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A theoretical and experimental investigation on limitations of pulsed laser drilling
Journal of Materials Processing Technology 183 (2007) 96–103
A theoretical and experimental investigation on limitations of pulsed laser drilling Konstantinos Salonitis, Aristidis Stournaras, George Tsoukantas, Panagiotis Stavropoulos, George Chryssolouris ∗ Laboratory for Manufacturing Systems and Automation, Department of Mechanical Engineering and Aeronautics, University of Patras, Patra 26100, Greece Received 28 April 2005; received in revised form 5 September 2005; accepted 20 September 2006
Abstract A theoretical and experimental investigation of the limitations of the pulsed laser drilling process is presented in this study. A theoretical model has been developed for simulating the process of drilling with medium irradiance laser beams. It takes into account the required time for reaching melting temperature as well as the melting and the subsequent removal of a volume of material during each laser pulse. The model estimates for a specific laser beam power, a maximum drill depth. The pulsing frequency of the laser beam has no effect on the maximum drill depth. A 1.8 kW CO2 laser has been used for the experimental verification of the theoretical predictions and the fine tuning of the model. © 2006 Elsevier B.V. All rights reserved. Keywords: Laser beam machining; Drilling; Process modeling
1. Introduction Laser drilling (Fig. 1) [1] has a variety of applications. Laser pulsed drilling, also referred to as percussion laser drilling, utilizes several pulses to create the hole. Holes with aspect ratio of up to 1:20 can be achieved. Laser drilling is based on the absorption of the laser energy by the workpiece material and the conversion of the photon energy into thermal energy. When the temperature exceeds that of the melting and/or vaporization, the workpiece material changes phase and the hole geometry is formed. If the laser irradiance is kept below a certain threshold (typically ca. 106 W/cm2 for steels) the workpiece material is melted and not vaporized. In that case, the hole is formed due to ejection of the melted material with the use of an assisting gas jet [2]. For laser irradiance values beyond the threshold value, the material is removed mainly due to vaporization. A number of attempts to simulate the laser drilling process have been reported in the literature utilizing analytical, numerical or empir-
∗
Corresponding author. Tel.: +30 2610 99 72 62; fax: +30 2610 99 77 44. E-mail addresses: [email protected] (K. Salonitis), [email protected] (A. Stournaras), [email protected] (G. Tsoukantas), [email protected] (P. Stavropoulos), [email protected] (G. Chryssolouris). 0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.09.031
ical methods. One of the first theoretical models for predicting the drill depth was presented in [1] for the case of continuous mode laser drilling. The drill depth was estimated as the maximum depth, where melting temperature was reached. The temperature field induced in the workpiece due to laser irradiation can be determined by using analytical [3] and numerical methods [4,5]. The erosion front velocity (Fig. 1) can also be predicted assuming that the absorbing layer moves at a constant velocity into the material [6]. In [7] a model was solved numerically for the direct estimation of the hole depth. More complex models have also been presented taking into consideration the presence of the three different phases (solid, melted and vaporized) in the irradiated area. In [8] such a model was presented having included the presence of two moving boundaries (liquid–vapor and solid–liquid) as well as the conduction heat loss. It was proved that the conduction heat loss has no significant effect on the vaporization process, whereas the thickness of the liquid layer is the most dominant factor. In [9] the laser drilling process was simulated numerically, taking into account the temperature and phase dependence of the thermal properties. In [10] the hole shape and size was predicted via a finite difference heat flow model. In [11] a model was presented for drilling holes with focused Gaussian laser beam, including the positioning of the focal plane, relative to the workpiece surface and the dependence of the drill profile, on laser beam divergence.
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Nomenclature ai
thermal diffusivity for temperature equal to Ti−1 (m2 /s) A irradiation area (m2 ) external area of finite melted volume in contact Acond with surrounding material (m2 ) Aconv area of finite melted volume exchanging heat with the environment (m2 ) fc correction factor h heat transfer coefficient (W/m2 K) ki thermal conductivity for temperature equal to Ti−1 (W/m K) L latent heat of fusion (J/kg) mi finite mass that changes phase during pulse i (kg) M beam quality parameter P laser beam power (W) Q/A power density (W/m2 ) Qcond heat conducted in the workpiece (W) Qconv heat lost by convection to air (W) Qin energy entering workpiece from laser beam in one laser pulse duration (W) QL energy consumed for changing phase (W) r laser beam radius (m) r0 laser beam radius at the workpiece surface (m) R reflectivity of the workpiece material s theoretical estimated drill depth (m) si theoretical melted depth during pulse i (m) st corrected theoretical drill depth (m) toff cooling time between each successive pulse (s) tp laser pulse duration (s) Ti temperature during heating of pulse i (K) temperature during cooling of pulse i (K) Ti T0 ambient temperature (K) T temperature difference between melted surface and surrounding air (K) T/s average temperature decrease rate (K/m) Greek letters δf position of the focal plane relative to the workpiece surface (m) λ laser wavelength (m)
Few theoretical attempts have been reported on the pulsed laser drilling: in [12] modeling of the three dimensional laser grooving for both continuous wave and pulsed laser beams was presented. The pulsed model derived was based on the transformation of the temporal pulse waveform into a spatial waveform for the case of the laser grooving process. The model was verified experimentally for composite materials. A one-dimension model was developed for predicting the geometry of the hole by considering the attenuation of the laser beam within the vapor through an average absorption coefficient in [13]. The pulsing mode of the laser source was taken into consideration through the Heaviside function. In [14] the pulsed laser drilling with high
Fig. 1. Laser drilling process schematic [1].
power density beams was theoretically investigated. The model did not take into account the melting of the workpiece material and assumed that the pulse power overcame the threshold energy required for vaporization. In the present work, a theoretical model is developed that is based on the assumption that during each laser pulse, a finite volume of material is melted and removed. However, this finite volume is not constant, since, while the erosion front is propagating into the material, the laser power density is reduced, due to the increasing distance from the focal plane. The model takes into account only the melting and not the vaporization of the workpiece material; therefore its applicability is limited to low to medium irradiance laser drilling operations. The model’s predictions are discussed and compared with experimental results. 2. Theoretical analysis The heating and subsequent cooling during each pulse, is first considered in order to estimate the required time for reaching melting temperature on the workpiece surface. The second step considers the melting of the finite volume occurring in each laser pulse. The model takes into account the de-focusing effect during the laser drilling process for the estimation of the melted depth during each pulse. 2.1. Estimation of the heating time required for melting For simplifying the analysis, it is assumed that the workpiece is a semi-infinite solid. As a result of the pulsed laser ablation, the workpiece surface is heated and subsequently cooled for a number of pulses, until the melting temperature has been reached. Considering the heating of the workpiece surface during pulse i, it is assumed that the workpiece temperature is at an initial after the cooling of i − 1 pulse. Once the laser temperature Ti−1 pulse has been applied, the workpiece surface is exposed to a constant surface heat flux due to laser irradiation, given from
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equation: (1 − R)P Q = A πr02
(1)
where Q/A is the power density of the surface heat flux, R the reflectivity of the workpiece material, P the laser beam power and r0 is the laser beam radius on the workpiece surface. The differential equation describing the heat transfer on the workpiece surface is: ∂ 2 Ti 1 ∂Ti = ∂z2 ai ∂t
(2)
where ai is the thermal diffusivity for temperature equal to Ti−1 . The initial and boundary conditions are: Ti (z, 0) = Ti−1 ∂Ti Q = − ki A ∂z z=0
(3)
where toff is the cooling time between each successive pulse and erf is the error function. The above system of equations is solved for repeating laser pulses, until the melting temperature has been reached.
(4) 2.2. Estimation of drilling depth
where ki is the thermal conductivity for temperature equal to Ti−1 . Eq. (3) represents the temperature of the workpiece surface after the cooling of i − 1 pulse and Eq. (4) refers to the heat flux due to laser irradiation. The solution of the differential equation gives the temperature field Ti (z, t) and can be determined analytically [15]. For t = tp , the laser becomes inactive and the surface temperature can be determined from equation: Q ai tp /π Ti = Ti−1 + 2 (5) A ki where tp is the laser pulse duration. The heating of the workpiece is followed by a subsequent cooling, which is described from the same differential equation, as with heating (Eq. (1)). Assuming that the workpiece has uniform temperature equal to that on the surface, the initial condition is given from Eq. (6). This assumption may result in a slight under-estimation of the time needed for melting. T (z, 0) = Ti
Fig. 2. Melted volume modeling.
(6)
Besides the heat conduction within the workpiece, heat is lost due to convection to the gas jet, on the workpiece surface, leading to the following boundary condition: ∂T hA(T0 − T )z=0 = − ki A (7) ∂z z=0 where h is the heat transfer coefficient and T0 is the ambient temperature. The above system of equations is solved analytically in [15] so as to derive the temperature field during the cooling. When equating time with the “off” time of the laser (t = toff ), the surface temperature (z = 0) is estimated as: √ h ai toff h2 ai toff Ti = Ti + (T0 − Ti ) 1− exp 1− erf ki ki2 (8)
Once the melting temperature has been reached, every additional pulse results in melting a finite volume of mass (Fig. 2). This volume can be assumed to be cylindrical, having a radius equal to that of the beam at the irradiated erosion front surface. The depth of the melted volume per pulse can be calculated once the melted mass has been determined. Furthermore, while the erosion front propagates into the workpiece, the beam radius changes due to the defocusing of the laser beam. The beam radius is calculated using Eq. (9) as a function of a number of factors and has to be estimated during each pulse. ⎡
2 ⎤1/2 λ(z + δ ) f ⎦ r = r0 ⎣1 + M 2 (9) πr02 where r is the laser beam radius at a depth z from the workpiece surface, M the beam quality parameter, λ the laser wavelength and δf is the position of the focal plane relative to the workpiece surface. 2.3. Estimation of drilling depth per pulse The material, liquefied during the laser “on” time, in each pulse, is assumed to be instantly ejected from the irradiated area and thus, no energy loss for raising the temperature of the molten pool occurs. However, energy is consumed due to convection to the surrounding environment (for example on the air jet flow that might exist for the enhancement of molten material ejection). Further energy is lost due to conduction from the molten pool to the surrounding unmelted material. The energy balance of the molten pool (Fig. 3) can be expressed by the equation: Qin = QL + Qcond + Qconv
(10)
where Qin is the energy that enters the workpiece in a pulse from the laser beam, QL the energy consumed for phase change, Qcond
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beam. According to Grigoryants [16], the critical power density for inducing material melting, varies from 104 to 107 W/cm2 depending on the material thermal properties and on the laser–material interaction time. For typical mild steels (as St.37) the critical power density is ca. 4 × 104 W/cm2 . Therefore, it can be assumed that should the power density become less than this threshold value, the energy input in the workpiece will result in a dynamic equilibrium, without imposing further material melting. 3. Theoretical results
Fig. 3. Energy balance on the molten pool.
the heat conducted in the workpiece and Qconv is the heat lost by convection to air. The energy per pulse that enters the workpiece is given by equation: Qin = (1 − R)Ptp
(11)
The energy consumed for changing the phase of mass m from solid to liquid is estimated as: QL = mi L
(12)
where L is the latent heat of fusion and mi is the cylindrical mass that changes phase during pulse i. The energy conducted to the surrounding workpiece can be estimated by assuming an average heat flux per conducting area: ∂T T Qcond = kT =Tm Acond tp = kT =Tm Acond tp ∂n avg s (13) where Acond is assumed to be the external area of the finite melted volume, being in contact with the surrounding material and (T/s) is the average temperature decrease rate. The energy lost due to convection to the surrounding air is given from equation: Qconv = (hAconv T )tp
(14)
where Aconv is the area of the finite volume that can exchange heat through convection and T is the temperature difference between the melted finite volume surface and the surrounding environment. The depth of the melted finite volume for each pulse is finally estimated from the system of Eqs. (10)–(14). Eq. (15) estimates the depth of the melted finite volume during pulse i. si =
(1 − R)P/πr − kT =Tm r(T/s) − hrT rL/tp + 2kT =Tm (T/s)
In order to solve the model, a numerical module was developed in Visual Basic. The flowchart of the module is presented in Appendix A. Utilizing the developed module, the model was solved for the case of a workpiece made of steel St.37 (Table 1) irradiated by a CO2 pulsed laser beam (Table 2). Thermal conductivity, thermal diffusivity, specific heat and density were temperature dependent properties (Fig. 4) [17] and thus, their values were iteratively calculated for each pulse. In order to estimate the amount of energy required for reaching the melting temperature, the number of pulses required for raising the surface temperature to the melting point was calculated for various values of power density and pulsing frequency (Fig. 5). The analysis revealed that the pulsing frequency has little effect on the amount of energy, required for reaching the melting point for a specific power density. This indicates that the laser “off” time is so small that the heated volume does not have sufficient time to cool. During the “off” time the heated volume remains essentially in constant temperature. In Fig. 6, the cool-
Table 1 Workpiece (St.37) material properties Chemical composition (wt%) Fe C Mn P S
Balance 0.2 0.4 0.04 0.05
Thermal properties Reflectivity Latent heat of fusion Critical power density Heat transfer coefficient Melting temperature Ambient temperature
0.9 2750 kJ/kg 4 × 104 W/cm2 50 W/m2 K 1808 K 298 K
(15)
where si is the theoretical melted depth during pulse i. 2.4. Estimation of the maximum depth per power input While the erosion front propagates into the workpiece, the power density is decreased due to the defocusing of the laser
Table 2 Laser source properties Wavelength Beam radius at the focal plane Beam index, K Mode quality, M2 = 1/K Pulse duration
10.6 m 0.16 mm 0.6 1.66 10 s
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Fig. 7. Estimation of the heating time as function of laser power density.
Fig. 4. St.37 thermal properties.
Fig. 5. Estimation of the required pulses for initiating melting of the workpiece.
Fig. 6. Temperature evolution during heating phase. In the detail the slight cooling during “off” time can be observed.
ing during laser “off” time when irradiating a St.37 workpiece is shown. The laser beam in this case induces power density equal to 1.4 × 105 W/cm2 with high pulsing frequency (10,000 Hz) at the workpiece surface. Additionally, the heat transfer coefficient is considered to be 10,000 W/m2 K, corresponding to an extremely high velocity and turbulent flow of the coolant gas that cannot practically be achieved in real laser drilling operation. However, it is evident from the figure that even for this exaggerating case the cooling frequency is still negligible when compared to that of the heating per pulse. As already mentioned the pulsing frequency determines the amount of energy entering the workpiece per time unit. For larger values of pulsing frequency, the irradiated workpiece surface will be heated up to the material’s melting temperature more rapidly (Fig. 7). The maximum depth that can be drilled with the specific CO2 laser (characteristics found in Table 2) on a St.37 workpiece was estimated based on the criterion of the minimum critical power density, required for melting, as described in the theoretical analysis. The analysis shows that for a specific power density, the pulsing frequency has little effect on the maximum drill depth. As in the case of heating the workpiece surface up to the melting temperature, the analysis shows that during the laser “off” time the erosion front does not have sufficient time to cool. The laser “off” time on the other hand, is of great significance when performing laser drilling operations with high intensity laser beams for the depressing of the laser plume, which however, is not within the scope of the present paper. Furthermore, in the case of through holes, percussion drilling instead of drilling with continuous laser beam, prevents contraction at the exit side from slag formation according to [2]. Fig. 8 presents the maximum drill depth as a function of the power density. The subsequent analysis step was the estimation of the hole depth as a function of the “laser–workpiece material” interaction time and the power density. In Fig. 9, the results for pulsing frequency equal to 500 Hz are depicted. As it can be seen, increasing power density results in deeper holes. Furthermore, as the interaction time is increased, the drill depth is increased, respectively.
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Fig. 8. Maximum drill depth as a function of laser power density. Fig. 10. Theoretical estimations of drill depth as function of the interaction time for various values of pulsing frequency. power that this laser could provide was 1800 W with pulsing frequency ranging from 10 to 10,000 Hz. Two sets of experiments have been conducted, the first one for checking the effect of the pulsing frequency on the maximum depth achieved and the second one for validating the effect of the laser power on the maximum depth and evaluating the model’s predictions. The second set of experiments was also used for “calibrating” the model. The specimens were 10 mm thick rods from St.37. Each one of the experimental drills was repeated five times. The specimens were sectioned along the drill axis, they were ground and the drill depth was measured with an optical microscope (Fig. 11). The average value of the five measured depths was used for comparison with the theoretical predictions. The depth was measured from the material surface to the innermost point of material damage.
4.1. Pulsing frequency effect
Fig. 9. Theoretical estimation of drill depth as a function of laser–material interaction time for various values of power density.
However, once the maximum depth is achieved, the additional energy, entering the workpiece, is entirely dissipated inside the irradiated material, without any further material removal as it has been also pointed out in the theoretical analysis. The pulsing frequency effect on the hole depth is presented in Fig. 10, for power density equal to 8.4 × 105 W/cm2 . The maximum depth that can be achieved with a specific power density does not depend on the pulsing frequency. However, the required interaction time for achieving the maximum drill depth is also different for large values of pulsing frequency; the maximum depth is achieved with less laser–workpiece material interaction time. Furthermore, for comparison reasons, the depth calculated with the model proposed in [1] for continuous mode laser drilling, is shown in Fig. 10.
For three values of the power density and for a number of pulsing frequencies, the laser drilling experiments were conducted with large interaction times so as to achieve the maximum drilling depth. The experimental results are presented in Fig. 12 verifying the theoretical predictions for pulsing frequencies larger than 100 Hz. At low pulsing frequencies (10–100 Hz), the depth measured at each specimen of the five ones processed with the same parameters, presented high deviation. This implies that the laser source at low pulsing frequency cannot reliably provide a beam with the specified laser power.
4. Experimental results In order to verify and calibrate the proposed theoretical model, a number of experiments have been conducted. A CO2 laser source providing a Gaussian laser beam, with characteristics described in Table 2, was used. The maximum
Fig. 11. Specimen No. 12 sectioned and measured.
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Fig. 12. Drill depth measurements as a function of pulsing frequency.
4.2. Maximum drill depth A number of experiments were further conducted for deriving the maximum depth for various power densities. In Fig. 13, the average measurements of the maximum depth is compared with the model’s predictions. The experimental data show a deviation between the relationship slopes and the analytical model predictions. It is shown that the theoretical model overestimates the drill depth, when compared with experimental measurements especially for laser drilling operations with high power densities. This deviation can be attributed to the assumptions of the model for simplifying the analysis. A portion of the energy that is irradiated onto the workpiece, is consumed for further increasing the temperature of the molten pool before this is ejected, which is considered negligible in the model. Furthermore, for laser drilling operations with higher power densities the material may be vaporized inducing a plasma plume above the erosion front that absorbs a part of the irradiated power. A third cause for model/experiment discrepancies can be related to the energy transfer to the gas jet through convection that is also considered negligible in the current study. However, the above energy reducing factors are contradicted by the fact that the coefficient of absorptivity of the workpiece material is temperature dependent and it is increased as the workpiece temperature is raised. In the analysis presented, the coefficient of absorptivity is assumed to have a constant value. In order to compensate these deviations, a linear correction factor has been introduced for calibrating the theoretical model, as shown in Eq. (16). s t = fc s
(16)
where st is the corrected theoretical depth, s the theoretically estimated depth and fc is the correction factor. The factor has been derived by matching the experimental measurements with the theoretical predictions and was found to be ca. 0.80. The comparison
Fig. 13. Comparison between model predictions and experimental results.
Fig. 14. Comparison between experimental results and corrected model predictions. of the experimental measurements and the corrected theoretical predicted drill depths is presented in Fig. 14.
5. Conclusions The limitations posed by the pulsing mode of the percussion laser drilling on the drill depth have been investigated theoretically and have been verified experimentally in the present paper. After examining the test data and the analytical model’s predictions, several conclusions can be drawn: • The irradiation time, required for reaching melting temperature on the workpiece surface depends on the pulsing frequency. • The maximum drill depth that can be achieved is not a function of the pulsing frequency when laser drilling with low to medium power densities. • The theoretical model overestimates the maximum drill depth for high power densities. This deviation is due to the model’s assumptions and has been thoroughly discussed. • In order to improve the accuracy of the drill depth predictions, a linear correction factor was introduced. The agreement between the experimental measurements and the theoretical estimations was improved significantly after having considered a correction factor equal to 0.80. The theoretical model introduced allows the prediction of the minimum power, required for achieving a hole of specified depth. This implies that the laser power optimization will enable the drilling of workpieces with reduced workpiece material damage such as HAZ, formation of tensile residual stresses, generation of cracks, excess oxidation, etc. The present paper introduces a model that does not take into consideration the vaporization of the workpiece material when this is irradiated with high power density. An enhanced model needs to be further developed including the vaporization of the material, the formation of plasma plume in the drilling cavity and
K. Salonitis et al. / Journal of Materials Processing Technology 183 (2007) 96–103
the attenuation of the laser beam within the plume. Furthermore, the applicability of the present model in laser drilling of other materials, such as aluminum and stainless steel grades needs to be further examined. Appendix A. Model solution module Flowchart of the module developed for the numerical simulation of the presented theoretical model.
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References [1] G. Chryssolouris, Laser Machining: Theory and Practice, Springler-Verlag, 1991. [2] LIA, Handbook of laser materials processing, J.F. Ready (Editor in chief), 2001. Magnolia Publishing Inc. [3] B.S. Yilbas, A.Z. Sahin, R. Davies, Laser heating mechanism including evaporation process initiating laser drilling, Int. J. Mach. Tools Manufac. 35 (7) (1995) 1047–1062. [4] J.C. Conde, F. Lusquinos, P. Gonzalez, B. Leon, M. Perez-Amor, Temperature distribution in laser marking, J. Laser Appl. 13 (3) (2001) 105–110. [5] R.K. Ganesh, A. Faghiri, Y. Hahn, A generalized thermal modeling for laser drilling process. II. Numerical simulation and results, Int. J. Heat Mass Transfer 40 (14) (1997) 3361–3373. [6] M. Allmen, Laser drilling velocity in metals, J. Appl. Phys. 47 (12) (1976) 5460–5463. [7] V.N. Tokarev, J. Lopez, S. Lazare, Modeling of high-aspect ratio microdrilling of polymers with UV laser ablation, Appl. Surf. Sci. 168 (2000) 75–78. [8] Y. Zhang, A. Faghri, Vaporization, melting and heat conduction in the laser drilling process 1999, Int. J. Heat Mass Transfer 42 (1999) 1775–1790. [9] R.K. Ganesh, A. Faghiri, Y. Hahn, A generalized thermal modeling for laser drilling process. I. Mathematical modeling and numerical methodology, Int. J. Heat Mass Transfer 40 (14) (1997) 3351–3360. [10] C.F. Cheng, Y.C. Tsui, T.W. Clyne, Application of a three-dimensional heat flow model to treat laser drilling of carbon fibre composites, Acta Mater. 46 (12) (1998) 4273–4285. [11] O.W. Radley, W.C. Swope, Laser drilling with focused Gaussian beams, J. Appl. Phys. 72 (1992) 3686–3696. [12] G. Chryssolouris, P. Sheng, W.C. Choi, Three dimensional laser machining of composite materials, J. Eng. Mater. Technol. 112 (1990) 387–392. [13] P. Solana, P. Kapadia, J.M. Dowden, P.J. Marsden, An analytical model for the laser drilling of metals with absorption within the vapor, J. Phys. D: Appl. Phys. 32 (1999) 942–952. [14] S. Tosto, Modeling and computer simulation of pulsed laser-induced ablation, J. Appl. Phys. A 68 (1999) 439–446. [15] J.P. Holman, Heat Transfer, 8th ed., McGraw-Hill Inc., 1997. [16] A. Grigoryants, Basics of Laser Material Processing, Mir Publishers, 1994. [17] ASM Handbook, vol. 1, 10th ed., ASM International, 1999.
A theoretical and experimental investigation on limitations of pulsed laser drilling Konstantinos Salonitis, Aristidis Stournaras, George Tsoukantas, Panagiotis Stavropoulos, George Chryssolouris ∗ Laboratory for Manufacturing Systems and Automation, Department of Mechanical Engineering and Aeronautics, University of Patras, Patra 26100, Greece Received 28 April 2005; received in revised form 5 September 2005; accepted 20 September 2006
Abstract A theoretical and experimental investigation of the limitations of the pulsed laser drilling process is presented in this study. A theoretical model has been developed for simulating the process of drilling with medium irradiance laser beams. It takes into account the required time for reaching melting temperature as well as the melting and the subsequent removal of a volume of material during each laser pulse. The model estimates for a specific laser beam power, a maximum drill depth. The pulsing frequency of the laser beam has no effect on the maximum drill depth. A 1.8 kW CO2 laser has been used for the experimental verification of the theoretical predictions and the fine tuning of the model. © 2006 Elsevier B.V. All rights reserved. Keywords: Laser beam machining; Drilling; Process modeling
1. Introduction Laser drilling (Fig. 1) [1] has a variety of applications. Laser pulsed drilling, also referred to as percussion laser drilling, utilizes several pulses to create the hole. Holes with aspect ratio of up to 1:20 can be achieved. Laser drilling is based on the absorption of the laser energy by the workpiece material and the conversion of the photon energy into thermal energy. When the temperature exceeds that of the melting and/or vaporization, the workpiece material changes phase and the hole geometry is formed. If the laser irradiance is kept below a certain threshold (typically ca. 106 W/cm2 for steels) the workpiece material is melted and not vaporized. In that case, the hole is formed due to ejection of the melted material with the use of an assisting gas jet [2]. For laser irradiance values beyond the threshold value, the material is removed mainly due to vaporization. A number of attempts to simulate the laser drilling process have been reported in the literature utilizing analytical, numerical or empir-
∗
Corresponding author. Tel.: +30 2610 99 72 62; fax: +30 2610 99 77 44. E-mail addresses: [email protected] (K. Salonitis), [email protected] (A. Stournaras), [email protected] (G. Tsoukantas), [email protected] (P. Stavropoulos), [email protected] (G. Chryssolouris). 0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.09.031
ical methods. One of the first theoretical models for predicting the drill depth was presented in [1] for the case of continuous mode laser drilling. The drill depth was estimated as the maximum depth, where melting temperature was reached. The temperature field induced in the workpiece due to laser irradiation can be determined by using analytical [3] and numerical methods [4,5]. The erosion front velocity (Fig. 1) can also be predicted assuming that the absorbing layer moves at a constant velocity into the material [6]. In [7] a model was solved numerically for the direct estimation of the hole depth. More complex models have also been presented taking into consideration the presence of the three different phases (solid, melted and vaporized) in the irradiated area. In [8] such a model was presented having included the presence of two moving boundaries (liquid–vapor and solid–liquid) as well as the conduction heat loss. It was proved that the conduction heat loss has no significant effect on the vaporization process, whereas the thickness of the liquid layer is the most dominant factor. In [9] the laser drilling process was simulated numerically, taking into account the temperature and phase dependence of the thermal properties. In [10] the hole shape and size was predicted via a finite difference heat flow model. In [11] a model was presented for drilling holes with focused Gaussian laser beam, including the positioning of the focal plane, relative to the workpiece surface and the dependence of the drill profile, on laser beam divergence.
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Nomenclature ai
thermal diffusivity for temperature equal to Ti−1 (m2 /s) A irradiation area (m2 ) external area of finite melted volume in contact Acond with surrounding material (m2 ) Aconv area of finite melted volume exchanging heat with the environment (m2 ) fc correction factor h heat transfer coefficient (W/m2 K) ki thermal conductivity for temperature equal to Ti−1 (W/m K) L latent heat of fusion (J/kg) mi finite mass that changes phase during pulse i (kg) M beam quality parameter P laser beam power (W) Q/A power density (W/m2 ) Qcond heat conducted in the workpiece (W) Qconv heat lost by convection to air (W) Qin energy entering workpiece from laser beam in one laser pulse duration (W) QL energy consumed for changing phase (W) r laser beam radius (m) r0 laser beam radius at the workpiece surface (m) R reflectivity of the workpiece material s theoretical estimated drill depth (m) si theoretical melted depth during pulse i (m) st corrected theoretical drill depth (m) toff cooling time between each successive pulse (s) tp laser pulse duration (s) Ti temperature during heating of pulse i (K) temperature during cooling of pulse i (K) Ti T0 ambient temperature (K) T temperature difference between melted surface and surrounding air (K) T/s average temperature decrease rate (K/m) Greek letters δf position of the focal plane relative to the workpiece surface (m) λ laser wavelength (m)
Few theoretical attempts have been reported on the pulsed laser drilling: in [12] modeling of the three dimensional laser grooving for both continuous wave and pulsed laser beams was presented. The pulsed model derived was based on the transformation of the temporal pulse waveform into a spatial waveform for the case of the laser grooving process. The model was verified experimentally for composite materials. A one-dimension model was developed for predicting the geometry of the hole by considering the attenuation of the laser beam within the vapor through an average absorption coefficient in [13]. The pulsing mode of the laser source was taken into consideration through the Heaviside function. In [14] the pulsed laser drilling with high
Fig. 1. Laser drilling process schematic [1].
power density beams was theoretically investigated. The model did not take into account the melting of the workpiece material and assumed that the pulse power overcame the threshold energy required for vaporization. In the present work, a theoretical model is developed that is based on the assumption that during each laser pulse, a finite volume of material is melted and removed. However, this finite volume is not constant, since, while the erosion front is propagating into the material, the laser power density is reduced, due to the increasing distance from the focal plane. The model takes into account only the melting and not the vaporization of the workpiece material; therefore its applicability is limited to low to medium irradiance laser drilling operations. The model’s predictions are discussed and compared with experimental results. 2. Theoretical analysis The heating and subsequent cooling during each pulse, is first considered in order to estimate the required time for reaching melting temperature on the workpiece surface. The second step considers the melting of the finite volume occurring in each laser pulse. The model takes into account the de-focusing effect during the laser drilling process for the estimation of the melted depth during each pulse. 2.1. Estimation of the heating time required for melting For simplifying the analysis, it is assumed that the workpiece is a semi-infinite solid. As a result of the pulsed laser ablation, the workpiece surface is heated and subsequently cooled for a number of pulses, until the melting temperature has been reached. Considering the heating of the workpiece surface during pulse i, it is assumed that the workpiece temperature is at an initial after the cooling of i − 1 pulse. Once the laser temperature Ti−1 pulse has been applied, the workpiece surface is exposed to a constant surface heat flux due to laser irradiation, given from
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equation: (1 − R)P Q = A πr02
(1)
where Q/A is the power density of the surface heat flux, R the reflectivity of the workpiece material, P the laser beam power and r0 is the laser beam radius on the workpiece surface. The differential equation describing the heat transfer on the workpiece surface is: ∂ 2 Ti 1 ∂Ti = ∂z2 ai ∂t
(2)
where ai is the thermal diffusivity for temperature equal to Ti−1 . The initial and boundary conditions are: Ti (z, 0) = Ti−1 ∂Ti Q = − ki A ∂z z=0
(3)
where toff is the cooling time between each successive pulse and erf is the error function. The above system of equations is solved for repeating laser pulses, until the melting temperature has been reached.
(4) 2.2. Estimation of drilling depth
where ki is the thermal conductivity for temperature equal to Ti−1 . Eq. (3) represents the temperature of the workpiece surface after the cooling of i − 1 pulse and Eq. (4) refers to the heat flux due to laser irradiation. The solution of the differential equation gives the temperature field Ti (z, t) and can be determined analytically [15]. For t = tp , the laser becomes inactive and the surface temperature can be determined from equation: Q ai tp /π Ti = Ti−1 + 2 (5) A ki where tp is the laser pulse duration. The heating of the workpiece is followed by a subsequent cooling, which is described from the same differential equation, as with heating (Eq. (1)). Assuming that the workpiece has uniform temperature equal to that on the surface, the initial condition is given from Eq. (6). This assumption may result in a slight under-estimation of the time needed for melting. T (z, 0) = Ti
Fig. 2. Melted volume modeling.
(6)
Besides the heat conduction within the workpiece, heat is lost due to convection to the gas jet, on the workpiece surface, leading to the following boundary condition: ∂T hA(T0 − T )z=0 = − ki A (7) ∂z z=0 where h is the heat transfer coefficient and T0 is the ambient temperature. The above system of equations is solved analytically in [15] so as to derive the temperature field during the cooling. When equating time with the “off” time of the laser (t = toff ), the surface temperature (z = 0) is estimated as: √ h ai toff h2 ai toff Ti = Ti + (T0 − Ti ) 1− exp 1− erf ki ki2 (8)
Once the melting temperature has been reached, every additional pulse results in melting a finite volume of mass (Fig. 2). This volume can be assumed to be cylindrical, having a radius equal to that of the beam at the irradiated erosion front surface. The depth of the melted volume per pulse can be calculated once the melted mass has been determined. Furthermore, while the erosion front propagates into the workpiece, the beam radius changes due to the defocusing of the laser beam. The beam radius is calculated using Eq. (9) as a function of a number of factors and has to be estimated during each pulse. ⎡
2 ⎤1/2 λ(z + δ ) f ⎦ r = r0 ⎣1 + M 2 (9) πr02 where r is the laser beam radius at a depth z from the workpiece surface, M the beam quality parameter, λ the laser wavelength and δf is the position of the focal plane relative to the workpiece surface. 2.3. Estimation of drilling depth per pulse The material, liquefied during the laser “on” time, in each pulse, is assumed to be instantly ejected from the irradiated area and thus, no energy loss for raising the temperature of the molten pool occurs. However, energy is consumed due to convection to the surrounding environment (for example on the air jet flow that might exist for the enhancement of molten material ejection). Further energy is lost due to conduction from the molten pool to the surrounding unmelted material. The energy balance of the molten pool (Fig. 3) can be expressed by the equation: Qin = QL + Qcond + Qconv
(10)
where Qin is the energy that enters the workpiece in a pulse from the laser beam, QL the energy consumed for phase change, Qcond
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beam. According to Grigoryants [16], the critical power density for inducing material melting, varies from 104 to 107 W/cm2 depending on the material thermal properties and on the laser–material interaction time. For typical mild steels (as St.37) the critical power density is ca. 4 × 104 W/cm2 . Therefore, it can be assumed that should the power density become less than this threshold value, the energy input in the workpiece will result in a dynamic equilibrium, without imposing further material melting. 3. Theoretical results
Fig. 3. Energy balance on the molten pool.
the heat conducted in the workpiece and Qconv is the heat lost by convection to air. The energy per pulse that enters the workpiece is given by equation: Qin = (1 − R)Ptp
(11)
The energy consumed for changing the phase of mass m from solid to liquid is estimated as: QL = mi L
(12)
where L is the latent heat of fusion and mi is the cylindrical mass that changes phase during pulse i. The energy conducted to the surrounding workpiece can be estimated by assuming an average heat flux per conducting area: ∂T T Qcond = kT =Tm Acond tp = kT =Tm Acond tp ∂n avg s (13) where Acond is assumed to be the external area of the finite melted volume, being in contact with the surrounding material and (T/s) is the average temperature decrease rate. The energy lost due to convection to the surrounding air is given from equation: Qconv = (hAconv T )tp
(14)
where Aconv is the area of the finite volume that can exchange heat through convection and T is the temperature difference between the melted finite volume surface and the surrounding environment. The depth of the melted finite volume for each pulse is finally estimated from the system of Eqs. (10)–(14). Eq. (15) estimates the depth of the melted finite volume during pulse i. si =
(1 − R)P/πr − kT =Tm r(T/s) − hrT rL/tp + 2kT =Tm (T/s)
In order to solve the model, a numerical module was developed in Visual Basic. The flowchart of the module is presented in Appendix A. Utilizing the developed module, the model was solved for the case of a workpiece made of steel St.37 (Table 1) irradiated by a CO2 pulsed laser beam (Table 2). Thermal conductivity, thermal diffusivity, specific heat and density were temperature dependent properties (Fig. 4) [17] and thus, their values were iteratively calculated for each pulse. In order to estimate the amount of energy required for reaching the melting temperature, the number of pulses required for raising the surface temperature to the melting point was calculated for various values of power density and pulsing frequency (Fig. 5). The analysis revealed that the pulsing frequency has little effect on the amount of energy, required for reaching the melting point for a specific power density. This indicates that the laser “off” time is so small that the heated volume does not have sufficient time to cool. During the “off” time the heated volume remains essentially in constant temperature. In Fig. 6, the cool-
Table 1 Workpiece (St.37) material properties Chemical composition (wt%) Fe C Mn P S
Balance 0.2 0.4 0.04 0.05
Thermal properties Reflectivity Latent heat of fusion Critical power density Heat transfer coefficient Melting temperature Ambient temperature
0.9 2750 kJ/kg 4 × 104 W/cm2 50 W/m2 K 1808 K 298 K
(15)
where si is the theoretical melted depth during pulse i. 2.4. Estimation of the maximum depth per power input While the erosion front propagates into the workpiece, the power density is decreased due to the defocusing of the laser
Table 2 Laser source properties Wavelength Beam radius at the focal plane Beam index, K Mode quality, M2 = 1/K Pulse duration
10.6 m 0.16 mm 0.6 1.66 10 s
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Fig. 7. Estimation of the heating time as function of laser power density.
Fig. 4. St.37 thermal properties.
Fig. 5. Estimation of the required pulses for initiating melting of the workpiece.
Fig. 6. Temperature evolution during heating phase. In the detail the slight cooling during “off” time can be observed.
ing during laser “off” time when irradiating a St.37 workpiece is shown. The laser beam in this case induces power density equal to 1.4 × 105 W/cm2 with high pulsing frequency (10,000 Hz) at the workpiece surface. Additionally, the heat transfer coefficient is considered to be 10,000 W/m2 K, corresponding to an extremely high velocity and turbulent flow of the coolant gas that cannot practically be achieved in real laser drilling operation. However, it is evident from the figure that even for this exaggerating case the cooling frequency is still negligible when compared to that of the heating per pulse. As already mentioned the pulsing frequency determines the amount of energy entering the workpiece per time unit. For larger values of pulsing frequency, the irradiated workpiece surface will be heated up to the material’s melting temperature more rapidly (Fig. 7). The maximum depth that can be drilled with the specific CO2 laser (characteristics found in Table 2) on a St.37 workpiece was estimated based on the criterion of the minimum critical power density, required for melting, as described in the theoretical analysis. The analysis shows that for a specific power density, the pulsing frequency has little effect on the maximum drill depth. As in the case of heating the workpiece surface up to the melting temperature, the analysis shows that during the laser “off” time the erosion front does not have sufficient time to cool. The laser “off” time on the other hand, is of great significance when performing laser drilling operations with high intensity laser beams for the depressing of the laser plume, which however, is not within the scope of the present paper. Furthermore, in the case of through holes, percussion drilling instead of drilling with continuous laser beam, prevents contraction at the exit side from slag formation according to [2]. Fig. 8 presents the maximum drill depth as a function of the power density. The subsequent analysis step was the estimation of the hole depth as a function of the “laser–workpiece material” interaction time and the power density. In Fig. 9, the results for pulsing frequency equal to 500 Hz are depicted. As it can be seen, increasing power density results in deeper holes. Furthermore, as the interaction time is increased, the drill depth is increased, respectively.
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Fig. 8. Maximum drill depth as a function of laser power density. Fig. 10. Theoretical estimations of drill depth as function of the interaction time for various values of pulsing frequency. power that this laser could provide was 1800 W with pulsing frequency ranging from 10 to 10,000 Hz. Two sets of experiments have been conducted, the first one for checking the effect of the pulsing frequency on the maximum depth achieved and the second one for validating the effect of the laser power on the maximum depth and evaluating the model’s predictions. The second set of experiments was also used for “calibrating” the model. The specimens were 10 mm thick rods from St.37. Each one of the experimental drills was repeated five times. The specimens were sectioned along the drill axis, they were ground and the drill depth was measured with an optical microscope (Fig. 11). The average value of the five measured depths was used for comparison with the theoretical predictions. The depth was measured from the material surface to the innermost point of material damage.
4.1. Pulsing frequency effect
Fig. 9. Theoretical estimation of drill depth as a function of laser–material interaction time for various values of power density.
However, once the maximum depth is achieved, the additional energy, entering the workpiece, is entirely dissipated inside the irradiated material, without any further material removal as it has been also pointed out in the theoretical analysis. The pulsing frequency effect on the hole depth is presented in Fig. 10, for power density equal to 8.4 × 105 W/cm2 . The maximum depth that can be achieved with a specific power density does not depend on the pulsing frequency. However, the required interaction time for achieving the maximum drill depth is also different for large values of pulsing frequency; the maximum depth is achieved with less laser–workpiece material interaction time. Furthermore, for comparison reasons, the depth calculated with the model proposed in [1] for continuous mode laser drilling, is shown in Fig. 10.
For three values of the power density and for a number of pulsing frequencies, the laser drilling experiments were conducted with large interaction times so as to achieve the maximum drilling depth. The experimental results are presented in Fig. 12 verifying the theoretical predictions for pulsing frequencies larger than 100 Hz. At low pulsing frequencies (10–100 Hz), the depth measured at each specimen of the five ones processed with the same parameters, presented high deviation. This implies that the laser source at low pulsing frequency cannot reliably provide a beam with the specified laser power.
4. Experimental results In order to verify and calibrate the proposed theoretical model, a number of experiments have been conducted. A CO2 laser source providing a Gaussian laser beam, with characteristics described in Table 2, was used. The maximum
Fig. 11. Specimen No. 12 sectioned and measured.
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Fig. 12. Drill depth measurements as a function of pulsing frequency.
4.2. Maximum drill depth A number of experiments were further conducted for deriving the maximum depth for various power densities. In Fig. 13, the average measurements of the maximum depth is compared with the model’s predictions. The experimental data show a deviation between the relationship slopes and the analytical model predictions. It is shown that the theoretical model overestimates the drill depth, when compared with experimental measurements especially for laser drilling operations with high power densities. This deviation can be attributed to the assumptions of the model for simplifying the analysis. A portion of the energy that is irradiated onto the workpiece, is consumed for further increasing the temperature of the molten pool before this is ejected, which is considered negligible in the model. Furthermore, for laser drilling operations with higher power densities the material may be vaporized inducing a plasma plume above the erosion front that absorbs a part of the irradiated power. A third cause for model/experiment discrepancies can be related to the energy transfer to the gas jet through convection that is also considered negligible in the current study. However, the above energy reducing factors are contradicted by the fact that the coefficient of absorptivity of the workpiece material is temperature dependent and it is increased as the workpiece temperature is raised. In the analysis presented, the coefficient of absorptivity is assumed to have a constant value. In order to compensate these deviations, a linear correction factor has been introduced for calibrating the theoretical model, as shown in Eq. (16). s t = fc s
(16)
where st is the corrected theoretical depth, s the theoretically estimated depth and fc is the correction factor. The factor has been derived by matching the experimental measurements with the theoretical predictions and was found to be ca. 0.80. The comparison
Fig. 13. Comparison between model predictions and experimental results.
Fig. 14. Comparison between experimental results and corrected model predictions. of the experimental measurements and the corrected theoretical predicted drill depths is presented in Fig. 14.
5. Conclusions The limitations posed by the pulsing mode of the percussion laser drilling on the drill depth have been investigated theoretically and have been verified experimentally in the present paper. After examining the test data and the analytical model’s predictions, several conclusions can be drawn: • The irradiation time, required for reaching melting temperature on the workpiece surface depends on the pulsing frequency. • The maximum drill depth that can be achieved is not a function of the pulsing frequency when laser drilling with low to medium power densities. • The theoretical model overestimates the maximum drill depth for high power densities. This deviation is due to the model’s assumptions and has been thoroughly discussed. • In order to improve the accuracy of the drill depth predictions, a linear correction factor was introduced. The agreement between the experimental measurements and the theoretical estimations was improved significantly after having considered a correction factor equal to 0.80. The theoretical model introduced allows the prediction of the minimum power, required for achieving a hole of specified depth. This implies that the laser power optimization will enable the drilling of workpieces with reduced workpiece material damage such as HAZ, formation of tensile residual stresses, generation of cracks, excess oxidation, etc. The present paper introduces a model that does not take into consideration the vaporization of the workpiece material when this is irradiated with high power density. An enhanced model needs to be further developed including the vaporization of the material, the formation of plasma plume in the drilling cavity and
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the attenuation of the laser beam within the plume. Furthermore, the applicability of the present model in laser drilling of other materials, such as aluminum and stainless steel grades needs to be further examined. Appendix A. Model solution module Flowchart of the module developed for the numerical simulation of the presented theoretical model.
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