Applications of Fractional Calculus to the Theory of Viscoelasticity

1984 ◽  
Vol 51 (2) ◽  
pp. 299-307 ◽  
Author(s):  
R. C. Koeller
Keyword(s):  
Integral Equation ◽  
Solid State ◽  
Fractional Calculus ◽  
Materials With Memory ◽  
Continuous Transition ◽  
Fluid State ◽  
Relaxation Functions ◽  
Theory Of Viscoelasticity ◽  
With Memory ◽  
Memory Parameter

The connection between the fractional calculus and the theory of Abel’s integral equation is shown for materials with memory. Expressions for creep and relaxation functions, in terms of the Mittag-Leffler function that depends on the fractional derivative parameter β, are obtained. These creep and relaxation functions allow for significant creep or relaxation to occur over many decade intervals when the memory parameter, β, is in the range of 0.05–0.35. It is shown that the fractional calculus constitutive equation allows for a continuous transition from the solid state to the fluid state when the memory parameter varies from zero to one.

A Theory Relating Creep and Relaxation for Linear Materials With Memory

2010 ◽  
Vol 77 (3) ◽  
Author(s):  
R. C. Koeller
Keyword(s):  
Experimental Data ◽  
Fractional Calculus ◽  
Creep Compliance ◽  
Complex Modulus ◽  
Relaxation Times ◽  
Relaxation Modulus ◽  
Experimental Methods ◽  
Generalized Function ◽  
Materials With Memory ◽  
With Memory

The purpose of this paper is to suggest a linear theory of materials with memory, which gives a description for the similarities resulting when the various analytical and experimental methods used to reduce the creep and relaxation data are imposed on the observational changes in curvature that take place in both the creep compliance and relaxation modulus graphs. On a Log-Log graph both have one, two, or at most three pairs of changes in curvature depending on whether the material is a fluid or solid. These changes in curvature have been observed in many experiments and various regions have been discussed and classified. Section 1 gives a few of the many applications of fractional calculus to physical problems. In Sec. 2 an equation that contains both integration and differentiation is presented using geometrical observations about the relationship between the changes in curvature in the relaxation modulus and creep compliance based on published experiments. In Sec. 3 the generalized function approach to fractional calculus is given. In Sec. 4 a mechanical model is discussed. This model is able to share experimental data between the creep and relaxation functions, as well as the real and imaginary parts of the complex compliance or the complex modulus. This theory shares information among these three experimental methods into a unifying theory for solid materials when the loads are within the linear range. Under a limiting case, this theory can account for flow so that the material need not return to its original shape after the load is removed. The theory contains one physical parameter, which is related to the speed of sound and a group of phenomenological parameters that are functions of temperature and the composition of the material. These phenomenological parameters are relaxation times and creep times. This theory differs from the classical polynomial constitutive equations for linear viscoelasticity. It is a special case of Rabotnov’s equations and Torvik and Bagley’s fractional calculus polynomial equations, but it imposes symmetry conditions on the stress and strain when the material is a solid. Sections 56 are comments and conclusions, respectively. No experimental results are given at this time since this paper presents the foundations of materials with memory as related to experimental data. The introduction of experimental data to fit this theory will result in the breakdown of an important part of this research.

A class of materials with memory

Meccanica ◽  
1972 ◽  
Vol 7 (1) ◽  
pp. 21-21
Author(s):  
G. Capriz
Keyword(s):  
Materials With Memory ◽  
With Memory

scholarly journals Asymptotic Periodicity for a Class of Partial Integrodifferential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Alejandro Caicedo ◽  
Claudio Cuevas ◽  
Hernán R. Henríquez
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Integral Equations ◽  
Integrodifferential Equations ◽  
Materials With Memory ◽  
Asymptotic Periodicity ◽  
With Memory ◽  
Equations With Delay

We study the existence of S-asymptotically ω-periodic solutions for a class of abstract partial integro-differential equations and for a class of abstract partial integrodifferential equations with delay. Applications to integral equations arising in the study of heat conduction in materials with memory are shown.

The Mechanics of Non-Linear Materials with Memory

1997 ◽  
pp. 1127-1136
Author(s):  
A. E. Green ◽  
R. S. Rivlin ◽  
A. J. M. Spencer
Keyword(s):  
Materials With Memory ◽  
Non Linear ◽  
With Memory

scholarly journals The use of fractional calculus for the optimal placement of electronic components on a linear array

2015 ◽  
Vol 28 (1) ◽  
pp. 77-84
Author(s):  
Mey de ◽  
Mariusz Felczak ◽  
Bogusław Więcek
Keyword(s):  
Integral Equation ◽  
Fractional Calculus ◽  
Forced Convection ◽  
Linear Array ◽  
The Other ◽  
Simple Solution ◽  
Optimal Placement ◽  
Critical Component ◽  
Electronic Components ◽  
One Dimensional

Cooling of heat dissipating components has become an important topic in the last decades. Sometimes a simple solution is possible, such as placing the critical component closer to the fan outlet. On the other hand this component will heat the air which has to cool the other components further away from the fan outlet. If a substrate bearing a one dimensional array of heat dissipating components, is cooled by forced convection only, an integral equation relating temperature and power is obtained. The forced convection will be modelled by a simple analytical wake function. It will be demonstrated that the integral equation can be solved analytically using fractional calculus.

Closure to “Thermo-Plastic Materials with Memory”

1974 ◽  
Vol 100 (6) ◽  
pp. 1255-1255
Author(s):  
J. Tinsley Oden ◽  
Der R. Bhandari
Keyword(s):  
Materials With Memory ◽  
Plastic Materials ◽  
With Memory

Thermodynamics of Materials with Memory

2021 ◽  
pp. 101-113
Author(s):  
Giovambattista Amendola ◽  
Mauro Fabrizio ◽  
John Golden
Keyword(s):  
Materials With Memory ◽  
With Memory
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