Viscoelasticity: An electrical point of view (original) (raw)
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Electrical analogous in viscoelasticity
Communications in Nonlinear Science and Numerical Simulation, 2014
In this paper, electrical analogous models of fractional hereditary materials are introduced. Based on recent works by the authors, mechanical models of materials viscoelasticity behaviour 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 behaviour can be obtained only with modal analysis. The use of electrical analogous in viscoelasticity can better reveal the real behaviour of fractional hereditary materials.
A discrete mechanical model of fractional hereditary materials
Meccanica, 2013
Fractional hereditary materials are characterized for the presence, in the stress-strain relations, of fractional-order operators with order β ∈ [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 β: (i) Elasto-Viscous (EV) materials for 0 ≤ β ≤ 1/2; (ii) Visco-Elastic (VE) materials for 1/2 ≤ β ≤ 1. 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 β ∈ [0, 1].
Notes on computational aspects of the fractional-order viscoelastic model
Journal of Engineering Mathematics
This paper deals with the computational aspect of the investigation of the relaxation properties of viscoelastic materials. The constitutive fractional Zener model is considered under continuous deformation with a jump at the origin. The analytical solution of this equation is obtained by the Laplace transform method. It is derived in a closed form in the terms of the Mittag-Leffler function. The method of numerical evaluation of the Mittag-Leffler function for arbitrary negative arguments which corresponds to physically meaningful interpretation is demonstrated. A numerical example is given to illustrate the effectiveness of this result.
Fractional hereditary materials are characterized for the presence, in the stress-strain relations, of fractional-order operators with order β ∈ [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 β: (i) Elasto-Viscous (EV) materials for 0 ≤ β ≤ 1/2; (ii) Visco-Elastic (VE) materials for 1/2 ≤ β ≤ 1. These two ranges correspond to different continuous mechanical models.
On fractional modelling of viscoelastic mechanical systems
Mechanics Research Communications, 2016
Since Leibniz's fractional derivative, introduced by Lazopoulos [1], has physical meaning contrary to other fractional derivatives, the viscoelastic mechanical systems are modelled with the help of Leibniz fractional derivative. The compliance and relaxation behaviour of the viscoelastic systems is revisited and comparison with the conventional systems and the existing fractional viscoelastic systems is presented.
Physics of Fluids
Soft materials such as gels, elastomers, and biological tissues have diverse applications in nature and technology due to their viscoelastic nature. These soft materials often exhibit complex rheology and display elastic and viscous characteristics when undergoing deformation. In recent years, fractional calculus has emerged as a promising tool to explain the viscoelastic behavior of soft materials. Scalar constants are primarily used to quantify viscoelastic elements such as springs and dashpots. However, in three-dimensional (3D) space, not all materials show the same elastic or viscoelastic properties in all directions, especially under elastic/viscoelastic wave propagation (or anisotropy). Though previously reported studies on viscoelastic models have explained a power-law decay of the memory functions, none of them explicitly explained the 3D complex modulus through a matrix notation. In this paper, we present a mathematical formulation that employs tensor algebra and fractiona...
Fractional calculus in viscoelasticity: An experimental study
Communications in Nonlinear Science and Numerical Simulation, 2010
Viscoelastic properties of soft biological tissues provide information that may be useful in medical diagnosis. Noninvasive elasticity imaging techniques, such as Magnetic Resonance Elastography (MRE), reconstruct viscoelastic material properties from dynamic displacement images. The reconstruction algorithms employed in these techniques assume a certain viscoelastic material model and the results are sensitive to the model chosen. Developing a better model for the viscoelasticity of soft tissue-like materials could improve the diagnostic capability of MRE. The well known ''integer derivative" viscoelastic models of Voigt and Kelvin, and variations of them, cannot represent the more complicated rate dependency of material behavior of biological tissues over a broad spectral range. Recently the ''fractional derivative" models have been investigated by a number of researchers. Fractional order models approximate the viscoelastic material behavior of materials through the corresponding fractional differential equations. This paper focuses on the tissue mimicking materials CF-11 and gelatin, and compares fractional and integer order models to describe their behavior under harmonic mechanical loading. Specifically, Rayleigh (surface) waves on CF-11 and gelatin phantoms are studied, experimentally and theoretically, in order to develop an independent test bed for assessing viscoelastic material models that will ultimately be used in MRE reconstruction algorithms.
Rheological representation of fractional order viscoelastic material models
Rheologica Acta, 2010
We develop rheological representations, i.e., discrete spectrum models, for the fractional derivative viscoelastic element (fractional dashpot or springpot). Our representations are generalized Maxwell models or series of Kelvin-Voigt units, which, however, maintain the number of parameters of the corresponding fractional order model. Accordingly, the number of parameters of the rheological representation is independent of the number of rheological units. We prove that the representations converge to the corresponding fractional model in the limit as the number of units tends to infinity. The representations extend to compound fractional derivative models such as the fractional Maxwell model, fractional Kelvin-Voigt model, and fractional standard linear solid. Computational experiments show that the rheological representations are accurate approximations of the fractional order models even for a small number of units.
Three-dimensional constitutive viscoelastic laws with fractional order time derivatives
Synopsis In this article the three-dimensional behavior of constitutive models containing fractional order time derivatives in their strain and stress operators is investigated. Assuming isotropic viscoelastic behavior, it is shown that when the material is incompressible, then the one-dimensional constitutive law calibrated either from shear or elongation tests can be directly extended in three dimensions, and the order of fractional differentiation is the same in all deformation patterns. When the material is viscoelastically compressible, the constitutive law in elongation involves additional orders of fractional differentiation that do not appear in the constitutive law in shear. In the special case where the material is elastically compressible, the constitutive laws during elongation and shear are different; however the order of fractional differentiation remains the same. It is shown that for an elastically compressible material, the four-parameter fractional solid—the rubbery, transition, and glassy model, which has been used extensively to approximate the elongation behavior of various polymers, can be constructed from the three-parameter fractional Kelvin—the rubbery transition model in shear and the elastic bulk modulus of the material. Some of the analytical results obtained herein with operational calculus are in agreement with experimental observations reported in the literature. Results on the viscoelastic Poisson behavior of materials described with the fractional solid model are presented and it is shown that at early times the Poisson function reaches negative values.
This is the first annual report to the U.S. Army Medical Research and Material Command for the three year project "Advanced Soft Tissue Modeling for Telemedicine and Surgical Simulation" supported by grant No. DAMD17-01-1-0673 to The Cleveland Clinic Foundation, to which the NASA Glenn Research Center is a subcontractor through Space Act Agreement SAA 3-445. The objective of this report is to extend popular one-dimensional (1D) fractional-order viscoelastic (FOV) materials models into their three-dimensional (3D) equivalents for finitely deforming continua, and to provide numerical algorithms for their solution.