Approximate determination of geometric parameters of the sag following the embedding of a paraboloid indenter into a half-space

The problem of the indenter embedding into a half-space is formulated and solved. A concept of the elastic contact is used, which implies the uniformity of the stress-strain state in a surface layer. The process of the surface layer formation after mechanical treatment is studied. Based on the general theory of indenter penetration into a half-space, numerical and analytical solutions, determining the length and depth of penetration of a paraboloid into elastoplastic space, are obtained. According to the theory of indentation, the problem is reduced to the use of the method of approximate determination of geometric parameters of a bead, when a paraboloid is being embedded into elastoplastic space. An exact solution to the formulated problem allows one to obtain analytical dependences for calculating the velocities and to study the stress-strain state of the material in the half-space surface layer during the processing. The reliability of the obtained analytical solution is confirmed by numerical calculations without introducing additional hypotheses.
Based on the analytical solution, the geometric parameters of the influx from the penetration depth are determined. Calculations can be performed at any embedding depth. Sag formation during the indenter embedding by 18 mm into plasticine specimens is considered as an example. It is shown that the rigid zone is insignificant or absent.