## 3 Experimental Results  
This vibration technique does not have any type of contact on the beam (see Figures 13 and 14) as compared to other fatigue methods such as four- and three-point bending. Vibration fatigue testing was performed at 32 Hz. All beams failed abruptly in a core shear failure with a debonding of the material similar to that observed in three-point bending tests, as shown in Figure 15. Due to the relatively high frequency and the catastrophic nature of the failure, shear crack growth could not be observed. The FEA vibration resonance frequency predictions, along with the average number of cycles to failure under vibration and three-point bending with their respective stress ratios, are shown in Table 2.  
In both three-point bending and vibration testing, specimens were repeatedly cycled at a given stress ratio until failure. The S-N curve that defines the number of cycles to failure is shown in Figure 16.  
Figure 14: Schematic outline of the experimental vibration setup. Figure 15: (**a**) Failure found in three-point bending tests; (**b**) failure found in vibration tests. The vibration fatigue and three-point bending S-N regression lines shown in Figure 16 are almost parallel, suggesting that the two techniques yield consistent results with the same material configuration. The observed linear behavior and data scatter in three-point bending and vibration testing is in agreement with previous results, as reported in the literature [1, 2, 12, 42]. The total face sheet thickness to core ratio in the three-point bending specimen was (\(\rm{f_{f}/t_{c}=0.08}\)). In the original S-N curve generated, the effect of contact stress was not considered in the three-point bending stress ratio calculation. It is clearly evident that the fatigue life predicted with vibration testing yields a longer life than that predicted by three-point bending at the same stress level. One possible explanation for this behavior can be attributed to the jig-beam contact (see Figure 11) used to hold the three-point bending specimens. The vibration testing does not contain any contact points (see Figure 13). The Von Mises centerline stresses were calculated with a typical three-point bending test formulation that does not consider the effects of contact stress [43]. By considering contact stress [44, 45], representative values of the Von Mises centerline stresses occurring in three-point bending can be estimated using Equation (2) [45]. \[P_{0}=\left(\frac{P^{\prime}E^{*}}{\pi R}\right)^{1/2} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stress Ratio** & \begin{tabular}{c} **Experimental** \\ **Result (Hz)** \\ \end{tabular} & \begin{tabular}{c} **Predicted** \\ **Frequency (Hz)** \\ \end{tabular} & \
\begin{tabular}{c} **Average Number of Cycles** \\ **to Failure (Vibrations)** \\ \end{tabular} &
\begin{tabular}{c} **Average Number of Cycles to** \\ **Failure (Three-Point Bending)** \\ \end{tabular} \\ \hline
0.77 & 32.5 & 31 & 5954 & 4507 \\ \hline
0.70 & 31.95 & 31 & 18,078 & 14,726 \\ \hline
0.62 & 32.25 & 31 & 85,461 & 66,741 \\ \hline
0.55 & 32.5 & 31 & 186,876 & 166,742 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental results from Vibration and Three-Point Bending tests. Figure 16: Comparison of sandwich composite fatigue life curves. Failures obtained during three-point bending test and vibration test. where \(P_{0}\) is the stress caused by the maximum pressure over the specimen. \(P^{\prime}\) is the applied force per unit length. \(R\) is the effective curvature of the loading pin including the damping rubbers, and \(E^{*}\) is the contact modulus and is determined from Equation (3) [45]. \[\frac{1}{E^{*}}=\frac{1-\
u_{1}^{2}}{E_{1}}+\frac{1-\
u_{2}^{2}}{E_{2}} \tag{3}\]  
In Equation (3), the modulus of elasticity and Poisson ratio for the damping rubber and foam are given by \(E_{1}\), \(\
u_{1}\), \(E_{2}\), and \(\
u_{2}\), respectively.