Francesco dell’Isola

Truss Modular Beams with Deformation Energy Depending on Higher Displacement Gradients

Until now, no third gradient theory has been proposed to describe the homogenized energy associated with a microscopic structure. In this paper, we prove that this is possible using pantographic-type structures. Their deformation energies involve combinations of nodal displacements having the form of second-order or third-order finite differences. We establish the Γ-convergence of these energies to second and third gradient functionals. Some mechanical examples are provided so as to illustrate the special features of these homogenized models.

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.

Linear elastic trusses leading to continua with exotic mechanical interactions

We study the statics of some trusses, i.e. networks of nodes linked by linear springs. The trusses are designed in such a way that a few number of floppy modes are present and remain when considering the homogenized limit of the truss. We then obtain linear elastic materials with exotic mechanical interactions which cannot be described in the classical framework of Cauchy stress theory. For aim of simplicity, the structures described here are two-dimensional. The extension to the 3D case does not present any difficulty.

A revival of electric analogs for vibrating mechanical systems aimed to their efficient control by PZT actuators

In this paper we revive classical methods (i.e. those needed to synthesize analog circuits for designing direct computers) in order to conceive novel devices which we call piezo-electro-mechanical (PEM) structural members. By means of two independent synthesis procedures, new circuital analogs for the uniformly damped Elastica and Kirchhoff–Love plate are found. These circuits are used as electrical wave guides gyroscopically coupled to the corresponding mechanical members to design electrically dissipative PEM systems. The concept on which these systems are based exploits piezoelectric transducers––uniformly distributed on the member––to transform strain energy into capacitive energy: this last will be subsequently dissipated using resistors, the optimal value of which is determined by a pole placement criterion. We underline that in PEM structures PZT transducers can be regarded, at the same time, as sensors and actuators. By means of analytical methods and numerical simulations, the electro-mechanical constitutive parameters of some PEM structures are determined and it is shown that they can be designed and may be technically feasible. Moreover it is shown that in PEM structures, also when mechanical dissipative phenomena are negligible, mechanical vibrations are efficiently damped by means of completely passive electrical circuits.

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

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.

A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials

In this paper, the phenomena of resorption and growth of bone tissue and resorption of the biomaterial inside a bicomponent system are studied by means of a numerical method based on finite elements. The material behavior is described by a poro-viscoelastic model with infiltrated voids. The mechanical stimulus that drives these processes is a linear combination of density of strain energy and viscous dissipation. The external excitation is represented by a bending load slowly variable with sinusoidal law characterized by different frequencies. Investigated aspects are the influence of the load frequency, of type of the stimulus and of the effective porosity on the time evolution of the mass densities of considered system.

Variational principles are a powerful tool also for formulating field theories

Variational principles and calculus of variations have always been an important tool for formulating mathematical models for physical phenomena. Variational methods give an efficient and elegant way to formulate and solve mathematical problems that are of interest for scientists and engineers and are the main tool for the axiomatization of physical theories.

Beyond Euler-Cauchy Continua: The structure of contact actions in N-th gradient generalized continua: a generalization of the Cauchy tetrahedron argument

The most general and elegant axiomatic framework on which continuum mechanics can be based starts from the Principle of Virtual Works (or Virtual Power). This Principle, which was most likely used already at the very beginning of the development of mechanics (see e.g. Benvenuto (1981), Vailati (1897), Colonnetti (1953), Russo (2003)), became after D’Alembert the main tool for an efficient formulation of physical theories. Also in continuum mechanics it has been adopted soon (see e.g. Benvenuto (1981), Salencon (1988), Germain (1973), Berdichevsky (2009), Maugin (1980), Forest (2006)). Indeed the Principle of Virtual Works becomes applicable in continuum mechanics once one recognizes that to estimate the work expended on regular virtual displacement fields of a continuous body one needs a distribution (in the sense of Schwartz). Indeed in the present paper we prove, also by using concepts from differential geometry of embedded Riemanniam manifolds, that the Representation Theorem for Distributions allows for an effective characterization of the contact actions which may arise in N-th order strain-gradient multipolar continua (as defined by Green and Rivlin (1964)), by univocally distinguishing them in actions (forces and n-th order forces) concentrated on contact surfaces, lines (edges) and points (wedges). The used approach reconsiders the results found in the pioneering papers by Green and Rivlin (1964)–(1965), Toupin (1962), Mindlin (1964)–(1965) and Casal (1961) as systematized, for second gradient models, by Paul Germain (1973). Finally, by recalling the results found in dell’Isola and Seppecher (1995)–(1997), we indicate how Euler-Cauchy approach to contact actions and the celebrated tetrahedron argument may be adapted to N-th order strain-gradient multipolar continua.

Intorno alle equazioni fondamentali del movimento di corpi qualsivogliono, considerati secondo la naturale loro forma e costituzione

Avviene non di rado che i nuovi ritrovamenti mediante i quali fu accresciuto qualche ramo delle Matematiche applicate, non appajano subito nel concetto e nella esposizione sgombri da superfluita o lungaggini. La complicazione de’ procedimenti analitici puo giungere anche a tale da non parer piu possibile l’andare innanzi: ed e invece allora che talvolta si scopre un punto di vista piu generale, si concentrano molte particolarita e si forma una teorica compendiosa e cosi bene assicurata da infondere lena per ulteriori progressi. Sarebbe desiderabile che questo avvenisse anche per le ultime aggiunte fatte da moderni Geometri alla Meccanica razionale: e quanto a me direi che il modo di riuscirvi l’abbiamo nelle nostre mani: resta da vedere se altri vorranno essere del mio avviso.

Some Cases of Unrecognized Transmission of Scientific Knowledge: From Antiquity to Gabrio Piola’s Peridynamics and Generalized Continuum Theories

The aim of this paper is to show some typical mechanisms in the transmission of scientific knowledge through the study of some examples. We will start by considering some ancient examples concerning Democritus, Heron, Galileo and the history of the theory of tides. Then we will mainly focus on the works of the Italian scientist Gabrio Piola (1794–1850). In particular: 1. we show clear similarities between Noll’s postulation of mechanics and the ‘ancient’ presentation by Piola of the ideas needed to found Analytical Continuum Mechanics; 2. we prove that non-local and higher gradient continuum mechanics were conceived (and clearly formulated) already in Piola’s works; 3. we explain the reasons of the unfortunate circumstances which caused the (temporary) erasure of the memory of many among Piola’s contributions to mechanical sciences. Moreover, we discuss how the theory which has recently been called peridynamics, i.e. a mechanical theory which assumes that the force applied on a material particle of a continuum depends on the deformation state of a neighbourhood of the particle, was first formulated in Piola’s works. In this way we argue that in the passage from one a cultural tradition to another the content of scientific texts may often be lost, and it is possible to find more recent sources which are scientifically more primitive than some more ancient ones.