Stefan Neukamm

AN OPTIMAL QUANTITATIVE TWO-SCALE EXPANSION IN STOCHASTIC HOMOGENIZATION OF DISCRETE ELLIPTIC EQUATIONS

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the

Derivation of a Homogenized Von-Kármán Plate Theory from 3d Nonlinear Elasticity

L^2

On the Commutability of Homogenization and Linearization in Finite Elasticity

-norm in probability of the \mbox{

Stochastic Homogenization of Nonconvex Discrete Energies with Degenerate Growth

H^1

Quantitative Homogenization in Nonlinear Elasticity for Small Loads

-norm} in space of this error scales like

Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

\epsilon

On the interplay of two-scale convergence and translation

, where

A regularity theory for random elliptic operators

\epsilon

Homogenization, linearization and dimension reduction in elasticity with variational methods

is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author.

Rigorous Derivation of a Homogenized Bending-Torsion Theory for Inextensible Rods from Three-Dimensional Elasticity

We rigorously derive a homogenized von-Karman plate theory as a Γ-limit from nonlinear three-dimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, three-dimensional plate with spatially periodic material properties. The functional features two small length scales: the period e of the elastic composite material, and the thickness h of the slender plate. We study the behavior as e and h simultaneously converge to zero in the von-Karman scaling regime. The obtained limit is a homogenized von-Karman plate model. Its effective material properties are determined by a relaxation formula that exposes a non-trivial coupling of the behavior of the out-of-plane displacement with the oscillatory behavior in the in-plane directions. In particular, the homogenized coefficients depend on the relative scaling between h and e, and different values arise for h ≪ e, e ~ h and e ≪ h.