NUMERICAL INVESTIGATION AND STATISTICAL ANALYSIS OF LASER BENDING OF TITANIUM SHEETS

5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India NUMERICAL INVESTIGATION AND STATISTICAL ANALYSIS OF LASER BENDING OF TITANIUM SHEETS K.Paramasivan 1*, Sandip Das 1, Dipten Misra 1, M. Sundar 2 1* School of Laser Science & Engineering, Jadavpur University, Kolkata-700032, parma.gce@gmail.com 1 School of Laser Science & Engineering, Jadavpur University, Kolkata-700032, sdas@mech.jdvu.ac.in 1 School of Laser Science & Engineering, Jadavpur University, Kolkata-700032, dipten@gmail.com 2 School of Mechanical and Manufacturing, Loughborough University, Loughborough, Leicestershire, LE11 3TUUK, UK, s.marimuthu@lboro.ac.uk Abstract This paper presents a numerical investigation and statistical analysis of laser bending of Titanium sheets. The aim of the present simulation is to identify the response through response surface methodology related to bending angle and characterize the effects of input parameters: laser power, spot diameter, scanning speed and plate thickness. The numerical simulations are carried out based on the four factors, central composite face centered design. The statistical software Design-Expert is used to create the design layout and to obtain the final regression equation. The numerical simulation is carried out by implementing a Gaussian surface heat flux, and convection radiation boundary conditions through a commercial finite element software COMSOL MULTIPHYSICS. Investigations reveal that bending angle increases with laser power and decreases with the increase in scanning speed, spot diameter and plate thickness. The optimum process parameters for the target bending angle are also found based on misimisation of processing time and operating energy. Keywords: Numerical investigation, Statistical analysis, Laser bending 1 Introduction The metal sheets employed in the laser forming process are mainly those of materials with good deformability at the ambient temperature namely, steel, titanium, aluminium, etc.,[1]. Titanium is very suitable for aerospace, marine, chemical, medical implant fabrication industries and sports. It has superior properties such as good strength, low weight, good corrosion resistance and stability of mechanical and chemical properties at elevated temperatures. Laser bending is a flexible technique that uses a defocused laser beam to generate a bend angle on components, by introducing residual thermal stresses in contrast to conventional bending processes, wherein, bending is achieved by application of external forces. Laser metal forming has diverse applications in shipbuilding, automotive, microelectronics and aerospace industries. Shen and Vollertsen [2] presented review article about a number of recent developments and new techniques in modeling of laser forming, including analytical models, numerical simulations and various empirical models. Wu and Ji [3] simulated the deformation field during laser forming of sheet metal. They have found that counter-bending, away from the laser source, in a thin sheet is greater than that in a thick sheet and the stresses changes severely with time and location in the zone being passed through by the laser beam and its surrounding zone from their results. Kyrsanidi et al.[4] investigated the laser forming process of metallic plates through numerical and experimental analyses. They also carried out numerical simulation for achieving sine shape plate by laser forming. Venkadeshwaran et al.[5] studied the deformation field in circular plate by using circle line heating. They have found that discrete section heating in symmetry with shifting in starting point of irradiation in P242 1

NUMERICAL INVESTIGATION AND STATISTICAL ANALYSIS OF LASER BENDING OF TITANIUM SHEETS subsequent passes reduces the undesired waviness. Jamil et al.[6] investigated the laser forming process by using different laser beam geometries, namely, rectangular, square, triangular while area of each beam is kept constant to transfer the same amount of energy into the specimen. A number of attempts have been made to simulate the laser bending process using finite numerical and experimental study with design of experiments (DOE) techniques. Zahrani and Marasi [7] experimentally investigated the effects of process parameters on the edge effect and longitudinal distortion in the laser bending process. Their results indicate that increasing sheet thickness, scan speed and laser power lead to the reduction of edge effect and longitudinal distortion responses, while the beam diameter should also have a minimum value. Maji et al. [8] presented experimental investigation on pulsed laser bending of sheet metal and statistical analysis to study the effect of process parameters. They made comparisons between continuous mode and pulsed mode laser bending to study the process efficiency in terms of energy input and the deformation thus generated. Their results showed that pulsed laser could produce more bending compared to continuous laser for the same total energy input. Gollo et al.[9] investigated the statistical analysis of parameter effects on bending angle by FEM and by experiments. Zahrani and Marasi [10] studied the effects of process parameters, namely, laser power, beam diameter, scan speed, sheet thickness and heating position on the bending angle from the free edge of the sheet. Laser bending process is systematically modeled and investigated, through sequential application of FEA and DoE techniques. In the present work, a three-dimensional FE model, for laser bending of titanium with a moving laser beam, has been constructed, initially. DoE along with regression analyses are then employed to plan the experiments and to develop regression models based on simulation results. Finally, a comparison is made between simulation results and predicted results. In the present study, optimum values of process parameters are identified to achieve specified bend angles with low processing time and low operating energy cost which are derived based on desirability function. The parameter combinations with highest desirability values are selected as the best laser forming conditions [11]. 2 Finite element simulation Fig. 1 shows a typical workpiece, scanning direction and fixed end of the plate. The simplysupported boundary conditions are applied at the fixed end of the plate, and the other end is free. The laser beam scanning path is at the middle of the plate along the y-axis. The laser beam is modeled as a moving surface heat flux with small time increment. A 3-D free tetrahedral mesh is used in the present numerical modeling. A non-uniform mesh pattern is used to minimize the simulation time and memory requirement by decreasing the total number of elements. Due to high heat flux involved in laser path, very fine meshes are used along the path of the laser beam. Fig. 1. schematic diagram showing the workpiece and irradiation path Coarse meshes are used in other parts of the plate. Fig. 2 shows, the meshes used for numerical modeling. Fig. 2 Finite element mesh used for numerical modeling. During laser forming, the transient temperature field is generated based on the mechanism of the heat conduction. The governing equation for heat conduction within the specimen can be written as follows: T ( r, t) ρ c = k r.( rt ) (1) t where ρ is the material density (kg/m 3 ), c is the specific heat (J/kg C), k is the thermal conductivity (W/m C), T(r,t) is the temperature (K), r is the coordinate (m) in the reference configuration, t is time (s) and r is the gradient operator. The moving heat flux q produced by the laser beam, is applied on the top surface of the sheet metal. In this work, laser beam is assumed to have a Gaussian distribution and expressed as follows: P242 2

5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India q 2AP 2r exp 2 πr R = 2 2 (2) where, A is the absorption coefficient, P is the laser power [W], R is the laser beam radius (m) and r is the distance (m) of a point from the centre of the laser beam. The material cooling phase is made through natural convection and radiation from its surfaces exposed to ambient air. The convection and radiation boundary conditions can be expressed as follows q = h( T To) (3) q conv rad s 4 s 4 o = εσ ( T T ) (4) where, h is the heat transfer coefficient, which is taken as 10 W/m 2 K, T s is the sheet metal surface temperature and T 0 is the ambient temperature, which is taken as 300K, ε is the emissivity, set as 0.6 and σ is the Stefan Boltzmann constant (5.6703 10-8 W/m 2 K 4 ). Commercially pure Titanium sheet is used as the workpiece material. Temperature dependent material properties are used in this modeling [12]. Elasto-plastic properties also used in the present analysis. The following assumptions are made to develop a model for simulation of the laser forming process using the finite element method; 1. Material properties of the workpiece are isotropic. 2. The distribution of laser intensity follows a Gaussian mode. 3. Within the workpiece, heat transfer takes place by conduction obeying Fourier's law and heat loss by free convection and radiation are considered from the surfaces of the sheet metal to the surrounding air. 4. Melting is not involved in the workpiece during laser forming process; phase changes and heat generation are neglected. 5. von-mises yield criterion is considered in the bending process. 3 Response surface methodology Response surface methodology is a set of mathematical and statistical techniques that is useful for empirical model and optimization. A model predicting the response for some independent input variables can be obtained by conducting experiments and applying regression analysis [13,14]. If all variables are assumed to be measurable, the response surface can be expressed as follows: y = f ( x1, x2, x3,... xn ) ± ε where y is the response, f is the function of response, ε is the experimental error, and ( x 1, x2, x3,... xn ) are independent parameters. The application of response surface method is to use a sequence of designed experiments to obtain an approximate relationship between a true response and a number of design variables, based on the observed data from the process or system. The response is generally obtained from real experiments or computer simulations. In the present work, response is collected from numerical simulation for laser forming of Titanium sheets after one laser pass. Table 1 Process parameters and their units and limits Parameter Unit Low actual High actual Laser power (P) W 450 650 Scanning speed (V) mm/s 5 15 Spot diameter (D) mm 1.5 2.5 Plate thickness (S) mm 1 2 The process parameters and their levels and units are presented in Table 1. The size of the sheet is 100 mm 50 mm. A central composite design matrix with four factors (laser power, scanning speed, laser spot diameter and sheet thickness) is considered. The central composite designs is run with α =1.82, where α is the distance of the axial points from the central point. The numerical simulations are carried out according to the design layout and the maximum bending angle as response is listed in Table 2. The maximum bending angle of the sheet is measured on top surface at the mid-point of the free end of the sheet having coordinates (100, 25). Fig. 3 shows the z- component of displacement on the top surface of the plate after cooling time. The adequacy measures R 2, adjusted R 2 and predicted R 2 are listed in Table 3, and it shows reasonable agreement and are close to 1,which indicate adequacy of the model. Fig. 3. Z-component of final displacement (mm) P242 3

NUMERICAL INVESTIGATION AND STATISTICAL ANALYSIS OF LASER BENDING OF TITANIUM SHEETS Sl. No Table 2 Design layout and calculated response Process Parameters P[W] V D [mm/s] [mm] S [mm] Response Bend angle (α b ) (deg) 1 450 5 2.5 2 0.095569 2 650 5 1.5 2 0.527679 3 550 10 2 1.5 0.250839 4 650 15 2.5 2 0.108976 5 650 15 1.5 1 0.697471 6 368 10 2 1.5 0.077235 7 550 19.1 2 1.5 0.148656 8 650 5 1.5 1 1.012655 9 450 15 1.5 2 0.112758 10 550 10 1.09 1.5 0.563658 11 550 10 2.91 1.5 0.113102 12 450 5 2.5 1 0.460856 13 550 10 2 2.41 0.114362 14 650 5 2.5 2 0.356375 15 450 5 1.5 2 0.192399 16 550 10 2 1.5 0.250839 17 550 10 2 1.5 0.250839 18 450 5 1.5 1 0.670445 19 650 15 2.5 1 0.412522 20 650 15 1.5 2 0.33426 21 550 10 2 1.5 0.250839 22 550 10 2 1.5 0.250839 23 450 15 2.5 2 0.250839 24 550 10 2 0.59 0.571449 25 550 0.9 2 1.5 0.537533 26 450 15 1.5 1 0.368865 27 550 10 2 1.5 0.250839 28 450 15 2.5 1 0.127196 29 650 5 2.5 1 0.588648 30 732 10 2 1.5 0.574084 The significant and non- significant terms are identified from analysis of variance table. From this table, Values of "Prob > F" less than 0.0500 indicate model terms are significant and values greater than 0.1000 indicate the model terms are not significant. In this case the effect of the laser power ( ), scanning velocity ( ), spot diameter (D), thickness ( ), the second order effect of laser power, scanning speed, spot diameter and thickness and the two level interaction of power and spot diameter ( D), and scanning velocity and plate thickness (V S) and spot diameter and plate thickness (D S) are the most significant model terms associated with the bend angle. In the regression equation, non significant terms are included as hierarchical terms. 3.1 Effect of process parameters on the responses The effects of parameters on the response are identified through the developed response surface models. Figs. 4 (a) and 4 (b) show the response contour plots of the effect of the interaction between process variables on the bend angle. From the figures, it can also be observed that the bending angle increases with laser power as more energy is delivered by a higher power beam. It also shows that bending angle decreases with plate thickness and spot diameter. Figs. 5 (a) and 5 (b) illustrate the combined effects of power and velocity. Both P and V show linear trends. From the figures, it can also be observed that the bending angle decreases with scanning speed. As scanning speed increases, interaction time between the workpiece and the laser beam decreases, resulting in reduced absorption of heat by the workpiece. Thus, the amount of bending decreases with increase in scanning speed. It also shows that bending angle decreases with plate thickness and increases with laser power. The final mathematical model for bending angle in terms of actual factors as determined by Design- Expert software is shown below: α b = 2.18771-8.42815 10-5 P - 0.061542 V - 0.64954 D - 1.00253 S - 1.97219 10-5 P V - 6.00993 10-4 P D - 2.41831 10-4 P S + 1.82918 10-3 V D + 0.013665 V S + 0.14753 D S + 2.83508 10-6 P 2 + 1.34457 10-3 V 2 + 0.12876 D 2 + 0.13423 S 2 Table 3 ANOVA results Standard deviation=0.044 R 2 =0.9820 Mean=0.34 Adjusted R 2 =0.9652 Coefficient of variation=12.69 Predicted R 2 =0.8937 Predicted residual error of sum of squares (PRESS)= 0.17 Adequate Precision=32.242 P242 4

5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India introduced in this numerical optimization. The first criterion is to achieve the required bend angle at relatively minimum processing time by maximising the scanning speed. The second criterion is to achieve the required bend angle at relatively low operating energy cost by minimising the line energy, i.e., by minimising laser power, P and maximising scanning speed, V. Tables 4 and 6 summarise these two optimization criteria. Within the region of process parameters considered, the optimum values to bend sheets of thickness 1 mm by 0.5, 0.75 and l degrees are identified and given in Tables 5 and 7. Table 4 The first criterion of numerical optimization Fig. 4 Contour plots showing the effects of (a) laser power and spot diameter on the bend angle (b) spot diameter and plate thickness on the bend angle Lower Upper Parameter Goal Importance is in P (W) range 450 650 3 V (mm/s) maximize 5 15 3 is in D (mm) range 1.5 2.5 3 equal to S (mm) =1 1 2 3 α b (deg) is target = 0.5,0.75,1 0.0360 1.0126 5 Table 5 Optimal bending condition based on the first criterion. Target bend angle P (W) V (mm/s) D (mm) S (mm) Desirability 0.5 611.71 15 1.76 1 1 0.5 645.51 15 1.93 1 1 0.5 647.60 15 1.94 1 1 0.75 650 15 1.5 1 0.946 0.75 649.94 14.96 1.5 1 0.945 1 650 15 1.5 1 0.784 1 649.01 15 1.5 1 0.782 Table 6 The second criterion of numerical optimization Fig. 5 Response plots showing the effects of (a) laser power and plate thickness on the bend angle (b) laser power and scanning speed on the bend angle Name Goal Lower Upper Importanc e 4.Optimization The aim of optimization is to find a required bending angle within the design space. Two criteria are P minimize 450 650 3 V maximize 5 15 3 D is in range 1.5 2.5 3 S equal to 1 2 3 P242 5

NUMERICAL INVESTIGATION AND STATISTICAL ANALYSIS OF LASER BENDING OF TITANIUM SHEETS α b (deg) =1 is target = 0.5,0.75,1 0.0360 1.0126 5 Table 7 Optimal bending condition based on the second criterion α b (deg) P (W) V (mm/s) D (mm) S (mm) Desirability 0.5 458.78 15 1.5 1 0.862 0.5 460.21 15 1.5 1 0.862 0.75 459.29 15 1.5 1 0.708 0.75 457.37 15 1.5 1 0.708 1 458.80 15 1.5 1 0.618 1 459.77 15 1.5 1 0.618 5 Conclusion The following conclusions can be drawn from this study based on the range of values of parameters considered. 1. Bend angle achieved changes with the heating conditions. It increase with power and decreases with velocity. 2. Bend angle decreases with increase in spot diameter due to the decrease in energy density. 3. Response surface methodology is used to study the relationship between input process parameters and the bend angle. The input variables considered are laser power, scanning velocity, laser spot diameter and sheet thickness. The regression equation is developed with RSM has been validated and found to be suitable for evaluating the bend angle within the parameter space considered. 4. The optimum process parameters for the target bending angle at relatively minimum processing time and low operating energy cost are also found based on desirability function. References Stephen, A. Esther, T. (2013), Experimental Investigation of Laser beam forming of Titanium and Statistical Analysis of the Effects of Parameters on Curvature, Proceedings of the International Multi Conference of Engineers and Computer Scientists, Vol. II, Hong Kong. Shen, H. Vollertsen, F. (2009), Modelling of Laser Forming - An Review, Computational Materials science, Vol.48, pp.834-840. Wu, S. Ji, Z. (2002), FEM simulation of the Deformation Field During the Laser Forming of Sheet Metal, Journal of Materials Processing Technology, Vol.121, pp.269-272 Kyrsanidi, A.K. Kermanidis, T.B. Pantelakis, S.G. (1999), Numerical and experimental Investigation of the Laser forming Process, Journal of Materials Processing Technology, Vol. 87, pp. 281-290. Venkadeshwaran, K. Das, S. Misra, D. (2010), Finite element simulation of 3-D laser forming by discrete section circle line heating, International Journal of Engineering, Science and Technology, 2(4), pp.163-175. Jamil, M.S. Sheikh, M.A. Li, L. (2011), A Study of the Effect of Laser Beam Geometries on Laser Bending of Sheet Metal by Buckling Mechanism, Optics and Laser Technology, Vol.43, pp. 183-193. Zahrani, G. Marasi, A. (2013), Experimental investigation of edge effect and longitudinal distortion in laser bending process, Optics & Laser Technology, Vol.45, pp.301 307. Maji, K. Pratihar, D. Nath, A.K. (2013), Experimental Investigations and Statistical Analysis of Pulsed Laser Bending of AISI304 Stainless Steel Sheet, Optics and Laser Technology, Vol. 49, pp. 18-27. Gollo, H. Mahdavian, S.M. Naeini, M. (2011). Statistical analysis of parameter effects on bending angle in laser forming process by pulsed Nd:YAG laser, Optics & Laser Technology, Vol.43, pp.475 482. Zahranil, E.G. Marasi, A, (2012), Modeling and optimization of laser bending parameters via response surface methodology, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 227(7), pp.1577 1584. Acherjee, B. Kuar, A. Mitra, S. Misra, D. Acharyya, S. (2012), Experimental investigation on laser transmission welding of PMMA to ABS via response surface modeling, Optics & Laser Technology, Vol. 44, pp. 1372 1383. Adamus, K. Kucharczyk, Z. Wojsyk, K. Kudla, K. (2013), Numerical analysis of electron beam welding of different grade titanium sheets, Computational Materials Science, Vol.77, pp. 286 294. Acherjee, B. Kuar, A. Mitra, S. Misra, D. (2012), Modeling of laser transmission contour welding process using FEA and DoE, Optics & Laser Technology, Vol.44, pp.1281 1289. Montgomery, D.C. (2001). Design and analysis of experiments.5th Ed. New York, Wiley. Design-Expert Software, v7, Stat-Ease, Inc., Minneapolis, USA. P242 6