{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:19:02Z","timestamp":1774628342940,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,17]],"date-time":"2022-03-17T00:00:00Z","timestamp":1647475200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12002088"],"award-info":[{"award-number":["12002088"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12002225"],"award-info":[{"award-number":["12002225"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12072301"],"award-info":[{"award-number":["12072301"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["2020J05103"],"award-info":[{"award-number":["2020J05103"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["2020J05102"],"award-info":[{"award-number":["2020J05102"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["2020J05101"],"award-info":[{"award-number":["2020J05101"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Education and scientific research foundation for young teachers in Fujian Province","award":["JAT190008"],"award-info":[{"award-number":["JAT190008"]}]},{"name":"Starting Grants of Fuzhou University","award":["GXRC-20017"],"award-info":[{"award-number":["GXRC-20017"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The present work aims to study the free vibration, buckling and post-buckling behaviors of bidirectional functionally graded (BDFG) microbeams. The material properties of a BDFG microbeam were varied continuously in both thickness and axial directions. Furthermore, four different kinds of material distribution function were taken into consideration, two of which were symmetrical in the thickness direction, and the remaining two were asymmetrical. Employing the Timoshenko beam theory and the consistent couple stress theory (CCST), the governing equations and associated boundary conditions of BDFG microbeams were formulated by Hamilton\u2019s principle. The differential quadrature method (DQM) and Newton\u2019s method were applied to solve the eigenvalue problems and buckling path, respectively. Finally, several parametric investigations were carried out to probe the influence of material distribution functions, length to thickness ratio, gradient indexes and size effect on the vibration and buckling behaviors of BDFG microbeam under different boundary conditions.<\/jats:p>","DOI":"10.3390\/sym14030602","type":"journal-article","created":{"date-parts":[[2022,3,20]],"date-time":"2022-03-20T21:37:17Z","timestamp":1647812237000},"page":"602","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["The Consistent Couple Stress Theory-Based Vibration and Post-Buckling Analysis of Bi-directional Functionally Graded Microbeam"],"prefix":"10.3390","volume":"14","author":[{"given":"Rong","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China"}]},{"given":"Hongbai","family":"Bai","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2175-3020","authenticated-orcid":false,"given":"Xiaochao","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/j.ijpharm.2018.01.030","article-title":"BioMEMS for biosensors and closed-loop drug delivery","volume":"544","author":"Coffel","year":"2018","journal-title":"Int. J. Pharm."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"112003","DOI":"10.1016\/j.sna.2020.112003","article-title":"A novel single axis capacitive MEMS accelerometer with double-sided suspension beams fabricated using \u03bc WEDM","volume":"309","author":"Tahmasebipour","year":"2020","journal-title":"Sens. Actuators A Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"103455","DOI":"10.1016\/j.ijnonlinmec.2020.103455","article-title":"Exploiting nonlinearity to enhance the sensitivity of mode-localized mass sensor based on electrostatically coupled MEMS resonators","volume":"121","author":"Lyu","year":"2020","journal-title":"Int. J. Nonlin. Mech."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"107100","DOI":"10.1016\/j.ijmecsci.2022.107100","article-title":"Investigation of size effect on band structure of 2D nano-scale phononic crystal based on nonlocal strain gradient theory","volume":"219","author":"Jin","year":"2022","journal-title":"Int. J. Mech. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"114923","DOI":"10.1016\/j.compstruct.2021.114923","article-title":"Static and dynamic modeling of functionally graded Euler-Bernoulli microbeams based on reformulated strain gradient elasticity theory using isogeometric analysis","volume":"280","author":"Dinachandra","year":"2022","journal-title":"Compos. Struct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/j.apm.2013.08.011","article-title":"Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory","volume":"38","author":"Roostai","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.mechrescom.2017.12.005","article-title":"Size-dependent post-buckling behaviors of geometrically imperfect microbeams","volume":"88","author":"Chen","year":"2018","journal-title":"Mech. Res. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"108376","DOI":"10.1016\/j.compositesb.2020.108376","article-title":"30 Years of functionally graded materials: An overview of manufacturing methods, Applications and Future Challenges","volume":"201","author":"Saleh","year":"2020","journal-title":"Compos. Part B Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/j.istruc.2021.12.004","article-title":"Functionally graded materials for marine risers by additive manufacturing for high-temperature applications: Experimental investigations","volume":"35","author":"Chandrasekaran","year":"2022","journal-title":"Structures"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4623","DOI":"10.1016\/j.jeurceramsoc.2020.05.061","article-title":"Hydroxyapatite\/bioactive glass functionally graded materials (FGM) for bone tissue engineering","volume":"40","author":"Luginina","year":"2020","journal-title":"J. Eur. Ceram. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"113596","DOI":"10.1016\/j.compstruct.2021.113596","article-title":"State of the art in functionally graded materials","volume":"262","author":"Boggarapu","year":"2021","journal-title":"Compos. Struct."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.ijengsci.2012.12.002","article-title":"Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory","volume":"64","author":"Reddy","year":"2013","journal-title":"Int. J. Eng. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1016\/j.compstruct.2012.11.020","article-title":"Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory","volume":"98","author":"Civalek","year":"2013","journal-title":"Compos. Struct."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4971","DOI":"10.1016\/j.apm.2011.12.035","article-title":"Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory","volume":"36","author":"Nateghi","year":"2012","journal-title":"Appl. Math. Model."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1016\/j.compstruct.2017.01.032","article-title":"Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory","volume":"165","author":"Li","year":"2017","journal-title":"Compos. Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1016\/j.ijmecsci.2016.03.006","article-title":"Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory","volume":"110","author":"Shariati","year":"2016","journal-title":"Int. J. Mech. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1016\/j.apm.2018.11.004","article-title":"Free vibration, buckling and dynamic stability of bi-directional FG microbeam with a variable length scale parameter embedded in elastic medium","volume":"67","author":"Chen","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1016\/j.compstruct.2019.01.080","article-title":"Free vibration analysis of bidirectional-functionally graded and double-tapered rotating micro-beam in thermal environment using modified couple stress theory","volume":"215","author":"Bhattacharya","year":"2019","journal-title":"Compos. Struct."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/j.ijmecsci.2018.12.001","article-title":"Static and dynamic analysis of the postbuckling of bi-directional functionally graded material microbeams","volume":"151","author":"Chen","year":"2019","journal-title":"Int. J. Mech. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.compstruct.2018.07.045","article-title":"Nonlinear bending, buckling and vibration of bi-directional functionally graded nanobeams","volume":"204","author":"Yang","year":"2018","journal-title":"Compos. Struct."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1016\/j.compstruct.2018.10.106","article-title":"Postbuckling analysis of bi-directional functionally graded imperfect beams based on a novel third-order shear deformation theory","volume":"209","author":"Lei","year":"2019","journal-title":"Compos. Struct."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijengsci.2016.03.001","article-title":"Buckling analysis of arbitrary two-directional functionally graded Euler\u2013Bernoulli nano-beams based on nonlocal elasticity theory","volume":"103","author":"Nejad","year":"2016","journal-title":"Int. J. Eng. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ast.2017.02.019","article-title":"Buckling analysis on the bi-dimensional functionally graded porous tapered nano-\/micro-scale beams","volume":"66","author":"Shafiei","year":"2017","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2496","DOI":"10.1016\/j.ijsolstr.2011.05.002","article-title":"Couple stress theory for solids","volume":"48","author":"Hadjesfandiari","year":"2011","journal-title":"Int. J. Solids Struct."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1016\/j.ijsolstr.2012.12.021","article-title":"Fundamental solutions for isotropic size-dependent couple stress elasticity","volume":"50","author":"Hadjesfandiari","year":"2013","journal-title":"Int. J. Solids Struct."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4097","DOI":"10.1007\/s00542-019-04351-0","article-title":"Nonlinear free vibration analysis of pre-actuated isotropic piezoelectric cantilever Nano-beams","volume":"25","author":"Vaghefpour","year":"2019","journal-title":"Microsyst. Technol."},{"key":"ref_29","first-page":"951","article-title":"A Size-dependent Bernoulli-Euler Beam Formulation based on a New Model of Couple Stress Theory","volume":"27","author":"Alashti","year":"2014","journal-title":"Int. J. Eng."},{"key":"ref_30","first-page":"663","article-title":"Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory","volume":"26","author":"Hadi","year":"2018","journal-title":"Steel Compos. Struct."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3165","DOI":"10.1007\/s00542-018-4216-4","article-title":"Pull-in behavior of functionally graded micro\/nano-beams for MEMS and NEMS switches","volume":"25","author":"Hosseini","year":"2019","journal-title":"Microsyst. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.ijmecsci.2014.12.009","article-title":"Size effects on nanomechanical behaviors of nanoelectronics devices based on consistent couple-stress theory","volume":"92","year":"2015","journal-title":"Int. J. Mech. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1016\/j.apm.2019.10.058","article-title":"Development of size-dependent consistent couple stress theory of Timoshenko beams","volume":"79","author":"Alavi","year":"2020","journal-title":"Appl. Math. Model."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"104085","DOI":"10.1016\/j.mechmat.2021.104085","article-title":"A unified size-dependent plate theory for static bending and free vibration analyses of micro- and nano-scale plates based on the consistent couple stress theory","volume":"162","author":"Wu","year":"2021","journal-title":"Mech. Mater."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"110896","DOI":"10.1016\/j.compstruct.2019.110896","article-title":"Nonlinear resonant behaviors of bi-directional functionally graded material microbeams: One-\/two-parameter bifurcation analyses","volume":"223","author":"Chen","year":"2019","journal-title":"Compos. Struct."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"115115","DOI":"10.1016\/j.compstruct.2021.115115","article-title":"A domain decomposition method based vibration analysis of BDFGs imperfect beams with arbitrary boundary conditions","volume":"284","author":"Chen","year":"2022","journal-title":"Compos. Struct."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"114142","DOI":"10.1016\/j.compstruct.2021.114142","article-title":"Imperfection sensitivity of nonlinear primary resonance behavior in bi-directional functionally graded porous material beam","volume":"271","author":"Chen","year":"2021","journal-title":"Compos. Struct."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1016\/j.compstruct.2017.10.087","article-title":"Nonlinear bending of a two-dimensionally functionally graded beam","volume":"184","author":"Li","year":"2018","journal-title":"Compos. Struct."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1007\/s00366-020-01080-1","article-title":"Nonlinear thermal buckling and postbuckling analysis of bidirectional functionally graded tapered microbeams based on Reddy beam theory","volume":"38","author":"Attia","year":"2022","journal-title":"Eng. Comput.-Ger."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/j.ijmecsci.2018.03.035","article-title":"A standard experimental method for determining the material length scale based on modified couple stress theory","volume":"141","author":"Li","year":"2018","journal-title":"Int. J. Mech. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2382","DOI":"10.1016\/j.jmps.2011.06.008","article-title":"Microstructure-dependent couple stress theories of functionally graded beams","volume":"59","author":"Reddy","year":"2011","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/j.compstruct.2016.04.034","article-title":"Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions","volume":"149","year":"2016","journal-title":"Compos. Struct."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.compstruct.2016.11.048","article-title":"Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams","volume":"161","author":"Yang","year":"2017","journal-title":"Compos. Struct."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/602\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:38:36Z","timestamp":1760135916000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/602"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,17]]},"references-count":43,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["sym14030602"],"URL":"https:\/\/doi.org\/10.3390\/sym14030602","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,17]]}}}