Thermoelastic instability in planar solidification

Abstract

Abstract--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.

Effect of Stefan number on thermoelastic instabilities in unidirectional solidification

“…Their results showed that increased specific heat reduces thc sensitivity of the contact pressure to perturbations in heat flux and hence might be expected to exert a stabilizing effect on the coupled problem. This hypothesis is examined in the present paper, which extends the analysis of Li and Barber [8] to materials of finite specific heat.…”

“…As explained in the Introduction, the solidified material is assumed to be linear elastic and hence the thermoelastic stress field can be determined using the procedure developed by Li and Barber [8]. Notice, however, that in their case the simplification c = 0 permitted the temperature field to be obtained in closed form, whereas in the present problem it is determined by a numerical algorithm and is therefore defined in discrete form.…”

“…Since the analysis is closely related to that of Li and Barber [8], only the essential steps are presented in the following derivations, the reader being referred to Ref. [8] for more details.…”

“…The first order stress components corresponding to this potential are then obtained by substituting into Eqns (3) of Ref. [8] in the form e = -mf'(y, t) sin(mx);…”

“…The other assumptions of Li and Barber [8] will be retained, including that of elastic behavior of the solidified material. In this connection, we note that newly solidified material will be deposited in a state of hydrostatic stress and the stress state at any given point only deviates from this condition as solidification proceeds.…”