Discriminating linear from nonlinear elastic damage using a nonlinear time reversal DORT method

The DORT method is a selective detection and focusing technique originally developed to detect defects and damages which induce linear changes of the elastic moduli. It is based on the time reversal (TR) where a signal collected from an array of transducers is time reversed and then back-propagated into the medium to obtain focusing on selected targets. TR is based on the principle of spatial reciprocity. Attenuation, dispersion, multiple scattering, mode conversion, etc. do not break spatial reciprocity. The presence of defects or damage, may cause materials to show nonlinear elastic wave propagation behavior that will break spacial reciprocity. Therefore the DORT method will not allow focusing on nonlinear elastic scatterers. This paper presents a new method for the detection and identification of multiple linear and nonlinear scatterers by combining nonlinear elastic wave spectroscopy, time reversal and DORT method. In the presence of nonlinear hysteretic elastic scatterers, forcing the solid with a harmonic excitation, the time reversal operator can be obtained not only at the fundamental frequency of excitation, but also at the odd harmonics. At the fundamental harmonic, either inhomogeneities and linear damages can be individually selected but only at odd harmonics nonlinear hysteretic elastic damages exist. A procedure was developed where by decomposing the operator at the odd harmonics, it was possible to focus on nonlinear scatterers and to differentiate them from the linear inhomogeneities. A complete mathematical nonlinear DORT formulation for 1 and 2D structures is presented. To model the presence of nonlinear elastic hysteretic scatterers a Preisach–Mayergoyz (PM) material constitutive model was used. Results relative to 1 and 2 dimensional structures are reported showing the capability of the method to focus and discern selectively linear and nonlinear scatterers. Furthermore, an analysis was conducted to study the influence of the number of sources and their location on the imaging process showing that using a higher numbers of sensors does not automatically bring to a minor uncoupled behaviour between the nonlinear targets.