Hertzian contact theory

Hertzian contact theory

<h2>Hertzian contact stress</h2> <p style="text-align: justify;">Hertzian 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 <a href="http://www.tribonet.org/wiki/elastohydrodynamic-lubrication-ehl/">Elastohydrodynamic lubrication theory</a> (this solution is known as <a href="http://www.tribonet.org/wiki/analytical-solution-of-reynolds-equation-grubins-approximation/">Grubin's solution</a>). 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].</p> 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: <ol> <li>Surface are infinitely large half-spaces.</li> <li>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)</li> <li>All the assumptions of the classical theory of elasticity apply (small strain, homogeneous material).</li> </ol> If there are only vertical forces acting on the surface, elastic deflection of the surface under applied pressure is given by the following relation: [math] \begin{eqnarray} \label{complete_sys1} 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} [/math] Here [math] u_z [/math] is the elastic deflection, [math] 1/{E'}= {1 - {\nu_1}^2}/{E_1} + {{\nu_2}^2}/{E_2} [/math] is the reduced elastic modulus, [math] {\nu_1}, {E_1},{\nu_2}, {E_2} [/math] are the Poisson's ratio and Young's modulus of the bodies, [math] p(x,y) [/math] 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: [math] \begin{eqnarray} \label{complete_sys1} p(x,y)=p_0(1-{r^2}/{a^2})^{1/2} \\ \end{eqnarray} [/math] where [math] r [/math] is the distance to the arbitrary point on the surface and  [math] a [/math] is the unknown parameter (which is called Hertz contact radius). Parameter [math] p_0 [/math] 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]: [math] \begin{eqnarray} \label{complete_sys1} u_z= \frac{\pi p_0}{4E'a}(2*{a^2} - {r^2}), r&lt;=a \\ \end{eqnarray} [/math] <h2></h2> [caption id="attachment_10034" align="aligncenter" width="930"]<img class="wp-image-10034 size-full" src="http://www.tribonet.org/wp-content/uploads/2017/10/Hertz.jpg" alt="spherical Hertz contact" width="930" height="316" /> Fig. 1. Sphere in contact with flat.[/caption] 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: [math] \begin{eqnarray} \label{complete_sys1} u_z={\delta} - \frac{r^2}{2R}, r&lt;=a \\ \end{eqnarray} [/math] where the local curvature of the sphere is approximated by the expression [math]{r^2}/{2R} \end{eqnarray} [/math]. By equating this expression to the expression for [math] u_z [/math] obtained earlier, the equations for the unknown parameters are obtained: [math] \begin{eqnarray} \label{complete_sys1} 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} [/math] where [math] F [/math] 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 <a href="http://www.tribonet.org/wiki/hertz-equations-for-elliptical-spherical-and-cylindrical-contacts/" target="_blank" rel="noopener">here</a> (this list includes solution for spherical, elliptical (point) contacts and cylindrical (line) contact). A Matlab code of Hertz solution is given <a href="http://www.tribonet.org/cmdownloads/hertz-contact-calculator/" target="_blank" rel="noopener">here</a>. The online calculators to obtain Hertz solution for a spherical (elliptical) case is given <a href="http://www.tribonet.org/online-hertz-calculator-line-contact/">here</a>, for a cylinder (line) contact case is given <a href="http://www.tribonet.org/hertz-pressure-calculator/">here</a>. Further overview of the case of contact of two spheres can be found <a href="http://www.tribonet.org/cmdownloads/contact-mechanics-overview-g-adams/" target="_blank" rel="noopener">here</a>. [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.