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A three-dimensional Airy stress function

Salvatore Federico, Mawafag F. Alhasadi, Shoji Imatani and Marcelo Epstein

Vol. 11 (2023), No. 3, 429–449
Abstract

The most general stress function in three dimensions is given by the symmetric Beltrami tensor, which consists of six scalar functions. In this brief communication, we explore the possibility of expressing a three-dimensional state of stress by means of a single scalar function, in the spirit of the Airy stress function in two dimensions. The price to pay for such a severe restriction is a loss of generality, in the sense that not every divergence-free stress field can be derived from the single stress function. Clearly, any such single scalar function gives rise to a particular case of the Beltrami tensor. One of these particular cases exhibits an interesting analogy with the tensor of inertia of a rigid body.

Keywords
Airy stress function, Beltrami stress function, Maxwell stress function, Morera stress function, tensor of inertia, three-dimensional
Mathematical Subject Classification
Primary: 74B05, 74G05
Milestones
Received: 4 March 2023
Revised: 16 June 2023
Accepted: 21 July 2023
Published: 15 November 2023

Communicated by Francesco dell'Isola
Authors
Salvatore Federico
Deptment of Mechanical and Manufacturing Engineering
University of Calgary
Calgary, AB
Canada
Department of Energy Conversion Science
Kyoto University
Sakyō-ku, Kyoto
Japan
Mawafag F. Alhasadi
Deptment of Mechanical and Manufacturing Engineering
University of Calgary
Calgary, AB
Canada
Shoji Imatani
Department of Energy Conversion Science
Kyoto University
Sakyō-ku, Kyoto
Japan
Marcelo Epstein
Deptment of Mechanical and Manufacturing Engineering
University of Calgary
Calgary, AB
Canada