The periodic unfolding method was introduced in 2002 [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104] (with the basic proofs [Proceedings of Narvik Conference 2004, GAKUTO Internat. Math. Appl. 24, Gakkōtosho, Tokyo, 2006, 119–136]). In present paper we go into all details include complete proofs, as well several new extensions developments. This approach is based on two distinct ideas, each leading to a ingredient. first idea change scale, which...

We give a comprehensive presentation of the periodic unfolding method for perforated domains, both when unit hole is compact subset open cell and this impossible to achieve. In order apply boundary-value problems with nonhomogeneous Neumann conditions on boundaries holes, properties boundary operator are also extensively studied. The paper concludes applications such examples reiterated unfolding.

This paper deals with the error estimate in problems of periodic homogenization. The methods used are those unfolding. We give upper bound distance between unfolded gradient a function belonging to $H1(\Omega)$ and space $\nabla_x H^1(\Omega)\oplus \nabla_y L^2(\Omega ; H^1_{per}(Y))$. These distances obtained thanks technical result presented Theorem 2.3: defect harmonic $H1(Y)$ is written help norms $H^{1/2}$ its traces diff erences on opposite faces cell $Y$. without any supplementary...

In a previous paper about homogenization of the classical problem diffusion in bounded domain with sufficiently smooth boundary, we proved that global error is order ε 1/2 . Now, for an open set Ω boundary [Formula: see text] and homogeneous Dirichlet or Neumann limit conditions, show any strongly included ε. If ⊂ ℝ n polygonal (n = 2) polyhedral 3) also give interior estimates.

We consider a three-dimensional composite material made of small inclusions periodically embedded into an elastic matrix; the whole structure presents strong heterogeneities between its different components. In general framework linearized elasticity we show that, when size microstructures tends to zero, limit homogeneous presents, for some wavelengths, negative "mass density" tensor. Hence are able rigorously justify existence forbidden bands, i.e., intervals frequencies in which there is...

The aim of this paper is to give a decomposition the displacements plates with very high thickness contrast. Estimates all terms respect norm strain tensor are obtained. Weighted Poincaré-Wirtinger and Korn inequalities also given.

In this paper, we study the asymptotic behavior of a structure made curved rods thickness 2δ when δ tends to 0. This is carried on within frame linear elasticity by using unfolding method. It based several decompositions displacements and passing limit in fixed domains. We show that any displacement sum an elementary rods-structure (e.r.s.d.) concerning rods' cross sections residual one related deformation section. The e.r.s.d. coincides with rigid body junctions. Any given two functions...

The paper is dedicated to the asymptotic investigation of textiles as an elasticity problem on beam structures. structure subjected a simultaneous homogenization and dimension reduction with respect behavior beams' thickness periodicity. Important for are contact conditions between beams, which yield multiple limits depending order. In this two limiting cases presented: linear case Leray--Lions-type problem.

The aim of this work is to study the asymptotic behavior a structure made plates thickness $2δ$ when $δ\to 0$. This carried on within frame linear elasticity by using unfolding method. It based several decompositions displacements and passing limit in fixed domains. We begin with studying plate. show that any displacement sum an elementary concerning normal lines middle surface plate residual linked these deformations. An respect variable $x$3. written $U(^x)+R(^x)\land x3e3$ where U...

An efficient technique is proposed for analyzing two- and three-dimensional lossy periodic composite materials, combining an asymptotic multiscale method with the unfolding method. The computed effective conductivities square cylinders cubes suspended in a host isotropic medium are compared to Maxwell-Garnett mixing formula predictions. electromagnetic field finite lattice of round exact calculated directly heterogeneous

Abstract We consider in this work general curved rods with a circular cross‐section of radius δ . Our aim is to study the asymptotic behaviour such as →0, framework linear elasticity according unfolding method. It consists giving some decompositions displacements rods, and then passing limit fixed domain. A first decomposition concerns elementary rod which characterize its translations rotations, residual related deformation cross‐section. The second middle‐line rod. prove that displacement...

This paper is devoted to investigate a few microscopic effects in the homogenization process of junction periodic family rods with thin plate elasticity. We focus on case where thickness tends zero faster than periodicity. As consequence studied effects, elastic coefficients membrane and bending limit problems for are modified. Moreover, we observe torsion homogenized “continuum” which depends curl displacement plate.

The elasticity problem for two domains separated by a heterogeneous layer of the thickness ε is considered. has an ε-periodic structure, ε≪1, including multiple cracks and contact between structural components. inclusions are surrounded can have rigid displacem ents. contacts described Signorini Tresca-friction conditions. In order to obtain preliminary estimates, modification Korn inequality ε-dependent periodic performed. An asymptotic analysis with respect ε→0 provided limit obtained,...

The paper is dedicated to the modeling of elasticity problem for a textile structure. made long and thin fibers, crossing each other in periodic pattern, forming woven canvas square domain. partially clamped. fibers cannot penetrate but can slide with respect in‐plane directions. sliding bounded by contact function, which chosen loose. partial clamp loose lead domain partitioning, different expected behaviors on four subdomains. homogenization via unfolding method, an additional dimension...

In this paper, we show that any displacement of a plate is the sum Kirchhoff–Love and two terms, one for shear warping. The then loaded so bending contribute same order magnitude to fiber rotation.