Francesco Maddalena - Academia.edu (original) (raw)

Papers by Francesco Maddalena

Nonlinearity, Oct 3, 2022

We study the dispersive properties of a linear equation in one spatial dimension which is inspire... more We study the dispersive properties of a linear equation in one spatial dimension which is inspired by models in peridynamics. The interplay between nonlocality and dispersion is analyzed in detail through the study of the asymptotics at low and high frequencies, revealing new features ruling the wave propagation in continua where nonlocal characteristics must be taken into account. Global dispersive estimates and existence of conserved functionals are proved. A comparison between these new effects and the classical local scenario is deepened also through a numerical analysis.

arXiv (Cornell University), Jul 13, 2022

In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily... more In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarilyshaped two-dimensional (2D) closed manifolds is proposed. When dealing with non parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise. Here, a routing procedure is implemented for computing geodesic distances by re-interpreting the triangular computational mesh as a non-oriented graph; thus returning a suitable and general method. Moreover, the time integration of the peridynamics equation is demanded to a P-(EC) k formulation of the implicit β-Newmark scheme. The convergence of the overall proposed procedure is questioned and rigorously proved. Its abilities and limitations are analyzed by simulating the evolution of a two-dimensional sphere. The performed numerical investigations are mainly motivated by the issues related to the insurgence of singularities in the evolution problem. The obtained results return an interesting picture of the role played by the nonlocal character of the integrodifferential equation in the intricate processes leading to the spontaneous formation of singularities in real materials.

Applying the concepts of Nonlinear Normal Modes and Limiting Phase Trajectories introduced by L. ... more Applying the concepts of Nonlinear Normal Modes and Limiting Phase Trajectories introduced by L. I. Manevitch in [46] to a two-dimensional mass-spring system, the authors propose a generalised method to tune a plane metamaterial and get the desirable resonant behaviour at short wavelengths. Indeed, the account of nonlinear coupling between the oscillators enables the localisation of energy leading the origin of a bandgap at short wavelengths regardless the existence of external disturbances. Moreover, further restrictions on the modes amplitude allow the observation of FermiPasta-Ulam-Tsingou recurrence and super-recurrence in the two-dimensional metamaterial. These findings can open the way to further research in order to improve efficiency and performance of resonant metamaterials.

Journal of the Mechanics and Physics of Solids, 2017

In this paper we study the diffusely observed occurrence of Fractality and Self-organized Critica... more In this paper we study the diffusely observed occurrence of Fractality and Self-organized Criticality in mechanical systems. We analytically show, based on a prototypical compressed tensegrity structure, that these phenomena can be viewed as the result of the contemporary attainment of mass minimization and global stability in elastic systems.

We show how some problems coming from different fields of applied sciences such as physics, engin... more We show how some problems coming from different fields of applied sciences such as physics, engineering, biology, admit a common variational formulation characterized by the competition of two energetic terms. We discuss related problems and techniques studied by the authors and collaborators in the recent past as well open problems and further possible research directions in these topics.

The paper studies the initial boundary value problem related to the dynamic evolution of an elast... more The paper studies the initial boundary value problem related to the dynamic evolution of an elastic beam interacting with a substrate through an elastic-breakable forcing term. This discontinuous interaction is aimed to model the phenomenon of attachement-detachement of the beam occurring in adhesion phenomena. We prove existence of solutions in energy space and exhibit various counterexamples to uniqueness. Furthermore we characterize some relavant features of the solutions, ruling the main effectes of the nonlinearity due to the elasic-breakable term on the dynamical evolution, by proving the linearization property according to <cit.> and an asymtotic result pertaining the long time behavior.

Journal of Remanufacturing, 2019

The increasing rate of production and diffusion of photovoltaic (PV) technologies for industrial ... more The increasing rate of production and diffusion of photovoltaic (PV) technologies for industrial and domestic applications urges improvement of the sustainability of their demanufacuring processes in order to reduce the amount of electronic wastes. Sustainability of demanufacturing processes concerns the reduction of energy consumption, the reduction of polluting substances as well as the reduction of the effort spent in recovery of the components. No optimal process exists so far, provided a number of different approaches have been devised. A promising choice relies on the use of thermo-mechanical treatments for inducing a delamination process where interfacial bonding between layers are weakened and, finally, broken inducing delamination of the layers. In this view, the paper presents a preliminary theoretical industrialization study. We introduce a mathematical model based of the equations of thermo-elasticity to prove the feasibility of the technological process; the results of a Finite Element (FE) Analysis are then discussed to show the validity of the new sustainable demanufacturing process endeavouring the delamination process. The analysis is performed searching the optimal thermally induced cycles at cryogenic temperatures.

IMA Journal of Applied Mathematics, 2009

We consider the problem of mass reduction for elastic bodies by appearance of cavities. In this w... more We consider the problem of mass reduction for elastic bodies by appearance of cavities. In this work, this problem is related to the minimization of a surface energy, depending on the stress tensor in the original equilibrium configuration. Special cases of mechanical interest are also analysed.

Continuum Mechanics and Thermodynamics, 2009

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2001

The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet bo... more The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet boundary conditions. An existence result was known so far for C 1 (∂) boundary dataû. We show here that the same result holds forû ∈ C 0,µ (∂) if µ > 1 2 and it cannot be extended to cover the case µ = 1 2. The proof is based on some geometric measure theoretic properties, in part introduced here, which are proved a priori to hold for all the possible minimizers.  2001 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-On savait que le minimum d'une fonctionnelle à discontinuité libre avec condition de Dirichlet sur le bord était atteint quand la donnée de bordû est C 1 (∂). Nous étendons ce résultat àû ∈ C 0,µ (∂) si µ > 1 2 et montrons qu'il n'est plus vrai pour µ = 1 2. Pour cela, nous démontrons des propriétés de théorie de la mesure géométrique a priori valides pour tout minimiseur de la fonctionnelle.  2001 Éditions scientifiques et médicales Elsevier SAS

Journal of Optimization Theory and Applications, 2010

We study the dispersive properties of a linear peridynamic equation in one spatial dimension. The... more We study the dispersive properties of a linear peridynamic equation in one spatial dimension. The interplay between nonlocality and dispersion is analyzed in detail through the study of the asymptotics at low and high frequencies, revealing new features ruling the wave propagation in continua where nonlocal characteristics must be taken into account. Global dispersive estimates and existence of conserved functionals are proved. A comparison between these new effects and the classical local scenario is deepen also through a numerical analysis.

Applied Mathematics Letters

This note provides a variational description of the mechanical effects of flexural stiffening of ... more This note provides a variational description of the mechanical effects of flexural stiffening of a 2D plate glued to an elastic-brittle or an elastic-plastic reinforcement. The reinforcement is assumed to be linear elastic outside possible free plastic yield lines or free crack. Explicit Euler equations and a compliance identity are shown for the reinforcement of a 1D beam.

The increasing rate of production and diffusion of photovoltaic (PV) technologies for industrial ... more The increasing rate of production and diffusion of photovoltaic (PV) technologies for industrial and domestic applications urges improvement of the sustainability of their demanufacturing processes in order to reduce the amount of electronic wastes. Sustainability of demanufacturing processes concerns the reduction of energy consumption, the reduction of polluting substances as well as the reduction of the effort spent in recovery of the components. There is not an optimal process so far, provided a number of different approaches have been devised (see e.g. [1, 2, 3]). A promising choice relies on the use of thermo-mechanical treatments for inducing a delamination process where interfacial bonding between layers are weakened and, finally, broken inducing separation of the layers [4]. In this paper we present a preliminary industrialization study, based on Finite Element (FE) Analysis, to prove the validity of the new sustainable demanufacturing process endeavouring the delamination ...

We study the dispersive properties of a linear peridynamic equation in one spatial dimension. The... more We study the dispersive properties of a linear peridynamic equation in one spatial dimension. The interplay between nonlocality and dispersion is analyzed in detail through the study of the asymptotics at low and high frequencies, revealing new features ruling the wave propagation in continua where nonlocal characteristics must be taken into account. Global dispersive estimates and existence of conserved functionals are proved. A comparison between these new effects and the classical local scenario is deepen also through a numerical analysis. Introduction A fundamental trait in the mathematical modeling of continuum physics relies in capturing the essential phenomena of a complex problem still keeping the technical difficulties as manageable as possible. A typical example of this strategy is provided by classical linear elasticity ([10]) which, in spite of its very long-lasting tradition, represents yet an unavoidable comparison term even for the more recent mechanical theories aime...

In this paper we analyze and compare two different models for adhesion phenomena, recently propos... more In this paper we analyze and compare two different models for adhesion phenomena, recently proposed by the authors. In the first approach [9] a feasible expression of the adhesion energy is suggested by the existence problem of partially detached equilibrium states. In the second model [10] the macroscopic energy is obtained by performing a multiscale analysis and it is deduced via a macroscopic Γ-limit of the energy at the scale of the microstructure. Interestingly, we obtain that the first model can be deduced by the second one as the limit case when the parameter measuring the relative stiffness of the adhesive layer and the beam diverges.

Journal of Peridynamics and Nonlocal Modeling

In this paper, we investigate, through numerical studies, the dynamical evolutions encoded in a l... more In this paper, we investigate, through numerical studies, the dynamical evolutions encoded in a linear one-dimensional nonlocal equation arising in peridynamics. The different propagation regimes ranging from the hyperbolic to the dispersive, induced by the nonlocal feature of the equation, are carefully analyzed. The study of an initial value Riemann-like problem suggests the formation of a singularity.

Milan Journal of Mathematics

The paper studies the initial boundary value problem related to the dynamic evolution of an elast... more The paper studies the initial boundary value problem related to the dynamic evolution of an elastic beam interacting with a substrate through an elastic-breakable forcing term. This discontinuous interaction is aimed to model the phenomenon of attachment-detachment of the beam occurring in adhesion phenomena. We prove existence of solutions in energy space and exhibit various counterexamples to uniqueness. Furthermore we characterize some relevant features of the solutions, ruling the main effects of the nonlinearity due to the elastic-breakable term on the dynamical evolution, by proving the linearization property according to Gérard (J Funct Anal 141(1):60–98, 1996) and an asymptotic result pertaining the long time behavior.

Calculus of Variations and Partial Differential Equations

We study an old variational problem formulated by Euler as Proposition 53 of his Scientia Navalis... more We study an old variational problem formulated by Euler as Proposition 53 of his Scientia Navalis by means of the direct method of the calculus of variations. Precisely, through relaxation arguments, we prove the existence of minimizers. We fully investigate the analytical structure of the minimizers in dependence of the geometric parameters and we identify the ranges of uniqueness and non-uniqueness.

Nonlinearity, Oct 3, 2022

We study the dispersive properties of a linear equation in one spatial dimension which is inspire... more We study the dispersive properties of a linear equation in one spatial dimension which is inspired by models in peridynamics. The interplay between nonlocality and dispersion is analyzed in detail through the study of the asymptotics at low and high frequencies, revealing new features ruling the wave propagation in continua where nonlocal characteristics must be taken into account. Global dispersive estimates and existence of conserved functionals are proved. A comparison between these new effects and the classical local scenario is deepened also through a numerical analysis.

arXiv (Cornell University), Jul 13, 2022

In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily... more In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarilyshaped two-dimensional (2D) closed manifolds is proposed. When dealing with non parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise. Here, a routing procedure is implemented for computing geodesic distances by re-interpreting the triangular computational mesh as a non-oriented graph; thus returning a suitable and general method. Moreover, the time integration of the peridynamics equation is demanded to a P-(EC) k formulation of the implicit β-Newmark scheme. The convergence of the overall proposed procedure is questioned and rigorously proved. Its abilities and limitations are analyzed by simulating the evolution of a two-dimensional sphere. The performed numerical investigations are mainly motivated by the issues related to the insurgence of singularities in the evolution problem. The obtained results return an interesting picture of the role played by the nonlocal character of the integrodifferential equation in the intricate processes leading to the spontaneous formation of singularities in real materials.

Applying the concepts of Nonlinear Normal Modes and Limiting Phase Trajectories introduced by L. ... more Applying the concepts of Nonlinear Normal Modes and Limiting Phase Trajectories introduced by L. I. Manevitch in [46] to a two-dimensional mass-spring system, the authors propose a generalised method to tune a plane metamaterial and get the desirable resonant behaviour at short wavelengths. Indeed, the account of nonlinear coupling between the oscillators enables the localisation of energy leading the origin of a bandgap at short wavelengths regardless the existence of external disturbances. Moreover, further restrictions on the modes amplitude allow the observation of FermiPasta-Ulam-Tsingou recurrence and super-recurrence in the two-dimensional metamaterial. These findings can open the way to further research in order to improve efficiency and performance of resonant metamaterials.

Journal of the Mechanics and Physics of Solids, 2017

In this paper we study the diffusely observed occurrence of Fractality and Self-organized Critica... more In this paper we study the diffusely observed occurrence of Fractality and Self-organized Criticality in mechanical systems. We analytically show, based on a prototypical compressed tensegrity structure, that these phenomena can be viewed as the result of the contemporary attainment of mass minimization and global stability in elastic systems.

We show how some problems coming from different fields of applied sciences such as physics, engin... more We show how some problems coming from different fields of applied sciences such as physics, engineering, biology, admit a common variational formulation characterized by the competition of two energetic terms. We discuss related problems and techniques studied by the authors and collaborators in the recent past as well open problems and further possible research directions in these topics.

The paper studies the initial boundary value problem related to the dynamic evolution of an elast... more The paper studies the initial boundary value problem related to the dynamic evolution of an elastic beam interacting with a substrate through an elastic-breakable forcing term. This discontinuous interaction is aimed to model the phenomenon of attachement-detachement of the beam occurring in adhesion phenomena. We prove existence of solutions in energy space and exhibit various counterexamples to uniqueness. Furthermore we characterize some relavant features of the solutions, ruling the main effectes of the nonlinearity due to the elasic-breakable term on the dynamical evolution, by proving the linearization property according to <cit.> and an asymtotic result pertaining the long time behavior.

Journal of Remanufacturing, 2019

The increasing rate of production and diffusion of photovoltaic (PV) technologies for industrial ... more The increasing rate of production and diffusion of photovoltaic (PV) technologies for industrial and domestic applications urges improvement of the sustainability of their demanufacuring processes in order to reduce the amount of electronic wastes. Sustainability of demanufacturing processes concerns the reduction of energy consumption, the reduction of polluting substances as well as the reduction of the effort spent in recovery of the components. No optimal process exists so far, provided a number of different approaches have been devised. A promising choice relies on the use of thermo-mechanical treatments for inducing a delamination process where interfacial bonding between layers are weakened and, finally, broken inducing delamination of the layers. In this view, the paper presents a preliminary theoretical industrialization study. We introduce a mathematical model based of the equations of thermo-elasticity to prove the feasibility of the technological process; the results of a Finite Element (FE) Analysis are then discussed to show the validity of the new sustainable demanufacturing process endeavouring the delamination process. The analysis is performed searching the optimal thermally induced cycles at cryogenic temperatures.

IMA Journal of Applied Mathematics, 2009

We consider the problem of mass reduction for elastic bodies by appearance of cavities. In this w... more We consider the problem of mass reduction for elastic bodies by appearance of cavities. In this work, this problem is related to the minimization of a surface energy, depending on the stress tensor in the original equilibrium configuration. Special cases of mechanical interest are also analysed.

Continuum Mechanics and Thermodynamics, 2009

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2001

The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet bo... more The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet boundary conditions. An existence result was known so far for C 1 (∂) boundary dataû. We show here that the same result holds forû ∈ C 0,µ (∂) if µ > 1 2 and it cannot be extended to cover the case µ = 1 2. The proof is based on some geometric measure theoretic properties, in part introduced here, which are proved a priori to hold for all the possible minimizers.  2001 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-On savait que le minimum d'une fonctionnelle à discontinuité libre avec condition de Dirichlet sur le bord était atteint quand la donnée de bordû est C 1 (∂). Nous étendons ce résultat àû ∈ C 0,µ (∂) si µ > 1 2 et montrons qu'il n'est plus vrai pour µ = 1 2. Pour cela, nous démontrons des propriétés de théorie de la mesure géométrique a priori valides pour tout minimiseur de la fonctionnelle.  2001 Éditions scientifiques et médicales Elsevier SAS

Journal of Optimization Theory and Applications, 2010

We study the dispersive properties of a linear peridynamic equation in one spatial dimension. The... more We study the dispersive properties of a linear peridynamic equation in one spatial dimension. The interplay between nonlocality and dispersion is analyzed in detail through the study of the asymptotics at low and high frequencies, revealing new features ruling the wave propagation in continua where nonlocal characteristics must be taken into account. Global dispersive estimates and existence of conserved functionals are proved. A comparison between these new effects and the classical local scenario is deepen also through a numerical analysis.

Applied Mathematics Letters

This note provides a variational description of the mechanical effects of flexural stiffening of ... more This note provides a variational description of the mechanical effects of flexural stiffening of a 2D plate glued to an elastic-brittle or an elastic-plastic reinforcement. The reinforcement is assumed to be linear elastic outside possible free plastic yield lines or free crack. Explicit Euler equations and a compliance identity are shown for the reinforcement of a 1D beam.

The increasing rate of production and diffusion of photovoltaic (PV) technologies for industrial ... more The increasing rate of production and diffusion of photovoltaic (PV) technologies for industrial and domestic applications urges improvement of the sustainability of their demanufacturing processes in order to reduce the amount of electronic wastes. Sustainability of demanufacturing processes concerns the reduction of energy consumption, the reduction of polluting substances as well as the reduction of the effort spent in recovery of the components. There is not an optimal process so far, provided a number of different approaches have been devised (see e.g. [1, 2, 3]). A promising choice relies on the use of thermo-mechanical treatments for inducing a delamination process where interfacial bonding between layers are weakened and, finally, broken inducing separation of the layers [4]. In this paper we present a preliminary industrialization study, based on Finite Element (FE) Analysis, to prove the validity of the new sustainable demanufacturing process endeavouring the delamination ...

We study the dispersive properties of a linear peridynamic equation in one spatial dimension. The... more We study the dispersive properties of a linear peridynamic equation in one spatial dimension. The interplay between nonlocality and dispersion is analyzed in detail through the study of the asymptotics at low and high frequencies, revealing new features ruling the wave propagation in continua where nonlocal characteristics must be taken into account. Global dispersive estimates and existence of conserved functionals are proved. A comparison between these new effects and the classical local scenario is deepen also through a numerical analysis. Introduction A fundamental trait in the mathematical modeling of continuum physics relies in capturing the essential phenomena of a complex problem still keeping the technical difficulties as manageable as possible. A typical example of this strategy is provided by classical linear elasticity ([10]) which, in spite of its very long-lasting tradition, represents yet an unavoidable comparison term even for the more recent mechanical theories aime...

In this paper we analyze and compare two different models for adhesion phenomena, recently propos... more In this paper we analyze and compare two different models for adhesion phenomena, recently proposed by the authors. In the first approach [9] a feasible expression of the adhesion energy is suggested by the existence problem of partially detached equilibrium states. In the second model [10] the macroscopic energy is obtained by performing a multiscale analysis and it is deduced via a macroscopic Γ-limit of the energy at the scale of the microstructure. Interestingly, we obtain that the first model can be deduced by the second one as the limit case when the parameter measuring the relative stiffness of the adhesive layer and the beam diverges.

Journal of Peridynamics and Nonlocal Modeling

In this paper, we investigate, through numerical studies, the dynamical evolutions encoded in a l... more In this paper, we investigate, through numerical studies, the dynamical evolutions encoded in a linear one-dimensional nonlocal equation arising in peridynamics. The different propagation regimes ranging from the hyperbolic to the dispersive, induced by the nonlocal feature of the equation, are carefully analyzed. The study of an initial value Riemann-like problem suggests the formation of a singularity.

Milan Journal of Mathematics

The paper studies the initial boundary value problem related to the dynamic evolution of an elast... more The paper studies the initial boundary value problem related to the dynamic evolution of an elastic beam interacting with a substrate through an elastic-breakable forcing term. This discontinuous interaction is aimed to model the phenomenon of attachment-detachment of the beam occurring in adhesion phenomena. We prove existence of solutions in energy space and exhibit various counterexamples to uniqueness. Furthermore we characterize some relevant features of the solutions, ruling the main effects of the nonlinearity due to the elastic-breakable term on the dynamical evolution, by proving the linearization property according to Gérard (J Funct Anal 141(1):60–98, 1996) and an asymptotic result pertaining the long time behavior.

Calculus of Variations and Partial Differential Equations

We study an old variational problem formulated by Euler as Proposition 53 of his Scientia Navalis... more We study an old variational problem formulated by Euler as Proposition 53 of his Scientia Navalis by means of the direct method of the calculus of variations. Precisely, through relaxation arguments, we prove the existence of minimizers. We fully investigate the analytical structure of the minimizers in dependence of the geometric parameters and we identify the ranges of uniqueness and non-uniqueness.