Partial contact between elastic surfaces with periodic profiles

The situation under consideration is that of a flat rigid plane being pressed down on an elastic body with a rough, but nominally flat, surface which takes the form of parallel ridges and troughs so that the elastic body is in a state of plane strain. The pressure distribution at the surface and the gradients of the deformed surface throughout the loading process from first contact to complete flattening can be derived directly from the real and imaginary parts of root M(x), where M(x) is a real function derived from the initial surface profile and the mean applied pressure at the interface, and x is an axis taken across the surface normal to the direction of the ridges and troughs. If the profile of the surface has a periodic form and can be represented by a Fourier series with a finite number of harmonics, then the basic form of M(x) contains a finite number of unknown parameters which are used to satisfy boundary conditions. The paper derives the basic form of M(x) and discusses the use of boundary conditions to complete the solution. The resulting solutions are exact and will enable many partial contact problems involving rough surfaces to be studied accurately and in detail.