href="#ulink_b67272e2-fc08-50be-812d-d103e8c5cbcb">Eq. (2.9) using k and α of the workpiece material (e.g. steel).
Step 3. Repeat Steps 1 and 2 for x = −1, 0, 1, 2, 3, 4, 5 cm.
Step 4. Sketch the temperature distribution T (x, 4, 0).
Step 5. Convert temperature distribution T (x, 4, 0) into thermal cycle T (t) by dividing x by V.
Figure 2.19 explains how to convert the calculated temperature distribution in Figure 2.18 to a thermal cycle. It is assumed that the workpiece in Figure 2.18 is long enough such that, with respect to the moving coordinate system, the temperature field does not change. The T – x plot, i.e. the temperature distribution along the welding direction, can be converted into T – t plot, namely, the thermal cycle, by calculating time t using t = (x − 0)/V. For instance, assume the travel speed of the heat source is V = 4 mm/s. At the point x = 2 cm, y = 4 cm, and z = 0 cm, t = (20 mm − 0)/(4 mm/s) = 5 s.
Figure 2.19 Converting the calculated temperature distribution in Figure 2.18 to thermal cycle, assuming the temperature field is steady with respect to the moving coordinate system.
The shape of the weld pool can be calculated such as at the top surface of the weld pool, i.e. at z = 0 cm. Let T L be the liquidus temperature of the workpiece material, e.g. 1530 °C for steel. The liquidus temperature of an alloy is equivalent to the melting point of pure metal T m, above which the workpiece material is melted completely. The procedure for calculating pool shape is illustrated using Eq. (2.9) for 3D heat flow and V = 2.4 mm/s, Q = 3200 W and T o = 25 °C.
Step 1. Let T = T L in Eq. (2.9). Calculate R at x = 0 cm and z = 0 cm from Eq. (2.9) by substituting into the equation the values of T o , V, Q, and the physical properties of the workpiece materials (e.g. steel) k and α.
Step 2. Find the value of y from R = (02 + y 2 + 02)1/2.
Step 3. Repeat steps 1 and 2 for x = 1 cm, 2 cm, etc. Use the calculated y values to construct the pool shape at the workpiece surface.
Let T H be the temperature at which solid‐state phase transformation occurs (e.g. 780 °C). The shape of the HAZ can be calculated, such as at the top surface of the weld pool, i.e. at z = 0 cm, following the same procedure for calculating the pool shape except that T = T H in Step 1. The distance in y‐direction between the weld pool shape (isotherm T L) and the HAZ shape (isotherm T H) is the width of the HAZ.
2.2.3 Adams' Equations
Adams [28] derived the following equations for calculating the peak temperature T p at the workpiece surface (z = 0) at a distance Y away from the fusion line (measured along the normal direction):
(2.10)
for two‐dimensional heat flow and
(2.11)
for three‐dimensional heat flow. Several other analytical solutions have also been derived for two‐dimensional [29–34] and three‐dimensional [31,35–37] welding heat flow.
2.3 Effect of Welding Conditions
Results of heat flow calculated by Rosenthal's 3D equation are shown in Figures 2.20 and 2.21. They show the calculated thermal cycles and temperature distributions at the top surface (z = 0) of a thick plate of 1018 steel for two different sets of heat input and welding speed. The pool boundary is assumed to be at the liquidus temperature T L = 1530 °C. The solid‐state transformation temperature responsible for the formation of the HAZ at T H = 780 °C. The eutectoid temperature 727 °C of the binary Fe‐C phase diagram might have been a better choice for 1018 steel. However, since the isotherm at 780 °C has already been calculated, it will be taken as T H as an approximation just for the purpose of illustration.
Figure 2.20 Calculated Rosenthal's three‐dimensional heat flow in 1018 steel: (a) thermal cycles; (b) isotherms. Welding speed: 2.4 mm/s; heat input: 3200 W; material: 1018 steel; liquidus temperature: 1530 °C; solid‐state transformation temperature: 780 °C.
Figure 2.21 Similar to Figure 2.20 but with faster welding speed of 6.2 mm/s and higher heat input of 5000 W, resulting in faster cooling rate and more elongated weld pool.
As can be seen in Figures 2.20a and 2.21a, the peak temperature of the thermal cycle at the centerline of the weld surface (i.e., at y = z = 0) is infinite. This error is caused by the singularity problem in Rosenthal's equations, that is, caused by the assumption of a point heat source at x = y = z = 0 (the origin of the coordinate system). Thus, near the center of the heat source (the origin), Rosenthal's equations are not reliable. Away from it, however, they can be more accurate. Because of their simplicity, Rosenthal's equations have been used despite the error.
In Figure 2.20 Q = 3200 W and V = 2.4 mm/s, and the Q/V ratio is 1333 J/mm. At the top surface of the workpiece, the weld pool is 9.5 mm wide and 12 mm long, and the length/width ratio is 1.26. In Figure 2.21 the power input is increased to Q = 5000 W and the travel speed increased to V = 6.2 mm/s. The Q/V ratio is 806 J/mm. At the top surface of the workpiece, the weld pool is 9.0 mm wide and 16 mm long, and the length/width ratio is 1.78.
