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Igor Velcic

Publications and source records attributed to Igor Velcic.

10 recordsLinked to original sources
arXiv2025

Homogenisation and spectral convergence of high-contrast convolution type operators

The paper deals with homogenisation problems for high-contrast symmetric convolution-type operators with integrable kernels in media with a periodic microstructure. We adapt the two-scale convergence method to nonlocal convolution-type operators and obtain the homogenisation result both for problems stated in the whole space and in bounded domains with the homogeneous Dirichlet boundary condition. Our main focus is on spectral analysis. We describe the spectrum of the limit two-scale operator and characterize the limit behaviour of the spectrum of the original problem as the microstructure period tends to zero. It is shown that the spectrum of the limit operator is a subset the limit of the spectrum of the original operator, and that they need not coincide.

math.AP↗
arXiv2016

On the simultaneous homogenization and dimension reduction in elasticity and locality of $Γ$-closure

On the example of linearized elasticity we provide a framework for simultaneous homogenization and dimension reduction in the setting of linearized elasticity as well as non-linear elasticity for the derivation of homogenized von Kármán plate and bending rod models. The framework encompasses even perforated domains and domains with oscillatory boundary, provided that the corresponding extension operator can be constructed. Locality property of $Γ$-closure is established, i.e. every energy density obtained by the homogenization process can be in almost every point be obtained as the limit of periodic energy densities.

math.AP↗
arXiv2014

On the derivation of homogenized bending plate model

We derive, via simultaneous homogenization and dimension reduction, the $Γ$-limit for thin elastic plates of thickness $h$ whose energy density oscillates on a scale $\eh$ such that $ \eh^2 \ll h\ll \eh$. We consider the energy scaling that corresponds to Kirchhoff's nonlinear bending theory of plates.

math.AP↗
arXiv2014

Homogenization of bending theory for plates; the case of elastic laminates

In this paper we study the homogenization effects on the model of elastic plate in the bending regime, under the assumption that the energy density (material) oscillates in the direction of thickness. We study two different cases. First, we show, starting from 3D elasticity, by means of $Γ$-convergence and under general (not necessarily periodic) assumption, that the effective behavior of the limit is not influenced by oscillations in the direction of thickness. In the second case, we study periodic in-plane oscillations of the energy density coupled with periodic oscillations in the direction of thickness. In contrast to the first case we show that there are homogenization effects coming also from the oscillations in the direction of thickness.

math.AP↗
arXiv2014

General homogenization of bending-torsion theory for inextensible rods from 3D elasticity

We derive, by means of Gamma-convergence, the equations of homogenized bending rod starting from $3D$ nonlinear elasticity equations. The main assumption is that the energy behaves like h^2 (after dividing by the order h^2 of vanishing volume) where h is the thickness of the body. We do not presuppose any kind of periodicity and work in the general framework. The result shows that, on a subsequence, we always obtain the equations of bending-torsion rod and identifies, in an abstract formulation, the limiting quadratic form connected with that model. This is a generalization from periodic to non-periodic homogenization of bending-torsion rod theory already present in the literature.

math.AP↗
arXiv2013

Derivation of a homogenized von-Karman shell theory from 3D elasticity

We derive the model of homogenized von Kármán shell theory, starting from three dimensional nonlinear elasticity. The original three dimensional model contains two small parameters: the oscillations of the material $\e$ and the thickness of the shell $h$. Depending on the asymptotic ratio of these two parameters, we obtain different asymptotic theories. In the case $h\ll\e$ we identify two different asymptotic theories, depending on the ratio of $h$ and $\e^2$. In the case of convex shells we obtain a complete picture in the whole regime $h\ll\e$.

math.AP↗
arXiv2012

Derivation of a homogenized nonlinear plate theory from 3d elasticity

We derive, via simultaneous homogenization and dimension reduction, the Gamma-limit for thin elastic plates whose energy density oscillates on a scale that is either comparable to, or much smaller than, the film thickness. We consider the energy scaling that corresponds to Kirchhoff's nonlinear bending theory of plates.

math.AP↗