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On tensor projections, stress or stretch vectors and their relations to Mohr's three circles

Klaus Heiduschke

Vol. 12 (2024), No. 2, 173–216
Abstract

The objective of this classroom note is to recall the relation between Mohr’s three circles and Cauchy’s stress definition of a (symmetric) tensor by its projection onto a unit vector n. In the case of Cauchy stress, the resulting projection vector is called (surface) traction and acts on the section plane with outward normal n. The normal component σ of the projection vector points in direction n normal to the section plane, and its shear component τ lies within the section plane. In Mohr’s normal σ and shear τ component plane, the 2.5-dimensional coordinate rotations about principal axes reflect Mohr’s three circles, and the tensor projections onto arbitrary unit vectors n cover the interspace between the three. The tensor projections and Mohr’s three circles can mathematically be applied to all symmetric second-order tensors. But only a few tensors (like the Cauchy stress or the stretch tensors) have a physical interpretation with respect to tensor projections. The most relevant of these stress or deformation tensors are discussed and presented with their associated stress or deformation (reciprocal) ellipsoids. In this context, the Cauchy–Green deformation tensors (i.e., the stretch tensors to the second power) have no physical interpretation in terms of tensor projections or additive trace-deviator separations.

Keywords
(interspace of) Mohr's three circles, (reciprocal) eigenvalue space, (reciprocal) stress/deformation ellipsoids, (reverse) tensor projections
Mathematical Subject Classification
Primary: 74A05, 74A10, 74B20
Milestones
Received: 29 October 2023
Accepted: 15 April 2024
Published: 7 May 2024

Communicated by Emilio Barchiesi
Authors
Klaus Heiduschke
Alumnus of Institut für Mechanik, ETH Zürich
Schaffhausen
Switzerland