Papers by Antonio Battista
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An analysis of nonlinear thin structures
The main theme of this thesis is the study of the mechanical behavior of thin slender structures ... more The main theme of this thesis is the study of the mechanical behavior of thin slender structures in the nonlinear domain. This thesis work is presented in the form of a collection of articles published during the Ph.D. and is divided into two parts. The first part deals with the analysis of nonlinear models of inextensible and extensible beams, generalizing on different aspects the beam models of Euler and Timoshenko. A theoretical study of the existence and uniqueness of solutions is completed by numerical simulations highlighting the existence of multiple solutions with the increase of the applied force. A numerical study of the multiplicity of solutions of an extensible beam model in large displacements is also carried out. The second part concerns the formal justification by asymptotic methods of an original membrane model presenting a multiplicity of solutions for particular loads, able to model the wrinkling of some very thin structures with a shear stress applied.
In this paper a 2D continuum model, thought as the homogenized limit of a microstructured pantogr... more In this paper a 2D continuum model, thought as the homogenized limit of a microstructured pantographic sheet, is studied. The microstructure is characterized by two families of parallel fibers, whose deformation measures account for bending, elongation and relative rotation of the fibers. The deformation energy density of the homogenized model depends on both first and second gradients of the displacement. Modal analysis is performed in order to assess the peculiarities of the dynamic behavior of higher gradient models, and in particular the difference, with respect to classical laminae, in the dependence of the eigenfrequencies on the stiffness.
Mechanics Research Communications, 2018
In this paper, we present a discrete system characterized by a geometrically-driven evolution law... more In this paper, we present a discrete system characterized by a geometrically-driven evolution law and very light computational costs. Potential applications to the simulation of the nonlinear deformation of extensible strings with null flexural stiffness, as well as to dynamics of granular media, are suggested.
Continuum Mechanics and Thermodynamics, 2018
In this work, a formal deduction of a two-dimensional membrane theory, similar to Landau-Lifshitz... more In this work, a formal deduction of a two-dimensional membrane theory, similar to Landau-Lifshitz model, is performed via an asymptotic development of the weak formulation of the three-dimensional equations of elasticity. Some interesting aspects of the deduced model are investigated, in particular the property of obtaining a hyperbolic equation for the out-of-plane displacement under a certain class of boundary conditions and loads. Some simple cases are analyzed to show the relevant aspects of the model and the phenomenology that can be addressed. In particular, it is shown how this mathematical formulation is capable to describe instabilities well known as wrinkling, often observed for the buckling of very thin membranes. Keywords Asymptotic methods • Dimensional analysis • Plate theory • Membrane theory • Hyperbolic problem Communicated by Francesco dell'Isola.
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2019
Nonlinear beam, Hencky bar-chain, discrete modelling MSC (2010) 00-xx "Frustra fit per plura quod... more Nonlinear beam, Hencky bar-chain, discrete modelling MSC (2010) 00-xx "Frustra fit per plura quod fieri potest per pauciora." (It is pointless to do with more what can be done with fewer.) In Summa Totius Logicae by William of Ockham It has been numerically observed and mathematically proven that for a clamped Euler's Elastica, which is uniformly loaded, there exist, in large deformations, some 'undocumented' equilibrium configurations which resemble a curled pending wire. Even if Elastica is one of the most studied model in mathematical physics, we could not find in the literature any description of an equilibrium like the one whose existence was forecast theoretically in [36]. In this paper, we prove that this kind of equilibrium configurations can be actually observed experimentally when using 'soft' beams. We mean with soft beams: Elasticae whose ratio between the applied load intensity and the bending stiffness is large enough. Moreover, we prove experimentally that such equilibrium configurations are actually stable, by observing their oscillations around the considered nonstandard equilibrium configuration. To describe theoretically such oscillations we consider, instead of a 'soft' Elastica model, directly a Hencky-type discrete model, i.e. a 'masses-springs' finite dimensional Lagrangian model. In this way we formulate, avoiding the use of an intermediate continuum model, a model for which numerical simulations can be performed without the introduction of any further discretization. In this way, we can also predict quantitatively the motions of soft beams, in the regime of large displacements and deformations. Postponing to future investigations more careful quantitative measurements, we report here that it was possible to get a rather promising qualitative agreement between observed motions and predictive numerical simulations.
Zeitschrift für angewandte Mathematik und Physik, 2019
In this paper, the general equilibrium equations for a geometrically nonlinear version of the Tim... more In this paper, the general equilibrium equations for a geometrically nonlinear version of the Timoshenko beam are derived from the energy functional. The particular case in which the shear and extensional stiffnesses are infinite, which correspond to the inextensible Euler beam model, is studied under a uniformly distributed load. All the global and local minimizers of the variational problem are characterized, and the relative monotonicity and regularity properties are established.
Zeitschrift für angewandte Mathematik und Physik, 2018
In the present paper we study a natural nonlinear generalization of Timoshenko beam model and sho... more In the present paper we study a natural nonlinear generalization of Timoshenko beam model and show that it can describe the homogenized deformation energy of a 1D continuum with a simple microstructure. We prove the well-posedness of the corresponding variational problem in case of a generic end load, discuss some regularity issues and evaluate the critical load. Moreover, we generalize the model so as to include an additional rotational spring in the microstructure. Finally, some numerical simulations are presented and discussed.
Continuum Mechanics and Thermodynamics, 2018
In this paper we account for the research eorts that have been started, for some among us, alread... more In this paper we account for the research eorts 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 rst stage of these eorts, 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 eorts i) in designing pantographic meta-materials, ii) in exploiting the modern technology of rapid prototyping, and iii) in the mechanical testing of many real prototypes. Among the key ndings that have been obtained there are the following ones: pantographic meta-materials i) undergo very large deformations while remaining in the elastic regime, ii) are very tough in resisting to damage phenomena, iii) exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties, iv) have superior strength to weight ratio, v) have predictable damage behavior, and vi) possess physical properties that are critically dictated by their geometry at the microlevel.
Frequency Shifts Induced by Large Deformations in Planar Pantographic Continua
Nanomechanics Science and Technology: An International Journal, 2015
The present article deals with the dynamic behavior of 2D continua representing the homogenized l... more The present article deals with the dynamic behavior of 2D continua representing the homogenized limit of microstructured pantographic systems, i.e., the structures in which two orders of fibers are interconnected by means of pivots. The strain energy density of the continuum model depends on the first and second gradient of the displacement. Numerical simulations on modal analysis are performed and the results are compared with the behavior of an orthotropic lamina of equal mass density. The results show the characteristic features of the dynamic behavior of higher gradient models, and in particular the difference, with respect to classical laminae, in the dependence of the eigenfrequencies on the stiffness
An Inverse Method to Get Further Analytical Solutions for a Class of Metamaterials Aimed to Validate Numerical Integrations
Advanced Structured Materials, 2017
We consider an isotropic second gradient elastic two-dimensional solid. Besides, we relax the iso... more We consider an isotropic second gradient elastic two-dimensional solid. Besides, we relax the isotropic hypothesis and consider a D4 orthotropic material. The reason for this last choice is that such anisotropy is the most general for pantographic structures, which exhibit attracting mechanical properties. In this paper we analyze the role of the external body double force \(m^{ext}\) on the partial differential equations and we subsequently revisit some analytical solutions that have been considered in the literature for identification purposes. The revisited analytical solutions will be employed as well for identification purposes in a further contribution.
Advanced Structured Materials, 2017
In this paper we use a simple discrete system in order to model deformation and fracture within t... more In this paper we use a simple discrete system in order to model deformation and fracture within the same theoretical and numerical framework. The model displays a rich behavior, accounting for different fracture phenomena, and in particular for crack formation and growth. A comparison with standard Finite Element simulations and with the basic Griffith theory of fracture is provided. Moreover, an 'almost steady' state, i.e. a long apparent equilibrium followed by an abrupt crack growth, is obtained by suitably parameterizing the system. The model can be easily generalized to higher order interactions corresponding, in the homogenized limit, to higher gradient continuum theories.
Zeitschrift für angewandte Mathematik und Physik, 2017
The present paper deals with the rigorous homogenization of a discrete system consisting of exten... more The present paper deals with the rigorous homogenization of a discrete system consisting of extensible rods linked by rotational springs. Specifically, a Γ-convergence result is proven for a sequence of discrete measure functionals En, describing the energy of the discrete system, toward the continuous energy functional for the extensible Euler beam model (Elastica) in large deformation regime. A relative compactness result for the sequence En is also proven. Moreover, numerical results are shown on the deformed shape and on the total energy of the system when the number of elements of the discrete system increases. The numerical convergence of the energy to a definite value is shown in two cases. The results provide rigorous justification of a very commonly used algorithm for the discretization of the extensible Euler beam, namely Hencky-type beam model.
In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions
Mathematics and Mechanics of Solids, 2016
The aim of the present paper is the analysis of a two-dimensional continuum with two families of ... more The aim of the present paper is the analysis of a two-dimensional continuum with two families of inextensible fibers that are orthogonal in the initial configuration. In the first part of the work, a new formulation is presented, in which the problem is reduced to a standard nonlinear constrained minimization, while in the second part of the work several numerical investigations are presented considering different boundary conditions with respect to standard symmetric bias extensional tests. The conceptual framework can be recognized in the researches by Pipkin and Rivlin on inextensible nets. Furthermore, an implicit version of the Rivlin representation of the generic placement for a two-dimensional sheet with two families of inextensible fibers is provided by considering the angles of the fiber directors as degrees of the freedom of the formulation. In this way the first gradient formulation is given in terms of two angle fields only.
Mathematics and Mechanics of Solids, 2016
A discrete system constituted of particles interacting by means of a centroid-based law is numeri... more A discrete system constituted of particles interacting by means of a centroid-based law is numerically investigated. The elements of the system move in the plane, and the range of the interaction can be varied from a more local form (firstneighbours interaction) up to a generalized nth order interaction. The aim of the model is to reproduce the behaviour of deformable bodies with standard (Cauchy model) or generalized (second gradient) deformation energy density. The numerical results suggest that the considered discrete system can effectively reproduce the behaviour of first and second gradient continua. Moreover, a fracture algorithm is introduced and some comparison between firstand secondneighbour simulations are provided.
Advanced Structured Materials, 2016
The new possibilities arisen in the last years in material manufacturing (3D-printing, electrospi... more The new possibilities arisen in the last years in material manufacturing (3D-printing, electrospinning, roll-to-roll processing, self-assembly, etc.) and the theoretical tools made available by generalized continuum mechanics are still far from achieving their full potential. The main thesis of the present paper is that it is necessary a multidisciplinary approach to address the emerging issues in metamaterials' design. Therefore, an improvement in the degree and the depth of the cooperation between scientists from different areas is required. The advancements needed in mechanics and physics of solids and fluids, mathematical and numerical modeling and advanced technology in material construction can be obtained only as a consequence of a synergic effort.
The mathematical modeling for bulk acoustic wave propagation velocities in transversely isotropic piezoelectric materials
Mathematics and Mechanics of Solids, 2015
The objective of this paper is to study the bulk acoustic wave (BAW) propagation velocities in tr... more The objective of this paper is to study the bulk acoustic wave (BAW) propagation velocities in transversely isotropic piezoelectric materials, aluminum nitride, zinc oxide, cadmium sulfide and cadmium selenide. The bulk acoustic wave velocities are computed for each direction by solving the Christoffel’s equation based on the theory of acoustic waves in anisotropic solids exhibiting piezoelectricity. These values are calculated numerically and implemented on a computer by Bisection Method Iterations Technique (BMIT). The modification of the bulk acoustic wave velocities caused by the piezoelectric effect are graphically compared with the velocities in the corresponding non-piezoelectric materials. The results obtained in this study can be applied to signal processing, sound systems and wireless communication in addition to the improvement of surface acoustic wave (SAW) devices and military defense equipment.
International Journal of Non-Linear Mechanics, 2016
The main theme of this thesis is the study of the mechanical behavior of thin slender structures ... more The main theme of this thesis is the study of the mechanical behavior of thin slender structures in the nonlinear domain. This thesis work is presented in the form of a collection of articles published during the Ph.D. and is divided into two parts. The first part deals with the analysis of nonlinear models of inextensible and extensible beams, generalizing on different aspects the beam models of Euler and Timoshenko. A theoretical study of the existence and uniqueness of solutions is completed by numerical simulations highlighting the existence of multiple solutions with the increase of the applied force. A numerical study of the multiplicity of solutions of an extensible beam model in large displacements is also carried out. The second part concerns the formal justification by asymptotic methods of an original membrane model presenting a multiplicity of solutions for particular loads, able to model the wrinkling of some very thin structures with a shear stress applied.Le thème pri...
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Papers by Antonio Battista