{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T22:09:12Z","timestamp":1783030152008,"version":"3.54.6"},"reference-count":45,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,7,28]],"date-time":"2018-07-28T00:00:00Z","timestamp":1532736000000},"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":["11632004"],"award-info":[{"award-number":["11632004"]}],"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":["11602040"],"award-info":[{"award-number":["11602040"]}],"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":["11502036"],"award-info":[{"award-number":["11502036"]}],"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":["11604033"],"award-info":[{"award-number":["11604033"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Program for International Science and Technology Cooperation Projects of Ministry of Science and Technology of China","award":["2016YFE0125900"],"award-info":[{"award-number":["2016YFE0125900"]}]},{"name":"Key Project of Natural Science Foundation of CQ CSTC","award":["cstc2017jcyjBX0063"],"award-info":[{"award-number":["cstc2017jcyjBX0063"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The symmetric zero-frequency mode induced by weak material nonlinearity during Lamb wave propagation is explored for the first time. We theoretically confirm that, unlike the second harmonic, phase-velocity matching is not required to generate the zero-frequency mode and its signal is stronger than those of the nonlinear harmonics conventionally used, for example, the second harmonic. Experimental and numerical verifications of this theoretical analysis are conducted for the primary S0 mode wave propagating in an aluminum plate. The existence of a symmetric zero-frequency mode is of great significance, probably triggering a revolutionary progress in the field of non-destructive evaluation and structural health monitoring of the early-stage material nonlinearity based on the ultrasonic Lamb waves.<\/jats:p>","DOI":"10.3390\/s18082451","type":"journal-article","created":{"date-parts":[[2018,7,30]],"date-time":"2018-07-30T11:55:08Z","timestamp":1532951708000},"page":"2451","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Interaction of Lamb Wave Modes with Weak Material Nonlinearity: Generation of Symmetric Zero-Frequency Mode"],"prefix":"10.3390","volume":"18","author":[{"given":"Xiaoqiang","family":"Sun","sequence":"first","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiangyan","family":"Ding","sequence":"additional","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Feilong","family":"Li","sequence":"additional","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shijie","family":"Zhou","sequence":"additional","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yaolu","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6444-8467","authenticated-orcid":false,"given":"Ning","family":"Hu","sequence":"additional","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"},{"name":"Key Disciplines Lab of Novel Micro-nano Devices and System, International R &amp; D Center of Micro-nano Systems and New Materials Technology, Chongqing University, Chongqing 400044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0840-1891","authenticated-orcid":false,"given":"Zhongqing","family":"Su","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7412-8216","authenticated-orcid":false,"given":"Youxuan","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jun","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mingxi","family":"Deng","sequence":"additional","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Shen, Y., and Giurgiutiu, V. (2012, January 16\u201319). Predictive simulation of nonlinear ultrasonics. Proceedings of the SPIE\u2014The International Society for Optical Engineering, San Diego, CA, USA.","DOI":"10.1117\/12.913950"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"214301","DOI":"10.1103\/PhysRevLett.118.214301","article-title":"Proof of Concept for an Ultrasensitive Technique to Detect and Localize Sources of Elastic Nonlinearity Using Phononic Crystals","volume":"118","author":"Miniaci","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3650","DOI":"10.1103\/PhysRevLett.68.3650","article-title":"Elastic interface waves propagating in a fracture","volume":"68","author":"Pyraknolte","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"124301","DOI":"10.1103\/PhysRevLett.90.124301","article-title":"Self-induced hysteresis for nonlinear acoustic waves in cracked material","volume":"90","author":"Moussatov","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"104301","DOI":"10.1103\/PhysRevLett.98.104301","article-title":"Interaction Dynamics of Elastic Waves with a Complex Nonlinear Scatterer through the Use of a Time Reversal Mirror","volume":"98","author":"Ulrich","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"275501","DOI":"10.1103\/PhysRevLett.107.275501","article-title":"Laser Excitation of a Fracture Source for Elastic Waves","volume":"107","author":"Blum","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.ndteint.2008.09.009","article-title":"A nonlinear-guided wave technique for evaluating plasticity-driven material damage in a metal plate","volume":"42","author":"Pruell","year":"2009","journal-title":"NDT E Int."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"044914","DOI":"10.1063\/1.4863639","article-title":"Creep damage characterization using nonlinear ultrasonic guided wave method: A mesoscale model","volume":"115","author":"Xiang","year":"2014","journal-title":"J. Appl. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2141","DOI":"10.1063\/1.3527959","article-title":"Experimental characterization of efficient second harmonic generation of Lamb wave modes in a nonlinear elastic isotropic plate","volume":"109","author":"Matlack","year":"2011","journal-title":"J. Appl. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2067","DOI":"10.1063\/1.2431467","article-title":"Experimental characterization of material nonlinearity using Lamb waves","volume":"90","author":"Bermes","year":"2007","journal-title":"Appl. Phys. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1134\/S1063771006050137","article-title":"Nonlinear lamb waves in a metal plate with defects","volume":"52","author":"Korobov","year":"2006","journal-title":"Acoust. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1063\/1.2811954","article-title":"Evaluation of plasticity driven material damage using Lamb waves","volume":"91","author":"Pruell","year":"2007","journal-title":"Appl. Phys. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"035003","DOI":"10.1088\/0964-1726\/18\/3\/035003","article-title":"Evaluation of fatigue damage using nonlinear guided waves","volume":"18","author":"Pruell","year":"2009","journal-title":"Smart Mater. Struct."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Sun, X., Liu, X., Liu, Y., Hu, N., Zhao, Y., Ding, X., Qin, S., Zhang, J., Zhang, J., and Liu, F. (2017). Simulations on Monitoring and Evaluation of Plasticity-Driven Material Damage Based on Second Harmonic of S\u2080 Mode Lamb Waves in Metallic Plates. Materials, 10.","DOI":"10.3390\/ma10070827"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2141","DOI":"10.1121\/1.3294714","article-title":"Characteristics of second harmonic generation of Lamb waves in nonlinear elastic plates","volume":"127","author":"Kim","year":"2010","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"134103","DOI":"10.1103\/PhysRevB.94.134103","article-title":"Nonlinear coda wave analysis of hysteretic elastic behavior in strongly scattering media","volume":"94","author":"Ouarabi","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"045023","DOI":"10.1088\/0964-1726\/25\/4\/045023","article-title":"Analytical and numerical studies of approximate phase velocity matching based nonlinear S0 mode Lamb waves for the detection of evenly distributed microstructural changes","volume":"25","author":"Wan","year":"2016","journal-title":"Smart Mater. Struct."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Wan, X., Tse, P.W., Xu, G., Tao, T., Liu, F., Chen, X., and Zhang, Q. (2015). FEM Simulation of Nonlinear Lamb Waves for Detecting a Micro-Crack in a Metallic Plate, Engineering Asset Management-Systems. Engineering Asset Management\u2014Systems, Professional Practices and Certification, Springer.","DOI":"10.1007\/978-3-319-09507-3_133"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"8528","DOI":"10.3390\/s140508528","article-title":"Numerical simulation of nonlinear Lamb waves used in a thin plate for detecting buried micro-cracks","volume":"14","author":"Wan","year":"2014","journal-title":"Sensors"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1016\/j.ultras.2013.09.023","article-title":"Modeling nonlinearities of ultrasonic waves for fatigue damage characterization: Theory, simulation, and experimental validation","volume":"54","author":"Hong","year":"2014","journal-title":"Ultrasonics"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4517","DOI":"10.1016\/j.jsv.2013.03.021","article-title":"On selection of primary modes for generation of strong internally resonant second harmonics in plate","volume":"332","author":"Liu","year":"2013","journal-title":"J. Sound Vib."},{"key":"ref_22","first-page":"231911","article-title":"Experimental investigation of symmetry properties of second harmonic Lamb waves","volume":"111","author":"Liu","year":"2012","journal-title":"J. Appl. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1088\/0964-1726\/21\/8\/085019","article-title":"Detection of thermal fatigue in composites by second harmonic Lamb waves","volume":"21","author":"Li","year":"2012","journal-title":"Smart Mater. Struct."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1016\/j.ndteint.2008.01.008","article-title":"Higher harmonic analysis of ultrasonic signal for ageing behaviour study of C-250 grade maraging steel","volume":"41","author":"Metya","year":"2008","journal-title":"NDT E Int."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4152","DOI":"10.1063\/1.1601312","article-title":"Analysis of second-harmonic generation of Lamb modes using a modal analysis approach","volume":"94","author":"Deng","year":"2003","journal-title":"J. Appl. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1063\/1.1753876","article-title":"Finite-amplitude ultrasonic waves in aluminum","volume":"3","author":"Breazeale","year":"1963","journal-title":"Appl. Phys. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3051","DOI":"10.1063\/1.369642","article-title":"Cumulative second-harmonic generation of Lamb-mode propagation in a solid plate","volume":"85","author":"Deng","year":"1999","journal-title":"J. Appl. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1016\/S0022-460X(02)01260-9","article-title":"Finite-amplitude waves in isotropic elastic plates","volume":"265","author":"Lima","year":"2003","journal-title":"J. Sound Vib."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"9780","DOI":"10.1103\/PhysRevB.35.9780","article-title":"Acoustic radiation-induced static strains in Solids","volume":"35","author":"Cantrell","year":"1987","journal-title":"Phys. Rev. B Condens. Matter"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1093","DOI":"10.1121\/1.1910442","article-title":"Calculation of Lattice-Parameter Changes with Hydrostatic Pressure from Third-Order Elastic Constants","volume":"41","author":"Thurston","year":"1967","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1760","DOI":"10.1121\/1.4817840","article-title":"Finite-size effects on the quasistatic displacement pulse in a solid specimen with quadratic nonlinearity","volume":"134","author":"Nagy","year":"2013","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3449","DOI":"10.1121\/1.3583501","article-title":"On the acoustic-radiation-induced strain and stress in elastic solids with quadratic nonlinearity (L)","volume":"129","author":"Qu","year":"2011","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1827","DOI":"10.1121\/1.3681922","article-title":"Pulse propagation in an elastic medium with quadratic nonlinearity (L)","volume":"131","author":"Qu","year":"2012","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_34","first-page":"1112","article-title":"Simplified experimental technique to extract the acoustic radiation induced static strain in solids","volume":"91","author":"Narasimha","year":"2007","journal-title":"Appl. Phys. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1282","DOI":"10.1063\/1.2902581","article-title":"Static displacement component and its significance in nonlinear ultrasonic measurements","volume":"975","author":"Narasimha","year":"2008","journal-title":"AIP Conf. Proc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1063\/1.2191428","article-title":"Experimental study of the acoustic radiation strain in solids","volume":"88","author":"Jacob","year":"2006","journal-title":"Appl. Phys. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1063\/1.4864877","article-title":"Reflection and diffraction corrections for nonlinear materials characterization by quasi-static pulse measurement","volume":"1581","author":"Nagy","year":"2014","journal-title":"AIP Conf. Proc."},{"key":"ref_38","unstructured":"Sun, X., Liu, X., and Hu, N. (2017, January 8\u201310). Interaction of guided waves in isotropic elastic plates with weak material nonlinearity: Sum-frequency, difference frequency, second harmonic and zero-frequency components. Proceedings of the 9th International Symposium on NDT in Aerospace, Xiamen, China. Available online: https:\/\/www.ndt.net\/article\/aero2017\/papers\/NDT2017-044_Interaction_of_guided_waves_in_isotropic_elastic_plates_with_weak_material_nonlinearity.pdf."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1088\/1361-665X\/aaafeb","article-title":"Numerical study on static component generation from the primary Lamb waves propagating in a plate with nonlinearity","volume":"27","author":"Wan","year":"2018","journal-title":"Smart Mater. Struct."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"932","DOI":"10.1016\/j.jsv.2009.01.027","article-title":"On the existence of antisymmetric or symmetric Lamb waves at nonlinear higher harmonics","volume":"323","author":"Srivastava","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"124909","DOI":"10.1063\/1.4729554","article-title":"Interaction of guided wave modes in isotropic weakly nonlinear elastic plates: Higher harmonic generation","volume":"111","author":"Lissenden","year":"2012","journal-title":"J. Appl. Phys."},{"key":"ref_42","unstructured":"Landau, L.D., and Lifshitz, E.M. (1986). Theory of Elasticity, Pergamon Press. [3rd ed.]."},{"key":"ref_43","unstructured":"Auld, B.A. (1990). Acoustic Fields and Waves in Solids, Ripol Classic."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Rose, J.L. (2014). Ultrasonic Guided Waves in Solid Media, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9781107273610"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.1016\/j.ultras.2014.04.009","article-title":"Nonlinear guided waves in plates: A numerical perspective","volume":"54","author":"Chillara","year":"2014","journal-title":"Ultrasonics"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/8\/2451\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:14:58Z","timestamp":1760195698000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/8\/2451"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,28]]},"references-count":45,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2018,8]]}},"alternative-id":["s18082451"],"URL":"https:\/\/doi.org\/10.3390\/s18082451","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,28]]}}}