## 2 Materials and Methods
### Instrumental Signal Measurements in the Two-Point Bending Tests  
Figure 2 shows a scheme of the electronic circuits used to measure the instrumental signal in the two-point bending tests. The change in the electrical resistivity (\(\Delta\)R) of the sensor when the strain gauge was being bent by the loading nose (see Figure 1b) was measured through a Wheatstone bridge (PW) because of its great precision [16; 23]. As the signal (or drop in voltage) obtained from the PW was very small, in the range of millivolts, an instrumentation amplifier was required for the signal conditioning. In addition, a fixed-gain differential-input amplifier (AIDR) with high common-mode rejection was developed. A Wheatstone bridge circuit (PW) in a quarter-bridge arrangement was used to register the changes in the electrical resistance of the sensor concerning its initial resistance (R\({}_{\text{o}}\)). The quarter-bridge configuration was balanced (V\({}_{\text{AB}}\) = 0) and showed equality of nominal resistance values [8]. Moreover, an inactive gauge was added to prevent changes in resistance for the initial resistance. Resistors in series, Rm, were added to each component of the PW to establish a 10-mA upper limit for the current flow through the sensor. The  
Figure 2: Scheme of the instrumental signal measurement during bending tests. resistor ratio \(\rm R_{m}/R_{o}\) = 3.25 was employed for the resistors. In practice, the change in resistivity \(\rm\Delta R/R_{o}\) is in the range of \(10^{-2}\), which is negligible relative to unity. The response of the proposed device (sensor) was indicated in the PW through a voltage drop, as Equation (1) illustrates. This voltage drop and the change in resistivity of the sensor are in a proportional relationship because of the lateral displacement of the sensor. The constant of proportionality was 0.588. \[\rm V_{AB}=\frac{\rm\Delta R/R_{o}}{\big{[}2^{\rm AR/R_{o}}+4(1+R_{m}/R_{o}) \big{]}}V_{r}\approx\frac{\rm\Delta R/R_{o}}{4(1+R_{m}/R_{o})}V_{r}=0.588\rm \Delta R/R_{o} \tag{1}\]  
The response of the sensor obtained by the PW is amplified by the AIDR circuit, wherein the voltage drops in nodes 3 and 4 must be the same (see Figure 2) to achieve a linear performance of the instrumentation amplifier. Therefore, the instrumental signal was obtained by applying [PERSON]'s law to each loop of the circuit shown in Figure 2, according to the output voltage given by Equation (2). \[\rm V_{s}=A_{1}V_{A}-A_{2}V_{B}, \tag{2}\]  
where \(A_{1}=\Big{[}\frac{R_{7}}{R_{4}}\frac{(R_{4}+R_{5})}{(R_{7}+R_{6})}\frac{(R_{ 2}+R_{3})}{R_{2}}+\frac{R_{1}R_{5}}{R_{2}R_{4}}\Big{]}\) and \(A_{2}=\Big{[}\frac{R_{7}}{R_{4}}\frac{(R_{4}+R_{5})}{(R_{7}+R_{6})}\frac{R_{ 3}}{R_{2}}+\frac{R_{5}}{R_{4}}\frac{(R_{1}+R_{2})}{R_{2}}\Big{]}\) are the equivalent voltage drop percentages in terminals A and B of the Wheatstone bridge, whereas the nominal resistance values of the amplifier are represented by \(\rm R_{i}\), with i = 1 to 7. To rearrange Equation (2) in terms of the voltage drops in differential mode, \(\rm V_{AB}=V_{A}-V_{B}\), and in common mode, \(\rm V_{mc}=(V_{A}+V_{B})/2\), the voltage drops of the instrumental signal, Vs, can be rewritten in terms of the common-mode gain, \(\rm G_{mc}\), and the gain in differential mode, \(\rm G_{AB}\), as Equation (3) illustrates:  
\[\rm V_{s}=G_{mc}V_{mc}+G_{AB}V_{AB}, \tag{3}\]  
where \(\rm G_{mc}=\Big{[}\big{(}1-\frac{R_{c}R_{c}}{R_{c}R_{c}}\big{)}\Big{/}\big{(} 1+\frac{R_{c}}{R_{c}}\big{)}\Big{]}\) and \(\rm G_{AB}=\Big{[}\big{(}1+\frac{R_{c}}{R_{c}}\big{)}\Big{(}\frac{R_{c}}{R_{ c}}+\frac{1}{2}\big{)}\Big{/}\big{(}1+\frac{R_{c}}{R_{c}}\big{)}+\frac{R_{c}}{R_{ 4}}\Big{(}\frac{R_{1}}{R_{c}}+\frac{1}{2}\Big{)}\Big{]}\). Given that the resistances \(\rm R_{5}R_{6}=R_{7}R_{4}\), the amplifier worked in a high common-mode rejection ratio, wherein \(\rm G_{mc}\) is null. Thus, the amplification factor, K, was estimated as \(\rm K=R_{5}/R_{4}=R_{7}/R_{6}\), and the average gain factor, G, was estimated as \(\rm G/2=R_{1}/R_{2}=R_{3}/R_{2}\).