## 2 Materials and Methods
### Dynamic Young's Modulus  
The dynamic Young's modulus was determined by impulse excitation of vibration technique described in the ASTM E 1876-01 [16]. This test method measures the fundamental resonant frequency of test specimens of suitable geometry by exciting them mechanically with an impulse tool, as shown in the flowchart of Figure 3. A ceramic piezoelectric transducer that is sensitive to a certain mechanical stimulus such as mechanical vibration was used and generates a potential difference when it undergoes a certain deformation. The signal analysis system is a digital oscilloscope connected to a laptop computer that uses the PicoScope 6 software (Pico(r) Technology, St. Neots, UK). Subsequently, using the OriginPro 2015 (OriginLab(r), Northampton, MA, USA) program, a signal analysis was performed using a fast Fourier transform (FFT). By knowing the resonant frequency of a given specimen as well as its dimensions and mass, it was possible, using Equation (3), to determine the dynamic Young's modulus.  
\[E_{d}=0.9465\left(\frac{mf_{f}^{2}}{b}\right)\left(\frac{L^{3}}{t^{3}}\right)T _{1} \tag{3}\]  
where: \(E_{d}\) is the dynamic Young's modulus (Pa), \(m\) the mass of the specimen (g), \(f_{f}\) the fundamental resonant frequency (Hz), \(t\) the thickness (mm), \(b\) the width (mm) and \(L\) the length of the specimen (mm). \(T_{1}\) is a correction factor that depends on the geometry of the specimen that was set equal to the unity, for the specimen dimensions used in the current work.  
Figure 3: Flowchart of the experimental protocol used to measure the dynamic [PERSON]’s modulus using the impulse excitation of vibration technique (the graphs show a real test for the Activa(tm) specimen after thermocycling).  
Before the exciting impulse, all the specimens were heated in an oven, Digitheat (Selecta, Barcelona, Spain), at 40 \({}^{\circ}\)C for 1 h. Each specimen was removed one by one from the oven and transported on a ceramic plate and tested immediately at a temperature of 37 \(\pm\) 2 \({}^{\circ}\)C. For each specimen, five determinations of the Young's modulus were achieved in order to estimate the mean and standard deviation.  
The ratio of the loss modulus to storage modulus (\(E^{\prime\prime}/E^{\prime}\)) in a viscoelastic material is defined as the tan \(\delta\) and provides information on the stress response under oscillatory deformation. This value was determined according to the following method, already used in previous papers [17]:  
The logarithmic decrement, \(\Delta\) can be calculate using Equation (4).  
\[\Delta=\frac{1}{j}ln\left(\frac{Xi}{Xi+j}\right)=\zeta w_{n}T_{a} \tag{4}\]  
where: \(X_{i}\) and \(X_{i+j}\) are the amplitude of two peaks of the free vibrating system, separated by \(j\) oscillations, \(\zeta\) the damping factor, \(w_{n}\) the system vibratory frequency and \(T_{a}\) the average period of the wave form that can be calculated through Equation (5).  
\[T_{a}=\frac{2\pi}{w_{d}}=\frac{2\pi}{w_{n}\sqrt{1-\zeta^{2}}} \tag{5}\]  
where \(w_{d}\) is the dumped frequency of the system. Considering Equations (4) and (5), the logarithmic decrement, \(\Delta\) is as follows:  
\[\Delta=\frac{2\pi\zeta}{\sqrt{1-\zeta^{2}}} \tag{6}\]  
For light dumping, Equation (6) can be simplified as follows (Equation (7)):  
\[\Delta=2\pi\zeta \tag{7}\]  
From Equations (4)-(7), the damping factor \(\zeta\) can be obtained and finally tan \(\delta\) can be obtained as \(2\zeta\).