By John A Goldak,Mehdi Akhlaghi,SpringerLink (Online service)
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Additional resources for Computational welding mechanics
Indeed, in welding dissimilar metals, it may be necessary to use four octants, each with independent values of a, b and c In cases where the fusion zone differs from an ellipsoidal shape, other models should be used for the flux and power dcnsity distribution. For example, in welds with a cross-section shaped as shown in Figure 2-10, four ellipsoid quadrants can be superimposed to more accurately model such welds. Figure 2-10: Cross-sectional weld shape of the fusion zone where a double ellipsoid is used to approximate the heat source (A), compound double ellipsoids must be superimposed to more accurately model such welds.
Ns 1 n m ) Figure 2-27: Residual longitudinal stress on the top surface vs. transverse distance from the weld centerline in HY-130 weld measured by Corrigan and computed by Hibbitt & Marcal  are shown in a). Note the lack of agreement between computed and measured stresses. The longitudinal stress shown in b) was computed by Oddy et al.  for HY-80 for three cases. The top curve ignores the phase transformation. The bottom curve includes the volume change due to the transformation. The middle curve includes the effect of transformation plasticity (In steel welds during the austenite-ferrite phase transformation the variations of stress and strain on the length scale of microstructure produce an important contribution to the plastic strain called transformation plasticity).
As many researchers have shown, Rosenthal's point or line heat source models are subject to a serious error for temperatures in or near the fusion zone (FZ) and heat affected zone (HAZ). The infinite temperature at the heat source assumed in this model and the temperature sensitivity of the material thermal properties increases the error as the heat source is approached. The effect of these assumptions and others on the accuracy of temperature distributions from the Rosenthal analysis has been discussed in detail by Myers et al.