# Stefan NeukammTechnische Universität Dresden | TUD · Faculty of Mathematics

Stefan Neukamm

Dr. rer. nat.

## About

48

Publications

2,777

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812

Citations

Citations since 2016

Introduction

Stefan Neukamm works at the Faculty of Mathematics at TU Dresden, Germany.
Research interests: applied analysis, homogenization, random heterogeneous materials, regularity in connection with homogenization, dimension reduction and homogenization in nonlinear elasticity.

## Publications

Publications (48)

The qualitative theory of stochastic homogenization of uniformly elliptic
linear (but possibly non-symmetric) systems in divergence form is
well-understood. Quantitative results on the speed of convergence, and on the
error in the representative volume method, like those recently obtained by the
authors for scalar equations, require a type of stoch...

We consider periodic homogenization of nonlinearly elastic composite materials. Under suitable assumptions on the stored energy function (frame indifference; minimality, non-degeneracy and smoothness at identity; $p\geq d$-growth from below), and on the microgeometry of the composite (covering the case of smooth, periodically distributed inclusions...

We study the continuum limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions $d\geq 2$. Since we are interested in energy functionals with random (stationary and ergodic) pair interactions, our problem corresponds to a stochastic homogenization problem. In the non-degenerate case, when the interactions satisfy a...

The aim of our work is to provide a simple homogenization and discrete-to-continuum procedure for energy driven problems involving stochastic rapidly-oscillating coefficients. Our intention is to extend the periodic unfolding method to the stochastic setting. Specifically, we recast the notion of stochastic two-scale convergence in the mean by intr...

We present an introduction to periodic and stochastic homogenization of ellip- tic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients in a unified approach based on Tartar's method of oscillating test functions. In partic- ular, we present a s...

We study the large-scale behavior of a small-strain lattice model for a network composed of elastoplastic springs with random material properties. We formulate the model as an evolutionary rate independent system (ERIS). In an earlier work we derived a homogenized continuum model, which has the form of linearized elastoplasticity, as evolutionary $...

We introduce a nonlinear, one-dimensional bending-twisting model for an inextensible bi-rod that is composed of a nematic liquid crystal elastomer. The model combines an elastic energy that is quadratic in curvature and torsion with a Frank-Oseen energy for the liquid crystal elastomer. Moreover, the model features a nematic-elastic coupling that r...

The presence of prestrain can have a tremendous effect on the mechanical behavior of slender structures. Prestrained elastic plates show spontaneous bending in equilibrium -- a property that makes such objects relevant for the fabrication of active and functional materials. In this paper we study microheterogeneous, prestrained plates that feature...

In this paper, we study an elastic bilayer plate composed of a nematic liquid crystal elastomer in the top layer and a nonlinearly elastic material in the bottom layer. While the bottom layer is assumed to be stress-free in the flat reference configuration, the top layer features an eigenstrain that depends on the local liquid crystal orientation....

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a...

In mechanical systems it is of interest to know the onset of fracture in dependence of the boundary conditions. Here we study a one-dimensional model which allows for an underlying heterogeneous structure in the discrete setting. Such models have recently been studied in the passage to the continuum by means of variational convergence ($\Gamma$-con...

We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $$\mathbb {R}^d$$ R d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $$\varep...

In this paper we study quantitative stochastic homogenization of a nonlinearly elastic composite material with a laminate microstructure. We prove that for deformations close to the set of rotations the homogenized stored energy function $W_{\rm hom}$ is $C^3$ and that $W_{\rm hom}$, the stress-tensor $DW_{\rm hom}$, and the tangent-moduli $D^2W_{\...

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikeli\'c and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as...

Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space app...

In this paper we investigate rods made of nonlinearly elastic, composite–materials that feature a micro-heterogeneous prestrain that oscillates (locally periodic) on a scale that is small compared to the length of the rod. As a main result we derive a homogenized bending–torsion theory for rods as \(\Gamma \)-limit from 3D nonlinear elasticity by s...

The limiting behaviour of a one‐dimensional discrete system is studied by means of Γ‐convergence. We consider a toy model of a chain of atoms. The interaction potentials are of Lennard‐Jones type and periodically or stochastically distributed. The energy of the system is considered in the discrete to continuum limit, i.e. as the number of atoms ten...

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension $d...

Nanochains of atoms, molecules and polymers have gained recent interest in the experimental sciences. This article contributes to an advanced mathematical modeling of the mechanical properties of nanochains that allow for heterogenities, which may be impurities or a deliberately chosen composition of different kind of atoms. We consider one-dimensi...

We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $\mathbb{R}^d$ with stationary law (i.e. spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $\varepsilon>0$, w...

We study the random conductance model on the lattice $\mathbb{Z}^d$, i.e.\ we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a qu...

Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space app...

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $\Lambda$-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equ...

In this paper we investigate rods made of nonlinearly elastic, composite--materials that feature a micro-heterogeneous prestrain that oscillates (locally periodic) on a scale that is small compared to the length of the rod. As a main result we derive a homogenized bending-torsion theory for rods as $\Gamma$-limit from 3D nonlinear elasticity by sim...

We consider periodic homogenization of nonlinearly elastic composite materials. Under suitable assumptions on the stored energy function (frame indifference; minimality, non-degeneracy and smoothness at identity; \(p\ge d\)-growth from below), and on the microgeometry of the composite (covering the case of smooth, periodically distributed inclusion...

We study quantitative periodic homogenization of integral functionals in the context of non-linear elasticity. Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; $p\geq d$-growth from below; and regularity of the microstructure), we show that in a neighbor...

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficien...

Extended abstract of an MFO-Report.

In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. We appeal to the notion of \mbox{$H$-convergence} introduced by Murat and Tartar. In our main result we establish an \mbox{$H$-compactness} result that applies to elliptic operators with measurable, uniformly elliptic coeffi...

The behavior of an electrochemical thin film under input voltage (potentiostatic) conditions is numerically investigated. Thin films are used in micro-batteries and proton-exchange-membrane fuel cells: these devices are expected to play a significant role in the next generation energy systems for use in vehicles as a replacement to combustion engin...

We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the "stiff" material, and a "soft" material that fills the remaining pores. We assume that the pores are of size 0 < ϵ ≪ 1 and are periodically distributed with period ϵ. We a...

Extended abstract of an MFO-Report.

We study the corrector equation in stochastic homogenization for a simplified
Bernoulli percolation model on $\mathbb{Z}^d$, $d>2$. The model is obtained
from the classical $\{0,1\}$-Bernoulli bond percolation by conditioning all
bonds parallel to the first coordinate direction to be open. As a main result
we prove (in fact for a slightly more gene...

Extended abstract of an MFO-Report

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...

We consider the corrector equation from the stochastic homogenization of
uniformly elliptic finite-difference equations with random, possibly
non-symmetric coefficients. Under the assumption that the coefficients are
stationary and ergodic in the quantitative form of a Logarithmic Sobolev
inequality (LSI), we obtain optimal bounds on the corrector...

We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewe...

We rigorously derive a homogenized von-Kármán 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...

We carry out the spatially periodic homogenization of nonlinear bending
theory for plates. The derivation is rigorous in the sense of
Gamma-convergence. In contrast to what one naturally would expect, our result
shows that the limiting functional is not simply a quadratic functional of the
second fundamental form of the deformed plate as it is the...

We rigorously derive a homogenized von-Kármán 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...

We present a multiscale asymptotic framework for the analysis of the
macroscopic behaviour of periodic two-material composites with high contrast in
a finite-strain setting. Our derivation starts with the geometrically nonlinear
description of a composite consisting of a stiff material matrix and soft,
periodically distributed inclusions, where the...

We present a rigorous derivation of a homogenized, bending-torsion theory for inextensible rods from three-dimensional nonlinear elasticity in the spirit of Γ -convergence. We start with the elastic energy functional associated with a nonlinear composite material, which in a stress-free reference configuration occupies a thin cylindrical domain wit...

In this note we prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a qua-dratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Müller and the second author by dropping their ass...

We study the effects of translation on two-scale convergence. Given a two-scale convergent sequence (uε(x))ε with two-scale limit u(x,y), then in general the translated sequence (u ε(x+t))ε is no longer two-scale convergent, even though it remains two-scale convergent along suitable subsequences. We prove that any two-scale cluster point of the tra...

We study non-convex elastic energy functionals associated to (spatially)
periodic, frame indifferent energy densities with a single non-degenerate
energy well at SO(n). Under the assumption that the energy density admits a
quadratic Taylor expansion at identity, we prove that the Gamma-limits
associated to homogenization and linearization commute....

Diese Arbeit beschäftigt sich mit der Entwicklung von effektiven Modellen für dünne, elastische Objekte, welche periodische Strukturen auf kleinen Längenskalen aufweisen. Als Hauptergebnis leiten wir eine homogenisierte Cosserat-Theorie für elastische Stäbe aus der nichtlinearen, dreidimensionalen Elastizitätstheorie her. Die Betrachtungen basieren...

## Projects

Projects (5)

The aim of the project is to understand (solution operators of) ordinary or partial differential equations if their coefficients converge in a topology weaker than the strong operator topology. Parts of the analysis are particularly devoted to both qualitative and quantitative statements in homogenisation problems. Aiming at a considerably general class of coefficients operators, we try to obtain statements also for nonlocal differential equations.