Applicability of Hertz Contact Theory in Large Indentation

Hertz contact theory, published by Heinrich Hertz in 1882, is one of the most fundamental theories in contact mechanics. The theory concerns about the contact of elastic bodies, and provides a mathematical model to describe the relationship between load, indentation depth, contact area and contact pressure. However, Hertz model should abide by assumptions of small indentation depth and contact region. Based on experiments, shallow depth indentation usually suffers from excessive noises, and it makes measurement inaccurate. Increasing the indentation depth is a way to avoid this problem. However, current studies did not point out the extension of large indentation in Hertz model. In this study, we concern influences of large indentation depth in Hertz model, and try to quantify the applied conditions. This study utilizes finite element method (FEM) software to construct a two-dimensional axisymmetric model and compute results of the contact of elastic bodies, including contact of rigid sphere and elastic half-space, finite thickness elastic substrate and elastic spheres in different radius. It attempts to extend Hertz model in large indentation. Based on simulation, load-indentation relationship conforms to Hertz model as the ratio of indentation depth and radius of rigid sphere reaches 0.66 in the contact of rigid sphere and elastic half-space. Result of nanoindentation with polydimethylsiloxane (PDMS) has the same trend with simulation. Moreover, when the thickness of elastic substrate becomes 12 times of the radius of the rigid sphere, load-indentation relationship conforms to Hertz model until the ratio of indentation depth and sphere radius becomes one. In addition, contacts of rigid and elastic sphere with radius ratio of elastic and rigid sphere more than 10 have identical load-indentation relationship by simulation. When radius ratio of elastic and rigid sphere becomes 10, load-indentation relationship conforms to Hertz model. Besides, this study also focuses on different surface characteristic of samples in nanoindentation. In conclusion, as long as the indentation depth of nanoindentation lies in the applicable region mentioned above, you will obtain an accurate measurement without noise.