Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams
Recent developments have shown that the widely used simplified differential model of Eringen’s nonlocal elasticity in nanobeam analysis is not equivalent to the corresponding and initially proposed integral models, the pure integral model and the two-phase integral model, in all cases of loading and...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-04-01
|
| Series: | Algorithms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1999-4893/15/5/151 |
| _version_ | 1797502176540491776 |
|---|---|
| author | Efthimios Providas |
| author_facet | Efthimios Providas |
| author_sort | Efthimios Providas |
| collection | DOAJ |
| description | Recent developments have shown that the widely used simplified differential model of Eringen’s nonlocal elasticity in nanobeam analysis is not equivalent to the corresponding and initially proposed integral models, the pure integral model and the two-phase integral model, in all cases of loading and boundary conditions. This has resolved a paradox with solutions that are not in line with the expected softening effect of the nonlocal theory that appears in all other cases. In addition, it revived interest in the integral model and the two-phase integral model, which were not used due to their complexity in solving the relevant integral and integro-differential equations, respectively. In this article, we use a direct operator method for solving boundary value problems for <i>n</i>th order linear Volterra–Fredholm integro-differential equations of convolution type to construct closed-form solutions to the two-phase integral model of Euler–Bernoulli nanobeams in bending under transverse distributed load and various types of boundary conditions. |
| first_indexed | 2024-03-10T03:29:12Z |
| format | Article |
| id | doaj.art-5aa685a9c1c443e3bf4726b1867d4f59 |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-03-10T03:29:12Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-5aa685a9c1c443e3bf4726b1867d4f592023-11-23T09:45:22ZengMDPI AGAlgorithms1999-48932022-04-0115515110.3390/a15050151Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli NanobeamsEfthimios Providas0Department of Environmental Sciences, Gaiopolis Campus, University of Thessaly, 415 00 Larissa, GreeceRecent developments have shown that the widely used simplified differential model of Eringen’s nonlocal elasticity in nanobeam analysis is not equivalent to the corresponding and initially proposed integral models, the pure integral model and the two-phase integral model, in all cases of loading and boundary conditions. This has resolved a paradox with solutions that are not in line with the expected softening effect of the nonlocal theory that appears in all other cases. In addition, it revived interest in the integral model and the two-phase integral model, which were not used due to their complexity in solving the relevant integral and integro-differential equations, respectively. In this article, we use a direct operator method for solving boundary value problems for <i>n</i>th order linear Volterra–Fredholm integro-differential equations of convolution type to construct closed-form solutions to the two-phase integral model of Euler–Bernoulli nanobeams in bending under transverse distributed load and various types of boundary conditions.https://www.mdpi.com/1999-4893/15/5/151integro-differential equationsVolterra–Fredholm equationsnonlocal boundary value problemsdecomposition of operatorsEringen’s nonlocal elasticityEuler–Bernoulli beams |
| spellingShingle | Efthimios Providas Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams Algorithms integro-differential equations Volterra–Fredholm equations nonlocal boundary value problems decomposition of operators Eringen’s nonlocal elasticity Euler–Bernoulli beams |
| title | Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams |
| title_full | Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams |
| title_fullStr | Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams |
| title_full_unstemmed | Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams |
| title_short | Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams |
| title_sort | closed form solution of the bending two phase integral model of euler bernoulli nanobeams |
| topic | integro-differential equations Volterra–Fredholm equations nonlocal boundary value problems decomposition of operators Eringen’s nonlocal elasticity Euler–Bernoulli beams |
| url | https://www.mdpi.com/1999-4893/15/5/151 |
| work_keys_str_mv | AT efthimiosprovidas closedformsolutionofthebendingtwophaseintegralmodelofeulerbernoullinanobeams |