The phenomenon of relaxation and oscillation is a joint event that is often encountered. These properties can occur in viscoelastic materials even though they do not co-occur. Because the characteristics of viscoelastic materials are difficult to describe using classical-order differential equations, in this study, fractional-order differential equations were used to model each of the relaxation and oscillation phenomena in viscoelastic materials with the help of the Laplace transform as a solution method. The solution obtained characterizes the phenomenon of memory effect as well as viscoelastic materials in general. In addition to this phenomenon, several other variables were also found to influence the related material motion dynamics.