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The aim of this paper is to decompose the displacements of thin-walled beams with rectangular cross-section. The decomposition is accompanied by estimates of all its terms with respect to the norm of the strain tensor. Korn's inequality is also given. [ABSTRACT FROM AUTHOR]

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

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

We consider a perforated half-cylindrical thin shell and investigate the limit behavior when the period and the thickness simultaneously go to zero. By using the decomposition of shell displacements presented in Griso [JMPA89 (2008) 199–223] we obtain a priori estimates. With the unfolding and rescaling operator we transform the problem to a reference configuration. In the end this yields a homogenized limit problem for the shell. [ABSTRACT FROM AUTHOR]

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

The elasticity problem for two domains separated by a heterogeneous layer of the thickness ε is considered. The layer has an ε-periodic structure, ε ≪ 1, including a multiple cracks and the contact between the structural components. The inclusions are surrounded by cracks and can have rigid displacements. The contacts are described by the Signorini and Trescafriction conditions. In order to obtain preliminary estimates, a modification of the Korn inequality for the ε-dependent periodic layer is performed. An asymptotic analysis with respect to ε → 0 is provided and the limit elasticity problem is obtained, together with the transmission condition across the interface. The periodic unfolding method is used to study the limit behavior. [ABSTRACT FROM AUTHOR]

This paper deals with the introduction of a decomposition of the deformations of curved thin beams, with section of order δ, which takes into account the specific geometry of such beams. A deformation v is split into an elementary deformation and a warping. The elementary deformation is the analog of a Bernoulli–Navier's displacement for linearized deformations replacing the infinitesimal rotation by a rotation in SO(3) in each cross section of the rod. Each part of the decomposition is estimated with respect to the L2 norm of the distance from gradient v to SO(3). This result relies on revisiting the rigidity theorem of Friesecke–James–Müller in which we estimate the constant for a bounded open set star-shaped with respect to a ball. Then we use the decomposition of the deformations to derive a few types of asymptotic geometrical behavior: large deformations of extensional type, inextensional deformations and linearized deformations. To illustrate the use of our decomposition in nonlinear elasticity, we consider a St Venant–Kirchhoff material and upon various scalings on the applied forces we obtain the Γ-limit of the rescaled elastic energy. We first analyze the case of bending forces of order δ2 which leads to a nonlinear extensible model. Smaller pure bending forces give the classical linearized model. A coupled extentional-bending model is obtained for a class of forces of order δ2 in traction and of order δ3 in bending. [ABSTRACT FROM AUTHOR]

Abstract: In this study we present first the main theorem of the unfolding method in linearized elasticity. Then we prove that every displacement of a thin structure (curved rod or shell) is the sum of an elementary displacement and a warping. Thanks to the previous theorem we obtain sharp estimates of the displacements of this decomposition. [Copyright &y& Elsevier]

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

In this paper, we study the asymptotic behavior of a structure made of curved rods of thickness 2δ when δ tends to 0. This study is carried on within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of the structure displacements and on the passing to the limit in fixed domains. We show that any displacement of a structure is the sum of an elementary rods-structure displacement (e.r.s.d.) concerning the rods' cross sections and a residual one related to the deformation of the cross section. The e.r.s.d. coincides with rigid body displacements in the junctions. Any e.r.s.d. is given by two functions belonging to $H^1(\mathcal{S};\mathbb{R}^3)$ where $\mathcal{S}$ is the skeleton structure (i.e. the set of rods with middle lines). One of this function $\mathcal{U}$ is the skeleton displacement, the other $\mathcal{R}$ gives the cross section rotation. We show that $\mathcal{U}$ is the sum of an extensional displacement and an inextensional one. We establish a priori estimates and then, we characterize the unfolded limits of the rods-structure displacements. Eventually, we pass to the limit in the linearized elasticity system and using all results in [6], where on one hand, we obtain a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem coupling the limit of inextensional displacement and the limit of the rod torsion angles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the asymptotic behavior of a structure made of plates of thickness 2δ when δ → 0. This study is carried out within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of displacements of the structure and on the passing to the limit in fixed domains. We begin by studying the displacements of a plate. We show that any displacement is the sum of an elementary displacement concerning the normal lines on the middle surface of the plate and a warping. An elementary displacement is linear with respect to the variable x3. It is written $\mathcal{U}(\hat{x})+\mathcal{R}(\hat{x})\wedge x_3 {\bf e}_3$ where $\mathcal{U}$ is a displacement of the mid-surface of the plate. We show a priori estimates and convergence results when δ → 0. We characterize the limits of the unfolded displacements of a plate as well as the limits of the unfolded strained tensor. Then, we extend these results to structures made of plates. We show that any displacement of a structure is the sum of an elementary displacement of each plate and of a residual displacement. The elementary displacements of the structure (e.p.s.d.) coincide with elementary rod displacements in the junctions. Any e.p.s.d. is given by two functions belonging to H1(S; ℝ3) where S is the skeleton of the structure (the set formed by the mid-surfaces of the plates constituting the surface). One of these functions, $\mathcal{U}$, is the skeleton displacement. We show that $\mathcal{U}$ is the sum of an extensional displacement and of an inextensional one. The first one characterizes the membrane displacements and the second one is a rigid displacement in the direction of the plates and it characterizes the flexion of the plates. Eventually, we pass to the limit as δ → 0 in the linearized elasticity system. On the one hand, we obtain a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem satisfied by the limit of inextensional displacements. [ABSTRACT FROM AUTHOR]

Abstract: We consider a set of elastic rods periodically distributed over a 3d elastic plate (both of them with axis ) and we investigate the limit behavior of this problem as the periodicity ε and the radius r of the rods tend to zero. We use a decomposition of the displacement field in the rods of the form where the principal part U is a field which is piecewise constant with respect to the variables (and then naturally extended on a fixed domain), while the perturbation remains defined on the domain containing the rods. We derive estimates of U and in term of the total elastic energy. This allows to obtain a priori estimates on u without solving the delicate question of the dependence, with respect to ε and r, of the constant in Korn''s inequality in a domain with such a rough boundary. To deal with the field , we use a version of an unfolding operator which permits both to rescale all the rods and to work on the same fixed domain as for U to carry out the homogenization process. The above decomposition also helps in passing to the limit and to identify the limit junction conditions between the rods and the 3d plate. [Copyright &y& Elsevier]