Evaluation of deflection in semi-infinite bodies by a differential method

The paper presents a procedure for evaluating the Laplacian of the deflection of a semiinfinite body subject to pressure loading using suitable quadrature expressions. Both line contact and point contact situations are considered. The validity of the treatments is verified by consideration of the Hertzian pressure distributions, and it is shown for each case that the deflection can be obtained numerically by solution of the resulting differential equation. The effect of the pressure distribution in this 'differential deflection' method is shown to be extremely localized in comparison with direct evaluation of the deflection. A companion paper clarifies how this important property can be exploited to enable a fully coupled approach to the elastohydrodynamic problem to be constructed without the need to consider the solution of the fully populated matrix that has hitherto been thought to be necessary. The technique developed in this paper thus forms the key to exploiting the acknowledged benefits of full coupling in these problems.