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Bernd Schmidt

Publications and source records attributed to Bernd Schmidt.

At least 19 recordsLinked to original sources
arXiv2025

Dimension reduction for time-dependent von K\'arm\'an rods

This paper aims to study the convergence of solutions in three-dimensional nonlinear elastodynamics for a thin rod as its cross section shrinks to zero for displacements that are comparable to the small radius of the rod. Assuming the existence of solutions and proper control of the torsional velocity, we show how these converge to the solutions of an effective dimensionally reduced model which is a version of the the time dependent von K\'arm\'an equations for a one-dimensional rod. In the presence of high-frequency torsional vibrations, energy can dissipate in the limit and we obtain additional contributions in the limiting equations.

math.AP↗
arXiv2025

Numerical simulation of a fine-tunable F\"oppl-von K\'arm\'an model for foldable and bilayer plates

A numerical scheme is proposed to identify low energy configurations of a F\"oppl-von K\'arm\'an model for bilayer plates. The dependency of the corresponding elastic energy on the in-plane displacement $u$ and the out-of-plane deflection $w$ leads to a practical minimization of the functional via a decoupled gradient flow. In particular, the energies of the resulting iterates are shown to be monotonically decreasing. The discretization of the model relies on $P1$ finite elements for the horizontal part $u$ and utilizes the discrete Kirchhoff triangle for the vertical component $w$. The model allows for analysing various different problem settings via numerical simulation: (i) stable low-energy configurations are detected dependent on a specified prestrain described by elastic material properties, (ii) curvature inversions of spherical and cylindrical configurations are investigated, (iii) elastic responses of foldable cardboards for different spontaneous curvatures and crease geometries are compared.

math.NA↗
arXiv2024

A derivation of the time dependent von K\'arm\'an equations from atomistic models

We derive the time-dependent von K\'arm\'an plate equations from three dimensional, purely atomistic particle models. In particular, we prove that a thin structure of interacting particles whose dynamics is governed by Newton's laws of motion is effectively described by the von K\'arm\'an equations in the limit of vanishing interatomic distance $\eps$ and vanishing plate thickness $h$. While the classical plate equations are obtained for $\eps \ll h \ll 1$, we find new plate equations for finitely many layers in the ultrathin case $\eps \sim h$.

math.AP↗
arXiv2024

A note on the Winterbottom shape

In this short note we review results on equilibrium shapes of minimizers to the sessile drop problem. More precisely, we study the Winterbottom problem and prove that the Winterbottom shape is indeed optimal. The arguments presented here are based on relaxation and the (anisotropic) isoperimetric inequality.

math.AP↗
arXiv2024

NLS approximation for a scalar FPUT system on a 2D square lattice with a cubic nonlinearity

We consider a scalar Fermi-Pasta-Ulam-Tsingou (FPUT) system on a square 2D lattice with a cubic nonlinearity. For such systems the NLS equation can be derived to describe the evolution of an oscillating moving wave packet of small amplitude which is slowly modulated in time and space. We show that this NLS approximation makes correct predictions about the dynamics of the original scalar FPUT system for the strain and the displacement variables.

math.DS↗
arXiv2023

Cooling power analysis of a small scale 4 K pulse tube cryocooler driven by an oil-free low input power Helium compressor

Here we report the performance of a small scale 4 K pulse tube cryocooler operating with a low input power reaching a minimum temperature of 2.2 K, as well as a cooling capacity of over 240 mW at 4.2 K. The compressor is air cooled and can be supplied by single phase power sockets. With an input power of about 1.3 kW the coefficient of performance reaches values of up to 185 mW/kW, which is among the highest currently reported values for small to medium power pulse tubes. The combination of an oil-free Helium compressor and low maintenance pulse tube cryocooler provides a unique miniaturized, energy efficient and mobile cooling tool for applications at 4 K and below.

cond-mat.mtrl-sci↗
arXiv2023

Most likely balls in Banach spaces: existence and non-existence

We establish a general criterion for the existence of convex sets of fixed shape as, e.g., balls of a given radius, of maximal probability on Banach spaces. We also provide counterexamples showing that their existence my fail even in some common situations.

math.FA↗
arXiv2022

A continuum model for brittle nanowires derived from an atomistic description by $Γ$-convergence

Starting from a particle system with short-range interactions, we derive a continuum model for the bending, torsion, and brittle fracture of inextensible rods moving in three-dimensional space. As the number of particles tends to infinity, it is assumed that the rod's thickness is of the same order as the interatomic distance. Fracture energy in the $Γ$-limit is expressed by an implicit cell formula, which covers different modes of fracture, including (complete) cracks, folds and torsional cracks. In special cases, the cell formula can be significantly simplified. Our approach applies e.g. to atomistic systems with Lennard-Jones-type potentials and is motivated by the research of ceramic nanowires.

math.AP↗
arXiv2022

A bending-torsion theory for thin and ultrathin rods as a $Γ$-limit of atomistic models

The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as $Γ$-limits of 3D atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness $h$ and interatomic distance $\varepsilon$. First, we set up a novel theory for ultrathin rods composed of finitely many atomic fibres ($\varepsilon\sim h$), which incorporates surface energy and new discrete terms in the limiting functional. This can be thought of as a contribution to the mechanical modelling of nanowires. Second, we treat the case where $\varepsilon\ll h$ and recover a nonlinear rod model $-$ the modern version of Kirchhoff's rod theory.

math.AP↗
arXiv2022

A Blake-Zisserman-Kirchhoff theory for plates with soft inclusions

We consider a two phase elastic thin film with soft inclusions subject to bending dominated deformations. The soft (void) phase may comprise asymptotically small droplets within the elastic matrix. We perform a dimension reduction analysis and obtain a novel `Blake-Zisserman-Kirchhoff' model that combines Kirchhoff's classical plate theory with Blake-Zisserman type surface energy contributions at cracks, folds and the boundary of voids.

math.AP↗
arXiv2022

Korn type Inequalities for Objective Structures

We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry group is a general space group we obtain a full discrete Korn inequality. For systems with non-trivial codimension our results provide an intrinsic rigidity estimate within the extended dimensions of the structure. As their continuum counterparts in elasticity theory, such estimates are at the core of energy estimates and, hence, a stability analysis for a wide class of atomistic particle systems.

math.AP↗
arXiv2022

Stability of Objective Structures: General Criteria and Applications

We develop a general stability analysis for objective structures, which constitute a far reaching generalization of crystal lattice systems. We show that these particle systems, although in general neither periodic nor space filling, allow for the identification of stability constants in terms of representations of the underlying symmetry group and interaction potentials. Our main results provide general stability criteria and second order energy bounds for equilibrium configurations. In particular, a general computational algorithm to test objective structures for their stability is derived. By way of example we show that our method can be applied to verify the stability of carbon nanotubes with chirality.

math.AP↗
arXiv2022

Eigendamage: an Eigendeformation model for the variational approximation of cohesive fracture -- a one-dimensional case study

We study an approximation scheme for a variational theory of cohesive fracture in a one-dimensional setting. Here, the energy functional is approximated by a family of functionals depending on a small parameter $0 < \varepsilon \ll 1$ and on two fields: the elastic part of the displacement field and an eigendeformation field that describes the inelastic response of the material beyond the elastic regime. We measure the inelastic contributions of the latter in terms of a non-local energy functional. Our main result shows that, as $\varepsilon \to 0$, the approximate functionals $Γ$-converge to a cohesive zone model.

math.AP↗
arXiv2022

Two phase models for elastic membranes with soft inclusions

We derive an effective membrane theory in the thin film limit within a two phase material model for a specimen consisting of an elastic matrix and soft inclusions. The soft inclusions may lead to the formation of cracks within the elastic matrix and the corresponding limiting models are described by Griffith type fracture energy functionals. We also provide simplified proofs of relaxation results for bulk materials.

math.AP↗
arXiv2021

Unique Minimizers and the Representation of Convex Envelopes in Locally Convex Vector Spaces

It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let $X$ be a locally convex Hausdorff space and $f \colon X \mapsto \left( - \infty , \infty \right]$ a function with compact sublevel sets and exhibiting some mildly superlinear growth. Then each tilted minimization problem \begin{equation} \label{eq. minimization problem} \min_{x \in X} f(x) - \langle x' , x \rangle_X \end{equation} admits at most one minimizer as $x'$ ranges over $\text{dom} \left( \partial f^* \right)$ if and only if the biconjugate $f^{**}$ is essentially strictly convex and agrees with $f$ at all points where $f^{**}$ is subdifferentiable. We prove this via a representation formula for $f^{**}$ that might be of independent interest.

math.OC↗
arXiv2020

Distribution of cracks in a chain of atoms at low temperature

We consider a one-dimensional classical many-body system with interaction potential of Lennard-Jones type in the thermodynamic limit at low temperature $1/β\in(0,\infty)$. The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with $N$ particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of $N\exp(- βe_\mathrm{surf}/2)$ with $e_\mathrm{surf}>0$ a surface energy. For the proof, the system is mapped to an effective model, which is a low-density lattice gas of defects. The results require conditions on the interactions between defects. We succeed in verifying these conditions for next-nearest neighbor interactions, applying recently derived uniform estimates of correlations.

math-ph↗
arXiv2020

Emergence of rigid Polycrystals from atomistic Systems with Heitmann-Radin sticky disk energy

We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the `sticky disk' interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of $Γ$-convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, i.e., on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. We further provide a fine analysis of the surface energies at grain boundaries both for vacuum-solid and solid-solid phase transitions. The latter relies fundamentally on a structure result for grain boundaries showing that due to the extremely brittle setup interpolating boundary layers near cracks are energetically not favorable.

cond-mat.stat-mech↗