scholarly journals On infinite order differential operators in fractional viscoelasticity

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Andrea Giusti
Keyword(s):  
Fractional Order ◽  
Constitutive Equations ◽  
Linear Viscoelasticity ◽  
Infinite Order ◽  
Differential Operators ◽  
Fractional Viscoelasticity ◽  
Viscoelastic Models ◽  
General Properties ◽  
Derivatives Of ◽  
Short Times

AbstractIn this paper we discuss some general properties of viscoelastic models defined in terms of constitutive equations involving infinitely many derivatives (of integer and fractional order). In particular, we consider as a working example the recently developed Bessel models of linear viscoelasticity that, for short times, behave like fractional Maxwell bodies of order 1/2.

Viscoelasticity of Fractional Order: New Restrictions on Constitutive Equations with Applications

2020 ◽  
Vol 20 (13) ◽  
pp. 2041011
Author(s):  
Teodor M. Atanacković ◽  
Marko B. Janev ◽  
Stevan Pilipović ◽  
Dora Seleši
Keyword(s):  
Fractional Order ◽  
Constitutive Equations ◽  
Linear Viscoelasticity ◽  
Fractional Derivatives ◽  
Second Law Of Thermodynamics ◽  
Weak Form ◽  
Second Law ◽  
Isothermal Conditions ◽  
Complex Order ◽  
Derivatives Of

In this paper, we analyze the restrictions on the coefficients in the constitutive equations of linear Viscoelasticity that follow from the Second Law of Thermodynamics under isothermal conditions. Especially, we analyze the constitutive equations in which fractional derivatives of real and complex order appear. We present the conditions that follow after application of the Bochner–Schwartz theorem. Conditions derived here, representing in certain cases a weak form of the Second law of Thermodynamics, are more general (weaker) than the classical Bagley–Torvik conditions widely used in Viscoelasticity Theory. Several examples that illustrate the theory are presented.

A Comparison Between Fractional-Order and Integer-Order Differential Finite Deformation Viscoelastic Models: Effects of Filler Content and Loading Rate on Material Parameters

2018 ◽  
Vol 10 (09) ◽  
pp. 1850099 ◽  
Author(s):  
Hesam Khajehsaeid
Keyword(s):  
Fractional Order ◽  
Differential Operators ◽  
Filler Content ◽  
Viscoelastic Model ◽  
Constitutive Models ◽  
Viscoelastic Behavior ◽  
Fractional Model ◽  
Material Parameters ◽  
Integer Order ◽  
Viscoelastic Models

Elastomers or rubber-like materials exhibit nonlinear viscoelastic behavior such as creep and relaxation upon mechanical loading. Differential constitutive models and hereditary integrals are the main frameworks followed in the literature for modeling the viscoelastic behavior at finite deformations. Regular differential operators can be replaced by fractional-order derivatives in the standard models in order to make fractional viscoelastic models. In the present paper, the relaxation behavior of elastomers is formulated both in terms of ordinary (integer-order) and fractional differential viscoelastic models. The derived constitutive equations are fitted to several experimental data to compare their efficiency in modeling the stress relaxation phenomenon. Specifically, a fractional viscoelastic model with one fractional dashpot (FD) is compared with two ordinary models including respectively one and two ordinary dashpots (OD). The models are compared in fitting accuracy, number of required material parameters and also variation of parameters from one compound to another to clarify the effects of filler content and deformation rate. It is shown that, the results of the ordinary model with one OD is not good at all. The fractional model with one FD and the ordinary model with two ODs provide good fittings for all compounds whereas the former uses only three parameters and the latter uses five material parameters. For the fractional model, the order of the Maxwell element and the associated relaxation time approximately remain the same for different compounds of each material at certain loading rates, but it is not the case for the ordinary differential models.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

Higher order Wirtinger inequalities

1980 ◽  
Vol 85 (1-2) ◽  
pp. 87-110 ◽  
Author(s):  
Kurt Kreith ◽  
Charles A. Swanson
Keyword(s):  
Boundary Conditions ◽  
Quadratic Forms ◽  
Differential Operators ◽  
Conjugate Point ◽  
Wirtinger Inequality ◽  
Even Order ◽  
Linear Differential ◽  
Natural Boundary Conditions ◽  
Derivatives Of ◽  
Admissible Functions

SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

scholarly journals An analytical solution for solving a new wave equation within Lorenzo-Hartley kernel

2019 ◽  
Vol 23 (6 Part B) ◽  
pp. 3739-3744
Author(s):  
Feng Gao
Keyword(s):  
Wave Equation ◽  
Analytical Solution ◽  
Fractional Order ◽  
New Wave ◽  
Liouville Type ◽  
Derivatives Of

In this article we investigate the general fractional-order derivatives of the Riemann-Liouville type via Lorenzo-Hartley kernel, general fractional-order integrals and the new general fractional-order wave equation defined on the definite domain with the analytical soluton.

Sign in / Sign up

Export Citation Format

Share Document