A bending-torsion theory for thin and ultrathin rods as a Γ-limit of atomistic models

2022/08/08 by Bernd Schmidt, Schmidt, Bernd, Jiří Zeman +1
Biochemistry, Genetics and Molecular Biology · Engineering · #49J45 #74K10 #Advanced Materials and Mechanics #Analysis of PDEs (math.AP) #Cellular Mechanics and Interactions #Elasticity and Material Modeling #FOS: Mathematics #FOS: Physical sciences #Materials Science (cond-mat.mtrl-sci) #Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

paper · pdf · doi:10.48550/arxiv.2208.04199

openalex publication_date 2022/08/08 · openalex created_date 2022/10/02 · openalex updated_date 2026/08/01

Abstract

The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as Γ-limits of 3D atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness h and interatomic distance ε. First, we set up a novel theory for ultrathin rods composed of finitely many atomic fibres (ε∼ h), which incorporates surface energy and new discrete terms in the limiting functional. This can be thought of as a contribution to the mechanical modelling of nanowires. Second, we treat the case where ε≪ h and recover a nonlinear rod model - the modern version of Kirchhoff's rod theory.

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