Francesco dell'isola

• PhD In Mathematical Physics Downlodable papers also at www.fdellisola.it
• Professor (Full) at University of L'Aquila

In this paper we exhibit a planar truss structure, which, once homogenised at a macro-level, will need to be modelled as a third gradient beam, i.e. as a one dimensional continuum whose deformation energy depends on the third derivative of its displacement. We call it Zigzaged Articulated Parallelograms with Articulated Braces structure (ZAPAB stru...

Mechanical metamaterials consist of specially engineered features designed to tailor and enhance the mechanical properties of their constituent materials. In this context, 2D pantographic fabrics have gained attention for their unique deformation behavior, providing remarkable resilience and damage tolerance. This study explores micro-metric metama...

Mechanical metamaterials consist of specially engineered features designed to tailor and enhance the mechanical properties of their constituent materials. In this context, 2D pantographic fabrics have gained attention for their unique deformation behavior, providing remarkable resilience and damage tolerance. This study explores micro-metric metama...

The present contribution proves that Kresling-patterned tubular origami metamaterials can exhibit a nonlinear buckling behavior in compression characterized by extremely significant and sudden twisting, as well as by extreme transverse dilation/contraction. It is proved that such an extreme buckling behavior can be achieved by tuning the ratio betw...

We consider deformations of an elastic body having initially a spherical shape. Assumed deformation energy depends on the first and second gradient of displacements. We apply an equatorial line density of dead loads, that are forces per unit line length directed in radial direction and applied along the equator of the sphere. We restrict ourselves...

Based on the papers published in the Special Issue of the scientific journal Axioms, here we present the Editorial Article “Computational Mathematics and Mathematical Physics”, the main topics of which include both fundamental and applied research in computational mathematics and differential equations of mathematical physics [...]

It is well known that “Physics and Symmetry/Asymmetry” is a topical Section of Symmetry [...]

We propose a variational approach that employs a generalized principle of virtual work to estimate both the mechanical response and the changes in living bone tissue during the remodeling process. This approach provides an explanation for the adaptive regulation of the bone substructure in the context of orthotropic material symmetry. We specifical...

We study, from a variational viewpoint, the asymptotic behavior of a planar beam with a periodic wavy shape when the amplitude and the wavelength of the shape tend to zero. We assume that the beam behaves, at the microscopic level, as a compressible Euler–Bernoulli beam and that the material properties have the same period as the geometry. We allow...

A continuum theory of pantographic lattices, based on second-grade elasticity, is presented. The proposed model is able to describe the mechanical behavior of a type of material structure made up of multiple layers of pantographic sheets connected with a third family of fibers. Thus, these materials are characterized by an orthogonal pattern of fib...

In this paper Piola transformations are found that relate the Eulerian and Lagrangian external loads which third gradient continua can sustain. As shown by Gabrio Piola and Paul Germain, the most effective postulation scheme in mechanics is based on the principle of virtual work and therefore continuum mechanics must be mathematically founded based...

Novel theories are needed for the discovery of innovative and exotic metamaterial and for their rational design. The current practice of mechanical analyses based upon moribund classical theories and experimental trial-error campaigns is caught in an inescapable vortex and illusion of inductive reasoning. The needed novel research paradigm is one i...

A dynamical study of small vibrations of an annular system made of an orthogonal network of fibers around the undeformed configuration is performed. Logarithmic spirals characterize the net of fibers connected by deformable cylinders. This particular arrangement is chosen because it produces a tough material with an economy of matter.

Numerical simulations of several planar failure modes of masonry structures are presented, based on the model and solving code from a recent hemivariational block-based model inspired from granular micromechanics (Tran et al. [33]). The numerical tests include a comparison with literature results for a constant shearing load, a parametric study of...

This paper tracks the development of lattice models that aim to describe linear elasticity of solids and the field equations of which converge asymptotically toward those of isotropic continua, thus showing the connection between discrete and continuum. In 1759, Lagrange used lattice strings/rod dynamics to show the link between the mixed different...

In this paper we start the analysis of the nonlinear dynamics of structural elements having an origami type micro-structure and micro-kinematics, also known as origami metamaterials. We use a finite dimensional Lagrangian system to explore, via numerical simulations, the overall behaviour of an origami beam. This provides some significant hints abo...

In this paper we show why the postulation scheme based on forces and couples is detrimental when designing novel metamaterials. Instead, the most effective postulation scheme for mechanics is that based on the Principle of Virtual Work formulated by, for example, d'Alembert, Lagrange, Piola, and Paul Germain. In fact, generalized continuum mechanic...

Duoskelion structures have been recently introduced by Barchiesi et al. (2021) as a proof-of-concept motif for a new class of metamaterials. The properties of these periodic beam-like chiral structural elements have been investigated, up to now, by means of a discrete model formulation whose predictions are obtained by numerical methods. In this pa...

Chiral effects in 2D Cosserat continuum model may arise from coupling between stretching deformations and the micro-rotation for hemitropic materials. An orthotropic 2D chiral Cosserat continuum is introduced to account for this type of coupling. The expected chiral effects are verified through an experimental effort using a specimen with granular...

We present and discuss the results of some numerical simulations dealing with wave motion in one-dimensional pantographic media, also known as pantographic beams. Specifically, the analysis is carried out in large displacements regime and pantographic beams are modeled by using a completely discrete approach. Nonlinear vibrations induced by a trans...

We postulate a deformation energy for describing the mechanical behavior of so called pantographic blocks, that is bodies constituted by stacking of N layers of pantographic sheets. We remark that the pantographic effect is limited in the plane of pantographic sheets and therefore only the second derivatives of transverse displacements along the pa...

In a previous work, we have shown that a granular micromechanics approach can lead to load path dependent continuum models. In the present work, we generalize such a micromechanical approach introducing an intrinsic 2nd gradient energy storage mechanism (resembling pantographic micromechanism), in the grain-grain interaction. Such a mechanism, repr...

Dynamics of pantographic sheets presents exotic aspects that deserve investigation. In this paper, we focus the attention on some possible modalities of non-linear wave propagation in planar pantographic sheets. We use a smaller length-scale lattice model, in which the beams and pivots constituting the sheet are described by constrained Euler–Berno...

We report a continuum theory for 2D strain gradient materials accounting for a class of dissipation phenomena. The continuum description is constructed by means of a (reversible) placement function and by (irreversible) damage and plastic functions. Besides, expressions of elastic and dissipation energies have been assumed as well as the postulatio...

An old and debated problem in Mechanics concerns the capacity of finite dimensional Lagrangian systems to describe dissipation phenomena. It is true that Helmholtz conditions determine not-always verifiable conditions establishing when a system of n second-order ordinary differential equations in normal form (nODEs) be the Lagrange equations derivi...

In this paper, we represent second-gradient internal work functionals in Lagrangian (referential) and Eulerian (spatial) descriptions, and we deduce the corresponding expressions for the Piola transformations of stress and double-stress tensors and of external forces and double-forces. We also derive, in both the Eulerian and Lagrangian description...

In this paper, we investigate the behavior of the strain gradient elasticity solutions around the sharp edges of the body loaded by the concentrated loads. We consider the general Mindlin-Toupin strain gradient elasticity (SGET) and several simplified models, including the dilatation gradient elasticity theory (DGET) and the couple stress theory (C...

Second-gradient continua are defined as those continua whose internal virtual work functionals depend on the first and second-gradient of the virtual displacement. These functionals can be represented either in Lagrangian (referential) or Eulerian (spatial) description thus defining respectively the Piola–Lagrange as well as the Cauchy–Euler stress...

In this contribution, a previously-introduced discrete model for studying the statics of duoskelion beam-like structures is extended to dynamics. The results of numerical simulations performed using such an extended model are reported to discuss the in-plane dynamic buckling of duoskelion structures under different loading and kine- matic boundary...

In this introductory chapter we present the motivations that prompted the authors and editors to work on this volume. The fil rouge followed in the discussion presented below is based on the following consideration: the study of the genesis of mathematical and mechanical theories does not have a merely philological purpose, but can influence and ev...

Describing the emerging macro-scale behavior by accounting for the micro-scale phenomena calls for microstructure-informed continuum models accounting properly for the deformation mechanisms identifiable at the micro-scale. Classical continuum theory, in contrast to the micromorphic continuum theory, is unable to take into account the effects of co...

In this chapter, we present the translation of the main excerpts of Heiberg’s Prolegomena to his Archimedes Edition. This text was originally written in Latin [Heiberg, J. L. (1910). Archimedis Opera omnia cum commentariis Eutocii: Vol. 1-3. In aedibus BG Teubneri.] and contains the evidence of interesting phenomena in the transmission of ancient s...

In this paper, the method of electric analog synthesis is applied to design a piezo-electro-mechanical arch able to show the capacity of multimodal damping. An electric-analog circuit is designed by using a finite number of lumped elements representing the equivalent of a curved beam. Spatial and frequency coherence conditions are proven to be veri...

We provide the proof of an existence and uniqueness theorem for weak solutions of the equilibrium problem in linear dilatational strain gradient elasticity for bodies occupying, in the reference configuration, Lipschitz domains with edges. The considered elastic model belongs to the class of so-called incomplete strain gradient continua whose poten...

We call nonlinear dilatational strain gradient elasticity the theory in which the specific class of dilatational second gradient continua is considered: those whose deformation energy depends, in an objective way, on the gradient of placement and on the gradient of the determinant of the gradient of placement. It is an interesting particular case o...

This contribution presents the results of a campaign of numerical simulations aimed at better understanding the propagation of longitudinal waves in pantographic beams within the large deformation regime. Initially, we recall the key features of a Lagrangian discrete spring model, which was introduced in previous works and that was extensively test...

This paper shows how the classical representation techniques for the solution of elasticity problems, based on the Green’s functions, can be generalized to second-gradient continua focusing on the specific case of pantographic lattices. As these last are strongly anisotropic, the fundamental solutions of isotropic second-gradient continua involving...

A Correction to this paper published: https://doi.org/10.1007/s00033-021-01480-3

The design of micro mechanical devices that can facilitate large but recoverable deformations requires a mechanical behavior that embosoms hyperelasticity. While multiphoton lithography is the epitome of microscale fabrication, the employed materials demonstrate a linear elastic response accompanied by limited ductility. In this study, we investiga...

In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkable potentialities. Indeed it allowed for the synthesis of second gradient 2D (nonlinear) continua: i.e. 2D shells whose deformation energy depends also on the second derivatives of displacements in the tangent directions to the reference configuration....

The concept of Industry 4.0 is defined as a common term for technology and the concept of new digital tools to optimize the manufacturing process. Within this framework of modular smart factories, cyber-physical systems monitor physical processes creating a virtual copy of the physical world and making decentralized decisions. This article presents...

In this paper, it is proven an existence and uniqueness theorem for weak solutions of the equilibrium problem for linear isotropic dilatational strain gradient elasticity. Considered elastic bodies have as deformation energy the classical one due to Lamé but augmented with an additive term that depends on the norm of the gradient of dilatation: onl...

The present paper is dedicated to numerical solving of three dimensional boundary-value problems in poroelastic formulation. Boundary element method and boundary integral equation method are applied to obtain Laplace domain solution of boundary-value problem. Modified Durbin’s algorithm of numerical inversion of Laplace transform is applied to perf...

The paper presents the results of dynamic testing of two wood species: lime-tree (Tilia europoea) and pine (Pinaceae). The dynamic compressive tests were carried out using the traditional Kolsky method in compression tests. The Kolsky method was modified for testing the specimen in a rigid limiting holder. In the first case, stress–strain diagrams...

The overall behaviour of an articulated beam structure constituted by elements arranged according to a specific chirality is studied. The structure as a whole, due to its slenderness and geometry, is called duoskelion beam. The name duoskelion is a neologism which is inspired by the Greek word ‌δύοσκέλιον‌ (two-legged). A discrete model for shearab...

Background: Since the mechanical behavior of pantographic metamaterials depends upon the properties of their microstructure, accurate descriptions of unit cells are needed. Objective: The present effort is motivated by this requirement to characterize the detailed deformation of unit cells formed of two orthogonal sets of 3 beams. Methods: Their de...

This Special Issue will present the latest achievements in several industrial application scenarios, leading to the so-called Industry 4.0 and the latest research related to the computational methods for a wide range of industrial applications. Both research and review articles focusing on new developments in the new digital industrial technology (...

Animation of Schrödinger's solution

Animation of Schrödinger's solution

Animation of Schrödinger's solution

The paper ‘ Zur Dynamik elastisch gekoppelter Punktsysteme’ by Schrödinger (1914) does not seem to have attracted the attention that it deserves. We translate it into English here and we discuss its results in detail, with a view to its possible influence in the modern theories of generalised continua. The clever solution found, in terms of Bessel...

Cosserat continuum model has attracted increasing interest for describing the mechanical behavior of microstructured solids. Existing formulation of Cosserat continuum model often overlooks chiral effects that arise from coupling between stretching deformations and the micro-rotation. Here, we introduce an extended Cosserat model that accounts for...

The equilibrium forms of pantographic blocks in a three-point bending test are investigated via both experiments and numerical simulations. In the computational part, the corresponding minimization problem is solved with a deformation energy derived by homogenization within a class of admissible solutions. To evaluate the numerical simulations, ser...

Within the six-parameter nonlinear shell theory we analyzed the in-plane rotational instability which occurs under in-plane tensile loading. For plane deformations the considered shell model coincides up to notations with the geometrically nonlinear Cosserat continuum under plane stress conditions. So we considered here both large translations and...

A second gradient theory for woven fabrics is applied to Kirchhoff-Love shell elements to analyze the mechanics of fiber reinforced composite materials. In particular, we assume a continuous distribution of the fibers embedded into the shell surface, accounting for additional in-plane flexural resistances within the hyperelastic regime. For the fin...

We consider the reduced constrained linear Cosserat continuum, a particular type of a Cosserat medium, for three different material behaviors or symmetries: the isotropic elastic case, a special type of elastic transversely isotropic case, and the isotropic viscoelastic case. Such continua, in which stresses do not work on rates of microrotation gr...

In the online Stanford Encyclopedia of Philosophy1 one finds an accurate account of formal modern model theory, with a careful description of the underlying philosophical and mathematical arguments. Model theory was developed, in a more recent époque, by Alfred Tarski. It appears though that (see the Forgotten Revolution by Lucio Russo [1]) Helleni...

The main physical laws of thermal–plastic deformation and fatigue damage accumulation processes in polycrystalline structural alloys under various regimes of cyclic thermal–mechanical loading are considered. Within the framework of mechanics of damaged media, a mathematical model is developed that describes thermal–plastic deformation and fatigue d...

In the present contribution, we address the following problem: is it possible to find a microstructure producing, at the macro-level and under loads of the same order of magnitude, a beam which can be both extensible and flexible? Using an asymptotic expansion and rescaling suitably the involved stiffnesses, we prove that a pantographic microstruct...

The derivation by variational asymptotic homogenization of a 2D-continuum model describing large elastic planar deformations of a discrete bi-pantographic structure is presented. A rectangular bi-pantographic specimen was additively manufactured and subjected to a bias extension test for macroscopic strains up to ca. 40%. The deformations of the bi...

Bi-pantographic fabrics are composed of two families of pantographic beams and correspond to a class of architectured materials that are described in plane as second-gradient 2D continua. On a discrete level, a pantographic beam is a periodic arrangement of cells and looks like an expanding barrier. The materialization of a bi-pantographic fabric m...

In the past years, many attempts have been made in order to model the process of bone remodeling. This process is complex, as it is governed by not yet completely understood biomechanical coupled phenomena. It is well known that bone tissue is able to self-adapt to different environmental demands of both mechanical and biological origin. The mechan...

The synthesis of a 1D full second gradient continuum was obtained by the design of so-called pantographic beam (see Alibert et al. Mathematics and Mechanics of Solids (2003)) and the problem of the synthesis of planar second gradient continua has been faced in several subsequent papers: in dell’ Isola et al. Zeitschrift für angewandte Mathematik un...

In this paper we present numerical solutions to a geometrically nonlinear version of the extensible Timoshenko beam model under distributed load. The particular cases in which: i) extensional stiffness is infinite (inextensible Timoshenko model), ii) shear stiffness is infinite (extensible Euler model) and iii) extensional and shear stiffnesses are...

In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradient-incomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-in...

We propose an estimate for the effective elastic properties of a new imagined two-phase bi-continuous composite material type with a so-called ``pantographic-inspired” (P-I) architecture in the sense of a matrix reinforcement which is a 3D fiber network capable of large, pantographic-like, deformations, owing to particular properties carried by the...

Recent advanced manufacturing techniques such as 3D printing have prompted the need for designing new multiscale architectured materials for various industrial applications. These multiscale architectures are designed to obtain the desired macroscale behavior by activating interactions between different length scales and coupling different physical...

There is a class of planar 1D-continua which can be described exclusively by their placement functions which in turn are curves in a two-dimensional space. In contrast to the Elastica for which the deformation energy depends on the projection of the second gradient to the normal vector of the placement function, i.e. the material curvature, the pro...

Arrow’s theorem, based on Condorcet’s ingenious intuition, is only a first important step towards a General Mathematical Theory of Social Structures. Yet, the understanding that has been made possible by this fundamental step forward has had a great impact on our understanding of a myriad of social and ethological phenomena. The present work has th...

Here, we try to present a preliminary theoretical explanation of the huge amount of phenomenological evidence, which was presented in the previous pages. We are aware of the fact that many other pages, like those which we wrote about the Theorem of Arrow in the second appendix, may be needed to explain in detail those mathematical ideas which we wi...

In this chapter, we want to discuss a masterpiece of political thought: the famous pamphlet by Étienne de La Boétie entitled Discours sur la servitude volontaire. We believe that the results by Arrow and Nash, together with the ideas of evolutionary etology, confirm the analysis developed by La Boétie.

For concluding this essay we want to explicitly state that the search for democracy should not be considered a hopeless endeavour, as it is proven by the evolution from the populistic Athenian Democracy into the Roman Republic and then into the modern Constitutional Democracies. It is conceivable to develop some mathematical tools to formulate the...

In this chapter we will discuss the behaviour of personalities whose actions had a negative effect on the life of the social groups which they dominated. Following the modern usage of the word, we decided, in this chapter, to use the word dictatorship with a negative meaning.

Even if we are aware of the fact that it may appear as a completely arbitrary differentiation, however, we will try to distinguish, in the following pages, between leadership and dictatorship, giving to the first word a slightly more positive meaning. In the present chapter some examples of “well-behaving” dictators will be described: we used the t...

The main characters of this chapter will be Yeroen, Nikkie, Luit, Mama and Kanzi, chimpanzees’and bonobos' dictators whose behaviour has been described by Frans De Waal. One could believe that what chimpanzees and bonobos are doing is not at all interesting for us. Instead, as De Waal is explaining so clearly in all his books, the observation of th...

How the gender of a dictator influences his/her way of managing power? In this chapter we argue that there is no true difference between a male and a female dictator.

In this chapter the main ideas underlying Condorcet's Conjecture and Arrow's Theorem are presented and some first applications to social sciences are discussed. The mathematical results are presented in an informal way to make the discussion accessible to the layman.

In this chapter we show how Arrow's Theorem can be applied to understand the structure of social groups and many biological, psycological and social phenomena.