Development and comparison of high order toroidal finite elements for calculating disc springs

The present manuscript deals with the development and the application of toroidal finite elements. These finite elements belong to the class of curvilinear elements and, therefore, a special element formulation is required. This means to express stress, strain, and displacement in curvilinear coordinates. Here, an 8-node quadrilateral element is developed which is described by an isoparametric interpolation. The development of the governing equations is one main topic of this contribution. Another main topic is to evaluate the quality and efficiency of these elements with respect to the calculation of disc springs. In this context, calculations with toroidal elements are checked against experimental data which consists of load-deflection-curves. In addition, the effects of friction and differences to the exact rectangular cross section are considered. With the help of Riks procedure, characteristic curves with negative spring rates are obtained.