Indentation response of soft viscoelastic matter with hard skin

The structure of a hard film mounted on a soft viscoelastic substrate exists widely in nature and in industrial production, in which the hard film plays an important role. In this study, in order to accurately describe the indentation behaviors of hard film/soft viscoelastic substrate systems and to separate the mechanical properties of the hard film from the overall indentation response, an analytical relationship between the indentation loads and displacements was derived and presented herein based on plate theory, Hankel transformation, the elastic-viscoelastic correspondence principle, and Boltzmann superposition principle. In addition, finite element simulations and experiments were performed to investigate the effectiveness and applicability of the theoretical model. The results showed that the analytical solution agreed extremely well with the experiments and simulations. By analyzing the analytical solution, it was found that, due to the presence of the hard film, the viscous behaviors of the hard film/soft viscoelastic substrate system depended not only on the viscous properties of the soft substrate but also on the elastic properties of the soft substrate. Surprisingly, the relaxation time of the hard film/soft viscoelastic substrate system became longer than that of the soft substrate, but the ratio of the decay modulus to the equilibrium modulus was lower than that of the soft substrate. The above findings remind us that it is impossible to simply judge whether the hard film inhibits or promotes the soft substrate viscous behaviors. Our research can help readers gain insights into the indentation response characteristics of hard film/soft viscoelastic substrate systems, and it provide a theoretical basis for the measurement of the mechanical properties of a hard film mounted on a soft viscoelastic substrate.