A modified Hertz model for finite spherical indentation inspired by numerical simulations

In this paper, a modified Hertz model inspired by numerical simulations is proposed to predict the contact response of a linearly elastic half-space under finite spherical indentations. The proposed contact model presents a theoretical fundamental to measure the Young's moduli of the soft materials based on the finite indentation tests. The axisymmetric finite element (FE) model is created, and it is first applied to simulate the infinitesimal spherical indentations. The findings show that the FE simulation results agree well with those predicted by the classical Hertz model, which verifies that the FE model is accurate to simulate the contact responses. The FE model is then used to simulate the finite spherical indentions. Based on the numerical results of the finite spherical indentation up to the indenter radius, the radius of the contact zone is reformulated using the exact shape of the indenter, while the classical Hertz model uses a second-order approximation. This modified expression for the contact radius of the contact zone is then applied to reformulate the maximum contact pressure as well as the pressure distribution in the contact zone. After that, the reformulated pressure distribution is utilized to derive the contact force. The numerical results show that the modified Hertz model can predict very well the contact response of the linearly elastic half-space under the finite spherical indentations. The paradox whether the classical Hertz model can be extended to finite indentation is also clarified.