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

Gabrio Piola's scientific papers have been underestimated in mathematical physics literature. Indeed, a careful reading of them proves that they are original, deep and far-reaching. Actually, even if his contribution to the 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 to and substantial influence on the work of many preeminent mechanicians, is in fact generally ignored. It has to be remarked that authors Capecchi and Ruta 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 were conceived already in Piola's works and to try to explain the reasons for the unfortunate circumstance which caused the erasure of the memory of this aspect of Piola's contribution. Some relevant differential relationships obtained in Piola (Memoria intorno alle equazioni fondamentali del movimento di corpi qualsivogliono considerati secondo la naturale loro forma e costituzione, 1846) are carefully discussed, as they are still 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.