The singly periodic beam theory of freezing front growth instability in pure metal castings due to Richmond et al., (1990) is extended here to the more general case of a doubly periodic plate. The casting is cooled with a doubly periodic heat flux which oscillates along two orthogonal axes of the mold/casting interface. The cooling profile induces nonuniform growth of the freezing front and hence a strain distribution in the casting that leads to contraction of the metal at specific locations along the mold interface. Metal contraction is studied through calculation of the interface pressure. Growth instability occurs when the pressure falls to zero beneath a thickness minimum, thereby signalling the nucleation of an air gap, and simultaneously increases beneath a thickness maximum thereby enhancing the initial nonuniformity of the freezing front. The significant differences between the beam and plate theories for a given pure metal are found in the more extreme stress levels which accumulate within the plate, the magnitude of the admissible uniform cooling term and hence predicted times to air gap nucleation, and casting thickness at air gap nucleation as well as interface pressures beneath thickness maxima. The doubly periodic plate theory permits larger uniform cooling and hence earlier air gap nucleation beneath thickness minima than does the singly periodic beam theory with more severe interface pressures beneath thickness maxima.
Growth Instability During Nonuniform Directional Solidification of a Pure Metal Plate
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Hector, L. G., Jr., Aksel, B., and Fridy, J. (June 1, 1991). "Growth Instability During Nonuniform Directional Solidification of a Pure Metal Plate." ASME. J. Appl. Mech. June 1991; 58(2): 326–333. https://doi.org/10.1115/1.2897189
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