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Effective energy integral functionals for thin films with bending moment in the Orlicz-Sobolev space setting

Włodzimierz Laskowski, Hôǹg Thái Nguyêñ (2014)

Banach Center Publications

In this paper we deal with the energy functionals for the elastic thin film ω ⊂ ℝ² involving the bending moments. The effective energy functional is obtained by Γ-convergence and 3D-2D dimension reduction techniques. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type and to satisfy the conditions Δ₂ and...

Effective energy integral functionals for thin films with three dimensional bending moment in the Orlicz-Sobolev space setting

Włodzimierz Laskowski, Hong Thai Nguyen (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider an elastic thin film ω ⊂ ℝ² with the bending moment depending also on the third thickness variable. The effective energy functional defined on the Orlicz-Sobolev space over ω is described by Γ-convergence and 3D-2D dimension reduction techniques. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type...

Equi-integrability results for 3D-2D dimension reduction problems

Marian Bocea, Irene Fonseca (2002)

ESAIM: Control, Optimisation and Calculus of Variations

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients α u ε | 1 ε 3 u ε bounded in L p ( Ω ; 9 ) , 1 < p < + . Here it is shown that, up to a subsequence, u ε may be decomposed as w ε + z ε , where z ε carries all the concentration effects, i.e. α w ε | 1 ε 3 w ε p is equi-integrable, and w ε captures the oscillatory behavior, i.e. z ε 0 in measure. In addition, if { u ε } is a recovering sequence then z ε = z ε ( x α ) nearby Ω .

Equi-integrability results for 3D-2D dimension reduction problems

Marian Bocea, Irene Fonseca (2010)

ESAIM: Control, Optimisation and Calculus of Variations

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients α u ε | 1 ε 3 u ε bounded in L p ( Ω ; 9 ) , 1 < p < + . Here it is shown that, up to a subsequence, u ε may be decomposed as w ε + z ε , where z ε carries all the concentration effects, i.e. α w ε | 1 ε 3 w ε p is equi-integrable, and w ε captures the oscillatory behavior, i.e. z ε 0 in measure. In addition, if { u ε } is a recovering sequence then z ε = z ε ( x α ) nearby Ω .

Estimation of vibration frequencies of linear elastic membranes

Luca Sabatini (2018)

Applications of Mathematics

The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2-dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to...

Eulerian formulation and level set models for incompressible fluid-structure interaction

Georges-Henri Cottet, Emmanuel Maitre, Thomas Milcent (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is devoted to Eulerian models for incompressible fluid-structure systems. These models are primarily derived for computational purposes as they allow to simulate in a rather straightforward way complex 3D systems. We first analyze the level set model of immersed membranes proposed in [Cottet and Maitre, Math. Models Methods Appl. Sci.16 (2006) 415–438]. We in particular show that this model can be interpreted as a generalization of so-called Korteweg fluids. We then extend this model...

Exact controllability of an elastic membrane coupled with a potential fluid

Scott Hansen (2001)

International Journal of Applied Mathematics and Computer Science

We consider the problem of boundary control of an elastic system with coupling to a potential equation. The potential equation represents the linearized motions of an incompressible inviscid fluid in a cavity bounded in part by an elastic membrane. Sufficient control is placed on a portion of the elastic membrane to insure that the uncoupled membrane is exactly controllable. The main result is that if the density of the fluid is sufficiently small, then the coupled system is exactly controllable....

Existence of a solution for a nonlinearly elastic plane membrane “under tension”

Daniel Coutand (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A justification of the two-dimensional nonlinear “membrane” equations for a plate made of a Saint Venant-Kirchhoff material has been given by Fox et al. [9] by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. This model, which retains the material-frame indifference of the original three dimensional problem in the sense that its energy density is invariant under the rotations of 3 , is equivalent to finding the critical points...

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