<p>Viscoelastic materials exhibit hysteresis under repetitive loading and unloading cycles. Although much research exists on viscoelastic constitutive relationships, this study aims to develop a procedure for estimating linear or nonlinear multi-element viscoelastic models based on hysteresis behaviour under a specified loading condition. A multi-element model based on a generic integral order time derivative operator is employed to represent the experimental hysteresis under cyclic displacement loading. The methodology develops linear and nonlinear models for viscoelastic semi-solids, including elastomeric vibration isolation elements and biomaterials such as pig cartilage from the lumbar spinal segment. Curve fitting via Genetic Algorithm solves a constrained optimization problem to extract model parameters. Initial results validate the model using experimental hysteresis data and predict the responses for varying loading frequencies and amplitudes. Additionally, higher-order nonlinear stiffness and damping parameters are simplified. The validated model effectively predicts dynamic behaviour, offering confidence in its application to structures incorporating such materials.</p> Graphical abstract <p></p>