High Precision Microscale Bending by Pulsed and CW Lasers

Transcription

1 X. Richard Zhang Xianfan Xu School of Mechanical Engineering, Purdue University, West Lafayette, IN High Precision Microscale Bending by Pulsed and CW Lasers This paper discusses high precision microscale laser bending and the thermomechanical phenomena involved. The use of a pulsed and a CW laser for microscale bending of ceramics, silicon, and stainless steel is demonstrated. For each laser, experiments are conducted to find out the relation between bending angles and laser operation parameters. Changes of the ceramics surface composition after laser irradiation are analyzed using an electron probe microanalyzer (EPMA). Results obtained by the pulsed and the CW laser are compared, and it is found that the CW laser produces more bending than the pulsed laser does. However, the pulsed laser causes much less surface composition change and thermomechanical damage to the targets. Numerical calculations based on the thermo-elasto-plastic theory are carried out and the results are used to explain the phenomena observed experimentally. DOI: / Introduction Laser bending or laser forming is a technique of using the energy from a laser beam to modify the curvature of sheet metals or hard materials. There are several mechanisms for laser bending, depending on the geometry and the thermophysical properties of the target material, and the laser processing parameters. Common laser bending mechanisms include the temperature gradient mechanism and the buckling mechanism 1. For the temperature gradient mechanism, only the surface layer is heated, thus, there is a temperature gradient with the highest temperature at the surface, and the residual stress and strain are concentrated in the near surface region. The residual strain is compressive, causing the target to bend toward the laser beam. This type of bending is preferred when a consistent bending direction is required. If laser heating is uniform throughout the thickness of the target, the target will bend just like a beam under compression. In this case, bending is caused by the buckling mechanism and the bending direction depends on the pre-curvature and the initial stress of the target. Applications of laser bending include ship body construction 2, removing welding distortion and straightening car body parts 3, and rapid prototyping 4,5. Laser bending was also studied for forming in space stations 6,7 and for bending of microelectronic components 8. Recently, Chen et al. 9 studied high precision laser bending for manufacturing computer components. They achieved a bending precision of sub-microradian, far exceeding those obtained by any other method. Numerical simulation using the finite element method is an effective way to study the influence of laser operating parameters and the target geometry on bending While continuous wave CW lasers are used in most laser bending operations, a 2D simulation of pulsed laser bending was also reported D models are more appropriate for predicting the actual pulsed laser bending process, however, the computation of 3-D pulsed laser bending is inhibited by the computer power. This is because in pulsed laser bending, thousands of laser pulses are irradiated onto the target. Therefore, it is extremely time-consuming to compute thermal and thermo-mechanical effects caused by all the pulses. Recently, Zhang et al. 15 developed an efficient, 3-D finite element method to calculate pulsed laser bending. In that method, only a fraction of the total laser pulses instead of all the laser pulses are calculated, thus the computation time is greatly reduced. Contributed by the Manufacturing Engineering Division for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received July 2001; Revised December Associate Editor: L. Yao. This paper presents high precision bending of ceramics, silicon, and stainless steel specimens using a pulsed laser and a CW laser. The relation between bending angles and laser operating parameters, such as the laser intensity, laser scanning speed, and number of scanning are studied experimentally. The dependence of bending on optical and thermophysical properties of the target material is illustrated. The surface composition of ceramic specimens before and after laser bending is analyzed using an electron probe microanalyzer EPMA. 3D numerical simulations of bending of stainless steel using a pulsed laser and a CW laser are performed. Experimental and numerical results are compared to demonstrate the different effects caused by the difference in the laser heating time. 1 Experiments For the experiments, a 2 W Nd:VA nanosecond pulsed laser and a 9 W CW fiber laser are used. The operation parameters of these two lasers are summarized in Table 1. For pulsed laser, the laser power level is controlled by using a polarizing beam splitter so that the pulse format can be kept the same. The laser beam diameter on targets is measured with a knife edge technique. The mean value of the beam diameter is given in Table 1. For the CW laser, beam diameters of 80 m and 40 m have been used in experiments. However, the numerical calculations are conducted for an 80 m diameter beam only. Figure 1 illustrates the experimental setup for performing laser bending as well as for measuring the bending angle. The laser beam is expanded by a beam expander and then focused onto the target using a focusing lens. One end of the target is clamped, and the distance between the scanning path and the clamped edge is about 5 mm. The focused laser beam scans the target in the y-direction, causing bending in the z-direction. A He-Ne laser beam is focused at the free end of the specimen to measure the bending angle. The reflected He-Ne laser beam is received by a position sensitive detector PSD with 1 m sensitivity in the position measurement. Bending of the specimen causes the reflected He-Ne laser beam to move across the PSD. The position change of He-Ne laser beam measured at the PSD surface is then converted to the bending angle of the specimen using geometrical calculations. The distance between the specimen and the PSD is set to 750 mm, resulting in an accuracy of the bending angle measurement of about 1.5 rad. The whole apparatus is set on a vibration-isolation table. Ceramics, silicon, and stainless steel AISI 301 sheets are used as specimens. The dimensions and some key material properties are given in Table 2. Before laser treatment, all the samples are polished and cleaned with acetone. The Al 2 O 3 /TiC ceramics is 512 Õ Vol. 125, AUGUST 2003 Copyright 2003 by ASME Transactions of the ASME

2 Table 1 Parameters of pulsed laser and CW laser Pulsed laser CW laser Laser wavelength 1.06 m 1.10 m Laser pulse full width 120 ns Laser pulse repetition 22 khz Laser maximum power 2.0 W 9.0 W Laser beam diameter 55 m 40,80 m used in computer hard disks as the material for the read/write head. Compositions of ceramic specimens are recorded with EPMA before and after laser treatment. High magnification photos of the surface topography and quantitative composition analyses of the surface area irradiated by the lasers are also obtained with EPMA. 2 Experimental Results and Discussion. Bending angles of the ceramic specimens are obtained at various laser processing conditions. Figures 2 and 3 compare results of pulsed and CW laser bending as a function of laser intensity and laser scanning speed, respectively. As expected, the bending angle increases when the input laser intensity increases, and decreases with an increase of the scanning speed. For the pulsed laser, the heat diffusion length in ceramics is about 1 m, and for the CW laser, the heat diffusion length is about 50 m at a scanning velocity of 130 mm/s. With such a small heat diffusion length in pulsed laser bending, bending is always controlled by the temperature gradient mechanism. In the experiments, it is found that the specimens always bend toward the laser beam. However, for CW laser bending, the bending direction could be controlled by different mechanisms depending on the scanning velocity. It is found that the specimen bends away from the laser beam due to the buckling mechanism at a laser intensity of W/cm 2 and a scanning speed lower than 26 mm/s. Fig. 2 Bending angle of ceramics as a function of laser intensity. For pulsed laser, vä3.25 mmõs, dä55 m; for CW laser, vä130 mmõs, dä40 m. Figure 4 indicates that for both pulsed and CW laser, scanning over the specimen surface repetitively at the same location would increase the total bending angle. However, the amount of additional bending decreases with the number of scanning lines. For the pulsed laser, the bending angle obtained by the second laser scan drops to about 25% of the angle obtained by the first scan, and no additional bending occurs after four scans. For the CW laser, the bending angle of the second scan is about 70% of the first scan, and no additional bending occurs after six scans. This is because bending depends on the initial stress/strain status in the specimen in this case, the residual stress/strain from a previous scan. The laser scan generates tensile residual stress 9, thus, it requires higher temperature to reach the compressive yield stress. On the other hand, the residual strain is compressive 9, and the area scanned by the laser beam becomes hardened which is known as the effect of strain hardening. Consequently, less additional compressive strains or bending can be obtained in the subsequent scans. Fig. 1 Experimental setup 1. Laser, 2. shutter, 3. polarizing beam splitter, 4. mirror, 5. beam expander, 6. x-y scanner, 7. specimen, 8. beam splitter, 9. position sensitive detector, 10. lens, 11. He-Ne laser Table 2 Specimen parameters and material properties at 300 K Specimen material Ceramics Al 2 O 3 /TiC Silicon Stainless steel 301 Specimen length mm Specimen width mm Specimen thickness mm Thermal diffusivity m 2 /s Optical absorption depth at N/A 1.1 m m Thermal expansion coefficient m/k Fig. 3 Bending angle of ceramics as a function of scanning speed. For pulsed laser, PÄ1.75Ã10 7 WÕcm 2, dä55 m; for CW laser, PÄ3.98Ã10 5 WÕcm 2, dä40 m. Journal of Manufacturing Science and Engineering AUGUST 2003, Vol. 125 Õ 513

3 Fig. 4 Additional bending angle of ceramics as a function of the number of laser scanning lines. For pulsed laser, PÄ1.75 Ã10 7 WÕcm 2, vä3.25 mmõs, dä55 m; for CW laser, PÄ3.98 Ã10 5 WÕcm 2, vä130 mmõs, dä40 m. There are many ways to compare bending angles induced by the pulsed laser and the CW laser. Here bending angles are compared when two adjacent scans do not influence each other in terms of the resulting bending angle. In practice, this information is useful when one intends to obtain a large bending angle with a minimum number of scans. The bending angle as a function of the separation distance between two scans is measured. As shown in Fig. 5, for the pulsed laser bending at a scanning speed of 3.25 mm/s and a laser intensity of W/cm 2, a minimum separation distance of 100 m is needed for not decreasing the bending angle. For the same separation distance, the scanning speed of the CW laser is 130 mm/s and its intensity is W/cm 2. However, the resulted bending angles from the two lasers are different. The bending angle obtained from the CW laser is about twice of that obtained from the pulsed laser. The backscattered electron BSE images of laser irradiated Al 2 O 3 /TiC ceramic specimens are shown in Fig. 6. TiC grains are Fig. 6 BSE images of ceramic specimen surface after a pulsed laser bending, PÄ1.75Ã10 7 WÕcm 2, vä13 mmõs, d Ä55 m, and b CW laser bending, PÄ3.98Ã10 5 WÕcm 2, v Ä130 mmõs, dä40 m. Fig. 5 Bending angle of ceramics as a function of distance between adjacent scanning lines. For pulsed laser, PÄ1.75 Ã10 7 WÕcm 2, vä3.25 mmõs, dä55 m; for CW laser, PÄ3.98 Ã10 5 WÕcm 2, vä130 mmõs, dä40 m. white irregular islands and alumina grains are the background sea. The bending angle obtained is 9 rad for the pulsed laser and 30 rad for the CW laser. After laser irradiation, the surface becomes gray. A few microcracks can be seen in Fig. 6 a in the pulsed laser-irradiated area. After the CW laser irradiation, an extensive gray color region appears as shown in Fig. 6 b.a30 m wide band of homogeneous material replaces the original ceramic composite material, and a curved, 1 m wide microcrack is located at the center of the scanning line, connected with several transverse microcracks. Formation of the gray substance is possibly due to diffusion of TiC into Al 2 O 3, and/or oxidation of TiC to form TiO 2 or TiAl 2 O 5. Quantitative analyses of the surface composition shown below will provide further explanations. Apparently, more material damages are produced by the CW laser than the pulsed laser, although a larger bending angle is obtained by the CW laser. Weight percentage changes of Al and Ti are also obtained using EPMA. Two sets of experiments are carried out. For the first set, the pulsed laser is used with an intensity of W/cm 2 and a scanning speed of 13 mm/s. Figure 7 a shows the element weight percentage change versus the number of scanning lines at the same location. It can be seen that the weight percent of Al decreases slightly with an increase in the number of scanning 514 Õ Vol. 125, AUGUST 2003 Transactions of the ASME

4 Fig. 8 Bending angle of silicon as a function of laser intensity at vä3.25 mmõs. For pulsed laser, dä55 m; For CW laser, d Ä40 m. Fig. 7 Al and Ti weight percent changes versus a number of pulsed laser scanning lines, PÄ1.75Ã10 7 WÕcm 2, vä13 mmõs, dä55 m, and b CW laser intensity, vä130 mmõs, d Ä40 m. lines. For the second set of data shown in Fig. 7 b the CW laser is used with a scanning speed of 130 mm/s. It can be seen that there is almost no change in the weight percent of Al when the laser intensity is increased from W/cm 2 to W/cm 2. The weight percent of Ti increases slightly from 23.8% to 24.8% when the CW laser intensity is increased from W/cm 2 to W/cm 2, and then decreases significantly to 17.0% at W/cm 2. The behavior of the Al 2 O 3 /TiC ceramics under laser irradiation can be qualitatively understood by analyzing its optical and thermal properties. In the ceramic specimen, the alumina grains are transparent to the laser irradiation, while the TiC grains absorb the laser energy. Heating of alumina is through heat conduction from the TiC grains to the alumina grains. On the other hand, alumina has a lower melting temperature 2345 K and vaporization temperature 3803 K compared to those of TiC 3413 K and 5093 K 16. It is estimated using a finite element calculation that during pulsed laser irradiation, the boiling temperature of TiC is reached at an intensity of about W/cm 2. Therefore, at the laser intensity of W/cm 2 used in this experiment, a certain amount of alumina and TiC grains are evaporated. TiO 2 and other oxides such as TiAl 2 O 5 can be produced due to oxidation. TiC and its oxides form a gray pattern on the surface as shown in Fig. 6 a. Overall, the evaporation and oxidation are weak because of the short heating time, which cause a small change in the element weight percentages. For the CW laser irradiation, at low laser intensity ( W/cm 2 ), the temperature reached could be lower than the vaporization temperature of TiC 5093 K but higher than the vaporization temperature of alumina 3803 K. Therefore, more alumina is evaporated and TiC and its oxides are left on the surface. Evaporation should be rather weak since the element percentage changes are small. On the other hand, at higher laser powers ( W/cm 2 ), a higher peak temperature is reached on the surface of the TiC grains, causing stronger evaporation and oxidation. However, it is noticed from Fig. 7 b that the slight increase in Al does not offset the large decrease in Ti, indicating other compounds are formed such as oxides. Forming TiO 2 from TiC would reduce the weight percentage of the Ti element. Thus, it is concluded that more oxides are formed when the laser power is high, resulting in reduced C and Ti element weight percentages. Figure 8 shows the bending angle of silicon as a function of power of the pulsed laser and the CW laser, respectively. Comparing Fig. 8 with Fig. 2, it can be seen that the bending angles of silicon are comparable or less than those of the ceramics specimen at most power levels, even the thickness of silicon specimen is only 57% of that of ceramics. According to material properties given in Table 2, the optical absorption depth of silicon at the laser wavelengths 1064 nm and 1100 nm is about 0.25 mm and that of ceramics is less than 100 nm. Also, the heat diffusion length of silicon is about 4 times longer than that of ceramics for the same laser parameters. The longer optical absorption depth and heat diffusion length of silicon result in a much lower peak temperature and smaller temperature gradient. Therefore, smaller bending angle of silicon is obtained when using identical laser parameters. Figure 9 shows the bending angle of stainless steel as a function of the laser intensity. The scanning speed of the pulsed laser is 195 mm/s. Surface melting occurs when the laser intensity is higher than W/cm 2 at this scanning speed. Compared with the pulsed laser bending of ceramics and silicon, it is found that comparable bending angles can be obtained for steel at a scanning speed about 60 times higher but only half of the laser intensity. This is because that the steel is much more ductile than ceramics and silicon. Journal of Manufacturing Science and Engineering AUGUST 2003, Vol. 125 Õ 515

5 Fig. 9 Bending angle of stainless steel as a function of laser intensity. For pulsed laser, vä195 mmõs, dä55 m;forcwlaser, vä8 mmõs, dä40 m. 3 Numerical Calculations CW and pulsed laser bending induced by the temperature gradient mechanism is calculated using 3-D finite element models. A thermal analysis and a stress analysis are conducted. The two analyses are treated as uncoupled since the heat dissipation due to deformation is negligible compared with the heat provided by the lasers. In an uncoupled thermo-mechanical model, a transient temperature field is obtained first in the thermal analysis, and is then used as a thermal loading in the subsequent stress analysis to obtain transient stress, strain, and displacement distributions. For the CW laser bending simulation, the laser beam is considered as a constant heat source moving at a constant speed. The time step in the calculation must be small enough so that the continuously moving laser flux can be calculated accurately. For pulsed laser bending, direct simulation using a 3-D model is impractical in terms of the computer power, since there are too many pulses along a single laser scanning line. An efficient method for simulating pulsed laser bending was recently developed by Zhang et al. 15, which is briefly described as follows. In a pulsed laser bending process, although a single laser pulse generates nonuniform stress and strain distributions, in practice, laser pulses with the same pulse energy, separated by a very small distance compared with the laser beam diameter are used. Thus, the laserinduced stress and strain vary little along the scanning direction. Also, the stress and strain fields induced by a laser pulse are contained within a short distance from the laser-irradiated area. Therefore, it is only necessary to calculate several laser pulses until the stress and strain fields in an x-z cross-section area are not changed by a new laser pulse see Fig. 1 and Fig. 10 for the coordinate system. Then, the residual strain field in this crosssection can be imposed onto the whole domain to calculate the deformation distribution. In other words, a strain field, which can be used to calculate displacements of the target after pulsed laser scanning, is generated by calculating only a fraction of the total pulses. The thermal analysis is based on solving the 3D heat conduction equation. The initial condition is that the whole specimen is at the room temperature 300 K. The laser flux is handled as a volumetric heat source decreasing exponentially from the target surface with an optical absorption depth of 20 nm. Using the transient temperature data obtained from the thermal analysis as thermal loading, the transient stress, strain, and displacement distributions are obtained by solving the quasi-static force equilibrium equations. The material is assumed to be linearly elasticperfectly plastic. The Von Mises yield criterion is used to model Fig. 10 Mesh for the 3D simulation of CW laser bending laser irradiates on the zä0 surface the onset of plasticity. The left edge of the specimen is completely constrained, and all other boundaries are force free. Material properties including thermal conductivity, thermal expansion coefficient, density, yield stress, and Young s modulus are considered as temperature dependent 17. For both CW and pulsed laser calculations, a dense mesh is used around the laser path and a coarse mesh is used outside the primary processing region. Transition elements are created to connect the dense mesh and the coarse mesh. Eight-node linear brick elements are used. The mesh used for the CW laser bending simulation is shown in Fig. 10. The Cartesian coordinate system is attached to the computational domain and the center of laser beam moves along the y-direction at x 0. The computational domain has the same dimensions of the steel sheet used in the experiments, 10 mm 1 mm 0.1 mm. It is constrained at the left edge (x 5 mm). The total element number is 20,790. The mesh for the pulsed laser bending simulation is similar except smaller elements are used and the total element number is 99,440. The step size between two adjacent laser pulses is 9 m, which is sufficiently small to provide uniform residual strain and stress along the y-direction. For each case, the same mesh is used for both thermal and stress analyses. The non-linear finite element solver, ABAQUS is employed for both CW and pulsed laser bending simulations. Only bending of stainless steel is simulated since more property data of stainless steel are available compared with the other two materials. The maximum temperatures obtained in all simulations are lower than the melting point of steel 1650 K. Details of calculations and experimental validations were provided elsewhere 12,15. Here the focus is on comparing pulse vs. CW laser heating, and explaining the difference between the two cases observed experimentally. Figure 11 shows the temperature distributions along the z-direction induced by the pulsed laser with the laser intensity of W/cm 2. Only the range from 0 to 10 m inthez direction is presented so that temperature variations can be seen clearly. The maximum temperature is obtained at the upper surface and reaches 890 K at t 87.7 ns. The heat propagation depth is around 4 m at 2.2 s and the temperature gradient during heating period is as high as 290 K/ m. This sharp temperature gradient results in the temperature gradient mechanism of bending. Figure 12 a shows the off-plane displacement w and the residual stress xx on the top surface of the specimen at the same laser intensity. The center of the laser beam is located at x 0. A V shape profile is obtained after laser scanning, with the valley located at about 10 m from the center of the scanning line. The positive off-plane displacement at the center of the scanning line 516 Õ Vol. 125, AUGUST 2003 Transactions of the ASME

6 Fig. 11 Transient temperature distributions along the z-direction induced by a laser pulse. PÄ1.54Ã10 6 WÕcm 2, d Ä55 m. is produced by thermal expansion along the positive z-direction. The stress xx is tensile and its value is around 1.04 GPa in the region within 15 m from the center. This agrees with the theoretical prediction that the tensile residual stress will be obtained near the center of the laser-irradiated area due to the thermal shrinkage during cooling 9. The total high stress region is about Fig. 13 Simulation results of CW laser bending simulation. P Ä1.59Ã10 5 WÕcm 2, vä8 mmõs, dä80 m. a Residual stress distributions along the x-direction; b residual stress and strain distributions along the z-direction. 30 m and is comparable to the radius of the laser beam 27.5 m. The residual stress and strain distribution along the thickness direction are shown in Fig. 12b. The high tensile stress is induced by the laser near the surface and it becomes compressive at depths greater than 0.6 m. The residual compressive strain xx reaches a minimum value of within 1 m from the surface, and then increases gradually with the depth. The compressive strain indicates that the bending is toward the laser beam. Figure 13 a shows the residual stress distribution along the x-direction induced by the CW laser. In order to compare with pulsed laser bending shown in Fig. 12 a only a region of 200 m instead of 5 mm along the x-direction is plotted. As shown in Fig. 13 a the largest residual stress is located at the center line (x 0) and the value is about 700 MPa, which is smaller than that induced by the pulsed laser 1.04 GPa. However, the total stressed zone in the x-direction is more than 100 m, much larger than the radius of the laser beam 40 m and the stressed zone induced by a pulsed laser. This is because the heat diffusion length of CW laser is much longer. Figure 13 b shows the peak tensile residual stress and strain distribution along the z-direction. It can be seen by comparing Fig. 13 b with Fig. 12 b that the residual stress and strain depth of CW laser bending are about 5 times larger than that of pulsed laser bending. The wider and deeper stressed and strained region explain the more visible mechanical damages observed in the CW laser experiments. Fig. 12 Simulation results of pulsed laser bending. PÄ1.54 Ã10 6 WÕcm 2, vä195 mmõs, dä55 m. a Residual stress and off-plane displacement distributions along the x-direction; b residual stress and strain distributions along the z-direction. 4 Conclusions This work demonstrated using pulsed and CW lasers for microscale bending of ceramic, silicon, and stainless steel samples. Experimental studies were conducted to find out relations between bending angles and laser operation parameters. Results obtained Journal of Manufacturing Science and Engineering AUGUST 2003, Vol. 125 Õ 517

7 by a pulsed and a CW laser were compared. For the ceramic specimen, when two adjacent laser scans do not influence each other, the CW laser produced more bending than the pulsed laser did. However, the pulsed laser caused much less surface composition change and thermomechanical damage. Numerical calculations using the thermo-elasto-plastic theory were conducted and the results are used to explain the phenomena observed experimentally. Acknowledgments Support of this work by the National Science Foundation is acknowledged. The authors are most grateful to Dr. Andrew C. Tam of IBM Almaden Research Center for his contribution to this work. The authors also thank SDL, Inc. for providing the CW fiber laser system, and Mr. Carl Hager of the Department of Earth and Atmospheric Sciences at Purdue University for his help on the EPMA measurements. Nomenclature P laser intensity d laser beam diameter on targets v laser scanning speed w off-plane displacement x, y, z Cartesian coordinates xx residual stress along the x-direction residual stress along the x-direction xx References 1 Geiger, M., and Vollertsen, F., 1993, The Mechanisms of Laser Forming, CIRP Ann., 42, pp Scully, K., 1987, Laser Line Heating, J. Ship Prod., 3, pp Geiger, M., Vollertsen, F., and Deinzer, G., 1993, Flexible Straightening of Car Body Shells by Laser Forming, Proceedings of NADDRG/DAE Sheet Metal Forum, Detroit, MI, pp Arnet, H., and Vollertsen, F., 1995, Extending Laser Bending for the Generation of Convex Shapes, Proc. Inst. Mech. Eng., 209, pp Vollertsen, F., Komel, I., and Kals, R., 1995, The Laser Bending of Steel Foils for Microparts by the Buckling Mechanism-A Model, Modelling and Simulation in Materials Science and Engineering, 3, pp Namba, Y., 1985, Laser Forming in Space, Proceedings of the International Conference of Lasers and Electro-Optics (ICALEO 85), Boston, MA, September Laser Institute of America, Orlando, pp Namba, Y., 1987, Laser Forming of Metals and Alloys, Proceedings of Laser Advanced Materials Processing (LAMP 87), Osaka, Japan, June High Temperature Society of Japan, Osaka, pp Verhoeven, E. C. M., de Bie, H. F. P., and Hoving, W., 2000, Laser Adjustment of Reed Switches: Micron Accuracy in Mass Production, Proceedings of the Laser Microfabrication Conference, ICALEO 2000, LIA Vol. 90, pp. B21 B30, Laser Institute of America, Orlando, FL. 9 Chen, G., Xu, X., Poon, C. C., and Tam, A. C., 1998, Laser-assisted Microscale Deformation of Stainless Steels and Ceramics, Opt. Eng., 37, pp Vollertsen, F., Geiger, M., and Li, W. M., 1993, FDM- and FEM-Simulation of Laser Forming: A Comparative Study, Advanced Technology of Plasticity 1993-Proceedings of the 4th International Conference on Technology of Plasticity, Z. R. Wang, Y. He, eds., Prentice-Hall, New York, pp Alberti, N., Fratini, L., and Micari, F., 1994, Numerical Simulation of the Laser Bending Processing by a Coupled Thermal Mechanical Analysis, Laser Assisted Net Shape Engineering, M. Geiger et al., eds., Meisenbach, Bamberg, pp Chen, G., and Xu, X., 2002, Experimental and 3D Finite Element Studies of CW Laser Forming of Thin Stainless Steel Sheets, ASME J. Manuf. Sci. Eng., 123, pp Bao, J. and Yao, Y. L., 1999, Analysis and Prediction of Edge Effects in Laser Bending, Proceedings of the International Conference of Lasers and Electro-Optics (ICALEO), San Diego, CA, November Laser Institute of America, Orlando, pp Chen, G., Xu, X., Poon, C. C., and Tam, A. C., 1999, Experimental and Numerical Studies on Microscale Bending of Stainless Steel with Pulsed Laser, ASME J. Appl. Mech., 66, pp Zhang, X., Chen, G., and Xu, X., 2002, Numerical Simulation of Pulsed Laser Bending, ASME J. Appl. Mech., 69, pp Swain, M. V., 1994, Structure and Properties of Ceramics in Materials Science and Technology: A Comprehensive Treatment, Vol. 11, VCH, New York. 17 Maykuth, D. J., 1980, Structural Alloys Handbook, Vol. 2, Metals and Ceramics Information Center, Battelee Columbus Laboratories, Columbus, Ohio, pp Õ Vol. 125, AUGUST 2003 Transactions of the ASME