Fibrillar Adhesion

What is modelled

A gecko-style pad is covered in fine fibrils, each ending in a small tip that sticks by van der Waals attraction. One tip is modelled as a sphere of radius R on a flat wall with the Johnson-Kendall-Roberts (JKR) contact; the pad is a large number of identical tips; and the whole pad is peeled from one end like a strip of tape.

Equations

• JKR pull-off of one tip Fc = (3/2) π R W, W the work of adhesion.
• JKR contact radius a³ = (3R / 4E*) [F + 3πW R + √(6πW R F + (3πW R)²)]; at zero load a&sub0;³ = 9πW R² / 2E*.
• Contact splitting: with a fixed pad area A and tip fill fraction f, the tip count is N = f A / (π R²), so Ftotal = N Fc = 3 f A W / (2R): finer tips, stronger pad.
• Kendall peel F = b Wapp / (1 − cosθ) with pad width b and apparent peel energy W_app = f W K, where K is an illustrative amplification for the energy spent bending and stretching fibrils (without it the bare van der Waals energy would predict peel forces far below real ones).

Checks run in the page

• The JKR pull-off found by bisection against 3/2 π R W.
• Contact splitting: N F_c against 3 f A W / 2R, and the doubling when R is halved.
• The Kendall peel energy balance F (1 − cosθ) / b against the apparent peel energy.
• The JKR contact radius at zero load against the closed form.

Honest simplifications

All tips are assumed identical, perfectly aligned and equally loaded, with no tip-tip contact, no roughness, no humidity effect and no shear or friction mode. The work of adhesion, modulus and fill fraction are generic values rather than measurements of a real animal or product, and the model predicts neither real-world adhesion nor repeated-use durability.

• Choose Coarse fibrils: the same pad area now grips about ten times less.
• Choose Steep peel and see why a toe is peeled from its edge, not pulled straight off.
• Raise the animal mass until the safety factor drops under 1 and the pad lets go.