{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:48:25Z","timestamp":1762508905105,"version":"build-2065373602"},"reference-count":71,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,7]],"date-time":"2022-12-07T00:00:00Z","timestamp":1670371200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11602204","2682022ZTPY081","SZDKF-202102","23NSFSC0849"],"award-info":[{"award-number":["11602204","2682022ZTPY081","SZDKF-202102","23NSFSC0849"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities of China","doi-asserted-by":"publisher","award":["11602204","2682022ZTPY081","SZDKF-202102","23NSFSC0849"],"award-info":[{"award-number":["11602204","2682022ZTPY081","SZDKF-202102","23NSFSC0849"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Open project of Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province","award":["11602204","2682022ZTPY081","SZDKF-202102","23NSFSC0849"],"award-info":[{"award-number":["11602204","2682022ZTPY081","SZDKF-202102","23NSFSC0849"]}]},{"name":"National Natural Science Foundation of Sichuan Province","award":["11602204","2682022ZTPY081","SZDKF-202102","23NSFSC0849"],"award-info":[{"award-number":["11602204","2682022ZTPY081","SZDKF-202102","23NSFSC0849"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The free vibration of isotropic gradient elastic thick non-rectangular microplates is analyzed in this paper. To capture the microstructure-dependent effects of microplates, a negative second-order gradient elastic theory with symmetry is utilized. The related equations of motion and boundary conditions are obtained using the energy variational principle. A closed-form solution is presented for simply supported free-vibrational rectangular microplates with four edges. A C1-type differential quadrature finite element (DQFE) is applied to solve the free vibration of thick microplates. The DQ rule is extended to the straight-sided quadrilateral domain through a coordinate transformation between the natural and Cartesian coordinate systems. The Gauss\u2013Lobato quadrature rule and DQ rule are jointly used to discretize the strain and kinetic energies of a generic straight-sided quadrilateral plate element. Selective numerical examples are validated against those available in the literature. Finally, the impact of various parameters on the free vibration characteristics of annular sectorial and triangular microplates is shown. It indicates that the strain gradient and inertia gradient effects can result in distinct changes in both vibration frequencies and mode shapes.<\/jats:p>","DOI":"10.3390\/sym14122592","type":"journal-article","created":{"date-parts":[[2022,12,8]],"date-time":"2022-12-08T02:17:57Z","timestamp":1670465877000},"page":"2592","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates"],"prefix":"10.3390","volume":"14","author":[{"given":"Bo","family":"Zhang","sequence":"first","affiliation":[{"name":"Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 611756, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1018-0087","authenticated-orcid":false,"given":"Cheng","family":"Li","sequence":"additional","affiliation":[{"name":"Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 611756, China"},{"name":"School of Rail Transportation, Soochow University, Suzhou 215131, China"},{"name":"School of Automotive Engineering, Changzhou Institute of Technology, Changzhou 213032, China"}]},{"given":"Limin","family":"Zhang","sequence":"additional","affiliation":[{"name":"Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 611756, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3609-4964","authenticated-orcid":false,"given":"Feng","family":"Xie","sequence":"additional","affiliation":[{"name":"School of Rail Transportation, Soochow University, Suzhou 215131, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.jsamd.2016.08.005","article-title":"High frequency modes meshfree analysis of Reissner\u2013Mindlin plates","volume":"1","author":"Bui","year":"2016","journal-title":"J. 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