Solving transient problems in soft Elasto-Hydrodynamic lubrication

Many important problems in elasto-hydrodynamic lubrication arise when one of the contacting bodies is elastically soft. An example is the squeezing of a thin liquid film between a rigid sphere/ cylinder and a soft elastic substrate, in which the sphere may not be sliding and be subjected only to normal loads (stationary contact), or it can be sheared at the same time and undergo steady or non-steady sliding. These problems are notoriously difficult to solve for soft solids under large loads and thin lubricant films. For this reason, indentation and sliding problems have usually been analyzed separately, and solutions for transient sliding problems are few. Here we develop a numerical method that avoids difficult-to-converge iterative methods and is able to solve both normal indentation and/or transient sliding lubrication problems. The scheme is fully automatic, stable, and efficient and requires only a linear matrix equation to be solved at every time step. We present solutions of two transient sliding problems: in the first of which, a rigid sphere undergoes transient normal lubricated contact followed by transient sliding on an elastic half space. In the second, a rigid cylinder undergoes transient lubricated sliding on an elastic foundation.