Stress formation in solidifying bodies. Solidification in a round continuous casting mold
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INTRODUCTION
The development of stresses in solidifying bodies and of the related deformations is the subject of considerable practical interest. Many mathematical models have been proposed in the literature. In one-dimensional models, the solidified shell is treated as a beam or as a cylindrical ring. Two-dimensional models have been applied to the continuous casting process. Finite element computations for two-dimensional or three- dimensional problems are presented in increasing number, usually being carried out with commercially available program packages. An extensive literature survey of the many approaches used is contained in Reference 1. The treatment of the mechanical behavior of a growing body is more complex than that in the conventional problem of stress formation in a body with fixed dimensions. This difference has not been clearly recognized by most previous authors. In a solidifying body, there is no stress-free reference state as in a normal body of fixed dimensions, which is stress-free, in all of its volume elements, in the isothermal state prior to heating or prior to cooling. During solidification, new layers are continuously added to the previously solidified shell. The new layer is, in its first instant, free of stress except for the effect of metallostatic pressure. Some seconds later it may already be loaded by forces due to shrinkage, caused by the cooling and mechanical interaction with the bulk of the solid shell. Hence, the different layers KLAUS SCHWERDTFEGER, Professor, Metallurgical Engineering, and Director, is with the Institut fu¨r Allgemeine Metallurgie, Technische Universita¨t Clausthal, D-38678, Robert-Koch-Str. 42, Clausthal-Zellerfeld, Germany. MITSURU SATO, formerly Scientist, Institut fu¨r Allgemeine Metallurgie, Technische Universita¨t Clausthal, is Senior Manager, Technical Development Planning Division, Nippon Steel Corporation, 20-1 Shintomi, Futtsu-Shi Chiba, 296 Japan. KARL-HERMANN TACKE, formerly Scientist, Institut fu¨r Allgemeine Metallurgie, Technische Universita¨t Clausthal, is Head, Metallurgical Department, Max-Planck-Institut fu¨r Eisenforschung, Max-Planck-Str. D-40237 Du¨sseldorf, Germany. Manuscript submitted March 28, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS B
of the solidifying body are stress free, except for the effect of metallostatic pressure, at different times, and there is always a stress profile in the solid shell during its growth. If the strand is completely solidified and isothermal at room temperature, a stress-free state has not developed either. In fact, it is a well-established practical experience that the socalled residual stresses remain in a casting after cooling to room temperature. In the stress theory, strain components are related to ‘‘displacements’’ in the conditions of compatibility. The displacement is the distance of a specified atom from the position it had when the whole body was stress free. Since in a solidifying body a state in which the whole body is stress-free does not exist, this conventional definition of
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