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Tribology & Contact Mechanics

JKR Adhesive Contact Simulator — Johnson-Kendall-Roberts Theory

Real-time computation of how surface energy γ enlarges the contact radius of soft elastic spheres and creates a pull-off force. Compare with Hertz to see adhesion at a glance.

Parameters
Sphere radius R
mm
Equivalent modulus E*
MPa
Surface energy γ
mJ/m²
Normal load F (negative = tension)
μN

Defaults model soft rubber (E*≈10 MPa) with a typical surface energy. Set F<0 to observe adhesive holding under tensile load.

Results
—
Hertz contact radius a_H (no adhesion)
—
JKR contact radius a_J (with adhesion)
—
Pull-off force F_pull-off = -3πRγ/2
—
a_J/a_H ratio (adhesion amplification)
Contact schematic + radius-vs-load curve

Left = sphere-plane adhesive contact (red line = JKR contact radius). Right = Hertz (blue) vs JKR (red), F_po dashed, yellow dot = current point.

Theory & Key Formulas

JKR theory accounts for adhesion (surface energy γ) between soft elastic spheres. The contact area is wider than Hertz and contact persists under tensile load.

Hertz contact radius (classical, no adhesion):

$$a_H^3 = \frac{3FR}{4E^*}$$

JKR contact radius (with adhesion):

$$a_J^3 = \frac{3R}{4E^*}\!\left[F + 3\pi R\gamma + \sqrt{6\pi R\gamma F + (3\pi R\gamma)^2}\right]$$

Pull-off force (minimum tension at which contact breaks):

$$F_\text{po} = -\frac{3}{2}\pi R\gamma$$

Contact radius at pull-off:

$$a_\text{po} = \left(\frac{9\pi R^2 \gamma}{8 E^*}\right)^{1/3}$$

Units: γ [N/m] = [J/m²]. Adhesion is most visible for soft materials (small E*), large γ and large R.

What is the JKR Adhesive Contact Simulator?

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Geckos can really walk on smooth walls without any suction cup or glue. How do they actually stick?
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That's a textbook case of JKR adhesion. Each gecko toe has millions of tiny hairs (setae), and every hair sticks to the surface by molecular-scale attraction. Hertz contact assumes no adhesion, but for soft and small objects the surface energy γ matters: contact happens even without pushing. In the simulator, push γ up to about 200 and watch the "JKR contact radius" stat grow noticeably.
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Wait — you can set the load F to a negative value? That means it's still stuck while being pulled?
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Exactly. With adhesion the contact doesn't break right away under tension. The minimum tensile force needed to detach is the pull-off force $F_\text{po}=-3\pi R\gamma/2$. Try F = -300 or -400 μN in the simulator — the "JKR contact radius" is still finite. Push below F_po and the contact physically separates, so the stat shows "—".
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Does the same adhesion effect work for hard steel balls too?
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That's the catch — JKR is meant for soft, large spheres. With large E* (metals or ceramics) a little adhesion can't overcome the elastic energy, so the result collapses back onto Hertz. The Tabor parameter μ_T formalises this, but as a rule of thumb: "rubber, gel, PDMS → JKR; steel, sapphire → Hertz". That's why this tool's E* range stops at 1000 MPa.
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On the right chart the blue (Hertz) and red (JKR) curves seem to merge as F gets bigger. Is that adhesion fading?
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Good catch. Once F is much larger than πRγ the adhesion term inside the JKR formula becomes negligible and the two predictions coincide. The gap is largest when F is small or negative, which is exactly when adhesion dominates. Practical rule: JKR matters for weak loads, small R and soft materials.

FAQ

JKR applies to soft, large spheres (rubber, gels) where adhesion is concentrated inside the contact zone. DMT (Derjaguin-Muller-Toporov) applies to stiff, small spheres and includes adhesion outside the contact area. Selection uses the Tabor parameter μ_T=(R·γ²/(E*²·z₀³))^(1/3): μ_T>5 favours JKR, μ_T<0.1 favours DMT, and Maugis-Dugdale bridges the intermediate range. This tool assumes the JKR regime.
AFM force curves give surface energy from the pull-off force F_po, but typical tip radii of 10-100 nm push the contact into the DMT regime (small Tabor parameter). For soft samples (PDMS, biological tissue, hydrogels) μ_T grows and JKR applies. Setting R=0.01 mm in this tool approaches AFM conditions. For purely hard samples use hertz-contact.html instead.
Yes, biological adhesion is a textbook JKR application. Gecko toes have millions of setae adhering via van der Waals forces (γ ≈ 30-50 mJ/m²). The JKR formula F_po=3πRγ/2 gives a tiny per-seta force, but the sheer count supports the whole body. Set R≈1 μm (i.e. 0.001 mm) and γ=40 in this tool to estimate the per-contact pull-off. For viscoelastic structural adhesives see adhesive-joint.html.
Usually no. When E* exceeds 1 GPa adhesion becomes negligible and Hertz contact is sufficient. This tool restricts E* to 0.1-1000 MPa for soft matter (rubber, gel, PDMS ≈ a few MPa). Use hertz-contact.html for metal contact and adhesive-joint.html for structural adhesive viscoelasticity. Adhesion dominates roughly when R·γ/(E*·a²) ≥ 0.01.

Real-World Applications

Biological adhesion (geckos, flies, frogs): Setae on gecko feet and toe pads on insects rely on JKR-style adhesion to climb smooth walls. The per-seta force is tiny, but millions of contacts add up. Biomimetic "gecko tapes" are designed directly with JKR formulas.

Soft robotics and micro-grippers: PDMS grippers handling MEMS or microchips switch between pick-up and release by tuning adhesion. JKR predicts the required contact radius and tensile force as functions of γ (surface treatment) and R (tip radius).

Tyre-road grip at low load: Rubber-asphalt micro-contact has a JKR-like adhesive component at low load. Plastic deformation and hysteresis dominate under high speed, but adhesion is non-negligible on wet or smooth surfaces.

AFM and nano-indentation: Extracting γ or E from force curves needs the right model among JKR, DMT or Maugis-Dugdale. Soft matter is fitted with JKR, while hard tips with small R fall into the DMT regime.

Common Misconceptions & Pitfalls

The most common misconception is to confuse JKR adhesion with industrial glue. JKR concerns molecular-scale surface forces (van der Waals, hydrogen bonding) with γ in the 20-200 mJ/m² range. Structural epoxy adhesives reach MPa-level peel strengths — three or more orders of magnitude higher. This tool predicts how adhesion changes the contact shape, not the bond strength of glues, which requires fracture-mechanics or viscoelastic models.

The second pitfall is assuming the pull-off force depends on E*. It doesn't: $F_\text{po}=-3\pi R\gamma/2$ has no E term. A softer or stiffer sphere with the same R and γ requires exactly the same tensile force to detach. Soft materials simply spread the contact wider; the detachment force itself is invariant. Try sweeping E* from 10 to 1000 MPa in the simulator — F_po stays put while a_J shrinks.

Third, do not apply JKR to hard contacts. With E* of several GPa the Tabor parameter is small and DMT or Hertz becomes appropriate. Set E*=1000 MPa here and the a_J/a_H ratio approaches 1: adhesion has vanished physically, not because the formula broke. JKR is at its best for rubber, gel, PDMS, biological tissue and hydrogels in the E*=0.1-100 MPa range.

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