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Convergence of equilibria for bending-torsion models of rods with inhomogeneities

Part of: Homogenization, determination of effective properties Elastic materials Thin bodies, structures Material properties given special treatment Equilibrium (steady-state) problems

Published online by Cambridge University Press:  24 January 2019

Matthäus Pawelczyk
Matthäus Pawelczyk*
Affiliation:
FB Mathematik, TU Dresden, 0 1062 Dresden, Germany (matthaeuspawelczyk@gmail.com)
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Abstract

We prove that, in the limit of vanishing thickness, equilibrium configurations of inhomogeneous, three-dimensional non-linearly elastic rods converge to equilibrium configurations of the variational limit theory. More precisely, we show that, as $h\searrow 0$, stationary points of the energy , for a rod $\Omega _h\subset {\open R}^3$ with cross-sectional diameter h, subconverge to stationary points of the Γ-limit of , provided that the bending energy of the sequence scales appropriately. This generalizes earlier results for homogeneous materials to the case of materials with (not necessarily periodic) inhomogeneities.

Keywords

elasticitydimension reductionhomogenizationconvergence of equilibria

MSC classification

Primary: 74K10: Rods (beams, columns, shafts, arches, rings, etc.)
Secondary: 74B20: Nonlinear elasticity 74G10: Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) 74E30: Composite and mixture properties 74Q05: Homogenization in equilibrium problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 233 - 260
Copyright
Copyright © Royal Society of Edinburgh 2019

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