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]. Structured continua, originated by the Cosserats [24, 25], represent another branch of continuum models and their study has lead to an established theory, see Capriz [26] for instance. Indeed, continua with (micro-)structure are optimal models for many objects in multiple fields of application: they may describe non-standard beams [27, 28], damaged structural elements [29], masonry [30, 31, 32], plasticity [33]. In addition, they may provide suitable frameworks for multi- field physics, such as piezoelectricity or the mechanics of mixtures or porous media: the various physical quantities entering the phenomena are seen simply as additional degrees of freedom in a generalized lagrangean system. Gabrio Piola (1794 1850) stood out among the Italian scholars in mechanics in the first half of 1800, though he was never in charge of a university chair. In a series of papers, published in Italian in some journals of almost no di diffusion outside Italy [34] [38], he was perhaps the first to present: a) a clear separation among kinematics, expressed by suitable constraint equations, and balance, expressed by Lagrange's virtual work [39], according to which inner actions are simply mechanical duals of suitable constraint equations; b) a clear statement that physical considerations on the constitution of inner actions lie beyond the position of kinematics and balance and are independent of them; c) an imaginary ideal state for any body, made up of a perfectly regular array of molecules, free of any stress; d) an imaginary intermediate con guration between the natural and the actual ones, so that constraint equations exist; and e) the possibility to obtain balance equations by considering a change in observer for the present con guration. These key points in Piola's principles of mechanics are put into evidence by Hellinger [41], and described with some depth in other studies on Piola's works [40, 47]. The aim of this work is, however, to stress the above said points, that seem quite original and basic for a more general theory of continua with respect to that by Cauchy, Poisson and their successors (among them Lamé [42] and Saint-Venant [43, 44]), well before the Cosserats', and with a very modern spirit......
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The complete works of Gabrio Piola: Volume I, 38, 2014, Advanced Structured Materials
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Giuseppe Ruta. Gabrio Piola and Balance Equations. The complete works of Gabrio Piola: Volume I, 38, 2014, Advanced Structured Materials. 〈hal-00955879〉

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