the tensor field aε = (aεαβγδ ) is given with the usual properties of symmetry

4

aεαβγδ = aεβαγδ = aεαβδγ = aεγδαβ ,

5

6

8 9

6

and satisfying

7

sup max aεαβγδ L∞ (Ω ) < ∞, ε αβγδ

αηαβ ηαβ aεαβγδ ηαβ ηγδ ,

(5.1)

F

7

10

14 15

m . a e(u), e(v) =

ε

j=0

16

20 21 22 23 24 25 26 27 28 29 30 31

37 38 39 40 41 42 43 44 45 46

O

16

ε

17 18 19 20

(5.2)

|Sε

([v]Sεj )ν and ([v]Sεj )τ respectively. By standard trace theorems, these jumps belong to H 1/2 (Sεj ). The tensor field

for non-negative real numbers

26

30 31 32

Mεj .

In case of Tresca friction, the functions Ψεj are actually explicitly given by

. j Gjε (x)|w| dσ(x), Ψε (w) = Sεj

25

29

O

Mεj w L1 (Sεj )

24

28

is the stress tensor associated to the deformation v (not to be confused with the surface measures dσ!). The functions gε0 and the gεj ’s are the original gaps (in the reference configuration), and the corresponding inequalities in the definition of Kε represent the non-penetration conditions. In case there is contact in the reference configuration, these functions are just 0. Consider also the family of convex functions Ψεj , 0 j m, where Ψε0 is non-negative and lower semi-continuous on H 1/2 (Sε0 ), while for j = 1, . . . , m, Ψεj is continuous on H 1/2 (Sεj ) and satisfies Ψεj (w)

23

27

3 . ε ε (v) = aαβγδ e(v)γδ σαβ

γ ,δ=1

21 22

C

36

15

N

35

14

The vector fields v are the admissible deformation fields with respect to the reference configuration Ωε∗ . We will denote by [v]Sεj the jump of the vector field across the surface Sεj . For j = 1, . . . , m, one has [v]Sεj = v j |Sεj − v 0 j . For simplicity, we may use the notations [vν ]Sεj and [vτ ]Sεj in place of

U

34

α,β ,γ ,δ=1

Let Kε be the convex set defined, for non-negative gεj on Sεj , by . Kε = v = v 0 , . . . , v m | v ∈ H 1 Ωε0 ; ΓD × · · · × H 1 Ωεm , vνj − vν0 gεj on Sεj and vν0 |S 0 gε0 .

32 33

aεαβγδ (x)e(u)γδ (x)e(v)αβ (x) dx.

D

19

Ωεj

13

TE

18

3

12

R EC

17

9

11

O

13

8

10

PR

12

with α independent of ε. Introduce the following symmetric bilinear form:

R

11

3

(5.3)

33 34 35 36 37 38 39 40 41 42 43 44

(5.4)

45 46




13

with Gjε bounded below by Mεj for j = 1, . . . , m.

1

2

6

m

j=0

7 8 9 10 11 12 13 14

Problem

Pε .

Find uε ∈

Kε

aε e(uε ), e(v − uε ) +

22 23 24 25 26 27 28 29 30 31 32 33 34

39 40 41 42 43 44 45 46

Ψεj [vτ ]Sεj − Ψεj (uε )τ S j ε

where ∂Ψεj denotes the subdifferential of Ψεj (here taken in the sense of the L2 (Sεj ) duality). The corresponding explicit Tresca conditions on the interface Sεj with the function Ψεj given in (5.4) are as follows: ⎧ ⎪ there exists a measurable function λjε on Sεj such that ⎪ ⎪ ⎨

(u ) j

+ λj

σ (ν)

= 0 a.e. on S j , ε τ Sε ε ε ε

τ

(5.7) j j

⎪ λ 0, G − σ (ν) 0, ε τ ⎪ ε ε

⎪ j j

⎩ λε Gε − σε (ν)τ = 0 a.e. on Sεj ,

C

j=0

+

14

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

where σε (ν) is the stress vector on the surface S j (whose normal is ν), and σε (ν)τ is its tangential component (see, for example [13] Section 10.3 for a detailed derivation). The first step in the proof of existence of a solution consists in obtaining a bound for minimizing sequences. We denote by u the general term of such a sequence. We use the generic notation C for constants which can be expressed independently of the data. We start with the usual estimate obtained for the variational inequality, by taking into account that v = 0 belongs to Kε , L (Ωεj )

10

15

The “strong formulation” of this problem is given here, with the notation σε for the stress tensor σ ε (uε ): ⎧ Find uε in Kε such that ⎪ ⎪ ∗ ⎪ ⎪ ⎪ − div σε = fε j in Ωε , ⎪ ⎨ (uε )ν j − gε 0, Sε 0, σε (ν)ν ⎪ ⎪ j ⎪ ⎪ (ν) σ ε ν (uε )ν Sεj − gε = 0, ⎪ ⎪ ⎩ σε (ν)τ ∈ ∂Ψεj (uε )τ S j on Sεj for j = 0, . . . , m,

m j 2 e u 2 α

9

13

(5.6)

PR

Ωε∗

fε (v − uε ) dx.

8

12

N

38

6

11

U

37

j=0

35 36

Kε ,

D

21

such that for every v ∈ m

5

7

TE

20

(5.5)

R EC

19

fε v dx.

R

18

Ωε∗

O

17

4

From the properties of convexity of the Ψεj , j = 0, . . . , m, the solutions of Pε are the same as that of the following problem:

15 16

F

5

j . 1 Eε (v) = aε e(v), e(v) + Ψε [vτ ]Sεj − 2

3

O

4

2

Problem Pε . Given fε = (fε0 , . . . , fεm ) in L2 (Ωε ) find a minimizer over Kε of the functional

O

3

m j=0

j

Ψε [uτ ]Sεj

Ωε0

fε0 u0 dx +

37 38 39 40 41 42 43

m j=1

36

Ωεj

fεj uj dx + C.

(5.8)

44 45 46


14 1 2 3 4



We now use (4.8) to control the first term on the right-hand side to get

fε0 u0 dx C fε0 L2 (Ω 0 ) e u0 L2 (Ω 0 ) . ε

Ωε0

1 2

(5.9)

ε

5

8 9 10 11

ε

Ωεj

which requires to control uj L2 (Ωεj ) .

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Tε (ujε )

44 45 46

ε

11 12 13 14 15 16

(5.10)

on the inside of ∂Y by Tε (uν )|S j (its trace on the outside of By the trace theorem, the outer trace is controlled by Tε (u) L2 (Ω ;H 1 (Y 0 )) . More

normal component of the trace of

0 +

j ),

17 18 19 20 21 22

and by ∂Y precisely, 0 T ε u ν j 2 C Tε u0 L2 (Ω ;H 1 (Y 0 )) C u0 H 1 (Ω 0 ) C e u0 L2 (Ω 0 ) . |S L (Ω×∂Y j )

23

Then, for the positive part of the normal component, using (4.3), one gets

28

j ),

Tε (gjε ).

j + T ε u ν j |S

L2 (Ω ;L1 (∂Y j ))

Tε gjε

L2 (Ω ;L1 (∂Y j ))

ε

+ C e u0

ε

L2 (Ωε0 )

.

ε

25 26 27

30 31 32 33 34 35 36

By (4.4) and (4.2) one observes that Tε [u]τ 2 2 C L (Ω ;L1 (∂Y j ))

24

29

Similarly, concerning the tangential components, one has j Tε uτ j 2 Tε [u]τ L2 (Ω ;L1 (∂Y j )) + C e u0 L2 (Ω 0 ) . |S L (Ω ;L1 (∂Y j ))

37

2 1 −1 Tε [u]τ 2 1 [u]τ L1 (S j ) . j ) = Cε L ( Ω×∂Y 3 ε ε |Y |

Combining all these inequalities with (4.11), we get the result.

2

38 39 40 41

42 43

10

O

PR

D

21

9

Proof. By the definition (5.2) of Kε , the jump of the normal component of u across Sεj is bounded above (the positive part of the by gjε . Applying the unfolding operator, it follows that one can control Tε (ujν )+ |S j

TE

20

R EC

19

R

18

O

17

C

16

N

15

8

Proposition 5.1. There exists a constant C such that for all u in H 1 (Ωεj ), j = 1, . . . , m, j u 2 j + ε∇uj 2 j C e u0 2 0 + εe uj 2 j L (Ωε ) L (Ωε ) L (Ωε ) L (Ωε ) + C Tε gjε L2 (Ω ;L1 (∂Y j )) + δj ε−1/2 [u]τ L1 (S j ) .

U

14

7

ε

12 13

6

F

7

4 5

The other terms of the right-hand side for 1 j m, are simply bounded as follows,

fεj uj dx fεj L2 (Ω j ) uj L2 (Ω j ) ,

O

6

3

42

2

Remark 5.2. Note that if one uses the L -norms, then ε √ Tε gj 2 C Tε gjε L2 (Ω×∂Y j )) = εC gjε L2 (S j ) . L (Ω ;L1 (∂Y j )) ε

43 44 45 46



D. Cioranescu et al. / Homogenization via unfolding in periodic elasticity with contact on closed and open cracks 1 2

15

Remark 5.3. An analogous estimate can be found in [12], but the estimate of Proposition 5.1 above is sharper (with ε instead of ε−1 ).

3

6

ε

Ωεj

7

+ Tε gjε

8 9 10 11

5 6

ε

ε

L2 (Ω ;L1 (∂Y j ))

4

+ δj ε

7

[uτ ]Sεj L1 (S j ) .

−1/2

(5.11)

ε

One can improve this last inequality, provided the restriction δj = 1) satisfies an extra condition.

fεj

to the non-locked domains (case

12

16 17 18 19 20 21 22 23 24

Y

Then

O

PR

15

Lemma 5.4. Suppose that = 1, . . . , m, is locally orthogonal to the rigid motions, that is

Tε fεj r dy = 0 for every r ∈ R and for a.e. x ∈ Ω. j

Ωεj

fεj uj dx Cεfεj

L2 (Ωεj )

j e u

L2 (Ωεj )

28 29 30

from which (5.13) follows in view of (5.14).

36 37 38 39 40 41 42 43 44 45 46

17 18

(5.13) rεj (x)

ε

2

L

+ (Ω 0 ) ε

j 2 e u 2 L

j (Ωε )

+

j=1

C fε L2 (Ω ∗ ) 2

ε

m ε 2 Tε gj 2 +

L (Ω ;L1 (∂Y j )

j=1

24

26 27 28 29

34 35 36 37 38 39

Ψεj [uτ ]Sεj

j=0

23

33

where Mεj are defined by (5.3) (and assumed to be strictly positive for cases (i) and (iii). Then there exists a constant C independent of ε such that for u in a minimizing sequence, 0 2 e u 2

22

32

(ii) or fεj satisfies condition (5.12), (iii) or fεj = fεj ,1 + fεj ,2 , with fεj ,1 satisfying condition (i) and fεj ,2 condition (ii),

m

21

31

fεj L2 (Ωεj ) is small enough (compared to C1 from (4.9)),

m

20

30

R

1

ε1/2 Mεj

19

25

O

(i) either

15 16

C

35

(5.12)

N

34

14

Proposition 5.5. Suppose that for non-locked domains (case δj = 1)

U

33

13

We write successively

j j 1 1 j j fε u dx = Tε fε Tε u dx dy = Tε fεj Tε uj − rεj (x) dx dy, |Y | Ω×Y j |Y | Ω×Y j Ωεj

31 32

12

R EC

27

10 11

. = MY j (Tε (uj )) in R such that Proof. By the Korn-W inequality in Y there exists a measurable j Tε u − rεj (x) 2 C e Tε uj L2 (Ω×Y j ) = Cεe uj L2 (Ω j ) . (5.14) L (Ω×Y j ) j,

25 26

.

D

14

TE

13

fεj , j

8 9

F

5

2 3

From (5.10) it follows that

fεj uj dx C fεj L2 (Ω j ) e u0 L2 (Ω 0 ) + εe uj L2 (Ω j )

O

4

1

40 41 42

43

.

(5.15)

44 45 46


16



1

Proof. For δj = 0, use (5.11) to get an upper bound for

2

fεj

12 13

Ωεj

14 15 16 17 18 19 20 21 22

m j 2 e u

+

ε

m j fε +C j=1

34

39 40

43 44 45 46

15 16 17 18 19 20

ε

21

j=0

22 23

ε

24 25

ε

26 27 28 29 30 31 32

2 . Tε gjε 2 D(ε) = fε 2L2 (Ω ∗ ) + . L (Ω ;L1 (∂Y j ) m

(5.17)

ε

j=1

33 34

C

35

Then under the hypotheses of Proposition 5.5, there exists a fixed constant C such that for u = (u0 , . . . , um ) element of a minimizing sequence

36

39

0 2 u

41 42

[uτ ]Sεj

14

N

38

13

U

37

Ψεj

(5.16)

R

Corollary 5.6. Set

35 36

m

12

Estimate (5.15) is obtained simply by multiple applications of the classical inequality ab a2 + 41 b2 (for any > 0). 2

O

33

ε

ε

11

j εe u L2 (Ω j ) + Tε gjε L2 (Ω ;L1 (∂Y j )) . L2 (Ω j )

30

32

j e u 2 j + Tε gjε 2 . + ε 2 0 1 j L (Ω ) L (Ω ) L (Ω ;L (∂Y ))

10

j=0

26

31

L2 (Ωεj )

0 e u

m 0 j fε 2 j C e u L2 (Ω 0 ) L (Ω )

25

29

L2 (Ωεj )

j=0

24

28

9

Combining (5.8) and (5.9), (5.11), (5.13) and (5.16) finally yields

23

27

+ C fεj

4

8

dx

ρΨεj (u)

3

7

fεj uj

2

6

for some ρ ∈ (0, 1). Consequently, we obtain

11

1

5

1 C 1/2 fεj ,1 L2 (Ω j ) [uτ ]Sεj L1 (S j ) ρΨεj [uτ ]Sεj ε ε ε

F

9 10

orthogonal to solid motions, use (5.13). For the case δj = 1 with = + we use the same j ,2 j j ,1 j estimate for Ωεj fε u dx, while Ωεj fε u dx is estimated by (5.11) with the following inequality:

O

8

fεj ,2 ,

O

7

fεj ,1

PR

6

fεj

D

5

fεj uj dx. Similarly, for the case δj = 1 with

TE

4

Ωεj

R EC

3

H 1 (Ωε0 )

m j 2 e u 2 +

L (Ωεj )

j=1

+

m

Ψεj

[uτ ]Sεj CD(ε),

41

+ Tε uj L2 (Ω ;H 1 (Y j )) CD(ε)(1 + ε)

L2 (Ω ;R)

j r ε

+ Tε uj L2 (Ω ;H 1 (Y j ))+L1 (Ω ;R)) L1 (Ω ;R)

for δj = 0 , ε CD(ε) 1 + j Mε

38

40

j=0

j rε

37

42

(5.18)

43 44

for δj = 1 .

45 46




17

Furthermore, the following estimate holds:

2

j u

3 4

H 1 (Ωεj ))

1

1 −3/2 . CD(ε) 1 + δj ε +ε 1+ j Mε

2

(5.19)

5

j Tε u

9 10

14 15

j Tε u − rεj (x) 2 C e Tε uj L2 (Ω×Y j ) L (Ω ;H 1 (Y j )) = Cεe uj L2 (Ω j ) CεD(ε).

17 18

ε

19

22

This, together with (5.20) proves the second estimate for rεj L2 (Ω ;R) . For δj = 1, (5.21) implies

D

21

j Tε u − rεj

23 24 25 26

j T ε u − Tε u 0 +

28

ν

29

Also, from (5.3),

j Tε u − Tε u 0

R

31 32

τ

C

j + rε

36

41 42 43 44 45 46

U

40

and

L1 (Ω×S j )

N

ν

37

39

ε

L2 (Ω ;L1 (S j )

j rε τ

εTε gεj

.

Ψ j (u). j ε

Mε

13 14 15 16 17

(5.21)

18 19 20 21 22

25 26 27 28 29 30 31 32 33 34

+ CεD(ε) + C Tε u0 |S j L2 (Ω×S j ) L2 (Ω ;L1 (S j )

35 36 37 38

L1 (Ω×S j )

12

24

From these three last estimates, it follows that

35

38

L1 (Ω×S j )

O

33 34

L2 (Ω ;L1 (S j ))

εTε gεj

11

23

On the other hand, from the definition of Kε we have the estimate

27

30

CεD(ε).

L1 (Ω×S j )

9 10

This implies the second estimate of (5.18) for Tε To prove the remaining estimates of (5.18), as in the beginning of the proof of Lemma 5.4, use the Korn-W inequality (in Y j ) to derive the existence of piecewise constant functions rεj with values in R satisfying (5.14). Then, (uj ).

16

20

(5.20)

O

13

ε

ε

8

O

12

+ εe uj L2 (Ω j ) + Tε gjε L2 (Ω ;L1 (∂Y j )) . L2 (Ω 0 )

PR

11

L2 (Ω ;H 1 (Y j ))

C e u0

7

F

8

6

TE

7

4 5

Proof. To obtain the first estimate of (5.18), apply (5.15) together with (4.8). For δj = 0, consider (5.10) rewritten in terms of Tε (uj ),

R EC

6

3

0 ε j C ε + j Ψε [uτ ]Sεj + Tε u |S j L2 (Ω×S j ) . Mε

39 40 41 42 43

Since by (4.8) and the trace theorem, 0 Tε u C e u0 L2 (Ω 0 ) , |S j L2 (Ω ;H 1 (S j )) ε

44 45 46


18

2 3 4 5 6 7 8


Lemma 4.4 implies the estimates of (5.18) for rεj L1 (Ω ;R) (recall that δj = 1). Reverting to uj , and making use of (5.21), this in turn implies the corresponding estimate of (5.18) for

Tε (uj ) L2 (Ω ;H 1 (Y j ))+L1 (Ω ;R)) . Finally, we note that the function rεj is in the finite dimensional space of ε-piecewise constant functions ε (i.e. constant on each ε[ x ]Y + εY ). Therefore by (4.3), on Ω ε j −3/2 j rε 2 ε rε L1 (Ω ;R) . L (Ω ;R) This, together with (5.21) again, proves (5.19).

16 17 18 19 20

Proof. Any minimizing sequence is bounded in Kε , and by weak lower semi-continuity, every weak limit point of such a sequence is a solution. Using formulation (5.6) for two solutions uε and u ˆε , one ˆ ε , uε − u ˆε ) = 0. It follows that e(uε − u ˆε ) = 0, which by (4.8), implies easily gets that aε (uε − u 0ε = 0, and ujε − u ˆjε ∈ R in each connected component of Ωεj . 2 u0ε − u

21

23

Remark 5.8. Uniqueness for the jth-components (for j = 1, . . . , m) cannot be obtained because of the lack of global strict convexity of the functional (5.5) on Kε .

TE

22

24

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

R EC

30

11 12 13 14 15 16 17 18 19 20 21 22 23

27 28 29 30 31

33

Ψεj

In this subsection we give hypotheses on fε , g j , Ψε0 and (j = 1, . . . , m) which allow to obtain uniform estimates with respect to ε for the solutions uε . First of all, we suppose that D(ε) (defined by (5.17)) is bounded uniformly with respect to ε. We also suppose that for all j with δj = 1 (non-locked domains), fε satisfies hypotheses of Proposition 5.5. Ψεj

6.1.1. Hypotheses on and (j = 0, . . . , m) To pass to the limit in the homogenization process, we need structural assumptions concerning the contact data which are more precise than those of Section 5 (see (5.17)), namely 1. Concerning the function gεj , there is a g j in L1 (S j ) with x j j 1Ω ε (x) gε (x) = εg ε

10

32

6.1. Boundedness hypotheses on the data

gj

6

26

R

29

5

25

The aim of this section is to pass to the limit as ε → 0 in problem (5.6). We will obtain a limit “homogenized” problem which is given in Theorem 6.10. For the proof we use the unfolding method and the results from Sections 4 and 5 above. We start by formulating the hypotheses we make in order to prove the homogenization result.

O

28

4

24

C

27

N

26

6. Homogenization

U

25

O

O

15

ε ε Corollary 5.7. For each ε > 0, there is a solution uε = (u0ε , . . . , um ε ) in K for the Problem P . j j j 0 0 Uniqueness holds only for uε and for uε − proj R uε , j = 1, . . . , m (equivalently for uε and for e(uε )).

PR

14

By standard minimizing arguments, we obtain the following result:

D

13

3

9

2

11 12

2

8

9 10

1

7

F

1


34 35 36 37 38 39 40 41 42 43 44

for x ∈ S j ,

45 46




19

so that

1

1

Tε gεj (x, y) = εg j (y)1Ω ε (x)

2 3

2

for (x, y) ∈ Ω × S j .

3

4

6 7 8 9

2. We assume that Ψε is a convex continuous function of its argument. More precisely, there is a positive normal convex integrand ψ 0 on Ω × H 1/2 (S 0 )2 which vanishes on 0, such that

1 . Ψε0 (w) = ψ 0 x, Tε (w) dx. ε Ω

F

5

4

0

10

13 14

3. For j = 1, . . . , m, we assume that j j 1 wτ , Ψε (w) = Θ ε

is of the form

where

16

17

Θj (v) =

18

Gj (x, y)|v| dx dσ(y),

20

for continuous and positive Gj ’s. Then,

21 22

24

27 28 29

Ω×S j

30

20 21 22 23

44 45 46

26 27 28 29

31 32 33

2

34 35 36 37

39

(6.1)

42 43

25

30

C

N

41

24

38

fε 1Λε → 0 strongly in L (Ω), Tε fε0 f 0 weakly in L2 Ω × Y 0 ,

U

40

19

6.1.2. Convergence hypotheses on fε From the boundedness hypotheses of proposition 5.5 it follows that, up to a subsequence (for ε), one can assume some weak convergences. We make the convergence hypotheses on fε explicit here. There exists some f 0 ∈ L2 (Ω × Y 0 ) with

38 39

18

O

33

37

17

We will still denote this extension by Θj . This choice of Ψεj corresponds to a Tresca friction coefficient ε−1 Gj (x, {x/ε}Y ).

32

36

16

R

31

35

15

R EC

26

34

14

It is further assumed that for all j = 1, . . . , m such that δj = 1, M j is strictly positive. Note that Θj has a natural extension to the space of bounded measures on Ω × S j which is continuous with respect to the strict topology on that space,3 namely

Gj d |v| , where d |v| is the total variation of the measure v.

25

9

13

TE

. M j = min Gj .

23

8

12

D

Ω×S j

19

7

11

PR

15

6

10

O

12

Ψεj

O

11

5

40 41 42

2

This hypothesis means that x → ψ 0 (x, ρ) is measurable for every ρ in H 1/2 (S 0 ) while for a.e. x ∈ Ω, the map ρ → ψ 0 (x, ρ) is positive and lower semi-continuous convex on H 1/2 (S 0 ). 3 This is the so-called “étroite” topology which combines the weak-∗ topology and the convergence of the total variation norm.

43 44 45 46


20 1 2 3



and for all j = 1, . . . , m, with δj = 0 (locked domains), there exist f j ∈ L2 (Ω × Y j ) with Tε fεj → f j

1

strongly in L2 Ω × Y j .

2

(6.2)

4 5 6 7 8

4

In the sequel F denotes the function . 0 j f 1Y j , F = MY f 1Y 0 +

5 6

(6.3)

F

9

2

which is the weak limit in L (Ω) of fε .

O

11 12

6.2. Convergence of the solutions of problem (5.6)

16 17

21 22 23 24 25 26 27

0 uε

H 1 (Ωε0 )

,

j Tε uε − Tε u0ε + 2 , ν L (Ω ;L1 (S j )) ν

1 e Tε ujε 2 , L (Ω×Y j ) ε

34 35 36 37 38 39 40 41 42 43 44 45 46

R

33

j rε

L1 (Ω ;R)

18 19

22 23 24 25 26 27 28 29

and

j Tε u ε

30

L1 (Ω ;R)+L2 (Ω ;H 1 (Y j )

.

O

32

1 Tε ujε − rεj 2 , L (Ω ;H 1 (Y j )) ε

1 Tε ujε − Tε u0ε 1 , τ τ L (Ω×S j ) ε

Proof. The result follows from the properties of the unfolding operator together with estimates (5.18) of Corollary 5.6 which are uniform bounds as soon as D(ε) is bounded uniformly with respect to ε. 2

C

31

As usual, in order to obtain a limit problem, one uses weak (or weak-∗) compactness to extract convergent subsequences (still using the notation ε which will now belong to an appropriate subsequence going to 0). Eventually, for the components of the solution of the limit problem which are unique, this will imply the convergence of the corresponding complete sequences. At this point, one can apply Theorem 2.13 of [3] to the sequence {u0ε }. We also need a variant of this theorem, which is adapted to the case of vector-valued functions, using the symetric gradient instead of the full gradient. 1 (Y 0 ) indicates the subspace of Y -periodic elements w of H 1 (Y 0 ) For simplicity, the notation Wper with MY 0 (w) = 0 (this notation is only used for j = 0 since the other sets Y j are all compact in Y so periodicity is meaningless).

N

30

and for δj = 1,

U

29

16

21

as well as for δj = 0, j j Tε uε 2 rε 2 , , L (Ω ;R) L (Ω ;H 1 (Y j )

28

15

20

D

20

TE

19

14

17

Proposition 6.1. Under the hypotheses of Sections 6.1 and 6.1.1, the following quantities are bounded uniformly in ε:

R EC

18

11

13

We start by proving that the uniform hypotheses of Sections 6.1, 6.1.1 and 6.1.2, in combination with the properties of the unfolding operator, allow to obtain uniform estimates with respect to ε of all the solutions uε of (5.6).

PR

15

10

12

O

13 14

7 8

δj =0

9 10

3

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46



D. Cioranescu et al. / Homogenization via unfolding in periodic elasticity with contact on closed and open cracks 1 2 3

21

Proposition 6.2 (Convergences for j = 0). Up to a subsequence, there exist 0 1 u0 ∈ H 1 (Ω; ΓD ), u 0 ∈ L2 Ω; Wper Y

1 2 3

4

8 9

12

15 16

6 7 8

where yM = y − MY 0 (y). Furthermore, 0 (i) Tε e u0ε e u0 + ey u

17

(ii)

18

0 1 0 S0 Tε u ε S 0 u ε

2

weakly in L Ω × Y

23 24 25 26 27 28 29

C

Moreover,

39

U

46

17

19 20

(6.6)

21 22

rεj u0

24 25 26 27

29 30 31 32 33 34

(6.7)

35 36 37

N

38

45

D

O

1 j j weakly in L2 Ω; W 1 Y j , Tε uε − rεj u ε j weakly in L2 Ω × Y j . Tε e ujε ey u

37

44

16

18

Proposition 6.3 (Convergences for j = 1, . . . , m). Up to a subsequence, there exists a function u j ∈ 2 1 j L (Ω; W (Y )) such that

36

43

(6.5)

28

35

42

15

,

We turn to the convergences for j = 1, . . . , m. The estimates of Proposition 6.1 provide weak/ weak-∗ convergences for sequences involving ujε and rεj .

34

41

14

23

33

40

13

R

32

12

Proof. Applying Theorem 3.12 of [3] to the sequence {u0ε } gives convergences (6.4). Convergence (6.5)(i) follows by taking the symmetric part of the last convergence of (6.4). The jump operator [·]S 0 is continuous from H 1 (Y 0 ) to H 1/2 (S 0 ). Applying it to the third convergence of (6.4) implies (6.5)(ii) and (6.6). 2

30 31

11

TE

22

on S 0 .

9 10

R EC

21

and consequently 0 u ν S 0 g 0

0

weakly in L2 Ω; H 1/2 S 0 ,

19 20

(6.4)

O

11

14

strongly in L2loc Ω; H Y 0 , weakly in L2 Ω; H 1 Y 0 ,

1 0 0 weakly in L2 Ω; H 1 Y 0 , ∇u0 · yM + u Tε uε − MY 0 Tε u0ε ε 0 weakly in L2 Ω × Y 0 , Tε ∇u0ε ∇u0 + ∇y u

10

13

5

1

F

7

O

6

PR

5

4

such that Tε u0e → u0 Tε u0ε u0

38

weakly in L2 (Ω; R) and strongly in L1 (Ω; R),

39

(6.8)

40 41

and Tε ujε u0 Tε ujε → u0

2

weakly in L Ω; H Y j , strongly in L2 Ω; H 1 Y j + L1 (Ω; R).

42

1

43

(6.9)

44 45 46


22

4 5

j Tε u − Tε u0ε + ν

6 7 8 9 10 11 12 13 14

L2 (Ω ;L1 (S j ))

and

τ L1 (Ω×S j )

ν

L2 (Ω ;L1 (S j ))

→0

22 23 24 25 26 27 28 29 30 31 32 33 34

In view of Lemma 4.4, this implies that − u L1 (Ω ;R) → 0, whence (6.8). Finally, (6.9) is a consequence of the convergence Tε (ujε ) − rεj L2 (Ω ;W 1 (Y j )) → 0.

rε0 = MY 0 Tε u0ε ,

43 44 45 46

17

2

1 j rε − rε0 β j weakly-∗ in M1 Ω × S j , ε j 1 j + β j − ∇u0 · yM − u 0 |S j Tε uε − Tε u0ε |S j u ε weakly-∗ in L2 Ω; H 1/2 S j + M1 (Ω; R),

18

u + β − ∇u · yM − u

U

j

j

0

19 20 21 22 23 24 25 26 27 28 29 30 31

(6.10)

32 33 34 35

C

N

42

16

Proposition 6.4. Under the hypotheses of Sections 6.1, there exists a measure β j in M1 (Ω; R) such that, up to a subsequence,

37

41

15

which is independent of y and converges to u0 in L2 (Ω) .

and so

40

0

14

The next step is to obtain a weak-∗ convergence for the jumps across S j for j 1. For this purpose, set

36

39

PR

rεj

35

38

13

D

21

12

TE

20

11

R EC

19

10

and for δj = 1, rεj − u0 τ L1 (Ω×S j ) → 0.

R

18

8

O

17

7

9

15 16

3

6

→ 0.

Recall that Tε (u0ε ) converges weakly to u0 in L2 (Ω; H 1 (Y 0 ). Therefore, j rε − u0 +

2

5

both converge to 0, as well as Tε (ujε )τ − Tε (u0ε )τ L1 (Ω×S j ) for δj = 1. Consequently, j rε − Tε u0ε

1

4

j Tε uε − rεj 2 , L (Ω ;H 1/2 (S j ))

F

3

Proof. The first convergence in (6.7) is just the application of the usual weak compactness criterion in the space L2 (Ω; H 1 (Y j )), the second convergence is obtained by applying the strain operator ey . From the estimates in Proposition 6.1 it follows that

O

2


O

1


36

0

37

g . j

ν|S j

j , rε0 and Proof. From the definitions of u 0 , u

(6.11)

38 39

rεj ,

40

the following convergences are straightforward:

1 0 0 weakly in L2 Ω; W 1 Y 0 , Tε uε − rε0 ∇u0 · yM + u ε 1 j j weakly in L2 Ω; W 1 Y j . Tε uε − rεj u ε

41 42 43

(6.12)

44 45 46



D. Cioranescu et al. / Homogenization via unfolding in periodic elasticity with contact on closed and open cracks 1 2 3 4 5 6 7 8

23

On the other hand, by estimates (??), ε1 Tε (ujε )ν − Tε (u0ε )ν L1 (Ω×S j ) is bounded and by the definition of Kε , ε1 (Tε (ujε )ν − Tε (u0ε )ν ) g j . Therefore ε1 (rεj ) − rε0 )ν )+ L1 (Ω×S j ) is also bounded. For δj = 1, we also have that ε1 Tε (ujε )τ − Tε (u0ε )τ L1 (Ω×S j ) is bounded. Consequently, ε1 (rεj − 0 rε )τ L1 (Ω×S j ) is bounded too. Using Lemma 4.4, we conclude that ε1 rεj − rε0 L1 (Ω×S j ) is itself bounded. Up to a subsequence, we can assume that ε1 (rεj − rε0 ) converges weekly ∗ in M1 (Ω × S j ) to some β j . Going back to ε1 (Tε (ujε ) − Tε (u0ε ))|S j , this, together with (6.12), imply (6.10)–(6.11). 2

9

16 17 18 19 20 21 22 23 24

27

1 |Y |

+

∗

Y m

m

−

j=1

38 39 40 41 42 43

Sj

F

G (x, y)| χ − χ j

Sj

G (x, y)| W (y) − W (y) j

j

0

0

|S j τ

|S j τ

46

18 19 20 21 22 23

O

26 27

O

| dσ(y)

| dσ(y) 0

29 30 31

(6.14)

32 33 34 35

39

Proposition 6.5. Let Ξ be the map . (x, y, U) −→ Ξ = ey χ0 , . . . , ey χm ,

41

C

for all W in K. By an argument similar to that used in the proof of the existence of a solution for (5.6), one shows that the corrector problem (6.14) has at least one solution χ. For similar reasons, there is not necessarily uniqueness of the solution, but the nonlinear map U → (ey (χ0 ), . . . , ey (χm )) is itself single valued. This map has properties which are stated below.

44 45

17

28

N

37

j

16

U

34

36

14

25

a U + ey (χ) ey (W ) − ey (χ) dy + ψ 0 x, W 0 τ S 0 − ψ 0 x, χ0 τ S 0 0

j=1

35

13

(6.13)

24

31

33

8

such that

30

32

7

15

28 29

6

By hypothesis (5.1), (a0 ) is bounded in L∞ (Ω × Y ∗ ) and α coercive. In order to state the main result, we introduce the so-called corrector problem: for every symmetric (which is the tensor U and for a.e. x ∈ Ω, find χ = (χ0 (x, ·), . . . , χm (x, ·)) in the closed convex set K set of local admissible deformations), defined as 0 . 1 = K v = (v0 , . . . , vm ) | v0 ∈ Wper Y , vj ∈ H 1 Y j , j ∈ {1, . . . , m}, [v0 ]ν|S0 g 0 , (vj − v0 )ν|Sj g j

25 26

5

12

PR

15

4

11

D

14

3

10

TE

13

We assume that there exists a tensor (a0 ) such that Tε aε → a0 a.e. in Ω × Y ∗ .

R EC

12

R

11

2

9

6.3. The corrector problem

O

10

1

36 37 38

40

42 43 44

is any solution of (6.14). where χ = (χ0 (x, ·), . . . , χm (x, ·)) in K

45 46


24

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

F

7

Definition 6.6. The map σ hom is defined by

. 1 hom a0 (x, y) U + Ξ(x, y, U) dy. σ (x, U) = |Y | Y ∗

j=1

27

The corresponding minimum is denoted as E

28

31 32 33 34

38 39 40 41 42 43 44 45 46

(x, U):

Sj

7 8 9 10

16 17 18 19 20 21 22 23 24 25

27 28 29 30 31 32 33 34

τ

35

One can check by a direct computation that E hom is convex with respect to U. It turns out that σ hom and E hom are very much connected. This is the subject of the next proposition.

N

37

j=1

6

26

U

36

C

35

5

15

R

30

4

14

Definition 6.7. For every symmetric tensor U and for a.e. x ∈ Ω, set

. 1 hom E (x, U) = a0 U + ey (χ) U + ey (χ) dy + ψ 0 x, χ0 τ |S 0 2|Y | Y ∗ m

+ Gj (x, y) d χj − χ0 |S j .

O

29

hom

3

13

τ

26

2

12

As a direct consequence of Proposition 6.5, the map σ hom (x, U) is a Caratheodory function on Ω × M3S (R) with values in M3S (R). the Going back to the corrector problem (6.14), one can see that it is equivalent to minimizing over K functional

1 a0 U + ey (W ) U + ey (W ) dy + ψ 0 x, W 0 τ |S 0 2|Y | Y ∗ m

+ Gj (x, y) d W j − W 0 |S j . Sj

1

11

O

6

O

5

Proof. If, for fixed x ∈ Ω, one considers a sequence Un in M3S (R) converging to some U and the associated χn , it is easy to show from the problem (CP)n that the sequence χn converges weakly in K to the solution of problem (CP) (all the terms converge except for the quadratic one where one uses the weak lower semi-continuity). This implies that the map U → Ξ is continuous from M3S (R) to w-L2 (Y ∗ ). A similar argument shows that if the map x → a0 is continuous, so is the map x → Ξ(x, U), again when using the weak topology of L2 (Y ∗ ). With a0 measurable, by Lusin’s theorem, this implies that the map x → Ξ(x, U) is weakly hence strongly measurable for every U. 2

PR

4

D

3

The map Ξ is a Caratheodory map from Ω ∗ × M3S (R) with values in w-L2 (Y ∗ ) (i.e. L2 (Y ∗ ) endowed of its weak topology).

TE

2


R EC

1


Proposition 6.8. The function E hom is a strictly convex Caratheodory function defined on Ω ×M3S (R). It is Gâteaux-differentiable with respect to its second argument and this derivative is the tensor σ hom (x, U). In other words, for a.e. x ∈ Ω, U and U in M3S (R),

E hom (U ) + σ hom (U), U − U E hom (U). In particular, the map U → σ hom (x, U) is (maximal) monotone. It is actually strictly monotone.

(6.15)

36 37 38 39 40 41 42 43 44 45 46




6 7 8 9 10 11

Combining this with the coercivity relation

1 U + Ξ − (U + Ξ) dy α 2 dy, a0 U + Ξ − (U + Ξ) U + Ξ − (U + Ξ) |Y | Y ∗ |Y | Y ∗ one gets

12

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0

(U − U) dy α a U + Ξ − (U + Ξ) |Y | Y∗

hom U − U α σ (U ) − σ hom (U), |Y |

39 40 41 42 43 44 45 46

U + Ξ − (U + Ξ)

Y∗

2

U + Ξ − (U + Ξ)

dy,

4 5 6 7 8 9 10

12 13

dy.

14 15 16 17

(6.16)

18 19

from which follows

32

Y∗

U − U) dy E hom (U) dy. a0 (U + Ξ)(

N

E

(U ) + σ

hom

22 23 24 25 26 27 28 29 30 31

34 35 36

By the definition of σ hom (U ), this implies hom

21

33

O

1 E hom (U ) + |Y |

U

38

Y∗

2

3

20

36 37

2

which shows that the map U → σ hom (x, U) is monotone. It is actually strictly monotone. Indeed, if the left-hand side of (6.16) vanishes, this implies that the ≡ U − U, which is independent of y. In particular, right-hand side also vanishes, implying that Ξ − Ξ 0 0 0 ey (χ − χ ) is constant in Y which is connected. Therefore, χ0 − χ 0 is an affine function of the variable 0 0 y in Y . But by definition, it also is Y -periodic and with mean value 0 in Y 0 . This implies that χ0 − χ hom is identically zero. Therefore, U − U = 0, which is the condition for strict monotonicity of σ . associated It remains to show that σ hom is the subdifferential of E hom . By definition of the minimizer χ one sees that (we drop the variable x for clarity) with U,

0 1 hom hom − a0 (U + Ξ)( U + Ξ) dy, + Ξ) E (U ) E (U) + a (U + Ξ)(U 2|Y | Y ∗

C

35

This implies that for a.e. x ∈ Ω

33 34

D

15

TE

14

1 |Y |

1

11

R EC

13

F

5

O

4

O

3

for Ξ(U), Proof. From (6.14) written for two tensors U and U and denoting for short Ξ for Ξ(U ) and Ξ it follows that

1 (Ξ − Ξ) dy 0. a0 U + Ξ − (U + Ξ) |Y | Y ∗

PR

2

R

1

25

(U), U − U E

hom

37

+ 1 (U) |Y |

38

U − U) dy. a (Ξ − Ξ)( 0

Y∗

. In this last inequality, replacing U by Ut = (1 − t)U + tU (for t ∈ (0, 1)), using the convexity inequality for E hom and then dividing by t gives

+ 1 E hom (U ) + σ hom (U), U − U E hom (U) a0 Ξ − Ξ(Ut ) (U − U) dy. |Y | Y ∗

39 40 41 42 43 44 45 46


26

5 6 7 8 9 10 11 12 13 14 15

2

Proposition 6.9. E hom is coercive.

3

Proof. Let show that E hom (U ) is bounded below by a multiple of |U|2 . Recall (see (5.1)) that the tensor (a0 ) is α coercive. Furthermore, the family of functions Ψεj , 0 j m is non-negative. Hence,

α

U + ey (χ) 2 dy. min E hom (U ) 1 (Y 0 ) 2|Y | χ∈Wper Y

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

and

. 0 1 u = u , U , . . . , Um ,

12 13 14

19 20

with u0 defined in Proposition 6.2, convergences (6.4). Note that by definition, j − e u0 . ey U j = ey u

21 22 23

By its definition, for almost every x ∈ Ω, the function u (x, ·) belongs to the following closed convex set of admissible unfolded deformations 0 . 1 = K (w0 , . . . , wm )|w0 ∈ L2 Ω; Wper Y , [w0 ]ν|S0 g 0 in L2 Ω; H 1/2 S 0 , for j ∈ [1, . . . , m], wj ∈ L2 Ω; W 1 Y j + M1 (Ω; R) , (6.18) (wj − w0 )ν|Sj g j in L2 Ω; H 1/2 S j + M1 (Ω) . Theorem 6.10. Assume that fε , g j , Ψε0 and Ψεj (j = 1, . . . , m) satisfy the hypotheses from Sections 6.1.1 and 6.1.2. Suppose furthermore that the assumption (6.13) on the tensor field Tε (aε ) together with the boundedness and coercivity of (a0 ) holds. Let uε ∈ Kε be a solution of Problem Pε and u0 ∈ H 1 (Ω; ΓD ) be the limit function given by Proposition 6.2. Then u0 is the unique minimizer over H 1 (Ω; ΓD ) of the functional

hom E x, e(v) − F v dx. (6.19) Ω

The limit problem can equivalently be written as 0 u ∈ H 1 (Ω; ΓD), 0 hom x, e u = F in Ω. − div σ

11

18

(6.17)

D

. j + β j − ∇u0 · yM Uj = u

TE

24

9

17

R EC

23

For j= 1, . . . , m, set

R

22

8

16

O

21

7

15

C

20

6

6.4. The main homogenization result

N

19

5

10

U

18

4

This minimum is a non-negative quadratic form of U on the finite dimensional space M3S (R) and defines an equivalent norm on it. To see that it is enough to show that it vanishes only for U = 0. But the proof is similar to that of the strict monotonicity of σ hom . Consequently, there is strictly positive constant 2 γ such that E hom (U) is bounded below by γα 2 |U| . 2

16 17

1

F

4

2

O

3

By the weak convergence of Ξ(Ut ) − Ξ to zero in L2 (Y ∗ ), one finally gets (6.15).

O

2


PR

1


24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

(6.20)

44 45 46




2 3 4 5 6 7 8 9 10

Proof. Problem (6.19) has a unique solutions because of the strict convexity of E hom or the strict monotonicity of σ hom . It can easily be seen that problem (6.20) is actually of generalized Leray–Lions type, so that it has one solution. It remains to show that the limits furnished by the convergences of Propositions 6.2–6.4 satisfy the homogenized limit problem. In the previous subsection, we have already established that u = 0 1 m ( u , U , . . . , U ) (defined in (6.17)) belongs to the set K defined by (6.18). The main point now is to pass to the limit in (5.6). To do so, we will use the following test function: x . , vε (x) = Φ(x) + εw x, ε

F

1

14

O

13

(this choice makes sense since where Φ is in C (Ω) and vanishes near ΓD , and w belongs to D(Ω; K) contains 0). K We have the obvious convergence 1

16

vε → Φ strongly in L (Ω). 2

17

20 21

We rewrite (5.6) in the form m Ψεj (vε )τ S j − aε e(uε ), e(vε ) + ε

22

ε

23

a e(uε ), e(uε ) +

24

30 31 32 33

so that

Tε e(vε ) → e(Φ)(x) + ey (w)(x, y)

34

38 39 40 41 42 43 44 45 46

Ωε∗

10

12 13 14

23

fε uε dx.

(6.21)

24 25 26 27 28 29 30 31

uniformly and in L Ω × Y 2

∗

32

33

.

34

By unfolding, and recalling Hypotheses 6.1 and 6.1.1, for ε small enough

35

C

37

−

9

22

36

m 1 ε Tε aε Tε e ujε Tε e vεj dx dy a e(uε ), e(vε ) = |Y | j j=0 Ω×Y

aε e(uε )e(vε ) dx, +

37

N

36

Sεj

8

21

38

U

35

(uε )τ

7

20

fε vε dx

6

19

R EC

29

R

28

Ψε

5

18

We study the terms appearing in this inequality. Note that x x + ey w x, , e(vε ) = e(Φ)(x) + εex w x, ε ε

O

27

j

j=0

25 26

m

Ωε∗

4

17

j=0

3

16

D

19

TE

18

2

15

PR

15

1

11

O

11 12

27

Λε

Ψεj (vε )τ S j = Θj wj (x, y) − w0 (x, y) τ |S j which is independent of ε, ε 0 j j j 1 Ψε (uε )τ S j = Θ Tε uε − Tε uε τ |S j . ε ε

39 40 41

(6.22)

42 43 44 45 46


28


By the convergences already shown and since and εy ( uj ) = ey (U j ) + e(u0 ), we have

1 2

4

Ω×Y ∗

5 6

5

aε (e(u

ε ), e(uε ))

and using the coercivity of the tensor

1 aε e(uε ), e(uε ) |Y |

m Ω×Y j

j=0

Tε aε Tε e ujε Tε e ujε dx dy + α

18 19 20 21 22 23 24

1 fε uε dx → |Y | Ωε0

0 0

|Y 0 | f u dx dy = |Y | Ω×Y 0 0 0

27

31

Ωεj

37 38 39 40 41 42

Ωε0

fε0 vε dx →

44

46

for δj = 1,

43

45

14 15 16 17 18

(6.23)

MY 0 f 0 u0 dx.

Ω

1 |Y |

f j uj dx dy =

Ω×Y j

for δj = 0,

|Y | |Y | 0

19 20 21 22 23 24 25 26 27

|Y |Y |

j|

28 29

MY j f j u0 dx. Ω

30 31 32

C

36

N

35

13

1

fεj L2 (Ωεj ) → 0, then Ωεj fεj ujε dx → 0. By (5.10), if δj = 1 and N/2−1 ε Mεj For δj = 1 and if condition (5.12) is satisfied, then the same convergence holds true due to (5.13), since ε fεj L2 (Ωεj ) → 0 . Similarly, one gets

U

34

fεj ujε dx →

O

32 33

12

For δj = 0 (locked domains), by Proposition 6.3

R

30

11

Let us now consider the right-hand side terms in (6.21). Recalling hypotheses (6.1) and (6.2) on fε , we first get the convergence

26

29

10

O

0 1 ε lim inf a e(uε ), e(uε ) a 0 e u 0 + ey u u) dx dy, e u + ey ( |Y | Ω×Y ∗ 0 0 0 lim inf Ψε [uε ]Sε0 Ψ u |S 0 , lim inf Ψεj [uε ]Sεj Ψ j U j − u 0 τ |S j .

25

28

Λε

9

PR

17

By the lower semi-continuity with respect to weak (or weak-∗) convergences,

e(uε ) 2 dx.

D

16

6

8

TE

15

lead

7

R EC

14

(aε ),

to

12 13

4

F

11

3

O

9

j a 0 e u 0 + ey u e(Φ)(x) + ey (w)(x, y) dx dy.

On the other hand, unfolding the term

8

10

1 2

1 lim aε e(uε ), e(vε ) = |Y |

3

7


33 34 35 36 37 38

MY 0 f 0 Φ dx,

Ω

39 40 41

Ωεj

Ωεj

fεj vε dx

→ 0,

fεj vε dx →

42

|Y |Y |

j|

Ω

43

MY j f j Φ dx.

44 45 46




29

Recall the notation Eε for the energy related to Problem Pε , (5.6), i.e.

1

2

2

m . ε Ψεj [uε ]Sεj . Eε (uε ) = a e(uε ), e(uε ) +

3 4

3 4

j=0

5

5

6

12 13

14

j=1

17 18 19 20

Here we used the formula

24

36 37

40

43

46

j=1

4

20 21 22 23 24 25 26 27

R

42

45

19

(6.24)

Ω

F u0 dx lim inf Eε − F u0 dx lim sup Eε − Ω Ω

1 a0 e u0 + ey ( u) e u0 + ey ( u) dx dy |Y | Ω×Y ∗

41

44

18

O

U

38 39

17

C

35

16

which is equivalent to (6.3). Recall that u belongs By a density argument,4 (6.24) holds for Φ ∈ H 1 (Ω, ΓD ) and for w in the set K. 0 Thus, with this choice for w and replacing Φ by u + Φ (which is still in H 1 (Ω, ΓD ) since u0 has to K. no jumps), yields

1 a0 e u0 + ey ( u) e(Φ)(x) + e u0 + ey ( u) dx dy |Y | Ω×Y ∗

m 0 j 0 j 0 +Ψ Θ U −u τ |S j − F Φ + u0 dx u τ |S 0 +

N

34

15

δj =0

27

33

14

R EC

26

32

13

|Y j | |Y 0 | MY 0 f 0 + MY j f j , F = |Y | |Y |

25

31

12

TE

22

30

11

Ω

j=1

21

29

10

PR

16

28

lim sup Eε − F u0 dx lim inf Eε − F u0 dx Ω Ω

1 a0 e u0 + ey ( u) e u0 + ey ( u) dx dy |Y | Ω×Y ∗

m 0 j 0 j 0 +Ψ Θ U −u τ |S j − F u0 dx. u τ |S 0 +

15

23

Ω

9

F

11

8

O

9 10

7

O

8

6

Taking the inf-limit in (5.6) and using the established convergences yield

1 a0 e u0 + ey ( u) e(Φ)(x) + ey (w) dx dy |Y | Ω×Y ∗

m 0 j 0 j 0 w τ |S 0 + +Ψ Θ w (y) − w (y) τ |S j − F Φ dx

D

7

Using regularization by convolution in x, this applies also to the various convex functions, making use of the continuity with respect to the strict topology for the Θj ’s.

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46


30


7 8 9

4

Subtracting I from each term in (6.25) gives 1 |Y |

5

a e u0 + ey ( u) e(Φ)(x) dx dy −

6

0

Ω×Y ∗

7

F Φ dx Ω

8

lim sup Eε − I lim inf Eε − I 0.

10

14 15 16

1 |Y |

Ω×Y ∗

Ω

18

lim Eε = I as well as α

26 27 28 29 30 31 32 33 34

39 40 41 42

0 w τ |S 0 − Ψ u τ |S 0 +Ψ 0 τ |S j 0. + Θj wj (y) − w0 (y)τ |S j − Θj U j − u

45 46

16 17 18

(6.28)

0

1 |Y |

Y∗

0 + ψ x, w0 τ |S 0 − ψ 0 x, u τ |S 0

+

j=1

−

m j=1

Sj

−

j

0

|S j τ

Sj

23 24 25

27 28 29 30

(6.29)

31 32 33 34

| dσ

Gj (x, y)| U j − u 0 |S j | dσ 0. τ

36 37 38 39 40

G (x, y)| w (y) − w (y)

22

35

0

j

21

26

0

a0 e u0 + ey ( u) ey (w) − ey ( u) dy

m

19 20

Clearly, (6.25) is equivalent to the pair (6.27) (6.29). Then in order to satisfy (6.29) it is enough to solve the following problem for a.e. x in Ω:

43 44

14 15

TE

Ω×Y ∗

N

38

13

(6.27)

a e u0 + ey ( u) ey (w) − ey ( u) dx dy 0

U

37

0

35 36

12

D

1 |Y |

R EC

25

R

24

11

u) in terms of u0 , while Equation (6.27) is the limit problem for u0 , provided we express each ey ( (6.28) is the convergence of the energy. u), consider (6.24) with Φ = u0 and substract the last term from the first To get the expression of ey ( one to get

O

23

9 10

Λε

C

22

e(uε ) 2 dx → 0 upon closer inspection! .

20 21

F Φ dx,

while, by choosing Φ = 0 in (6.26), one gets

17

19

a0 e u0 + ey ( u) e(Φ)(x) dx dy =

O

13

(6.26)

O

Since Φ can be replaced by ±Φ in (6.26), this gives the identity

12

PR

11

2 3

F

6

(6.25)

Ω

j=1

3

1

. F u0 dx = I.

0

2

5

m 0 +Ψ Θj U j − u 0 τ |S j − u τ |S 0 +

1

4


41 42 43 44 45 46




2 3 4 5 6 7

This is the same as the (6.14) where U is replaced by e(u0 ) so that ey ( u) is nothing else than Ξ(x, y; e(u0 )(x)). Plugging this into (6.27), gives the variational formulation for the homogenized problem (6.20), i.e. ⎧ ⎨ Find u0 ∈ H 1 (Ω; ΓD ) ∀v ∈ H 1 (Ω; ΓD )

σ hom x, e u0 e(v) dx = F v dx. ⎩ Ω

Ω

8

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

O

O

PR

18

ε

Ω×S j

D

17

Proof. This is just a restatement of (6.28) by taking into account (6.23).

2

TE

16

3 4 5 6 7

10 11 12 13 14 15 16 17 18 19 20 21 22

Corollary 6.12. The following strong convergences hold: 0 Tε e(uε ) |Ω×Y 0 → e u0 + ey u strongly in L2 Ω × Y 0 , j Tε e(uε ) |Ω×Y j → ey u strongly in L2 Ω × Y j for j = 1, . . . , m,

23

as well as

e(uε ) 2 dx → 0.

29

R EC

15

R

14

Proposition 6.11. Under the hypotheses of the preceding sections, one has the convergences

1 ε lim a e(uε ), e(uε ) = a0 e u0 + ey ( u) e u0 + ey ( u) dx dy, |Y | Ω×Y ∗ 0 |S 0 , lim Ψε0 (uε )τ |S 0ε = Ψ 0 u

lim Ψεj (uε )τ S j = Ψ j U j − u 0 τ |S j = Gj (x, y) d U j − u 0 τ |S j .

Λε

O

13

2

9

6.5. Convergence of the energy and correctors

Proof. By (6.22),

24 25 26 27 28

30 31 32 33 34

m

1 Tε aε Tε e ujε Tε e ujε dx dy a e(uε ), e(uε ) = |Y | j j=0 Ω×Y

aε e(uε )e(uε ) dx. + ε

C

12

1

8

2

N

11

U

9 10

The proof of the main theorem is now complete.

F

1

31

Λε

The result follows by passing to the limit by using Lemma 4.9 of [3], Assumption (6.13) and the convergences established in Proposition 6.2. 2 Classically in the unfolding method (see [4] and [3]) the convergences of Corollary 6.12 imply the following corrector result:

35 36 37 38 39 40 41 42 43 44 45 46


32 1 2 3 4 5



Corollary 6.13. Under the hypotheses of the preceding sections, as ε → 0, 0 e(uε ) − e u0 − Uε ey u j e(uε ) − Uε ey u

L2 (Ωεj )

1 2

L2 (Ωε0 )

→ 0,

3 4

→ 0 for j = 1, . . . , m,

5

6 7

6

where Uε is the right inverse of Tε .

7

8

F

9 10

8

One can also obtain correctors for uε itself making use of Q1 interpolates. We refer to [4] and [3] for such correctors.

O

11 12

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

O

PR

D

20

TE

19

R EC

18

R

17

O

16

[1] R.P. Agarwal, S. Ding and C. Nolder, Inequalities for Differential Forms, Springer Verlag, New York, 2009. [2] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. Stud. Math. Appl., Vol. 5, North Holland, 1978. [3] D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal. 44(2) (2012), 718–760. [4] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Sér. I 335 (2002), 99–104. [5] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal. 40(4) (2008), 1585–1620. [6] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Univ. Press, Oxford, 1999. [7] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Applied Mathematical Sciences, Vol. 136, Springer Verlag, New York, 1999. [8] A. Damlamian, An elementary introduction to periodic unfolding, in: Multi Scale Problems and Asymptotic Analysis, Vol. 24, A. Damlamian, D. Lukkassen, A. Meidell and A. Piatnitski, eds, GAKUTO International Series Math. Sci. Appl., 2005, pp. 119–136. [9] G. Duvaut and J.L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. [10] C. Eck and J. Jaru˘sek, Existence results for the static contact problem with Coulomb friction, Mathematical Models and Methods in Applied Sciences 8 (1998), 445–468. [11] M. Heida, Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities, Applicable Analysis, DOI: 10.1080/00036811.2011.567191. [12] H.-K. Hummel, Homogenization for heat transfer in polycrystals with interfacial resistances, Applicable Analysis 75(3,4) (2000), 403–424. [13] N. Kikuchi and J.T. Oden, Contact Problems in Elasticity, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1988. [14] A. Mikelic, M. Shillor and R. Tapiero, Homogenization of an elastic material with soft inclusions in frictionless contact. Recent advances in contact mechanics, Math. Comput. Modelling 28 (1998), 4–8, 287–307. [15] O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, NorthHolland, Amsterdam, 1992. [16] E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory, Springer Verlag, Berlin, 1980. [17] G.A. Yosifian, On some homogenization problems in perforated domains with nonlinear boundary conditions, Applicable Analysis 65(3,4) (1997), 257–288. [18] G.A. Yosifian, Some homogenization problems for the system of elasticity with nonlinear boundary conditions in perforated domains, Applicable Analysis 71(1,4) (1998), 379–411.

C

15

N

14

U

13

References

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41

41

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Homogenization via unfolding in periodic elasticity with contact on ...

Homogenization via unfolding in periodic elasticity with contact on ...

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