## 3 Results
### Theoretical Approach  
The CNT-reinforced adhesive film was modeled as a nanocomposite with an homogeneous CNT distribution. The electrical mechanisms inside a nanoparticle network are given by the intrinsic electrical resistance of the CNTs themselves, the contact resistance between nanotubes and the tunneling resistance between near CNTs. In this particular case, the variation of electrical resistance due to applied strain are mainly dominated by the tunneling effect as intrinsic and contact resistance are assumed to be invariable with applied strain as stated in other studies [54]. Therefore, the changes in the electrical resistance can be divided into two terms, the first one, correlated to the variations due to the applied strain, which depends on the changes due to the tunneling effect between adjacent nanoparticles and the second one, which is correlated to the breakage of electrical pathways due to the effect of damage accumulation, as shown in the schematics of Figure 6, leading to the following expression:  
\[\Delta R_{total}=\Delta R_{tunnel}+\Delta R_{damage} \tag{1}\]  
The tunneling effect can be calculated by using the well-known Simmons formula for the tunneling resistance [55]. It has been proved to be an effective way to predict the electromechanical response of a strained CNT network as it is the most dominant electrical transport mechanism. It takes several aspects such as the polymer type and the contact area between adjacent CNTs:  
\[R_{tunnel}=At\cdot e^{bt} \tag{2}\]  
where \(A\) and \(b\) are two constants depending on the CNT geometry and matrix barrier characteristics and \(t\) is the tunneling distance, which changes with the applied strain. The changes due to damage accumulation are not easy to model. There are many studies investigating this effect by proposing different damage evolution laws [56; 57], but the particularities of the tested systems make the damage calculation very difficult. Therefore, damage accumulation is estimated by comparing the measured changes in the electrical resistance and the known tunneling effect  
\[\Delta R_{damage}=\Delta R_{measured}-\Delta R_{tunnel} \tag{3}\]  
For this purpose, the initial tunneling distance is calculated as the distance that best fits the initial changes of the electrical resistance, where no damage is supposed. Figure 7 shows the comparison between the theoretical line, taking only the tunneling effect and the experimental measurements for aged and non-aged samples with \(1\) wt% SDS into account. The pattern areas indicate the differences between the theoretical and the experimental ones being, thus, the damage accumulation during the SLS test. Figure 6: Schematics of electromechanical behavior in a SLS test showing the increase of tunneling distance and crack nucleation inside the adhesive. The non-aged sample shows significant irregular behavior. The threshold for damage accumulation was observed at \(\sim\)0.25 mm displacement. After this point, the evolution of damage accumulation was very irregular. This can be explained due to the brittle mechanisms dominating the mechanical behavior of the adhesive. Secondary cracks start to nucleate and then they coalesce around the main crack [58], inducing a higher breakage of electrical paths in a similar way than that observed in other studies for fatigue testing [59]. This nucleation was not uniform so the unstable damage accumulation was explained. In the case of 1-month-aged samples, this damage accumulation starts to take place at 1 mm of displacement, that is, much later than in the non-aged specimen. This is in good agreement with the stated conclusions from water uptake and LSS measurements, as the induced plasticity in the first stages of water uptake avoids the early crack nucleation inside the adhesive joint. After that, damage accumulation takes places in a sudden way, that is, the cracks start to nucleate and then immediately coalesce.