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At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola
TL;DR: In this paper, the authors show that non-local and higher gradient continuum mechanics was conceived already in Piola's works and explain the reasons of the unfortunate circumstance which caused the erasure of the memory of this aspect of Piola contribution.
Abstract: Gabrio Piola's scientific papers have been underestimated in the mathematical-physics literature. Indeed a careful reading of them proves that they are original, deep and far reaching. Actually -even if his contribution to 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 and substantial influence on the work of many preminent mechanicians, is in fact generally ignored. It has to be remarked that authors [10] 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 was conceived already in Piola's works and to try to explain the reasons of the unfortunate circumstance which caused the erasure of the memory of this aspect of Piola's contribution. Some relevant differential relationships obtained in Piola [Piola, 1845-6] are carefully discussed, as they are still nowadays 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.
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TL;DR: In this article, the authors considered a discrete spring model for extensible beams and proposed a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model.
Abstract: The aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call ‘pantographic structures’. The interest in these materials was increased by the possibilities opened by the diffusion of technology of three-dimensional printing. They can be regarded, once choosing a suitable length scale, as families of beams (also called fibres) interconnected to each other by pivots and undergoing large displacements and large deformations. There are, however, relatively few ‘ready-to-use’ results in the literature of nonlinear beam theory. In this paper, we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model. The homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems. Furthermore, we consider pantographic structures, find the corresponding homogenized second gradient deformation energies and study some planar problems. Numerical solutions for these two-dimensional problems are obtained via minimization of energy and are compared with some experimental measurements, in which elongation phenomena cannot be neglected.
333 citations
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...A472:20150790 ................................................... been recently renamed as Peridynamics [37]....
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...been recently renamed as Peridynamics [37]....
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TL;DR: In this paper, a comprehensive review on the development of higher-order continuum models for capturing size effects in small-scale structures is presented, mainly focusing on the size-dependent beam, plate and shell models developed based on the nonlocal elasticity theory, modified couple stress theory and strain gradient theory.
275 citations
TL;DR: A review of the state of the art in the study of mechanical metamaterials is given in this article, where the very attractive property of having a microstructure capable of determining exotic and specific properties is discussed.
Abstract: In this paper, we give a review of the state of the art in the study of mechanical metamaterials. The very attractive property of having a microstructure capable of determining exotic and specifica...
266 citations
University of the South, Toulon-Var1, University of L'Aquila2, Medical University of Warsaw3, Warsaw University of Technology4, University of California, Berkeley5, Sapienza University of Rome6, University of Sassari7, Roma Tre University8, University of Lyon9, Gdańsk University of Technology10, University of Kansas11, University of Catania12, University of Cagliari13, University of La Rochelle14, University of Lorraine15, Technical University of Berlin16, University of Paris17, Fraunhofer Institute for High-Speed Dynamics, Ernst-Mach-Institut18, Université Paris-Saclay19
TL;DR: P pantographic metamaterials undergo very large deformations while remaining in the elastic regime, are very tough in resisting to damage phenomena, and exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties.
Abstract: In this paper, we account for the research efforts that have been started, for some among us, already since 2003, and aimed to the design of a class of exotic architectured, optimized (meta) materials. At the first stage of these efforts, as it often happens, the research was based on the results of mathematical investigations. The problem to be solved was stated as follows: determine the material (micro)structure governed by those equations that specify a desired behavior. Addressing this problem has led to the synthesis of second gradient materials. In the second stage, it has been necessary to develop numerical integration schemes and the corresponding codes for solving, in physically relevant cases, the chosen equations. Finally, it has been necessary to physically construct the theoretically synthesized microstructures. This has been possible by means of the recent developments in rapid prototyping technologies, which allow for the fabrication of some complex (micro)structures considered, up to now, to be simply some mathematical dreams. We show here a panorama of the results of our efforts (1) in designing pantographic metamaterials, (2) in exploiting the modern technology of rapid prototyping, and (3) in the mechanical testing of many real prototypes. Among the key findings that have been obtained, there are the following ones: pantographic metamaterials (1) undergo very large deformations while remaining in the elastic regime, (2) are very tough in resisting to damage phenomena, (3) exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties, (4) have superior strength to weight ratio, (5) have predictable damage behavior, and (6) possess physical properties that are critically dictated by their geometry at the microlevel.
264 citations
Cites background from "At the origins and in the vanguard ..."
...It is possible to state that some of these theories were already known at least two centuries ago [7,8]....
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TL;DR: In this paper, the authors present a numerical code implementing directly the discrete Hencky-type model which is robust enough to solve the problem of the determination of equilibrium configurations in the large deformation and displacement regimes.
Abstract: Hencky (Uber die angenaherte Losung von Stabilitatsproblemen im Raum mittels der elastischen Gelenkkette. Ph.D. thesis, Engelmann, 1921) proposed a discrete model for elasticae by introducing rigid bars and rotational springs. Hencky (Proc R Soc Lond A Math Phys Eng Sci 472(2185), 2016) approach has been introduced to heuristically motivate the need of second gradient continua. Here, we present a novel numerical code implementing directly the discrete Hencky-type model which is robust enough to solve the problem of the determination of equilibrium configurations in the large deformation and displacement regimes. We apply this model to study some potentially applicable problems, and we compare its performances with those of the second gradient continuum model. The numerical evidence presented supports the conjecture that Hencky-type converges to second gradient model.
224 citations
Cites background or methods from "At the origins and in the vanguard ..."
...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....
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..., [44,45,66]) then the corresponding Euler–Lagrange conditions were found in order to be able to apply for their solutions all the methods which are made available by mathematical analysis....
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References
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Book•
01 Dec 1979
TL;DR: Spivak's comprehensive introduction to differential geometry as discussed by the authors takes as its theme the classical roots of contemporary differential geometry, and explains why it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely to rigorize the concepts of classical differential geometry.
Abstract: Spivak's Comprehensive introduction takes as its theme the classical roots of contemporary differential geometry. Spivak explains his Main Premise (my term) as follows: "in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary; there is no point in introducing the curvature tensor without explaining how it was invented and what it has to do with curvature". His second premise concerns the manner in which the historical material should be presented: "it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely in order to rigorize the concepts of classical differential geometry". Here, Spivak is addressing "a dilemma which confronts anyone intent on penetrating the mysteries of differential geometry". On the one hand, the subject is an old one, dating, as we know it, from the works of Gauss and Riemann, and possessing a rich classical literature. On the other hand, the rigorous and systematic formulations in current use were established relatively recently, after topological techniques had been sufficiently well developed to provide a base for an abstract global theory; the coordinate-free geometric methods of E. Cartan were also a major source. Furthermore, the viewpoint of global structure theory now dominates the subject, whereas differential geometers were traditionally more concerned with the local study of geometric objects. Thus it is possible and not uncommon for a modern geometric education to leave the subject's classical origins obscure. Such an approach can offer the great advantages of elegance, efficiency, and direct access to the most active areas of modern research. At the same time, it may strike the student as being frustratingly incomplete. As Spivak remarks, "ignorance of the roots of the subject has its price-no one denies that modern formulations are clear, elegant and precise; it's just that it's impossible to comprehend how any one ever thought of them." While Spivak's impulse to mediate between the past and the present is a natural one and is by no means unique, his undertaking is remarkable for its ambitious scope. Acting on its second premise, the Comprehensive introduction opens with an introduction to differentiable manifolds; the remaining four volumes are devoted to a geometric odyssey which starts with Gauss and Riemann, and ends with the Gauss-Bonnet-Chern Theorem and characteristic classes. A formidable assortment of topics is included along the way, in which we may distinguish several major historical themes: In the first place, the origins of fundamental geometric concepts are investigated carefully. As just one example, Riemannian sectional curvature is introduced by a translation and close exposition of the text of Riemann's remarkable paper, Über die Hypothesen, welche der Geometrie zu Grunde
3,840 citations
3,323 citations
TL;DR: In this paper, a peridynamic formulation for the basic equations of continuum mechanics is proposed, and the propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived.
Abstract: Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the "peridynamic" formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.
2,842 citations
TL;DR: HAL as discussed by the authors is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, which may come from teaching and research institutions in France or abroad, or from public or private research centers.
Abstract: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Elastic materials with couple-stresses R. Toupin
2,574 citations
TL;DR: In this article, a theory of non-local elasticity is presented via the vehicles of global balance laws and the second law of thermodynamics via the use of a localized Clausius-Duhem inequality and a variational statement of Gibbsian global thermodynamics.
2,201 citations
"At the origins and in the vanguard ..." refers background or methods in this paper
...One can easily recognize by comparing, for example, the presentation in [54] with (12b) that in the works by Piola the functional (∫ B (X , X̄ , ρ)δρ2μ(X̄ ) dX̄ ) (N1) is assumed to satisfy a slightly generalized version of what in [54, p. 34] is called the ‘smooth neighbourhood hypothesis’ which reads as follows (in Eringen’s work the symbol V is used with the same meaning as our symbol B, X ′ is used instead of X̄ , x instead of χ , t′ denotes a time instant, the symbol () ,Ki denotes the partial derivatives with respect to the Kith coordinate of X , and we assume the convention of sums over repeated indices): Suppose that in a region V0 ⊂ V , appropriate to each material body, the independent variables admit Taylor series expansions in X ′ − X in V0 […] terminating with gradients of order P, Q, etc., x(X ′, t′) = x(t′) + (X ′K1 − XK1) x,K1 (t′) + . . . + 1 P ( X ′K1 − XK1 ) . . . ( X ′KP − XKP ) x,K1...KP (t ′), and […]....
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...In the work by Piola [7] the homogenized theory which is deduced by means of the identification of powers in the discrete micro-model and in the continuous macro-model can be called (in the language used by Eringen [48, 49]) a non-local theory....
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...A more detailed discussion about the eventual novelties contained in the formulation of peridynamics when compared with for example Eringen’s non-local continuum mechanics is postponed to further investigations....
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...In Eringen [48, 49, 54], the non-local continuum mechanics is founded on a postulation based on principles of balance of mass, linear and angular momentum, energy and entropy....
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...Many non-local continuum theories have been formulated since the first formulation by Piola: we cite here for instance [48, 49, 54, 55]....
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