A note on $3$d-$1$d dimension reduction with differential constraints

Carolin Kreisbeck

doi:10.3934/dcdss.2017003

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2017, Volume 10, Issue 1: 55-73. Doi: 10.3934/dcdss.2017003

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A note on $3$d-$1$d dimension reduction with differential constraints

Carolin Kreisbeck

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany¹

Received: March 31, 2015

Revised: May 31, 2015

Published: February 2018

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Abstract

Starting from three-dimensional variational models with energies subject to a general type of PDE constraint, we use Γ-convergence methods to derive reduced limit models for thin strings by letting the diameter of the cross section tend to zero. A combination of dimension reduction with homogenization techniques allows for addressing the case of thin strings with fine heterogeneities in the form of periodically oscillating structures. Finally, applications of the results in the classical gradient case, corresponding to nonlinear elasticity with Cosserat vectors, as well as in micromagnetics are discussed.

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Mathematics Subject Classification: Primary: 49J45; Secondary: 35E99, 74K15, 74Q05.

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Figure 1. Transformation of $\Omega_\varepsilon $ into $\Omega_1$ and rescaling of the differential constraints.

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Figure 2. Examples of heterogeneous strings. a) Fibered heterogeneities ($f$ constant in $y_d$); b) Fine material layers ($f$ constant in $y'$).

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