# What Happened on September 30, 1822, and What Were its Implications for the Future of Continuum Mechanics?

## Abstract

This contribution offers a discussion about the notion of stress in a general continuum as initially proposed in a magisterial paper by Cauchy in 1822 (but published only in 1828) without using arguments involving molecules. This is here presented in its historical context. Cauchy’s view is the currently accepted view among mechanicians and engineers although attempts (including by Navier and Cauchy himself) to start from a molecular description in the manner of Newton and Laplace were constantly offered in both nineteenth and twentieth centuries. The discussion introduces other secondary stress definitions such as those by Piola, Kirchhoff, and more recently Eshelby. The question naturally arises of what happens with the possibility to introduce other internal forces such as hyperstresses (in so-called gradient theories) and couple stresses (e.g., in Cosserat continua), and whether some introduced stresses have associated with them a meaningful boundary condition. Also pondered is the question whether one can identify a stress concept in physical approaches still considering interactions between point particles (lattice dynamics, kinetic theory, nonlocal theory, statistical-mechanics approach). The chapter is concluded by a more in depth discussion of the notion of stress-energy-momentum, culminating in that of pseudo-tensor of energy-momentum in gravitation theory.

## Keywords

Cauchy stress Piola-Kirchhoff stress Eshelby stress Energy-momentum tensor Natural boundary conditions## Notes

### Acknowledgments

Heartful thanks go to Dr Martine Rousseau in Paris and Professor James Casey in Berkeley for their critical careful reading of this contribution that led to much improvement and readability. Mme Florence Greffe (“Conservateur du Patrimoine”) from the Archives Library of the Paris Academy of Science is to be thanked for her definite help in providing the “birth certificate” of continuum mechanics.

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