**3.2 Study of a low-frequency ultrasonic device**

216 Acoustic Waves – From Microdevices to Helioseismology

type of process. Many models have been proposed to explain the phenomenon of aggregation. The most important ones are those of Flory (Flory, 1953), Stockmayer (Stockmayer, 1943), Case (Case, 1960), Gupta (Gupta et al., 1979), Eichinger (Eichinger, 1981), Allsopp (Allsopp, 1981) and San Biagio (San Biagio et al., 1990). In most cases the phenomenon is described by the classical theory as a "particular case of percolation" and the two-dimensional growth of the network according to Caylay's tree. Other studies including those of De Gennes (De Gennes, 1989) and Stauffer (Stauffer, 1981, 1985) describe the phenomenon of random aggregation and the problems of percolation and gelation. However, the different characteristics of the macromolecular chain-making system can be evaluated according to Clerc (Clerc et al., 1983), using for example a Monte-Carlo simulation, predicting the influence of the characteristics of the starting solution and the

In fact, the gelation process is a transition from an entirely soluble system to a heterogeneous two-phase system: composed of an insoluble entity (infinite-size macromolecule) and a soluble phase. This transition is accompanied by radical changes in some physical properties of the medium. Below the gelation "point", the viscosity of the

To study this phenomenon, several physical measurement techniques exist i.e. optical, thermal, rheological and acoustic (Nassar, 1997). However, sampling and sensitivity to a limited range of physical properties are often drawbacks. Consequently, different techniques are required to explore an entire process with the difficulty of bringing together the heterogeneous data provided by these techniques. This is, for example, the case of optical methods which are penalized by the opacity of the substances analyzed as well as the size of the molecules formed in relation to the wavelength. Thermal methods are insensitive to the mechanical characteristics of the medium. The fragility of some gels (milk gel) limits rheological techniques. In many cases, several analytical techniques exist, but

To develop further instrumentation in order to understand and to quantify the modification process of media in real conditions, a low-frequency ultrasonic technique using sensors with

gelation conditions on the structure and the arrangement of the masses.

they are only used in the laboratory.

Fig. 1. Basic principle

medium increases and the medium ceases to flow by developing an elasticity.

The aim of the study was to define and develop optimal ultrasonic instrumentation to understand the phenomenon and quantify the viscoelastic properties of changing media.

The usual ultrasonic characterization techniques are generally based on the use of a resonant piezoelectric transducer in thickness mode. As the resonant frequency of a transducer is inversely proportional to its size, it is greater for low frequencies around 100 KHz. Some researchers like Degertekin (Degertekin & Khury-Yakub, 1996a, 1996b, 1996c), Shuyu (Shuyu, 1996, 1997) and Nikolovski (Nikolovski & Royer, 1997) used this physical principle, but associated a tapered volume with the ceramic components to concentrate the mechanical energy.

The aim of this part of the work was to obtain a low frequency acoustic point source to generate a spherical wave in the medium. To do this a different procedure from that traditionally used in classic sensor design was implemented. A new technique was used which consisted in setting in resonance the entire mechanical structure of a reduced-size unit through the contact of an extremely pointed end with the material to be analyzed. In order to behave like an acoustic point source, the size of the point was smaller than the wavelength in the medium.

The first part presents a theoretical analysis of low-frequency ultrasonic resonators, beaming a spherical wave in the medium. The choice of a triangular shaped resonator and its mechanical behavior will be assessed and the study completed by a numerical approach based on the application of the finite elements method to characterize all the resonator vibration modes and visualize the corresponding distortions when the structure is excited. As the analytical results were in good agreement with the numerical results, they were applied to the whole triangular-shaped sensor to validate the findings experimentally. The resonance mode frequencies determined by the numerical calculation were then correlated with the electrical impedance measurements.

### **3.3 Study and design**

### **3.3.1 Analytical approach**

For a possible analytical analysis, the structure of a standard ultrasonic sensor is based on a simple triangle shape (Figure 2).

The propagation of longitudinal waves in the triangular part of the sensor was studied to determine the resonance frequency of the elongation mode and the velocity amplification

Low Frequency Acoustic Devices for Viscoelastic Complex Media Characterization 219

While an analytical study can only take into consideration one particular mode of vibration of the triangular part of the sensor, a numerical study based on the finite elements method can determine all the vibrating modes of these parts as well as those of the realised sensor. For a real structure; whole sensor included a binding rod and a triangular truncated part (Figure 4), the displacement differential equations were solved with a continuous regime, taking into account the boundary conditions at the surfaces. The materials were defined by Young's modulus E, Poisson's coefficient ν and density ρ. The results presented below were applied without loss and they were compared to the characteristics of the longitudinal mode determined by the analytical calculations. This comparison was also made for the triangular

For our study, the ANSYS analysis software was used. The sensors used were made essentially of piezoelectric material. A source of excitation was engraved in the general structure of the vibrating element (triangle part), providing mechanical continuity without any break (Figure 4a). This type of engraving was considered as it has been demonstrated (Nassar, 1997) that for the same longitudinal mode, the amplification ratio at the ends is **71** times bigger when there is one engraved source providing mechanical continuity with the

vibrating element so that one source can impact the structure by gluing (Figure 3b).

Fig. 4. From left to right: (a) engraved source; (b) embedded source; (c) elongation mode

Longitudinal resonance

Electrical impedance measurements 59 kHz

Table 1. Comparative study. Analytical and numerical approach

Table 2. Impact of the nature and the location of the source

Analytical calculation 60132 2.16 Numerical calculation **57611** 2.32

Frequency (Hz) 57729 **57611**  Standardised velocity ratio 1.4 10 -2 1.00

Table 1 show the resonance frequencies and the vibration velocity transformation ratio (|v(0) /v(L)|) and table 2 present the difference rate of this ratio according to the position of

frequency Hz

sensors Embedded source Engraved source

Velocity ratio v(0)/v(L) at the extremities

**3.3.2 Numerical approach** 

sensor which was studied as a whole.

the excitation source

ratio between the ends. The analysis is based on an extension of Ensminger's (Ensminger, 1960) theory.

According to figure 2, the x section is written:

$$\mathbf{S} = \mathbf{e} \cdot \ell(\mathbf{x}) \text{ Then } \mathbf{S} = \mathbf{e} \cdot \ell\_1(\mathbf{x}\_1 + \mathbf{x}) / \mathbf{x}\_1 \tag{1}$$

Fig. 2. Basic analytical shape

Ensminger studied the propagation of a wave in extensional mode in a cone with no loss of which the lateral dimensions were short in comparison with the length. In the case of a triangular shape, this equation takes the following form:

$$
\frac{
\partial^2 v
}{
\partial \mathbf{x}^2
} + \frac{1}{
\left(\mathbf{x}\_1 + \mathbf{x}\right)
} \frac{
\partial v
}{
\partial \mathbf{x}
} + \frac{
\partial^2}{
\mathbf{c}^2
} v = 0
\tag{2}
$$

Where:

v is the velocity of the particles, ω is the pulsation and c is the velocity of the longitudinal wave in the material making up the vibratory element.

On the basis of the dimensions given in figure 3, the solution to this differential equation leads to an approximate velocity amplification ratio between the two extremities (Nassar, 1997); |v(0) /v(L)| = 1/ 0.46 = 2.16 for a resonance frequency: f = 60 KHz.

Fig. 3. Basic shape. Triangular sensor of thickness e = 1 mm, L = 32 mm, 1 = 2 mm and <sup>2</sup> = 16 mm
