# Role of Piezoelectric Elements in Finding the Mechanical Properties of Solid Industrial Materials
## Using Piezoelectricity to Calculate Solid-Material Properties
### Modulus of Elasticity
To generate a longitudinal wave in a tree trunk has dimensions (1.2 m \(\times\) 50 mm \(\times\) 50 mm), the authors in [89] made a hole in the tree trunk to insert PE in order to study the relationship between \(E\) with the longitudinal wave. This study proved a positive relationship between these two parameters. Authors in [90] calculated the \(E\) for different types of granite according to another formula: \(E(GPa)=4\;\times\;T\;\times\;L^{2}\;\times\;F_{I}^{2}\;\times\;\rho\) where \(T\) is a factor depending on the relative specimen dimensions, \(L\) is specimen length, \(F_{I}\) is a resonance frequency in Hz, and \(\rho\) is the density of granite. To achieve the conditions of this equation, an Erudite MK3 test system (CNS Farnell) was used. This device worked in the range 1 kHz to 100 kHz to receive an optimization for the acoustic technique depending on the measure of the essential mode for the longitudinal wave resonance frequency. This measure was based on the use of PE as a receiver of the vibrational longitudinal wave that was changed to a voltage wave and then emitted to Erudite MK3. As indicated in Section 2.3, authors in [52] declared that \"the tangent of the curve in the plateau of the elastic rebound after impact is proportional to the size of dynamic elastic modulus of indentation\", as shown in Figure 5b. A general technique as was described previously in [52] and the traditional technique in [91] have been recently regarded as techniques in finding \(E\) and \(\
u\). [PERSON] et al. [92; 93] noted that there is a big difference between the standard values of \(E\), according to ASTM from one side, and the values of \(E\) calculated from TOF tests, for a group of metals which are the refractory metals, from another side. To address this issue, [PERSON]. [92; 93] provided a new relationship between pressure transmission coefficient (\(PTC\)) and \(E\times\rho\) of the test specimen (\(TS\)) as shown in Equations (6)-(8) to calculate \(E\) for refractory and non-refractory metals:  
\[Z_{TS}=C_{L\;for\;TS}\times\rho\;for\;TS \tag{6}\]  
\[PTC\;=\;\frac{(39.9Z_{TS})}{(1.5\;+\;Z_{TS})(Z_{TS}\;+\;9.97612)} \tag{7}\]  
The following equations were derived:  
\[E=\;\frac{-5.66211\;+\;73.0387 PTC-\;275.856 PTC^{2}\;+\;425.253 PTC^{3}-\;234.885 PTC^{4}}{\rho}\;\times\;10^{7} \tag{8a}\]  
Figure 9: Testing system of the modulus of elasticity using a piezoelectric transducer as an actuator and sensor. The Equation (8a) can be used for the solid materials that have an acoustic impedance (\(Z_{TS}\)) equals or less than 51 \(\times\) 10\({}^{6}\) kg/m\({}^{2}\) s (\(Z_{TS}\geq\) 51 \(\times\) 10\({}^{6}\) kg/m\({}^{2}\) s). The Equation (8b) is used for the materialshave values ofz\({}_{TS}\)within range of 51 \(\times\) 10\({}^{6}\)\(\leq\)\(Z_{TS}\)\(\leq\) 29 \(\times\) 10\({}^{6}\) kg/m\({}^{2}\) s:  
\[E\ =\ \frac{-6.86293\ +\ 58.1587 PTC-\ 102.833 PTC^{2}+\ 52.8154 PTC^{3}}{\rho}\times\ 10^{6} \tag{8b}\]  
While for values of \(Z_{TS}\) are around \(Z_{TS}\leq\) 29 \(\times\) 10\({}^{6}\) kg/m\({}^{2}\) the equation is  
\[E\ =\ \frac{2.55909-8.2542 PTC+\ 9.8896 PTC^{2}-\ 5.196 PTC^{3}+\ 1.0075 PTC^{4}}{\rho} \times\ 10^{8} \tag{8c}\]  
According [92; 93], the set of Equation (8) depends on the value of _PTC_ only to find the value of \(E\) where, \(\rho\) is already known. _PTC_ is based on the calculation of \(Z_{TS}\) as shown in Equation (7) and this \(Z_{TS}\) is equals to \(\rho\times\) C\({}_{L}\) as shown in Equation (6). Therefore, the calculation of E according to the method proposed by [93] depends on measuring \(C_{L}\) only, while the TOF method needs measuring two types of waves (\(C_{L}\) and \(C_{S}\)) to calculation E as shown in Equation (5), where \(v\ =\ \frac{C_{L}^{2}-2C_{L}^{2}}{2\left(C_{L}^{2}-C_{S}^{2}\right)}\). Figure 10 illustrates the difference between the proposed method by [PERSON] et al. and the TOF method represented by one of the OLYMPUS company products, 38 DL PLUS, that uses this technique.