consequences of different definitions of bending curvature on nonlinear dynamics of beams

Abstract Beam theories may be grouped in two broad categories, namely induced or intrinsic theories. In the former, beam models are obtained as exact consequences of three-dimensional theory, making use either of asymptotic expansions in a slenderness parameter or projections of three-dimensional elasticity on certain function spaces, while beams are inherently one-dimensional bodies in the latter category. Although induced theories show a clear connection between three- and one-dimensional representations, they are often more demanding with respect to intrinsic ones, in which a finite number of strain parameters, depending on just one space variable, characterizes the motion of beams in an internally consistent way and without a direct linkage to three-dimensional material properties. Hence, as a consequence, intrinsic theories do not provide any structure for constitutive equations and, at least in principle, different choices can be allowed. A typical example of this fact is represented by the one-dimensional relationship between the bending moment and the beam curvature, since for this latter two notions are admissible. Indeed, both are adopted in the literature and, apparently, preferring one to the other is only related to the predictive capability of the ensuing model. The arising question is about possible differences in both static and dynamic responses of beams, when one or the other definition of curvature is selected

Moving mass over a viscoelastic system: asymptotic behaviours and insights into nonlinear dynamics

Moving masses are of interest in many applications of structural dynamics, soliciting in the last decades a vast debate in the scientific literature. However, despite the attention devoted to the subject, to the best of the authors’ knowledge, there is a lack of analysis about the fate of a movable mass when it rolls or slips with friction on a structure. With the aim of elucidating the dynamics of the simplest paradigm of this system and to investigate its asymptotic response, we make reference to a two-degree-of-freedom model made of an elastically vibrating carriage surmounted by a spherical mass, facing the problem both theoretically and experimentally. In case of linear systems, the analytical solutions and the laboratory tests performed on ad hoc constructed prototypes highlighted a counterintuitive asymptotic dynamics, here called binary: in the absence of friction at the interface of the bodies’ system, the mass holds its initial position or, if nonzero damping acts, at the end of the motion it is in a position that exactly recovers the initial relative distance carriage–sphere. While the first result might be somewhat obvious, the second appears rather surprising. Such a binary behaviour is also confirmed for a Duffing-like system, equipped with cubic springs, while it can be lost when non-smooth friction phenomena occur, as well as in the case of elastic springs restraining the motion of the sphere. The obtained analytical results and the numerical findings, also confirmed by experimental evidences, contribute to the basic understanding of the role played by the damping parameters governing the systems’ dynamics with respect to its asymptotic behaviour and could pave the way for designing active or passive vibration controllers of interest in engineering

Numerical experiments in 2D variational fracture

In the present work we present some results of numerical experiments obtained with a variationalmodel for quasi-static Griffith-type brittle fracture. Essentially the analysis is based on a recent formulation byFrancfort and Marigo the main difference being the fact that we rely on local rather than on globalminimization. Propagation of fracture is obtained by minimizing, in a step by step process, a form of energythat is the sum of bulk and interface terms. To solve the problem numerically we adopt discontinuous finiteelements based on variable meshes and search for the minima of the energy through descent methods. We use asort of mesh dependent relaxation of the interface energy to get out of small energy wells. The relaxationconsists in the adoption of a carefully tailored cohesive type interface energy, tending to the Griffith limit as themesh size tends to zero

Numerical experiments in 2D variational fracture

In the present work we present some results of numerical experiments obtained with a variationalmodel for quasi-static Griffith-type brittle fracture. Essentially the analysis is based on a recent formulation byFrancfort and Marigo the main difference being the fact that we rely on local rather than on globalminimization. Propagation of fracture is obtained by minimizing, in a step by step process, a form of energythat is the sum of bulk and interface terms. To solve the problem numerically we adopt discontinuous finiteelements based on variable meshes and search for the minima of the energy through descent methods. We use asort of mesh dependent relaxation of the interface energy to get out of small energy wells. The relaxationconsists in the adoption of a carefully tailored cohesive type interface energy, tending to the Griffith limit as themesh size tends to zero

Computational Modeling of The Mechanics of Energy Harvesters Based On Tensegrity Solar Façades

This work is focused on the computational design of tensegrity shading systems of energy efficient buildings which mitigate air conditioning consumption and optimize the energy performance of the building. It is concerned with the design of active solar façade screens based on lightweight tensegrity units, which are easily integrated with energy harvesting piezoelectric cables. The unit cells of the analyzed screens are controlled by tensioning and releasing selected cables of the structure. A numerical procedure simulates the dynamics of the analyzed tensegrity façades, by considering the opening and closure motions of the screens, and the vibrations produced by the action of dynamic wind forces. The energy harvesting ability of the proposed façade is numerically estimated

Numerical applications of free discontinuity problems to folding and fracture

The aim of the thesis is the application of new numerical methods based on the theory of free discontinuities of E. De Giorgi to challenging fields of relevant practical interest in engineering, such as folding of thin walled tubes and propagation of fracture in brittle solids. Both the mathematical models to describe folding and fracture are based on the minimization of two competing energy forms: a volume and a surface energy. The variational formulation leads to the minimization (under displacement boundary conditions) of a functional F(K;u), where K is the set of discontinuous points of u. In order to perform the numerical search for the minimum of F, a powerful open code (developed by K. Brakke) named Surface Evolver has been adapted according to the purposes of the research at hand. Since the energy is non-convex the solution obtained through the algorithm is strongly dependent on the initial point. Starting from an initial faceted surface, the numerical code evolves the surface towards minimum energy through the nonlinear conjiugate gradient method