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NDT methods for interface characterization
Various nonlinear processes have been studied in the context of nonlinear characterization: nonlinear resonances, harmonic generation, non-linear wave-modulation spectroscopy, sub-harmonics… It is often observed that the measurements of nonlinear parameters from these nonlinear processes are more sensitive to the presence of an interface than linear elastic parameters (measured through linear vibro-acoustic methods), especially at early damage states . Driven by industrial needs to improve the detectability of defects, nonlinear characterization methods have been the subject of a considerable number of research work. These methods can be grouped together under the abbreviation NEWS (Nonlinear Elastic Wave Spectroscopy) . They differ by the nonlinear phenomenon exploited.
NL resonance spectroscopy
One of the non-linear phenomena caused by the presence of microscopic heterogeneities and defects distributed in volume of the material is the variation of the resonant frequency when increasing the excitation amplitude . This phenomenon is directly related to the presence of heterogeneities, thus it can be used to quantify the damage level. This method is referred to as nonlinear resonance ultrasound spectroscopy NRUS  .
The method consists of evaluating one or more frequency peak shifts while increasing the amplitude. In fact, a sample is subjected to progressive damage induced by excitation around its resonance frequencies. The amplitude is increased gradually and the new resonance frequency is evaluated at each iteration. The resonance frequency shift indicates the presence of damage. Van Den Abeele and al. applied this technique to artificial slate slabs , which were progressively damaged by successive impacts in the centre of the slab. The shift in resonance frequency is obtained when the excitation increases and is higher in the case of damaged sample. Figure 6 gives an example of an experimental result.
Generation of higher harmonics
The phenomenon of higher harmonic generation is one of the oldest phenomena in nonlinear acoustics. The first experimental demonstration of this phenomenon in solids was reported in 1963 by Breazeale and Thompson  and by Gedroits and Krasilnikov . In these two studies, the generation of harmonics is the result of classical nonlinearity. In 1965, Hikata, Chick and Elbaum  showed that the phenomenon of higher harmonics generation increases considerably in the presence of defects. This work studies the detection of dislocation defects in metallic materials. Yermilin and al.  reported the first studies of harmonic generation in the case of fatigue cracks in 1973 and by Buck and al.  in 1976. These initially consider an artificial crack made by two aluminium blocks in contact. In a second step, the test was carried out on aluminium specimen AL2024 with a real fatigue crack. In both cases, the nonlinear interaction between a longitudinal wave at normal incidence and the pre-stressed contact interface was investigated. The results of this interaction show that the amplitude of the generated second harmonic presents a maximum as a function of the applied compression, which was confirmed by Richardson’s analytical model  published in 1979. A similar result was obtained by Taehyung Nam and al  using a longitudinal wave with oblique incidence on an interface of contact between two aluminium blocks. The reflected wave is analysed according to the normal stress applied. For the two angles (22.5° and 45°), the amplitude of the second harmonic goes through a maximum but with different amplitudes. The optimal of second harmonic amplitude as a function of the applied compression is characteristic of the nonlinear contact. Further details will be discussed in the chapter 3.
Another possibility to detect the damage is to study the evolution of the nonlinear parameter 𝛽∗=𝐴2𝐴12 as a function of the applied load. Although this definition of the nonlinear parameter comes from the classical nonlinear acoustics theory, it is also used to assess nonlinearity caused by an interface. Buck and al.  show that this parameter reaches a maximum value when the applied compression is zero, and decreases exponentially as the applied compression increases. Biwa and al. obtained similar results again on aluminium blocks . However, the evolution of the second harmonic shows a slight hysteresis effect depending on the applied load . Lee and Jhang studied the evolution of the nonlinear parameter along a fatigue crack in an aluminium specimen . A constant pressure is maintained on the transducers by a pneumatically controlled system to ensure a stable and repeatable measurement. For constant compression applied to the specimen, the nonlinear parameter 𝛽∗decreases progressively as it approaches the crack tip, and becomes constant once the point of the crack has been surpassed. Due to the stress concentration at the end of the crack, the nonlinear mechanisms are less activated, which limits the nonlinearity of the interface. For a given position, the parameter 𝛽∗ decreases exponentially as the applied load increases. Another example of fatigue crack detection in a thin high resistance aluminium specimen using the nonlinear parameter 𝛽∗ has been reported by Morris and al .
Several experimental works have studied the evolution of the nonlinear parameter 𝛽∗ as a function of the number of fatigue cycles in different materials, such as aluminium alloys , nickel-based super alloys , carbon steel , titanium alloys and stainless steel . The overall results show that the nonlinear parameter 𝛽∗ is sensitive to the onset of fatigue damage. For example, Frouin and al  show that 𝛽∗ starts to increase from 40% of the fatigue life. Classical techniques do not indicate the onset of a defect until 80 or 90% of the fatigue life .
Second harmonic measurement is also used to monitor ageing damage in materials such as aluminium alloys , titanium alloys , ferritic steels  and stainless steels . The results of all experiments show a sensitivity of the nonlinear parameter 𝛽∗ as a function of the treatments during thermal ageing. For example, Yost and Cantrell  showed an increase of about 10% of 𝛽∗ during the heat treatment of Al-2024.
Interface stiffness preliminary characterization
The contact stiffness of the tested samples was previously determined by experimental measurements, as reported in . The proposed methodology for the estimation of normal stiffness is based on the dynamic contribution of the interface into the dynamic response of the mechanical system. A combined numerical and experimental approach is proposed for an indirect identification of the contact stiffness parameter. From an experimental point of view,
the setup (Tribobrake) has been designed and used to perform dynamic tests. Afterwards, a 3D finite element model, updated and representative of the experimental set-up, has been used to compare the numerical and experimental results, for estimating the normal contact stiffness between the considered contact surfaces.
Thus, the comparison between the results from the numerical model and the frequency response function obtained from experiments allows retrieving the value of the normal contact stiffness. In fact, a parametrical numerical modal analysis, as a function of the contact stiffness, is performed in order to meet the natural frequency of the investigated mode with the one measured experimentally. The numerical value of the contact stiffness that allows for a matching between experimental and numerical natural frequencies is retained as the estimated value of the interface stiffness. From preliminary dynamic tests at different contact pressures, the contact stiffness has been estimated between 0.14 MPa and 1 MPa.
Table 2 shows the results for contact stiffness as a function of the average contact pressure for the (AL)-(AL) and (AL)-PMMA interface , with surface roughness of Ra = 1 μm.
The data highlight how the contact stiffness increases with the rise in the average contact pressure. The contact stiffness values range from 1.15 x1012 Pa/m to 2.46 x1012 Pa/m for the AL-AL interface and from 3×1011 to 6×1011 for a AL-PMMA interface, when the contact pressure increases from 0.14 MPa to 1 MPa.
1D modelling of the experimental system
A one-dimensional numerical model of the experimental set-up is implemented (Figure 12). It consists of the guide bar Ω1 and the tested aluminium sample Ω2, modelled by unidimensional deformable bodies.
Two contacting interfaces are considered: the first at x=𝑥𝐶1 between the guide bar Ω1 and the tested sample Ω2, and the second at x=𝑥𝐶2 between the tested sample and the frame, considered as infinitely rigid in the model (tribometer disc). The model parameters are provided in Table 3. Note that, in reality, the cross-sections of the guide bar Ω1 and aluminium sample Ω2 are different. Numerically, an equivalent 1D-model with the same cross-section S2 is considered, corresponding to the contact surface S2, by modifying the corresponding Young Modulus.
This model is a simplification of the two-interface model (see Figure 12), where the system is reduced to a half-space model in contact with a rigid contact interface. The reflection of a normal incident plane wave at a contact interface is considered, as shown in Figure 13. The numerical model is defined for an homogeneous, isotropic elastic half-space, defined as Ω, and assumed to be initially in contact with a rigid wall at 𝑥=0. 𝛤𝑐 denotes the parts of the boundary where the solid and the wall are in contact. Here, a transparent boundary 𝜕Ω𝑡 is defined in order to model an infinite medium. It is assumed that an initial static compressive stress 𝜎0=−𝑝0 prevails. A source, placed at 𝑥=−𝐿, generates an incident stress, in Ω, denoted 𝜎𝑖. The maximal value of |𝜎𝑖(𝑡)| is noted 𝜎𝑖𝑚𝑎𝑥.
Table of contents :
CHAPTER 1: OVERVIEW OF MECHANICAL MODELS OF ROUGH CONTACT
1.2 MODELING OF CONTACT INTERFACES
1.2.1. BILINEAR MODELS
1.2.2. CONTACT LAWS
126.96.36.199. Contact unilateral
188.8.131.52. Rough interface models in compression
184.108.40.206. Rough interface modelling with a traction-free condition
220.127.116.11. Models involving adhesion
1.3 NDT METHODS FOR INTERFACE CHARACTERIZATION
1.3.1. NL RESONANCE SPECTROSCOPY
1.3.2. GENERATION OF HIGHER HARMONICS
1.3.3. NL SPECTROSCOPY BY WAVE MODULATION
1.4 POSITIONING OF THE THESIS WORK
CHAPTER 2: MATERIALS AND METHODS
2.1 DESCRIPTION OF THE METHODOLOGY
2.2 EXPERIMENTAL TOOLS
2.2.1 EXPERIMENTAL SET-UP
2.2.2 INTERFACE STIFFNESS PRELIMINARY CHARACTERIZATION
2.3 NUMERICAL IMPLEMENTATION TOOLS
2.3.1 1D MODELLING OF THE EXPERIMENTAL SYSTEM
2.3.2 SIMPLIFIED MODEL
2.3.4 NUMERICAL IMPLEMENTATION
2.4 CONCLUDING REMARKS
CHAPTER 3: NONLINEAR ANALYSIS OF THE RCCM CONTACT LAW WITH ADHESION
3.1 VALIDATION OF THE NUMERICAL MODEL
3.2 INTERFACE DYNAMIC BEHAVIOUR: COMPARISON BETWEEN RCCM CONTACT LAW AND UNILATERAL LAW
3.3 PARAMETRIC ANALYSIS
3.3.1 INFLUENCE OF THE CONTACT STIFFNESS ON THE NONLINEAR SIGNATURE OF THE RCCMLAW
3.3.2 INFLUENCE OF THE FREQUENCY ON THE NONLINEAR SIGNATURE OF THE RCCM CONTACT LAW
3.3.3 INFLUENCE OF THE DECOHESION ENERGY ON THE NONLINEAR SIGNATURE OF THE RCCM CONTACT LAW
3.3.4 INFLUENCE OF THE VISCOSITY ON THE NONLINEAR SIGNATURE OF THE RCCM CONTACT LAW
3.1. IDENTIFICATION OF CONTACT LAW PARAMETERS
3.4 CONCLUDING REMARKS
CHAPTER 4: NORMAL CONTACT STIFFNESS IN COMPRESSION
4.1 APPROACH FOR THE NONLINEAR STIFFNESS ANALYSIS
4.2 NEW CONTACT LAW: MODIFIED POWER-LAW
4.3 EXPERIMENTAL AND NUMERICAL COMPARISON
4.3.1 DYNAMIC RESPONSE OF THE CONTACT SYSTEM
4.3.2 NONLINEAR RESPONSE OF THE INTERFACE
4.4 CONCLUDING REMARKS
CHAPTER 5: CONTACT LAW IN COMPRESSION AND TRACTION
5.1 WAVE SCATTERING CONFIGURATION
5.2 VALIDATION OF THE NUMERICAL MODEL
5.2.1 ANALYTICAL SOLUTION FOR LINEAR SPRING WITH AND WITHOUT CONTACT LOSS
18.104.22.168 Without contact loss
22.214.171.124 With contact loss
5.2.2 NUMERICAL AND ANALYTICAL COMPARISON
5.3 RESULTS AND DISCUSSION
5.3.1 LINEAR SPRING IN COMPRESSION
5.3.2 NONLINEAR STIFFNESS IN COMPRESSION
126.96.36.199 Undamaged interface
188.8.131.52 Damaged interface
5.4 CONCLUDING REMARKS
6.1 ORIGINAL MAIN CONTRIBUTIONS
6.2 OUTLINES AND FUTURE STEPS