arXiv subjects

Carolin Kreisbeck

Publications and source records attributed to Carolin Kreisbeck.

24 records · Page 2Linked to original sources

arXiv2018

Characterizations of symmetric polyconvexity

Symmetric quasiconvexity plays a key role for energy minimization in geometrically linear elasticity theory. Due to the complexity of this notion, a common approach is to retreat to necessary and sufficient conditions that are easier to handle. This article focuses on symmetric polyconvexity, which is a sufficient condition. We prove a new characterization of symmetric polyconvex functions in the two- and three-dimensional setting, and use it to investigate relevant subclasses like symmetric polyaffine functions and symmetric polyconvex quadratic forms. In particular, we provide an example of a symmetric rank-one convex quadratic form in 3d that is not symmetric polyconvex. The construction takes the famous work by Serre from 1983 on the classical situation without symmetry as inspiration. Beyond their theoretical interest, these findings may turn out useful for computational relaxation and homogenization.

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arXiv2016

Homogenization of layered materials with rigid components in single-slip finite crystal plasticity

We determine the effective behavior of a class of composites in finite-strain crystal plasticity, based on a variational model for materials made of fine parallel layers of two types. While one component is completely rigid in the sense that it admits only local rotations, the other one is softer featuring a single active slip system with linear self-hardening. As a main result, we obtain explicit homogenization formulas by means of Γ-convergence. Due to the anisotropic nature of the problem, the findings depend critically on the orientation of the slip direction relative to the layers, leading to three qualitatively different regimes that involve macroscopic shearing and blocking effects. The technical difficulties in the proofs are rooted in the intrinsic rigidity of the model, which translates into a non-standard variational problem constraint by non-convex partial differential inclusions. The proof of the lower bound requires a careful analysis of the admissible microstructures and a new asymptotic rigidity result, whereas the construction of recovery sequences relies on nested laminates.

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arXiv2015

Heterogeneous thin films: Combining homogenization and dimension reduction with directors

We analyze the asymptotic behavior of a multiscale problem given by a sequence of integral functionals subject to differential constraints conveyed by a constant-rank operator with two characteristic length scales, namely the film thickness and the period of oscillating microstructures, by means of $Γ$-convergence. On a technical level, this requires a subtile merging of homogenization tools, such as multiscale convergence methods, with dimension reduction techniques for functionals subject to differential constraints. One observes that the results depend critically on the relative magnitude between the two scales. Interestingly, this even regards the fundamental question of locality of the limit model, and, in particular, leads to new findings also in the gradient case.

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arXiv2014

Thin-film limits of functionals on A-free vector fields

This paper deals with variational principles on thin films subject to linear PDE constraints represented by a constant-rank operator $\mathcal{A}$. We study the effective behavior of integral functionals as the thickness of the domain tends to zero, investigating both upper and lower bounds for the $Γ$-limit. Under certain conditions we show that the limit is an integral functional and give an explicit formula. The limit functional turns out to be constrained to $\mathcal{A}_0$-free vector fields, where the limit operator $\mathcal{A}_0$ is in general not of constant rank. This result extends work by Bouchitte, Fonseca and Mascarenhas [J. Convex Anal. 16 (2009), pp. 351--365] to the setting of $\mathcal{A}$-free vector fields. While the lower bound follows from a Young measure approach together with a new decomposition lemma, the construction of a recovery sequence relies on algebraic considerations in Fourier space. This part of the argument requires a careful analysis of the limiting behavior of the rescaled operators $\mathcal{A}_ε$ by a suitable convergence of their symbols, as well as an explicit construction for plane waves inspired by the bending moment formulas in the theory of (linear) elasticity. We also give a few applications to common operators $\mathcal{A}$.

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arXiv2013

Asymptotic spectral analysis in semiconductor nanowire heterostructures

Mathematical settings in which heterogeneous structures affect electron transport through a tube-shaped quantum waveguide are studied, highlighting the interaction between heterogeneities and geometric parameters like curvature and torsion. First, the macroscopic behavior of a nanowire made of composite fibers with microscopic periodic texture is analyzed, which amounts to determining the asymptotic behavior of the spectrum of an elliptic Dirichlet eigenvalue problem with finely oscillating coefficients in a tube with shrinking cross section. A suitable formal expansion suggests that the effective one-dimensional limit problem is of Sturm-Liouville type and yields the explicit formula for the underlying potential. In the torsion-free case, these findings are made rigorous by performing homogenization and 3d-1d dimension reduction for the two-scale problem in a variational framework by means of Gamma-convergence. Second, waveguides with non-oscillating inhomogeneities in the cross section are investigated. This leads to explicit criteria for propagation and localization of eigenmodes.

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arXiv2012

Another approach to the thin-film Gamma-limit of the micromagnetic free energy in the regime of small samples

The asymptotic behavior of the micromagnetic free energy governing a ferromagnetic film is studied as its thickness gets smaller and smaller compared to its cross section. Here the static Maxwell equations are treated as a Murat's constant-rank PDE constraint on the energy functional. In contrast to previous work this approach allows to keep track of the induced magnetic field without solving the magnetostatic equations. In particular, the mathematical results of Gioia and James [Proc. R. Soc. Lond. A 453 (1997), pp. 213-223] regarding convergence of minimizers are recovered by giving a characterization of the corresponding Gamma-limit.

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