Nonlinear elastic membranes involving the p-Laplacian operator

Applied Mathematics Letters, 2010

In this note we give some remarks and improvements on our recent paper [5] about an optimization problem for the p-Laplace operator that were motivated by some discussion that we had with Prof. Cianchi.

2022

We study a nonlinear generalization of a free boundary problem that arises in the context of thermal insulation. We consider two open sets Ω ⊆ A, and we search for an optimal A in order to minimize a non-linear energy functional, whose minimizers u satisfy the following conditions: ∆ p u = 0 inside A \ Ω, u = 1 in Ω, and a nonlinear Robin-like boundary (p, q)-condition on the free boundary ∂A. We study the variational formulation of the problem in SBV, and we prove that, under suitable conditions on the exponents p and q, a minimizer exists and its jump set satisfies uniform density estimates.

UDC 519.6

Computers & Structures, 2004

emis.ams.org

In this work we consider a semilinear evolution problem which we pose as follows: Let Ω and Ωm be two open bounded connected subsets of R2 with sufficiently smooth boundary ∂Ω and ∂Ωm so that Ωm ⊂⊂ Ω. Let Ωp := Ω\Ωm and Γ1 := ∂Ωm. We decompose ∂Ω in two ...

Eastern-European Journal of Enterprise Technologies

A geometrically and physically nonlinear model of a membrane cylindrical shell, which has been built and tested, describes the behavior of a airbag made of fabric material. Based on the geometrically accurate relations of "strain-displacement", it has been shown that the equilibrium equations of the shell, written in terms of Biot stresses, together with boundary conditions acquire a natural physical meaning and are the consequences of the principle of virtual work. The physical properties of the shell were described by Fung’s hyper-elastic biological material because its behavior is similar to that of textiles. For comparison, simpler hyper-elastic non-compressible Varga and Neo-Hookean materials, the zero-, first-, and second-order materials were also considered. The shell was loaded with internal pressure and convergence of edges. The approximate solution was constructed by an spectral method; the exponential convergence and high accuracy of the equilibrium equations in...

Journal of Elasticity, 2004

The relation between the classical formulation of linear elastic problems in displacements and the stress formulation proposed by Pobedria is studied. It is shown that if the Navier and Pobedria differential operators are elliptic then corresponding boundary value problems are equivalent. The values of parameters for which Pobedria's boundary value problem has the Fredholm property are found. The homogeneous Pobedria's system is considered as a spectral problem with Poisson's ratio as a spectral parameter. The points of the essential spectrum are found and classified. The example of solving Pobedria's system for the Lamé problem for a spherical shell is presented.

Acta Mechanica, 1983

This paper deals with the concept of homogeneity within the framework of hyperelastic anisotropic membranes. A frame field, i.e. an orthonormal set of vectors lying in the tangent plahe at each point of the membrane, is used to represent observers regarded as equivalent for comparing material response. The notions of homogeneity and unidirectional homogeneity are formulated in this setting and conditions required for a given strain energy function to define a homogeneous material are derived. The paper concludes with a discussion of certain additional features which illustrate the concepts and which arise out of special choices of the strain energy function.

Contemporary Mathematics, 2000

In this paper, continuing our earlier article [CGIKO], we study qualitative properties of solutions of a certain eigenvalue optimization problem. Especially we focus on the study of the free boundary of our optimal solutions on general domains.

Applied Mathematics Letters, 2010

In this note we give some remarks and improvements on a recent paper of us [3] about an optimization problem for the p−Laplace operator that were motivated by some discussion the authors had with Prof. Cianchi.

Control and cybernetics

In this paper, we introduce four new classes of open sets in general Euclidean space R^N. It is shown that every class of open sets is compact under the Hausdorff distance. The result is applied to a shape optimization problem of p-Laplacian equation. The existence of the optimal solution is presented.

Journal of Elasticity, 2009

For very thin shell-like structures it is common to ignore bending effects and model the structure using simple membrane theory. However, since the thickness of the membrane is not modeled explicitly in simple membrane theory it is not possible to use the three-dimensional strain energy function directly. Approximations must be introduced like the assumptions of: no thickness changes, generalized plane stress or incompressibility. In contrast, the theory of a Cosserat generalized membrane uses the three-dimensional strain energy function directly, it includes both thickness changes and shear deformation and it allows contact conditions to be formulated on the interface of the membrane with another body instead of on the middle surface of the membrane. A specific nonlinear contact problem is used to study these effects and comparison is made with solutions of a hierarchy of theories which include different levels of deformation through the thickness of the membrane and different formulations of the contact conditions. The results indicate that within the context of a simple membrane the assumption of generalized plane stress is best for this problem and that a generalized contact condition extends the range validity of the simple membrane solution to thicker membranes.

Communications in Nonlinear Science and Numerical Simulation, 2007

In this work, we are interested in obtaining an approximated numerical solution for the model of vibrating elastic membranes with moving boundary. The model is an extension of Kirchhoff's model, which takes into account the change of size during the vibration. We apply the Finite Element Method with a Finite Difference Method in time to obtain an approximated numerical solution. Some numerical experiments are presented to show the effect of moving boundary effects in vibrating elastic membranes.