Crush Contact Stress: How to Calculate Pressure Between Two Touching Surfaces

Hertzian contact stress theory, developed by Heinrich Hertz, allows us to determine the stress and deformation when two curved surfaces come into contact. The theory uses a set of equations that consider material properties (like Young's modulus and Poisson's ratio), geometry (radii of curvature), and applied forces to calculate contact pressure and the size of the contact area. It assumes homogenous, isotropic materials obeying Hooke's Law, with small deformations and only normal stresses at the contact surface. However, it doesn't account for tangential forces or friction, which could be significant in some real-world scenarios.

When dealing with a sphere on a flat plane, we can still utilize the sphere-on-sphere formulas from Hertzian contact stress theory. The radius of the flat surface is considered to approach infinity, which simplifies the equations. Using this simplification, you can calculate the maximum contact stress and the radius of the contact area. The largest shear stress doesn't occur at the surface, but at a depth z = 0.47a, where 'a' is the contact area radius, and its magnitude is max τ = 0.31·σο, where σο is the maximum contact stress. Keep in mind this only accounts for normal forces and does not consider any friction.

In the scenario of a cylinder on a cylinder under Hertzian contact stress theory, the contact area forms a rectangle with width 2a and length l. The contact stress is distributed semi-circularly across this width. To calculate the maximum contact stress (σο), you'd use the formula σο = √(F*E / (2πrl(1-v^2))), where F is the applied force, E is the effective Young's modulus, r is the effective radius, l is the length of the cylinder, and v is Poisson's ratio. The half-width of the contact area (a) is calculated as a = √(8Fr(1-v^2) / (πEl)). If the cylinder has a limited length, 'q = F/l' represents a uniformly distributed linear load, which factors into the stress calculations. The theory simplifies the stress calculation, assuming the cylinders are perfectly aligned and no other forces are in play.

For curved surfaces on a flat plane, Hertzian contact stress theory predicts that the pressure distribution forms an ellipse, following an ellipsoid pattern. The maximum contact stress (σο) is calculated as σο = 1.5|F / (πab)|, where F is the applied force, and a and b are the semi-major and semi-minor axes of the elliptical contact area, respectively. Determining 'a' and 'b' often requires solving more complex equations involving the geometry of the curved surface and the material properties, which is not explicitly covered.

When materials are different, Hertzian contact stress calculations require an effective Young's modulus (E) is calculated using the individual moduli (E1 and E2) as follows: E = 2E1E2 / (E1 + E2). Poisson's ratio (v) is often taken as 0.3 for many materials. The effective Young’s modulus is an important part of the calculations. This approach assumes that both materials deform elastically and the contact area remains small compared to the radii of curvature.