{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T00:01:30Z","timestamp":1774569690254,"version":"3.50.1"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,2,24]],"date-time":"2018-02-24T00:00:00Z","timestamp":1519430400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2019,1]]},"DOI":"10.1007\/s00366-018-0594-6","type":"journal-article","created":{"date-parts":[[2018,2,24]],"date-time":"2018-02-24T07:22:16Z","timestamp":1519456936000},"page":"215-228","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Non-classical plate model for FGMs"],"prefix":"10.1007","volume":"35","author":[{"given":"Marzieh","family":"Alizadeh","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0059-6479","authenticated-orcid":false,"given":"A. M.","family":"Fattahi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,2,24]]},"reference":[{"key":"594_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruct.2011.06.024","author":"R Ansari","year":"2011","unstructured":"Ansari R, Gholami R, Sahmani S (2011) Free vibration of size-dependent functionally graded microbeams based on a strain gradient theory. Compos Struct. \n                    https:\/\/doi.org\/10.1016\/j.compstruct.2011.06.024","journal-title":"Compos Struct"},{"key":"594_CR2","doi-asserted-by":"publisher","first-page":"1435","DOI":"10.1016\/j.matdes.2010.08.046","volume":"32","author":"M Asghari","year":"2011","unstructured":"Asghari M, Rahaeifard M, Kahrobaiyan MH, Ahmadian MT (2011) The modified couple stress functionally graded Timoshenko beam formulation. Mater Des 32:1435\u20131443","journal-title":"Mater Des"},{"key":"594_CR3","doi-asserted-by":"publisher","first-page":"906","DOI":"10.1016\/j.engstruct.2009.12.046","volume":"32","author":"M Batista","year":"2010","unstructured":"Batista M (2010) An elementary derivation of basic equations of the Reissner and Mindlin plate theories. Eng Struct 32:906\u2013909","journal-title":"Eng Struct"},{"key":"594_CR4","doi-asserted-by":"publisher","first-page":"1551","DOI":"10.1016\/S0022-5096(03)00050-4","volume":"51","author":"I Chasiotis","year":"2003","unstructured":"Chasiotis I, Knauss WG (2003) The mechanical strength of polysilicon films: part 2. Size effects associated with elliptical and circular perforations. J Mech Phys Solids 51:1551\u20131572","journal-title":"J Mech Phys Solids"},{"key":"594_CR5","first-page":"89","volume":"2","author":"AC Eringen","year":"1964","unstructured":"Eringen AC, Suhubi ES (1964) Nonlinear theory of simple microelastic solid-I. Int J Eng Sci 2:89\u2013203","journal-title":"Int J Eng Sci"},{"key":"594_CR6","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1016\/0020-7225(64)90017-5","volume":"2","author":"AC Eringen","year":"1964","unstructured":"Eringen AC, Suhubi ES (1964) Nonlinear theory of simple microelastic solid-II. Int J Eng Sci 2:389\u2013404","journal-title":"Int J Eng Sci"},{"key":"594_CR7","doi-asserted-by":"publisher","first-page":"4703","DOI":"10.1063\/1.332803","volume":"54","author":"AC Eringen","year":"1983","unstructured":"Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703\u20134710","journal-title":"J Appl Phys"},{"key":"594_CR8","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/j.compstruct.2009.05.008","volume":"91","author":"ME Fares","year":"2009","unstructured":"Fares ME, Elmarghany MK, Atta D (2009) An efficient and simple refined theory for bending and vibration of functionally graded plates. Compos Struct 91:296\u2013305","journal-title":"Compos Struct"},{"key":"594_CR9","doi-asserted-by":"publisher","first-page":"1825","DOI":"10.1016\/0022-5096(93)90072-N","volume":"41","author":"NA Fleck","year":"1993","unstructured":"Fleck NA, Hutchinson JW (1993) Phenomenological theory for strain gradient effects in plasticity. J Mech Phys Solids 41:1825\u20131857","journal-title":"J Mech Phys Solids"},{"key":"594_CR10","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\u2014theory and experiment. Acta Metall Mater 42:475\u2013484","journal-title":"Acta Metall Mater"},{"key":"594_CR11","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1016\/j.sna.2004.02.019","volume":"112","author":"YQ Fu","year":"2004","unstructured":"Fu YQ, Du HJ, Huang WM, Zhang S, Hu M (2004) TiNi-based thin films in MEMS applications: a review. J Sensors Actuators A 112:395\u2013408","journal-title":"J Sensors Actuators A"},{"key":"594_CR12","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/j.compstruct.2006.01.012","volume":"79","author":"M Ganapathi","year":"2007","unstructured":"Ganapathi M (2007) Dynamic stability characteristics of functionally graded materials shallow spherical shells. Compos Struct 79:338\u2013343","journal-title":"Compos Struct"},{"key":"594_CR13","doi-asserted-by":"publisher","first-page":"1006","DOI":"10.1080\/01495730802250714","volume":"31","author":"DJ Hasanyan","year":"2008","unstructured":"Hasanyan DJ, Batra RC, Harutyunyan RC (2008) Pull-in instabilities in functionally graded microthermoelectromechanical systems. J Therm Stresses 31:1006\u20131021","journal-title":"J Therm Stresses"},{"key":"594_CR14","doi-asserted-by":"publisher","first-page":"342","DOI":"10.1016\/j.compstruct.2010.09.008","volume":"93","author":"LL Ke","year":"2011","unstructured":"Ke LL, Wang YS (2011) Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct 93:342\u2013350","journal-title":"Compos Struct"},{"issue":"7","key":"594_CR15","doi-asserted-by":"publisher","first-page":"1387","DOI":"10.1016\/j.physe.2011.03.009","volume":"43","author":"LL Ke","year":"2011","unstructured":"Ke LL, Wang YS, Wang ZD (2011) Thermal effect on free vibration and buckling of size-dependent microbeams. Phys E\u00a043(7):1387\u20131393","journal-title":"Phys E"},{"issue":"1","key":"594_CR16","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1016\/j.ijengsci.2010.12.008","volume":"50","author":"LL Ke","year":"2011","unstructured":"Ke LL, Wang YS, Yang J, Kitipornchai S (2011) Nonlinear free vibration of size-dependent functionally graded microbeams. Int J Eng Sci 50(1):256\u2013267","journal-title":"Int J Eng Sci"},{"key":"594_CR17","first-page":"17","volume":"67","author":"WT Koiter","year":"1964","unstructured":"Koiter WT (1964) couple stresses in the theory of elasticity I and II. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 67:17\u201344","journal-title":"Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen"},{"key":"594_CR18","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1016\/j.ijengsci.2008.08.008","volume":"47","author":"S Kong","year":"2009","unstructured":"Kong S, Zhou S, Nie Z, Wang K (2009) Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int J Eng Sci 47:487\u2013498","journal-title":"Int J Eng Sci"},{"key":"594_CR19","doi-asserted-by":"publisher","first-page":"3784","DOI":"10.1557\/JMR.1999.0512","volume":"14","author":"DCC Lam","year":"1999","unstructured":"Lam DCC, Chong ACM (1999) Indentation model and strain gradient plasticity law for glassy polymers. J Mater Res 14:3784\u20133788","journal-title":"J Mater Res"},{"key":"594_CR20","doi-asserted-by":"publisher","first-page":"1477","DOI":"10.1016\/S0022-5096(03)00053-X","volume":"51","author":"DCC Lam","year":"2003","unstructured":"Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477\u20131508","journal-title":"J Mech Phys Solids"},{"key":"594_CR21","doi-asserted-by":"publisher","first-page":"3063","DOI":"10.1088\/0957-4484\/17\/12\/042","volume":"17","author":"Z Lee","year":"2006","unstructured":"Lee Z, Ophus C, Fischer LM, Nelson-Fitzpatrick N, Westra KL, Evoy S (2006) Metallic NEMS components fabricated from nanocomposite Al-Mo films. Nanotechnology 17:3063\u20133070","journal-title":"Nanotechnology"},{"key":"594_CR22","doi-asserted-by":"publisher","first-page":"2789","DOI":"10.1002\/nme.1869","volume":"69","author":"Y Liu","year":"2007","unstructured":"Liu Y, Soh C-K (2007) Shear correction for Mindlin type plate and shell elements. Int J Num Methods Eng 69:2789\u20132806","journal-title":"Int J Num Methods Eng"},{"key":"594_CR23","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/s00707-011-0480-4","volume":"220","author":"HM Ma","year":"2011","unstructured":"Ma HM, Gao XL, 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"},{"key":"594_CR24","doi-asserted-by":"publisher","first-page":"1060","DOI":"10.1088\/0960-1317\/15\/5\/024","volume":"15","author":"AW McFarland","year":"2005","unstructured":"McFarland AW, Colton JS (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15:1060\u20131067","journal-title":"J Micromech Microeng"},{"key":"594_CR25","unstructured":"Mindlin RD (1951) American Society of Mechanical Engineers. J Appl Mech 73"},{"key":"594_CR26","doi-asserted-by":"publisher","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 11:415\u2013448","journal-title":"Arch Ration Mech Anal"},{"key":"594_CR27","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/BF00248490","volume":"16","author":"RD Mindlin","year":"1964","unstructured":"Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51\u201378","journal-title":"Arch Ration Mech Anal"},{"key":"594_CR28","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1016\/0020-7683(65)90006-5","volume":"1","author":"RD Mindlin","year":"1965","unstructured":"Mindlin RD (1965) Second gradient of strain and surface tension in linear elasticity. Int J Solids Struct 1:417\u2013438","journal-title":"Int J Solids Struct"},{"key":"594_CR29","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/0020-7683(68)90036-X","volume":"4","author":"RD Mindlin","year":"1968","unstructured":"Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4:109\u2013124","journal-title":"Int J Solids Struct"},{"key":"594_CR30","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1006\/jsvi.1996.0885","volume":"202","author":"NG Stephen","year":"1997","unstructured":"Stephen NG (1997) Mindlin plate theory: best shear coefficient and higher spectra validity. J Sound Vib 202:539\u2013553","journal-title":"J Sound Vib"},{"key":"594_CR31","volume-title":"Theory of elasticity","author":"SP Timoshenko","year":"1970","unstructured":"Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York","edition":"3"},{"key":"594_CR32","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1007\/BF00253050","volume":"17","author":"RA Toupin","year":"1964","unstructured":"Toupin RA (1964) Theory of elasticity with couple stresses. Arch Ration Mech Anal 17:85\u2013112","journal-title":"Arch Ration Mech Anal"},{"key":"594_CR33","doi-asserted-by":"publisher","first-page":"2757","DOI":"10.1016\/j.ijsolstr.2009.03.004","volume":"46","author":"GC Tsiatas","year":"2009","unstructured":"Tsiatas GC (2009) A new Krichhoff plate model based on a modified couple stress theory. Int J Solids Struct 46:2757\u20132764","journal-title":"Int J Solids Struct"},{"key":"594_CR34","first-page":"399","volume":"8","author":"I Vardoulakis","year":"1998","unstructured":"Vardoulakis I, Exadaktylos G, Kourkoulis SK (1998) Bending of marble with intrinsic length scales: a gradient theory with surface energy and size effects. J Phys IV 8:399\u2013406","journal-title":"J Phys IV"},{"key":"594_CR35","doi-asserted-by":"publisher","first-page":"838","DOI":"10.1016\/S0141-0296(00)00092-4","volume":"23","author":"CM Wang","year":"2001","unstructured":"Wang CM, Lim GT, Reddy JN, Lee KH (2001) Relationships between bending solutions of Reissner and Mindlin plate theories. Eng Struct 23:838\u2013849","journal-title":"Eng Struct"},{"key":"594_CR36","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1016\/j.euromechsol.2009.12.005","volume":"29","author":"B Wang","year":"2010","unstructured":"Wang B, Zhao J, Zhou S (2010) A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur J Mech A\/Solids 29:591\u2013599","journal-title":"Eur J Mech A\/Solids"},{"key":"594_CR37","doi-asserted-by":"publisher","first-page":"255","DOI":"10.4028\/www.scientific.net\/MSF.492-493.255","volume":"4992\u2013493","author":"A Witvrouw","year":"2005","unstructured":"Witvrouw A, Mehta A (2005) The use of functionally graded Poly-SiGe layers for MEMS applications. Mater Sci Forum 4992\u2013493:255\u2013260","journal-title":"Mater Sci Forum"},{"key":"594_CR38","doi-asserted-by":"publisher","first-page":"2044","DOI":"10.1016\/j.ijengsci.2010.04.010","volume":"48","author":"W Xia","year":"2010","unstructured":"Xia W, Wang L, Yin L (2010) Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int J Eng Sci 48:2044\u20132053","journal-title":"Int J Eng Sci"},{"key":"594_CR39","doi-asserted-by":"publisher","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 et al (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731\u20132743","journal-title":"Int J Solids Struct"},{"key":"594_CR40","doi-asserted-by":"crossref","unstructured":"Fattahi AM, Sahmani S (2017) Size dependency in the axial postbuckling behavior of nanopanels made of functionally graded material considering surface elasticity. Arab J Sci Eng 1\u201317","DOI":"10.1007\/s13369-017-2600-5"},{"issue":"6","key":"594_CR41","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1088\/2053-1591\/aa6bc0","volume":"4","author":"S Sahmani","year":"2017","unstructured":"Sahmani S, Fattahi AM (2017) An anisotropic calibrated nonlocal plate model for biaxial instability analysis of 3D metallic carbon nanosheets using molecular dynamics simulations. Mater Res Express 4(6):1\u201314","journal-title":"Mater Res Express"},{"issue":"10","key":"594_CR42","doi-asserted-by":"publisher","first-page":"5079","DOI":"10.1007\/s00542-017-3345-5","volume":"23","author":"AM Fattahi","year":"2017","unstructured":"Fattahi AM, Safaei B (2017) Buckling analysis of CNT-reinforced beams with arbitrary boundary conditions. Microsyst Technol 23(10):5079\u20135091","journal-title":"Microsyst Technol"},{"issue":"10","key":"594_CR43","doi-asserted-by":"publisher","first-page":"5105","DOI":"10.1007\/s00542-017-3316-x","volume":"23","author":"S Sahmani","year":"2017","unstructured":"Sahmani S, Fattahi AM (2017) Thermo-electro-mechanical size-dependent postbuckling response of axially loaded piezoelectric shear deformable nanoshells via nonlocal elasticity theory. Microsyst Technol 23(10):5105\u20135119","journal-title":"Microsyst Technol"},{"issue":"10","key":"594_CR44","doi-asserted-by":"publisher","first-page":"5121","DOI":"10.1007\/s00542-017-3377-x","volume":"23","author":"AM Fattahi","year":"2017","unstructured":"Fattahi AM, Sahmani S (2017) Nonlocal temperature-dependent postbuckling behavior of FG-CNT reinforced nanoshells under hydrostatic pressure combined with heat conduction. Microsyst Technol 23(10):5121\u20135137","journal-title":"Microsyst Technol"},{"issue":"10","key":"594_CR45","doi-asserted-by":"publisher","first-page":"4717","DOI":"10.1007\/s00542-016-3220-9","volume":"23","author":"S Sahmani","year":"2016","unstructured":"Sahmani S, Fattahi AM (2016) Size-dependent nonlinear instability of shear deformable cylindrical nanopanels subjected to axial compression in thermal environments. Microsyst Technol 23(10):4717\u20134731","journal-title":"Microsyst Technol"},{"issue":"2","key":"594_CR46","first-page":"719","volume":"55","author":"AM Fattahi","year":"2017","unstructured":"Fattahi AM, Najipour A (2017) Experimental study on mechanical properties of PE\/CNT composites. J Theor Appl Mech 55(2):719\u2013726","journal-title":"J Theor Appl Mech"},{"key":"594_CR47","doi-asserted-by":"crossref","unstructured":"Azizi S, Safaei B, Fattahi AM, Tekere M (2015) Nonlinear vibrational analysis of nanobeams embedded in an elastic medium including surface stress effects. Adv Mater Sci Eng 1\u20137","DOI":"10.1155\/2015\/318539"},{"key":"594_CR48","doi-asserted-by":"publisher","first-page":"4179","DOI":"10.1166\/jctn.2015.4334","volume":"12","author":"S Azizi","year":"2015","unstructured":"Azizi S, Fattahi AM, Kahnamouei JT (2015) Evaluating mechanical properties of nanoplatelet reinforced composites undermechanical and thermal loads. Compu Theor Nanosci 12:4179\u20134185","journal-title":"Compu Theor Nanosci"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00366-018-0594-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-018-0594-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-018-0594-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,2,23]],"date-time":"2019-02-23T19:35:05Z","timestamp":1550950505000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00366-018-0594-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,24]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["594"],"URL":"https:\/\/doi.org\/10.1007\/s00366-018-0594-6","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,2,24]]},"assertion":[{"value":"1 October 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 February 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 February 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}