A Numerical Study of the Temperature Gradient Mechanism in Laser Forming Using Different Laser Beam Geometries

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1 Lasers in Eng., Vol. 22, pp Reprints available directly from the publisher Photocopying permitted by license only 2011 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group A Numerical Study of the Temperature Gradient Mechanism in Laser Forming Using Different Laser Beam Geometries M.S. Che Jamil 1,2, *, M.A. Sheikh 1 and L. Li 1 1 Laser Processing Research Centre (LPRC), School of Mechanical Aerospace and Civil Engineering, University of Manchester, Manchester, M60 1QD, UK 2 School of Mechanical Engineering, Universiti Sains Malaysia, Pulau Pinang 11800, Malaysia Laser forming has attracted considerable attention as a viable technique to form sheet metal by thermal residual stresses. Many numerical and experimental investigations of laser forming processes were carried out to understand the mechanisms and the effects of various parameters on the characteristics of the formed parts. The objective of this work is to investigate the effect of different beam geometries on laser bending process of metal sheets, which is dominated by temperature gradient mechanism (TGM). In this paper a comprehensive thermal and structural finite element analysis is conducted to investigate the effect that these laser beam geometries have on the process and the final product characteristics. To achieve this, the temperature distribution, deformation, plastic strains and stresses produced by different beam geometries are compared numerically. The findings suggest that beam geometry could be an important controlling parameter for bending angle, edge effect and bend radius. Keywords: Laser forming, bending, beam geometries, finite element (FE) method, temperature gradient mechanism (TGM), sheet metal 1 INTRODUCTION In recent years laser forming has emerged as a viable technique to form sheet metal by thermal residual stress. The important element in laser forming process is the material which is scanned with a defocused laser beam such that *Corresponding author: Tel: +44 (0) ; Fax: +44 (0) ; mohdsabri. chejamil@postgrad.manchester.ac.uk 413

2 414 M.S. Che Jamil et al. laser-material interaction causes localized heating of the surface without melting. Laser forming idea was first discovered by Kitamura [1] who successfully bent a 22 mm thick steel plates using a 15 kw CO 2 laser. Since then many research works have been carried out in this area for applications in aerospace, automotive, shipbuilding, and electronic industries. There are many advantages of laser forming compared to conventional sheet bending. Among these are design flexibility, production of complex shapes (which is not achievable by conventional methods), forming of thick plates and the possibility of rapid prototyping. Dearden and Edwardson [2] in their review described recent developments and techniques of laser forming for both micro and macro scale applications. Recently, Shen and Vollertsen [3], in their review on laser forming, described many recent developments and new techniques in modelling of laser forming, including analytical models, numerical simulations and empirical models. Modelling of laser beam forming is a complex and demanding task due to its highly non-linear and transient nature. Ju and Wu [4] have done a comparison between finite element method (FEM) and finite difference method (FDM) simulations on laser forming of sheet metal and studied the temperature fields during the process. Kyrsanidi et al [5] have numerically investigated laser forming by temperature gradient mechanism (TGM) with the results validated by a series of experiments. These results were then used by Zhang and Michaleris [6] in their investigations using Lagrangian and Eulerian formations of the FEM for modelling laser forming process. Sichun and Zhong [7] and Shi [8] have numerically investigated temperature gradient mechanism of thin plates irradiated with a straight line laser beam. Chen et al [9] and Zhang [10] have also numerically investigated the deformation behaviour of laser path which is along a curve rather than in a straight line on sheet metal. Hu et al [11] investigated the buckling instability of laser sheet forming and concluded that buckling mechanism is activated when there is an insignificant temperature gradient through the thickness. Laser forming is based on the earlier flame bending used for ship construction. One of the advantages of laser forming compared to conventional flame bending is its ability to accurately control the size and geometry of the heat source. In the area of optics, many studies [12, 13] explain the methods and design approaches to produce a variety of beam shapes such as line, rectangular, star, D-shape, annular, cross, etc. Currently, whilst there are many studies on laser parameters such as power, feed rate and beam size etc, there is very limited work on the effects of different beam geometries on laser bending. One possible method of varying the temperature distribution, and hence the strain and stress distribution, without changing the input power or the scanning speed, is by modifying the geometry of laser beams. Safdar [14] has investigated the effect of beam geometries on various laser material processes such as laser transformation hardening, laser surface heating, laser melting of

3 Temperature Gradient Mechanism in Laser Forming 415 mild steel and tube bending. Another interesting study by Jamil et al [14] focused on the effect of beam geometries on laser forming dominated by buckling mechanism (BM). This paper describes the results of numerical investigations of the effect of five beam geometries, on laser bending process of sheet metal which is dominated by th etgm. To date, many forming mechanisms have been suggested [8, 11, 15-19]. The most extensively studied and reported in literature are TGM and the buckling mechanism (BM). In particular, the TGM has been extensively studied and reported in literature. When the material surface is irradiated with a laser beam, a fraction of the laser energy is absorbed into the material. Hence the laser energy is deposited on a very thin layer of material [20]. The layers of the material close to the irradiated surface are at higher temperature than those away from the surface, thus creating a steep temperature gradient within the thickness of the material. Hence the material layers close to the surface expand more. This differential thermal expansion produces thermal stresses in the material leading to counter-bending of the sheet away from the laser beam. The yield strength and the flow stress of the material decreases with increasing temperature, as shown in Figure 1. When the generated thermal stress exceeds the yield stress of the material, further expansion of the heated region results in into compressive plastic deformation. After the laser scan, the surface is rapidly cooled which causes thermal contraction. The thermal contraction results in the local shortening of the surface layers thus causing the sheet to bend towards the laser beam, as illustrated in Figure 2. Figure 1 Effect of temperature on the flow stress of a material [21].

4 416 M.S. Che Jamil et al. Figure 2 TGM of laser bending for (a) the heating process and (b) the cooling process [11]. 2 NUMERICAL MODELLING 2.1 Finite element equations The finite element equation for transient heat transfer analysis can be expressed as [22] C T K T Qt (1) [ ] +[ ]{ }= { () } where [C] is the specific heat matrix, T is the time derivative of temperature, [K] is the heat conductivity matrix, {T} is temperature and {Q(t)} is the heat flux. The material used in this model is (D36) which is shipbuilding steel. Temperature dependency of the material properties was considered including Poisson s ratio, elastic modulus and yield stress, as shown in Table 1. The nonlinear transient dynamic structural equation based on FEM can be written in the matrix form as [ MT ( )]{ u'' ()} t + [ DT ( )]{ ut '()} + [ KT ( )]{ ut ()} = { Ft ()} + { F th ()} t (2) where [M(T)] is the temperature dependent mass matrix,[d(t)] is the temperature dependent damping matrix, [K(T)] is the temperature dependent stiffness matrix,{f(t)} is the external load vector, {F th (t)} is the thermal load vector, {u"(t)} is the acceleration vector, {u'(t)} is the velocity vector and {u(t)} is the nodal displacement vector. For the case of transient thermal stresses generated due to laser scanning with no damping or external forces/ loadings, Equation (2) can be reduced to {F th (t)} can be evaluated by using [ KT ( )]{ ut ()} = { F th ()} t (3) ( ) { F } [ B] E T T th T α = δdv 1-2v vol (4)

5 Temperature Gradient Mechanism in Laser Forming 417 TABLE 1 Material properties of the ship building steels (D36) used in this work [23]. Temperature (K) Young s modulus (MPa) Tangent modulus (MPa) Poison s ratio Yield stress (MPa) Specific heat (J/kg o C) 427 Thermal conductivity (W/m o C) 35.1 Density (kg/m 3 ) 7860 Coefficient of thermal expansion (K -1 ) where [B] is the strain displacement matrix, δ = [ ] T, E is the elastic modulus, v is Poisson s ratio and T is the temperature difference. Total strains can be evaluated from nodal displacements by using { ε}= [ B]{ u } (5) where {u} is the nodal displacement vector. The difference between total strain and thermal strain gives mechanical strain which consists of plastic and elastic strain components and are expressed as th pl el {}-{ ε ε }= { ε }+{ ε } (6) where {ε} is the total strain vector, {ε el } is the elastic strain vector, {ε pl } is the plastic strain vector and {ε th } is the thermal strain vector. 2.2 Specimen material and dimensions In the present numerical study, the commercial non-linear finite element code, ANSYS is used. The model parameters were: dimension of the plate mm 3 thickness; plate material - shipbuilding (D36) [23]; laser power 1.5 kw; laser beam velocity 0.3 m/min and laser beam diameter 16 mm. The problem is symmetric and therefore only half of the plate is modelled with symmetric boundary conditions, as shown in Figure 3. Eight noded three-dimensional (3-D) brick elements (ANSYS: type SOLID70) with thermal conduction capability are used for the thermal analysis. For structural analysis, ANSYS: SOLID185 type elements are used which are defined by

6 418 M.S. Che Jamil et al. Figure 3 Boundary conditions for the half plate modelled. Figure 4 FE meshing for the half plate modelled. eight nodes having three degrees of freedom (u x, u y, u z ) at each node. The element has plasticity, large deflection, and large strain capabilities [24]. The finite element (FE) mesh is shown in Figure 4. The laser heating was modelled as moving heat flux with ANSYS Parametric Design Language (APDL) used to incorporate the laser beam motion. The laser beam intensity is assumed to be constant over the beam area. All exposed surfaces are subjected to convection but heat losses due to radiation are not considered as they are very small. 2.3 Laser beam geometries To investigate the effect of various beam geometries on the laser bending process, five beam geometries with different dimensions along the axial and transverse directions of the scan were chosen as shown in Figure 5. The area of each beam was kept constant at 200 mm 2 to maintain the power intensity and hence the same amount of energy transferred into the specimen.

7 Temperature Gradient Mechanism in Laser Forming 419 Figure 5 Laser beam geometries with similar effective area used in the FE simulations. The basic assumptions for the model are: (i) Material is isotropic; (ii) Laser intensity distribution is assumed to be uniform; (iii) Bauschinger s effect is neglected; (vi) Von-Misses criterion is used for plastic yielding; and (v) Energy dissipation due to plastic deformation is neglected when compared with the energy involved in the thermal process. 3 RESULTS AND DISCUSSION 3.1 Thermal analysis Figure 6 shows temperature against time for all the five beam geometries measured at the centre of scanning paths (0, 6, 75). For each beam, the temperature starts rising approximately when the beam enters the reference point. The temperature increases rapidly until the beam leaves the point and the temperature starts dropping. From Figure 6 it is seen that the beams that have a longer dimension in the scanning direction with respect to its lateral dimension, produce higher temperatures despite of the fact that the power intensity and the beam effective area is similar for all beams. This is mainly due to the longer beam-material interaction for the beams with longer dimensions in the scanning direction. The duration of the beam-material interaction, given by t s, is the ratio of the axial length, d ax, to the scanning speed, v (t s = d ax /v). Beam REC-V, which has the longest axial length, gives the highest temperature of 1240 o C; however, its lateral temperature spread (along the x direction) is narrower when compared to the other beams. 3.2 Structural analysis Stress and strain behaviour To investigate the effect of different beam geometries on stress and strain behaviour of the material, stress-strain curves are plotted. Stresses and strains in the x-direction are chosen due to its significant contribution to the bending

8 420 M.S. Che Jamil et al. Figure 6 Temperature versus time at reference point (0, 6, 75). formation. The strain, ε x, consists of plastic and elastic component. The measurement is made on top surface (see Figure 3) at the middle of the scanning path (0, 6, 75). A typical stress-strain path during a laser scan is shown in Figure 7. Generally, the steps could be divided into four stages which are; pre-heating, heating, post-heating and cooling. The stages are explained as follows. Preheating. This is the stage when laser scan starts and moves towards the measured point. The plate is assumed to be free from any residual stresses and strains and hence the plot starts at origin (0, 0) in Figure 7. At the beginning of the scan, the stress and strain are tensile. This is due to the expansion of the heated region behind the measured point. The maximum tensile stress is produced just before the laser beam reaches the measured point. Heating. This is the stage when the measured point is heated by the beam. During heating, the stress rapidly changes into compression. This is because the material expansion is restricted by the surrounding material (path 1-a). With increasing temperature during the heating stage, the material flow stress reduces (step a-b) due to the effect of the temperature-dependent material properties (see Figure 1). Compressive plastic strain continues to develop at constant flow stress (step b-2). Post-heating. This is the stage when laser beam leaves the reference point and moves towards the exit path. At this point, the temperature starts to drop. As the temperature drops, the thermal strain reduces leaving plastic and residual elastic strain (path 2-c). In addition, compressive stress reduces and becomes tensile (path 2-3). Cooling. This is the period after the laser scan path is completed. During this stage, the temperature and thermal strain continue to drop. The compres-

9 Temperature Gradient Mechanism in Laser Forming 421 Figure 7 Typical stress-strain path during a laser scan. sive plastic strain in x-direction increases slightly due to the thermal shrinkage in both the z- and y-direction. Figure 8 shows the stress-strain curve for rectangular and square beams measured at the reference point. The purpose of this plot is to show the effect of beam width and length on stress and strain. Triangular beams are purposely omitted since its width changes across its length. Figure 8 indicates that REC-V has the highest tensile stress at the end of the pre-heating stage with a value of 375 MPa, followed by SQUARE and REC-H at about 100 MPa. This is due to the fact that REC-V, has the longest longitudinal dimension (25 mm) which is causing more material to expand along the heated path behind the measured point. Therefore, a high tensile stress is induced at the reference point. During heating, the highest temperature is attained by REC-V. This causes higher thermal strains and larger material expansions. The expansion is restricted by the surrounding material and causes high compressive stress. It is noticeable from Figure 8 that REC-V has the lowest residual tensile stress in the x-direction at the end of the cooling period with a value of 57 MPa. This is followed by 98 MPa for SQUARE and 143 MPa for REC-H. It has been established that lower pre-stresses on the heated surface produce more compressive plastic strains and improve bending angle [25]. This is an advantage for REC-V beam if it is used in multiple scans laser forming where it can provide lower pre-stress for subsequent scans. It can be seen from Figure 8 that the magnitude of plastic strain for REC-V is quite high and close to the SQUARE beam. On the other hand, REC-H has the lowest plastic strain. This is due to its lower temperature distribution (maximum temperature < 1000 o C) and hence higher flow stress. REC-H also has the highest residual stress at the end of the cooling period. The large distortion (edge effect) along the scanning edge is a possible reason for this.

10 422 M.S. Che Jamil et al. Figure 8 Stress-strain curve for rectangular and square beams measured at reference point (0, 6, 75) Plate deformation Table 2 shows the results of the displacements for each beam. The SQUARE beam has the largest average bending angle followed by the REC-H, TRI-F, REC-V and TRI-R. The displacements of the free edge were measured at both the beam entrance side, UY i (150, 6, 0), and the beam exit side, UY f (150, 6, 150) to observe the edge effects. For all cases, higher displacement occurs on the beam exit side rather than the entrance side. This agrees with general findings and has been established to be caused by a higher temperature at the exit path rather than the entrance [26]. From Table 2 it is interesting to note that REC-V and TRI-F both produce the lowest percentage angle variation compared to the rest of the beams. This could definitely help in cases where high tolerance accuracy is needed especially when a small angle adjustment is required in a single pass scan. REC-H produces the highest edge effect with 10.7% angle variation between the entrance and the exit angle, followed by TRI-R with 9.3%. From this result, it could be concluded that a beams with a narrow leading edge such as REC-V and TRI-F produces less bending angle variation. On the other hand, beam with wider leading edge such as REC-H and TRI-R produces higher bending angle variation. Another type of edge effects is the curvature of the scanning line which is shown in Figure 9. All beams produce concave profiles. Again, REC-H produces the highest edge effect with the maximum displacement of 0.14 mm. Interestingly, both the triangular beams, TRI-F and TRI-R, produce lower distortion along the scanning line with the maximum displacements around 0.07 mm. However TRI-R produces a variation of 9.3% in the bending angle along the scanning line, while TRI-F produces only 5.4% variation (see Table 2). This leads to the conclusion that TRI-F is the better option for lower edge effects.

11 Temperature Gradient Mechanism in Laser Forming 423 Table 2 Results of displacement and bending angle. Parameter Unit REC-H SQUARE REC-V TRI-F TRI-R Beam entrance side, UY i mm Beam exit side, UY f mm UY=UY f -UY i mm % variation (Edge effect) 10.7% 6.9% 4.5% 5.4% 9.3% UY ave mm Average bending angle, α b Figure 9 Vertical displacement along the scanning path ((0, 6, 0) to (0, 6, 150)). Figure 10 shows the plastic strain distribution at the end of the cooling period for the rectangular and square beams. As can be seen, the width of the plastically deformed zone on the irradiated surface is about the width of the laser beam. This is the cause for different deformation behaviour from different beam geometries. The sum of plastic strains is zero at any given time based on the assumption of constant volume during plastic deformation, defined by dε + dε + dε = 0 (7) x y z Generally, the materials under the spot scan shrink in the x-direction (transverse) and expand in the thickness direction. Small shrinkage also occurs in the z-direction (laser beam movement direction) which contributes to the edge effect.

12 424 M.S. Che Jamil et al. Figure 10 Plastic strain distribution at t = 100 s for (a) REC-H, (b) SQUARE and (c) REC-V.

13 Temperature Gradient Mechanism in Laser Forming 425 Figure 11 shows the plastic strain, ε x, measured in the middle of the plate from the top surface (0, 6, 75) to the bottom (0, 0, 75). Generally, for the TGM processes one would expect the plastic strain, ε x, to be higher at the top surface and gradually decreased through the thickness which is true for all beams in Figure 11 except REC-V. For REC-V the maximum plastic strain occurs at 2 mm below the top surface which suggests that an upsetting mechanism (UM) occurs besides TGM. This explains the reason for the low bending angle for this beam, despite having the highest maximum temperature and plastic strain Bend radius One of the most crucial factors influencing the quality in laser bending is the bend radius. In conventional sheet metal bending, there is a range of achievable bend radii, which depend on sheet thickness, modulus of elasticity and yield strength [27]. If the bend radius is too small, the material at the outside of the bend tends to crack or fracture. On the other hand, if the radius is too large the bend will be very hard to control and will spring back erratically. To investigate the quality of the bend radius, curvatures across the bend are measured. Curvature is a reciprocal of the radius of curvature (k = 1/r). Zero curvature means a straight line, while a large curvature means an arc with a small bending radius. Figure 12 shows the curvature measured at the bottom surface at the middle of the scanning track (plane z = 75 mm). Distance along the horizontal axis is measured from the plane of symmetry (x=0) or the bending edge. The plot shows that REC-V has the highest curvature of 6 m - 1 at the bending edge (x = 0). The curvature drops rapidly and becomes Figure 11 Plastic strain through the thickness direction.

14 426 M.S. Che Jamil et al. Figure 12 Curvature versus distance (measured at bottom surface on plane z = 75 mm). nearly zero at distance 6 mm. This shows that a small bend radius is achievable by a narrower beam width. On the contrary, REC-H has the lowest curvature with a value of around 1 m -1. The curvature is consistent along the first 5 mm from the axis of symmetry, before it slowly reduces and becomes nearly zero at 10 mm. The unchanged curvature denotes a circular arc with a constant bend radius which is normally used in product design. For the SQUARE beam, as expected, the curvature lies in between REC-V and REC-H. The bend radius for each beam is directly related to its plastic strain distribution. Small radius for REC-V is due to high compressive plastic strain developed in a localized region around the 8 mm width beam, as shown in Figure 10(c). On the other hand, a larger and consistent bend radius for REC-H is produced by the wider plastic strain spread along the transverse direction. These results indicate that laser beam geometry could play a vital role in controlling the desired bend radius in laser forming. The subject of bending radius in a single scan laser forming has rarely been discussed in previous research works. These findings suggest that beam geometry could provide a degree of control on bending angle, edge effect and bend radius. 4 SUMMARY AND CONCLUSIONS A numerical study of the effects of five different beam geometries on laser bending of sheet metal, dominated by temperature gradient mechanism (TGM), has been carried out. From the thermal analysis, it was established that the maximum temperature of the specimen depends strongly on the dura-

15 Temperature Gradient Mechanism in Laser Forming 427 tion of the beam-material interaction. Higher temperatures influence the development of higher plastic strain due to the temperature-dependent yield stress. Beam REC-V, despite having the highest temperature and plastic strain, has the small bending angle due to the presence of the upsetting mechanism together with TGM. The width of the plastically deformed zone which is related to the transverse width of the beam, contributes to the different plate deformation behaviours by different beams. For the parameters tested, the SQUARE beam produces the highest bending angle followed by REC-H, TRI-F, REC-V and TRI-R. In terms of the bending angle variations along the scanning path, REC-V and TRI-F produce smaller variations of 4.5 and 5.4%, respectively. REC-H and TRI-R, on the other hand, produce higher variations in the angle of 10.7 and 9.3%, respectively. It could be concluded that the beams with a narrow leading edge are preferable to produce less bending angle variations compared to the ones with wider leading edge. In terms of the scanning line curvatures, REC-H produces the highest distortion with maximum displacement of mm. On the other hand, both triangular beams produce lower distortion with the maximum displacement of around 0.07 mm. Considering the two types of edge effects, TRI-F is seen as the best beam for applications where high accuracy tolerance is needed especially when a small angle adjustment is required in a single pass scan. A study of bending curvature using rectangular and square beams shows that bend radius is significantly influenced by beam geometry. A wider beam produces bending with a large bend radius, while a narrower beam produces a smaller bend radius. This study forms the basis for further investigations on the effects of different laser beam geometries in laser forming. It is established that laser beam geometry is an important controlling parameter in laser forming and should be exploited further. REFERENCES [1] Kitamura N. Technical Report of Joint Project on Material Processing by High Power Laser. JWES-TP-8302 (1983), [2] Dearden G. and Edwardson S.P. Some recent developments in two and three-dimensional laser forming for macro and micro applications. Journal of Optics A: Pure and Applied Optics 5 (2003), S8 S15. [3] Shen H. and Vollertsen F. Modelling of laser forming - A review. Computational Material Science 46 (2009), [4] Ji Z. and Wu S. FEM simulation of the temperature field during the laser forming of sheet metal. Journal of Materials Processing Technology 74(1 3) (1998), [5] Kyrsanidi A.K., Kermanidis T.B. and Pantelakis S.G. Numerical and experimental investigation of the laser forming process. Journal of Materials Processing Technology 87 (1999), [6] Zhang L., Reutzel E.W., and Michaleris P. Finite element modeling discretization requirements for the laser forming process. International Journal of Mechanical Sciences 46(4) (2004),

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