Massimiliano Zingales

Exact mechanical models of fractional hereditary materials

Fractional Viscoelasticity is referred to materials, whose constitutive law involves fractional derivatives of order β∈R such that 0≤β≤1. In this paper, two mechanical models with stress-strain relation exactly restituting fractional operators, respectively, in ranges 0≤β≤1/2 and 1/2≤β≤1 are presented. It is shown that, in the former case, the mechanical model is described by an ideal indefinite massless viscous fluid resting on a bed of independent springs (Winkler model), while, in the latter case it is a shear-type indefinite cantilever resting on a bed of independent viscous dashpots. The law of variation of all mechanical characteristics is of power-law type, strictly related to the order of the fractional derivative. Because the critical value 1/2 separates two different behaviors with different mechanical models, we propose to distinguish such different behavior as elasto-viscous case with 0≤β≤1/2 and visco-elastic case for 1/2≤β≤1. The motivations for this different definitions as well as the comp...

Fractional differential equations and related exact mechanical models

The aim of the paper is the description of fractional-order differential equations in terms of exact mechanical models. This result will be archived, in the paper, for the case of linear multiphase fractional hereditariness involving linear combinations of power-laws in relaxation/creep functions. The mechanical model corresponding to fractional-order differential equations is the extension of a recently introduced exact mechanical representation (Di Paola and Zingales (2012) [33] and Di Paola et al. (2012) [34]) of fractional-order integrals and derivatives. Some numerical applications have been reported in the paper to assess the capabilities of the model in terms of a peculiar arrangement of linear springs and dashpots.

Power-law hereditariness of hierarchical fractal bones

In this paper, the authors introduce a hierarchic fractal model to describe bone hereditariness. Indeed, experimental data of stress relaxation or creep functions obtained by compressive/tensile tests have been proved to be fit by power law with real exponent 0 ⩽ β ⩽1. The rheological behavior of the material has therefore been obtained, using the Boltzmann-Volterra superposition principle, in terms of real order integrals and derivatives (fractional-order calculus). It is shown that the power laws describing creep/relaxation of bone tissue may be obtained by introducing a fractal description of bone cross-section, and the Hausdorff dimension of the fractal geometry is then related to the exponent of the power law.

The mechanically based non-local elasticity: an overview of main results and future challenges

The mechanically based non-local elasticity has been used, recently, in wider and wider engineering applications involving small-size devices and/or materials with marked microstructures. The key feature of the model involves the presence of non-local effects as additional body forces acting on material masses and depending on their relative displacements. An overview of the main results of the theory is reported in this paper.

Digital simulation of multivariate earthquake ground motions

In this paper a new generation procedure of multivariate earthquake ground motion is presented. The technique takes full advantage of the decomposition of the power spectral density matrix by means of its eigenvectors. The application of the method to multivariate ground accelerations shows some very interesting physical properties which allows one to obtain significant reduction of the computational effort in the generation of sample functions relative to multivariate earthquake ground motion processes. Copyright

Fractional calculus in solid mechanics: local versus non-local approach

Several enriched continuum mechanics theories have been proposed by the scientific community in order to develop models capable of describing microstructural effects. The aim of the present paper is to revisit and compare two of these models, whose common denominator is the use of fractional calculus operators. The former was proposed to investigate damage in materials exhibiting a fractal-like microstructure. It makes use of the local fractional derivative, which turns out to be a powerful tool to describe irregular patterns such as strain localization in heterogeneous materials. On the other hand, the latter is a non-local approach that models long-range interactions between particles by means of the Marchaud fractional derivative. Analogies and differences between the two models are outlined and discussed. PACS number: 62.20D

Fractional mechanical model for the dynamics of non-local continuum

In this chapter, fractional calculus has been used to account for long-range interactions between material particles. Cohesive forces have been assumed decaying with inverse power law of the absolute distance that yields, as limiting case, an ordinary, fractional differential equation. It is shown that the proposed mathematical formulation is related to a discrete, point-spring model that includes non-local interactions by non-adjacent particles with linear springs with distance-decaying stiffness. Boundary conditions associated to the model coalesce with the well-known kinematic and static constraints and they do not run into divergent behavior. Dynamic analysis has been conducted and both model shapes and natural frequency of the non-local systems are then studied.

Finite-Element Formulation of a Nonlocal Hereditary Fractional-Order Timoshenko Beam

AbstractA mechanically-based nonlocal Timoshenko beam model, recently proposed by the authors, hinges on the assumption that nonlocal effects can be modeled as elastic long-range volume forces and moments mutually exerted by nonadjacent beam segments, which contribute to the equilibrium of any beam segment along with the classical local stress resultants. Long-range volume forces/moments linearly depend on the product of the volumes of the interacting beam segments, and on pure deformation modes of the beam, through attenuation functions governing the space decay of nonlocal effects. This paper investigates the response of this nonlocal beam model when viscoelastic long-range interactions are included, modeled by Caputo’s fractional derivatives. The finite-element method is used to discretize the pertinent fractional-order equations of motion. Closed-form solutions are obtained for creep tests by typical tools of fractional calculus. Numerical results are presented for various nonlocal parameters.

Contrasting probabilistic and anti-optimization approaches in an applied mechanics problem

Probabilistic and non-probabilistic, anti-optimization analyses of uncertainty are contrasted in this study. Specifically, the comparison of these two competing approaches is conducted for an uniform column, with initial geometric imperfection, subjected to an impact axial load. The reliability of the column is derived for the cases when the initial imperfections posses either (a) uniform probability density, (b) truncated exponential density or (c) generic truncated probability density. The problem is also analyzed in the context of an interval analysis. It is shown that in, the most important near-unity reliability range these two approaches tend to each other. Since the interval analysis constitutes a much simpler procedure than the probabilistic approach, it is argued that the former is advantageous over the latter in some circumstances.

A viscoelastic model for the long-term deflection of segmental prestressed box girders

Most of segmental prestressed concrete box girders exhibit excessive multidecade deflections unforeseeable by past and current design codes. To investigate such a behavior, mainly caused by creep and shrinkage phenomena, an effective finite element (FE) formulation is presented in this article. This formulation is developed by invoking the stationarity of an energetic principle for linear viscoelastic problems and relies on the Bazant creep constitutive law. A case study representative of segmental prestressed concrete box girders susceptible to creep is also analyzed in the article, that is, the Colle Isarco viaduct. Its FE model, based on the aforementioned energetic formulation, was successfully validated through the comparison with monitoring field data. As a result, the proposed 1D FE model can effectively reproduce the past behavior of the viaduct and predict its future behavior with a reasonable run time, which represents a decisive factor for the model implementation in a decision support system.