{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:48:45Z","timestamp":1760237325580,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T00:00:00Z","timestamp":1585785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Polish National Science Centre","award":["OPUS 14 No. 2017\/27\/B\/ST8\/01330"],"award-info":[{"award-number":["OPUS 14 No. 2017\/27\/B\/ST8\/01330"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Vibrations of single-layered graphene sheets subjected to a longitudinal magnetic field are considered. The Winkler-type and Pasternak-type foundation models are employed to reproduce the surrounding elastic medium. The governing equation is based on the modified couple stress theory and Kirchhoff\u2013Love hypotheses. The effect of the magnetic field is taken into account due to the Lorentz force deriving from Maxwell\u2019s equations. The developed approach is based on applying the Ritz method. The proposed method is tested by a comparison with results from the existing literature. The numerical calculations are performed for different boundary conditions, including the mixed ones. The influence of the material length scale parameter, the elastic foundation parameters, the magnetic parameter and the boundary conditions on vibration frequencies is studied. It is observed that an increase of the magnetic parameter, as well as the elastic foundation parameters, brings results closer to the classical plate theory results. Furthermore, the current study can be applied to the design of microplates and nanoplates and their optimal usage.<\/jats:p>","DOI":"10.3390\/sym12040515","type":"journal-article","created":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T13:39:34Z","timestamp":1585834774000},"page":"515","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Ritz Method in Vibration Analysis for Embedded Single-Layered Graphene Sheets Subjected to In-Plane Magnetic Field"],"prefix":"10.3390","volume":"12","author":[{"given":"Olga","family":"Mazur","sequence":"first","affiliation":[{"name":"Biomechanics and Mechatronics, Department of Automation, Lodz University of Technology, 90-924 Lodz, Poland"},{"name":"Department of Applied Mathematics, National Technical University Kharkiv Polytechnic Institute, Kyrpychova Str 2, 61002 Kharkiv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0387-921X","authenticated-orcid":false,"given":"Jan","family":"Awrejcewicz","sequence":"additional","affiliation":[{"name":"Biomechanics and Mechatronics, Department of Automation, Lodz University of Technology, 90-924 Lodz, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1016\/j.spmi.2012.03.009","article-title":"High-performance ZnO nanoflake moisture sensor","volume":"51","author":"Bu","year":"2012","journal-title":"Superlattices Microstruct."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1016\/j.materresbull.2012.10.047","article-title":"Hieu NCrystalline mesoporous tungsten oxide nanoplate monoliths synthesized by directed soft template method for highly sensitive NO2 gas sensor applications","volume":"48","author":"Hoa","year":"2013","journal-title":"Mater. Res. Bull."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1557\/mrs2004.14","article-title":"Bio-resorbable nanoceramics for gene and drug delivery","volume":"29","author":"Kriven","year":"2004","journal-title":"Mrs Bull."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.bios.2012.11.029","article-title":"Fabrication of large-scale gold nanoplate films as highly active SERS substrates for label-free DNA detection","volume":"43","author":"Bi","year":"2013","journal-title":"Biosens. Bioelectron."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1016\/j.solmat.2010.02.004","article-title":"Heat transfer enhancement of paraffin wax using graphite foam for thermal energy storage","volume":"94","author":"Zhong","year":"2010","journal-title":"Sol. Energy Mater. Sol. Cells"},{"key":"ref_6","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. Mech. Phys. Solids"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1557\/JMR.2001.0146","article-title":"Torsion and bending of micron-scaled structures","volume":"16","author":"Chong","year":"2001","journal-title":"J. Mater. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5109","DOI":"10.1016\/S1359-6454(98)00153-0","article-title":"A microbend test method for measuring the plasticity length scale","volume":"46","author":"Evans","year":"1998","journal-title":"Acta Mater."},{"key":"ref_9","unstructured":"Delphenich, D.H. (1909). Theory of Deformable Bodies, Scientific Library A. Herman and Sons."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mindlin, R.D., and Tiersten, H.F. (1962). Effects of couple-stresses in linear elasticity. Arch. Ration Mech. Anal., 11.","DOI":"10.1007\/BF00253946"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/BF00253945","article-title":"Elastic materials with couple stresses","volume":"11","author":"Toupin","year":"1962","journal-title":"Arch. Ration Mech. Anal."},{"key":"ref_12","first-page":"17","article-title":"Couple stresses in the theory of elasticity, I and II","volume":"67","author":"Koiter","year":"1964","journal-title":"Proc. K Ned. Akad. Wet B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4703","DOI":"10.1063\/1.332803","article-title":"On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves","volume":"54","author":"Eringen","year":"1983","journal-title":"J. Appl. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2731","DOI":"10.1016\/S0020-7683(02)00152-X","article-title":"Couple stress based strain gradient theory for elasticity","volume":"39","author":"Yang","year":"2002","journal-title":"Int. J. Solids Struct."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2757","DOI":"10.1016\/j.ijsolstr.2009.03.004","article-title":"A new Kirchhoff plate model based on a modified couple stress theory","volume":"46","author":"Tsiatas","year":"2009","journal-title":"Int. J. Solids Struct."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/S0894-9166(10)60040-7","article-title":"Vibration analysis of microscale plates based on modified couple stress theory","volume":"23","author":"Yin","year":"2010","journal-title":"Acta Mech. Solida Sin."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3807","DOI":"10.1007\/s00707-015-1437-9","article-title":"Size-dependent vibration of a microplate under the action of a moving load based on the modified couple stress theory","volume":"226","author":"Simsek","year":"2015","journal-title":"Acta Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1016\/j.physe.2010.11.005","article-title":"The size-dependent vibration analysis of micro-plates based on a modified couple stress theory","volume":"43","author":"Jomehzadeh","year":"2011","journal-title":"Physica E"},{"key":"ref_19","unstructured":"Sapountzakis, E.J. (2010). A Microstructure-Dependent Orthotropic Plate Model Based on a Modified Couple Stress Theory Recent Developments in Boundary Element Methods, National Technical University of Athens."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2373","DOI":"10.1016\/j.asej.2017.05.003","article-title":"Linear free vibration of micro-\/nano-plates with cut-out in thermal environment via modified couple stress theory and Ritz method","volume":"9","author":"Ziaee","year":"2018","journal-title":"Ain Shams Eng. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1007\/s00707-014-1249-3","article-title":"Size effect on the static, dynamic and buckling analysis of orthotropic Kirchhoff-type skew micro-plates based on a modified couple stress theory: Comparison with the nonlocal elasticity theory","volume":"226","author":"Tsiatas","year":"2015","journal-title":"Acta Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.matdes.2012.06.002","article-title":"Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory","volume":"42","author":"Civalek","year":"2012","journal-title":"Mater. Des."},{"key":"ref_23","first-page":"285","article-title":"Ritz Solution of Buckling and Vibration Problem of nanoplates embedded in an elastic medium","volume":"35","author":"Bastami","year":"2017","journal-title":"Sigma J. Eng. Nat. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1590\/S1679-78252014000300005","article-title":"Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium","volume":"11","author":"Mohammadi","year":"2014","journal-title":"Lat. Am. J. Solids Struct."},{"key":"ref_25","first-page":"128","article-title":"Small Scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial in-plane pre-load via nonlocal elasticity theory","volume":"4","author":"Mohammadi","year":"2012","journal-title":"J. Solid Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/j.compscitech.2004.11.011","article-title":"Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium","volume":"65","author":"Behfar","year":"2005","journal-title":"Compos. Sci. Technol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"6034","DOI":"10.1063\/1.1616638","article-title":"Enhancement of thermal and electrical properties of carbon nanotube polymer composites by magnetic field processing","volume":"94","author":"Choi","year":"2003","journal-title":"J. Appl. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4166","DOI":"10.1016\/j.tsf.2011.02.007","article-title":"Magnetic field enhances the graphitized structure and field emission effect of carbon nanotubes","volume":"519","author":"Lee","year":"2011","journal-title":"Thin Solid Film."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"155436","DOI":"10.1103\/PhysRevB.81.155436","article-title":"Effect of a magnetic field on the two-phonon Raman scattering in graphene","volume":"81","author":"Faugeras","year":"2010","journal-title":"Phys. Rev. B Condens. Matter Mater. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1021\/nl802810g","article-title":"Room-temperature ferromagnetism of graphene","volume":"9","author":"Wang","year":"2009","journal-title":"Nano Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.compstruct.2012.09.005","article-title":"In-plane magnetic field affected transverse vibration of embedded single-layer graphene sheets using equivalent nonlocal elasticity approach","volume":"96","author":"Murmu","year":"2013","journal-title":"Compos. Struct."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3511","DOI":"10.1007\/s12206-014-0811-1","article-title":"Revisiting the free transverse vibration of embedded single-layer graphene sheets acted upon by an in-plane magnetic field","volume":"28","author":"Kiani","year":"2014","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_33","first-page":"193","article-title":"2D-Magnetic Field and Biaxiall In-Plane Pre-Load Effects on the Vibration of Double Bonded Orthotropic Graphene Sheets","volume":"5","author":"Maboudi","year":"2013","journal-title":"J. Solid Mech."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2165","DOI":"10.1007\/s00707-017-1815-6","article-title":"Forced transverse vibrations of an elastically connected nonlocal orthotropic double-nanoplate system subjected to an in-plane magnetic field","volume":"228","author":"Atanasov","year":"2017","journal-title":"Acta Mech."},{"key":"ref_35","unstructured":"Vol\u2019mir, A.S. (1972). Nonlinear Dynamics of Plates and Shells, Nauka."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1842","DOI":"10.1016\/j.compstruct.2011.01.020","article-title":"Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation","volume":"93","author":"Baferani","year":"2011","journal-title":"Compos. Struct."},{"key":"ref_37","first-page":"815","article-title":"Nonlinear free vibration analysis of functionally graded plate resting on elastic foundation in thermal environment using higher-order shear deformation theory","volume":"26","author":"Parida","year":"2019","journal-title":"Sci. Iran. B"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/S0029-8018(97)00022-X","article-title":"Vibrations of a rectangular orthotropic plate with a free edge: A comparison of analytical and numerical results","volume":"25","author":"Rossi","year":"1998","journal-title":"Ocean Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/515\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:14:43Z","timestamp":1760174083000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/515"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,2]]},"references-count":38,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040515"],"URL":"https:\/\/doi.org\/10.3390\/sym12040515","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,4,2]]}}}