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

By Francesco dell'Isola, Ugo Andreaus, Luca Placidi. 2013

523 citations recorded by the source corpus.

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.

Subfields: Materials Chemistry, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/46878c4052d2e30adc95a4c2740c57abaa572eac

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In Materials Chemistry

In Mechanics of Materials

Citations in the corpus

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions (2017, methodology): “The procedure which is used in the aforementioned papers can be summarized as follows (see [5, 19]): i) one postulates the principle of virtual work, i.”
Strain gradient and generalized continua obtained by homogenizing frame lattices (2018, background): “Indeed materials with such energies are seldom found in nature [12], [13] and are expected to have a very special behavior [32], [36], [39], [60].”
Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium (2016, unknown): “By means of this procedure Piola manages to formulate (again for the first time) the theory of generalised continua and also the continuum models which have been recently renamed as Peridynamics (see [36]).”
Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives (2017, background): “Somebody who, with open mind, reads Mindlin’s paper immediately after having read Piola [1, 3, 40] cannot avoid remarking on the striking equality of vision and purpose.”
Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models (2016, methodology): “The works of Piola on generalized continua (see [44]) and what has been later rediscovered and called peridynamics (see [45]) remained for a long time nearly unknown.”