# Unloading Model of Elastic–Plastic Half-Space Contacted by an Elastic Spherical Indenter

## TL;DR

This paper introduces a new analytical model for predicting the unloading behavior of an elastic–plastic half-space indented by a spherical indenter.

## Contribution

The model accurately predicts residual indentation and curvature after unloading using an elastic modulus ratio.

## Key findings

- The recovered deformation depends on the elastic modulus ratio after full unloading.
- The proposed model conforms to Hertzian-type contact force and indentation relationships.
- Numerical simulations validate the model's accuracy across various indentations and material properties.

## Abstract

A new unloading contact model of an elastic–perfectly plastic half-space indented by an elastic spherical indenter is presented analytically. The recovered deformation of the elastic indenter and the indented half-space has been found to be dependent on the elastic modulus ratio after fully unloading. The recovered deformation of the indented half-space can be calculated based on the deformation of the purely elastic indenter. The unloading process is assumed to be entirely elastic, and then the relationship of contact force and indentation can be determined based on the solved recovered deformation and conforms to Hertzian-type. The model can accurately predict the residual indentation and residual curvature radius after fully unloading. Numerical simulations are performed to demonstrate the assumptions and the unloading model. The proposed unloading model can cover a wide range of indentations and material properties and is compared with existing unloading models. The cyclic behavior including loading and unloading can be predicted by combining the proposed unloading law with the existing contact loading model. The combined model can be employed for low-velocity impact and nanoindentation tests and the comparison results are in good agreement.