Indentation tests have been widely used to evaluate the mechanical properties of industrial and biological materials in many studies. In this study, we obtained an analytical solution for the axisymmetric contact problem for a viscoelastic layer bonded to a rigid substrate and indented by a flat‐ended cylindrical punch using the elastic–viscoelastic correspondence principle. The analytical solution was obtained from a method using an infinite system of simultaneous equations instead of a conventional method using the second kind of Fredholm equation. Relaxation indentation tests were conducted for silicon rubber, polyurethane gel (a soft material used for elastography), and articular cartilage from a bovine femoral head. The articular cartilage was tested as obtained and then tested again after treatment by collagenase. The viscoelastic models in this study were a three‐element model (standard linear solid model) and a five‐element model (generalized Maxwell model), and the viscoelastic parameters for the models were determined by fitting the analytical solution to results for applied load as it varied over time during relaxation indentation tests. The results suggested that the five‐element model was more appropriate than the three‐element model for reproducing the relaxation indentation behavior of silicon rubber, polyurethane gel, and articular cartilage. Furthermore, the viscoelastic parameters for treated articular cartilage clearly decreased compared with those for unaltered articular cartilage. These results suggest that the method for evaluating the time‐dependent mechanical properties of articular cartilage based on analytical solutions of relaxation indentation tests could be a useful tool for evaluating the state of degenerated cartilage.