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Peter Hornung

Publications and source records attributed to Peter Hornung.

17 recordsLinked to original sources
arXiv2025

Babuška's paradox in a nonlinear bending-folding model

The Babuška or plate paradox concerns the failure of convergence when a domain with curved boundary is approximated by polygonal domains in linear bending problems with simply supported boundary conditions. It can be explained via a boundary integral representation of the total Gaussian curvature that is part of the Kirchhoff--Love bending energy. It is shown that the paradox also occurs for a nonlinear bending-folding model which enforces vanishing Gaussian curvature. A simple remedy that is compatible with simplicial finite element methods to avoid incorrect convergence is devised.

math.NA↗
arXiv2022

Modeling and simulation of thin sheet folding

The article addresses the mathematical modeling of the folding of a thin elastic sheet along a prescribed curved arc. A rigorous model reduction from a general hyperelastic material description is carried out under appropriate scaling conditions on the energy and the geometric properties of the folding arc in dependence on the small sheet thickness. The resulting two-dimensional model is a piecewise nonlinear Kirchhoff plate bending model with a continuity condition at the folding arc. A discontinuous Galerkin method and an iterative scheme are devised for the accurate numerical approximation of large deformations.

math.NA↗
arXiv2021

Stability of boundary conditions for the Sadowsky functional

It has been proved by the authors that the (extended) Sadowsky functional can be deduced as the Gamma-limit of the Kirchhoff energy on a rectangular strip of height $\varepsilon$, as $\varepsilon$ tends to 0. In this paper we show that this Gamma-convergence result is stable when affine boundary conditions are prescribed on the short sides of the strip. These boundary conditions include those corresponding to a Möbius band.

math.AP↗
arXiv2021

Deformation of Framed Curves

We consider curves $γ: [0, 1]\to\mathbb{R}^3$ endowed with an adapted orthonormal frame $r : [0, 1]\to SO(3)$. We are interested in the cases where the frame is constrained, in the sense that one of its `curvatures' (i.e., off-diagonal elements of $r'r^T$) is prescribed. One example is the Frénet frame. In order to deform such constrained framed curves without spoiling the constraint, we proceed in two stages. First we deform the frame $r$ in a way that is naturally compatible with the differential constraint, by interpreting it in terms of parallel transport on the sphere. The deformation of the base curve $γ$ is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications.

cond-mat.mtrl-sci↗
arXiv2020

On Material Optimisation for Nonlinearly Elastic Plates and Shells

This paper investigates the optimal distribution of hard and soft material on elastic plates. In the class of isometric deformations stationary points of a Kirchhoff plate functional with incorporated material hardness function are investigated and a compliance cost functional is taken into account. Under symmetry assumptions on the material distribution and the load it is shown that cylindrical solutions are stationary points. Furthermore, it is demonstrated that the optimal design of cylindrically deforming, clamped rectangular plates is non trivial, i.e. with a material distribution which is not just depending on one axial direction on the plate. Analytical results are complemented with numerical optimization results using a suitable finite element discretization and a phase field description of the material phases. Finally, using numerical methods an outlook on the optimal design of non isometrically deforming plates and shells is given.

math.NA↗
arXiv2017

$C^{1,α}$ $h$-principle for von Kármán constraints

Exploiting some connections between the system $\nabla v\otimes\nabla v + 2$ sym $\nabla w = A$ and the isometric immersion problem in two dimensions, we provide a simple construction of $C^{1,α}$ convex integration solutions for the former from the corresponding result for the latter.

math.AP↗
arXiv2017

One-dimensional von Kármán models for elastic ribbons

By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von Kármán plates, passing to the limit as the width of the plate goes to zero. The one-dimensional model found starting from the "linearized" von Kármán energy corresponds to that of a linearly elastic beam that can twist but can deform in just one plane; while the model found from the von Kármán energy is a non-linear model that comprises stretching, bendings, and twisting. The "constrained" von Kármán energy, instead, leads to a new Sadowsky type of model.

math.AP↗
arXiv2016

A variational model for anisotropic and naturally twisted ribbons

We consider thin plates whose energy density is a quadratic function of the difference between the second fundamental form of the deformed configuration and a "natural" curvature tensor. This tensor either denotes the second fundamental form of the stress-free configuration, if it exists, or a target curvature tensor. In the latter case, residual stress arises from the geometrical frustration involved in the attempt to achieve the target curvature: as a result, the plate is naturally twisted, even in the absence of external forces or prescribed boundary conditions. Here, starting from this kind of plate energies, we derive a new variational one-dimensional model for naturally twisted ribbons by means of Gamma-convergence. Our result generalizes, and corrects, the classical Sadowsky energy to geometrically frustrated anisotropic ribbons with a narrow, possibly curved, reference configuration.

math.AP↗
arXiv2016

Material Optimization for Nonlinearly Elastic Planar Beams

We consider the problem of an optimal distribution of soft and hard material for nonlinearly elastic planar beams. We prove that under gravitational force the optimal distribution involves no microstructure and is ordered, and we provide numerical simulations confirming and extending this observation.

math.NA↗
arXiv2014

Nonlinear bending theories for non Euclidean plates

Thin growing tissues (such as plant leaves) can be modelled by a bounded domain $S\subset R^2$ endowed with a Riemannian metric $g$, which models the internal strains caused by the differential growth of the tissue. The elastic energy is given by a nonlinear isometry-constrained bending energy functional which is a natural generalization of Kirchhoff's plate functional. We introduce and discuss a natural notion of (possibly non-minimising) stationarity points. We show that rotationally symmetric immersions of the unit disk are stationary, and we give examples of metrics $g$ leading to functionals with infinitely many stationary points.

cond-mat.soft↗
arXiv2014

Another remark on constrained von Karman theories

We study the natural non-flat version of the so-called "constrained von Karman" theory for thin nonlinearly elastic films. We prove that every (admissible) radially symmetric out-of-plane displacement on the unit disk is a stationary point. This allows us to construct data leading to constrained von Karman functionals which have infinitely many stationary points.

math.AP↗
arXiv2013

Derivation of a homogenized von-Karman shell theory from 3D elasticity

We derive the model of homogenized von Kármán 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. In the case $h\ll\e$ we identify two different asymptotic theories, depending on the ratio of $h$ and $\e^2$. In the case of convex shells we obtain a complete picture in the whole regime $h\ll\e$.

math.AP↗
arXiv2012

Derivation of a homogenized nonlinear plate theory from 3d elasticity

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.

math.AP↗
arXiv2012

Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells

We perform a detailed analysis of first order Sobolev-regular infinitesimal isometries on developable surfaces without affine regions. We prove that given enough regularity of the surface, any first order infinitesimal isometry can be matched to an infinitesimal isometry of an arbitrarily high order. We discuss the implications of this result for the elasticity of thin developable shells.

math.AP↗