Thermoelastic instability in planar solidification

doi:10.1016/0020-7403(91)90051-4

International Journal of Mechanical Sciences

Volume 33, Issue 12, 1991, Pages 945-959

https://doi.org/10.1016/0020-7403(91)90051-4 Get rights and content

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.

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Cited by (25)

Effect of coating material on the growth instability in solidification of pure metals on a coated planar mold of finite thickness
2016, International Journal of Solids and Structures
Citation Excerpt :
In the present model, however, the stress fields in the solidified shell, coating layer and mold must be determined simultaneously by using the same procedure. Readers can refer to the Ref. (Li and Barber, 1991) for more details. Eqs. (116) and (118) serve to determine residual stress function and position of the moving boundary as a function of time.
In this study, a theoretical basis of thermomechanical instability during the solidification of pure metals on a coated planar mold is presented. This study extends the previous works by taking into account the presence of a deformable coating layer on the inner mold surface. The thermal and mechanical problems are assumed to be coupled through the pressure dependent thermal contact resistances at the shell/coating and coating/mold interfaces. On the other hands, the thermal capacitance of solidified shell, coating layer and mold materials are neglected for the sake of simplicity. A linear perturbation method is used to reduce the spatial dimension of the problem. The model leads to two coupled differential equations for the shell thickness perturbation and residual stress which are solved numerically. The results document the variation of the perturbed solidification front as a function of ratio between the thermal conductivities of the shell and coating materials for combinations of other process parameters such as the combination of shell and mold materials, the values of coating thickness and coupling rates. In case of weak coupling, the maximum magnitude of this perturbation decreases regardless of the value of coating thickness and shell-mold material combinations when the thermal conductivity of the coating material is decreased. In case of strong couplings however, a critical thermal conductivity ratio between shell and coating materials which leads to a formation of greatest or lowest perturbation in the solidification front depending on the values of other process parameter has been found.
Sinusoidal perturbation solution for solidification of pure materials on a planar mold of finite thickness
2008, International Journal of Thermal Sciences
A linear perturbation method is used to solve two-dimensional heat conduction problem in which a liquid becomes solidified by heat transfer to a plane mold of finite thickness. Heat flux drawn from the lower surface of the mold is approximately uniform, but contains a small sinusoidal perturbation in one space dimension. Numerical results are obtained for the consequent sinusoidal perturbation in the solid/melt boundary as a function of time. Analytical results are obtained for the limiting case in which diffusivities of the solidified shell and the mold materials are infinitely large. These results are then compared with the numerical predictions to establish the validity of the model and the numerical approach. The results document the effects of both solidified shell and the mold material diffusivities on the growth of a perturbation in a nominally plane solidification front. It is demonstrated that the magnitude of this perturbation increases with the diffusivity of the mold, and this effect of mold diffusivity is overweighed by the diffusivity of the casting material. The influence of other process parameters such as the mold thickness, thermal contact resistance at the mold–shell interface, and thermal conductivity ratio on the growth of solidified shell thickness is also investigated in detail.
Effect of Stefan number on thermoelastic instabilities in unidirectional solidification
1994, International Journal of Mechanical Sciences
During the casting process, thermoelastic distortion of the partially solidified material affects the contact pressure at the solid/mold interface, which in turn can affect the thermal contact resistance, thus coupling the heat transfer and thermomechanical problems. This coupled system has the potential for instability. In this paper, the effect of Stefan number on the stability of unidirectional solidification is investigated, under the simplifying assumption that the solidified material is linear elastic. The Stefan number is a measure of the influence of thermal capacity on the solution and previous analyses have generally been restricted to the case of zero Stefan number, corresponding to a solidifying material of zero thermal capacity.
This generalization necessitates a numerical solution, which is here implemented using the finite difference method. However, since the growth of the perturbation is linear, the two-dimensional stability problem is reduced to two one-dimensional numerical problems which can be solved sequentially.
The results show that, in all cases, an initial sinusoidal perturbation grows to a maximum amplitude in the solidification front and then decays, the maximum being reached when the mean solidified layer thickness is about half the wavelength of the perturbation.
In general, increasing Stefan number has a stabilizing effect on the process. This effect is most noticeable in cases where the zero Stefan number approximation predicts substantial growth of an initial perturbation, e.g. where the thermal contact resistance is very sensitive to pressure.
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2017, Metallurgical and Materials Transactions B: Process Metallurgy and Materials Processing Science
Short time perturbation solution for thermoelastic instabilities in planar solidification
2012, Advanced Materials Research

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^†: Present address: ALCOA laboratories, Alcoa Center, PA 15069, U.S.A.

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Thermoelastic instability in planar solidification

Abstract

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

J. Mech. Phys. Solids

Thermal contact resistance in a vacuum environment

ASME J. Heat Transfer

The contact of nominally flat surfaces

Metal-mold interfacial heat transfer

Metall. Trans.