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Igor Velčić

Publications and source records attributed to Igor Velčić.

At least 19 recordsLinked to original sources
arXiv2026

Homogenization of elasto-plastic plate equations with vanishing hardening

We study the asymptotic behavior of thin heterogeneous elastoplastic plates in the framework of linearized elastoplasticity, focusing on the regime where the plate thickness vanishes much faster than the characteristic scale of the material's periodic microstructure. In contrast to earlier analyzes that required restrictive geometric assumptions on admissible yield surfaces, our approach accommodates general relations between phases without imposing any specific ordering. The analysis proceeds in two main steps. First, we rigorously derive a heterogeneous plate model with both isotropic and kinematic hardening through a dimension reduction procedure based on evolutionary $Γ$-convergence. This result extends existing plate models for homogeneous materials to the heterogeneous setting and allows for general forms of hardening and dissipation potentials. In the second step, we perform two-scale homogenization while simultaneously letting the hardening tend to zero. This process yields an effective elasto-perfectly plastic plate model and, crucially, provides a characterization of the dissipation potential at the interfaces between different phases. The resulting dissipation functional takes the form of a non-local inf-convolution of the traces of plastic strains on both sides of the interface, reflecting the Kirchhoff-Love structure of admissible displacements.

math.AP↗
arXiv2025

Quantitative estimates for high-contrast random media

This paper studies quantitative homogenization of elliptic equations with random, uniformly elliptic coefficients that vanish in a union of random holes. Assuming an upper bound on the size of the holes and a separation condition between them, we derive optimal bounds for the regularity radius $r_*$ and suboptimal growth estimates for the corrector. These results are key ingredients for error analysis in stochastic homogenization and serve as crucial input for recent developments in the double-porosity model, such as those by Bonhomme, Duerinckx, and Gloria (arXiv:2502.02847).

math.AP↗
arXiv2025

High-contrast random systems of PDEs: homogenisation and spectral theory

We develop a qualitative homogenisation and spectral theory for elliptic systems of partial differential equations in divergence form with highly contrasting (i.e., non uniformly elliptic) random coefficients. The focus of the paper is on the behaviour of the spectrum as the heterogeneity parameter tends to zero; in particular, we show that in general one doesn't have Hausdorff convergence of spectra. The theoretical analysis is complemented by several explicit examples, showcasing the wider range of applications and physical effects of systems with random coefficients, when compared with systems with periodic coefficients or with scalar operators (both random and periodic).

math.AP↗
arXiv2025

Poroelastic plate model obtained by simultaneous homogenization and dimension reduction

In this paper, the starting point of our analysis is a coupled system of linear elasticity and Stokes equation. We consider two small parameters: the thickness $h$ of the thin plate and the pore scale $\varepsilon(h)$ which depends on $h$. We will focus specifically on the case when the pore size is comparatively small relative to the thickness of the plate. The main goal here is derive a model of a poroelastic plate, starting from the $3D$ problem as $h$ goes to zero, using simultaneous homogenization and dimension reduction techniques. The obtained model generalizes the poroelastic plate model derived by A. Mikelić et. al. in 2015 using dimension reduction techniques from $3D$ Biot's equations in the sense that it also covers the case of contacts of poroelastic and (poro)elastic plate as well as the evolution equation with inertial term.

math.AP↗
arXiv2024

High-contrast random composites: homogenisation framework and new spectral phenomena

We develop a framework for multiscale analysis of elliptic operators with high-contrast random coefficients. For a general class of such operators, we provide a detailed spectral analysis of the corresponding homogenised limit operator. Under some lenient assumptions on the configuration of the random inclusions, we fully characterise the limit of the spectra of the high-contrast operators in question, which unlike in the periodic setting is shown to be different to the spectrum of the homogenised operator. Introducing a new notion of the relevant limiting spectrum, we describe the connection between these two sets.

math.AP↗
arXiv2023

Eigenfunctions localised on a defect in high-contrast random media

We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrum of a high-contrast random operator. We consider a family of elliptic operators $\mathcal{A}^\varepsilon$ in divergence form whose coefficients are random, possess double porosity type scaling, and are perturbed on a fixed-size compact domain (a defect). Working in the gaps of the limiting spectrum of the unperturbed operator $\hat{\mathcal{A}}^\varepsilon$, we show that the point spectrum of $\mathcal{A}^\varepsilon$ converges in the sense of Hausdorff to the point spectrum of the limiting two-scale operator $\mathcal{A}^\mathrm{hom}$ as $\varepsilon \to 0$. Furthermore, we prove that the eigenfunctions of $\mathcal{A}^\varepsilon$ decay exponentially at infinity uniformly for sufficiently small $\varepsilon$. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of $\mathcal{A}^\mathrm{hom}$.

math.SP↗
arXiv2023

Operator-norm resolvent estimates for thin elastic periodically heterogeneous rods in moderate contrast

We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing the thin infinite elastic rod with material coefficients which periodically highly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the thickness of the rod and the period of material oscillations. These two parameters are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under some additional assumptions, as well as in the general case where these two sorts of displacements intertwine inseparably.

math.AP↗
arXiv2023

Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: the limiting regimes

We identify effective models for thin, linearly elastic and perfectly plastic plates exhibiting a microstructure resulting from the periodic alternation of two elastoplastic phases. We study here both the case in which the thickness of the plate converges to zero on a much faster scale than the periodicity parameter and the opposite scenario in which homogenization occurs on a much finer scale than dimension reduction. After performing a static analysis of the problem, we show convergence of the corresponding quasistatic evolutions. The methodology relies on two-scale convergence and periodic unfolding, combined with suitable measure-disintegration results and evolutionary $Γ$-convergence.

math.AP↗
arXiv2022

Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure

An effective model is identified for thin perfectly plastic plates whose microstructure consists of the periodic assembling of two elastoplastic phases, as the periodicity parameter converges to zero. Assuming that the thickness of the plates and the periodicity of the microstructure are comparably small, a limiting description is obtained in the quasistatic regime via simultaneous homogenization and dimension reduction by means of evolutionary $Γ$-convergence, two-scale convergence, and periodic unfolding.

math.AP↗
arXiv2022

Spectral and evolution analysis of composite elastic plates with high contrast

We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.

math.AP↗
arXiv2022

Sharp operator-norm asymptotics for thin elastic plates with rapidly oscillating periodic properties

We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions, in the asymptotic regime when the period and the plate thickness are of the same order of smallness. Assuming that the displacement gradients of the points of the plate are small enough for the equations of linearised elasticity to be a suitable approximation of the material response, such as the case in e.g. acoustic wave propagation, we derive a class of "hybrid", homogenisation dimension-reduction, norm-resolvent estimates for the plate, under different energy scalings with respect to the plate thickness.

math.AP↗
arXiv2021

Microscopical Justification of the Winterbottom problem for well-separated Lattices

We consider the discrete atomistic setting introduced in \cite{PiVe1} to microscopically justify the continuum model related to the \emph{Winterbottom problem}, i.e., the problem of determining the equilibrium shape of crystalline film drops resting on a substrate, and relax the rigidity assumption considered in \cite{PiVe1} to characterize the \emph{wetting} and \emph{dewetting regimes} and to perform the \emph{discrete to continuum passage}. In particular, all results of \cite{PiVe1} are extended to the setting where the distance between the reference lattices for the film and the substrate is not smaller than the optimal bond length between a film and a substrate atom. Such optimal film-substrate bonding distance is prescribed together with the optimal film-film distance by means of two-body atomistic interaction potentials of Heitmann-Radin type, which are both taken into account in the discrete energy, and in terms of which the wetting-regime threshold and the effective expression for the wetting parameter in the continuum energy are determined.

math.AP↗
arXiv2020

Microscopical justification of Solid-State Wetting and Dewetting

The continuum model related to the Winterbottom problem, i.e., the problem of determining the equilibrium shape of crystalline drops resting on a substrate, is derived in dimension two by means of a rigorous discrete-to-continuum passage by $Γ$-convergence of atomistic models taking into consideration the atomic interactions of the drop particles both among themselves and with the fixed substrate atoms. As a byproduct of the analysis effective expressions for the drop surface anisotropy and the drop/substrate adhesion parameter appearing in the continuum model are characterized in terms of the atomistic potentials, which are chosen of Heitmann-Radin sticky-disc type. Furthermore, a threshold condition only depending on such potentials is determined distinguishing the wetting regime, where discrete minimizers are explicitly characterized as configurations contained in a layer with a one-atom thickness, i.e., the wetting layer, on the substrate, from the dewetting regime. In the latter regime, also in view of a proven conservation of mass in the limit as the number of atoms tends to infinity, proper scalings of the minimizers of the atomistic models converge (up to extracting a subsequence and performing translations on the substrate surface) to a bounded minimizer of the Winterbottom continuum model satisfying the volume constraint.

math.AP↗
arXiv2017

Stochastic homogenisation of high-contrast media

Using a suitable stochastic version of the compactness argument of [V. V. Zhikov, 2000. On an extension of the method of two-scale convergence and its applications. Sb. Math., 191(7--8), 973--1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled "macroscopic" and "microscopic" components of the field, giving rise to an analogue of the "Zhikov function", which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit.

math-ph↗
arXiv2011

Periodically wrinkled plate of the Föppl-von Kármán type

In this paper we derive, by means of $Γ$-convergence, the periodically wrinkled plate model starting from three dimensional nonlinear elasticity. We assume that the thickness of the plate is $h^2$ and that the mid-surface of the plate is given by $(x_1,x_2) \to (x_1,x_2,h^2θ(\per))$, where $θ$ is $[0,1]^2$ periodic function. We also assume that the strain energy of the plate has the order $h^8=(h^2)^4$, which corresponds to the Föppl-von Kármán model in the case of the ordinary plate. The obtained model mixes the bending part of the energy with the stretching part.

math.AP↗
arXiv2011

Derivation of the nonlinear bending-torsion model for a junction of elastic rods

In this paper we derive the one-dimensional bending-torsion equilibrium model modeling the junction of straight rods. The starting point is a three-dimensional nonlinear elasticity equilibrium problem written as a minimization problem for a union of thin rod-like bodies. By taking the limit as the thickness of the 3D rods goes to zero, and by using ideas from the theory of $Γ$-convergence, we obtain that the resulting model consists of the union of the usual one-dimensional nonlinear bending-torsion rod models which satisfy the following transmission conditions at the junction point: continuity of displacement and rotation of the cross-sections and balance of contact forces and contact couples.

math.AP↗
arXiv2011

Shallow shell models by Gamma convergence

In this paper we derive, by means of $Γ$-convergence, the shallow shell models starting from non linear three dimensional elasticity. We use the approach analogous to the one for shells and plates. We start from the minimization formulation of the general three dimensional elastic body which is subjected to normal volume forces and free boundary conditions and do not presuppose any constitutional behavior. To derive the model we need to propose how is the order of magnitudes of the external loads related to the thickness of the body $h$ as well as the order of the "geometry" of the shallow shell. We analyze the situation when the external normal forces are of order $h^α$, where $α>2$. For $α=3$ we obtain the Marguerre-von Kármán model and for $α>3$ the linearized Marguerre-von Kármán model. For $α\in (2,3)$ we are able to obtain only the lower bound for the $Γ$-limit. This is analogous to the recent results for the ordinary shell models.

math.AP↗