At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola

TL;DR

This paper highlights Gabrio Piola's overlooked pioneering contributions to non-local and higher gradient continuum mechanics, emphasizing their foundational role and the reasons for their historical neglect.

Contribution

It demonstrates that Piola's early work already contained concepts of non-local and higher gradient mechanics, which are still relevant and often ignored in modern literature.

Findings

Piola's differential relationships are foundational for modern continuum mechanics.

Piola's contributions predate and influence Levi-Civita's theory of Connection.

Historical reasons led to the neglect of Piola's work in this area.

Abstract

Gabrio Piola's scientific papers have been underestimated in the mathematical-physics literature. Indeed a careful reading of them proves that they are original, deep and far reaching. Actually -even if his contribution to mechanical sciences is not completely ignored- one can undoubtedly say that the greatest part of his novel contributions to mechanics, although having provided a great impetus and substantial influence on the work of many preminent mechanicians, is in fact generally ignored. It has to be remarked that authors [10] dedicated many efforts to the aim of unveiling the true value of Gabrio Piola as a scientist; however, some deep parts of his scientific results remain not yet sufficiently illustrated. Our aim is to prove that non-local and higher gradient continuum mechanics was conceived already in Piola's works and to try to explain the reasons of the unfortunate circumstance which caused the erasure of the memory of this aspect of Piola's contribution. Some relevant differential relationships obtained in Piola [Piola, 1845-6] are carefully discussed, as they are still nowadays too often ignored in the continuum mechanics literature and can be considered the starting point of Levi-Civita's theory of Connection for Riemannian manifolds.