Mathematical Physics


Cauchy Tetrahedron Argument and the Proofs for the Existence of Stress Tensor, a Comprehensive Review, Challenges, and Improvements

Authors: Ehsan Azadi

Cauchy in 1822 presented the idea of traction vector that contains both the normal and tangential components of the internal surface forces per unit area and gave the tetrahedron argument to prove the existence of stress tensor. These great achievements form the main part of the foundation of continuum mechanics. During nearly two centuries, some versions of tetrahedron argument and a few other proofs for the existence of stress tensor are presented in every text in continuum mechanics, fluid mechanics, and the related subjects. In this article, we show the birth, importance, and location of these Cauchy's achievements, then by presenting the formal tetrahedron argument in detail, for the first time we extract some fundamental challenges. These conceptual challenges are related to the result of applying the conservation of linear momentum to any mass element and the order of its surface and volume terms, the definition of traction vectors on the surfaces that pass through the same point, the limiting and approximating processes in the derivation of stress tensor, and some others. In a comprehensive review, we present the different tetrahedron arguments and the proofs for the existence of stress tensor, consider the challenges in each one, and classify them in two general approaches. In the first approach that is followed in most texts, the traction vectors do not define exactly on the surfaces that pass through the same point so, most of the challenges hold. But in the second approach, the traction vectors are defined on the surfaces that pass exactly through the same point, so some of the related challenges are removed. We also represent the improved works of Hamel and Backus, and show that the original work of Backus removes most of the challenges. This article shows that the foundation of continuum mechanics is not a finished subject and there are still some fundamental challenges.

Comments: 34 pages

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Submission history

[v1] 2017-07-06 13:56:20

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