Manfred Hahn and Rafael D. Jarzabek present two new analytical 3D solutions in continuum mechanics. These analytical solutions can be used to verify numerical solution methods. This essential begins by outlining the historical development of analytical solution methods in continuum mechanics for 2D and 3D spaces and the resulting problems. Due to the mathematical difficulties that caused the advancement of analytical methods to stagnate several decades ago, and because users preferred numerical methods with the development of computers, only numerical solutions for specific problems were continuously calculated. However, today mathematics has progressed, allowing for the discovery of new analytical solutions. To this end, the authors present the original solution approaches and compare them with Pagano's new solution method.