{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T18:51:00Z","timestamp":1760986260363,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,3,16]],"date-time":"2024-03-16T00:00:00Z","timestamp":1710547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["MCA"],"abstract":"<jats:p>Carbon nanotubes are widely used as material reinforcement in diverse fields of engineering. Being that their contribution is significant to improving the mean properties of the resulting materials, it is important to assess the influence of the variability on carbon nanotubes\u2019 material and geometrical properties to structures\u2019 responses. This work considers functionally graded plates constituted by an aluminum continuous phase reinforced with single-walled or multi-walled carbon. The nanotubes' weight fraction evolution through the thickness is responsible for the plates\u2019 functional gradient. The plates\u2019 samples are simulated considering that only the nanotubes\u2019 material and geometrical characteristics are affected by uncertainty. The results obtained from the multiple regression models developed allow us to conclude that the length of the nanotubes has no impact on the maximum transverse displacement of the plates in opposition to the carbon nanotubes\u2019 weight fraction evolution, their internal and external diameters, and the Young\u2019s modulus. The multiple regression models developed can be used as alternative prediction tools within the domain of the study.<\/jats:p>","DOI":"10.3390\/mca29020022","type":"journal-article","created":{"date-parts":[[2024,3,18]],"date-time":"2024-03-18T04:25:15Z","timestamp":1710735915000},"page":"22","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Variability on Functionally Graded Plates\u2019 Deflection Due to Uncertainty on Carbon Nanotubes\u2019 Properties"],"prefix":"10.3390","volume":"29","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2642-4947","authenticated-orcid":false,"given":"Alda","family":"Carvalho","sequence":"first","affiliation":[{"name":"CIMOSM, Centro de Investiga\u00e7\u00e3o em Modela\u00e7\u00e3o e Otimiza\u00e7\u00e3o de Sistemas Multifuncionais, ISEL\/IPL, Instituto Superior de Engenharia de Lisboa, 1959-007 Lisboa, Portugal"},{"name":"DCeT, Universidade Aberta & CEMAPRE\/REM, ULisboa & CIMOSM, ISEL, 1269-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3733-6619","authenticated-orcid":false,"given":"Ana","family":"Martins","sequence":"additional","affiliation":[{"name":"CIMOSM, Centro de Investiga\u00e7\u00e3o em Modela\u00e7\u00e3o e Otimiza\u00e7\u00e3o de Sistemas Multifuncionais, ISEL\/IPL, Instituto Superior de Engenharia de Lisboa, 1959-007 Lisboa, Portugal"},{"name":"CIMA\u2014Research Centre for Mathematics and Applications & CIMOSM, ISEL, 7000-671 \u00c9vora, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3682-6472","authenticated-orcid":false,"given":"Ana F.","family":"Mota","sequence":"additional","affiliation":[{"name":"CIMOSM, Centro de Investiga\u00e7\u00e3o em Modela\u00e7\u00e3o e Otimiza\u00e7\u00e3o de Sistemas Multifuncionais, ISEL\/IPL, Instituto Superior de Engenharia de Lisboa, 1959-007 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4452-5840","authenticated-orcid":false,"given":"Maria A. R.","family":"Loja","sequence":"additional","affiliation":[{"name":"CIMOSM, Centro de Investiga\u00e7\u00e3o em Modela\u00e7\u00e3o e Otimiza\u00e7\u00e3o de Sistemas Multifuncionais, ISEL\/IPL, Instituto Superior de Engenharia de Lisboa, 1959-007 Lisboa, Portugal"},{"name":"IDMEC, IST-Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1115\/1.2777164","article-title":"Modeling and analysis of functionally graded materials and structures","volume":"60","author":"Birman","year":"2007","journal-title":"Appl. Mech. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1007\/s42823-019-00068-2","article-title":"Carbon nanotubes: Synthesis, properties and engineering applications","volume":"29","author":"Gupta","year":"2019","journal-title":"Carbon Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"11509","DOI":"10.1016\/j.mseb.2021.115095","article-title":"A review on carbon nanotube: An overview of synthesis, properties, functionalization, characterization, and the application","volume":"268","author":"Rathinavel","year":"2021","journal-title":"Mater. Sci. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.compstruct.2014.09.041","article-title":"Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review","volume":"120","author":"Liew","year":"2015","journal-title":"Compos. Struct."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.compositesa.2018.08.010","article-title":"CNT-reinforced Metal and Steel Nanocomposites: A Comprehensive Assessment of Progress and Future Directions","volume":"114","author":"Radhamani","year":"2018","journal-title":"Compos. Part A Appl. Sci. Manuf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"092002","DOI":"10.1088\/2053-1591\/ab3175","article-title":"A Comprehensive Review on Analysis of Nanocomposites: From Manufacturing to Properties Characterization","volume":"6","author":"Vinyas","year":"2019","journal-title":"Mater. Res. Express"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"110339","DOI":"10.1016\/j.engstruct.2020.110339","article-title":"Functionally graded graphene reinforced composite structures: A review","volume":"210","author":"Zhao","year":"2020","journal-title":"Eng. Struct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4183","DOI":"10.1007\/s00542-021-05211-6","article-title":"Carbon nanotubes: A review on properties. synthesis methods and applications in micro and nanotechnology","volume":"27","author":"Shoukat","year":"2021","journal-title":"Microsyst. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"116075","DOI":"10.1016\/j.compstruct.2022.116075","article-title":"Functionally graded carbon nanotubes reinforced composite structures: An extensive review","volume":"299","author":"Soni","year":"2022","journal-title":"Compos. Struct."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Valencia Murillo, C.E., Gutierrez Rivera, M.E., and Celaya Garcia, L.D. (2023). Thermal\u2013Structural Linear Static Analysis of Functionally Graded Beams Using Reddy Beam Theory. Math. Comput. Appl., 28.","DOI":"10.3390\/mca28040084"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kurpa, L., Pellicano, F., Shmatko, T., and Zippo, A. (2024). Free Vibration Analysis of Porous Functionally Graded Material Plates with Variable Thickness on an Elastic Foundation Using the R-Functions Method. Math. Comput. Appl., 29.","DOI":"10.3390\/mca29010010"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Alghanmi, R.A., and Aljaghthami, R.H. (2024). A Four-Variable Shear Deformation Theory for the Static Analysis of FG Sandwich Plates with Different Porosity Models. Math. Comput. Appl., 29.","DOI":"10.3390\/mca29020020"},{"key":"ref_13","unstructured":"Ruoff, R.S., and Lorents, D.C. (1996). Carbon Nanotubes, Elsevier."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1016\/S0022-3697(99)00376-5","article-title":"Mechanical and physical properties on carbon nanotube","volume":"61","author":"Xie","year":"2000","journal-title":"J. Phys. Chem. Solids"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1555","DOI":"10.1016\/j.compositesa.2005.02.006","article-title":"Thermo-mechanical properties of randomly oriented carbon\/epoxy nanocomposites","volume":"36","author":"Fidelusa","year":"2005","journal-title":"Compos. Part A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1016\/j.compscitech.2011.04.002","article-title":"The effect of the aspect ratio of carbon nanotubes on their effective reinforcement modulus in an epoxy matrix","volume":"71","author":"Martone","year":"2011","journal-title":"Compos. Sci. Technol."},{"key":"ref_17","unstructured":"Peng, H., Li, Q., and Chen, T. (2017). Industrial Applications of Carbon Nanotubes. Micro and Nano Technologies, Elsevier."},{"key":"ref_18","first-page":"7842413","article-title":"Dynamic Response of Soft Core Sandwich Beams with Metal-Graphene Nanocomposite Skins","volume":"2017","author":"Loja","year":"2017","journal-title":"Shock Vib."},{"key":"ref_19","unstructured":"Eringen, A.C. (2002). Nonlocal Continuum Field Theories, Springer."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2237","DOI":"10.1007\/s11831-021-09652-0","article-title":"A Review of Available Theories and Methodologies for the Analysis of Nano Isotropic, Nano Functionally Graded, and CNT Reinforced Nanocomposite Structures","volume":"29","author":"Garg","year":"2022","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"23511","DOI":"10.1063\/1.2833431","article-title":"Nonlocal continuum theories of beams for the analysis of carbon nanotubes","volume":"103","author":"Reddy","year":"2008","journal-title":"J. Appl. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kong, L., Zhang, B., and Li, C. (2022). Thermal Buckling and Postbuckling Behaviors of Couple Stress and Surface Energy-Enriched FG-CNTR Nanobeams. Symmetry, 14.","DOI":"10.3390\/sym14112228"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1128","DOI":"10.1016\/j.compstruct.2011.10.006","article-title":"A nonlinear modified couple stress-based third-order theory of functionally graded plates","volume":"94","author":"Reddy","year":"2012","journal-title":"Compos. Struct."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.compstruct.2017.03.067","article-title":"Size-dependent analysis of homogeneous and functionally graded microplates using IGA and a non-classical Kirchhoff plate theory","volume":"172","author":"Liu","year":"2017","journal-title":"Compos. Struct."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Thanh, C.-L., Vu-Huu, T., Phung-Van, P., Nguyen-Xuan, H., and Abdel Wahab, M. (2018, January 28\u201329). Size-Dependent Analysis for FG-CNTRC Nanoplates Based on Refined Plate Theory and Modified Couple Stress. Proceedings of the 1st International Conference on Numerical Modelling in Engineering, NME 2018, Ghent, Belgium. Lecture Notes in Civil Engineering, 20.","DOI":"10.1007\/978-981-13-2405-5_19"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Zhang, B., Li, C., Zhang, L., and Xie, F. (2022). Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates. Symmetry, 14.","DOI":"10.3390\/sym14122592"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Sotiropoulos, D.G., and Tserpes, K. (2022). Interval-Based Computation of the Uncertainty in the Mechanical Properties and the Failure Analysis of Unidirectional Composite Materials. Math. Comput. Appl., 27.","DOI":"10.3390\/mca27030038"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Yin, S., Qin, H., and Gao, Q. (2022). An Efficient Orthogonal Polynomial Method for Auxetic Structure Analysis with Epistemic Uncertainties. Math. Comput. Appl., 27.","DOI":"10.3390\/mca27030049"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bouwer, J.M., Wilke, D.N., and Kok, S. (2023). Spatio-Temporal Gradient Enhanced Surrogate Modeling Strategies. Math. Comput. Appl., 28.","DOI":"10.3390\/mca28020057"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3040","DOI":"10.1038\/s41467-019-10959-7","article-title":"Strength of carbon nanotubes depends on their chemical structures","volume":"10","author":"Takakura","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.compstruct.2016.05.019","article-title":"Metamodel-based approach for stochastic free vibration analysis of functionally graded carbon nanotube reinforced plates","volume":"152","author":"Friswell","year":"2016","journal-title":"Compos. Struct."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1750071","DOI":"10.1142\/S1758825117500715","article-title":"Uncertain Buckling and Sensitivity Analysis of Functionally Graded Carbon Nanotube-Reinforced Composite Beam","volume":"9","author":"Pouresmaeeli","year":"2017","journal-title":"Int. J. Appl. Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1080\/15376494.2016.1191100","article-title":"Assessing the influence of material and geometrical uncertainty on the mechanical behavior of FGM plates","volume":"24","author":"Carvalho","year":"2017","journal-title":"Mech. Adv. Mater. Struct."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Rosa, R.S.B., Loja, M.A.R., and Carvalho, A.C.J.V.N. (2018). Toward variability characterization and statistic models constitution for the prediction of exponentially graded plates\u2019 static response. J. Compos. Sci., 2.","DOI":"10.3390\/jcs2040059"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.compositesb.2018.03.043","article-title":"Stochastic dynamic analysis of twisted functionally graded plates","volume":"147","author":"Karsh","year":"2018","journal-title":"Compos. Part B Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"106315","DOI":"10.1016\/j.tws.2019.106315","article-title":"Machine learning aided stochastic structural free vibration analysis for functionally graded bar-type structures","volume":"144","author":"Wang","year":"2019","journal-title":"Thin-Walled Struct."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"105738","DOI":"10.1016\/j.ast.2020.105738","article-title":"Probabilistic stability analysis of functionally graded graphene reinforced porous beams","volume":"98","author":"Gao","year":"2020","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"111356","DOI":"10.1016\/j.engstruct.2020.111356","article-title":"The influence of mechanical uncertainties on the free vibration of functionally graded graphene-reinforced porous nanocomposite shells of revolution","volume":"228","author":"Baghlani","year":"2021","journal-title":"Eng. Struct."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1002\/pen.760160512","article-title":"The Halpin-Tsai Equations: A Review","volume":"16","author":"Halpin","year":"1976","journal-title":"Polym. Eng. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"808","DOI":"10.3934\/matersci.2016.3.808","article-title":"Assessing the static behavior of hybrid CNT-metal-ceramic composite plates","volume":"3","author":"Costa","year":"2016","journal-title":"AIMS Mater. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0263-8223(93)90147-I","article-title":"An evaluation of equivalent-single-layer and layerwise theories of composite laminates","volume":"25","author":"Reddy","year":"1993","journal-title":"Compos. Struct."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Reddy, J.N. (2003). Mechanics of Laminated Composite Plates and Shells, CRC Press. [2nd ed.].","DOI":"10.1201\/b12409"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.finel.2010.12.001","article-title":"Finite element analysis of functionally graded plates under transverse load","volume":"47","author":"Singha","year":"2011","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Mota, A.F., Loja, M.A.R., Barbosa, J.I., and Rodrigues, J.A. (2020). Porous Functionally Graded Plates: An Assessment of the Influence of Shear Correction Factor on Static Behavior. Math. Comput. Appl., 25.","DOI":"10.3390\/mca25020025"},{"key":"ref_45","unstructured":"Zienkiewicz, O.C., and Taylor, R.L. (2000). The Finite Element Method, Butterworth-Heinemann. [5th ed.]."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1093\/biomet\/38.1-2.159","article-title":"Testing for Serial Correlation in Least Squares Regression","volume":"38","author":"Durbin","year":"1951","journal-title":"Biometrika"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1287","DOI":"10.2307\/1911963","article-title":"A Simple Test for Heteroskedasticity and Random Coefficient Variation","volume":"47","author":"Breusch","year":"1979","journal-title":"Econometrica"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"James, G., Witten, D., Hastie, T., and Tibshirani, R. (2013). An Introduction to Statistical Learning, Springer.","DOI":"10.1007\/978-1-4614-7138-7"},{"key":"ref_49","unstructured":"Siegel, S., and Castellan, N.J. (1988). Nonparametric Statistics for the Behavioral Sciences, Mcgraw-Hill Book Company. [2nd ed.]."},{"key":"ref_50","unstructured":"Hollander, M., and Wolfe, D.A. (2013). Nonparametric Statistical Methods, John Wiley & Sons."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.compstruct.2017.06.048","article-title":"Levy solution for bending analysis of functionally graded sandwich plates based on four variable plate theory","volume":"177","author":"Demirhan","year":"2017","journal-title":"Compos. Struct."}],"container-title":["Mathematical and Computational Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2297-8747\/29\/2\/22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:14:46Z","timestamp":1760105686000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2297-8747\/29\/2\/22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,16]]},"references-count":51,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["mca29020022"],"URL":"https:\/\/doi.org\/10.3390\/mca29020022","relation":{},"ISSN":["2297-8747"],"issn-type":[{"type":"electronic","value":"2297-8747"}],"subject":[],"published":{"date-parts":[[2024,3,16]]}}}