Hertzcontacttheorie

Hertzcontacttheorie, or Hertz contact theory, is a foundational framework in contact mechanics that describes the elastic contact between smooth, non-adhesive bodies under small deformations. Developed by Heinrich Hertz in 1881, it provides relationships among load, geometry, indentation, and material properties for contacts such as a sphere on a plane or two cylinders in contact.

For two bodies with radii R1 and R2 at the contact, the reduced radius is R* = (R1 R2)/(R1 + R2). For isotropic, linear-elastic materials with Young’s moduli E1 and E2 and Poisson’s ratios ν1 and ν2, the reduced modulus is E* = 1 / [(1 − ν1^2)/E1 + (1 − ν2^2)/E2]. The normal load F and the indentation δ are related by F = (4/3) E* sqrt(R*) δ^(3/2). The contact radius a satisfies a^2 = R* δ, and equivalently F = (4/3) E* a^3 / R*.

The contact pressure distribution is p(r) = p0 sqrt(1 − (r^2/a^2)) for 0 ≤ r ≤ a, with maximum p0 = (3F)/(2π a^2). The contact area is A = π a^2. These relations allow prediction of deformation, contact area, and pressure for various geometries, most commonly sphere-on-flat contacts.

Limitations include the assumptions of purely elastic, isotropic materials, smooth surfaces, frictionless contact, and small deformations, with no adhesion. Extensions address adhesion (JKR and DMT theories) and tangential loading (Mindlin theory), providing broader applicability in tribology and indentation testing.