Gabrio Piola and Balance Equations

Abstract

The origin of modern continuum mechanics dates back to Cauchy [1–8] and Poisson [9, 10], who investigated linear elastic solids and fluids subject to infinitesimal displacements. This is stated in the well known monographs on history of mechanics by Todhunter and Pearson [11], Dugas [12], Timoshenko [13], Benvenuto [14]. We find some more hints on the origins of the theory of elasticity also in the recent contributions by Capecchi et al. [15, 16, 17]. Cauchy and Poisson imagined natural bodies as constituted by very small particles of matter interacting by central forces. However, they derived continuum field equations by suitable analytical tricks, and eventually Cauchy adopted only continuous functions to describe the regions of ambient space filled by a huge number of particles very close to each other. Since then, continuum mechanics has influenced all basic studies on theoretical and applied mechanics, enlarging both its scopes and range of applications: electro-magnetism and heat/work are only two of them. Examples of continuum mechanics in these fields are provided by the pioneering works by Green [18] and Thomson [19, 20]; comprehensive expositions are the well known ones by Truesdell and Toupin [21] and Truesdell and Noll [22]; a more recent handbook is that by Gurtin et al. [23].