Francesco Paolo Pinnola

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.

Linear and nonlinear fractional hereditary constitutive laws of asphalt mixtures

The aim of this paper is to propose a fractional viscoelastic and viscoplastic model of asphalt mixtures using experimental data of several tests such as creep and creep recovery performed at different temperatures and at different stress levels. From a best fitting procedure it is shown that both the creep one and recovery curve follow a power law model. It is shown that the suitable model for asphalt mixtures is a dashpot and a fractional element arranged in series. The proposed model is also available outside of the linear domain but in this case the parameters of the model depend on the stress level.

On the Dynamics of Fractional Visco-Elastic Beams

With increasing advanced manufacturing process, viscoelastic materials are very attractive for mitigation of vibrations, provided that you may have advanced studies for capturing the realistic behavior of such materials. Experimental verification of the visco-elastic behavior is limited to some well-known low order models as the Maxwell or Kelvin models. However, both models are not sufficient to model the visco-elastic behavior of real materials, since only the Maxwell type can capture the relaxation tests and the Kelvin the creep tests, respectively. Very recently, it has been stressed that the most suitable model for capturing the viscoelastic behavior is the spring-pot, characterized by a fractional constitutive law. Based on this assumption, the quasi-static behavior has been investigated very recently, however for noise control there is a need of exploiting the dynamic behavior of such a fractional visco-elastic beam. The present paper introduces the dynamic response of fractional visco-elastic Euler-Bernoulli beam under dynamic loads.

Analysis of multi-degree-of-freedom systems with fractional derivative elements of rational order

In this paper a novel method based on complex eigenanalysis in the state variables domain is proposed to uncouple the set of rational order fractional differential equations governing the dynamics of multi-degree-of-freedom system. The traditional complex eigenanalysis is appropriately modified to be applicable to the coupled fractional differential equations. This is done by expanding the dimension of the problem and solving the system in the state variable domain. Examples of applications are given pertaining to multi-degree-of-freedom systems under both deterministic and stochastic loads.

Viscoelasticity: An electrical point of view

Time dependent hereditary properties of complex materials are well described by power-laws with real order exponent. This experimental observation and analogous electrical experiments, yield a description of these properties by using fractional-order operators. In this paper, elasto-viscous and visco-elastic behaviors of fractional order hereditary materials are firstly described by using fractional mathematical operators, based on recent work of some of the authors. Then, electrical analogous models are introduced. Viscoelastic models have elastic and viscous components which can be obtained by combining springs and dashpots: these models can be equivalently viewed as electrical circuits, where the spring and dashpot are analogous to the capacitance and resistance, respectively. The proposed models are validated by using modal analysis. The use of electrical analogous in viscoelasticity can better reveal the real behavior of fractional hereditary materials.

A numerical assessment of the free energy function for fractional-order relaxation

In this paper the authors discuss the free energy function of fractional hereditary materials. The evaluation of the free energy has been obtained from a mechanical model that represents, exactly, the power-law relaxation of the material. The mechanical model separates, exactly, the elastic and the viscous phases, yielding the stored energy of the material that corresponds to the Staverman-Schwarz stress based free energy. Some numerical approximations of the free energy function in terms of the discretized rheological model have been reported in the paper.