Asymptotic Analysis - Volume 104, issue 1-2

Authors: Chaussade-Beaudouin, Marie | Dauge, Monique | Faou, Erwan | Yosibash, Zohar

Article Type: Research Article

Abstract: Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lamé system) are determined by an asymptotic analysis as the thickness (2 ε ) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar model that depends on two parameters: the angular frequency k and the half-thickness ε . Optimizing k for each chosen ε , we find power laws for k in function of ε that provide the smallest eigenvalues of the scalar reductions. Corresponding eigenpairs generate quasimodes for the …3D Lamé system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lamé system. Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as ε tends to 0. Its angular frequency exhibits a power law relation of the form k = γ ε − β with β = 1 4 in the parabolic case (cylinders and trimmed cones), and the various β s 2 5 , 3 7 , and 1 3 in the elliptic case. For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented. Show more

Keywords: Lamé, Koiter, asymptotic analysis, scalar reduction

DOI: 10.3233/ASY-171426

Citation: Asymptotic Analysis, vol. 104, no. 1-2, pp. 1-47, 2017

Authors: Ait-Mahiout, Karima | Alves, Claudianor O.

Article Type: Research Article

Abstract: This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems − Δ Φ u + ϕ ( | u | ) u = h ( ε x ) f ( u ) in  R N , u ∈ W 1 , Φ ( R N ) , where Φ ( t ) = ∫ 0 | t | ϕ ( s ) s d …s is an N -function, Δ Φ is the Φ-Laplacian operator, ε > 0 and h , f are continuous functions. In the proof of our main result we have used Variational methods, Ekeland’s variational principle and some properties of the Nehari manifolds. Show more

Keywords: Variational methods, quasilinear problems, Orlicz–Sobolev space

DOI: 10.3233/ASY-171428

Citation: Asymptotic Analysis, vol. 104, no. 1-2, pp. 49-66, 2017

Authors: Cherdantsev, Mikhail | Cherednichenko, Kirill | Neukamm, Stefan

Article Type: Research Article

Abstract: We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the “stiff” material, and a “soft” material that fills the remaining pores. We assume that the pores are of size 0 < ε ≪ 1 and are periodically distributed with period ε . We also assume that the stiffness of the soft material degenerates with rate ε 2 γ , γ > 0 , so that the contrast between the two materials becomes infinite …as ε ↓ 0 . We study the homogenisation limit ε ↓ 0 in a low energy regime, where the displacement of the stiff component is infinitesimally small. We derive an effective two-scale model, which, depending on the scaling of the energy, is either a quadratic functional or a partially quadratic functional that still allows for large strains in the soft inclusions. In the latter case, averaging out the small scale-term justifies a single-scale model for high-contrast materials, which features a non-linear and non-monotone effect describing a coupling between microscopic and the effective macroscopic displacements. Show more

Keywords: High-contrast homogenisation, nonlinear elasticity, two-scale Γ-convergence

DOI: 10.3233/ASY-171430

Citation: Asymptotic Analysis, vol. 104, no. 1-2, pp. 67-102, 2017