## 4 Analytical and Numerical Solutions
As seen from Figure 8 these stresses become nearly constant radially through the cylinder for \(\omega=300\) rad/sec and low rotational velocities. Figure 9 represents the trends of radial strain rates \(\dot{\epsilon}_{r}\) radially through cylinder for \(\omega=300,\ \omega=500,\ \text{and}\ \omega=1000\) rad/sec. All radial strain rates are higher (in magnitude) at the inner periphery of the cylinder and vary with parallel curves radially through the cylinder. It can be observed from Figure 9 that radial strain rate increase is potentially much higher at lower rotational speed, as can be seen from its trend line with maximum difference between initial and final values. In a pressurized cylinder, the internal or external pressure creates hoop stresses, which are tangential stresses along the circumference of the cylinder. These hoop stresses generate forces that cause radial expansion of the cylinder's cross-section. As we move from the inner radius to the outer radius, the magnitude of the hoop stress decreases due to the principles of mechanics [39]. Figure 10 shows variations of tangential strain rates \(\dot{\epsilon}_{t}\) radially through cylinder for \(\omega=300,\ \omega=500,\ \text{and}\ \omega=1000\) rad/sec. All tangential strain rates are higher (in magnitude) at the inner periphery of the cylinder and vary with parallel curves radially through the cylinder. The tangential strain rates decrease progressively as rotational velocity increases. The percent decrease in tangential strain rate is higher for 1000 rad/sec rotational speed in comparison to lower rotational speeds. Moreover, the results show reasonable compliance between FEM and analytical solutions. When internal pressure is applied to a thick-walled cylinder, the pressure generates hoop stresses (tangential stresses) along the circumference of the cylinder. At smaller radii (closer to the inside surface), the material experiences a higher internal pressure, resulting in higher hoop stresses. For larger radii (closer to the outside surface), the internal pressure acting on the material decreases, leading to lower hoop stresses [40]. It is important to state that in creep modeling, it is common to attain or encounter very high exponential coefficients. In some cases, the noticed value could reach about 10 power \(-90\), especially in cases where the material investigated is relatively brittle, the values of strain are extremely small, and the load time history covers a very long period of time, as happened in our case, but in general, when dealing with normal materials and shorter periods of load time history, the average reported values are around 10 power \(-13\)[9; 15; 16; 43]. Figure 11 shows variations of equivalent creep strain rates \(\dot{\epsilon}_{e}\) radially through cylinder for \(\omega=300,\ \omega=500,\ \text{and}\ \omega=1000\) rad/sec. All equivalent creep strain rates are higher (in magnitude) at the inner periphery of the cylinder and vary with parallel curves radially through the cylinder. They decrease gradually as rotational velocity increases. Finally, Figure 12 displays the linear trend of varying equivalent creep strain with respect to time in this time region at all different radii of the cylinder. Because of using the Norton Constitutive model, which is valid in secondary creep region only (linear region behavior), the figure above can be explained. The equivalent creep strain is maximum for a lower radius (0.3 m) and decreases as the radius increases from 0.3 to 0.45 m. It can be credited to higher stress concentration at lower radius. Moreover, in regions with lower radii, the material might experience more pronounced microstructural alterations, which can influence the creep behavior and lead to a higher creep strain rate [41; 42].