(8) η π = +⎛⎝ ⎜ Equation 8 permits an estimate of the temperature variation (T) with position (x) behind the heat source. Thus, these equations can be combined to obtain an expression between the fraction liquid (fL) and distance along the centerline of the weld x (9) Gd η π This expression is useful because it permits direct estimation of both the variation in fraction liquid with location and the size of the crack-susceptible region of the mushy zone as a function of alloy parameters (To, mL, h, k, Cogd) and welding parameters (To, P). The solid + liquid region will exist where the temperature is between TL and Te, and the fraction liquid will vary from 1 at T = TL to fe at T = Te. Figure 28 shows calculated fL – x curves for the Gd-containing alloys evaluated here. The curves have all been shifted so that the reference point is taken as distance from the solid/liquid interface where fL = 1 (instead of distance from the heat source). This shift permits more direct comparison between curves of different nominal Gd concentrations. The terminal fraction eutectic values (fe) calculated to form at the edge of the mushy zone for each alloy are also noted in the figure. It should be noted that the variation in temperature with position calculated through Equation 8 is not expected to be highly accurate due to the assumptions invoked to arrive at the Rosenthal solution, most notably that the thermal properties are assumed constant with temperature and the heat flow via convection in the melt pool is ignored. However, the objective here is to evaluate the influence of nominal composition on the solid + liquid zone characteristics and resultant weldability under identical heat flow conditions. In view of this, it is only required to have an estimate of the functional form of the temperature variation with the mushy zone. As demonstrated by the good comparison between experimental and calculated values of ΔT and fe, the influence of nominal composition on mushy zone characteristics can be accurately captured with these equations. This is also confirmed by noting that the experimentally determined maximum crack lengths shown in Fig. 24 and the calculated mushy zone sizes shown in Fig. 28 are of similar size. Solidification cracks are not expected to propagate through the high temperature region of the mushy zone where the liquid fraction is high because solid-solid boundaries have not yet begun to form in these regions, but rather are constrained to a region within the mushy zone. Thus, the maximum crack length values should be less than the size of the mushy region, and this trend is certainly observed by the maximum crack lengths shown in Fig. 27 and the mushy zone sizes shown in Fig. 28. More importantly, the calculated fL – x curves, combined with the experimental weldability results, aid in the development of a more detailed understanding on the influence of Gd on the mushy zone characteristics and resultant cracking susceptibility. At the lowest Gd concentration (0.46 wt-%), the mushy zone is the largest because the solidification temperature range is the widest. In addition, much of the mushy zone is occupied by relatively low fL values. For example, the fraction liquid only varies from 0.03 at the edge of the mushy zone (at x ≈ 1.4 mm) to 0.07 at a distance of 0.5 mm behind the solid/liquid interface, (i.e., ~ 65% of the mushy zone is occupied by 0.03 < fL < 0.07). This liquid fraction range is high enough to cause moderate cracking, but too low for appreciable backfilling and healing of the cracks. The mushy zone size for the 1.01 wt-% Gd alloy is slightly reduced due to the smaller solidification temperature range. However, the higher amount of liquid present in the trailing edge of the mushy zone aggravates cracking by preventing the formation of solid/solid boundaries, and the terminal fraction liquid value of 0.07 is too low to permit healing of cracks by backfilling. This suggests that, at a terminal value of fe ≈ 0.07 and below, the liquid from the molten pool is cutoff from cracks that form in the mushy zone because the solid dendritic morphology is well developed. T T p p 2 hx ⎞⎠ ⎟ f m C T T p 2 hx L l o o p = − − + WELDING JOURNAL 43-s WELDING RESEARCH Fig. 27 — Influence of Gd concentration on the cracking susceptibility, amount of terminal eutectic constituent, and solidification temperature range in Alloy C-4. Fig. 28 — Calculated fL - x curves for various Gd- containing C-4 alloys.
Welding Journal | February 2014
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