Toward fractional gradient elasticity
The use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the power law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one dimension. The...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2014-05-01
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| Series: | Journal of the Mechanical Behavior of Materials |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/jmbm-2014-0006 |
| _version_ | 1818730932487585792 |
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| author | Tarasov Vasily E. Aifantis Elias C. |
| author_facet | Tarasov Vasily E. Aifantis Elias C. |
| author_sort | Tarasov Vasily E. |
| collection | DOAJ |
| description | The use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the power law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one dimension. The second involves the Riesz fractional derivative in three dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case, stress equilibrium in a Caputo elastic bar requires the existence of a nonzero internal body force to equilibrate it. In the second case, in a Riesz-type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed. |
| first_indexed | 2024-12-17T23:09:38Z |
| format | Article |
| id | doaj.art-d765123cb5ef45ab8a93c1c22f24d2ab |
| institution | Directory Open Access Journal |
| issn | 0334-8938 2191-0243 |
| language | English |
| last_indexed | 2024-12-17T23:09:38Z |
| publishDate | 2014-05-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Journal of the Mechanical Behavior of Materials |
| spelling | doaj.art-d765123cb5ef45ab8a93c1c22f24d2ab2022-12-21T21:29:10ZengDe GruyterJournal of the Mechanical Behavior of Materials0334-89382191-02432014-05-01231-2414610.1515/jmbm-2014-0006Toward fractional gradient elasticityTarasov Vasily E.0Aifantis Elias C.1Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, RussiaLaboratory of Mechanics and Materials, Aristotle University of Thessaloniki, Thessaloniki 54124, GreeceThe use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the power law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one dimension. The second involves the Riesz fractional derivative in three dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case, stress equilibrium in a Caputo elastic bar requires the existence of a nonzero internal body force to equilibrate it. In the second case, in a Riesz-type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.https://doi.org/10.1515/jmbm-2014-0006fractional derivativefractional elasticitygradient elasticity45.10.hj62.20.dc81.40.jj |
| spellingShingle | Tarasov Vasily E. Aifantis Elias C. Toward fractional gradient elasticity Journal of the Mechanical Behavior of Materials fractional derivative fractional elasticity gradient elasticity 45.10.hj 62.20.dc 81.40.jj |
| title | Toward fractional gradient elasticity |
| title_full | Toward fractional gradient elasticity |
| title_fullStr | Toward fractional gradient elasticity |
| title_full_unstemmed | Toward fractional gradient elasticity |
| title_short | Toward fractional gradient elasticity |
| title_sort | toward fractional gradient elasticity |
| topic | fractional derivative fractional elasticity gradient elasticity 45.10.hj 62.20.dc 81.40.jj |
| url | https://doi.org/10.1515/jmbm-2014-0006 |
| work_keys_str_mv | AT tarasovvasilye towardfractionalgradientelasticity AT aifantiseliasc towardfractionalgradientelasticity |