Jaw couplings have become an indispensable part for industrial applications. Besides the compensation of radial, axial and angular misalignments, they are crucial for the reduction of torque shocks in drive systems. However, due to the complex material behavior of the elastic gear rim, which is typically made of thermoplastic polyurethane (TPU), there are few numerical models to describe the behavior of jaw couplings. In the present work, the material behavior of the gear rim is obtained by uniaxial loading‐unloading tests performed at different deformation rates and temperatures. By using experimental data, a hyperviscoelastic material model was fitted and a numerical model of a jaw coupling was realized in the finite element tool ABAQUS. Based on this model, (i) the relationship between torsion and torque as well as (ii) the deformation behavior of the gear rim was determined.