# Analysis of Elastic Nonlinearity Using Continuous Waves: Validation and Applications
## 6 Conclusions  
We discuss here the efficiency, robustness and limitations of an approach recently proposed [45] and able to measure the nonlinearity in velocity and attenuation coefficient and separate their contributions. This method, based on a semi-analytical solution of the problem valid for quasi one-dimensional samples, is a faster and simpler alternative to standard approaches, such as  
Figure 10: Experimental behaviour of \(\Delta c/c_{0}\) and \(\Delta\alpha/\alpha_{0}\) vs. amplitude, for the four investigated samples. The same colours are used in the two subplots and are related to the samples reported in the legend.  
nonlinear resonant ultrasound spectroscopy [51], nonlinear elastic wave spectroscopy [12] or Dynamic AcoustoElastic Testing [52].  
The optimisation and definition of the limits of the MoDaNE approach are discussed by using the method to invert virtual data (resulting from numerical simulations). Different nonlinear cases were investigated and results show that the method allows measuring independently nonlinearity in velocity and attenuation coefficient and therefore to monitor their dependence on strain. The approach is extremely efficient in determining the coupling between modulus/damping nonlinearities and velocity/attenuation nonlinearities, which are normally neglected in other approaches. The robustness of the approach in determining the functional dependence of velocity and attenuation on strain has been demonstrated and the importance of a correct calibration has been quantified as well.  
As far as quantification of nonlinear parameters is concerned, some approximations/hypothesis have to be added. Since ultrasonic measurements, in which multiple reflections are present, detect signals resulting from the superposition of waves travelling along different paths, the nonlinear signature is intrinsically averaged over time and space. In the case of pure power law nonlinearities, assuming measured velocities and attenuation to be dependent on averaged quadratic strains, the discussed method provides an excellent approach to quantify correctly the nonlinear parameters. In more complex situations, e.g., when the nonlinear dependence is not quadratic or when conditioning is present [53], other approaches must be developed.  
The work presented here opens interesting perspectives in the field of non-destructive testing and characterisation of materials. Detection and correct quantification of elastic nonlinearity is indeed of great importance in view of applications of the several results obtained in the laboratory and reported in the literature. Furthermore, a method such as the one proposed and validated here allows easy and fast measurements, providing information on both velocity and attenuation coefficient. Extracting correctly from this information the corresponding physical nonlinear parameters (e.g., damaged-related nonlinearity in modulus and damping) is fundamental in order to achieve a classification of materials or damage types into classes exhibiting different nonlinear elastic responses.  
Conceptualisation, A.S.G. and M.S.; methodology, A.D.B. and M.S.; simulations, A.D.B.; data analysis, A.D.B., A.S.G., M.S. and M.T.; experiments and data analysis, A.D.B. and M.T.; and writing, A.D.B., A.S.G., M.S., and M.T.  
This research received no external funding  
The authors declare no conflict of interest.