Figure 5-17 shows the behavior of four constitutive models. Their properties are shown in Tables 5-2 and 5-3. Two models use rate independent plasticity, one uses rate dependent plasticity and one uses linear viscous plasticity. The test is uniaxial and the strain rate is 0.005 5_1in the time interval [0, 4] seconds. The strain rate is zero in the time interval [4, 10]. The strain rate is -0,005 s 1 in the time interval [10, 13]. The strain rate is zero in the time interval [13, 25]. Note that in the rate independent models there is no stress relaxation. In the rate dependent model the stress relaxes toward the deformation resistance. In the linear viscous model, the stress decays toward zero. The viscosity used in Figure 5-17 is larger than that used in the welding test. If the viscosity was not increased, the stress would appear to relax instantly in the time scale used in Figure 5-17.

Time (sec)

Stress (MPa) Displacement (mm)

Time (sec)

Figure 5-17: The response of two rate independent materials (RIP), a rate dependent material (RDP) and a linear viscous material (LVP) to sequence of strain rates is shown [36].

Table 5-2: Material properties for rate independent plasticity (RIP) and linear viscous plasticity (LVP) RIP-1 RIP-2 LVP Young’s Modulus (GPa) 200 200 200 Poisson Ratio 0.3 0.3 0.3 Hardening Modulus (GPa) 2 2 0 Yield Stress (MPa) 200 100 — Viscosity (GPa. s) -- - 200

Table 5-3: Material properties for rate dependent plasticity (RDP) Material Parameter Value Material Parameter Value A( - S-1) 6.346* 1011 3.25 AG (J/mole) 3.1235xl05 m 0.1956 s (Pa) 1.251xl08 n 0.06869 К (Pa) 3.0931х 109 1 1.5 a(m) 3.50x 10~10

In the weld analyses a prescribed temperature weld heat source was used [43] to model the arc.

Three and ten seconds after starting the weld Figures 5-18, 5-19,

5- 20 and 5-21 show the temperature distribution, sub-domains for each constitutive equation, effective plastic strain, pressure and several stress components. Plate dimensions 5 *5*0.2 cm. Weld pool dimensions 1.5 (long) *1.5 (wide) *0.5 (deep) cm. Welding speed 6 ipm (0.25 cm/s). Maximum temperature was 2000 °С and solidus temperature was 1500°C. A 15*15*1 element mesh was used. Although the mesh is relatively coarse, it does demonstrate the expected behavior. The high compressive ridge in front of the weld is in the rate independent plasticity sub-domain. It is possible that the yield strength used should be corrected for this strain rate. The longitudinal strain pulling material back from the weld pool is another interesting feature.

Other cases were also analyzed. They differed only in weld speed. The slow weld speed was 6 ipm and the fast speed was 20 ipm (0.25 and 0.83 cm/s), Figure 5-22. In the 20 ipm weld the mesh is not fine enough to resolve the thermal shock and the constitutive equations change directly from rate independent to linear viscous. The longitudinal strain pulling material back from the weld pool is an interesting feature.

Due to a slight change in mesh distribution on the left and right side of the weld (element row 8 or 7) the temperature field on the left and right side is not symmetric. It is shown that it causes no problems in the analysis by switching between constitutive equations.

Т*!ПрЄГОІІІ>« ГС) A-42 В • 200 с /во Г) 1*М	ConMihsl^vft Мгуірі ■Ц Lineai Viscous flH Пвіг 0»-;>огчівііі L_JH. iv, ІчЗитгчІнпг
ENoctive Ида be Strain	Pressure (Pa)
А -0.05
8-0.2	-Эе»ле
С -0.5	■L *96*07
0-07	F^T ^
	1___ L - »**oe yfs.
^ * /
г)	Y і /
Х (jfj у 1 J 1	V ^ /
лЧ / / і 1	- /

Figure 5-18: At three seconds after starting the weld, the transient temperature, constitutive equation type, effective plastic strain and pressure are shown.

Figure 5-19: At three seconds after starting the weld, (Jxx, <7yy, Gxy and <7yz are shown.

Eflectrve Plastic Strain A -0.1 3-0.6 С -1.0

Pressure (Pa)

Figure 5-20: At ten seconds after starting the weld, the transient temperature, constitutive equation type, effective plastic strain and pressure are shown.

Figure 5-21: At ten seconds after starting the weld, <7^,(7 ,<X shown.

Constitutive Model

Constitutive Model

I Linear Viscous Ram Dependent [pat* Independent

Pressure (Pa)

2e+08

I Linear Viscous Rale Cepervdent I Rate Independent

»6e*0/

Figure 5-22: At ten seconds after starting the weld, the transient temperature, constitutive equation type, effective plastic strain and pressure are shown for the 6 and 20 ipm welding speeds.

The weld direction is the ^-direction, the upward normal on the plate is the z-direction and the x-direction is transverse to the weld. Figure 5-23 shows the displacements transverse to the weld at various distances from one node, (1/3 cm) from the weld centerline extends from nodes 5 to 9 in the slow weld and from nodes 3 to 15 in the fast weld.

Distance (node) 20 ipm

Figure 5-23: The abscissas are nodes numbered from left to right. The distance between nodes is 1/3 cm. When the weld arc reaches the center of the plate, the weld pool is centered at node 8 and extends from node 5.5 to node 10.5 on the plate. At this instant the transverse displacements along lines parallel to the weld at distances of 1, 2, 4 and 8 nodes from the weld centerline are shown. The welding speeds of 6 and 20 ipm are shown above each plot.

6 ipm

Figure 5-24 shows the transverse stress, <7XX, to the weld on the weld centerline. Chihoski was not able to measure stress. Data of this type would be also useful to compare with Sigmajig test data.

6 ipm

20 ipm Figure 5-24: The welding speeds are 6 and 20 ipm. When the weld arc reaches the center of the plate, the transverse stress (7XX is shown along the weld line

Also the compressive stress region in Figure 5-24 is larger for the faster weld. This is the effect observed by Chihoski and suggests the faster weld would be less susceptible to hot cracking because more of the region susceptible to hot cracking is in compression. The qualitative agreement is as good as could be expected because the welds are quite different.

This model appears to be useful step towards quantitative analysis of stress and strain in the region susceptible to hot cracking in welds. A more realistic quantitative analysis of hot cracking will require a finer mesh to resolve the thermal shock in front of the weld pool. A larger plate is needed to provide more constraint to the weld pool region.

We next consider one or more algorithms for computing flow lines or streamlines associated with a steady state weld.