Theoretical methodology

A theoretical model of the gap nucleation process during pure metal solidification on a deformable mold of finite thickness is presented. Both surfaces of the mold follow a sinusoidal lay for which the ratio of the amplitude to the wavelength, or aspect ratio, is much less than one. This makes the aspect ratio a convenient perturbation parameter for the thermal and mechanical problems since it is indicative of the spatial variation in the surfaces. The thermal and mechanical fields are coupled along the upper surface of the mold through a pressure-dependent thermal contact resistance. The main goal of the model is to develop a means for examining the contact pressure along the mold-shell interface and how variation of the mold surface wavelength affects the time and location of gap nucleation. Gaps, which result from irregular distortion of the shell due to the modest variation of the mold surface geometry, are assumed to nucleate when the contact pressure locally falls to zero. The model leads to two coupled differential equations for the shell thickness and contact pressure perturbations which are solved with a numerical scheme. Using a series solution methodology, it is shown that the contact pressure perturbation predicted by the present model reduces to that for a rigid, perfectly conducting mold (which was considered in another work) in the limit of zero mold thickness. In the companion paper, we specifically examine various combinations of pure materials acting either as the shell or the mold material. The concept of a critical wavelength, which separates those wavelengths that lead to gap nucleation at the crests, from those that lead to gap nucleation at the troughs, is introduced. The potential for development of design criteria for mold surface topographies using the present theoretical model as a limiting solution for finite element models of more complex casting processes is discussed. [S0021-8936(00)03201-3]