Eldorado Collection:http://hdl.handle.net/2003/382252021-09-25T17:05:31Z2021-09-25T17:05:31ZDimension reduction for elastoplastic rods and homogenization of elastoplastic latticesPoelstra, Klaas Hendrikhttp://hdl.handle.net/2003/404742021-08-18T22:12:00Z2021-01-01T00:00:00ZTitle: Dimension reduction for elastoplastic rods and homogenization of elastoplastic lattices
Authors: Poelstra, Klaas Hendrik
Abstract: We derive effective equations for periodic lattices of linearly elastoplastic rods in
the limit of both infinitesimal periodicity and infinitesimal relative width of the
rods. For this derivation we use the method of evolutionary Γ-convergence for
quadratic rate-independent systems.
As a first step towards this goal we derive effective equations for a single
rod. After introducing appropriate scalings, the main difficulty lies in the proof
of Γ-convergence for the stored energy. For the study of periodic lattices we
then introduce the notion of periodic graph frameworks, discuss infinitesimal
rigidity properties of such frameworks and define a notion of two-scale
convergence. The stored energy of a lattice of rods ist just the sum of the
energies of the individual rods, coupled by boundary conditions at the nodes.
For this energy we again prove Γ-convergence. In the presence of volume loads
we observe qualitatively different behaviour depending on the relative rate of
convergence of the periodicity parameter and the thickness parameter.2021-01-01T00:00:00ZHigher integrability for variational integrals with non-standard growthSchäffner, Mathiashttp://hdl.handle.net/2003/402562021-06-16T22:11:48Z2021-04-09T00:00:00ZTitle: Higher integrability for variational integrals with non-standard growth
Authors: Schäffner, Mathias
Abstract: We consider autonomous integral functionals of the form
F[u]:=∫Ωf(Du)dx where u:Ω→RN,N≥1,
where the convex integrand f satisfies controlled (p, q)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of F
assuming qp<1+2n−1, n≥3. This improves earlier results valid under the more restrictive assumption qp<1+2n.2021-04-09T00:00:00ZDerivation of a homogenized bending–torsion theory for rods with micro-heterogeneous prestrainBauer, RobertNeukamm, StefanSchäffner, Mathiashttp://hdl.handle.net/2003/400772021-03-11T23:10:19Z2020-05-29T00:00:00ZTitle: Derivation of a homogenized bending–torsion theory for rods with micro-heterogeneous prestrain
Authors: Bauer, Robert; Neukamm, Stefan; Schäffner, Mathias
Abstract: 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 Γ-limit from 3D nonlinear elasticity by simultaneous homogenization and dimension reduction under the assumption that the prestrain is of the order of the diameter of the rod. The limit model features a spontaneous curvature–torsion tensor that captures the macroscopic effect of the micro-heterogeneous prestrain. We devise a formula that allows to compute the spontaneous curvature–torsion tensor by means of a weighted average of the given prestrain. The weight in the average depends on the geometry of the composite and invokes correctors that are defined with help of boundary value problems for the system of linear elasticity. The definition of the correctors depends on a relative scaling parameter γ, which monitors the ratio between the diameter of the rod and the period of the composite’s microstructure. We observe an interesting size-effect: For the same prestrain a transition from flat minimizers to curved minimizers occurs by just changing the value of γ. Moreover, in the paper we analytically investigate the microstructure-properties relation in the case of isotropic, layered composites, and consider applications to nematic liquid–crystal–elastomer rods and shape programming.2020-05-29T00:00:00ZOn time-harmonic Maxwell’s equations in periodic mediaUrban, Maikhttp://hdl.handle.net/2003/392102020-07-22T01:40:51Z2019-01-01T00:00:00ZTitle: On time-harmonic Maxwell’s equations in periodic media
Authors: Urban, Maik
Abstract: In this thesis we study the propagation of time-harmonic
electromagnetic waves through periodic media for two different
regimes. In the first part, we consider a periodic medium in a bounded
domain with a period that is much smaller than the wavelength of the
electromagnetic wave hitting the medium. The medium is a periodic
assembly of conducting microstructures and void space. In order to
describe the effective behaviour of the field propagating through this
medium, we homogenise the time-harmonic Maxwell equations. There is a
vast literature on homogenising Maxwell's equations under rather
restrictive assumptions on the (conducting) microstructure. Using a
new averaging method---the so-called geometric average---allows us to
consider a large class of microstructures that have not been treated
before in the literature. We derive the effective Maxwell equations
for two cases: perfectly conducting microstructures and highly
conductive microstructures.
The second part of this thesis is concerned with the propagation of
electromagnetic waves in a closed, unbounded and periodic
waveguide. In this part we assume that the wavelength of the fields
and the period of the medium are of the same order. Imposing suitable
assumptions on the geometry and the fields, Maxwell's equations reduce
to a scalar Helmholtz equation. We truncate the waveguide to obtain a
bounded domain and replace the radiation condition at infinity
appropriately. In order to establish the existence of a solution to
the Helmholtz equation in this bounded periodic waveguide, we derive a
limiting absorption principle for sesquilinear forms.
Using this principle, we show that up to an at most countable set of
singular frequencies there exists a unique solution to the Helmholtz
equation in the bounded waveguide that satisfies the replacement of
the radiation condition at infinity.2019-01-01T00:00:00Z