{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T01:19:20Z","timestamp":1777339160049,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,3,7]],"date-time":"2019-03-07T00:00:00Z","timestamp":1551916800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Parameters to measure nonlinearity in polymethylmethacrylate (PMMA) and carbon fiber reinforced polymer (CFRP) materials have been determined with nonlinear ultrasound (NLUS). The nonlinear parameter    \u03b2    has been determined using the variation of the Finite Amplitude Method (FAM) with harmonic generation. Using this as a reference, the first contribution of this work consists of deducting the experimental configuration necessary to measure this nonlinear parameter in a correct and feasible way. Excitation level, frequency of the wave generated, number of cycles analysed and the distances transducer-specimen and specimen-hydrophone have been determined in both materials. The second contribution is a semi-analytical model that allows to obtain the nonlinear parameter in materials by removing water contribution and considering geometric and viscous attenuation, using the data obtained in an immersion tank. Finally, an application of this model has been carried out in PMMA in order to determinate the nonlinear parameter in this material. From the results, we confirm that the configuration determined in this paper to obtain the parameter    \u03b2    decreases the noise in the measurements.<\/jats:p>","DOI":"10.3390\/s19051156","type":"journal-article","created":{"date-parts":[[2019,3,8]],"date-time":"2019-03-08T04:58:35Z","timestamp":1552021115000},"page":"1156","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Experimental Configuration to Determine the Nonlinear Parameter \u03b2 in PMMA and CFRP with the Finite Amplitude Method"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3196-5935","authenticated-orcid":false,"given":"Antonio","family":"Callejas","sequence":"first","affiliation":[{"name":"Department of Structural Mechanics, University of Granada, 18071 Granada, Spain"},{"name":"Instituto de Investigaci\u00f3n Biosanitaria, ibs.GRANADA, 18012 Granada, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9239-294X","authenticated-orcid":false,"given":"Guillermo","family":"Rus","sequence":"additional","affiliation":[{"name":"Department of Structural Mechanics, University of Granada, 18071 Granada, Spain"},{"name":"Instituto de Investigaci\u00f3n Biosanitaria, ibs.GRANADA, 18012 Granada, Spain"},{"name":"Excellence Research Unit, \u201cModelling Nature\u201d (MNat), University of Granada, 18071 Granada, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3066","DOI":"10.1177\/0954406217704222","article-title":"Non-linear methods based on ultrasonic waves to analyse disbonds in single lap joints","volume":"231","author":"Scarselli","year":"2017","journal-title":"Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s10921-014-0227-y","article-title":"Frequency Dependence of Second-Harmonic Generation in Lamb Waves","volume":"33","author":"Matsuda","year":"2014","journal-title":"J. Nondestruct. Eval."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1070\/PU1971v013n06ABEH004281","article-title":"Nonlinear phenomena in the propagation of elastic waves in solids","volume":"13","author":"Zarembo","year":"1971","journal-title":"Physics-Uspekhi"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"214106","DOI":"10.1063\/1.2937838","article-title":"Evaluation of B\/A nonlinear parameter using an acoustic self-calibrated pulse-echo method","volume":"92","author":"Haumesser","year":"2008","journal-title":"Appl. Phys. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/s10921-013-0213-9","article-title":"Pulse-Echo Harmonic Generation Measurements for Non-destructive Evaluation","volume":"33","author":"Best","year":"2014","journal-title":"J. Nondestruct. Eval."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4529","DOI":"10.1143\/JJAP.46.4529","article-title":"Real time detection of second-harmonic components generated from plastic-deformed metal rod using double-layered piezoelectric transducer","volume":"46","author":"Fukuda","year":"2007","journal-title":"Jpn. J. Appl. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.wavemoti.2012.08.009","article-title":"The generation of second harmonic waves in an isotropic solid with quadratic nonlinearity under the presence of a stress-free boundary","volume":"50","author":"Bender","year":"2013","journal-title":"Wave Motion"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/BF01037881","article-title":"Nonlinear acoustic methods of crack diagnostics","volume":"38","author":"Sutin","year":"1995","journal-title":"Radiophys. Quantum Electron."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"064108","DOI":"10.1103\/PhysRevB.79.064108","article-title":"Analysis of elastic nonlinearity using the scaling subtraction method","volume":"79","author":"Bruno","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/s11665-009-9532-5","article-title":"Damage characterization and real-time health monitoring of aerospace materials using innovative NDE tools","volume":"19","author":"Matikas","year":"2010","journal-title":"J. Mater. Eng. Perform."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1007\/978-1-4613-0383-1_176","article-title":"Linear and Nonlinear Ultrasonic Properties of Fatigued 400Cb Stainless Steel","volume":"15A","author":"Na","year":"1996","journal-title":"Rev. Prog. Quant. Nondestruct. Eval."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1080\/01418619408244346","article-title":"Acoustic harmonic generation from fatigue-induced dislocation dipoles","volume":"69","author":"Cantrell","year":"1994","journal-title":"Philos. Mag. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/S0963-8695(00)00063-3","article-title":"Nonlinear acoustic interaction on contact interfaces and its use for nondestructive testing","volume":"34","author":"Donskoy","year":"2001","journal-title":"Ndt E Int."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1016\/S0041-624X(02)00186-5","article-title":"CAN: An example of nonclassical acoustic nonlinearity in solids","volume":"40","author":"Solodov","year":"2002","journal-title":"Ultrasonics"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1002\/j.1538-7305.1935.tb00688.x","article-title":"Extraneous frequencies generated in air carrying intense sound waves","volume":"14","author":"Thuras","year":"1935","journal-title":"Bell Syst. Tech. J."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Burgos, D.A.T., Mujica, L.E., and Rodellar, J. (2015). Emerging Design Solutions in Structural Health Monitoring Systems, IGI Global.","DOI":"10.4018\/978-1-4666-8490-4"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1121\/1.1918142","article-title":"Determination of the Nonlinearity Parameter B\/A for Water and m-Xylene","volume":"34","author":"Adler","year":"1962","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1016\/j.ultrasmedbio.2006.10.008","article-title":"Finite Amplitude Measurements of the Nonlinear Parameter B\/A for Liquid Mixtures Spanning a Range Relevant to Tissue Harmonic Mode","volume":"33","author":"Wallace","year":"2007","journal-title":"Ultrasound Med. Biol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1121\/1.1909806","article-title":"Parameter of nonlinearity in fluids. II","volume":"38","author":"Coppens","year":"1965","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1012","DOI":"10.1016\/j.ultras.2013.01.008","article-title":"Determination of acoustical nonlinear parameter beta of water using the finite amplitude method","volume":"53","author":"Pantea","year":"2013","journal-title":"Ultrasonics"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1121\/1.1432978","article-title":"Measurement of the B\/A nonlinearity parameter under high pressure: Application to water","volume":"111","author":"Plantier","year":"2002","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1121\/1.391277","article-title":"Ultrasonic investigation of the nonlinearity parameter B\/A in biological media","volume":"76","author":"Gong","year":"1984","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1210","DOI":"10.1121\/1.385704","article-title":"Ultrasonic determination of the nonlinearity parameter B\/A for biological media","volume":"69","author":"Law","year":"1981","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Sturtevant, B.T., Pantea, C., and Sinha, D.N. (2012). Determination of the Acoustic Nonlinearity Parameter in Liquid Water up to 250 \u00b0C and 14 MPa. Ultrason. Sympos., 285\u2013288.","DOI":"10.1109\/ULTSYM.2012.0070"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"077133","DOI":"10.1063\/1.4926974","article-title":"Simultaneous evaluation of acoustic nonlinearity parameter and attenuation coefficients using the finite amplitude method","volume":"5","author":"Zhang","year":"2015","journal-title":"AIP Adv."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1080\/09349847.2016.1158889","article-title":"Assessment of Acoustic Nonlinearity Parameters Using an Optimized Data-Fitting Method with Multi-Gaussian Beam Model-Based Diffraction Correct","volume":"27","author":"Jeong","year":"2016","journal-title":"Res. Nondestruct. Eval."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1121\/1.4999328","article-title":"Calibration of focused ultrasonic transducers and absolute measurements of fluid nonlinearity with diffraction and attenuation corrections","volume":"142","author":"Li","year":"2017","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"EL237","DOI":"10.1121\/1.5029299","article-title":"Acoustic nonlinearity parameter measurements in a pulse-echo setup with the stress-free reflection boundary","volume":"143","author":"Jeong","year":"2018","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.ijnonlinmec.2007.12.007","article-title":"Non-linear acoustic measurements to assess crack density in trabecular bone","volume":"43","author":"Renaud","year":"2008","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1861","DOI":"10.1016\/j.ultras.2013.12.002","article-title":"Effects of experimental configuration on the detection threshold of hysteretic elastic nonlinearity","volume":"54","author":"Idjimarene","year":"2014","journal-title":"Ultrasonics"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1088\/0370-1301\/62\/2\/307","article-title":"The absortion of ultrasonic waves in liquids and its relation to molecular constitution","volume":"62","author":"Pikerton","year":"1949","journal-title":"Proc. Phys. Soc."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/5\/1156\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:37:00Z","timestamp":1760186220000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/5\/1156"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,7]]},"references-count":32,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["s19051156"],"URL":"https:\/\/doi.org\/10.3390\/s19051156","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,7]]}}}