Three-dimensional probabilistic simulation of solidification grain structures: Application to superalloy precision casti
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I.
INTRODUCTION
THE control o f the grain structure is o f primary importance in many solidification processes. This is particularly the case in superalloy turbine blade castings, for which the grain structure has a direct influence on the high-temperature mechanical and corrosion resistances. The Bridgman-type directional solidification process has lead to a considerable improvement in the uniaxial creep and thermomechanical fatigue properties o f airfoils.l~J Its high cost and limited application, however, have motivated further improvements o f the classical "equiaxed" casting technology, where liquid metal is simply poured into a preheated mold produced by the lost w a x process. The transverse section microstructure o f an equiaxed low pressure turbine blade, designed for aeroengine application, is shown in Figure 1. Two methods may be adopted for the control o f the grain size and shape in such specimens via the optimization o f the process parameters (e.g., cluster design, gating, pouring temperature and r a t e , preheating temperature, inoculation conditions, etc.). The first method involves a large amount o f data and rules acquired from past experiments accumulated in a general database (i.e., "expert system"). Although this experiment-based approach is needed in Ch.-A. GANDIN, Graduate Student, and M. R A P P A Z , Professor, are with the D6partement des MatEriaux, E c o l e Polytechnique F6dErale de Lausanne, MX-G Ecublens, 1015 Lausanne, Switzerland. R. TINTILLIER, F o u n d r y Research Engineer, is with the D6partment Mat6riaux et Proc6d6s-Direction Technique. Soci6t6 Nationale d'Etude ct de Construction de Moteurs d'Aviation, 9 2 2 3 0 Gennevilliers, France. Manuscript submitted June 22, 1992. METAI.LURGICAL
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any case, it does not give much insight into the physical mechanisms responsible for grain structure formation in superalloy castings. In order to limit the n u m b e r o f experimental trials, the expert system route must be complemented by a theoretical approach based upon physical models describing the heterogeneous nucleation and growth o f grains during cooling o f the melt. This second approach is described in the present article. In thin wall castings, the size o f the grains is o f the order o f the specimen thickness (Figure 1). An unambiguous identification o f the grains nucleated at the mold wall (outer equiaxed region t2'31) and o f those formed in the bulk o f the liquid (equiaxed region) is thus not easy. Indeed, in thin specimens, the transition from the outer equiaxed region to the columnar zone o f castings does not necessarily occur before equiaxed grains eventually form in the bulk o f the liquid. Under such conditions, the heterogeneous nucleation o f grains at the mold wall and in the bulk o f the liquid, the grain growth, and the grain selection all take place simultaneously and the crystallographic orientation o f the grains plays a significant role. Deterministic micro-macroscopic models o f solidificationI~j couple the heat-flow equation governing th
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