# Analysis of Elastic Nonlinearity Using Continuous Waves: Validation and Applications
## 1 Introduction  
Materials with a very different microstructure, including consolidated [1; 2; 3; 4; 5] and unconsolidated [6] granular media or damaged composites and metallic materials [7; 8; 9; 10; 11], share a very similar elastic behaviour, in which hysteresis plays a crucial role. Their nonlinear response to an ultrasonic excitation is governed by a combination of effects due to velocity dependence on strain and attenuation nonlinearity [12; 13; 14; 15]. Furthermore, even more important are effects due to conditioning, relaxation and memory [16; 17], i.e., the dependence of velocity and attenuation on time and maximum strain (history).  
Qualitatively, the nonlinear behaviour of such materials looks very similar. On the contrary, their microstructure, which should suggest the physical mechanisms responsible of their elastic behaviour, is very different. Consolidated granular materials feature grain contacts, which may give rise to sliding [18] and/or static friction [19], together with adhesion mechanisms linked to the nonlinear behaviour due to the presence of water at the interfaces between microcracks faces [20; 21]. On the other hand, in damaged composites, open or partially open cracks, responsible for clapping [22; 23] or dislocations [24], are more likely to be the cause of the observed nonlinearities. Furthermore, the nonlinear elastic response increases dramatically in damaged materials. Again, the behaviour remains qualitatively the same, but the effects of the damaging processes, e.g., in consolidated granular media, could be very different: formation of open cracks for mechanical or thermal damaging [2; 25; 26; 27], chemical reactions with expansive compounds in carbonation, corrosion or salt reactions [28; 29; 30], or even healing of cracks in some cases [31; 32].  
Consequently, it is hard to think of a single physical mechanism responsible for the elastic nonlinearity observed and, thus, it could be important to highlight differences in nonlinear materials with different microstructures, either intrinsic or damaged induced [33; 34; 35]. Nevertheless, to achieve this purpose, a finer analysis is necessary, which requires developing novel experimental approachesbeyond those existing in the literature, i.e., harmonics and sidebands analysis [36; 37], determination of the resonance frequency shift [25; 38], quantification of the loss of proportionality [39; 40], analysis of subharmonics [41] and others [42; 43].  
The development of a method which could cope with experimental and physical constraints is required, remaining at the same time able to separate nonlinearity due to damping from that due to elastic modulus [44; 45]. In this regard, a method has been recently proposed [45], which makes use of a semianalytical analysis of experimental data obtained with a monochromatic continuous waves (cw) excitation in one dimensional samples.  
The novelty of this contribution is manifold. Indeed, the aim of the present work is: (i) to confirm the correctness of the proposed method over a wide range of nonlinearity conditions (validation); (ii) to quantify the sensibility of the approach (quantification); and (iii) to establish the limits of its applicability (optimisation and robustness).  
In the next section, we first present the approach to measure velocity and attenuation coefficient as a function of amplitude, by using a monochromatic continuous wave excitation. The approach is validated by analysing its efficiency in the treatment of synthetic data obtained using a numerical approach [46], also discussed in Section 2. Validation of the efficiency in separating nonlinearities, due to modulus or damping, is presented in Section 3. The validity of the approximations, needed to comply with experimental constraints when applying this approach, is discussed in Section 4. Finally, experimental results are shown in Section 5 to prove the suitability of the method to discriminate between different nonlinearities in different media.