Intrinsic Nonlinear Elasticity: An Exterior Calculus Formulation

Ramy Rashad*, Andrea Brugnoli, Federico Califano, Erwin Luesink, Stefano Stramigioli

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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Abstract

In this paper, we formulate the theory of nonlinear elasticity in a geometrically intrinsic manner using exterior calculus and bundle-valued differential forms. We represent kinematics variables, such as velocity and rate of strain, as intensive vector-valued forms, while kinetics variables, such as stress and momentum, as extensive covector-valued pseudo-forms. We treat the spatial, material and convective representations of the motion and show how to geometrically convert from one representation to the other. Furthermore, we show the equivalence of our exterior calculus formulation to standard formulations in the literature based on tensor calculus. In addition, we highlight two types of structures underlying the theory: first, the principal bundle structure relating the space of embeddings to the space of Riemannian metrics on the body and how the latter represents an intrinsic space of deformations and second, the de Rham complex structure relating the spaces of bundle-valued forms to each other.

Original languageEnglish
Article number84
JournalJournal of nonlinear science
Volume33
Issue number5
DOIs
Publication statusPublished - Oct 2023

Keywords

  • Bundle-valued forms
  • Exterior calculus
  • Geometric mechanics
  • Nonlinear elasticity
  • UT-Hybrid-D

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