{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T02:14:14Z","timestamp":1762049654087,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T00:00:00Z","timestamp":1651104000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Recent developments have shown that the widely used simplified differential model of Eringen\u2019s 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 nth order linear Volterra\u2013Fredholm integro-differential equations of convolution type to construct closed-form solutions to the two-phase integral model of Euler\u2013Bernoulli nanobeams in bending under transverse distributed load and various types of boundary conditions.<\/jats:p>","DOI":"10.3390\/a15050151","type":"journal-article","created":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T12:06:01Z","timestamp":1651147561000},"page":"151","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams"],"prefix":"10.3390","volume":"15","author":[{"given":"Efthimios","family":"Providas","sequence":"first","affiliation":[{"name":"Department of Environmental Sciences, Gaiopolis Campus, University of Thessaly, 415 00 Larissa, Greece"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1016\/S0022-5096(03)00053-X","article-title":"Experiments and theory in strain gradient elasticity","volume":"51","author":"Lam","year":"2003","journal-title":"J. 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