[Paper Review] A threshold-force model for adhesion and mode I fracture

Srivatsan Hulikal, Kaushik Bhattacharya+1 more|arXiv (Cornell University)|Jun 10, 2016

Adhesion, Friction, and Surface Interactions6 references3 citations

TL;DR

This paper proposes a discrete spring model with a tensile threshold force to simulate adhesive contact and mode I fracture, showing that scaling the threshold force as $1/\sqrt{r}$ (matching Linear Elastic Fracture Mechanics) yields macroscopically size-independent behavior. The model reproduces key JKR adhesion scalings—linear dependence of maximum tensile force on fracture energy and sphere radius—enabling direct calibration of the threshold force in terms of material properties like fracture energy, Young's modulus, and Poisson's ratio.

ABSTRACT

We study the relation between a threshold-force based model at the microscopic scale and mode I fracture at the macroscopic scale in a system of discrete interacting springs. Specifically, we idealize the contact between two surfaces as that between a rigid surface and a collection of springs with long-range interaction and a constant tensile threshold force. We show that a particular scaling similar to that of crack-tip stress in Linear Elastic Fracture Mechanics leads to a macroscopic limit behavior. The model also reproduces the scaling behaviors of the JKR model of adhesive contact. We determine how the threshold force depends on the fracture energy and elastic properties of the material. The model can be used to study rough-surface adhesion.

Motivation & Objective

• To establish a connection between microscopic threshold-force behavior in discrete springs and macroscopic mode I fracture mechanics.
• To determine the scaling of the threshold force with discretization size that yields a macroscopically size-independent response.
• To validate the model by reproducing the scaling laws of the JKR adhesion model, particularly the linear dependence of maximum tensile force on fracture energy and sphere radius.
• To derive an analytical expression for the threshold force in terms of material properties such as fracture energy, shear modulus, and Poisson's ratio.
• To enable numerical simulation of adhesive contact with long-range elastic interactions, especially for rough surfaces.

Proposed method

• Model a deformable surface as a set of discrete, normally oriented springs with long-range elastic interactions governed by the Boussinesq and Love solutions for half-space elasticity.
• Implement a tensile threshold force $F_{\text{th}}$ per spring; if exceeded, the spring detaches and force drops to zero.
• Use nondimensionalization with length scale $L^*$ and force scale $F^*$ to derive a dimensionless compliance kernel $\bar{C}_{ij}$, with $1/\bar{r}_{ij}$ for $i \neq j$ and $3.8/\bar{\Delta}$ for $i = j$.
• Apply the Fast Multipole Method to efficiently compute long-range interactions in large systems.
• Use a predictor-corrector algorithm to update spring deformations and forces under displacement-controlled loading, with dynamic contact status updates at each time step.
• Calibrate the threshold force via a scaling $F_{\text{th}} \propto \Delta^{\alpha}$, where $\Delta$ is the discretization spacing, and identify $\alpha = 0.75$ as the critical value for macroscopic invariance.

Experimental results

Research questions

• RQ1 What scaling of the threshold force with discretization size leads to a macroscopically size-independent response in a discrete spring model?
• RQ2 How does the threshold-force model reproduce the key scaling laws of the JKR adhesion model?
• RQ3 Can the model’s threshold force be related to fundamental material properties such as fracture energy and elastic moduli?
• RQ4 Does the model’s behavior align with predictions from Linear Elastic Fracture Mechanics, particularly the $1/\sqrt{r}$ stress singularity at crack tips?
• RQ5 Can the model be extended to simulate viscoelastic effects or mixed-mode fracture by modifying the interaction kernel or adding degrees of freedom?

Key findings

• The macroscopic response becomes independent of discretization size only when the threshold force scales as $\Delta^{0.75}$, matching the $1/\sqrt{r}$ singularity in Linear Elastic Fracture Mechanics.
• The model reproduces the JKR scaling: the maximum tensile force increases linearly with both fracture energy and sphere radius, confirming consistency with adhesive contact theory.
• The maximum tensile force scales as the square of the effective stress intensity factor $\bar{K}_{\text{th}}$, with $\bar{F}_{\text{max}} = 7.6 \bar{K}_{\text{th}}^2 \bar{R}$, validating the fracture mechanics analogy.
• The threshold force parameter $K_{\text{th}}$ is analytically related to material properties via $K_{\text{th}} = \pi \sqrt{\frac{3G\gamma}{7.6(1-\nu)}}$, enabling direct calibration from measurable material parameters.
• The model is robust under different interaction kernels: size independence breaks when long-range elasticity is removed or replaced with local interactions, confirming the role of elastic interaction range.
• The model can be extended to time-dependent behavior by making the kernel $C_{ij}(t)$ time-dependent, enabling simulation of viscoelastic dissipation and rate-dependent effects.