Igor Velcic

Igor Velcic
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Mathematics - Analysis of PDEs (13)

Publications Authored By Igor Velcic

We use the notion of stochastic two-scale convergence to solve the problem of stochastic homogenization of the elastic plate in the bending regime. Read More

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 K\'arm\'an 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 $\Gamma$-closure is established, i. Read More

We prove smoothness of $H^2$ isometric immersions of surfaces endowed with a smooth Riemannian metric of positive Gauss curvature. We use this regularity result to rigorously derive homogenized bending models of convex shells from three-dimensional nonlinear elasticity. Read More

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 $\Gamma$-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. Read More

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. Read More

Starting from 3D elasticity equations we derive the model of the homogenized von K\'arm\'an plate by means of $\Gamma$-convergence. This generalizes the recent results, where the material oscillations were assumed to be periodic. Read More

We derive, via simultaneous homogenization and dimension reduction, the $\Gamma$-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. Read More

We derive the model of homogenized von K\'arm\'an 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. Read More

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. Read More

In this paper we derive, by means of $\Gamma$-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 $(x_1,x_2) \to (x_1,x_2,h^2\theta(\per))$, where $\theta$ 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 F\"oppl-von K\'arm\'an model in the case of the ordinary plate. Read More

In this paper we derive, by means of $\Gamma$-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. Read More

We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use the approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity and do not presuppose any constitutional behavior. To derive the model, by means of $\Gamma$-convergence, we need to propose how is the order of strain energy related to the thickness of the body $h$. Read More

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 $\Gamma$-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. Read More