Variational principles in numerical practice

Variational principles represent a general framework for determining the mechanical state of a system, by identifying its motion as a minimum of a pertinent functional. Moreover, finite element methods are naturally based on variational principles and provide a very powerful tool for numerically solving many mechanical as well as other multi-physics problems. The purpose of the present note is to illustrate some recent applications with special reference to biomechanics and dissipation in quasi-brittle materials and piezo-electromechanical structures, in order to confirm the validation and to highlight the bright prospects of this method

Dynamic problems for metamaterials: Review of existing models and ideas for further research

Metamaterials are materials especially engineered to have a peculiar physical behaviour, to be exploited for some well-specified technological application. In this context we focus on the conception of general micro-structured continua, with particular attention to piezoelectromechanical structures, having a strong coupling between macroscopic motion and some internal degrees of freedom, which may be electric or, more generally, related to some micro-motion. An interesting class of problems in this context regards the design of wave-guides aimed to control wave propagation. The description of the state of the art is followed by some hints addressed to describe some possible research developments and in particular to design optimal design techniques for bone reconstruction or systems which may block wave propagation in some frequency ranges, in both linear and non-linear fields. (C) 2014 Elsevier Ltd. All rights reserved

Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model

A linear elastic second gradient orthotropic two-dimensional solid that is invariant under (Formula presented.) rotation and for mirror transformation is considered. Such anisotropy is the most general for pantographic structures that are composed of two identical orthogonal families of fibers. It is well known in the literature that the corresponding strain energy depends on nine constitutive parameters: three parameters related to the first gradient part of the strain energy and six parameters related to the second gradient part of the strain energy. In this paper, analytical solutions for simple problems, which are here referred to the heavy sheet, to the nonconventional bending, and to the trapezoidal cases, are developed and presented. On the basis of such analytical solutions, gedanken experiments were developed in such a way that the whole set of the nine constitutive parameters is completely characterized in terms of the materials that the fibers are made of (i.e., of the Young’s modulus of the fiber materials), of their cross sections (i.e., of the area and of the moment of inertia of the fiber cross sections), and of the distance between the nearest pivots. On the basis of these considerations, a remarkable form of the strain energy is derived in terms of the displacement fields that closely resembles the strain energy of simple Euler beams. Numerical simulations confirm the validity of the presented results. Classic bone-shaped deformations are derived in standard bias numerical tests and the presence of a floppy mode is also made numerically evident in the present continuum model. Finally, we also show that the largeness of the boundary layer depends on the moment of inertia of the fibers

Virtual spring damper method for nonholonomic robotic swarm self-organization and leader following

In this paper, we demonstrate a method for self-organization and leader following of nonholonomic robotic swarm based on spring damper mesh. By self-organization of swarm robots we mean the emergence of order in a swarm as the result of interactions among the single robots. In other words the self-organization of swarm robots mimics some natural behavior of social animals like ants among others. The dynamics of two-wheel robot is derived, and a relation between virtual forces and robot control inputs is defined in order to establish stable swarm formation. Two cases of swarm control are analyzed. In the first case the swarm cohesion is achieved by virtual spring damper mesh connecting nearest neighboring robots without designated leader. In the second case we introduce a swarm leader interacting with nearest and second neighbors allowing the swarm to follow the leader. The paper ends with numeric simulation for performance evaluation of the proposed control method

Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycloidal metamaterial

AbstractA nonlinear elastic model for nets made up of two families of curved fibers is proposed. The net is planar prior to the deformation, but the equilibrium configuration that minimizes the total potential energy can be a surface in the three-dimensional space. This elastic surface accounts for the stretching, bending, and torsion of the constituent fibers regarded as a continuous distribution of Kirchhoff rods. A specific example of fiber arrangement, namely a cycloidal orthogonal pattern, is examined to illustrate the predictive abilities of the model and assess the limit of applicability of it. A numerical micro–macro-identification is performed with a model adopting a standard continuum deformable body at the level of scale of the fibers. A few finite element simulations are carried out for comparison purposes in statics and dynamics, performing modal analysis. Finally, a topology optimization problem has been carried out to change the macroscopic shear stiffness to enlarge the elastic regime and reduce the risk of damage without excessively losing bearing capacity

King post truss as a motif for internal structure of (meta)material with controlled elastic properties

One of the most interesting challenges in the modern theory of materials consists in the determination of those microstructures which produce, at the macro-level, a class of metamaterials whose elastic range is many orders of magnitude wider than the one exhibited by ‘standard’ materials. In Dell’Isola et al. (2015 Zeitschrift für angewandte Mathematik und Physik 66, 3473- 3498. (doi: 10.1007/s00033-015-0556-4)), it was proved that, with a pantographic microstructure constituted by ‘long’ microbeams it is possible to obtainmetamaterials whose elastic range spans up to an elongation exceeding 30%. In this paper, we demonstrate that the same behaviour can be obtained bymeans of an internal microstructure based on a king post motif. This solution shows many advantages: it involves only microbeams; all constituting beams are undergoing only extension or compression; all internal constraints are terminal pivots. While the elastic deformation energy can be determined as easily as in the case of long-beam microstructure, the proposed design seems to have obvious remarkable advantages: it seems to be more damage resistant and therefore to be able to have a wider elastic range; it can be realized with the same three-dimensional printing technology; it seems to be less subject to compression buckling. The analysis which we present here includes: (i) the determination of Hencky-type discrete models for king post trusses, (ii) the application of an effective integration scheme to a class of relevant deformation tests for the proposed metamaterial and (iii) the numerical determination of an equivalent second gradient continuum model. The numerical tools which we have developed and which are presented here can be readily used to develop an extensive measurement campaign for the proposed metamaterial

Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: Current and upcoming applications

This review paper intends to gather and organize a series of works which discuss the possibility of exploiting the mechanical properties of distributed arrays of piezoelectric transducers. The concept can be described as follows: on every structural member one can uniformly distribute an array of piezoelectric transducers whose electric terminals are to be connected to a suitably optimized electric waveguide. If the aim of such a modification is identified to be the suppression of mechanical vibrations then the optimal electric waveguide is identified to be the 'electric analog' of the considered structural member. The obtained electromechanical systems were called PEM (PiezoElectroMechanical) structures. The authors especially focus on the role played by Lagrange methods in the design of these analog circuits and in the study of PEM structures and we suggest some possible research developments in the conception of new devices, in their study and in their technological application. Other potential uses of PEMs, such as Structural Health Monitoring and Energy Harvesting, are described as well. PEM structures can be regarded as a particular kind of smart materials, i.e. materials especially designed and engineered to show a specific andwell-defined response to external excitations: for this reason, the authors try to find connection between PEM beams and plates and some micromorphic materials whose properties as carriers of waves have been studied recently. Finally, this paper aims to establish some links among some concepts which are used in different cultural groups, as smart structure, metamaterial and functional structural modifications, showing how appropriate would be to avoid the use of different names for similar concepts. © 2015 - IOS Press and the authors

The influence of different geometries of matrix/scaffold on the response of a bone and resorbable material mixture with voids

A 2-D dimensional sample made of natural bone tissue and artificial bioresorbable material is numerically investigated in order to study the influence of different geometries of the assemblage of matrix and scaffold. With the specific tools of the Mixture theory we consider the solid matrix with evolving apparent mass densities (rb for bone and rm for material) to describe bone tissue synthesis and resorption when a bio-resorbable material of the kind used in bone reconstruction is present (see e.g. [1, 2, 3]). To take porosity effects in account the adopted model is derived from the Nunziato-Cowin theory developed for porous solids in which the matrix material is linearly elastic and the interstices are void of material. In detail, to describe the mechanical phenomena which influence the porosity variation we introduce, following the aforementioned theory [4], an independent kinematic degree of freedom, namely the change in volume fraction from the reference volume fraction, namely x = (rb+rm)/rMax-(rb R+rmR)/rMax, with rMax the maximal density without pores. It is well established that exercise results in increased bone mass, while unloading due to immobilization, bedrest, and weightlessness results in bone atrophy. The strains induced by external loads are sensed by mechanoreceptors, primarily on osteocytes which essentially transduce the mechanical signals into biological signals. These biological signals are able to trigger bone remodeling by directing osteoblast activity and osteoclastic resorption. [1] T. Lekszycki and F. dell’Isola. A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. J. Applied Math and Mech. (ZAMM), 92:426–444, 2012. [2] A. Madeo, T. Lekszycki, and F. dell’Isola. A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery. Comptes Rendus Mécanique, 339:625–640, 2011. [3] A. Madeo, D. George, T. Lekszycki, M. Nierenberger, and Y. Rémond. A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodeling. Comptes Rendus M´ecanique, 340:575–589, 2012. [4] S. C. Cowin and J. W. Nunziato. Linear elastic materials with voids. J. Elasticity, 13:125–147, 1983

Braid Groups on Triangulated Surfaces and Singular Homology

Let $\Sigma_g$ denote the closed orientable surface of genus $g$ and fix an arbitrary simplicial triangulation of $\Sigma_g$. We construct and study a natural surjective group homomorphism from the surface braid group on $n$ strands on $\Sigma_g$ to the first singular homology group of $\Sigma_g$ with integral coefficients. In particular, we show that the kernel of this homomorphism is generated by canonical braids which arise from the triangulation of $\Sigma_g$. This provides a simple description of natural subgroups of surface braid groups which are closely tied to the homology groups of the surfaces $\Sigma_g$

Edge effects in Hypar nets

Edge effects in hyperbolic paraboloidal nets are analyzed using a model that features elastic resistance of the fibers of the net to flexure and twist in addition to the extensional elasticity of the conventional membrane theory of networks.; Les effets de bord dans les réseaux paraboloïdaux hyperboliques sont analysés à l'aide d'un modèle présentant la résistancé elastique des fibres du réseaù a la flexion et à la torsion, en plus de l'élasticité en extension de la théorie conventionnelle des membranes pour les réseaux