{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T11:11:45Z","timestamp":1770981105057,"version":"3.50.1"},"reference-count":88,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T00:00:00Z","timestamp":1663632000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T00:00:00Z","timestamp":1663632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Ho Chi Minh City Open University"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2023,2]]},"DOI":"10.1007\/s00366-022-01726-2","type":"journal-article","created":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T12:07:34Z","timestamp":1663675654000},"page":"993-1018","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":49,"title":["Modified nonlocal couple stress isogeometric approach for bending and free vibration analysis of functionally graded nanoplates"],"prefix":"10.1007","volume":"39","author":[{"given":"Quoc-Hoa","family":"Pham","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3762-4226","authenticated-orcid":false,"given":"Phu-Cuong","family":"Nguyen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Van Ke","family":"Tran","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qui X.","family":"Lieu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Trung Thanh","family":"Tran","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,20]]},"reference":[{"key":"1726_CR1","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/j.engstruct.2014.01.029","volume":"64","author":"TP Vo","year":"2014","unstructured":"Vo TP, Thai H-T, Nguyen T-K, Maheri A, Lee J (2014) Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng Struct 64:12\u201322. https:\/\/doi.org\/10.1016\/j.engstruct.2014.01.029","journal-title":"Eng Struct"},{"issue":"5","key":"1726_CR2","doi-asserted-by":"publisher","first-page":"2324","DOI":"10.1016\/j.matdes.2009.12.006","volume":"31","author":"M Asghari","year":"2010","unstructured":"Asghari M, Ahmadian MT, Kahrobaiyan MH, Rahaeifard M (2010) On the size-dependent behavior of functionally graded micro-beams. Mater Des 31(5):2324\u20132329. https:\/\/doi.org\/10.1016\/j.matdes.2009.12.006","journal-title":"Mater Des"},{"key":"1726_CR3","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1016\/j.compstruct.2015.11.010","volume":"156","author":"LC Trinh","year":"2016","unstructured":"Trinh LC, Vo TP, Osofero AI, Lee J (2016) Fundamental frequency analysis of functionally graded sandwich beams based on the state space approach. Compos Struct 156:263\u2013275.\u00a0https:\/\/doi.org\/10.1016\/j.compstruct.2015.11.010","journal-title":"Compos Struct"},{"issue":"1","key":"1726_CR4","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/s11012-013-9780-1","volume":"49","author":"TP Vo","year":"2013","unstructured":"Vo TP, Thai H-T, Nguyen T-K, Inam F (2013) Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica 49(1):155\u2013168. https:\/\/doi.org\/10.1007\/s11012-013-9780-1","journal-title":"Meccanica"},{"issue":"1\u20133","key":"1726_CR5","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1002\/(SICI)1097-0207(20000110\/30)47:1\/3<663::AID-NME787>3.0.CO;2-8","volume":"47","author":"JN Reddy","year":"2000","unstructured":"Reddy JN (2000) Analysis of functionally graded plates. Int J Numer Methods Eng 47(1\u20133):663\u2013684","journal-title":"Int J Numer Methods Eng"},{"issue":"5","key":"1726_CR6","doi-asserted-by":"publisher","first-page":"3269","DOI":"10.1016\/j.apm.2012.08.008","volume":"37","author":"H-T Thai","year":"2013","unstructured":"Thai H-T, Vo TP (2013) A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Appl Math Model 37(5):3269\u20133281. https:\/\/doi.org\/10.1016\/j.apm.2012.08.008","journal-title":"Appl Math Model"},{"key":"1726_CR7","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/j.compositesb.2014.05.012","volume":"66","author":"V-H Nguyen","year":"2014","unstructured":"Nguyen V-H, Nguyen T-K, Thai H-T, Vo TP (2014) A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates. Compos B Eng 66:233\u2013246. https:\/\/doi.org\/10.1016\/j.compositesb.2014.05.012","journal-title":"Compos B Eng"},{"issue":"1","key":"1726_CR8","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1016\/j.apm.2005.03.009","volume":"30","author":"AM Zenkour","year":"2006","unstructured":"Zenkour AM (2006) Generalized shear deformation theory for bending analysis of functionally graded plates. Appl Math Model 30(1):67\u201384. https:\/\/doi.org\/10.1016\/j.apm.2005.03.009","journal-title":"Appl Math Model"},{"key":"1726_CR9","volume":"156","author":"S Li","year":"2020","unstructured":"Li S, Zheng S, Chen D (2020) Porosity-dependent isogeometric analysis of bi-directional functionally graded plates. Thin-Walled Struct 156:106999","journal-title":"Thin-Walled Struct"},{"issue":"1","key":"1726_CR10","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.ijmecsci.2010.10.002","volume":"53","author":"S Hosseini-Hashemi","year":"2011","unstructured":"Hosseini-Hashemi S, Fadaee M, Atashipour SR (2011) A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates. Int J Mech Sci 53(1):11\u201322","journal-title":"Int J Mech Sci"},{"issue":"3\u20135","key":"1726_CR11","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1016\/j.jsv.2008.06.025","volume":"319","author":"X Zhao","year":"2009","unstructured":"Zhao X, Lee YY, Liew KM (2009) Free vibration analysis of functionally graded plates using the element-free kp-Ritz method. J Sound Vib 319(3\u20135):918\u2013939","journal-title":"J Sound Vib"},{"issue":"11","key":"1726_CR12","doi-asserted-by":"crossref","first-page":"3019","DOI":"10.1016\/j.compstruct.2011.04.028","volume":"93","author":"H Nguyen-Xuan","year":"2011","unstructured":"Nguyen-Xuan H, Tran LV, Nguyen-Thoi T, Vu-Do HC (2011) Analysis of functionally graded plates using an edge-based smoothed finite element method. Compos Struct 93(11):3019\u20133039","journal-title":"Compos Struct"},{"key":"1726_CR13","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.ast.2014.04.005","volume":"36","author":"S Chakraverty","year":"2014","unstructured":"Chakraverty S, Pradhan K (2014) Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions. Aero Sci Technol 36:132\u2013156","journal-title":"Aero Sci Technol"},{"key":"1726_CR14","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.compositesb.2018.02.030","volume":"144","author":"A Karamanl\u0131","year":"2018","unstructured":"Karamanl\u0131 A, Vo TP (2018) Size dependent bending analysis of two directional functionally graded microbeams via a quasi-3D theory and finite element method. Compos B Eng 144:171\u2013183","journal-title":"Compos B Eng"},{"key":"1726_CR15","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.compstruct.2015.11.066","volume":"39","author":"CH Thai","year":"2016","unstructured":"Thai CH, Zenkour AM, Abdel-Wahab M, Nguyen-Xuan H (2016) A simple four-unknown shear and normal deformations theory for functionally graded isotropic and sandwich plates based on isogeometric analysis. Compos Struct 39:77\u201395","journal-title":"Compos Struct"},{"key":"1726_CR16","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1016\/j.dt.2020.06.001","volume":"17","author":"TT Tran","year":"2021","unstructured":"Tran TT, Pham QH, Nguyen-Thoi T (2021) Static and free vibration analyses of functionally graded porous variable-thickness plates using an edge-based smoothed finite element method. Defence Technol 17:971\u2013986","journal-title":"Defence Technol"},{"key":"1726_CR17","first-page":"8278743","volume":"2020","author":"TT Tran","year":"2020","unstructured":"Tran TT, Pham QH, Nguyen-Thoi T (2020) An edge-based smoothed finite element for free vibration analysis of functionally graded porous (FGP) plates on elastic foundation taking into mass (EFTIM). Math Probl Eng 2020:8278743","journal-title":"Math Probl Eng"},{"key":"1726_CR18","doi-asserted-by":"crossref","first-page":"966","DOI":"10.1016\/j.compstruct.2015.08.102","volume":"134","author":"M Sobhy","year":"2015","unstructured":"Sobhy M (2015) A comprehensive study on FGM nanoplates embedded in an elastic medium. Compos Struct 134:966\u2013980","journal-title":"Compos Struct"},{"key":"1726_CR19","doi-asserted-by":"crossref","unstructured":"Tran TT, Pham Q-H, Nguyen-Thoi T (2020) Dynamic analysis of functionally graded porous plates resting on elastic foundation taking into mass subjected to moving loads using an edge-based smoothed finite element method. Shock Vib 2020","DOI":"10.1155\/2020\/8853920"},{"key":"1726_CR20","volume":"13","author":"P-C Nguyen","year":"2022","unstructured":"Nguyen P-C, Pham QH, Tran TT, Nguyen-Thoi T (2022) Effects of partially supported elastic foundation on free vibration of FGP plates using ES-MITC3 elements. Ain Shams Eng J 13:101615","journal-title":"Ain Shams Eng J"},{"issue":"4","key":"1726_CR21","doi-asserted-by":"crossref","first-page":"527","DOI":"10.3390\/sym11040527","volume":"11","author":"H-N Nguyen","year":"2019","unstructured":"Nguyen H-N, Canh TN, Thanh TT, Ke TV, Phan V-D, Thom DV (2019) Finite element modelling of a composite shell with shear connectors. Symmetry 11(4):527","journal-title":"Symmetry"},{"key":"1726_CR22","doi-asserted-by":"crossref","unstructured":"Pham Q-H, Tran TT, Tran VK, Nguyen P-C, Nguyen-Thoi T, Zenkour AM , Bending and hygro-thermo-mechanical vibration analysis of a functionally graded porous sandwich nanoshell resting on elastic foundation. Mech Adv Mater Struct 1\u201321","DOI":"10.1080\/17455030.2023.2177500"},{"issue":"3740","key":"1726_CR23","doi-asserted-by":"publisher","first-page":"2911","DOI":"10.1016\/j.cma.2009.04.011","volume":"198","author":"F Tornabene","year":"2009","unstructured":"Tornabene F (2009) Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput Methods Appl Mech Eng 198(3740):2911\u20132935. https:\/\/doi.org\/10.1016\/j.cma.2009.04.011","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1726_CR24","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1016\/j.compositesb.2015.08.048","volume":"83","author":"JL Mantari","year":"2015","unstructured":"Mantari JL (2015) Refined and generalized hybrid type quasi-3d shear deformation theory for the bending analysis of functionally graded shells. Compos B Eng 83:142\u2013152. https:\/\/doi.org\/10.1016\/j.compositesb.2015.08.048","journal-title":"Compos B Eng"},{"key":"1726_CR25","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1016\/j.compositesb.2013.02.025","volume":"50","author":"J Torabi","year":"2013","unstructured":"Torabi J, Kiani Y, Eslami MR (2013) Linear thermal buckling analysis of truncated hybrid FGM conical shells. Compos B Eng 50:265\u2013272. https:\/\/doi.org\/10.1016\/j.compositesb.2013.02.025","journal-title":"Compos B Eng"},{"issue":"23","key":"1726_CR26","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1016\/j.sna.2004.02.019","volume":"112","author":"Y Fu","year":"2004","unstructured":"Fu Y, Du H, Huang W, Zhang S, Hu M (2004) TiNi-based thin films in MEMS applications: a review. Sens Actuators A 112(23):395\u2013408. https:\/\/doi.org\/10.1016\/j.sna.2004.02.019","journal-title":"Sens Actuators A"},{"issue":"5418","key":"1726_CR27","doi-asserted-by":"publisher","first-page":"1340","DOI":"10.1126\/science.284.5418.1340","volume":"284","author":"RH Baughman","year":"1999","unstructured":"Baughman RH, Cui C, Zakhidov AA, Iqbal Z, Barisci JN, Spinks GM, Wallace GG, Mazzoldi A, Rossi DD, Rinzler AG, Jaschinski O, Roth S, Kertesz M (1999) Carbon nanotube actuators. Science 284(5418):1340\u20131344. https:\/\/doi.org\/10.1126\/science.284.5418.1340","journal-title":"Science"},{"issue":"1","key":"1726_CR28","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1166\/jctn.2008.003","volume":"5","author":"K-T Lau","year":"2008","unstructured":"Lau K-T, Cheung H-Y, Lu J, Yin Y-S, Hui D, Li H-L (2008) Carbon nanotubes for space and bioengineering applications. J Comput Theor Nanosci 5(1):23\u201335. https:\/\/doi.org\/10.1166\/jctn.2008.003","journal-title":"J Comput Theor Nanosci"},{"issue":"9","key":"1726_CR29","doi-asserted-by":"publisher","first-page":"2521","DOI":"10.1016\/j.actamat.2004.01.043","volume":"52","author":"KM Liew","year":"2004","unstructured":"Liew KM, He XQ, Wong CH (2004) On the study of elastic and plastic properties of multi-walled carbon nanotubes under axial tension using molecular dynamics simulation. Acta Mater 52(9):2521\u20132527. https:\/\/doi.org\/10.1016\/j.actamat.2004.01.043","journal-title":"Acta Mater"},{"issue":"12","key":"1726_CR30","doi-asserted-by":"publisher","first-page":"3046","DOI":"10.1021\/jp015591+","volume":"106","author":"SJV Frankland","year":"2002","unstructured":"Frankland SJV, Caglar A, Brenner DW, Griebel M (2002) Molecular simulation of the influence of chemical cross-links on the shear strength of carbon nanotube polymer interfaces. J Phys Chem B 106(12):3046\u20133048. https:\/\/doi.org\/10.1021\/jp015591+","journal-title":"J Phys Chem B"},{"issue":"2","key":"1726_CR31","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1016\/0956-7151(94)90502-9","volume":"42","author":"NA Fleck","year":"1994","unstructured":"Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42(2):475\u2013487. https:\/\/doi.org\/10.1016\/0956-7151(94)90502-9","journal-title":"Acta Metall Mater"},{"issue":"14","key":"1726_CR32","doi-asserted-by":"publisher","first-page":"5109","DOI":"10.1016\/S1359-6454(98)00153-0","volume":"46","author":"JS Stlken","year":"1998","unstructured":"Stlken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46(14):5109\u20135115. https:\/\/doi.org\/10.1016\/S1359-6454(98)00153-0","journal-title":"Acta Mater"},{"issue":"1","key":"1726_CR33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0020-7225(72)90070-5","volume":"10","author":"AC Eringen","year":"1972","unstructured":"Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10(1):1\u201316","journal-title":"Int J Eng Sci"},{"key":"1726_CR34","first-page":"1","volume":"2021","author":"Q-H Pham","year":"2021","unstructured":"Pham Q-H, Nguyen P-C, Tran TT, Nguyen-Thoi T (2021) Free vibration analysis of nanoplates with auxetic honeycomb core using a new third-order finite element method and nonlocal elasticity theory. Engine with Comput 2021:1\u201319","journal-title":"Engine with Comput"},{"key":"1726_CR35","volume":"290","author":"Q-H Pham","year":"2022","unstructured":"Pham Q-H, Nguyen P-C, Tran TT (2022) Dynamic response of porous functionally graded sandwich nanoplates using nonlocal higher-order isogeometric analysis. Compos Struct 290:115565.","journal-title":"Compos Struct"},{"issue":"12","key":"1726_CR36","doi-asserted-by":"crossref","first-page":"1825","DOI":"10.1016\/0022-5096(93)90072-N","volume":"41","author":"NA Fleck","year":"1993","unstructured":"Fleck NA, Hutchinson JW (1993) A phenomenological theory for strain gradient effects in plasticity. J Mech Phys Solids 41(12):1825\u20131857","journal-title":"J Mech Phys Solids"},{"issue":"1","key":"1726_CR37","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/BF00253945","volume":"11","author":"RA Toupin","year":"1962","unstructured":"Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11(1):385\u2013414","journal-title":"Arch Ration Mech Anal"},{"issue":"10","key":"1726_CR38","doi-asserted-by":"crossref","first-page":"2731","DOI":"10.1016\/S0020-7683(02)00152-X","volume":"39","author":"F Yang","year":"2002","unstructured":"Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731\u20132743","journal-title":"Int J Solids Struct"},{"issue":"1","key":"1726_CR39","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/s00161-015-0413-x","volume":"28","author":"X-L Gao","year":"2016","unstructured":"Gao X-L, Zhang GY (2016) A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects. Continuum Mech Thermodyn 28(1):195\u2013213","journal-title":"Continuum Mech Thermodyn"},{"key":"1726_CR40","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1016\/j.ijengsci.2007.04.004","volume":"45","author":"JN Reddy","year":"2007","unstructured":"Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45:288\u2013307","journal-title":"Int J Eng Sci"},{"issue":"11","key":"1726_CR41","doi-asserted-by":"crossref","first-page":"1507","DOI":"10.1016\/j.ijengsci.2010.09.020","volume":"48","author":"JN Reddy","year":"2010","unstructured":"Reddy JN (2010) Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int J Eng Sci 48(11):1507\u20131518","journal-title":"Int J Eng Sci"},{"key":"1726_CR42","doi-asserted-by":"crossref","first-page":"3225","DOI":"10.1098\/rspa.2007.1903","volume":"463","author":"P Lu","year":"2007","unstructured":"Lu P, Zhang PQ, Lee HP, Wang CM, Reddy JN (2007) Non-local elastic plate theories. Proc. R. Soc. A 463:3225\u20133240","journal-title":"Proc. R. Soc. A"},{"issue":"1\u20132","key":"1726_CR43","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.jsv.2009.04.044","volume":"326","author":"R Ahababaei","year":"2009","unstructured":"Ahababaei R, Reddy JN (2009) Nonlocal third-order shear deformation plate theory with application to bending and vibration of plate. J Sound Vib 326(1\u20132):277\u2013289","journal-title":"J Sound Vib"},{"issue":"9","key":"1726_CR44","doi-asserted-by":"crossref","first-page":"1950","DOI":"10.1016\/j.physe.2012.05.024","volume":"44","author":"N Satish","year":"2012","unstructured":"Satish N, Narendar S, Gopalakrishnan S (2012) Thermal vibration analysis of orthotropic nanoplate based on nonlocal continuum mechanics. Phys E 44(9):1950\u20131962","journal-title":"Phys E"},{"key":"1726_CR45","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1016\/j.compstruct.2012.07.006","volume":"95","author":"P Malekzadeh","year":"2013","unstructured":"Malekzadeh P, Shojaee M (2013) Free vibration of nanoplates based on a nonlocal two-variable refined plate theory. Compos Struct 95:443\u2013452","journal-title":"Compos Struct"},{"key":"1726_CR46","doi-asserted-by":"crossref","first-page":"7047","DOI":"10.1016\/j.apm.2013.02.015","volume":"37","author":"AR Shahidi","year":"2013","unstructured":"Shahidi AR, Anjomshoa A, Shahidi SH, Kamrani M (2013) Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method. Appl Math Model 37:7047\u20137061","journal-title":"Appl Math Model"},{"key":"1726_CR47","first-page":"8786373","volume":"2020","author":"VK Tran","year":"2020","unstructured":"Tran VK, Tran TT, Phung MV, Pham QH, Nguyen-Thoi T (2020) A finite element formulation and nonlocal theory for the static and free vibration analysis of the sandwich functionally graded nanoplates resting on elastic foundation. J Nanomater 2020:8786373","journal-title":"J Nanomater"},{"issue":"3","key":"1726_CR48","first-page":"1593","volume":"7","author":"TL Doan","year":"2021","unstructured":"Doan TL, Le PB, Tran TT, Trai VK, Pham QH (2021) Free vibration analysis of functionally graded porous nano-plates with different shapes resting on elastic foundation. J Appl Comput Mech 7(3):1593\u20131605","journal-title":"J Appl Comput Mech"},{"key":"1726_CR49","volume":"26","author":"QH Pham","year":"2021","unstructured":"Pham QH, Tran VK, Tran TT, Nguyen-Thoi T, Nguyen PC (2021) A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation. Case Studies Therm Eng 26:101170","journal-title":"Case Studies Therm Eng"},{"issue":"3","key":"1726_CR50","doi-asserted-by":"crossref","first-page":"1788","DOI":"10.1016\/j.aej.2021.06.082","volume":"61","author":"QH Pham","year":"2021","unstructured":"Pham QH, Tran TT, Tran VK, Nguyen PC, Nguyen-Thoi T (2021) Free vibration of functionally graded porous non-uniform thickness annular-nanoplates resting on elastic foundation using ES-MITC3 element. Alex Eng J 61(3):1788\u20131802","journal-title":"Alex Eng J"},{"key":"1726_CR51","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1007\/s00366-020-01107-7","volume":"38","author":"VK Tran","year":"2022","unstructured":"Tran VK, Pham QH, Nguyen-Thoi T (2022) A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations. Eng Comput 38:1465\u20131490","journal-title":"Eng Comput"},{"issue":"1","key":"1726_CR52","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/BF00253946","volume":"11","author":"RD Mindlin","year":"1962","unstructured":"Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11(1):415\u2013448","journal-title":"Arch Ration Mech Anal"},{"issue":"5","key":"1726_CR53","doi-asserted-by":"publisher","first-page":"904","DOI":"10.1007\/s00033-006-6073-8","volume":"59","author":"SK Park","year":"2007","unstructured":"Park SK, Gao X-L (2007) Variational formulation of a modified couple stress theory and its application to a simple shear problem. Zeitschrift fr angewandte Mathematik und Physik 59(5):904\u2013917. https:\/\/doi.org\/10.1007\/s00033-006-6073-8","journal-title":"Zeitschrift fr angewandte Mathematik und Physik"},{"issue":"2","key":"1726_CR54","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1016\/j.compstruct.2010.09.008","volume":"93","author":"L-L Ke","year":"2011","unstructured":"Ke L-L, Wang Y-S (2011) Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct 93(2):342\u2013350","journal-title":"Compos Struct"},{"key":"1726_CR55","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1016\/j.compstruct.2012.09.011","volume":"96","author":"CMC Roque","year":"2013","unstructured":"Roque CMC, Fidalgo DS, Ferreira AJM, Reddy JN (2013) A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method. Compos Struct 96:532\u2013537","journal-title":"Compos Struct"},{"key":"1726_CR56","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/j.compstruct.2014.11.065","volume":"123","author":"H-T Thai","year":"2015","unstructured":"Thai H-T, Vo TP, Nguyen T-K, Lee J (2015) Size-dependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory. Compos Struct 123:337\u2013349","journal-title":"Compos Struct"},{"issue":"13","key":"1726_CR57","doi-asserted-by":"crossref","first-page":"2757","DOI":"10.1016\/j.ijsolstr.2009.03.004","volume":"46","author":"GC Tsiatas","year":"2009","unstructured":"Tsiatas GC (2009) A new Kirchhoff plate model based on a modified couple stress theory. Int J Solids Struct 46(13):2757\u20132764","journal-title":"Int J Solids Struct"},{"issue":"5","key":"1726_CR58","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/S0894-9166(10)60040-7","volume":"23","author":"L Yin","year":"2010","unstructured":"Yin L, Qian Q, Wang L, Xia W (2010) Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech Solida Sin 23(5):386\u2013393","journal-title":"Acta Mech Solida Sin"},{"key":"1726_CR59","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/s00707-011-0480-4","volume":"220","author":"HM Ma","year":"2011","unstructured":"Ma HM, Gao X-L, Reddy JN (2011) A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech 220:217\u2013235","journal-title":"Acta Mech"},{"issue":"3","key":"1726_CR60","doi-asserted-by":"crossref","first-page":"1128","DOI":"10.1016\/j.compstruct.2011.10.006","volume":"94","author":"JN Reddy","year":"2012","unstructured":"Reddy JN, Kim J (2012) A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos Struct 94(3):1128\u20131143","journal-title":"Compos Struct"},{"issue":"12","key":"1726_CR61","doi-asserted-by":"crossref","first-page":"3664","DOI":"10.1016\/j.compstruct.2012.04.019","volume":"94","author":"JN Reddy","year":"2012","unstructured":"Reddy JN, Berry J (2012) Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress. Compos Struct 94(12):3664\u20133668","journal-title":"Compos Struct"},{"issue":"9","key":"1726_CR62","doi-asserted-by":"crossref","first-page":"2973","DOI":"10.1007\/s00707-015-1370-y","volume":"226","author":"J Kim","year":"2015","unstructured":"Kim J, Reddy JN (2015) A general third-order theory of functionally graded plates with modified couple stress effect and the von karman nonlinearity: theory and finite element analysis. Acta Mech 226(9):2973\u20132998","journal-title":"Acta Mech"},{"key":"1726_CR63","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.compstruct.2015.04.033","volume":"130","author":"L He","year":"2015","unstructured":"He L, Lou J, Zhang E, Wang Y, Bai Y (2015) A size-dependent four variable refined plate model for functionally graded microplates based on modified couple stress theory. Compos Struct 130:107\u2013115","journal-title":"Compos Struct"},{"issue":"1","key":"1726_CR64","doi-asserted-by":"crossref","first-page":"1636","DOI":"10.1016\/j.compositesb.2012.09.065","volume":"45","author":"H-T Thai","year":"2013","unstructured":"Thai H-T, Kim S-E (2013) A size-dependent functionally graded reddy plate model based on a modified couple stress theory. Compos B Eng 45(1):1636\u20131645","journal-title":"Compos B Eng"},{"key":"1726_CR65","doi-asserted-by":"crossref","first-page":"112634","DOI":"10.1016\/j.compstruct.2020.112634","volume":"251","author":"CH Thai","year":"2020","unstructured":"Thai CH, Ferreira AJM, Phung-Van P (2020) A nonlocal strain gradient isogeometric model for free vibration and bending analyses of functionally graded plates. Compos Struct 251:112634","journal-title":"Compos Struct"},{"issue":"13","key":"1726_CR66","doi-asserted-by":"crossref","first-page":"1962","DOI":"10.1016\/j.ijsolstr.2011.03.006","volume":"48","author":"H Askes","year":"2011","unstructured":"Askes H, Aifantis EC (2011) Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct 48(13):1962\u20131990","journal-title":"Int J Solids Struct"},{"issue":"3","key":"1726_CR67","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/BF01176650","volume":"94","author":"KP Soldatos","year":"1992","unstructured":"Soldatos KP (1992) A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech 94(3):195\u2013220","journal-title":"Acta Mech"},{"key":"1726_CR68","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.ijmecsci.2015.11.002","volume":"105","author":"MA Attia","year":"2016","unstructured":"Attia MA, Mahmoud FF (2016) Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories. Int J Mech Sci 105:126\u2013134","journal-title":"Int J Mech Sci"},{"key":"1726_CR69","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1007\/s00366-019-00742-z","volume":"36","author":"F Ebrahimi","year":"2020","unstructured":"Ebrahimi F, Barati MR, Civalek \u00d6 (2020) Application of Chebyshev-Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures. Eng Comput 36:953\u2013964","journal-title":"Eng Comput"},{"issue":"13","key":"1726_CR70","doi-asserted-by":"crossref","first-page":"1121","DOI":"10.1080\/15376494.2017.1365979","volume":"25","author":"F Ebrahimi","year":"2017","unstructured":"Ebrahimi F, Barati MR (2017) A modified nonlocal couple stress-based beam model for vibration analysis of higher-order FG nanobeams. Mech Adv Mater Struct 25(13):1121\u20131132","journal-title":"Mech Adv Mater Struct"},{"key":"1726_CR71","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.physe.2014.05.025","volume":"63","author":"EM Miandoab","year":"2014","unstructured":"Miandoab EM, Pishkenari HN, Yousefi-Koma A, Hoorzad H (2014) Polysilicon nano-beam model based on modified couple stress and Eringen\u2019s nonlocal elasticity theories. Phys E 63:223\u2013228","journal-title":"Phys E"},{"key":"1726_CR72","first-page":"107754632110646","volume":"2022","author":"H Kumar","year":"2022","unstructured":"Kumar H, Mukhopadhyay S (2022) Size-dependent thermoelastic damping analysis in nanobeam resonators based on Eringen\u2019s nonlocal elasticity and modified couple stress theories. J Vib Control 2022:10775463211064689","journal-title":"J Vib Control"},{"key":"1726_CR73","doi-asserted-by":"crossref","DOI":"10.1016\/j.physb.2020.412623","volume":"600","author":"F Attar","year":"2021","unstructured":"Attar F, Khordad R, Zarif A, Modabberasl A (2021) Application of nonlocal modifed couple stress to study of functionally graded piezoelectric plates. Physica B 600:412623","journal-title":"Physica B"},{"key":"1726_CR74","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1140\/epjp\/i2017-11458-0","volume":"132","author":"R Sourki","year":"2017","unstructured":"Sourki R, Hosseini SA (2017) Coupling effects of nonlocal and modified couple stress theories incorporating surface energy on analytical transverse vibration of a weakened nanobeam. Eur Phys J Plus 132:184","journal-title":"Eur Phys J Plus"},{"issue":"39\u201341","key":"1726_CR75","doi-asserted-by":"crossref","first-page":"4135","DOI":"10.1016\/j.cma.2004.10.008","volume":"194","author":"TJR Hughes","year":"2005","unstructured":"Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Engrg 194(39\u201341):4135\u20134195","journal-title":"Comput Methods Appl Mech Engrg"},{"issue":"1\u20135","key":"1726_CR76","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1002\/nme.2968","volume":"87","author":"MJ Borden","year":"2011","unstructured":"Borden MJ, Scott MA, Evams JA, Hughes TJR (2011) Isogeometric finite element data structures based on Bezier extraction of NURBS. Int J Numer Methods Eng 87(1\u20135):15\u201347","journal-title":"Int J Numer Methods Eng"},{"key":"1726_CR77","doi-asserted-by":"publisher","first-page":"222","DOI":"10.1016\/j.compositesb.2014.04.001","volume":"64","author":"H Nguyen-Xuan","year":"2014","unstructured":"Nguyen-Xuan H, Tran LV, Thai CH, Kulasegaram S, Bordas S (2014) Isogeometric analysis of functionally graded plates using a refined plate theory. Compos B Eng 64:222\u2013234. https:\/\/doi.org\/10.1016\/j.compositesb.2014.04.001","journal-title":"Compos B Eng"},{"issue":"8","key":"1726_CR78","doi-asserted-by":"publisher","first-page":"644","DOI":"10.1016\/j.cagd.2010.06.006","volume":"27","author":"AV Vuong","year":"2010","unstructured":"Vuong AV, Heinrich C, Simeon B (2010) ISOGAT: a 2d tutorial MATLAB code for isogeometric analysis. Comput Aided Geomet Des 27(8):644\u2013655. https:\/\/doi.org\/10.1016\/j.cagd.2010.06.006","journal-title":"Comput Aided Geomet Des"},{"key":"1726_CR79","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1016\/j.compstruct.2018.03.021","volume":"192","author":"QX Lieu","year":"2018","unstructured":"Lieu QX, Lee S, Kang J, Lee J (2018) Bending and free vibration analyses of in-plane bi- directional functionally graded plates with variable thickness using isogeometric analysis. Compos Struct 192:434\u2013451","journal-title":"Compos Struct"},{"key":"1726_CR80","doi-asserted-by":"crossref","first-page":"1064","DOI":"10.1080\/15376494.2018.1430273","volume":"26","author":"QX Lieu","year":"2018","unstructured":"Lieu QX, Lee D, Kang J, Lee J (2018) NURBS-based modeling and analysis for free vibration and buckling problems of in-plane bi-directional functionally graded plates. Mech Adv Mater Struct 26:1064\u20131080","journal-title":"Mech Adv Mater Struct"},{"key":"1726_CR81","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.compstruct.2014.12.021","volume":"123","author":"P Phung-Van","year":"2015","unstructured":"Phung-Van P, Abdel-Wahab M, Liew KM, Bordas SPA, Nguyen-Xuan H (2015) Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory. Compos Struct 123:137\u2013149","journal-title":"Compos Struct"},{"key":"1726_CR82","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.cma.2017.12.015","volume":"332","author":"Y Kiani","year":"2018","unstructured":"Kiani Y (2018) Isogeometric large amplitude free vibration of graphene reinforced laminated plates in thermal environment using NURBS formulation. Comput Methods Appl Mech Eng 332:86\u2013101","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1726_CR83","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.matcom.2015.05.008","volume":"117","author":"VP Nguyen","year":"2015","unstructured":"Nguyen VP, Anitescu C, Bordas SPA, Rabczuk T (2015) Isogeometric analysis: an overview and computer implementation aspects. Math Comput Simulat 117:89\u2013116","journal-title":"Math Comput Simulat"},{"issue":"5\u20138","key":"1726_CR84","first-page":"374","volume":"199","author":"M Bischoff","year":"2010","unstructured":"Bischoff M, Echter R (2010) Numerical efficiency, locking and unlocking of NURBS finite elements. Comput Methods Appl Mech Eng 199(5\u20138):374\u2013382","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1726_CR85","doi-asserted-by":"crossref","first-page":"904","DOI":"10.1016\/j.cma.2016.10.002","volume":"313","author":"HX Nguyen","year":"2017","unstructured":"Nguyen HX, Nguyen TN, Abdel-Wahab M, Bordas SPA, Nguyen-Xuan H, Vo TP (2017) Isogeometric analysis for functionally graded microplates based on modified couple stress theory. Comput Methods Appl Mech Eng 313:904\u2013940","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1726_CR86","doi-asserted-by":"crossref","first-page":"2731","DOI":"10.1016\/S0020-7683(02)00152-X","volume":"39","author":"F Yang","year":"2002","unstructured":"Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731\u20132743","journal-title":"Int J Solids Struct"},{"key":"1726_CR87","doi-asserted-by":"crossref","first-page":"5934","DOI":"10.1016\/j.apm.2014.03.036","volume":"38","author":"A Anjomshoa","year":"2014","unstructured":"Anjomshoa A, Shahidi AR, Hassani B, Jomehzadeh E (2014) Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory. Appl Math Model 38:5934\u20135955","journal-title":"Appl Math Model"},{"key":"1726_CR88","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.07.021","author":"N-T Nguyen","year":"2015","unstructured":"Nguyen N-T, Hui D, Lee J, Nguyen-Xuan H (2015) An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Methods Appl Mech Eng. https:\/\/doi.org\/10.1016\/j.cma.2015.07.021","journal-title":"Comput Methods Appl Mech Eng"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-022-01726-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-022-01726-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-022-01726-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,26]],"date-time":"2023-03-26T22:03:04Z","timestamp":1679868184000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-022-01726-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,20]]},"references-count":88,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["1726"],"URL":"https:\/\/doi.org\/10.1007\/s00366-022-01726-2","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,20]]},"assertion":[{"value":"24 February 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 July 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 September 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there are no conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}}]}}