- Email: [email protected]
FE simulation of laser curve bending of sheet metals
Journal of Materials Processing Technology 184 (2007) 157–162
FE simulation of laser curve bending of sheet metals Peng Zhang a,∗ , Bin Guo a , De-Bin Shan a , Zhong Ji b a
Harbin Institute of Technology, Harbin 150001, PR China b Shandong University, JiNan 250061, PR China
Received 9 February 2006; received in revised form 14 November 2006; accepted 18 November 2006
Abstract Laser bending is a highly flexible sheet metal forming technique. For the production of complex shaped or spatially curved parts, it is more convenient and efficient to apply curved irradiation paths instead of linear paths. In this paper, a finite element model of heat flux based on scanning path described with B-spline curve was built. Then, FE simulation of laser beam scanning on the forming sheet metals was carried out. And transient temperature fields, displacement fields, stress fields and strain fields were investigated. The proposed model can be used to forecast laser curve bending of sheet metal conveniently. The simulated results show that: (1) there is a good agreement between the FE simulations based on the proposed model and the experiments; (2) the peak temperatures of the upper surface increase when the laser power or the path curvature increases, but decrease when the laser spot diameter or the scanning velocity increases. The peak temperatures increase roughly when the laser energy density increases; (3) the laser curve bending produces a significant change of distortion under the same conditions of deformation. The warped curvature increases when the laser energy density or the path curvature increases. © 2006 Elsevier B.V. All rights reserved. Keywords: Laser bending; Sheet metal; Scanning path; FE simulation
1. Introduction Laser bending is a new type of flexible manufacturing process for sheet metals. The mechanism is the forming caused by thermal stress resulted from irradiation of laser beam scanning [1]. In the laser bending process, no external forces or dies are required, so it can be used to adjust welded constructions and form the prototypes, such as the samples of cars and airplanes. One of the main advantages is that laser bending is the accumulative forming under the condition of thermal state. Furthermore, laser bending is capable of shaping hard or brittle metals, such as Ti-alloys, while traditional methods proved to be either inapplicable or highly labor consuming. As a consequence of the thermal induced forming process, no springback occurs in the cooling phase after the laser is switched off. This effect is mainly responsible for the high achievable working accuracies, especially in the case of controlled laser bending [1–3]. In addition, laser bending can be used for compound process with forming, cutting and welding easily. Thomson has successfully used laser compound processes to form the car door in high precision [4,5].
Mostly, the applications mentioned above have hitherto been focused on laser forming along linear irradiation paths. For the complex shaped parts, it is more convenient and efficient to apply curved irradiation paths instead of linear paths. Hennige has used laser curve bending processes to form a spherical dome instead of previous linear bending processes [6]. Chen has studied the process of laser curve bending for Ti–6Al–4V alloy sheets [12]. However, the process of laser curve bending and the effect of processing parameters have not been systemically studied. Numerical simulation is a strong tool of analyzing the mechanism and optimizing the processing parameters of laser bending. Many research have been conducted on the FE simulation of laser bending [7–10]. So, in this study, a finite element model of heat flux based on process route described with B-spline curve was built, temperature fields, displacement fields, stress fields and strain fields under provided conditions were analyzed according to the FE simulation results. Additionally, the effect of processing parameters on the temperature fields and displacement fields were also investigated. 2. FE model of heat flux
∗
Corresponding author. Tel.: +86 451 86402775. E-mail address: [email protected] (P. Zhang).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.11.017
Fig. 1 shows a schematic diagram of the laser curve bending process of sheet metals. In the coordinates, the general
158
P. Zhang et al. / Journal of Materials Processing Technology 184 (2007) 157–162
Fig. 1. Schematic of laser bending process.
mathematical model of the three-dimensional transient temperature can be written as: ∂T ∂T ∂ ∂T ∂ ∂T ∂ ρc λ + λ + λ +Q (1) = ∂x ∂x ∂y ∂y ∂z ∂z ∂t where T is the temperature (◦ C), which is a function of coordinate and time t (s), ρ the material density (kg/mm3 ), c the heat capacity (J/kg ◦ C), λ the thermal conductivity (W/mm ◦ C) and Q is the source of heat being expressed as heat generation per unit volume (J/mm3 ). However, this term can be neglected especially as it is much smaller than the heat-transfer terms. If T0 is the temperature of the environment, the initial condition is: T |t=0 = T0
(2)
During the simulation, the thermal load can be given in the form of the heat flux density that obeys a normal distribution as follows: 2AP 2r 2 (3) I= exp − πR2 R2 where I is the thermal flux density of the laser beam, A the absorption coefficient on the sheet metal surface, P the laser beam output power, R the radius of the laser beam irradiated to the surface of the sheet metal, and r is the distance from the laser beam center. In the focused laser spot, the beam power P is input as an external heat flux vector: k
∂T = −Iz ∂z
(4)
The other boundary condition is as follows: K
∂T = h(T − T0 ) + kr (T − T0 ) ∂n
(5)
where n expresses the direction of the surface, h is the natural convection exchange coefficient, and kr = εσ(T 2 + T0 2 )(T + T0 ), where ε is the surface emissivity and σ is the Boltzmann constant. Based on the thermal elastic–plastic FEM, A threedimensional coupled thermal–mechanical model for numerical simulation was used. And the heat flux model based on scanning path was built with the development of the finite element code
Fig. 2. Flow chart of heat model based on scanning path.
ANSYS. The flow chart of heat flux model is shown in Fig. 2. The scanning path was designed by parameterized curve and described by B-spline curve. A B-spline curve P(t), can be defined as: P(t) =
n
Pi Ni,k (t)
(6)
i=0
where the Pi (i = 0, 1, . . ., n) are the control points, k is the order of the polynomial segments of the B-spline curve. Order k means that the curve is made up of piecewise polynomial segments of degree k − 1, the Ni,k (t) (i = 0, 1, . . ., n) are the normalized Bspline blending functions. They are described by the order k and by a non-decreasing sequence of real numbers. {ti : i = 0, 1, . . ., n + k} normally called the “knot sequence”. The Ni,k functions can be described as follows: 1 ti ≤ x ≤ ti+l Ni,1 (t) = (7) 0 otherwise and if k > 1, Ni,k (t) =
t − ti ti+k − t Ni,k−l (t) + Ni+l,k−l (t) ti+k−l − ti ti+k − ti+l
(8)
where {t0 , t1 , . . ., tn+k } is a non-decreasing sequence of knots, and k is the order of the curve. The order k is independent of the number of control points (n + 1). The normalized B-spline blending function Ni,k (t) is positive if and only if t ∈ [ti , ti+k ]. In the B-spline curve, many control points can be used flexibly, and the degree of the polymonial segments can be restricted. These functions are difficult to calculate directly for a general knot sequence. However, if the knot sequence is uniform, it is quite straightforward to calculate these functions. Assume that {t0 , t1 , . . ., tn } is a uniform knot sequence. This will simplify the calculation of the blending functions. And the scanning path can be described exactly by B-spline curve. Simulation of the laser bending process is complex and difficult. Therefore, analytical simulation tends to be case specific. The basic assumptions of the proposed model can be summarized as:
P. Zhang et al. / Journal of Materials Processing Technology 184 (2007) 157–162
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(i) The laser beam is at a steady speed and energy follows a Gaussian distribution. (ii) The material is isotropic. (iii) Variation of material parameters with temperature can be obtained by means of linear interpolation. (iv) The laser bending is being performed under the melting temperature of the material. 3. Simulation results and discussion The laser bending process of Ti–6Al–4V alloy sheet was studied. The length of the sheet was 50 mm, the width (along the scanning direction) was 40 mm and the thickness was 0.8 mm. The material physical property parameters were taken from literature [11]. The processing parameters were 1000 W for laser power, 66.67 mm/s for scanning velocity and 5.2 mm for laser spot diameter. Four control points in the scanning path were given as P1 (0, 40), P2 (13.33, 43.72), P3 (26.67, 43.72) and P4 (40, 40). 3.1. Transient temperature fields Fig. 3 shows the temperature distribution on the upper surface. Note that there is a pre-heated zone in front of the laser spot and a “tail” behind the laser spot because of the heat transfer. At the edge of the sheet metal the peak temperatures are higher than that in the middle, the reason is that the heat flux conducts inefficiently. As can be seen in Fig. 4, radiation of the laser beam yields to a rapid temperature increase at the irradiated surface, which leads to high temperature gradients between the upper surface and the lower surface. 3.2. Displacement fields Fig. 5 shows the distributions of displacement on the upper surface of the sheet in the z-direction. It can be seen that a noticeable counter-bending occurs due to thermal expansion in the heated zone when the laser beam has just scanned the edge of the sheet. Along with the laser spot moving away from the sheet edge, the heated surface begins to shrinkage and the lower surface begins to expand due to heat transfer, which causes the deformation in the direction of the laser beam. The final displacement can be gained while cooling to room temperature. The final sheet edge is warped and dissymmetric due to laser curve scanning, as shown in Fig. 6. Fig. 3. The temperature distribution along with time.
3.3. Stress and strain fields The simulated temperature distributions referred above, the yield stress and plastic strain distributions displayed in Figs. 7 and 8. It can be seen that compressive stresses occur in the heated zone because the materials around the heated zone are cold, and slight tensile stresses occur in the neighbouring areas. The upper surface keeps compressive strain during the process of heating. And there exists a large strain difference between the upper surface and the lower surface in the three
directions in the heated zone, which leads to the 3D bending deformations. 3.4. Comparison between simulated and experimental results The experimental results can be utilized to evaluate the simulation of laser bending process based on the proposed FE model.
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Fig. 4. The temperature peak of the upper and lower surface along with time.
Fig. 5. The z-displacement of a node along with time.
Fig. 9 shows the simulated results during the laser bending process and the experimental results obtained from literature [12]. The bending angle, which can be obtained through the average value of several points’ bending angle, can be used to scale the distortion of sheet [12]. As shown in Fig. 6, the results demonstrate good correlation. 3.5. Effects of processing parameters on peak temperature In fact, the laser forming is a thermo-mechanical coupling process, but solving the coupling problem is too complicated.
Fig. 6. z-Displacement of the free end of sheet metal.
Fig. 7. The stress distribution at t = 0.05 s.
The main mechanism of the laser bending process is the temperature gradient mechanism. Fig. 4 shows a steep thermal gradient into the material that results in a differential thermal expansion through the thickness due to the rapid heating of the sheet surface by a laser beam. For the spatial sheet, the temperature gradients are mainly concerned with the peak temperatures of laser-irradiated surface. So, the peak temperature was chosen as the evaluation parameter. And the parameters on the temperature fields of the laser power, the laser spot diameter, the scanning velocity and the scanning path were investigated systemically using the FE simulation. Figs. 10–12 show the processing parameters that affect on the temperature fields including laser power, spot diameter and scanning velocity. The peak temperatures increase when the laser power increases. The peak values decrease when the laser spot diameter or the scanning velocity increases. But the relationships between peak temperatures and processing parameters are non-linear. Laser power P, scanning velocity v and laser spot diameter d are coupled into the laser energy density (I = P/dv) and the effect of energy density on the peak temperature were investigated in Fig. 13. It can be observed that the peak temperatures increase roughly when the laser energy density increases.
P. Zhang et al. / Journal of Materials Processing Technology 184 (2007) 157–162
161
Fig. 10. Variation of peak temperature with laser power.
Fig. 11. Variation of temperature with spot diameter.
Fig. 8. The strain distribution at t = 0.0.05 s.
Fig. 12. Variation of temperature with scanning velocity.
Fig. 9. Comparison between FE simulation and experiment.
Fig. 13. The effect of laser energy density on peak temperature and warped curvature.
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be used to evaluate the extent of warped distortion. As shown in Fig. 13, it can be concluded that the warped curvature of distortion increases when the laser energy density increases. As shown in Fig. 15, the warped curvature increases when the path curvature increases. The reason is that shearing strength cause the sheet 3D deformation during the laser curve scanning. 4. Conclusions
Fig. 14. The peak temperature of linear and curvature scanning.
Fig. 15. The effect of path curvature on peak temperature and warped curvature.
The convex polygon is one of B-spline’s characteristics. The curvature of a point in the convex curve can be defined as: α dα 1 = = K = lim (9) ds r s→0 s where α is the tangent angle, s is the arc length, r = |[(1 + y2 )3/2 ]/y | can be defined as the radius of the tangent circle. During the FE simulation, the coordinate (xn , yn ) of a series of points in the scanning path can be gained from the FE model. Then yn and yn can be obtained using differential calculation, and the average curvature of the curve can be gained. Fig. 14 shows the peak temperature of linear and curvature scanning with time. It can be observed that the peak temperatures remains constant and the 3D transient field come to a quasi steady state after scanning a distance of about two spots. At the edge of the sheet metal, the peak value is higher than that in the middle. The reason is that the heat conducts weaker at the edge of sheet. As shown in Fig. 15, the peak temperatures increase when the path curvature increases. 3.6. Effects of processing parameters on distortion As mentioned above, the warped distortion is a distinct phenomenon of laser curve bending. The warped curvature, which is calculated through the z-displacement of the free end, can
(1) A finite element model of heat flux based on scanning path described with B-spline curve was built with the development of ANSYS. The proposed model can be used to make the process simulation of scanning path manageable. (2) FE simulation of transient temperature fields, displacement fields, stress fields and strain fields produced by laser beam scanning on the forming sheet metals were carried out. The simulated results are in good agreement with experimental results. (3) The peak temperatures of the upper surface and lower surface increase with the increase of the laser power, but decrease with the increase of the laser spot diameter or the scanning velocity. The peak temperatures increase roughly with the increase of energy density. The peak temperatures in the steady state increase when the path curvature increases. (4) The laser curve bending produces a significant change of distortion under the same conditions of deformation. The warped curvature increases with the increase of the laser energy density or the path curvature. References [1] M. Geiger, Synergy of laser material processing and metal forming, Ann. CIPP 43 (2) (1993) 563–570. [2] H. Frackiewicz, Z. Mucha, Sheets and cups by laser forming (in German), Laser-Praxis (1990) 111–113. [3] A. Rief, M. Geiger, Process development for combined laser beam welding and cutting (in German), DVS-Ber. Bd. 135 (1991) 87–94. [4] G. Thomson, M. Pridham, Prototype and part manufacture using laser forming with feedback control. IMC-14, Trinity Coll. Dublin Set (1997) 753–761. [5] G. Thomson, Improvements to laser forming through process control refinements, Opt. Laser Technol. (30) (1998) 141–146. [6] T. Hennige, Development of irradiation strategies for 3D-laser forming, J. Mater. Process. Technol. 103 (2000) 102–108. [7] F. Vollertsen, M. Geiger, W.M. Li, FDM-and-FEM simulation of laser forming: a comparative study, in: Proceedings of the 4th ICTP, Beijing, 1993, pp. 1793–1798. [8] J. Zhong, S. Wu, FEM simulations of temperature field during the laser forming of sheet metal, J. Mater. Process. Technol. 74 (1998) 89–95. [9] S. Wu, J. Zhong, FEM simulation of the deformation field during the laser forming of sheet metal, J. Mater. Process. Technol. 121 (2002) 269–272. [10] An.K. Kyrsanidi, Th. Bkermanidis, Sp.G. Pantelakis, Numerical and experimental investigation of the laser forming process, J. Mater. Process. Technol. 87 (1999) 281–290. [11] B. Rodney, W. Gerhard, Materials Properties Handbook: Titanium Alloy, ASM International, Materials Park, OH, 1994. [12] D.J. Chen, S.C. Wu, M.Q. Li, Studies on laser forming of Ti–6Al–4V alloy sheet, J. Mater. Process. Technol. 152 (2004) 62–65.
FE simulation of laser curve bending of sheet metals Peng Zhang a,∗ , Bin Guo a , De-Bin Shan a , Zhong Ji b a
Harbin Institute of Technology, Harbin 150001, PR China b Shandong University, JiNan 250061, PR China
Received 9 February 2006; received in revised form 14 November 2006; accepted 18 November 2006
Abstract Laser bending is a highly flexible sheet metal forming technique. For the production of complex shaped or spatially curved parts, it is more convenient and efficient to apply curved irradiation paths instead of linear paths. In this paper, a finite element model of heat flux based on scanning path described with B-spline curve was built. Then, FE simulation of laser beam scanning on the forming sheet metals was carried out. And transient temperature fields, displacement fields, stress fields and strain fields were investigated. The proposed model can be used to forecast laser curve bending of sheet metal conveniently. The simulated results show that: (1) there is a good agreement between the FE simulations based on the proposed model and the experiments; (2) the peak temperatures of the upper surface increase when the laser power or the path curvature increases, but decrease when the laser spot diameter or the scanning velocity increases. The peak temperatures increase roughly when the laser energy density increases; (3) the laser curve bending produces a significant change of distortion under the same conditions of deformation. The warped curvature increases when the laser energy density or the path curvature increases. © 2006 Elsevier B.V. All rights reserved. Keywords: Laser bending; Sheet metal; Scanning path; FE simulation
1. Introduction Laser bending is a new type of flexible manufacturing process for sheet metals. The mechanism is the forming caused by thermal stress resulted from irradiation of laser beam scanning [1]. In the laser bending process, no external forces or dies are required, so it can be used to adjust welded constructions and form the prototypes, such as the samples of cars and airplanes. One of the main advantages is that laser bending is the accumulative forming under the condition of thermal state. Furthermore, laser bending is capable of shaping hard or brittle metals, such as Ti-alloys, while traditional methods proved to be either inapplicable or highly labor consuming. As a consequence of the thermal induced forming process, no springback occurs in the cooling phase after the laser is switched off. This effect is mainly responsible for the high achievable working accuracies, especially in the case of controlled laser bending [1–3]. In addition, laser bending can be used for compound process with forming, cutting and welding easily. Thomson has successfully used laser compound processes to form the car door in high precision [4,5].
Mostly, the applications mentioned above have hitherto been focused on laser forming along linear irradiation paths. For the complex shaped parts, it is more convenient and efficient to apply curved irradiation paths instead of linear paths. Hennige has used laser curve bending processes to form a spherical dome instead of previous linear bending processes [6]. Chen has studied the process of laser curve bending for Ti–6Al–4V alloy sheets [12]. However, the process of laser curve bending and the effect of processing parameters have not been systemically studied. Numerical simulation is a strong tool of analyzing the mechanism and optimizing the processing parameters of laser bending. Many research have been conducted on the FE simulation of laser bending [7–10]. So, in this study, a finite element model of heat flux based on process route described with B-spline curve was built, temperature fields, displacement fields, stress fields and strain fields under provided conditions were analyzed according to the FE simulation results. Additionally, the effect of processing parameters on the temperature fields and displacement fields were also investigated. 2. FE model of heat flux
∗
Corresponding author. Tel.: +86 451 86402775. E-mail address: [email protected] (P. Zhang).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.11.017
Fig. 1 shows a schematic diagram of the laser curve bending process of sheet metals. In the coordinates, the general
158
P. Zhang et al. / Journal of Materials Processing Technology 184 (2007) 157–162
Fig. 1. Schematic of laser bending process.
mathematical model of the three-dimensional transient temperature can be written as: ∂T ∂T ∂ ∂T ∂ ∂T ∂ ρc λ + λ + λ +Q (1) = ∂x ∂x ∂y ∂y ∂z ∂z ∂t where T is the temperature (◦ C), which is a function of coordinate and time t (s), ρ the material density (kg/mm3 ), c the heat capacity (J/kg ◦ C), λ the thermal conductivity (W/mm ◦ C) and Q is the source of heat being expressed as heat generation per unit volume (J/mm3 ). However, this term can be neglected especially as it is much smaller than the heat-transfer terms. If T0 is the temperature of the environment, the initial condition is: T |t=0 = T0
(2)
During the simulation, the thermal load can be given in the form of the heat flux density that obeys a normal distribution as follows: 2AP 2r 2 (3) I= exp − πR2 R2 where I is the thermal flux density of the laser beam, A the absorption coefficient on the sheet metal surface, P the laser beam output power, R the radius of the laser beam irradiated to the surface of the sheet metal, and r is the distance from the laser beam center. In the focused laser spot, the beam power P is input as an external heat flux vector: k
∂T = −Iz ∂z
(4)
The other boundary condition is as follows: K
∂T = h(T − T0 ) + kr (T − T0 ) ∂n
(5)
where n expresses the direction of the surface, h is the natural convection exchange coefficient, and kr = εσ(T 2 + T0 2 )(T + T0 ), where ε is the surface emissivity and σ is the Boltzmann constant. Based on the thermal elastic–plastic FEM, A threedimensional coupled thermal–mechanical model for numerical simulation was used. And the heat flux model based on scanning path was built with the development of the finite element code
Fig. 2. Flow chart of heat model based on scanning path.
ANSYS. The flow chart of heat flux model is shown in Fig. 2. The scanning path was designed by parameterized curve and described by B-spline curve. A B-spline curve P(t), can be defined as: P(t) =
n
Pi Ni,k (t)
(6)
i=0
where the Pi (i = 0, 1, . . ., n) are the control points, k is the order of the polynomial segments of the B-spline curve. Order k means that the curve is made up of piecewise polynomial segments of degree k − 1, the Ni,k (t) (i = 0, 1, . . ., n) are the normalized Bspline blending functions. They are described by the order k and by a non-decreasing sequence of real numbers. {ti : i = 0, 1, . . ., n + k} normally called the “knot sequence”. The Ni,k functions can be described as follows: 1 ti ≤ x ≤ ti+l Ni,1 (t) = (7) 0 otherwise and if k > 1, Ni,k (t) =
t − ti ti+k − t Ni,k−l (t) + Ni+l,k−l (t) ti+k−l − ti ti+k − ti+l
(8)
where {t0 , t1 , . . ., tn+k } is a non-decreasing sequence of knots, and k is the order of the curve. The order k is independent of the number of control points (n + 1). The normalized B-spline blending function Ni,k (t) is positive if and only if t ∈ [ti , ti+k ]. In the B-spline curve, many control points can be used flexibly, and the degree of the polymonial segments can be restricted. These functions are difficult to calculate directly for a general knot sequence. However, if the knot sequence is uniform, it is quite straightforward to calculate these functions. Assume that {t0 , t1 , . . ., tn } is a uniform knot sequence. This will simplify the calculation of the blending functions. And the scanning path can be described exactly by B-spline curve. Simulation of the laser bending process is complex and difficult. Therefore, analytical simulation tends to be case specific. The basic assumptions of the proposed model can be summarized as:
P. Zhang et al. / Journal of Materials Processing Technology 184 (2007) 157–162
159
(i) The laser beam is at a steady speed and energy follows a Gaussian distribution. (ii) The material is isotropic. (iii) Variation of material parameters with temperature can be obtained by means of linear interpolation. (iv) The laser bending is being performed under the melting temperature of the material. 3. Simulation results and discussion The laser bending process of Ti–6Al–4V alloy sheet was studied. The length of the sheet was 50 mm, the width (along the scanning direction) was 40 mm and the thickness was 0.8 mm. The material physical property parameters were taken from literature [11]. The processing parameters were 1000 W for laser power, 66.67 mm/s for scanning velocity and 5.2 mm for laser spot diameter. Four control points in the scanning path were given as P1 (0, 40), P2 (13.33, 43.72), P3 (26.67, 43.72) and P4 (40, 40). 3.1. Transient temperature fields Fig. 3 shows the temperature distribution on the upper surface. Note that there is a pre-heated zone in front of the laser spot and a “tail” behind the laser spot because of the heat transfer. At the edge of the sheet metal the peak temperatures are higher than that in the middle, the reason is that the heat flux conducts inefficiently. As can be seen in Fig. 4, radiation of the laser beam yields to a rapid temperature increase at the irradiated surface, which leads to high temperature gradients between the upper surface and the lower surface. 3.2. Displacement fields Fig. 5 shows the distributions of displacement on the upper surface of the sheet in the z-direction. It can be seen that a noticeable counter-bending occurs due to thermal expansion in the heated zone when the laser beam has just scanned the edge of the sheet. Along with the laser spot moving away from the sheet edge, the heated surface begins to shrinkage and the lower surface begins to expand due to heat transfer, which causes the deformation in the direction of the laser beam. The final displacement can be gained while cooling to room temperature. The final sheet edge is warped and dissymmetric due to laser curve scanning, as shown in Fig. 6. Fig. 3. The temperature distribution along with time.
3.3. Stress and strain fields The simulated temperature distributions referred above, the yield stress and plastic strain distributions displayed in Figs. 7 and 8. It can be seen that compressive stresses occur in the heated zone because the materials around the heated zone are cold, and slight tensile stresses occur in the neighbouring areas. The upper surface keeps compressive strain during the process of heating. And there exists a large strain difference between the upper surface and the lower surface in the three
directions in the heated zone, which leads to the 3D bending deformations. 3.4. Comparison between simulated and experimental results The experimental results can be utilized to evaluate the simulation of laser bending process based on the proposed FE model.
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P. Zhang et al. / Journal of Materials Processing Technology 184 (2007) 157–162
Fig. 4. The temperature peak of the upper and lower surface along with time.
Fig. 5. The z-displacement of a node along with time.
Fig. 9 shows the simulated results during the laser bending process and the experimental results obtained from literature [12]. The bending angle, which can be obtained through the average value of several points’ bending angle, can be used to scale the distortion of sheet [12]. As shown in Fig. 6, the results demonstrate good correlation. 3.5. Effects of processing parameters on peak temperature In fact, the laser forming is a thermo-mechanical coupling process, but solving the coupling problem is too complicated.
Fig. 6. z-Displacement of the free end of sheet metal.
Fig. 7. The stress distribution at t = 0.05 s.
The main mechanism of the laser bending process is the temperature gradient mechanism. Fig. 4 shows a steep thermal gradient into the material that results in a differential thermal expansion through the thickness due to the rapid heating of the sheet surface by a laser beam. For the spatial sheet, the temperature gradients are mainly concerned with the peak temperatures of laser-irradiated surface. So, the peak temperature was chosen as the evaluation parameter. And the parameters on the temperature fields of the laser power, the laser spot diameter, the scanning velocity and the scanning path were investigated systemically using the FE simulation. Figs. 10–12 show the processing parameters that affect on the temperature fields including laser power, spot diameter and scanning velocity. The peak temperatures increase when the laser power increases. The peak values decrease when the laser spot diameter or the scanning velocity increases. But the relationships between peak temperatures and processing parameters are non-linear. Laser power P, scanning velocity v and laser spot diameter d are coupled into the laser energy density (I = P/dv) and the effect of energy density on the peak temperature were investigated in Fig. 13. It can be observed that the peak temperatures increase roughly when the laser energy density increases.
P. Zhang et al. / Journal of Materials Processing Technology 184 (2007) 157–162
161
Fig. 10. Variation of peak temperature with laser power.
Fig. 11. Variation of temperature with spot diameter.
Fig. 8. The strain distribution at t = 0.0.05 s.
Fig. 12. Variation of temperature with scanning velocity.
Fig. 9. Comparison between FE simulation and experiment.
Fig. 13. The effect of laser energy density on peak temperature and warped curvature.
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P. Zhang et al. / Journal of Materials Processing Technology 184 (2007) 157–162
be used to evaluate the extent of warped distortion. As shown in Fig. 13, it can be concluded that the warped curvature of distortion increases when the laser energy density increases. As shown in Fig. 15, the warped curvature increases when the path curvature increases. The reason is that shearing strength cause the sheet 3D deformation during the laser curve scanning. 4. Conclusions
Fig. 14. The peak temperature of linear and curvature scanning.
Fig. 15. The effect of path curvature on peak temperature and warped curvature.
The convex polygon is one of B-spline’s characteristics. The curvature of a point in the convex curve can be defined as: α dα 1 = = K = lim (9) ds r s→0 s where α is the tangent angle, s is the arc length, r = |[(1 + y2 )3/2 ]/y | can be defined as the radius of the tangent circle. During the FE simulation, the coordinate (xn , yn ) of a series of points in the scanning path can be gained from the FE model. Then yn and yn can be obtained using differential calculation, and the average curvature of the curve can be gained. Fig. 14 shows the peak temperature of linear and curvature scanning with time. It can be observed that the peak temperatures remains constant and the 3D transient field come to a quasi steady state after scanning a distance of about two spots. At the edge of the sheet metal, the peak value is higher than that in the middle. The reason is that the heat conducts weaker at the edge of sheet. As shown in Fig. 15, the peak temperatures increase when the path curvature increases. 3.6. Effects of processing parameters on distortion As mentioned above, the warped distortion is a distinct phenomenon of laser curve bending. The warped curvature, which is calculated through the z-displacement of the free end, can
(1) A finite element model of heat flux based on scanning path described with B-spline curve was built with the development of ANSYS. The proposed model can be used to make the process simulation of scanning path manageable. (2) FE simulation of transient temperature fields, displacement fields, stress fields and strain fields produced by laser beam scanning on the forming sheet metals were carried out. The simulated results are in good agreement with experimental results. (3) The peak temperatures of the upper surface and lower surface increase with the increase of the laser power, but decrease with the increase of the laser spot diameter or the scanning velocity. The peak temperatures increase roughly with the increase of energy density. The peak temperatures in the steady state increase when the path curvature increases. (4) The laser curve bending produces a significant change of distortion under the same conditions of deformation. The warped curvature increases with the increase of the laser energy density or the path curvature. References [1] M. Geiger, Synergy of laser material processing and metal forming, Ann. CIPP 43 (2) (1993) 563–570. [2] H. Frackiewicz, Z. Mucha, Sheets and cups by laser forming (in German), Laser-Praxis (1990) 111–113. [3] A. Rief, M. Geiger, Process development for combined laser beam welding and cutting (in German), DVS-Ber. Bd. 135 (1991) 87–94. [4] G. Thomson, M. Pridham, Prototype and part manufacture using laser forming with feedback control. IMC-14, Trinity Coll. Dublin Set (1997) 753–761. [5] G. Thomson, Improvements to laser forming through process control refinements, Opt. Laser Technol. (30) (1998) 141–146. [6] T. Hennige, Development of irradiation strategies for 3D-laser forming, J. Mater. Process. Technol. 103 (2000) 102–108. [7] F. Vollertsen, M. Geiger, W.M. Li, FDM-and-FEM simulation of laser forming: a comparative study, in: Proceedings of the 4th ICTP, Beijing, 1993, pp. 1793–1798. [8] J. Zhong, S. Wu, FEM simulations of temperature field during the laser forming of sheet metal, J. Mater. Process. Technol. 74 (1998) 89–95. [9] S. Wu, J. Zhong, FEM simulation of the deformation field during the laser forming of sheet metal, J. Mater. Process. Technol. 121 (2002) 269–272. [10] An.K. Kyrsanidi, Th. Bkermanidis, Sp.G. Pantelakis, Numerical and experimental investigation of the laser forming process, J. Mater. Process. Technol. 87 (1999) 281–290. [11] B. Rodney, W. Gerhard, Materials Properties Handbook: Titanium Alloy, ASM International, Materials Park, OH, 1994. [12] D.J. Chen, S.C. Wu, M.Q. Li, Studies on laser forming of Ti–6Al–4V alloy sheet, J. Mater. Process. Technol. 152 (2004) 62–65.