{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T07:29:01Z","timestamp":1773559741973,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,5,27]],"date-time":"2022-05-27T00:00:00Z","timestamp":1653609600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12072004"],"award-info":[{"award-number":["12072004"]}]},{"name":"National Natural Science Foundation of China","award":["11872082"],"award-info":[{"award-number":["11872082"]}]},{"name":"National Natural Science Foundation of China","award":["KZ202110005005"],"award-info":[{"award-number":["KZ202110005005"]}]},{"name":"Beijing Municipal Education Commission","award":["12072004"],"award-info":[{"award-number":["12072004"]}]},{"name":"Beijing Municipal Education Commission","award":["11872082"],"award-info":[{"award-number":["11872082"]}]},{"name":"Beijing Municipal Education Commission","award":["KZ202110005005"],"award-info":[{"award-number":["KZ202110005005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Functionally graded materials (FGM) have received extensive attention in recent years due to their excellent mechanical properties. In this research, the theoretical process of calculating the propagation characteristics of Lamb waves in FGM sandwich plates is deduced by combining the FGM volume fraction curve and Legendre polynomial series expansion method. In this proposed method, the FGM plate does not have to be sliced into multiple layers. Numerical results are given in detail, and the Lamb wave dispersion curves are extracted. For comparison, the Lamb wave dispersion curve of the sliced layer model for the FGM sandwich plate is obtained by the global matrix method. Meanwhile, the FGM sandwich plate was subjected to finite element simulation, also based on the layered-plate model. The acoustic characteristics detection experiment was performed by simulation through a defocusing measurement. Thus, the Lamb wave dispersion curves were obtained by V(f, z) analysis. Finally, the influence of the change in the gradient function on the Lamb wave dispersion curves will be discussed.<\/jats:p>","DOI":"10.3390\/s22114052","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T02:30:06Z","timestamp":1653964206000},"page":"4052","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Lamb Waves Propagation Characteristics in Functionally Graded Sandwich Plates"],"prefix":"10.3390","volume":"22","author":[{"given":"Jie","family":"Gao","sequence":"first","affiliation":[{"name":"Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianbo","family":"Zhang","sequence":"additional","affiliation":[{"name":"The 3th Research Institute of China Electronics Technology Group Corporation, Beijing 100015, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3479-4385","authenticated-orcid":false,"given":"Yan","family":"Lyu","sequence":"additional","affiliation":[{"name":"Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guorong","family":"Song","sequence":"additional","affiliation":[{"name":"Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cunfu","family":"He","sequence":"additional","affiliation":[{"name":"Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,27]]},"reference":[{"key":"ref_1","first-page":"213","article-title":"Proceedings of the international symposium on functionally gradient materials","volume":"34","author":"Kumakawa","year":"1992","journal-title":"J. Am. Ceram. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.ijengsci.2016.05.008","article-title":"Size-dependent behavior of functionally graded anisotropic composite pate","volume":"106","author":"Guo","year":"2016","journal-title":"Int. J. Eng. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.paerosci.2015.07.001","article-title":"Talha, Recent development in modeling and analysis of functionally graded materials and structures","volume":"79","year":"2015","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Sha, G., and Lissenden, C.J. (2021). Modeling Magnetostrictive Transducers for Structural Health Monitoring: Ultrasonic Guided Wave Generation and Reception. Sensors, 21.","DOI":"10.3390\/s21237971"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"105953","DOI":"10.1016\/j.ultras.2019.105953","article-title":"Modeling guided wave propagation in functionally graded plates by state-vector formalism and the Legendre polynomial method","volume":"99","author":"Jie","year":"2019","journal-title":"Ultrasonics"},{"key":"ref_6","first-page":"366","article-title":"Dispersion of Lamb Waves in Layered Plates","volume":"3","author":"Zhu","year":"1998","journal-title":"J. Vib. Eng."},{"key":"ref_7","first-page":"199","article-title":"Propagation of Lamb Wave in a Functionally Graded Plate with a Layered Model","volume":"35","author":"Wu","year":"2016","journal-title":"Appl. Acoust."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6383","DOI":"10.1016\/S0020-7683(99)00236-X","article-title":"One-dimensional model for designing functionally graded materials to manage stress waves","volume":"37","author":"Bruck","year":"2000","journal-title":"Int. J. Solids. Struct."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/j.compstruct.2006.08.009","article-title":"On calculating dispersion curves of waves in a functionally graded elastic plate","volume":"81","author":"Chen","year":"2007","journal-title":"Compos. Struct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3419","DOI":"10.1063\/1.369699","article-title":"Legendre polynomial approach for modeling free-ultrasonic waves in multilayered plates","volume":"85","author":"Lefebvre","year":"1999","journal-title":"J. Appl. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2498","DOI":"10.1109\/TUFFC.2012.2482","article-title":"Wave propagation in the circumferential direction of general multilayered piezoelectric cylindrical plates","volume":"59","author":"Yu","year":"2012","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/j.ultras.2010.10.003","article-title":"Love waves in functionally graded piezoelectric materials by stiffness matrix method","volume":"51","author":"Salah","year":"2011","journal-title":"Ultrasonics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s00707-009-0143-x","article-title":"An analysis of surface acoustic wave propagation in functionally graded plates with homotopy analysis method","volume":"208","author":"Gao","year":"2009","journal-title":"Acta. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Taylor, R.L., and Zienkiewicz, O.C. (2005). The Finite Element Method, McGraw-Hill. [6th ed.].","DOI":"10.1016\/B978-075066431-8.50196-X"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.jsv.2003.04.004","article-title":"Evaluation of modal density and group velocity by a finite element method","volume":"273","author":"Finnveden","year":"2004","journal-title":"J. Sound Vib."},{"key":"ref_16","first-page":"145","article-title":"Studying the Propagating of Surface Wave Traveling in Functional Gradient by Laser Ultrasonic","volume":"2","author":"Cheng","year":"2011","journal-title":"Acta Acustica."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1115\/1.1467094","article-title":"Isoparametric Graded Finite Elements for Nonhomogeneous Isotropic and Orthotropic Materials","volume":"69","author":"Kim","year":"2002","journal-title":"J. Appl. Mech."},{"key":"ref_18","first-page":"31","article-title":"Research on Finite Element Methods for Functional Graded Material","volume":"27","author":"Zhang","year":"2007","journal-title":"Aircr. Design"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"331","DOI":"10.4028\/www.scientific.net\/KEM.183-187.331","article-title":"Analysis of a Crack in a Functionally Gradient Interface Layer under Static and Dynamic Loading","volume":"183\u2013187","author":"Wang","year":"2000","journal-title":"Key Eng. Mater."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"111319","DOI":"10.1016\/j.compstruct.2019.111319","article-title":"Modeling guided wave propagation in multi-layered anisotropic composite laminates by state-vector formalism and the Legendre polynomials","volume":"228","author":"Gao","year":"2019","journal-title":"Compos. Struct."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1109\/58.393096","article-title":"Matrix techniques for modeling ultrasonic waves in multilayered media","volume":"42","author":"Lowe","year":"1995","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mart\u00ednez-Pa\u00f1eda, E. (2019). On the finite element implementation of functionally graded materials. Materials, 12.","DOI":"10.3390\/ma12020287"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1115\/1.1328089","article-title":"Use of graded finite elements to model the behavior of nonhomogeneous materials","volume":"67","author":"Santare","year":"2000","journal-title":"J. Appl. Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"102754","DOI":"10.1016\/j.wavemoti.2021.102754","article-title":"Legendre orthogonal polynomial method in calculating reflection and transmission coefficients of fluid-loaded functionally gradient plates","volume":"104","author":"Gao","year":"2021","journal-title":"Wave Motion"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1007\/s10921-014-0259-3","article-title":"Elastic Properties Inversion of an Isotropic Plate by Hybrid Particle Swarm-Based-Simulated Annealing Optimization Technique from Leaky Lamb Wave Measurements Using Acoustic Microscopy","volume":"33","author":"Lyu","year":"2014","journal-title":"J. Nondestr. Eval."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/11\/4052\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:19:30Z","timestamp":1760138370000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/11\/4052"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,27]]},"references-count":25,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["s22114052"],"URL":"https:\/\/doi.org\/10.3390\/s22114052","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,27]]}}}