[Paper Review] A simple model for elastic and viscoelastic punch indentation problems with experimental validation

Abdelaziz Sameur, H.P. Yin+2 more|ArXiv.org|Feb 2, 2008

Adhesion, Friction, and Surface Interactions14 references3 citations

TL;DR

This paper presents a simple analytical model for predicting contact force in elastic and viscoelastic punch indentation using geometric and material properties. It accurately predicts force from penetration depth for spherical, conical, and pyramidal punches, with experimental validation in both elastic and viscoelastic regimes using integral operators for time-dependent behavior.

ABSTRACT

This paper presents an analytical model of punctual elastic contact between a rigid body of arbitrary geometry and a plane surface. A simple analytical model is developed in order to evaluate the contact force knowing the volume of interpenetration, the surface and the perimeter of the base of this volume and the mechanical characteristics of surfaces in contact. Analytical and experimental validations are made for this model in the case of simple shapes (spherical, conical and pyramidal). Next, an approach for the resolution in case of contact between a rigid body and a viscoelastic plane is presented. The elastic constants are replaced by an integral operator corresponding to the viscoelastic stress-strain relation. At last, the viscoelastic punctual contact is studied analytically and validated experimentally.

Motivation & Objective

• To develop a simplified analytical model for elastic contact between a rigid punch of arbitrary shape and a plane surface.
• To enable prediction of contact force based on penetration volume, base area, perimeter, and material properties.
• To extend the elastic model to viscoelastic materials using an integral operator representation of the stress-strain relation.
• To validate the model experimentally across different punch geometries and material behaviors.
• To provide a computationally efficient yet accurate tool for indentation analysis in engineering and material science applications.

Proposed method

• Formulates a contact force model based on the volume of interpenetration, base area, and perimeter of the contact zone.
• Uses Hertzian contact theory principles adapted for arbitrary punch geometries via geometric and material input parameters.
• Replaces elastic constants with an integral operator to model viscoelastic behavior, capturing time-dependent deformation.
• Applies the correspondence principle to transform the viscoelastic problem into an equivalent elastic one using relaxation functions.
• Employs experimental indentation tests with spherical, conical, and pyramidal punches on elastic and viscoelastic materials.
• Compares analytical predictions with experimental force-displacement data to validate model accuracy.

Experimental results

Research questions

• RQ1 Can a simple analytical model predict contact force in elastic indentation using only geometric and material inputs?
• RQ2 How accurately does the model predict force for different punch shapes (spherical, conical, pyramidal) under elastic loading?
• RQ3 Can the model be extended to viscoelastic materials using an integral operator formulation of the stress-strain relation?
• RQ4 How well does the analytical prediction match experimental measurements in viscoelastic indentation?
• RQ5 What is the role of penetration volume, base area, and perimeter in determining contact force across different geometries?

Key findings

• The model accurately predicts contact force for spherical, conical, and pyramidal punches in elastic indentation with minimal input parameters.
• Experimental validation shows good agreement between predicted and measured force-displacement responses across all tested geometries.
• The viscoelastic extension of the model successfully captures time-dependent behavior using an integral operator representation.
• The model maintains computational simplicity while achieving high accuracy in both elastic and viscoelastic regimes.
• The inclusion of penetration volume, base area, and perimeter enables robust force prediction without requiring complex finite element simulations.
• The correspondence principle applied via relaxation functions allows effective modeling of viscoelastic response with minimal modification to the elastic framework.