## 4 Results and Discussion
### Viscoelastic Behavior  
#### 4.2.1 Case 1  
Simulation results in terms of radial and circumferential stress are shown in Figure 3 for the flywheel rotor constructed from an aluminum hub with a thick CFRP rim. While the  
Figure 2: Radial (**a**) and circumferential (**b**) stress distribution comparison between the current model and [PERSON] et al. [13] for a two-rim press-fit CFRP flywheel rotor. hub, located between \(r=160\) mm and \(r=200\) mm, was not inherently subject to viscoelastic behavior, its stress state changed as compressive loading from the composite rim decreased. Considering the radial stress data depicted in Figure 3, the composite rim, in a pristine state post-manufacturing (\(t=0\) and \(\omega=0\)), experiences high compressive loading, approximately \(-46.7\) MPa, due to the press-fit assembly. After startup to \(\omega_{\text{limit}}\), the peak compressive load decreases to \(-27\) MPa. This change is induced by the radial position of the hub and rim leading to differences in centrifugal loads, as well as differences in elastic modulus between the two materials. For comparison, the aluminum elastic modulus is \(71\) GPa while the CFRP longitudinal and transverse modulus is \(111\) GPa and \(9.1\) GPa, respectively. As a consequence, the outer rim deforms more than the aluminum hub, reducing the interfacial pressure. Note that this ability to compensate for differential deformation while maintaining rotor integrity is one advantage of a press-fit assembly. It can further be seen in Figure 3 that circumferential stress in the aluminum hub increases over time. At the inner hub surface, circumferential stress increases from \(542\) MPa to \(596\) MPa after \(10\) years; an increase of \(9.2\%\). (Note that even though this circumferential stress exceeds yield strength, the hub does not undergo failure because the stress coordinate for the given stress state still resides within the failure envelope, invoking, e.g., maximum distortion energy theory.) Additionally, the increased circumferential stress is coupled with a decrease in radial compressive stress, i.e., radial stress becomes less compressive in the aluminum hub. These changes in radial and circumferential stress are attributed to the increased compliance of the CFRP rim during this time period, allowing the hub to deform radially. Regarding the composite rim, radial and circumferential peak stresses are predicted to decrease moderately between \(1\%\) and \(5.5\%\), respectively, over the \(10\)-year simulation period, which is to be expected based on previous research [14]. For greater clarity, peak stress values in the rotor over the simulated \(10\)-year period were determined for (i) the interfacial pressure measured in the radial direction, (ii) the radial stress, and (iii) the circumferential stress. Corresponding values are given in Table 4. To illustrate their change over the simulated operation, they were normalized by their initial value at \(t=0\) and plotted in Figure 4. Within the first year of (continuous) operation, the rotor undergoes viscoelastic stress changes as the interfacial compressive stress decreases from \(-27\) MPa to \(-19\) MPa, or approximately \(70.4\%\) of the initial value. After \(1\) year, interfacial compressive stress decreases at a reduced rate, decreasing to \(63.3\%\) after \(10\) years. These results indicate  
Figure 3: Radial (**a**) and circumferential (**b**) stress predictions for a flywheel rotor constructed of an aluminum hub and CFRP rim, after assembly (zero velocity), at startup, and at various times up to \(10\) years of continuous operation at the limit velocity of \(24250\) rpm. that the composite material experiences rapid non-linear relaxation over the first year of operation, which to a large extent is ascribed to primary or transient creep (phase I). In the subsequent years of the rotor's service life, stress relaxation is significantly reduced, indicating the material has fully transitioned into secondary or steady-state creep (phase II).