Derivation of effective theories for thin 3D nonlinearly elastic rods with voids

Friedrich M, Kreutz L, Zemas K (2024)

Publication Type: Journal article

Publication year: 2024

Journal

Mathematical Models & Methods in Applied Sciences World Scientific Publishing

Book Volume: 34

Pages Range: 723-777

Journal Issue: 4

DOI: 10.1142/S0218202524500131

Abstract

We derive a dimension-reduction limit for a three-dimensional rod with material voids by means of Γ-convergence. Hereby, we generalize the results of the purely elastic setting [M. G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-convergence, Calc. Var. Partial Differential Equations 18 (2003) 287-305] to a framework of free discontinuity problems. The effective one-dimensional model features a classical elastic bending-torsion energy, but also accounts for the possibility that the limiting rod can be broken apart into several pieces or folded. The latter phenomenon can occur because of the persistence of voids in the limit, or due to their collapsing into a discontinuity of the limiting deformation or its derivative. The main ingredient in the proof is a novel rigidity estimate in varying domains under vanishing curvature regularization, obtained in [M. Friedrich, L. Kreutz and K. Zemas, Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces, preprint (2021), arXiv:2107.10808].

Authors with CRIS profile

Manuel Friedrich Professur für Angewandte Mathematik (Angewandte Analysis)

Involved external institutions

Rheinische Friedrich-Wilhelms-Universität Bonn DE Germany (DE) Technische Universität München (TUM) DE Germany (DE)

How to cite

APA:

Friedrich, M., Kreutz, L., & Zemas, K. (2024). Derivation of effective theories for thin 3D nonlinearly elastic rods with voids. Mathematical Models & Methods in Applied Sciences, 34(4), 723-777. https://doi.org/10.1142/S0218202524500131

MLA:

Friedrich, Manuel, Leonard Kreutz, and Konstantinos Zemas. "Derivation of effective theories for thin 3D nonlinearly elastic rods with voids." Mathematical Models & Methods in Applied Sciences 34.4 (2024): 723-777.

BibTeX: Download