4.4 Development of Numerical Model for Laser Bending of Magnesium Alloy M1A Using FEM
4.4.3 Solution parameters
In the present work, for solving the thermo-mechanical laser bending problem, the automatic (self-adaptive) time step algorithm was used to select the time increment in a step. This is based on the tolerance in the maximum temperature change allowed in a time increment. In automatic time step algorithm, the increment is adjusted based on the maximum temperature change in a time step and the corresponding convergence rate.
Table 4.7. Sensitivity analysis for optimal time step increment.
S. No. Maximum change in
temperature in a step Bend angle Change in bend
angle % CPU time (hrs.)
1 5 1.091540862 - 14.72
2 10 1.09166224 0.01 7.13
3 20 1.091665752 0.00 2.61
4 30 1.091255551 -0.04 1.81
5 50 1.089384586 -0.17 1.27
6 70 1.087188941 -0.20 1.02
7 100 1.08720017 0.00 1.01
The effect of maximum permissible temperature change in a step was studied by carrying out the sensitivity analysis for a typical process condition: laser power=300 W, scan
speed=1000 mm/min and beam diameter 3.87 mm. The maximum temperature change in a step was varied from 5 to 100 °C and its effect on the bend angle was analyzed. This is shown in Table 4.7. It can be observed that the maximum temperature change in a step marginally affects the bend angle, however it significantly affects the simulation time. The maximum temperature change in a step was chosen from 10‒50 °C. The maximum and minimum time steps size were taken as 0.02 and 0.0005 second, respectively. The equations solved in coupled thermo- mechanical analysis are equations of motion and heat conduction equation. These equations were solved by Full Newton technique. In this technique, the target stiffness matrix is evaluated at each iteration. This helps in getting proper convergence even when the guess value is far away from the solution. Also, the stiffness matrix is more accurate in each iteration which provides a better prediction accuracy. However, computationally, it is not as efficient as Modified Newton method, in which sometimes, convergence is not achieved (Bathe 1996).
4.5. A Case Study on Numerical Simulation of Laser Bending Process Using FEM After the development of numerical model, it was tested for the predictions/results. The worksheet geometry, material properties, meshing and solution scheme were implemented as discussed in Section 4.4. In the heated region, elements of size 0.5 mm × 0.5 mm were taken, and in the outer region a coarse biased mesh with total five number of elements was modeled.
The bias ratio of the biased mesh was five. Total four equidistance elements were taken in the thickness direction. As per this meshing scheme, the geometry was discretized into 8320 number of elements and 10935 number of nodes. The solution was carried out with automatic time step with maximum 10 °C increment in temperature in a step. The maximum time step was taken as 0.2 second, and the minimum time step was taken as 0.005 second.
Figure 4.7. Temperature contour induced by the laser scan.
This case study was carried out for a typical process condition: laser power=300 W, scan speed=1000 mm/min and laser beam diameter=3.87 mm. The thermo-mechanical
numerical model was able to predict temperature distribution, stress-strain distributions and distortions. Figure 4.7 shows the temperature contours obtained during laser scan. It can be seen that the highest temperature occurs in the beam diameter region. The temperature distribution is symmetric about the irradiation line.
The increase in temperature generates thermal stresses in the heated region. Figure 4.8 shows contours of the various components of thermal stresses induced due to the laser scan. It can be seen that thermal stresses are not symmetric about the irradiation line. It is due to clamping on one side the worksheet. The clamping provides external mechanical constraint which results in the generation of thermal stresses.
Figure 4.8. Thermal stresses contour induced by laser scan.
When induced thermal stresses exceed temperature dependent flow stress, the plastic deformation occurs. The contours for various components of plastic strains induced due to laser scan are shown in Figure 4.9. It can be seen that the plastic deformation is concentrated in the irradiated region. It justifies the use of fine mesh in the heated region and coarse biased mesh in the outer region.
Figure 4.9. Plastic strains contour induced by laser scan.
Figure 4.10. Laser bent worksheet.
The plastic strain generated due to laser scan produces distortions in the heated region.
These distortions result in the worksheet bending. The bending profile generated by laser scan is shown in Figure 4.10.
The bend angle generated by laser scanning was measured by taking two points on each side of the scan line. These points formed two lines on both sides of the laser scanning line and the bend angle was measured between these two lines. Details of the bend angle measurement scheme is given in Section 3.3.3. The bend angle was not uniform along the laser scan line, and it varied from one end to another end as shown in Figure 2.5. This is called edge effect. Details of the edge effect are discussed in Section 2.2. The edge effect can be quantified by measuring the bend angle at five equidistance positions along the scanning line as shown in Figure 4.11.
In the present work, the edge effect was quantified by calculating the relative variation in bend angle (RVBA) per unit length measured at five equidistant positions along the scan line as
1 1 max min
(mm ) =
average
RVBA L
, (4.29)
where
max ,
min and a v e r a g e are the maximum, minimum and average bend angle along the scanning line respectively, and L is the length of worksheet. The edge effect is more when the value of RVBA is higher, and vice-versa.Figure 4.11. Bend angle positions to calculate the edge effect.
The bend angle predicted by using the numerical simulation was compared with that obtained in experiment. Details of the experimental procedure are reported in Chapter 3. In experiments three trials were carried out for the process condition: P=300 W, V=1000 mm/min and D=3.87 mm. Bend angles of about 1.007º, 1.106º and 1.094º was obtained in the three trials respectively, and the average of these three values (1.069º) was considered as the experimental result. It was observed that the numerical bend angle has an error of about 2.4% in comparison with that of the experimental bend angle. Therefore, it can be concluded that the numerical model predicts the bend angle quite accurately for the chosen process condition.
The case study shows that numerical model can simulate the laser bending process well.
It is capable to predict the bend angle and edge effect based on the temperature distribution, stress-strain distributions, and distortions obtained during numerical computations. Predictions of the developed numerical model are validated with those obtained in the experimental studies for various cases of laser bending such as single scan, curvilinear, multi-scan, and with moving pre-displacement. Parametric studies have also been carried out using the developed numerical model. These are presented at length in the next chapters.