{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,29]],"date-time":"2026-03-29T01:28:13Z","timestamp":1774747693902,"version":"3.50.1"},"reference-count":74,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,17]],"date-time":"2020-04-17T00:00:00Z","timestamp":1587081600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012189","name":"Megagrants","doi-asserted-by":"publisher","award":["14.Z50.31.0046"],"award-info":[{"award-number":["14.Z50.31.0046"]}],"id":[{"id":"10.13039\/501100012189","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The fundamental motivation of this research is to investigate the effect of flexoelectricity on a piezoelectric nanobeam for the first time involving internal viscoelasticity. To date, the effect of flexoelectricity on the mechanical behavior of nanobeams has been investigated extensively under various physical and environmental conditions. However, this effect as an internal property of materials has not been studied when the nanobeams include an internal damping feature. To this end, a closed-circuit condition is considered taking converse piezo\u2013flexoelectric behavior. The kinematic displacement of the classical beam using Lagrangian strains, also applying Hamilton\u2019s principle, creates the needed frequency equation. The natural frequencies are measured in nanoscale by the available nonlocal strain gradient elasticity model. The linear Kelvin\u2013Voigt viscoelastic model here defines the inner viscoelastic coupling. An analytical solution technique determines the values of the numerical frequencies. The best findings show that the viscoelastic coupling can directly affect the flexoelectricity property of the material.<\/jats:p>","DOI":"10.3390\/sym12040643","type":"journal-article","created":{"date-parts":[[2020,4,21]],"date-time":"2020-04-21T05:48:52Z","timestamp":1587448132000},"page":"643","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":72,"title":["On the Dynamics of a Visco\u2013Piezo\u2013Flexoelectric Nanobeam"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7356-2168","authenticated-orcid":false,"given":"Mohammad","family":"Malikan","sequence":"first","affiliation":[{"name":"Department of Mechanics of Materials and Structures, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-233 Gdansk, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8128-3262","authenticated-orcid":false,"given":"Victor A.","family":"Eremeyev","sequence":"additional","affiliation":[{"name":"Department of Mechanics of Materials and Structures, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-233 Gdansk, Poland"},{"name":"Research and Education Center \u201cMaterials\u201d, Don State Technical University, Gagarina sq., 1, 344000 Rostov on Don, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1088\/0031-8949\/2007\/T129\/041","article-title":"Flexoelectricity: Strain gradient effects in ferroelectrics","volume":"T129","author":"Ma","year":"2007","journal-title":"Phys. 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