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

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

A discrete mechanical model of fractional hereditary materials

Fractional hereditary materials are characterized for the presence, in the stress-strain relations, of fractional-order operators with order beta a[0,1]. In Di Paola and Zingales (J. Rheol. 56(5):983-1004, 2012) exact mechanical models of such materials have been extensively discussed obtaining two intervals for beta: (i) Elasto-Viscous (EV) materials for 0a parts per thousand currency sign beta a parts per thousand currency sign1/2; (ii) Visco-Elastic (VE) materials for 1/2a parts per thousand currency sign beta a parts per thousand currency sign1. These two ranges correspond to different continuous mechanical models. In this paper a discretization scheme based upon the continuous models proposed in Di Paola and Zingales (J. Rheol. 56(5):983-1004, 2012) useful to obtain a mechanical description of fractional derivative is presented. It is shown that the discretized models are ruled by a set of coupled first order differential equations involving symmetric and positive definite matrices. Modal analysis shows that fractional order operators have a mechanical counterpart that is ruled by a set of Kelvin-Voigt units and each of them provides a proper contribution to the overall response. The robustness of the proposed discretization scheme is assessed in the paper for different classes of external loads and for different values of beta a[0, 1]

Fractional multiphase hereditary materials: Mellin Transforms and Multi-Scale Fractances

The rheological features of several complex organic natural tissues such as bones, muscles as well as of complex artificial polymers are well described by power-laws. Indeed, it is well-established that the time-dependence of the stress and the strain in relaxation/creep test may be well captured by power-laws with exponent β ∈ [0, 1]. In this context a generalization of linear springs and linear dashpots has been introduced in scientific literature in terms of a mechanical device dubbed spring-pot. Recently the authors introduced a mechanical analogue to spring-pot built upon a proper arrangements of springs and dashpots that results in Elasto-Viscous (EV) materials, as β ∈ [0, 1/2] and Visco-Elastic ones, as β ∈ [1/2, 1]. In this paper the authors will discuss the rheological description of the presence of multiple material phases that is highlighted by a linear combination of power-laws in the relaxation function G(t) with different exponents. Such rehological model is represented by a linear combination of fractional derivatives with different order and the inverse relations have been formulated in terms of the complex method Mellin transform. Additionally an alternative representation of direct and inverse relations of multi-phase fractional hereditary materials based on the exact mechanical description of spring-pot element will be discussed in the course of the paper

Prestress and experimental tests on fractional viscoelastic materials

Creep and/or Relaxation tests on viscoelastic materials show a power-law trend. Based upon Boltzmann superposition principle the constitutive law with a power-law kernel is ruled by the Caputo's fractional derivative. Fractional constitutive law posses a long memory and then the parameters obtained by best fitting procedures on experimental data are strongly influenced by the prestress on the specimen. As in fact during the relaxation test the imposed history of deformation is not instantaneously applied, since a unit step function may not be realized by the test machine. Aim of this paper, it is shown that, the experimental procedure, and in particular the initial ramp to reach the constant stress (or strain) strongly influences the best fitting procedure and the coefficients of the power-law

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 viscoelastic 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

Electrical analogous in viscoelasticity

In this paper, electrical analogous models of fractional hereditary materials are introduced. Based on recent works by the authors, mechanical models of materials viscoelasticity behavior are firstly approached by using fractional mathematical operators. Viscoelastic models have elastic and viscous components which are obtained by combining springs and dashpots. Various arrangements of these elements can be used, and all of these viscoelastic models can be equivalently modeled 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. Moreover, a comparison with numerical experiments based on finite difference time domain method shows that, for long time simulations, the correct time behavior can be obtained only with modal analysis. The use of electrical analogous in viscoelasticity can better reveal the real behavior of fractional hereditary materials