High-Damping Natural Rubber (HDNR) bearings are widely employed for seismic isolation. These bearings are characterized by a remarkable nonlinear behaviour and often by a degrading cyclic response, induced by the addition of filler to enhance its dissipation capacity. This latter phenomenon, denoted as stress-softening or Mullins effect, can significantly influence the nonlinear dynamic response of isolated structures leading to a variability of the seismic response of isolated buildings. Moreover, the behaviour of HDNR bearings may produce an amplification of response, due to higher modes contribution, which can affect the performance of non-structural components and equipment. Models providing an accurate description of the nonlinear beha...