# Comparative Analysis of Viscoelastic Models Involving Fractional Derivatives of Different Orders

• Volume: 10, Issue: 2, page 111-121
• ISSN: 1311-0454

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## Abstract

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Mathematics Subject Classification: 74D05, 26A33In this paper, a comparative analysis of the models involving fractional derivatives of di®erent orders is given. Such models of viscoelastic materials are widely used in various problems of mechanics and rheology. Rabotnov's hereditarily elastic rheological model is considered in detail. It is shown that this model is equivalent to the rheological model involving fractional derivatives in the stress and strain with the orders proportional to a certain positive value less than unit. In the scienti¯c literature such a model is referred to as Koeller's model. Inversion of Rabotnov's model developed by himself based on algebra of operators results in similar rheological dependences. Inversion of Koeller's model carried out using Miller's theorem coincides inherently with Rabotnov's inversion procedure.∗ This paper has been partially supported by the Russian Foundation for Basic Research under Grant No. 05-08-17936.

## How to cite

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Rossikhin, Yuriy, and Shitikova, Marina. "Comparative Analysis of Viscoelastic Models Involving Fractional Derivatives of Different Orders." Fractional Calculus and Applied Analysis 10.2 (2007): 111-121. <http://eudml.org/doc/11320>.

@article{Rossikhin2007,
abstract = {Mathematics Subject Classification: 74D05, 26A33In this paper, a comparative analysis of the models involving fractional derivatives of di®erent orders is given. Such models of viscoelastic materials are widely used in various problems of mechanics and rheology. Rabotnov's hereditarily elastic rheological model is considered in detail. It is shown that this model is equivalent to the rheological model involving fractional derivatives in the stress and strain with the orders proportional to a certain positive value less than unit. In the scienti¯c literature such a model is referred to as Koeller's model. Inversion of Rabotnov's model developed by himself based on algebra of operators results in similar rheological dependences. Inversion of Koeller's model carried out using Miller's theorem coincides inherently with Rabotnov's inversion procedure.∗ This paper has been partially supported by the Russian Foundation for Basic Research under Grant No. 05-08-17936.},
author = {Rossikhin, Yuriy, Shitikova, Marina},
journal = {Fractional Calculus and Applied Analysis},
keywords = {74D05; 26A33},
language = {eng},
number = {2},
pages = {111-121},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Comparative Analysis of Viscoelastic Models Involving Fractional Derivatives of Different Orders},
url = {http://eudml.org/doc/11320},
volume = {10},
year = {2007},
}

TY - JOUR
AU - Rossikhin, Yuriy
AU - Shitikova, Marina
TI - Comparative Analysis of Viscoelastic Models Involving Fractional Derivatives of Different Orders
JO - Fractional Calculus and Applied Analysis
PY - 2007
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 10
IS - 2
SP - 111
EP - 121
AB - Mathematics Subject Classification: 74D05, 26A33In this paper, a comparative analysis of the models involving fractional derivatives of di®erent orders is given. Such models of viscoelastic materials are widely used in various problems of mechanics and rheology. Rabotnov's hereditarily elastic rheological model is considered in detail. It is shown that this model is equivalent to the rheological model involving fractional derivatives in the stress and strain with the orders proportional to a certain positive value less than unit. In the scienti¯c literature such a model is referred to as Koeller's model. Inversion of Rabotnov's model developed by himself based on algebra of operators results in similar rheological dependences. Inversion of Koeller's model carried out using Miller's theorem coincides inherently with Rabotnov's inversion procedure.∗ This paper has been partially supported by the Russian Foundation for Basic Research under Grant No. 05-08-17936.
LA - eng
KW - 74D05; 26A33
UR - http://eudml.org/doc/11320
ER -

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