Meanwhile, we discuss the effect of the modelling parameter on viscoelasticity. In addition to the relaxation moduli and the creep compliances, we compute closed form expressions of the memory functions, impulse fluidities impulse response functions and impulse strainrate response functions of the generalized fractional derivative maxwell fluid, the generalized fractional derivative kelvinvoigt element and their. Pdf application of the fractional derivative kelvinvoigt model for. Fractional derivative heat conduction modeling based on. The fractional derivative kelvinvoigt model of viscoelasticity involving the timedependent poissons operator has been studied not only for the case of a timeindependent bulk modulus, but. Jun 28, 2019 in this paper, we focus on a generalized singular fractional order kelvinvoigt model with a nonlinear operator.
Table 3 presents the fitting parameters of creep responses corresponding to equal stresses for kelvinvoigt and fractional derivative kelvinvoigt models eq. This stressstrain relation has three parameters that can be obtained by fitting real data, namely, the stiffness, the pseudoviscosity and the fractional. Aditya jaishankaraditya jaishankar fractional calculus the murky bits the many definitions fractional derivative of a constant is not zero using the rl definition, while it is always zero using caputo definition makes caputo much more amenable to physical problems. Substituting eqs and into eq 26, the constitutive equation of the model eq 36 is obtained. Despite generalized would be a better option, the name fractional is used for denoting this kind of derivative. May 17, 2012 the fractional kelvin voigt model for rayleigh surface waves in viscoelastic fgm infinite half space mechanics research communications, vol. Impact of a viscoelastic sphere against an elastic kirchhoff. Creep response and stress relaxation of fractional kelvin voigt model are described but not derived since derivation exceeds the topics of this article. The fractional derivative kelvinvoigt model of viscoelasticity involving the timedependent poissons operator has been studied not only for the case of a timeindependent bulk modulus, but also when the volumetric relaxation is taken into account.
Central to these calculations is the fractional derivative of the dirac delta function which makes possible the extraction of singularities embedded in the fractional derivatives of the. Modeling power law absorption and dispersion in viscoelastic. In this paper, a fractional kelvinvoigt viscoelastic constitutive model was proposed to study timedependent settlement of soil foundations. Power law and fractional calculus model of viscoelasticity. Fractional calculus is the branch of calculus that generalizes the derivative of a function to noninteger order, allowing calculations such as deriving a function to 12 order. With different combinations of the thermal elements, the time. Substituting eqs and into eq 26, the constitutive equation of. Feb 11, 2020 in addition to the relaxation moduli and the creep compliances, we compute closed form expressions of the memory functions, impulse fluidities impulse response functions and impulse strainrate response functions of the generalized fractional derivative maxwell fluid, the generalized fractional derivative kelvin voigt element and their. Arbitrary attached mass is taken into account via internal boundary conditions. Oct 26, 2019 nonlinear forced vibrations of a fractional viscoelastic pipe conveying fluid exposed to the timedependent excitations is investigated in the present work. A singular fractional kelvinvoigt model involving a. The viscoelastic response of hydropolymers, which include gelatin phantom and glandular breast tissue, may be accurately characterized with as few as three parameters using the kelvinvoigt fractional derivative kvfd modeling approach. Fast iterative solvers for fractional di erential equations. Fractional kelvinvoigt model is also set up in chapter 2.
The paper proposes a fractional generalization of the maxwell and kelvinvoigt rheological models for a description of dynamic behavior of biopolymer materials. Koeller 1984 derived the stress relaxation function, with a time dependence t in the function for the kelvinvoigt fractional derivative kvfd model. Feb 06, 2007 read finite element analysis of vibrating linear systems with fractional derivative viscoelastic models, journal of sound and vibration on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Primary and secondary resonances in pipes conveying fluid. Fractional generalizations of maxwell and kelvinvoigt models. Settlement analysis of viscoelastic foundation under vertical.
By using analytic techniques, the uniqueness of solution and an iterative scheme converging to the unique solution are established, which are very helpful to govern the process of the kelvinvoigt model. Pdf the fractional derivative kelvinvoigt model of viscoelasticity involving the timedependent poissons operator has been studied not only. Comparison between classical kelvinvoigt and fractional derivative kelvinvoigt models in prediction of linear viscoelastic behaviour of waste activated sludge science of the total environment, vol. The development of the model is based on inparallel combination of fractional maxwell model and fractional. Fractional derivative models for ultrasonic characterization.
Introduction viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing. Assumed that the viscoelastic material of the beam obeys the kelvin voigt fractional derivative constitutive relationship, the governing equation of the axially moving viscoelastic beam is established by the d alembert principle. At the same time, the corresponding eigenvalue problem is studied and the. In this paper, we focus on a generalized singular fractional order kelvinvoigt model with a nonlinear operator. Waves propagation in a fractional viscoelastic continuum. Modeling power law absorption and dispersion in viscoelastic solids using a splitfield and the fractional laplaciana bradley e. Read finite element analysis of vibrating linear systems with fractional derivative viscoelastic models, journal of sound and vibration on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. In addition, it is possible to obtain the fractional derivative heat conduction equations from the corresponding transport equations. Other applications of fractional calculus are viscoelasticity for example using the kelvinvoigt fractional derivative model 3, 4, electrical circuits 5, 6, electroanalytical chemistry 7 or image processing 8. Our representations are generalized maxwell models or series of kelvinvoigt units, which, however, maintain the number of parameters of the corresponding fractional order model. Analysis of complex modal characteristics of fractional.
Response of viscoelastic rotating beams with kelvinvoigt fractional derivatives. The fractional kelvinvoigt model for rayleigh surface waves in viscoelastic fgm infinite half space mechanics research communications, vol. Love plate in a fractional derivative kelvinvoigt medium. Based on galerkin method, the fractional ordinary differential equations of simply supported beam are obtained. Creep, relaxation and viscosity properties for basic fractional. Generalized fractional derivative anisotropic viscoelastic. With the increase of problems using fractional differential equations there is corresponding interest in the development and study of accurate, fast, and reliable numerical methods that allow their solution.
Isotropic linear and nonlinear fractional derivative constitutive relations are formulated and examined in terms of many parameter generalized kelvin models and are analytically extended to cover general anisotropic homogeneous or nonhomogeneous as well as functionally graded viscoelastic material behavior. The simplest models of viscoelasticity involving fractional. In this study, classical kelvin voigt model, commonly used for sludge in literature, were compared with fractional derivative kelvin voigt model regarding their predictive ability for describing. A numerical method for calculating fractional derivative. The nonlinear geometric partial differential equations due to stretching effect have been.
Effects of fractional derivative on dynamic behavior, as well as. Storage and loss modulus are derived for periodic oscillations. In this study, classical kelvinvoigt model, commonly used for sludge in literature, were compared with fractional derivative kelvinvoigt model regarding their predictive ability for describing. With the increase of problems using fractional di erential equations there is cor. Comparison between classical kelvinvoigt and fractional derivative kelvinvoigt models in prediction of linear viscoelastic behaviour of waste. The basic models are those that generalize via derivatives of fractional order the classical mechanical models characterized by two, three and four parameters, that we refer to as kelvinvoigt, maxwell, zener, antizener and burgers. In this paper, a fractional kelvin voigt viscoelastic constitutive model was proposed to study timedependent settlement of soil foundations. May 17, 2012 comparison between classical kelvin voigt and fractional derivative kelvin voigt models in prediction of linear viscoelastic behaviour of waste activated sludge science of the total environment, vol. In this study axial waves propagation analysis will be faced in a fractional viscoelastic continuum in the simple monodimensional case.
In the last years, this subject has been studied in two di erent ways, though close. In this notes, we will give a brief introduction to fractional calculus. Vibration analysis of an axially moving viscoelastic beam. Fractional calculus is a collection of relatively littleknown mathematical results concerning generalizations of differentiation and integration to noninteger orders. We then develop methods where the imaging parameter can be interpreted in terms of the biphasic properties.
Voigt model, the maxwell model, and the zener model, are presented. We have presented a numerical algorithm to model ultrasound in biologic tissues based on a generalization of the kelvinvoigt model to the case of fractional time derivatives of the strain. Nov 24, 2015 further modification of the kelvin voigt model by replacing the elastic hooke element with another scottblair component allowed to obtain the fractional kelvin voigt model with two builtin springpots fkvm2 fig 4c. Attention is focused in particular on the primary and secondary resonances with the kelvinvoigt fractional order constitutive relationship model. We consider a generalization of the kelvinvoigt rheology to the case of rational orders of differentiation, the socalled kelvinvoigt fractionalderivative kvfd constitutive equation, and introduce a novel modeling method to solve the wave equation by means of the grunwaldletnikov approximation and the staggered fourier pseudospectral. Index termsconstitutive equation, distributed order derivative, fractional calculus, response. Identification of parameters of maxwell and kelvinvoigt. 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. Wave simulation in biologic media based on the kelvinvoigt. A short introduction to fractional calculus introduction. For the system, set the number of mesh points, small variables, and nonuniform meshing of kelvinvoigt viscoelastic material beam with fractional derivative, when,, and, and the response curve of different points on the viscoelastic beam is as. While these results have been accumulated over centuries in various branches of.
Kelvin voigt fractional derivative model 28,63, electrical circuits 25,47, electroanalytical chemistry 58, or image processing 48. Accordingly, the number of parameters of the rheological. The fractional mechanical models containing the maxwell and kelvinvoigt elements are graphically discussed with the laplace transform. Creep, relaxation and viscosity properties for basic. Wave simulation in biologic media based on the kelvin. Creep response and stress relaxation of fractional kelvinvoigt model are described but not derived since derivation exceeds the topics of this article.
As this takes place, the viscoelastic features of the impactor are described by the fractional derivative standard linear solid model. Pdf wave simulation in biologic media based on the. Cox department of medical physics and biomedical engineering, university college london, gower street. The fractional derivative kelvinvoigt model of viscoelasticity with and without volumetric relaxation. The rst approach is probabilistic and we think it is the rst step a mathematician has to do to build and investigate. Pdf the fractional derivative kelvinvoigt model of. Comparison between classical kelvinvoigt and fractional. This paper deals with parameter identification for basic and generalized kelvinvoigt and maxwell models for fluid viscous dampers. The viscoelastic response of hydropolymers, which include gelatin phantom and glandular breast tissue, may be accurately characterized with as few as three parameters using the kelvin voigt fractional derivative kvfd modeling approach. Pdf identification of the parameters of the kelvinvoigt and the. Rheological representation of fractional order viscoelastic. Further modification of the kelvinvoigt model by replacing the elastic hooke element with another scottblair component allowed to obtain the fractional kelvinvoigt model with two builtin springpots fkvm2 fig 4c.
A fractional calculus model of viscoelastic stator supports. A numerical method is proposed for calculating the fractional order derivative and successfully resolving the integrand singularity problem based on zhangshimizu algorithm. Nonlinear forced vibrations of a fractional viscoelastic pipe conveying fluid exposed to the timedependent excitations is investigated in the present work. The simple kelvinvoigt model seems to be more appropriate. Equivalent integral constitutive relations, which are computationally more powerful. Wave simulation in biologic media based on the kelvinvoigt fractional derivative stressstrain relation. Such a problem is ruled by a fractional differential equation of secondorder in the axial displacement function whose solution will be.
Parameters identification method for viscoelastic dielectric. This approach directly assimilates viscoelasticity into the system dynamics, since the. The identification procedure gives the best mechanical parameters by minimizing a suitable objective function that represents a measure of difference between analytical and experimental applied forces. The paper proposes a fractional generalization of the maxwell and kelvin voigt rheological models for a description of dynamic behavior of biopolymer materials. With the increase of problems using fractional di erential equations there is corresponding. Pdf the fractional derivative kelvinvoigt model of viscoelasticity. And then a method is developed to calculate the twice nonlinear fractional derivative, numerical examples demonstrate the numerical method with high precision and good. Assumed that the viscoelastic material of the beam obeys the kelvinvoigt fractional derivative constitutive relationship, the governing equation of the axially moving viscoelastic beam is established by the d alembert principle. Here, we limited our study to application of fractional order kelvinvoigt model, which is introduced into the motion equation of nonlocal eulerbernoulli beam. Fractional kelvin voigt model is also set up in chapter 2. Kelvinvoigt vs fractional derivative model as constitutive. It was found that the rheological models of maxwelltype do not work in the case of modeling.
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