10` eme Congr` es Franc ¸ais d’Acoustique Lyon, 12-16 Avril 2010 Nonlinear Ultrasonic Spectroscopy Using ESAM and DORT Symbiosis Sarka VEJVODOVA 1 , Zdenek PREVOROVSKY 1 , Serge DOS SANTOS 2 1 Institute of Thermomechanics AS CR, v.v.i., Dolejskova 5, CZ-18200 Prague 8, Czech Republic, e-mail: sv@it.cas.cz, zp@it.cas.cz 2 ENI Val de Loire, Unit´ e Mixte de Recherche ” Imagerie et Cerveau ” INSERM U930 - CNRS ERL 3106, Universit´ e Franc ¸ois Rabelais, Rue de la Chocolaterie BP 3410, F-41034 Blois cedex, France, serge.dossantos@univ-tours.fr Combination of DORT (D´ ecomposition de l’Op´ erateur du Retournement Temporel) and ESAM (Excitation Sym- metry Analysis Method) is presented as a powerful tool for detection and localization of defects by analysis of their nonlinear signature [1, 2]. The third order nonlinear responses are extracted from ESAM eigen-excitations and constitute the multistatic data matrix associated with the singular value decomposition. The symbiosis of ESAM-DORT provides a normalization process of singular values associated with scatterers, validated in simu- lations and experiments of acoustic propagation [3]. Piezoelectric transducers are used for both excitation and data acquisition. Transmitters consequently emit the excitation signals and corresponding responses are measured by the array of receivers. The amplitude and phase of excitation signals is variable as to separate the nonlinear parts of the measured signal. ESAM signal pre-processing is used for nonlinear parts extraction. Separated signal records form nonlinear multistatic data matrix. DORT method is applied on the data matrix to separate echoes of defects in the tested medium. Data obtained from DORT method are used for evaluation of nonlinear parameters corresponding to separated defects and also for their localization. The procedure is completed by visualization of nonlinear signatures of detected defects which is referred to as pseudotomographic imaging. This ESAM-DORT approach is compared to AE method and seems to improve the performance of the TR-NEWS methods, which can be applied to the tomography of structural defects. Introduction Methods based on elastic wave propagation provide a very effective tool in non-destructive testing (NDT) for detec- tion and quantification of damage in various materials. The most recent trend is to link time reversal (TR) and nonlinear (NEWS = Nonlinear Elastic Wave Spectroscopy) approaches of ultrasonic signal processing [4, 5]. On one hand, time reversal offers a bulk of advantages concerning signal-to- noise ratio enhancement, auto-focusing and heterogeneous medium compensation [6], on the other hand, nonlinear ap- proach permits evaluation and quantification of nonlinear be- havior typically generated by thin cracks and other defects like delamination or inhomohegeneities [7], which are the major subject of interest of latest NDT, since they are hardly detectable by standard linear methods. In this paper a new approach of TR-NEWS link will be presented. TR process is represented by DORT method (French acronym for D´ ecomposition de l’Op´ erateur de Re- tournement Temporel, i.e. decomposition of time reversal operator), an approximative method which allows separation of invariants of time reversal process, i.e. the Green’s func- tions (pure echoes) corresponding to well-resolved scatterers present it the tested sample [1]. The nonlinear signature of detected scatterers is evaluated by ESAM (Excitation Sym- metry Analysis Method). ESAM is used to extract nonlinear terms from the acoustic response. In order to obtain these terms, different excitations are used and corresponding re- sponses are treated according to the properties of point group C 3 [2]. Combination of ESAM and DORT allows us formu- late quantified nonlinear parameters associated to different scatterers or zones of the tested sample. With regard to the sample geometry and positions of the transducers used for the data acquisition, the parameter values can be visualized to obtain an image of damaged zones. 1 ESAM-DORT Analysis 1.1 ESAM signal pre-processing Medium with tiny cracks generally shows an enhanced 3 rd order nonlinearity which could be extracted with the ESAM signal processing approach using optimized excitations [2]. ESAM uses nonlinear signature coming from the direct acoustic response as a pre-processing tool for the extraction of nonlinear signature. If scatterers’ responses are supposed to have cubic ex- pansion in strain, the nonlinear response can be written as y(t) = N 1 x(t) + N 2 x 2 (t) + N 3 x 3 (t), where N 1 , N 2 and N 3 are nonlinear parameters. With use of the properties of point group C 3 and its irreducible representation, ESAM permits extraction of these parameters. The responses y E , y ǫ , y ǫ * to the excitations x E = x(t), x ǫ (t) = x(t)e 2iπ 3 , x ǫ * (t) = x(t)e - 2iπ 3 enable extraction of the cubic term s 3 (t) = N 3 x 3 (t) by s 3 (t) = N 3 x 3 (t) = y E (t) + y ǫ (t) + y ǫ * (t) 3 . (1) Term s 3 (t) extracted by ESAM corresponds to the nonli- near behavior of the system, but also it still depends on the original excitation x(t) and its powers. Therefore, to obtain the unique coefficient N 3 , the excitation dependent parts must be eliminated. In order to realize it, cubic energy related to the excitation, given by E 03 =  ∞ -∞ | x 3 (t)| 2 dt, (2)