Thermoelastic Instability in Planar Solidification

-A linear perturbation method is used to examine the stability of a unidirectional solidification problem in which a liquid, initially at the melting temperature, becomes solidified by heat transfer across a pressure-dependent thermal contact resistance to a plane mold. The contact pressure will be influenced by thermal distortion in response to the instantaneous temperature field in the solidified shell. The heat transfer and thermal stress problems are therefore coupled through the boundary conditions. The temperature and stress fields are assumed to consist of a unidirectional component and a small spatially-sinusoidal perturbation which can vary with time. Analysis of the thermoelastic problem for the solidified shell leads to an ordinary differential equation relating the perturbation in heat flux at the mold/casting interface and the corresponding perturbation in contact pressure. A second equation relating the same two variables is obtained by linear perturbation of the relation for heat conduction across the thermal contact resistance. These are then reduced to a single equation which is solved numerically. The results show that a small initial perturbation will grow substantially during the solidification process if the thermal contact resistance is very sensitive to pressure. 1. I N T R O D U C T I O N During the casting process, an initially liquid mass of material is caused to solidify by reducing its temperature below the melting point by heat transfer from its surfaces, which are in contact with a surrounding mold. There will be some thermal contact resistance at the mold/solid interface, since the surfaces are rough on the microscopic scale and generally carry adherent contaminant and oxide films. This thermal contact resistance has formed the subject of many investigations, both experimental [1, 2] and theoretical [3, 4] and it is generally agreed that it is very sensitive to contact pressure, principally because, being rough, the surfaces make contact only at the peaks of microscopic asperities and the number and size of these "actual contact areas" increases with load [5]. Initially, the contact pressure at the mold/casting interface will be determined by the hydrostatic pressure in the liquid, but as solidification proceeds, the temperature gradient through the solidified shell will induce thermoelastic distortion and hence influence the contact pressure. Ho and Pehlke [6, 7] deduced values of thermal contact conductance from temperature measurements during solidification and found that the conductance falls during the process. This effect is most probably attributable to shrinkage of the casting, resulting in a reduced contact pressure at the casting mold interface. If the initial hydrostatic pressure is insufficiently high, an air gap may develop at this interface, causing a substantial reduction in interface conductance I-6, 8]. The heat transfer and thermal stress problems are therefore coupled through the boundary conditions. This coupled process is potentially unstable and is cited by Richmond and Huang [8] as a possible explanation for the long-wavelength perturbations which are sometimes observed in the nominally planar solidification front in unidirectional solidification [9, 10]. Such perturbations are undesirable in the manufacturing process since the associated non-uniform thermal fields can cause cracks to develop during solidification [11]. Richmond et al. 1-12] recently developed an idealized analysis of this problem, in which a constant heat flux across the interface is taken to have a small prescribed spatiallysinusoidal perturbation. They found that the resulting thermal distortion increased the contact pressure in the regions where the heat flux is greatest. This indicates that the t Present address: ALCOA laboratories, Alcoa Center, PA 15069, U.S.A.