Hertz contact theory

Hertz contact of two spheres
Fig. 3. Contact of two spheres
Tribology Wikipedia > Hertz contact theory

Hertz contact theory is a classical theory of contact mechanics and is a very useful tool for engineers and researchers. Even though the derivation of the theory is relatively difficult, the final solution is a set of simple analytical equations relating the properties of the system to the developed stress. Hertz theory was also successfully applied to get a first analytical solution of Elastohydrodynamic lubrication theory (this solution is known as Grubin’s solution). Here, the main equations of the theory are considered, while the full derivation and the description can be found in the classical contact mechanics books [1,2].

Hertz contact theory is derived from the analytical solution of elasticity theory equations (as discussed by Timoshenko and Goodier in [2]) under half-space approximation:

  1. Surface are infinitely large half-spaces.
  2. Pressure profile is parabolic (which assumes that the shape of the bodies in contact can also be approximated well with parabolic shapes, e.g., sphere, ellipse or a cylinder)
  3. All the assumptions of the classical theory of elasticity apply (small strain, homogeneous material).

If there are only vertical forces acting on the surface, elastic deflection of the surface under applied pressure is given by the following relation:

(1)    \begin{eqnarray*} u_z(x,y) = \frac{2\pi}{E'} \int\int \frac{p(x',y')}{\sqrt{(x-x')^2+(y-y')^2}}dx'dy' \\ \end{eqnarray*}

Here  u_z is the elastic deflection,  1/{E'}= {1 - {\nu_1}^2}/{E_1} + {{\nu_2}^2}/{E_2} is the reduced elastic modulus,  {\nu_1}, {E_1},{\nu_2}, {E_2} are the Poisson’s ratio and Young’s modulus of the bodies,  p(x,y) is the contact pressure.

If the pressure profile is arbitrary, this equation does not lead to the analytical solution. However, Hertz solution is obtained under the assumption of a parabolic pressure distribution, which is a very good approximation for spherical,elliptical or cylindrical bodies in contact:

(2)    \begin{eqnarray*} p(x,y)=p_0(1-{r^2}/{a^2})^{1/2} \\ \end{eqnarray*}

where  r is the distance to the arbitrary point on the surface and   a is the unknown parameter (which is called Hertz contact radius). Parameter  p_0 is also unknown (it is called maximum Hertz pressure). Substituting this into the equation for deflection leads to the following expression for Hertzian pressure  [3]:

(3)    \begin{eqnarray*} u_z= \frac{\pi p_0}{4E'a}(2*{a^2} - {r^2}), r<=a \\ \end{eqnarray*}

spherical Hertz contact
Fig. 1. Sphere in contact with flat.

For a rigid sphere penetrating an elastic half-space as shown in Fig.1, the elastic deformation of the initially flat surface within the contact is given by the following equation:

(4)    \begin{eqnarray*} u_z={\delta} - \frac{r^2}{2R}, r<=a \\ \end{eqnarray*}

where the local curvature of the sphere is approximated by the expression

{r^2}/{2R} \end{eqnarray} .

By equating this expression to the expression for  u_z obtained earlier, the equations for the unknown parameters are obtained:

(5)    \begin{eqnarray*} a= \frac{\pi p_0R}{2E'} \\ \delta = \frac{\pi ap_0}{2E'} \\ p_0= \frac{2}{\pi}E'{\delta/R}^{1/2}\\ F = \frac{4}{3}E'{R}^{1/2}{\delta}^{1/2}\\ \end{eqnarray*}

where  F is the applied load.

Hertz theory briefly described is applicable for the case of spherical, cylindrical and elliptical contacts. List of all expressions of the Hertz contact theory is given here (this list includes solution for spherical, elliptical (point) contacts and cylindrical (line) contact).

A Matlab code of Hertz solution is given here.

The online calculators to obtain Hertz solution for a spherical (elliptical) case is given here, for a cylinder (line) contact case is given here.

[1] Contact Mechanics, K. Johnson, http://www.ewp.rpi.edu/hartford/~ernesto/S2015/FWLM/Books_Links/Books/Johnson-CONTACTMECHANICS.pdf

[2] Theory of Elasticity, S.P. Timoshenko, J.N. Goodier, https://engineering.purdue.edu/~ce597m/Handouts/Theory%20of%20elasticity%20by%20Timoshenko%20and%20Goodier.pdf

[3] Contact Mechanics and Friction, V. Popov.

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