[Paper Review] Contact and Quasi-Static Impact of a Dissipationless Mechanical Model

Hiroto Kuninaka, Hisao Hayakawa|arXiv (Cornell University)|Apr 29, 2005

Adhesion, Friction, and Surface Interactions1 references3 citations

TL;DR

This paper proposes a dissipationless two-dimensional spring-mass model with defect particles and free boundary conditions to reproduce Hertzian contact mechanics at equilibrium and quasi-static impact behavior at low speeds. By eliminating explicit energy dissipation, the model successfully reproduces the Hertzian force–deformation relation and the quasi-static theory's prediction for restitution coefficient vs. impact speed, demonstrating that irreversibility can emerge from reversible microscopic dynamics without external damping.

ABSTRACT

Collisions and contacts of elastic materials are numerically and theoretically investigated. Using a two-dimensional spring-mass model with defect particles under the free boundary condition, we reproduce the Hertzian contact theory at equilibrium and the quasi-static theory for low speed impacts.

Motivation & Objective

• To derive macroscopic elastic laws—Hertzian contact and quasi-static impact—directly from a purely mechanical, reversible microscopic model.
• To resolve the failure of previous dissipationless models in reaching equilibrium under contact forces.
• To reconcile numerical simulations with theoretical predictions for low-speed impacts without introducing explicit damping.
• To investigate the origin of irreversibility in finite-degree-of-freedom systems from reversible dynamics.

Proposed method

• A 2D mass-spring lattice model with 1099 mass points for the disk and 1269 for the wall, using nonlinear springs with $ k_a $ and $ k_b \propto k_a / R^2 $.
• Contact forces are modeled via a short-range exponential potential $ F \propto \exp(-a l_s) $, where $ l_s $ is the distance to the nearest wall spring, with $ a = 500/R $.
• The system is evolved using a fourth-order symplectic integrator with $ dt = 10^{-3} R/c $, preserving energy and time-reversal symmetry.
• Defect particles are introduced to break perfect periodicity and suppress recurrent energy exchange, enabling relaxation to equilibrium.
• The model uses free boundary conditions, avoiding fixed constraints that could induce artificial oscillations.
• Elastic moduli (Young’s modulus $ E $, Poisson’s ratio $ \nu $) are computed via perturbation methods on isotropic compression and shear, yielding $ E = 0.773k_a $, $ \nu = 0.336 $.

Experimental results

Research questions

• RQ1 Can a purely mechanical, dissipationless model reproduce the Hertzian contact law between elastic bodies at equilibrium?
• RQ2 Does the quasi-static theory accurately predict the restitution coefficient for low-speed impacts in a reversible system?
• RQ3 Why do previous dissipationless models fail to reach equilibrium under contact forces, and how can this be corrected?
• RQ4 Can irreversibility in macroscopic contact and impact phenomena emerge from reversible microscopic dynamics?
• RQ5 What is the role of defect particles and free boundary conditions in enabling relaxation and equilibrium in such models?

Key findings

• The model successfully reproduces the Hertzian contact relation $ \delta \propto \frac{P}{\pi E^*} \left[ \ln\left( \frac{4\pi E^* R}{P} \right) - 1 - \nu \right] $ at equilibrium, confirming the absence of artificial dissipation.
• For low-speed impacts ($ v/c \leq 0.007 $), the simulation data for the restitution coefficient $ e $ match the quasi-static theory prediction, validating its applicability in this regime.
• At higher speeds ($ v/c > 0.007 $), the discrepancy between simulation and quasi-static theory arises due to excitation of internal vibrational modes, invalidating the quasi-static assumption.
• The characteristic time $ \tau_0 = 0.011 R/c $ extracted from the data aligns with the theoretical estimate in the quasi-static framework.
• The computed elastic moduli are $ E = 0.773k_a $ and $ \nu = 0.336 $, consistent with the expected mechanical behavior of the lattice.
• Preliminary results indicate that gravity affects the restitution coefficient, suggesting a non-trivial role in quasi-static impact dynamics.