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Lagrangian and Eulerian formulations of second-grade elasticity via convected coordinates

Roberto Fedele and David J. Steigmann
Vol. 13 (2025), No. 3, 377–389
DOI: 10.2140/memocs.2025.13.377
Abstract

The transformation between the Lagrangian and Eulerian descriptions of the equilibrium equations for second-grade elastic materials is reconsidered in the setting of a convected-coordinate formulation of the relevant kinematics. The third-order contortion tensor, equivalent to the strain gradient and representing the change of the Levi-Civita connection induced by deformation, is adopted as the basic descriptor of the refined kinematics associated with the second-gradient theory.

Keywords
second-grade elasticity, Lagrangian formulation, Eulerian formulation, convected coordinates, contortion tensor
Mathematical Subject Classification
Primary: 74B20
Secondary: 74G99
Milestones
Received: 6 June 2025
Revised: 28 July 2025
Accepted: 13 August 2025
Published: 6 September 2025

Communicated by Francesco dell'Isola
Authors
Roberto Fedele
Department of Civil and Environmental Engineering (DICA)
Politecnico di Milano
20133 Milan
Italy
David J. Steigmann
Department of Mechanical Engineering
University of California
Berkeley, CA 94720
United States
© 2025 MSP (Mathematical Sciences Publishers).