Modeling the fluid-flow-induced stress and collapse in a dendritic network
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I.
INTRODUCTION
THE mechanical stability of dendrites has been the focus of many investigations over the past decades. Theories of nucleation have been proposed based on the assumption that thermal or mechanical effects can cause rupture of the dendrite arms.[1,2] Most studies and theories so far have been directed toward single dendrites or dendrite arms, but the mechanical rigidity of the dendritic network in the later stages of solidification is also of importance for a full understanding of solidification and casting. In cases where equiaxed dendrites develop and grow, a coherent dendritic network is usually established at solid fractions between 10 and 30 pct.[3,4] The dendrite coherency point therefore indicates the point from which strength starts to develop and also where stresses are exchanged between the dendrites. A stress-strength relationship was introduced to account for hot tearing that develops in the very final stages of solidification when tensile stresses and strains of thermal and mechanical origin exceed the strength of a dendritic network with low melting films on the grain boundaries.[5] From as early as 1936, researchers, such as Vero[6] and Bishop et al.,[7] started to develop tensile testing equipment for testing of the mechanical properties of partially solidified alloys. More recently, Dahle and Arnberg[8,9] reported results of measurements of the development of shear strength during solidification of some common aluminum alloys as a function of morphology and fraction solid. These results were later related to the porosity distribution observed in plate castings of the same alloys by Dahle et al.[10] Strength started to develop at the dendrite coherency point and the maximum measurable strength in this apparatus was 50 kPa, which was commonly reached at solid fractions between 35 and 90 pct depending on the alloy. Campbell[11,12]
introduced the concept of burst feeding as a possible mechanism to facilitate mass transport in a solidifying, shrinking, metal alloy. Burst feeding involves the collapse of a dendritic network on a microscopic or macroscopic scale. In a ‘‘macroscopic’’ burst, occurring at lower solid fractions, a collapsed network can be compacted as depicted in Reference 10, yielding pockets of porosity under certain conditions. The concept of burst feeding has not received much attention since it was introduced, although it may be important in understanding porosity formation in castings.[10] During solidification, the interdendritic permeability decreases with increasing solid fraction,[13] consequently implying an increasing pressure gradient within the mushy zone. The pressure gradient and flow of the liquid means that stresses are created in the solid part of the mushy zone as well. The solidification shrinkage, the thermal contraction of the solid, the dendrite-dendrite interactions in the dendritic network, and the governing pressure differential for flow cause development of larger stresses in the solid as the flow resistance (and fraction solid) increases. Pilling
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