Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives

TLDR

The principle of virtual work has guided continuum mechanics since Euler, with Piola pioneering higher‑gradient continua and later Mindlin, Sedov, and Toupin extending the framework to model complex architectured materials, while alternative postulations are less effective. The present work provides a historical and theoretical overview of higher‑gradient continua and outlines future research perspectives. Mindlin and Toupin’s framework successfully models the complex behavior of architectured, advanced, multiscale, or microstructured (meta)materials.

Abstract

Since the first studies dedicated to the mechanics of deformable bodies (by Euler, D’Alembert, Lagrange) the principle of virtual work (or virtual velocities) has been used to provide firm guidance to the formulation of novel theories. Gabrio Piola dedicated his scientific life to formulating a continuum theory in order to encompass a large class of deformation phenomena and was the first author to consider continua with non-local internal interactions and, as a particular case, higher-gradient continua. More recent followers of Piola (Mindlin, Sedov and then Richard Toupin) recognized the principle of virtual work (and its particular case, the principle of least action) as the (only!) firm foundation of continuum mechanics. Mindlin and Toupin managed to formulate a conceptual frame for continuum mechanics which is able to effectively model the complex behaviour of so-called architectured, advanced, multiscale or microstructured (meta)materials. Other postulation schemes, in contrast, do not seem able to be equally efficient. The present work aims to provide a historical and theoretical overview of the subject. Some research perspectives concerning this theoretical approach are outlined in the final section.