Hertz Sphere Contact

Mathematical Formulation

When an elastic sphere indents a flat surface by a small depth delta, a circular contact patch forms. The contact radius a and the resulting normal force F depend on the geometry and the effective elastic modulus E*. The formulas are derived assuming small strains, frictionless surfaces, and that the contact radius is much smaller than the sphere radius (half-space assumption).

The key equations governing this interaction are:

• Normal Force: F = (4/3) E* sqrt(R) delta^(3/2)
• Contact Radius: a = sqrt(R delta)
• Peak Pressure: p0 = 3F / (2pi a^2) = (2E* / pi) sqrt(delta / R)
• Tangent Stiffness: kt = dF/ddelta = 2 E* sqrt(R delta)
• Stored Energy: U = (2/5) F delta

At delta = 0, all outputs continuously limit to zero. Notably, the force increases proportionally to delta^(3/2), indicating a stiffening effect as the contact area grows, unlike a linear spring where force is directly proportional to displacement.

Worked Example

Let us calculate the parameters for a specific case: an effective modulus E* = 10 GPa, a sphere radius R = 4 mm, and an indentation delta = 4 micrometers.

1. Convert to SI Units:
   E* = 10 x 10^9 Pa
   R = 4 x 10^-3 m
   delta = 4 x 10^-6 m
2. Calculate Contact Radius (a):
   a = sqrt(4e-3 * 4e-6) = sqrt(16e-9) = 126.491 micrometers
3. Calculate Normal Force (F):
   F = (4/3) * (10e9) * sqrt(4e-3) * (4e-6)^(3/2)
   F = 6.746 N
4. Calculate Peak Pressure (p0):
   p0 = (3 * 6.74619) / (2 * pi * (126.49e-6)^2) = 201.317 MPa
5. Calculate Tangent Stiffness (kt):
   kt = 2 * (10e9) * sqrt(4e-3 * 4e-6) = 2.529 x 10^6 N/m
6. Calculate Stored Energy (U):
   U = (2/5) * 6.746 * 4e-6 = 1.079 x 10^-5 J

Experiment

Use the Hertz Sphere Contact module to observe how the contact mechanics evolve.

1. Start with the default settings: Effective modulus of 10 GPa, Sphere radius of 4 mm, and Indentation of 4 micrometers. Confirm the calculated values match the worked example above.
2. Observe the visual representation. The red line represents the contact radius, and the shaded red region illustrates the semi-elliptical pressure distribution. The force-indentation graph on the left shows your current state on the non-linear curve.
3. Change the Indentation micrometers to 9. Notice how the normal force and contact radius increase, and observe the point moving along the stiffening force-indentation curve.
4. Change the Effective modulus GPa to 100. Observe the significant increase in normal force and peak pressure required to maintain the same indentation into a stiffer material.

Limitations

This module assumes ideal linear elastic behavior and small strains, which means it does not account for plastic deformation or material yielding that would likely occur at high peak pressures. It also assumes frictionless, non-adhesive contact between perfectly smooth surfaces. The half-space assumption requires the contact radius to be much smaller than the sphere radius (here, the largest delta/R ratio is 0.009, which is suitably small).

References

• MIT FUNdaMENTALs Topic 9: Contact Mechanics

These resources provide the mathematical derivation and physical context of Hertzian contact, including the stiffness calculation and small-strain limitations.