## 2 Theoretical Approach
### Piezoelasticity under the Influence of Strain and Temperature Variations  
Ultrason transducers of the piezoelectric ceramic types play a crucial role in this work, as they propagate ultrasonic waves [40]. These transducers convert a short electrical pulse in the UT into acoustic waves. This conversion is made possible by the unique property of piezoelectricity, found in certain crystalline materials like quartz and perovskite and ceramics based on barium titanate, lead zirconate titanate, or PZT [41; 42]. These ceramics, belonging to the ferroelectric materials class, exhibit a spontaneous polarization in a temperature range. This polarization can be reversed with the application of an external electric field. However, they are isotropic when newly produced and present a macroscopic orientation of spontaneous polarization [42; 43]. For this reason, to be used as piezoelectric sensors, they need to be polarized under the application of high electric fields (kV/mm) at a high temperature, around 90 \({}^{\circ}\)C, so it is possible to choose the direction of the macroscopic polarization. This versatility allows for various applications, from medical imaging to industrial testing. Thus, the main characteristic of piezoelectric ceramics is that they present an electrical polarization when the material is subjected to deformation or mechanical load. In other words, electric charges and potentials arise when applying mechanical stresses and strains. This phenomenon is called the direct piezoelectric effect. It is also observed that the applied electrical energy is transformed into mechanical energy, called the inverse piezoelectric effect [41]. Piezoelectric materials are exhibited with about 20 to 32 classes of crystals. They are always associated with non-center-symmetric crystals [44], with the mechanical relation being symmetric and the electrical connection being antisymmetric. All piezoelectric effects occur in materials that have an asymmetric lattice in such a way that it is possible to produce electric dipole moments while having some deformation. Therefore, all piezoelectrics are anisotropic [45]. In addition, PZT transducers have specific natural frequencies associated with their elastic properties and geometry [46]. The natural frequency \(\omega_{n}\) for undamped systems is a relation between the stiffness and the inertia of the system. In under-damped vibration systems, the natural under-damped frequencies \(\omega_{d}\) are harmonic responses as a function of the undamped natural frequency \(\omega_{n}\). The damping factor \(\zeta\) is represented in Equation (1) [47]. \[\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}} \tag{1}\]  
The \(\zeta\) is the damping factor, a dimensionless variable with a value of \(0\leq\zeta\leq 1\). The closer \(\zeta\) is to 1, the greater the damping, which means the faster the system energy will dissipate [46]. In addition to this basic concept for natural frequencies, it is known that piezoelectric ceramic disks vibrate basically in two fundamental modes, the radial mode and the longitudinal mode [48]. The radial mode generates mechanical vibrations mostly in the radial direction. In contrast, the longitudinal mode generates mechanical vibrations mainly in the longitudinal direction. Several techniques are used to collect the ultrasonic response, such as pitch-catch [26]. In pitch-catch procedures, separate UT transmitters and receivers are used, each in a middle zone. The UT transmitter transforms the electrical pulse into an ultrasonic wave propagating through the material. The UT receiver--preferably as aligned as possible with the UT transmitter--performs the reverse process, transforming the ultrasonic wave into an electrical pulse [40]. The physical model used for data transfer is illustrated in Figure 1.  
Figure 1a presents the model used in the tests, in which an electrical pulse, named \(V\), excites one of the UTs coupled to the steel plate or the steel-water-steel assembly (Figure 1b).