[Paper Review] A mechanically-derived contact model for adhesive elastic-perfectly plastic particles. Part I: Utilizing the method of dimensionality reduction

William Zunker, Ken Kamrin|arXiv (Cornell University)|Sep 13, 2023

Adhesion, Friction, and Surface InteractionsEngineering3 citations

TL;DR

This paper presents a mechanics-based contact model for adhesive elastic-perfectly plastic spherical particles using the method of dimensionality reduction (MDR), enabling accurate simulation of elastic, plastic, and adhesive contact regimes with minimal physical inputs. The model captures multi-neighbor dependent effects via volume conservation during plastic deformation and achieves high accuracy in predicting force, contact area, displacement, and particle volume across elastic-to-plastic transitions, validated against finite element simulations and analytical theories including Hertz and JKR.

ABSTRACT

In this two part series, we present a contact model able to capture the response of interacting adhesive elastic-perfectly plastic particles under a variety of loadings. In Part I, we focus on elastic through fully-plastic contact with and without adhesion. For these contact regimes the model is built upon the method of dimensionality reduction which allows the problem of a 3D axisymmetric contact to be mapped to a semi-equivalent problem of a 1D rigid indenter penetrating a bed of independent Hookean springs. Plasticity is accounted for by continuously varying the 1D indenter profile subject to a constraint on the contact pressure. Unloading falls out naturally, and simply requires lifting the 1D indenter out of the springs and tracking the force. By accounting for the incompressible nature of this plastic deformation, the contact model is able to capture multi-neighbor dependent effects such as increased force and formation of new contacts. JKR type adhesion is recovered seamlessly within the method of dimensionality reduction by simply allowing the springs to stick to the 1D indenter's surface. Because of the mechanics-focused formulation of the contact model, only a few physical inputs describing the interacting particles are needed: particle radius, Young's modulus, Poisson ratio, yield stress, and effective surface energy. The contact model is validated against finite element simulations and analytic theory, including Hertz's contact law and the JKR theory of adhesion. These comparisons show that the proposed contact model is able to accurately capture plastic displacement, average contact pressure, contact area, and force as a function of displacement for contacts as well as particle volume within the elastic to fully-plastic regimes.

Motivation & Objective

• To develop a physically grounded, parameter-minimal contact model for adhesive elastic-perfectly plastic spherical particles suitable for discrete element method (DEM) simulations.
• To extend the method of dimensionality reduction (MDR) to include plasticity and adhesion in 3D axisymmetric contacts, enabling unified treatment of elastic and fully-plastic regimes.
• To capture multi-neighbor dependent effects such as increased contact force and radial expansion due to incompressible plastic deformation.
• To seamlessly incorporate JKR-type adhesion into the MDR framework by allowing springs to 'stick' to the indenter based on effective surface energy.
• To validate the model against finite element simulations and analytical theories (Hertz, JKR) across a wide range of material properties and loading conditions.

Proposed method

• Mapping 3D axisymmetric contact problems to a 1D equivalent problem involving a rigid indenter penetrating a bed of independent Hookean springs via the method of dimensionality reduction (MDR).
• Using an elliptical indenter profile in the 1D space to unify treatment of elastic and fully-plastic regimes, with plasticity enforced by dynamically adjusting the indenter's aspect ratio under a pressure constraint derived from FEM simulations.
• Implementing unloading by lifting the 1D indenter from the springs, which naturally recovers the unloading response and force-displacement hysteresis.
• Incorporating adhesion by allowing the springs to stick to the indenter surface, with separation governed by the effective surface energy, thus recovering JKR-type behavior.
• Enforcing volume conservation of plastic deformation to model particle radius expansion and multi-neighbor dependent effects such as increased force and contact area.
• Mapping 1D results back to 3D space using analytical transformations to recover force, contact radius, displacement, and average pressure.

Experimental results

Research questions

• RQ1 How can the method of dimensionality reduction (MDR) be extended to model plastic deformation in adhesive elastic-perfectly plastic spherical contacts?
• RQ2 What is the appropriate 1D indenter profile and evolution rule to accurately represent the transition from elastic to fully-plastic contact under axisymmetric loading?
• RQ3 How can adhesion be consistently incorporated into the MDR framework to recover JKR-type behavior, even after significant plastic deformation?
• RQ4 To what extent does volume conservation during plastic deformation lead to multi-neighbor dependent effects such as increased contact force and area?
• RQ5 How well does the proposed MDR-based contact model predict force, contact area, displacement, and particle volume compared to finite element simulations and analytical theories?

Key findings

• The contact model accurately captures the evolution of plastic displacement, force, contact area, average contact pressure, and particle volume across elastic-to-plastic transitions, with strong agreement to finite element simulations.
• The model successfully recovers Hertz’s contact law in the elastic regime and JKR theory of adhesion in the adhesive regime, demonstrating consistency with established analytical solutions.
• The transition between elastic and fully-plastic regimes is continuous in both force and contact area, achieved by smoothly varying the indenter profile under a pressure constraint.
• The inclusion of volume conservation during plastic deformation enables the model to predict multi-neighbor dependent effects such as radial expansion and increased contact force, even without relative particle center displacement.
• The model remains valid for $E/Y \approx 10$ as a conservative lower bound, with no upper limit, extending into the rigid plastic regime.
• Adhesion is effectively modeled within MDR by allowing spring-stick behavior, and the JKR solution is recovered even after significant plastic deformation, confirming robustness.