## 5.1. LOAD-DISPLACEMENT RELATIONSHIP IN CONTACT ACOUSTIC NONLINEARITY UNDER ARBITRARY OBLIQUE EXCITATION

The idea [126] behind this separation can be illustrated as follows. Suppose the tangential loading increases while keeping constant normal compression $N = N(a)$ corresponding to a certain normal displacement $a$. The bodies in contact move away from each other under load in both normal and tangential directions. In addition, the tangential receding $\tilde{b}$, in contrast to the normal one, $a$, is accompanied by partial slip. According to the MMD based on the Coulomb friction law, $\tilde{b}$ is not allowed to grow infinitely. Once the maximum value $\tilde{b}_{max} = \theta\mu a$ is achieved, the bodies can not deform anymore and a total sliding process develops when the very last stick point belonging to one face separates from its neighbor at the opposite one. The tangential displacement between those points is denoted $b_0$ and corresponds to the contribution from total sliding. In other words, $b_0$ is a reference point mismatch, in some sense.

<table><caption>FIGURE 5.6 – Three possible contact states. For each case, the following information is supplied: conditions under which the case occurs, solutions for components and , solutions for forces $N$ and $T$, and memory diagrams.</caption><thead><tr><th>Crack state</th><th>If</th><th>Then</th><th>Memory diagram</th></tr></thead><tbody><tr><td>(i) Contact loss</td><td>$a < 0$</td><td>$\tilde{b} := 0 \quad b_0 := b$<br>$T = N = 0$</td><td></td></tr><tr><td>(ii) Total sliding</td><td>$a \ge 0$<br>$|\tilde{b}| \ge \theta\mu a$</td><td>$\tilde{b} := \pm\theta\mu a \quad b_0 := b - \tilde{b}$<br>$T = \pm\mu N$</td><td></td></tr><tr><td>(iii) Partial slip</td><td>$a \ge 0$<br>$|\tilde{b}| < \theta\mu a$</td><td>$b_0 := b_0 \quad \tilde{b} := b - b_0$<br>$T = \text{MMD}(\tilde{b})$</td><td></td></tr></tbody></table>

Fig. 5.6 provides an overview of the full concept in case both normal and tangential displacements $a$ and $b$ evolve in an arbitrary manner. The scheme explains how to calculate the forces $N$ and $T$ for any given value of displacements $a$ and $b$. In order to do so, the tangential displacement components $b_0$ and $\tilde{b}$, together with the corresponding memory diagram should be updated. The updating operation for
the tangential displacement components is denoted using the assignment operator « := », which means that values obtained at the previous time step are overwritten by new ones [126].

<figure><img src="image_73.png" /><figcaption>FIGURE 5.7 – (a) Tangential force as a function of tangential displacement and normal force (b) in case of particular displacement histories shown in (c).</figcaption></figure>

A typical result provided by the MMD with some particular normal and tangential displacement histories is illustrated in fig. 5.7. In this example $a = 1.3+1.9 \sin(1.7t)+0.9 \sin(2.4t)$, $b = \sin(t) + 1.5 \sin(1.4t) + 0.8 \sin(2.1t)$. Note that since the tangential force is a history-dependent function of two arguments $a$ and $b$, its behavior is more complicated than typical patterns for uni-variate hysteresis with closed loops. The curve is composed of several fragments marked in fig. 5.7 a with different colours : blue stands for total sliding in which $T = \pm\mu N$, red stands for partial slip, and green is for contact loss.

The same regimes can be seen in fig. 5.7 b visualizing the link between the normal and tangential forces. As one can see, total sliding indeed corresponds to fragments $T = \pm\mu N$. Parts of curves $T = T(N)$ located in between the delimiting lines $T = \pm\mu N$ are related to the partial slip regime, while the case of contact loss corresponds to the point (0,0). This behavior is in complete agreement with the Coulomb’s friction
law.

The algorithm in fig. 5.6 completes the description of the contact model. It is formally valid for axisymmetric bodies; however, under some limitations, can be considered as a good approximation for contact of rough surfaces as well. Indeed, following the Greenwood and Williamson approach [127], all asperities can be considered as spherical or axisymmetric. Even if they are not, i.e. individual microcontacts can have arbitrary shapes, in many cases it is reasonable to consider isotropic roughness when along all possible directions parallel to the global surface plane all statistical properties of roughness are the same (one of exceptions is rough surfaces obtained by grinding). In the isotopic case, each microcontact can be effectively replaced by a pair of equivalent axisymmetric bodies for which the above theory is applicable.

Finally, we will list the assumptions that were used explicitly or implicitly to construct the method.

1. Loading is in one plane only (i.e. in 2D); 3D extensions are discussed in [128].

2. Only shift is considered; torsion and rolling are ignored.

3. The Coulomb friction law with friction coefficient $\mu$ is postulated for contact stress fields.

4. The model is quasi-static: differences between static and dynamic friction are not considered, inertial behavior in the material in the vicinity of the contact zone is neglected.

5. Plasticity and adhesion are ignored.

6. The normal to the contact area (or all normals to all microcontact areas for rough surfaces) does not rotate and stays in the same direction.

7. The normal solution is a known bi-unique function $N = N(a)$ independent of the tangential loading.

8. Dissimilarity effects are neglected.

9. The reduced elastic friction principle is valid for the considered geometry (exact for axisymmetric bodies and reasonable approximation for isotropic rough surfaces).

In the next section, the Method of Memory Diagrams (automated Hertz-Mindlin solution) will be used for computing forces excited by the Lamb waves at CAN location.
## 5.2 Modeling for contact acoustic nonlinearities excited by the Lamb waves

In this section, contact forces excited by the Lamb wave for two types of contact acoustic nonlinearities will be calculated from the known wave displacements: for the Hertz-Mindlin CAN (HM-CAN) and for rough surface CAN (RS-CAN). Here the HM-CAN corresponds to the physical model for damage used in the experiments that represents a contact between a sphere and a plate, whereas the RS-CAN refers to a more realistic case where a defect is a crack or a loose joint with rough surfaces. Both CANs produce nonlinear force reaction on acoustic excitation; the obtained forces as functions of time can be interpreted as a secondary waves' source located at small area (practically a point) on the plate surface. In what follows we focus on the interaction of CANs and the Lamb wave in more detail.

### 5.2.1 Hertz-Mindlin CAN

In the considered geometrically idealized experimental case, a small steel ball pressed against a top surface of an aluminum plate is exited by the A0 Lamb mode. Then the coordinates of the ball's center of mass supposed to be zero when the bodies are unstrained are given as a superposition of wave displacements and contact displacements:

$$ \begin{cases} x = b + u_x = 0 \\ y = a + u_y. \end{cases} \qquad (5.16) $$

The former equation takes into account that horizontal motion is prohibited by a device positioning the sphere. This taboo comes from the experiment in which sphere's horizontal motion on the plate activates acoustic emission due to friction and makes the defect detection method reference-based due to the varying sphere's coordinate. The latter equation in the system (5.16) is to be combined with the equation of motion:

$$ m \frac{\partial^2 y}{\partial t^2} = N_0 - N(a) \qquad (5.17) $$

with $m$, the mass of the sphere. Here $N_0$ denotes precompression force that has two components, elastic prestress and gravity, and is positive for the y-axis directed downward, whereas the normal contact reaction force $N(a)$ is directed upward. The geometry of the problem is illustrated in fig 5.8b.
<figure><img src="image_74.png" /><figcaption>(a) Small ball pressed against a plate is exited by the Lamb wave (plate's deformation is highly exaggerated). (b) Zoom on the ball with the normal contact displacement shown.</figcaption></figure>

<figure><img src="image_75.png" /><figcaption>FIGURE 5.8 – Model of HM-CAN.</figcaption></figure>
The equation of motion has to be supplemented by the calculation of the normal and tangential contact forces given by

$$ \left\{ \begin{aligned} N(a) &= \frac{4}{3} E^* R^{*1/2} a^{3/2} \\ T(b, a) &= T_{MMD}(b, a), \end{aligned} \right. \qquad (5.18) $$

where $T_{MMD}$ is an automated analytical solution given by the Method of Memory Diagrams described before in section 5.1.2.

Besides, we have to specify explicitly the known solution Eq. (1.3) for the A0 Lamb wave. This stationary solution is fully harmonic in time $\forall t \in (-\infty, +\infty)$. However, in order to solve the equation of motion, one has to set initial conditions in terms of coordinates or contact displacements which are a priori unknown in the stationary wave regime. To avoid this difficulty, we assume the initial condition $a = a_0$, with $a_0$, contact displacement produced by force $N_0$, i.e. in the absence of wave. To comply with zero wave displacements at $t = 0$, the wave form is considered with a formal « ramp function » that equals zero at the initial moment and then gradually reaches 1 :

$$ \left\{ \begin{aligned} u_x &= u_x^{\text{Lamb}} \cdot r(t) \\ u_y &= u_y^{\text{Lamb}} \cdot r(t) \\ r(t) &= 1 - e^{-(\frac{t}{t_{\text{ramp}}})^2}. \end{aligned} \right. \qquad (5.19) $$

A formal requirement on the ramp duration $t_{ramp} \gg 1/f$, where f denotes acoustic excitation frequency, guarantees that the distortion introduced by using this technique is negligible.

Due to the presence of the Hertz nonlinearity and of the hysteretic friction force, the equation of motion Eq. (5.17) together with Eq. (5.18) have to be solved numerically. As far as initial problems for ordinary differential equations are concerned, at least two numerical approaches are commonly used : the Runge-Kutta and the Adams-Bashforth [129] methods. The both are precise and efficient enough and assume higher orders of approximation (e.g. 4th, 5th). However, the former one involves expressions to be calculated in between of time discretization points $t_j$, while the latter one is fully based on calculations at points $t_j$ only. Since the MMD does not directly provide any values in between of time discretization points, the Adams-Bashforth method is more appropriate in that case. Having this in mind, we rewrite the equations (5.17) and (5.16) to be solved in a form suitable for Adams-Bashforth
method of the 5th order application :

$$ \left\{ \begin{aligned} \frac{\partial y}{\partial t} &= y_d \\ \frac{\partial y_d}{\partial t} &= \frac{1}{m} (N_0 - N|_{a=y-u_y(t)}) \end{aligned} \right. \qquad (5.20) $$

in which the first time derivative $y_d$ of the y-coordinate is defined explicitly. The numerical calculations are performed in accordance to the following scheme :

$$ \left\{ \begin{aligned} y_{j+5} &= y_{j+4} + dt \sum_{i=0}^{4} k_{i+1} f(t_{j+5-i}, y_{j+5-i}) \\ y_1 &= a_0, \end{aligned} \right. \qquad (5.21) $$

where $k_1 = \frac{1901}{720}$, $k_2 = -\frac{1387}{360}$, $k_3 = \frac{109}{30}$, $k_4 = -\frac{637}{360}$, $k_5 = \frac{251}{720}$, $dt$ is a time step with $f(t_i, y_i) = 0$ for $i = 1, 2, 3, 4$. The latter equation indicates precompression at the initial moment of time.

Note that for making a new step in time the method requires knowledge of all variables on five previous time steps. This means that for five first steps of the algorithm, some of these previous values are not defined. However, since the procedure starts with zero initial conditions, we assume all such variables to be 0.

Now we know how to calculate forces exerted at HM-CAN from the known Lamb wave-driven displacement. Calculation results for the HM-CAN excited by the Lamb wave obtained via the above method are considered later in section 5.2.3. Here we first concentrate on an analogous description for the RS-CAN.

### 5.2.2 Rough surface CAN

The obvious difference between the case of real damage and the model HM-CAN is that for real damage no additional mass has to be considered. The only inertial behavior for contact of rough surfaces can be associated with thin layers of material comprising surface roughness that can react on bulk load with some delay. However, the corresponding relaxation time is related to microstructure and is therefore much smaller than the acoustic period and thus is negligible. The quasi-static approximation for the boundary conditions at rough surfaces of real damage remains valid for any acoustic excitation of interest.

In the experiments and simulations for the HM-CAN the ball is prestressed. A real crack can be prestressed either at least due to the following reason. During cracking, an external action is applied that separates crack faces by a distance significantly exceeding the atomic size. Upon releasing the action the crack faces tend to return
back to the unstrained state, but due to micro-distortions of local surface shapes they can not do so with the atomic precision. The resulting shapes' mismatch acts as a thin layer of additional material « inserted » between crack faces which actually plays a role of prestress.

Implementing the above considerations, we assume the governing equation in the following form :

$$ \left\{ \begin{array}{l} a = a_0 - u_y \\ N = K a^2 \\ x = b + u_x(t) = 0 \\ T = T_{MMD}(b, a), \end{array} \right. \qquad (5.22) $$

where $K = C^2 A_n$, and $C = 6 \cdot 10^{10} Pa^{1/2}m^{-1}$ is Biwa's empirical constant.

Here no mass is present so that the equation of motion is replaced by calculation of instantaneous reaction of the contact on the action of the Lamb wave. Displacement $a_0$ is responsible here for the prestress effect.

A graphical representation for the considered system is given in fig. 5.9a. The no-mass formulation corresponds to a contact between a surface and a small (much less then the wavelength) unmovable block whose coordinate y always stays equal 0 and therefore is not introduced. Certainly, other configurations are possible, but the considered one fully corresponds to Eq. (5.22) and is geometrically similar to the HM-CAN case. Here $A_n$ is the nominal contact area (see section 5.1.1) and $C$ is a constant [113] that can be either estimated from contact microgeometry parameters by using the model discussed in [125] or measured experimentally for a real contact in a way similar to Biwa et al. [113].

### 5.2.3 Comparison of reactions of both CANs on acoustic excitation

Here it is appropriate to discuss the compliance between the model and real damage CANs. Obviously, the powers in the normal reaction law are not the same in these cases (3/2 and 2). However, by selecting proper parameters, the normal load-displacement curves can be numerically matched. This procedure is required to quantitatively justify relevance of the model defect in the experiment (HM-CAN) as approximation of the real damage (RS-CAN). Also there is an unknown constant $A_n$ in the expression for the force generated by RS-CAN. The choice of this constant is
<figure><img src="image_76.png" /><figcaption>(a) Two rough surfaces in contact exited by the Lamb wave (plate's deformation is highly exaggerated). (b) Zoom on the RS-CAN.</figcaption></figure>

<figure><img src="image_77.png" /><figcaption>FIGURE 5.9 – Model of RS-CAN.</figcaption></figure>
based on the equality of the initial normal displacements $a_0$ caused by precompression for both HM-CAN and RS-CAN. An example of numerical match between normal reaction in the both models is shown in fig. 5.10 in which the HM-CAN and the RS-CAN responses are observed to be qualitatively similar. The RS-CAN curve is matched to HM-CAN curve with the least squares method. Accepting the value of $C = 6 \cdot 10^{10} Pa^{1/2}m^{-1}$ measured by Biwa et al [113] for two aluminum blocks as a reference, we get the radius of nominal contact area $A_n \approx 6.2410^{-9} m^2$ that corresponds to the contact size of

$$r_{contact} \approx 4 \cdot 10^{-5} m.$$

This means that in both cases the contact size is much less than all macroscopic geometry dimensions and thus can be considered as punctual. Also it is the same order of magnitude as a scattering cross-section detection threshold $\sigma_{th} = 0.065$ mm estimated in section 3.1 of chapter 3. Diameter of the nominal contact area 8 mm is greater than $\sigma_{th}$, which confirms that the CAN can be detected.

<figure><img src="image_78.png" /><figcaption>FIGURE 5.10 – The normal load-displacement curve for the HM-CAN and a close curve for RS-CAN with matched parameters for $m^* = 414$.</figcaption></figure>

Despite similar normal responses, the reaction of the considered CANs on the acoustic excitation can be different. The reason is in dynamic effects that arise in the HM-CAN with the finite mass but does not occur in the case of massless RS-CAN.
In figs 5.11 and 5.12 we illustrate cases of various strength of the dynamic effects that consist in appearance of low-frequency modulation of the force response and in a general loss of periodicity with $1/f$ period. In fig. 5.11 they are almost negligible while in fig. 5.12 their magnitude is significant.

It is easy to quantify the expected dynamic effect level by calculating of the dimensionless mass $m^*$ appearing in the dimensionless form of Eqs. (5.17), (5.16), (5.18), (5.19) for the HM-CAN :

Here

$$m^* = \frac{a_0 m f^2}{N_0} \qquad (5.23)$$

is a combination of the inverse normal stiffness $a_0/N_0$ and the known parameter $mf^2$. To reduce the dynamic effects and make the HM model more adequate to real damage with no inertia, one has to increase the dimensionless mass $m^*$ by increasing the real mass, exciting the plate with a higher frequency, or having a less stiff contact. In figs 5.11 and 5.12, the parameter $m^*$ equals 414 and 155, respectively.

It is also important to mention that increasing prestress decreases overall non-linearity of the problem. Nevertheless, the secondary waves are generated even for high prestress levels making it possible to detect localized CAN position via methods developed in chapter 2.
<figure><img src="image_79.png" /><figcaption>FIGURE 5.11 – Normal (a) and tangential (b) forces applied to the plate by the HM-CAN and RS-CAN with matched normal responses $N(a)$. Here $m^* = 414$ indicating weak dynamic effects. Frequency $f=200$ kHz, plate thickness $2d=5$ mm, radius of the sphere $R=9$ mm, no precompression, $B_{amp} = 0.1$ nm. (c) Normal displacement produced by the HM-CAN and RS-CAN with matched normal responses $N(a)$.</figcaption></figure>
<figure><img src="image_80.png" /></figure>

FIGURE 5.12 – Normal (a) and tangential (b) forces applied to the plate by the HM-CAN and RS-CAN with matched normal responses $N(a)$. Here $m^* = 155$ that corresponds to moderate dynamic effects. Frequency $f=200$ kHz, plate thickness $2d=3$ mm, radius of the sphere $R=5$ mm, no precompression, $B_{amp} = 0.15$ nm. This HM-CAN mimics real damage only qualitatively. (c) Normal displacement produced by the HM-CAN and RS-CAN with matched normal responses $N(a)$.
## 5.3 Numerical modeling for elastic waves interacting with the CAN in a thin plate

### 5.3.1 Modeling for Lamb waves using finite difference and finite element methods

This section is concerned with modeling for wave propagation in a thin plate in the purpose of calculating the normal and tangential displacements activating the CAN. In the experiments, the first flexural mode A0 is primarily excited whose analytical form is known in the fully harmonics case where $t \in (-\infty, +\infty)$. In practice, some non-stationary transient wave processes are always present at least due to a finite observation time. In addition, the numerical tool to be developed has to be flexible enough in terms of the choice of excitation. These features motivate us to build up a numerical description for the propagating waves, despite the existence of the analytical solution.

In many cases, nowadays numerical modeling for waves in solid structures is based on the finite element method. However, for learning purposes it is interesting to implement the classical finite difference method as well and then compare their efficiencies. Thus, in this section, we set the problem geometry for the both methods and implement the finite element and finite difference solutions in the case corresponding to a pure Lamb mode where the exact solution is known. Then the tested algorithms and codes can be used for modeling any transient wave process when the analytical solution is cumbersome.

Unfortunately, the performance of available computers is not sufficient for an adequate description of real experimental geometry (plate of $L \times w \times 2d$ dimensions suspended on elastic strings). Due to performance limitations we consider a $L \times 2d$ fragment in the 2D case in which geometric parameters $L, d$, and $\lambda_{Lamb}$ (Lamb wavelength of A0 mode) are selected in accordance to the following requirements.

First of all, the representative fragment should be long enough to mimic the plate as an object with high geometric aspect ratio, i.e. $L \gg d$, $L \gg \lambda_{Lamb}$. Secondly, the experimental condition of preferential A0 mode excitation engenders a requirement in terms of the $\lambda_{Lamb}$ and the plate half-thickness $d$. Indeed, the theoretical number of the antisymmetric Lamb modes that can travel in a plate of a certain thickness is given by Eq. (1.1) [19], therefore, if $[\frac{2d}{\lambda_{Lamb}}] = 0$, then $N_a = 1$, namely, only A0 mode is present.
## 5.3. NUMERICAL MODELING FOR ELASTIC WAVES INTERACTING WITH THE CAN IN A THIN PLATE

These requirements are satisfied by selection of the following excitation and geometric parameters: $d = 2.5$ mm, $f = 200$ kHz, $L = 5\lambda_{Lamb} \approx 5$ cm. The $\lambda_{Lamb} = \frac{2\pi}{k_{A0}}$, where wavenumber $k_{A0}$ is to be determined in an implicit way as a root of an algebraic equation (1.2) or equivalently:

$$f(k) = (2k^2 - k_t^2)^2 \sinh \sqrt{(k^2 - k_l^2)} d \cosh \sqrt{k^2 - k_t^2} d - 4k^2 \sqrt{(k^2 - k_l^2)} \sqrt{(k^2 - k_t^2)} \cosh \sqrt{(k^2 - k_l^2)} d \sinh \sqrt{(k^2 - k_t^2)} d = 0 \quad (5.24)$$

that is convenient to solve by a bisection method i.e. by repeatedly dividing the interval containing the root. Indeed, taking an initial interval $k_{A0} \in [1, 50000] m^{-1}$ ($f(1)$ is positive while $f(50000)$ is negative), after 100 bisections only we localize the root $k_{A0} = 540.1073572879936 m^{-1}$ with the precision of $10^{-13} m^{-1}$.

After locating the root corresponding to the Lamb wavelength, we proceed to numerical solving of the following wave propagation problem:

$$\left\{
\begin{aligned}
\frac{\partial v_x}{\partial t} &= \frac{1}{\rho} \left( \frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{xy}}{\partial y} \right) \\
\frac{\partial v_y}{\partial t} &= \frac{1}{\rho} \left( \frac{\partial \sigma_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} \right) \\
\frac{\partial \sigma_{xx}}{\partial t} &= (\lambda + 2\mu) \frac{\partial v_x}{\partial x} + \lambda \frac{\partial v_y}{\partial y} \\
\frac{\partial \sigma_{yy}}{\partial t} &= (\lambda + 2\mu) \frac{\partial v_y}{\partial y} + \lambda \frac{\partial v_x}{\partial x} \\
\frac{\partial \sigma_{xy}}{\partial t} &= \mu \left( \frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x} \right).
\end{aligned}
\right.
\quad (5.25)$$

In this equations ($v_x, v_y$) is the 2D-velocity vector, ($\sigma_{xx}, \sigma_{yy}, \sigma_{xy}$) is the stress tensor, $\lambda$ and $\mu$ are Lamé coefficients. $L$ is the plate length, $d$ is a half of the plate's thickness.

Equation (5.25) describes the A0 Lamb wave if supplemented by the initial conditions in the form:

$$\left\{
\begin{aligned}
v_x(t=0) &= v_x^{\text{Lamb}}(t=0) \\
v_y(t=0) &= v_y^{\text{Lamb}}(t=0) \\
\sigma_{xx}(t=0) &= \sigma_{xx}^{\text{Lamb}}(t=0) \\
\sigma_{yy}(t=0) &= \sigma_{yy}^{\text{Lamb}}(t=0) \\
\sigma_{xy}(t=0) &= \sigma_{xy}^{\text{Lamb}}(t=0)
\end{aligned}
\right.
\quad (5.26)$$

and the boundary conditions:

$$\left\{
\begin{aligned}
\sigma_{yy}(x=0) &= \sigma_{yy}^{\text{Lamb}}(x=0) \\
v_x(x=0) &= v_x^{\text{Lamb}}(x=0)
\end{aligned}
\right.
\quad (5.27)$$
and

$$ \left\{ \begin{array}{l} \sigma_{yy}(x = L) = \sigma_{yy}^{\text{Lamb}}(x = L) \\ v_x(x = L) = v_x^{\text{Lamb}}(x = L) \end{array} \right. \qquad (5.28) $$

at the left and right edges of the plate. The boundary conditions at the top and bottom faces are stress-free and are given by

$$ \left\{ \begin{array}{l} \sigma_{xy}(y = \pm d) = 0 \\ \sigma_{yy}(y = \pm d) = 0. \end{array} \right. \qquad (5.29) $$

Here variables with the superscript « Lamb » defined in Eqs. (1.3), (1.4), (1.5) correspond to the analytical solution for the A0 Lamb wave. It is straightforward to check that the solution in Eqs. (1.3), (1.4), (1.5) exactly satisfies the above equation Eq. (5.25). The geometry together with boundary conditions are illustrated in fig. 5.13.

<figure><img src="image_81.png" /><figcaption>FIGURE 5.13 – Geometry and boundary conditions corresponding to the A0 Lamb mode in a thin plate.</figcaption></figure>

### a) Finite difference formulation

In the finite difference method, the derivatives in Eq. (5.25) are to be substituted with finite differences. They can be one-sided or centered which corresponds to the first or second order of approximation, respectively. The choice of the latter case results in a staggered grid formulation [130] in which some variables are defined at odd grid nodes while the others are defined at the even ones. This scheme is illustrated in fig. 5.14.

In particular, $v_x$ is defined only on even y-layers and odd x-layers, $v_y$ - on odd y-layers and even x-layers, $\sigma_{xx}$ and $\sigma_{yy}$ - on even y-layers and even x-layers, and $\sigma_{xy}$ - on odd y-layers and odd x-layers. In fig. 5.14 time stepping is shown for $v_x$ and
<figure><img src="image_82.png" /><figcaption>FIGURE 5.14 – Staggered grid formulation of the finite difference method for Eq. (5.25). (a) Each spacial layer contains nodes with odd and even numbers at which different variables are defined. (b) Examples of time stepping to the new time layer (variables marked with « + ») from the current one (unmarked variables) and the previous one (marked with « - »).</figcaption></figure>
$\sigma_{xy}$. For other variables the procedure is analogous. Here label « + » stands for new time layer, label « - » is for the previous one.

Introducing the grids $x_i, y_j, t_l$ in space and in time via expressions :

$$ egin{cases} x_i = (i - 1)\Delta x \\ y_j = -d + (j - 1)\Delta y \\ t_l = (l - 1)\Delta t \end{cases} $$

with $i = 1, .., N_x, j = 1, .., N_y, l = 1, .., N_t$ and steps :

$$ egin{cases} \Delta x = \frac{L}{N_x - 1} \\ \Delta y = \frac{2d}{N_y - 1} \\ \Delta t = \frac{T_{comp}}{N_t - 1}, \end{cases} $$

we denote variables on this grid as $v_x(i, j)$, $v_y(i, j)$ for x- and y- velocity components respectively, $\sigma_{xx}(i, j)$, $\sigma_{xy}(i, j)$, $\sigma_{yy}(i, j)$ for stress tensor components. $T_{comp}$ here stands for computation time.

Using the above scheme of time stepping, it is easy to write the corresponding discretized equations which take the form :

$$ egin{cases} v_x(i, j)^+ = v_x(i, j)^- + \frac{\Delta t}{\rho \Delta x} (\sigma_{xx}(i+1, j) - \sigma_{xx}(i-1, j)) + \frac{\Delta t}{\rho \Delta y} (\sigma_{xy}(i, j+1) - \sigma_{xy}(i, j-1)) \\ v_y(i, j)^+ = v_y(i, j)^- + \frac{\Delta t}{\rho \Delta x} (\sigma_{xy}(i+1, j) - \sigma_{xy}(i-1, j)) + \frac{\Delta t}{\rho \Delta y} (\sigma_{yy}(i, j+1) - \sigma_{yy}(i, j-1)) \\ \sigma_{xx}(i, j)^+ = \sigma_{xx}(i, j)^- + (\lambda + 2\mu) \frac{\Delta t}{\Delta x} (v_x(i+1, j) - v_x(i-1, j)) + \lambda \frac{\Delta t}{\Delta y} (v_y(i, j+1) - v_y(i, j-1)) \\ \sigma_{yy}(i, j)^+ = \sigma_{yy}(i, j)^- + (\lambda + 2\mu) \frac{\Delta t}{\Delta y} (v_y(i, j+1) - v_y(i, j-1)) + \lambda \frac{\Delta t}{\Delta x} (v_x(i+1, j) - v_x(i-1, j)) \\ \sigma_{xy}(i, j)^+ = \sigma_{xy}(i, j)^- + \mu \frac{\Delta t}{\Delta y} (v_x(i, j+1) - v_x(i, j-1)) + \mu \frac{\Delta t}{\Delta x} (v_y(i, j+1) - v_y(i, j-1)). \end{cases} $$

To represent the boundary conditions, it is convenient to take odd total numbers of points $N_x$ and $N_y$. This selection fully corresponds to the boundary conditions (5.27), (5.28) at the plate edges written here in the form :

$$ egin{cases} \sigma_{xy}(1, j) = \sigma_{xy}^{\text{Lamb}}(x_1, y_j, t_l) \\ v_x(1, j) = v_x^{\text{Lamb}}(x_1, y_j, t_l) \\ \sigma_{xy}(N_x, j) = \sigma_{xy}^{\text{Lamb}}(x_{N_x}, y_j, t_l) \\ v_x(N_x, j) = v_x^{\text{Lamb}}(x_{N_x}, y_j, t_l). \end{cases} \quad (5.30) $$

The stress-free boundary conditions (5.29) at the plate top and bottom surfaces should be replaced by the ones conforming to the staggered grid formulation :
$$
\left\{
\begin{aligned}
v_y(i, 1)^+ &= v_y(i, 1)^- + \frac{2\Delta t}{\rho\Delta y}\sigma_y y(i, 2) \\
\sigma_x y(i, 1) &= 0 \\
v_y(i, N_y)^+ &= v_y(i, N_y)^- - \frac{2\Delta t}{\rho\Delta y}\sigma_{yy}(i, N_y - 1) \\
\sigma_{xy}(i, N_y) &= 0.
\end{aligned}
\right.
\qquad (5.31)
$$

The initial conditions on the staggered grid read :

At $t=0$ :

$$
\left\{
\begin{aligned}
v_x(i, j)^- &= v_x^{\text{Lamb}}(x_i, y_j, 0) \\
v_y(i, j)^- &= v_y^{\text{Lamb}}(x_i, y_j, 0) \\
\sigma_{xx}(i, j)^- &= \sigma_{xx}^{\text{Lamb}}(x_i, y_j, 0) \\
\sigma_{xy}(i, j)^- &= \sigma_{xy}^{\text{Lamb}}(x_i, y_j, 0) \\
\sigma_{yy}(i, j)^- &= \sigma_{yy}^{\text{Lamb}}(x_i, y_j, 0)
\end{aligned}
\right.
\qquad (5.32)
$$

At $t = \Delta t$ :

$$
\left\{
\begin{aligned}
v_x(i, j) &= v_x^{\text{Lamb}}(x_i, y_j, \Delta t) \\
v_y(i, j) &= v_y^{\text{Lamb}}(x_i, y_j, \Delta t) \\
\sigma_{xx}(i, j) &= \sigma_{xx}^{\text{Lamb}}(x_i, y_j, \Delta t) \\
\sigma_{xy}(i, j) &= \sigma_{xy}^{\text{Lamb}}(x_i, y_j, \Delta t) \\
\sigma_{yy}(i, j) &= \sigma_{yy}^{\text{Lamb}}(x_i, y_j, \Delta t).
\end{aligned}
\right.
\qquad (5.33)
$$

The considered time stepping is explicit i.e. each value at the text time step is obtained independently, in contract to implicit schemes in which they are coupled so that an additional equation should be solved. For explicit schemes, the Courant-Friedrichs-Lewy [131] conditions should be required :

$$
c_{\text{Lamb}} \Delta t < \sqrt{(\Delta x^2 + \Delta y^2)}, \qquad (5.34)
$$

where $c_{Lamb} = \frac{\omega}{k_{A0}}$.

The comparison of the numerical solution via finite differences with the exact one will be done later. Now we turn to modeling for Lamb wave propagation using the finite element method.

b) Finite element solution

The finite element solution is performed with the help of a standard commer-
cial package COMSOL Multiphysics that uses two second-order equations for the
displacement components instead of Eq. (5.25) :

$$
\left\{
\begin{aligned}
\rho \frac{\partial^2 u_x}{\partial t^2} &= (\lambda + \mu) \frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial x \partial y} + \mu \frac{\partial^2 u_x}{\partial y^2} \\
\rho \frac{\partial^2 u_y}{\partial t^2} &= (\lambda + \mu) \frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial x \partial y} + \mu \frac{\partial^2 u_y}{\partial y^2}.
\end{aligned}
\right.
\qquad (5.35)
$$
The package accepts boundary conditions formulated in terms of displacements such as

$$
\left\{
\begin{aligned}
u_x(x = 0) &= u_x^{\text{Lamb}}(x = 0, y, t) \\
u_y(x = 0) &= u_y^{\text{Lamb}}(x = 0, y, t) \\
u_x(x = L) &= u_x^{\text{Lamb}}(x = L, y, t = 0) \\
u_y(x = L) &= u_y^{\text{Lamb}}(x = L, y, t = 0),
\end{aligned}
\right.
\qquad (5.36)
$$

as well as stress-free boundary conditions as in Eq. (5.29).

The initial conditions are specified at $t = 0$ in the usual form for displacements and velocities :

$$
\left\{
\begin{aligned}
u_x(t = 0) &= u_x^{\text{Lamb}}(x, y, t = 0) \\
u_y(t = 0) &= u_y^{\text{Lamb}}(x, y, t = 0) \\
\frac{\partial u_x}{\partial t} &= v_x^{\text{Lamb}}(x, y, t = 0) \\
\frac{\partial u_y}{\partial t} &= v_y^{\text{Lamb}}(x, y, t = 0).
\end{aligned}
\right.
\qquad (5.37)
$$

The code automatically meshes the calculation domain using triangular elements (in our case) and uses a built-in solver to get all acoustic fields in the domain at any moment of time. The details are given in the following paragraph in which the comparison with the finite difference solution is discussed.

c) Stationary Lamb wave simulated via the finite difference and finite element methods

In this section, we make a comparison between the finite difference and finite element methods in terms of performance. This comparison is possible due to the existence of the exact solution Eq. (1.3) for the A0 Lamb mode. The deviation between the exact solution and the numerical ones expressed as a normalized root mean square (rms) error is an obvious performance measure for the both methods. To be more specific, the performance is a definite error magnitude achieved during certain time of calculation on the same computer.

There is, however, a technical difficulty associated with the fact that the both methods include discretization in space and in time, and the most efficient ratio between space and time discretization steps is a priori unknown. Indeed, finer space mesh and smaller time step usually both have positive effect on the calculation error but are related to higher calculation times. This aspect deserves an additional optimization.
To do so, the following simple technique can be used. We use different but fixed spatial meshes for the both methods and perform a series of computations for different time steps keeping other parameters unchanged. First we start with a very coarse time stepping and gradually increase the time discretization quality. In finite element computing, the calculation error immediately drops as long as the time step decreases, then the gain becomes smaller and finally almost disappears for very fine time grids. In this final situation the quality of numerical solution is mostly affected by the quality of space meshing. The optimal performance is obtained when the gain from refining the time grid starts saturating. A better estimation can be obtained by varying both space and time discretizations.

However, the opposite effect can occur when explicit finite difference calculations are used. A key feature in this case is the presence of Courant-Friedrichs-Lewy condition [131]

$$CFL \equiv \frac{|c|\Delta t}{\Delta x} < 1 \qquad (5.38)$$

that determines the stability of calculations. Here c is the velocity, Δx and Δt are spatial and time steps, respectively. In other words, setting too large time step en-genders error accumulation during subsequent time steps. A simple Von Neumann stability analysis of wave equation [132] shows that calculations are most precise when the condition Eq. (5.38) is just fulfilled; the precision worsens when CFL is considerably less than 1.

It is instructive to cite these arguments here for completeness. For the simplest 1D wave equation

$$\frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} \qquad (5.39)$$

the corresponding wave dispersion relationship is obtained by substituting $u = u_0e^{i(\omega t - kx)}$ into Eq. (5.39)

$$-\frac{\omega^2}{c^2} u_0 e^{i(\omega t - kx)} = -k^2 u_0 e^{i(\omega t - kx)}$$

with the obvious result $k = \pm \frac{\omega}{c}$. The same harmonic solution substituted to the discretized equation

$$\frac{1}{c^2} \frac{u_j^+ - 2u_j + u_j^-}{\Delta t^2} = \frac{u_{j+1} - 2u_j + u_{j-1}}{\Delta x^2} \qquad (5.40)$$
or substituting $u_j = u_0e^{i(\omega t - kj\Delta x)}$ into Eq. (5.40) leads to another result :

$$ \frac{\cos(\omega\Delta t) - 1}{c^2\Delta t^2} = \frac{\cos(k\Delta x) - 1}{\Delta x^2}. \qquad (5.41) $$

### (a) Finite difference method

<table><tr><td>time step, ns</td><td>3</td><td>2.9</td><td>2.5</td><td>1.7</td><td>1.25</td><td>1</td><td>0.8</td></tr></table>

<figure><img src="image_83.png" /></figure>

### (b) Finite element method

<table><tr><td>time step, ns</td><td>62.5</td><td>50</td><td>41.7</td><td>35.7</td><td>31.25</td><td>27.8</td><td>25</td><td>22.7</td><td>20.8</td><td>19.2</td><td>17.9</td><td>16.7</td><td>15.6</td><td>14.7</td><td>13.9</td><td>13.16</td><td>12.5</td></tr></table>

<figure><img src="image_84.png" /></figure>

FIGURE 5.15 – Rms error of finite difference (a) and finite elements (b) method comparing to analytical solution for different time meshes as a function of computation time.

It is straightforward to check that the dispersion relationship Eq. (5.41) exactly corresponds to $k = \frac{\omega}{c}$ for

$$ \Delta x = c\Delta t \qquad (5.42) $$

and is different otherwise. The condition Eq. (5.42) requires that the parameter *CFL* be close to 1 i.e. the explicit finite difference scheme works best at the stability threshold. In other words, refining time discretization while keeping $\Delta x$ constant is
useless in this case. The calculation error can be decreased by simultaneous reducing $\Delta t$ and $\Delta x$ that obviously improve approximation of the finite difference scheme.

Fig. 5.15 illustrates the predicted effects and gives an idea on the comparative performance of the finite element and finite difference methods. The latter one is implemented in Fortran and Matlab having in this case a relative performance factor of 2.7 (Fortran wins).

Whilst Matlab finite differences realization and COMSOL finite element code show comparable performances, the finite element method is undoubtedly gains in terms of geometric flexibility. This feature underlies our decision to use the finite element method for further calculations.

### 5.3.2 Secondary waves generated by contact acoustic nonlinearity in a thin plate

This section is concerned with the propagation problem for secondary waves excited by the CAN that plays a role of a punctual force source. The geometry of the problem is same as for primary Lamb waves in the section 5.3.1 and is illustrated in fig. 5.16. A substantial difference is related to the boundary conditions. In the pri-

<figure><img src="image_85.png" /><figcaption>FIGURE 5.16 – Geometry of the secondary wave generation problem. The CAN is excited by the displacements obtained in section 5.3.1 and now, in turn, generates secondary waves via application of nonlinear punctual forces $N(t)$ and $T(t)$ calculated as explained in section 5.2.</figcaption></figure>

mary wave propagation problem, the boundary conditions corresponded to the exact Lamb solution. Since they are already satisfied, the conditions at the plate edges
for the secondary wave problem represent zero displacements (fixed boundaries). In our example, this formulation helps avoid activating the movement of the plate as a whole. Such movement would occur in a situation when the plate is excited by stresses or velocities without fixing displacements at least at some point of the boundary. In reality, the plate is suspended with elastic strings which are not considered in the present numerical model.

<figure><img src="image_86.png" /><figcaption>FIGURE 5.17 – Secondary waves emitted by Hertz-Mindlin CAN in a thin plate.</figcaption></figure>

A typical profile of the secondary wave excited by forces depicted in fig. 5.11 is shown in fig. 5.17. To see more features in the secondary wave, we have considered a long sequence of the force excursions comprising 300 periods of the primary Lamb wave (cf. 100 periods in fig. 5.11 for the forces). It can be immediately noticed that a quasi-periodic force protocol excites a long series of reverberations from the edges of the small plate fragment (fig. 5.17). The multiply reflected wave interferes with itself producing a complex non-periodic signal. Besides the fundamental frequency imposed by the primary Lamb mode, it contains higher harmonics as well as low-frequency modulations. The character of the signal is determined by the fact that the secondary wave generation is asymmetric even though the secondary source is located exactly at the middle of the plate.

In practice, the damage localization algorithm deals with first arrivals rather than reverberated waves. Certainly, the complexity of the process shown in fig. 5.17 is due to multiple reverberations in the considered confined geometry that is usually avoided in real measurements (chapter 2). However, physical nature of the underlying
processes remains the same and corresponds to the interaction between acoustic waves and a frictional contact.

<figure><img src="image_87.png" /><figcaption>FIGURE 5.18 – Spatial Fourier analysis of secondary waves : displacement (a) $u_x$ and (b) $u_y$ as functions of wavenumber $k_x$ on the top surface of the plate containing the contact acoustic nonlinearity. The dominant component is A0 Lamb wave ($k_x \approx 540.1 \text{ 1/m}$) that the plate was excited with.</figcaption></figure>
In the considered example, the secondary wave is about $10^{-4}$ times weaker in amplitude than the primary one. It is important to stress that even such a weak magnitude of the secondary signal is sufficient for successful damage locating in experiments.

It would be instructive to see the structure of spatial spectrum of secondary waves given at a certain (last) moment of time. The result of the 1D spatial Fourier transform as a function of spatial frequency $k_x$ corresponding to x-coordinate on the top surface is depicted in fig. 5.18. A0 Lamb wave with $k = k_{A0} \approx 540$ 1/m remains dominant in the secondary wave spectrum. Peak of A0 mode has finite width, at least because the signal is limited in space by the plate boundaries. A0 is the slowest Lamb mode, which implies the largest wavenumber $k_{A0}$, as confirmed in fig. 5.18. There are also other spatial frequencies in the spectrum with $k < k_{A0}$. In contrast to the primary A0 mode selectively excited in our numerical modeling, these modes are generated altogether in a non-selective way. Note that the spatial frequency step $\Delta k$ equals $\pi/L$ and can not be reduced without changing the plate length. This factor does not allow to see the spectral structure of these modes in detail.

### 5.3.3 Prototype of numerical tool accompanying SHM experiments for detecting damage in thin plates

As it was mentioned, there exists a substantial difference in geometric parameters between the simplified case described via the developed modeling tool and real SHM experiments in thin plates. Due to this reason, the developed tool can be seen as a prototype for a future more advanced numerical toolbox capable of imitating realistic secondary waves emitted by damage that contain information for damage locating algorithms. However, the principal stages of the modeling procedure are similar for the prototype and for a prospective working toolbox. The prototype tool comprises the following five steps.

1. Generation of stationary A0 Lamb wave or of a transitional wave process at a certain frequency in a fragment of a thin plate. This algorithm is implemented using COMSOL Multiphysics; data for the normal $u_y(t)$ and tangential $u_x(t)$ displacement wave components are exported at a point corresponding to the defect location.

2. Displacement time data previously obtained on a relatively coarse grid suitable for solving time-consuming wave equation are interpolated on a finer grid for solving ordinary differential equations describing the CAN behavior at the next step.
3. The equation of motion in ordinary derivatives for a HM-CAN excited by wave displacements is numerically solved using the Adams-Bashforth algorithm in the purpose of obtaining the normal $N(t)$ and tangential $T(t)$ contact forces applied to the plate at the defect position. In the case of RS-CAN, these forces are calculated directly from the known contact displacements $a(t)$ and $b(t)$ without solving of equation of motion. In the both situations, the load-displacement relationships for frictional contact are obtained with the use of the Method of Memory Diagrams. The corresponding algorithms are implemented in Fortran.

4. Technical step consisting in preparation of force data for importing to COMSOL at the next stage.

5. Solving the problem for secondary waves emitted by a punctual force with the normal $N(t)$ and tangential $T(t)$ components applied at the defect position (programmed in COMSOL). Same geometry as at step 1 is considered.

Conclusions

The principal result of the modeling effort presented in this chapter is the creation of a modeling tool capable of describing the dynamic CAN response to acoustic excitation and the subsequent generation of the secondary waves to be used in SHM damage locating procedures. The contact response is modeled here with the use of modern semi-analytical approaches based on the Hertz-Mindlin contact theory which takes into account friction and friction-induced hysteresis. Thus, in some sense, the proposed model bridges the gap between acoustics and contact mechanics and provides a description for processes underlying NDT/SHM experiments in acoustically excited structures containing inner frictional contacts.

At the same time, the developed numerical tool should be seen as a prototype of a real working toolbox that has to accompany SHM experiments in the future. The reason for that is in a considerable number of geometric simplifications accepted in the analysis. In particular, instead of realistic damaged plate-like structures we have considered a 2D small fragment of a plate interacting with an artificial CAN (such as small sphere pressed against the plate similar to the one in real experiments in chapter 2). On the one hand, the presence of inertia associated with the sphere make the response dynamic which is not pertinent to crack-like defects having no mass. On the other hand, the plate fragment is assumed to have fixed edges to avoid
the movement of the plate as a whole. These boundary conditions entail multiple
reflections of the secondary wave incompatible with the use of the damage locating
algorithms (chapter 2). However, despite these features following from experimental
or modeling simplifications, physical nature of the considered processes is represented
in an adequate manner.

In particular, we have arrived to the following specific conclusions.

1. There exist physical conditions under which the model CAN (sphere-plate contact or HM-CAN) behaves similarly to crack-like damage (RS-CAN) in terms of mechanical response to acoustic excitation. Namely, the dimensionless mass introduced in Eq. (5.23) must be much higher than 1. Higher dimensionless mass provides the similarity between the both responses at a longer time interval.

2. The developed numerical tool adequately describes physical nature of the CAN-wave interaction, but the considered test geometry significantly differs from the experimental one.

3. Modern FEM-based software provide generally higher but comparable efficiency with respect to the classical finite difference implementations.

This study can be substantially improved in the future; the related perspectives
include:

1. Account for geometries closer to real SHM experiments (higher geometric aspect ratios of the computation domain, realistic suspensions/fixing of the tested structure, excitation signals similar to the applied pump-probe scheme, etc). These improvements would require significantly higher computation resources.

2. Parameters of the used CAN models can be matched to behaviors of real defects.

3. Mutual compromises from both modeling and experimental sides can help reach a configuration which, on the one hand, assumes the application of the damage locating algorithms and visualizing damage, and, on the other hand, adequate modeling is possible. This effort would result in establishing a detailed theory and experiment agreement in NDT/SHM problems.

4. 3D-modeling can bring the model closer to the current experimental configuration and open the opportunity to numerically estimate the damage scattering cross-section, as well as to establish its dependence on CAN mechanical properties. Indeed, in 3D it is possible to do by analyzing secondary waves generated by a CAN. However, this change would require higher computer performance.
General conclusions and perspectives

As mentioned in introduction, the main goal of this thesis was to develop ambient noise based reference-free acoustic damage localization technique for structural health monitoring in thin plates capable of identifying contact acoustic nonlinera- rities modulated by low frequency vibrations. In this work, we particularly focused on the experimental tests with an artificial CAN in pump-probe configuration. We studied scattering characteristics of this artificial CAN and performed modeling for its nonlinear interaction with acoustic waves in a thin aluminum plate. Preliminary localization results have been obtained in experiments with white noise of kHz range along with low frequency harmonic vibrations. Image contrast increases with the growth of the defect scattering cross-section. Strain induced by pump wave is pro- portional to vertical displacement, i.e. it is mechanically linear regime. Strain caused by the probe wave appeared to be three orders of magnitude smaller than strain caused by pump wave.

In chapter 2, an active reference-free damage localization algorithm in thin plates based on a combination of repetitive probing with back-propagation of signals is described. This algorithm appeared to be capable of detecting an artificial CAN mimicked with a steel sphere pressed against an aluminum plate in pump-probe experiments. The algorithm’s working range is at few Hertz for pump wave and at tens kHz for probe wave. Two variants of experiment have been done : when a pump wave is synchronized with a probe wave (synchronized mode) and when it is not (non-synchronized mode). For non-synchronized experiments, random sign compensation technique was proposed to improve contrast of localization images. Contrast was improved by up to 15 times comparing to incoherent imaging of the initial algorithm. Derivation of formula for a contrast established relationship with a CAN scattering cross-section. For synchronized experiments, coherent imaging based
on classical synchronous detection was introduced. Contrast was improved by 3 to 5 times comparing to incoherent imaging of the initial algorithm. The proposed algorithm is robust for various emitter positions. The closer the emitter to the defect and to the center of the plate, the better localization image is obtained. The algorithm functions for different defect locations including situations when it is close to plate's boundaries. The parameters (probe and pump frequencies, emitter position) for high contrast images have been empirically optimized. Strain caused by probe wave ~$10^{-8}$, which is three orders lower than for pump wave. Strain induced by pump wave is proportional to displacement, i.e. our experiment remains in linear regime. Nonlinearity of experiment is not in stress-strain dependency but essentially in pump-probe interaction with a CAN.

Chapter 3 provides quantitative study of an artificial CAN scattering characteristics and image contrast in the experiment described in chapter 2. It is shown that the contrast depends monotonically on a defect scattering cross-section, and is directly proportional to it at small scattering cross-sections. This statement has been confirmed by experimental measurements, numerical simulation of scattering in infinite plate and theoretical derivation for active static experiments. Scattering cross-section of an artificial CAN is experimentally estimated under infinite plate approximation and isotropic scatterer assumption for the central excitation frequency. CAN with scattering cross-section from the order of hundredths of a millimeter to several millimeters can be resolved by the proposed localization algorithm. In pump-probe experiment, image contrast can be improved by increasing pump amplitude or changing the defect strength, i.e. angle between ruler and the plate. Number of receivers and sources also affect image contrast according to the obtained theoretical formula.

In the fourth chapter preliminary passive experiments are discussed which constitutes a proof of concept of passive SHM in environment subjected to low frequency vibrations. Algorithm developed in chapter 2 is applied for pump-probe experiment with white noise from 1 to 30 kHz as a probe wave. Relative noise level is estimated for the preliminary passive experiment with unsuccessful detection, the conclusion from that was to increase number of noise sources, which is done in the next chapter about passive dynamic experiments. Differential signals are replaced with differential correlations of short windows and are subjected to back-propagation procedure and coherent summation. Dependencies of image contrast on the window duration, number of receivers and noise sources, sources distribution are qualitatively studied.
Preliminary localization images are obtained. Interaction between an artificial CAN with white noise is weak compared to residual noise, therefore, the algorithm is sensitive to many parameters and in the current setup configuration is hard to be detected. Further developments are suggested to decrease relative noise level by optimizing noise sources and receivers distribution, modifying technique of summation over loading states, conducting tests on larger plate.

The principal result of chapter 5 is the development of a modeling tool capable of describing the dynamic CAN response to acoustic excitation and the subsequent generation of the secondary waves to be used in SHM damage locating procedures. Hertz-Mindlin contact response to acoustic wave is compared to a crack response. First, semi-analytical algorithm accounting for friction was used to obtain force-displacement relationship at the contact for both sphere pressed against the plate (Hertz-Mindlin contact) and crack (planar contact model) in a thin plate excited by Lamb A0 mode. Then these forces were applied as a point load on the plate top surface and resulting acoustic field was measured by solving wave equation with a finite element method. An artificial model CAN can be approximated by a crack under particular physical conditions, namely, prestress, Lamb waves amplitude and frequency, plate's thickness, and mass of the sphere. The developed numerical tool adequately describes physical nature of the CAN-wave interaction, but the considered test geometry significantly differs from the experimental one.

In perspective, geometries closer to real SHM experiments can be considered (higher geometric aspect ratios of the computation domain, realistic suspensions/fixing of the tested structure, excitation signals similar to the applied pump-probe scheme, etc). These improvements would require significantly higher computation resources. Parameters of the used CAN models can be matched to behaviors of real defects. Mutual compromises from both modeling and experimental sides can help reach a configuration which, on the one hand, assumes the application of the damage locating algorithms and visualizing damage, and, on the other hand, adequate modeling is possible. 3D-modeling can bring the model closer to the current experimental configuration and open the opportunity to numerically estimate the damage scattering cross-section, as well as to establish its dependence on CAN mechanical properties. However, this change would require higher computer performance. These efforts would result in establishing a detailed theory and experiment agreement in NDT/SHM problems.

We envisage to extend this method to a more realistic situation with using the
secondary noise sources (see fig. 5.19) developed by G. Lacerra, F. Massi, E. Chatelet (our project partners in LaMCoS, Lyon) [133] to convert a part of low-frequency plate modal vibrations into high-frequency noise and then apply the proposed localization algorithm to this converted noise. Similar experiments for passive static experiments were already done by L. Chehami et al. [89]. Applying such conversion devices is of interest, because this is the way to improve spatial stationarity of the noise sources (here probe signal), and also because the noise common in nature, vehicles and structures in service mostly exists in low-frequency range.

<figure><img src="image_88.png" /><figcaption>FIGURE 5.19 – Artificial secondary source : mass-spring resonator with a frictional beam.</figcaption></figure>

Another perspective of this work is to modify the proposed localization algorithm
for passive pump, i.e. instead of low frequency continuous harmonic modulation,
expose the plate to the low frequency noise. In this thesis, we studied cases of active
pump/active probe and active pump/passive probe. Using passive pump would be
a real challenge, and constitutes another step towards completely passive structural
health monitoring.

In this work, passive baseline-free localization algorithm is applied for model artificial contact acoustic nonlinearity. The next step is to try it on more realistic contact defects such as cracks, delaminations, notches, etc. Also it would be possible to quantify and optimize experimental parameters more, if the parameters of experiment were similar to parameters in our wave-CAN interaction modeling tool, since we would be able to predict the interaction regime.

The ultimate long term goal of this study is to apply the developed passive
localization in the field tests to find locations of real flaws in plate-like structures in
service such as a wing of an aircraft.
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Abstract

In structural health monitoring, acoustic methods without use of a reference intact sample are
nowadays of great interest due to variability of environmental and operational conditions. This thesis
presents an acoustic reference-free method for detecting and localizing defects in thin plates. The
proposed approach is based on pump-probe measurements and a differential signal processing
algorithm. The pump-probe scheme here means the simultaneous excitation of low-frequency
vibrations and high-frequency acoustic pulses generated by a piezoelectric transducer and recorded by
an array of transducers of the same kind. The role of pumping vibrations is in imposing changes to
loading conditions for a defect (here a model Hertzian contact i.e. a small ball pressed against the
tested plate) while the ultrasonic Lamb wave pulses probe the defect in various loading states. Then a
differential signal processing algorithm based on backpropagation of A0 Lamb mode is used to detect
and locate the defect. Two detection modes have been studied: the mode requiring the synchronization
between the pump and probe signals, and the non-synchronous mode. In the latter case, a correction
with the use of ad hoc weight coefficients improves the detection. Further, a parametric study was
performed by computing the image contrast as a function of the defect scattering cross-section that
allowed us to determine a minimum detection threshold. The application of the method to passive
imaging (i.e. using white noise instead of probe pulses) produced encouraging results for the price of
increasing the number of receivers. Finally, numerical modeling for the interaction of an ultrasonic
wave with a Hertz-Mindlin frictional contact was performed in 2D to compare the model defect (ball
pressed against the plate) with a more realistic one (crack).

**Key words:** structural health monitoring, Lamb waves, Green's function, noise correlation, passive imaging, baseline-free localization, beamforming, pump-probe experiment, Hertz-Mindlin contact, method of memory diagrams.

Résumé

Le contrôle-qualité de matériaux par méthodes acoustiques sans référence suscite aujourd'hui un grand intérêt en contrôle santé intégré (CSI) au vu de la variabilité des conditions environnementales et opérationnelles. Dans ce manuscrit, nous proposons une méthode de détection synchrone/non synchrone de défaut soumis à des sollicitations basses fréquences, permettant une localisation sans état hypothétique "sans défaut". Une série de tests "pompe-sonde" a été menée sur une plaque d'aluminium. Un pot vibrant est fixé sur cette dernière afin de la mettre en vibration à une fréquence de quelques Hz. Une bille solidaire d'une lame en acier est pressée contre la plaque (contact Hertzien) et joue le rôle d'un défaut (reproduisant ainsi un contact tel qu'on peut le rencontrer par exemple entre deux fronts d'une fissure). Un réseau de transducteurs piézoélectriques est utilisé pour faire l'imagerie ultrasonore. Le principe de la détection repose sur la modulation du contact entre la plaque et la bille par la vibration pompe basse fréquence. Des mesures ultrasonores répétées (onde sonde) permettent ainsi d'interroger le défaut dans plusieurs états de sollicitation. Un algorithme d'imagerie différentielle permet alors de mettre en évidence et localiser le défaut. Deux modes de détection ont été testés : 1) mode synchrone entre la pompe et les signaux ultrasonores et 2) mode non synchrone. Dans ce dernier cas, une correction par des coefficients de pondération ad-hoc permet d'améliorer la détection. Une étude paramétrique a été également réalisée en reliant le contraste des images en fonction de la section de diffusion de défaut. Ceci a permis de définir un seuil minimal de détection d'un défaut de type contact variable. L'application de cette méthode à l'imagerie passive par corrélation de bruit a donné des résultats encourageants, à condition d'augmenter la densité du réseau de capteurs ultrasonores. Enfin, des simulations numériques en 2D, modélisant l'interaction d'une onde ultrasonore avec un contact de frottement type "Hertz-Mindlin" ont été réalisées afin de comparer le défaut actuel (bille pressée sur la plaque) avec un défaut réaliste (une fissure).

**Mots clés** : contrôle santé intégré, ondes de Lamb, fonction de Green, corrélation du bruit, imagerie passive, localisation sans référence, beamforming, expérience pompe-sonde, contact de Hertz-Mindlin, méthode des diagrammes de mémoire.