Fractional differential equations and related exact mechanical models

Paola, Mario Di; Pinnola, Francesco Paolo; Zingales, Massimiliano

doi:10.1016/j.camwa.2013.03.012

Cited by 82 publications

(39 citation statements)

References 28 publications

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“…in [21] it is proposed that the damper force depends not from the first derivate of the displacement, that is the velocity │ │, but from the fractional derivative of the displacement; the order of derivation is comprised between 0 and 1. This approach derives from the recent studies about visco-elasticity that use the fractional calculus to represent the behaviour of viscoelastic systems in general.…”

Section: Model For Fluid Viscous Dampersmentioning

confidence: 99%

Solutions for the Design and Increasing of Efficiency of Viscous Dampers

Alotta

Cavaleri

Paola

et al. 2016

TOBCTJ

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Abstract:In last decades many strategies for seismic vulnerability mitigation of structures have been studied and experimented; in particular energy dissipation by external devices assumes a great importance for the relative simplicity and efficacy. Among all possible approaches the use of fluid viscous dampers are very interesting, because of their velocity-dependent behaviour and relatively low costs. Application on buildings requires a specific study under seismic excitation and a particular attention on structural details. Nevertheless seismic codes give only general information and in most case the design of a such protection systems results difficult; this problem is relevant also in Italy where last seismic code (NTC 2008) on one hand has increased the performance level to be attributed to buildings, especially public and strategic, on the other hand does not give a guide for the design of external dissipation devices as fluid viscous dampers or others. In this study, after a general description of fluid viscous dampers and the criteria of applying on buildings, a simplified procedure for the design of these devices is discussed also by means an application.

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Section: Model For Fluid Viscous Dampersmentioning

confidence: 99%

Solutions for the Design and Increasing of Efficiency of Viscous Dampers

Alotta

Cavaleri

Paola

et al. 2016

TOBCTJ

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“…mechanical description of the associated rheological devices. An efficient and exact representation of springpot devices has been recently obtained [15,41] and it will be called in the next section. We assume that the mechanical parameters of the model, namely the elastic modulus k(z) and the viscosity coefficient c(z) decay with power-law with the axial coordinate z as:…”

Section: Bone Hereditariness: the Power-law Rheological Modelmentioning

confidence: 99%

“…Validation and challenges of the mechanical equivalent representation of FHM have been discussed in previous papers [42,41] continuum mechanical model has been discretized into a mechanical fractance. Introducing a finite discretization grid of the z axis into point z j = (j 1) z, j = 1, 2, .…”

Section: The Discrete Equivalent Representation Of Fhmmentioning

confidence: 99%

Power‐law hereditariness of hierarchical fractal bones

Deseri

Paola

Zingales

et al. 2013

Numer Methods Biomed Eng

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SUMMARYIn 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   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 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. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 INTRODUCTIONMathematical models of material behavior are fundamental for optimization and for reliable design of engineered devices. Furthermore, the key issue is to be able to track down the multiscale behavior from the nano-to the macrolevel, with particular regard to biological and bioinspired materials.Indeed, the mechanical interactions among biomedical devices and biological tissues play a keyrole for the optimization of physiological functionality of such devices owing to the reduction of immunologic response. In this regard, it is clear that the detailed knowledge of the features of biological tissues at the different scales and their interactions with the devices is a crucial step to optimize the mechanical and physical response of the compound.Physical parameters of biological tissues usually investigated in scientific literature involve stiffness, strength, toughness, permeability, porosity, thermal conductivity among others [1,2,3].Besides these important features, mineralized bone tissues must provide load carrying capabilities and they exhibit a marked time-dependent behavior under applied loads. In this context, the term hereditariness is usually used in the sense that the actual response of bone material in terms of stress/displacement depends on previously applied stress/strain. This feature is macroscopically detectable by stress relaxation and creep observed in classical traction/compression mechanical tests. During a relaxation test, the imposed strain is held constant and a measure of the stress is monitored showing that it is a decreasing function of time; similarly, in a creep test an imposed constant stress is applied and a continuous monitoring of the strain is considered showing that it is an increasing function of time. Both these tests highlight the hereditariness feature of such material; the past undergone stress or strain history influence the future response of the specimen. A similar time-dependent behavior also arises in mineralized tissues as ligaments and tendons. Indeed the high stiffness (but highly brittleness) of the hydroxyapatite cryst...

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“…They were used in the mathematical modeling of systems and processes occurring in many engineering and scientific disciplines; for instance, see [1][2][3][4][5][6][7]. On the other hand, for studying the turbulent flow in a porous medium, Leibenson [8] introduced the model of a differential equation with the p-Laplacian operator.…”

Section: Introductionmentioning

confidence: 99%