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Modeling of laser heat source considering light scattering during laser transmission welding
Modeling of laser heat source considering light scattering during laser transmission welding Huixia Liu, Wei Liu, Xuejiao Zhong, Baoguang Liu, Dehui Guo, Xiao Wang PII: DOI: Reference:
S0264-1275(16)30327-6 doi: 10.1016/j.matdes.2016.03.052 JMADE 1530
To appear in: Received date: Revised date: Accepted date:
22 January 2016 29 February 2016 10 March 2016
Please cite this article as: Huixia Liu, Wei Liu, Xuejiao Zhong, Baoguang Liu, Dehui Guo, Xiao Wang, Modeling of laser heat source considering light scattering during laser transmission welding, (2016), doi: 10.1016/j.matdes.2016.03.052
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ACCEPTED MANUSCRIPT Modeling of laser heat source considering light scattering during laser transmission welding
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Huixia Liu*, Wei Liu, Xuejiao Zhong, Baoguang Liu, Dehui Guo, Xiao Wang
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School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China Abstract: The scattering effect of laser-transparent part has significant influence on the intensity profile of heat source during laser transmission welding (LTW). The
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purpose of the present study is to propose a method for modeling the laser heat source considering light scattering. The knife-edge experiment is served to obtain the
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normalized power flux distribution (NPFD) without considering light scattering, and the non-contact line scanning experiment is used to describe the NPFD considering light scattering. Subsequently, the energy transformation algorithm is presented to
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transform the normalized line energy intensity into the normalized point energy
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intensity. Then the heat source considering light scattering can be modeled based on the distribution of point energy intensity. Compared with the heat source without
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considering light scattering, the intensity profile of heat source considering light scattering shows a wider width and a lower peak height. Especially when the laser-transparent part contains reinforcements, the difference of intensity profile is
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more evident than the unreinforced laser-transparent part. This indicates that light scattering has a significant influence on the laser intensity at the weld interface. The proposed method for modeling the heat source, considering light scattering, can optimize the laser source in the numerical simulation of LTW. Keywords: Laser transmission welding; Light scattering; Laser beam profile; Energy transformation algorithm; Heat source considering light scattering
1. Introduction Laser transmission welding (LTW) is a technique increasingly being used for joining laser-transparent and laser-absorbent thermoplastic materials [1–3]. Along
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Corresponding author, E-mail address: [email protected] (H. Liu). Tel.: +86 0511 88797998; fax: +86 0511 88780276. 1
ACCEPTED MANUSCRIPT with the deepening investigations of LTW, the numerical simulation of LTW plays a more and more important role in predicting the conditions for the onset of welding, the temperature contours or even the start of thermal degradation [4–8]. During the
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process of laser beam transmission through the laser-transparent part, the laser light is scattered by crystal structures, reinforcing agent, pigments, filler, etc. This causes the
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difference of laser energy distribution before and after transmission through laser-transparent part. Therefore, light scattering has significant effect on the laser intensity distribution at the weld interface.
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Many investigations have involved in the measurements of transmission, reflection and light scattering. The main research methods for light scattering include
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pinhole approach, knife-edge technique, non-contact scanning method and analytical model, etc. Van Gemert et al. [9] used the diffusion approximation in one-dimension
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to solve the radiative transport equation in a slab. The absorption and scattering
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coefficients were obtained from the measurements of the diffuse transmission, collimated transmission and diffuse reflection. Alexander-Katz [10] presented an
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analytical expression for gloss and an experimental method to achieve the correlation length with a glossmeter was proposed. Tsilingiris [11] developed an analysis for the comparative evaluation of the total infrared transmission of various polymer material
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films. And the results indicated that the total transmission of a polymer film is determined by the specific spectral transmission characteristics of the materials. Plass et al. [12] presented an approach to the high-resolution knife-edge laser beam profile, and the inversion algorithm used to obtain the beam profile from knife-edge data was presented. Roundy [13] described the general state of the art of laser beam profile analysis, and many quantitative measurements (such as non-electronic tools, electronic measurement method, etc.) were made on the laser beam profile. Becker and Potente [14] used a pinhole and a power meter to assess the measurement of power flux (power per unit area) distribution at the weld interface after transmission through the unreinforced polypropylene.
Ishii et al. [15] estimated the
keV-submicron-ion-beam width using a knife-edge approach, and the beam width measurement system was constructed based on the flat-top model. Mayboudi et al. [16] 2
ACCEPTED MANUSCRIPT used the knife-edge and pinhole methods to measure the laser beam power profiles. Zak et al. [17] proposed a novel experimental technique to measure the scattering of the laser beam caused by the laser-transparent part in LTW. And the transverse
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normalized power flux distribution (T-NPFD) (also described as the transverse distribution of power flux within the laser beam cross-section) can be defined as the
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power per unit width normalized. Aden et al. [18] investigated the effect of additives on the weld seam properties, and the optical properties (scattering coefficient, absorption coefficient, and anisotropy factor) of the weldments were obtained. Chen
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et al. [19] presented an analytical model to describe the energy transmission in LTW of light scattering polymers. And the normalized power flux distribution model of
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the transmitted power from the discretized input beam was created. Xu et al. [20] studied the effect of thickness, glass fiber and crystallinity on the distribution of laser
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light. And an experimental technique was used to assess the transmitted energy
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distribution.
The above studies have explored the measurements of transmission and
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reflection, the phenomenon of light scattering when the laser beam propagates through the laser-transparent part. These investigations complement and perfect the studies on optical properties of polymer, and it has a significant influence on the
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temperature simulation for LTW incorporating scattering effect. Generally, the laser heat source is described as the absorbed light energy (absorbed by welding material) within the scope of laser spot. And the light energy can be defined by using the laser intensity, heat flux, or heat generation rate, etc. The heat source causes the materials temperature rise, and leads to the material melting and joining at the interface. Some investigations have involved in the laser intensity or heat source incorporating light scattering effect. Ilie et al. [21] proposed a method to quantify the scattering phenomena in the semi-transparent polymers by connecting the optical properties and the laser intensity spatial distribution into the medium. Then the numerical simulation of LTW was carried out using the intensity distribution (served to define the heat source), and the simulated results were validated by infrared thermography. Azhikannickal et al. [22] employed a transmission measurement method refraction, 3
ACCEPTED MANUSCRIPT absorption and reflection of the laser light. A model, presented to predict transmission as a function of thickness and laser incident angle, was used to validate the accuracy of experimental results. Aden et al. [23] reported the laser welding experiments and
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measurements of optical polymer properties (transmittance, reflectance and collimated transmittance) for thermoplastic polypropylene PP and Polyamide PA66 as
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well as for thermoplastic elastomer PEBAX. The results were characterized by measuring the heat affected zone, and the scattering of laser radiation influenced the welding results considerably. Asséko et al. [24] presented the analytical and
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numerical models for laser beam scattering in the thermoplastics composites. Also, the laser intensity at the weld interface was described and optimized for good
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weldability. These studies have investigated the energy distribution at the weld interface (considering light scattering), and the scattering effect is incorporated in the
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temperature simulation of LTW. However, the laser intensity distribution has not been
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modeled using the T-NPFD of the laser beam. Thus the purpose of present investigation is to propose a method to model the laser intensity of the heat source
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based on the T-NPFD. And since the T-NPFD can be used to describe the 1-D line energy intensity distribution along the weld width direction, an energy transformation algorithm is proposed to transform the 1-D line energy intensity into the 2-D point
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energy intensity. Then the laser intensity of the heat source can be modeled in terms of the point energy intensity. Furthermore, the effect of light scattering on the laser intensity is investigated and the heat source considering light scattering is modeled. In the present study, the purpose is to propose a method for modeling the heat source considering light scattering. The knife-edge experiment is used to describe the unscattered (without considering light scattering) T-NPFD within the laser beam cross-section. The non-contact line scanning experiment is presented to determine the scattered (considering light scattering) T-NPFD after transmission through three different laser-transparent parts (2 mm PA66, 2.5 mm PA66 and 2 mm PA66GF30). And the direct-scattered model, an analytical model that describes the energy transmission in LTW of light scattering polymers, is served to validate whether the scattered energy profile (intensity distribution of laser source considering light 4
ACCEPTED MANUSCRIPT scattering) follows Gaussian distribution. Since the T-NPFD can be used to describe the 1-D line energy intensity distribution along the weld width direction [17], an energy transformation algorithm is proposed to transform the 1-D line energy
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intensity (power per unit width of the laser beam normalized, 1/mm) into the 2-D
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point energy intensity (power per unit area of the laser beam normalized, 1/mm2).
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And the Gaussian fitting theory is served to determine the intensity distribution of the complete laser beam. Since the heat source is defined as the absorbed light energy, we cannot describe it only using the modeled laser intensity. The laser and material
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properties (such as laser efficiency, transmittance, reflectivity, etc.) are also considered when we model the heat source based on the laser intensity. Moreover, the
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heat source considering light scattering is also modeled in terms of the laser intensity incorporating light scattering effects. Compared with the unscattered energy profile
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(intensity distribution of laser source without considering light scattering), the effect
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of light scattering on the heat source is analyzed and discussed. And for the glass fiber reinforced laser-transparent part, the effect of glass fiber on the light scattering and
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the heat source is also investigated. The proposed method for modeling the heat source considering light scattering will have great significance on improving the
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accuracy and reliability of numerical simulation in LTW.
2. Experimental equipment and materials 2.1. Experimental equipment The power meter used in the knife-edge approach is manufactured by GENTEC-EO. The power meter mainly consists of a power meter probe (UP19K-30H-H5) and a power indicator (MAESTRO). The DILAS Compact 130/140 semiconductor continuous laser at 980±10nm wavelength is used in the present study. The maximum power of the laser is 130W, and the optical fiber transmission is used. A three-axis motion system is used for the welding experiment and its measurement range is W300× L300× H200 mm. The pneumatic clamping device is used to ensure the enough clamping force, which can be calculated and transformed from the 5
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2.2. Experimental materials
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The materials used in the present study are DuPont Zytel® 101L NC010
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polyamide 66 (PA66) manufactured by injection molding. The milky white and pure PA66 is used as the laser-transparent part. The black PA66 is used as the laser-absorbent part, where 0.2 wt.% carbon black (CB) is served as absorbent. All the
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specimens are cleaned using KQ3200E ultrasonic cleaning machine before welding, and then the specimens are placed in the drying oven for about 36h. The dimensions
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of the experimental materials are as follows
(1) Milky white PA66: 20×50×2 mm, 20×50×2.5 mm. (2) Milky white PA66GF30 (PA66 with 30% by weight of glass fiber
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reinforcements): 20×50×2 mm.
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(3) Black PA66 (laser-absorbent part): 20×50×2.5 mm.
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In order to investigate the effect of light scattering on the heat source, the transmittance of K9 glass ( Tg ), the transmittance of transparent PA66 ( Tr ) and the reflectivity of opaque PA66 ( Ro ) are measured by Cary 5000 UV Vis NIR absorption
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spectrometer manufactured by Varian. Each optical parameter is measured by means of three measurements taken on average. Note that, the reflectivity represents the total reflectivity which includes surface reflection and body reflection. Thus the reflectivity increases with the increase of material thickness. The average measurements for optical parameters at the wavelength = 980 nm are shown in Table 1. Table 1 Optical parameters of experimental materials at 980 nm wavelength Polymer name
Thickness (mm)
Milky white PA66
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Milky white PA66
2.5
Transmittance (%) 82.944 81.345 82.071 70.893 72.146 72.061
Mean value 82.12 Mean value 71.70 6
Reflectivity (%) 7.158 7.370 7.462 14.982 17.541 16.137
Mean value 7.33 Mean value 16.22
Absorptivity (%) 10.789 9.837 11.024 11.629 12.301 12.310
Mean value 10.55 Mean value 12.08
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35.690 34.718 35.972
Mean value 35.46
19.539 20.473 20.738
Mean value 20.25
43.678 44.821 44.371
Mean value 44.29
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Milky white PA66GF30
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3. Experiments and method
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In order to investigate the effect of light scattering on the laser intensity, the knife-edge and non-contact line scanning experiments are carried out respectively. The T-NPFD without considering light scattering is described by the knife-edge
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approach; the T-NPFD considering light scattering is obtained by using the non-contact line scanning method. After knowing the T-NPFD of the laser beam, an
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energy transformation algorithm (discussed in appendix A) is proposed to transform the 1-D line energy intensity into the 2-D point energy intensity. And then the laser
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heat source can be modeled in terms of the point energy intensity. During the
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modeling process, the following hypothesis should be made: 1) The intensity distribution of the laser beam follows the symmetrical Gaussian
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distribution after transmission through the laser-transparent part. 2) The heat source is modeled as the surface heat flux per unit area (W/mm2) on the top surface of laser-absorbent part.
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3) The heat source considering light scattering can be defined on the basis of the modeled laser intensity in the joining zone.
4) Isotropic material properties.
3.1. Knife-edge experiment The knife-edge approach is simplicity, ability to be carried out with relatively inexpensive equipment, and tolerance to high laser power. The knife-edge experiment is most commonly used to measure the energy profile of the laser beam. The knife-edge approach presented in this paper is used to determine the T-NPFD profile in the absence of light scattering [12,16]. In order to obtain the accurate results, the knife-edge experiment is carried out at the same position of the top surface of laser-absorbent part. Therefore, the working distance is given at 117mm in the 7
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knife-edge experiment, which is showing in Fig.1.
Fig. 1. Schematic of experimental setup for knife-edge experiment
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During the process of knife-edge experiment, the knife-edge incrementally
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moves a sharp linear edge underneath a laser beam and recording the power by means
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of power meter. The knife-edge is moved along y-axis in d steps, and this movement happens from the point where the laser beam is completely obscured to the point where it is completely revealed while recording the power by means of a power
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meter [16]. The recorded power increases from zero to full power as the knife-edge moving along y-axis. Here, the power of each position can be assumed to be Pi , and the power reading increases from the position yi to yi d . The T-NPFD (in unit of 1/length) can be described as the quantity of interest NPFD ( x, y) (in unit of 1/area) perpendicular to the weld line direction (i.e., along y-axis). And in the present knife-edge approach, the T-NPFD along y-axis can be defined as [16]: ( y)
Pi PE d
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where ( y) is the T-NPFD of the laser output (without considering light scattering);
PE is the laser power. For this experiment, a steel sheet placed underneath the 8
ACCEPTED MANUSCRIPT knife-edge is served for shielding the power meter from excessive radiation. And the knife-edge is moved incrementally with a 0.1 mm step size ( d =0.1mm).
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3.2. Non-contact line scanning experiment
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The knife-edge approach is mainly served to measure the power profile of the
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laser beam, and the knife-edge moves perpendicular to the direction of propagation of the laser beam in this measurement. If the laser beam is not the direct light, the measured results of knife-edge experiment will be incorrect. After transmission
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through the laser-transparent part, the laser light is scattered, and the scattered laser beam will not be perpendicular to the moving direction of the knife-edge. Therefore,
through the laser-transparent part.
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we cannot use this method to measure the beam power profile after transmission
The non-contact line scanning technique uses the idea of keeping the two parts
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slightly separated by thin shims in order to avoid their joining during the weld in
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order to facilitate quick examination of the joint interface [17]. This technique relies
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on the measurement of line widths for a sequence of scanning lines by a laser at progressively increasing power. Therefore, the non-contact line scanning experiment can be used to measure the scattered laser light after transmission through the
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laser-transparent part. And the T-NPFD considering light scattering can be described using the non-contact line scanning method. During the non-contact line scanning experiment, the laser-transparent part is separated from the laser-absorbent part (used as the sensor polymer) by parallel 0.5 mm thick metal shims to avoid contact between the parts after heating. The schematic of non-contact line scanning experiment is shown in Fig. 2. A sequence of parallel laser line scans at a constant speed ( v ) are made over the surface of laser-transparent PA66. The laser line scan starts incrementally with the lowest laser power ( Plow ), when no melt line appears, until the first (thinnest) melt line is generated at a threshold laser power P0 . The thinnest melt width observed is w0 , and the laser power ( P ) will be increased up to the highest power possible without significant 9
ACCEPTED MANUSCRIPT degradation of PA66. Then, a series of laser power Pk (k 1, 2, , n) for each scan line will be presented to obtain a sequence of scan lines, and the melt width wk
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( k 1, 2, , n ) on the top surface of laser-absorbent PA66 for each laser power level
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( Pk ) is recorded.
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Fig. 2. Schematic of non-contact line scanning experiment [17,19]
This line scanning experiment allows one to measure a series of melt widths on the basis of the different laser scanning powers. For a symmetrical laser beam (symmetrical about y=0), P0 is the critical power at a specific speed, when the laser-absorbent PA66 just begins to melt. Once the laser power increases from P0 to
Pk , the line width is increased from w0 to wk correspondingly. The energy intensity at P0 is equated with that at the edge of the weld line width ( y wk 2 ) at power Pk , as given by [17]:
* (
wk P * (0) ) 0 2 Pk
(2) 10
ACCEPTED MANUSCRIPT where * ( wk 2) is the T-NPFD of the transmitted laser beam at different position along y-axis, which is corresponding to the experimental values of wk 2 . * (0) is a
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scaling factor defined by the requirement that the integral of * ( y) along y direction must be equal to 1.
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In order to investigate the laser-transparent part’s scattering effect and the laser intensity distribution at the weld interface, milky white PA66 with a range of thickness and glass fiber contents are used as the laser-transparent parts in the
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non-contact line scanning experiment. The unreinforced PA66 with 0.2 wt.% CB is used as the laser-absorbent part. The working distance (117mm) is consistent with
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that of the knife-edge approach. The process parameters of the non-contact line scanning experiment are given in Table 2.
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material 2mm PA66 2.5mm PA66 2mm PA66GF30
Power range
Laser scanning speed
Line energy range
(mm/s) 25 25 10
(J/mm)
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Table 2 Process parameters of the non-contact line scanning experiment
(W)
12～43 17～53 28～60
0.48～1.72 0.68～2.12 2.8～6
For the purpose of obtaining accurate results, the non-contact scanning
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experiments are performed three times for each set of process parameters. And the final measured line widths are the average value of these three experiments ( wk (wk1 wk2 wk3 ) 3 , k 1, 2, , n ). Fig. 3 shows a typical image of the laser-absorbent PA66 after a sequence of scanning lines transmission through 2 mm laser-transparent PA66 at a speed of 25 mm/s.
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Fig. 3. Laser scanning lines produced on laser-absorbent PA66 after transmission through 2 mm
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transparent PA66 at a speed of 25 mm/s (a) original image; (b) zoomed in
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Although the scattered T-NPFD at any position can be calculated from Eq. (2), the ratio of P0 Pk at high and low position of y is difficult to obtain using the non-contact line scanning experiment. Thus we cannot conclude that the energy
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intensity profile of laser beam follows the Gaussian distribution. In order to validate whether the scattered T-NPFD follows the Gaussian distribution, the direct-scattered model is used in this paper. And for this analytical model, the process of modeling and solution is shown in appendix B. On the basis of the direct-scattered model, the fitted values of the scattering parameters , and * (0) are given in Table 3. It can be seen that the scattering coefficient ( ) of the three different laser-transparent materials are 0.65, 0.72 and 0.91 respectively. This indicates that the scattering coefficient of laser-transparent part will be increased with the material thickness. And moreover, the presence of glass fiber reinforcing agent apparently increases the scattering coefficient of the laser-transparent part. 12
ACCEPTED MANUSCRIPT Table 3 Light scattering parameters of different laser-transparent materials
Laser-transparent material
1 (0)
R-square
2mm PA66 2.5mm PA66 2mm PA66GF30
0.65 0.72 0.91
0.93 1.098 1.26
1.6578 1.8248 2.23
0.99 0.998 0.9977
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3.3. Method for modeling heat source
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According to the knife-edge and non-contact line scanning experiments, the T-NPFD without considering light scattering is obtained by the knife-edge approach,
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and the T-NPFD considering light scattering is described using the non-contact line scanning method. As we have discussed in the introduction, the T-NPFD can be also
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defined as the 1-D line energy intensity distribution along the weld width direction.
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Thus an energy transformation algorithm is proposed to transform the 1-D line energy intensity (power per unit width of the laser beam normalized, 1/mm) into the 2-D
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point energy intensity (power per unit area of the laser beam normalized, 1/mm2). The description and discussion of the energy transformation algorithm is given in Appendix A. Then the point energy intensity, transformed from the line energy
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intensity, can be used as the original data for modeling the heat source. The previous investigations for temperature simulation showed that the laser intensity can be generally assumed to be following the symmetrical Gaussian distribution [25-33]. Also, we have assumed that the laser intensity of the scattered light follows the Gaussian distribution. Therefore, the laser intensity can be fitted in terms of the Gaussian fitting theory, which is discussed in appendix C. For the reason that the laser energy distribution is assumed to be symmetrical about the center of laser spot, the coordinate of the symmetrical center ( t ) can be simplified as zero ( t 0 ) when we model the laser heat source. According to the Gaussian fitting theory and the above simplification, the normalized laser intensity, which is shown in Fig. 4, can be described using the point energy intensity. Then the laser heat source can be modeled and the laser-transparent 13
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part’s scattering effect on the heat source is analyzed and discussed.
Fig. 4. Modeling process of the normalized laser intensity
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4. Results and discussion
4.1. Heat source without considering light scattering
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According to the computational formula of T-NPFD without considering light
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scattering (see Eq. (1)), the T-NPFD profile along the Y direction can be obtained in
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the absence of the laser-transparent part’s scattering effect, which is shown in Fig. 5. It can be seen that the T-NPFD profile follows the Gaussian distribution. And we can assume that the complete energy profile of the laser beam follows the Gaussian
spot.
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distribution, and the laser intensity profile is symmetrical about the center of laser
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Fig. 5. Y-axis unscattered T-NPFD profile from knife-edge approach
Knowing the unscattered T-NPFD, the energy transformation algorithm can be
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used to transform the line energy intensity i into the point energy intensity j . Then the normalized intensity profile can be obtained on the basis of j
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( j 1, 2, , n ) and the Gaussian fitting theory. The fitted results are shown in Fig. 6. It can be seen that the goodness of fit is very good, and the Gaussian distribution can
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be served to accurately describe the laser intensity distribution. During the process of modeling the heat source, we assume that the radius of laser spot can be defined as the effective heating radius of laser source. The area within the laser spot will be loaded heat flux, which is following the distribution of laser intensity (modeled in Fig. 6). And the area outside the laser spot will not be given heat flux. In the absence of light scattering, for any laser power setting PE , the laser heat source ( I 0 ) within the scope of the laser spot ( x 2 y 2 R ) can be modeled on the basis of the laser intensity (described in Fig. 6), which is given by:
I 0 0.6615Pa e
x2 y 2 0.64692
(3)
where x and y are the coordinates of each position within the scope of the laser spot, which are relative to the center of laser spot; Pa is the laser power on the top surface 15
ACCEPTED MANUSCRIPT of laser-absorbent PA66, and the computational formula of Pa is as follows:
Pa PETrgTr (1 r )
(4)
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where is defined as the laser efficiency; Trg is defined as the transmittance of K9;
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Tr is defined as the transmittance of laser-transparent PA66; r is defined as the
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reflectivity of laser-absorbent PA66.
Fig. 6. Fitted results of intensity profile without considering light scattering (a) fitted Gaussian curve along Y–Z plane; (b) normalized intensity profile of unscattered laser light 16
ACCEPTED MANUSCRIPT 4.2. Heat source considering light scattering The T-NPFD of the laser beam considering light scattering can be described after
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transmission through the laser-transparent part. After transmission through 2 mm
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PA66, 2.5 mm PA66 and 2 mm PA66GF30 respectively, the experimental and
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modeled T-NPFD profiles are obtained by the non-contact line scanning method and the direct-scattered model, which are shown in Fig. 7. It can be seen that the agreement between the experimental and modeled T-NPFD is very well, and the
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Gaussian distribution can be served to describe the laser energy distribution at the weld interface. Furthermore, the scattered T-NPFDs are compared to the unscattered
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(without considering light scattering) T-NPFD obtained by the knife-edge approach. From the comparison, it can be seen that the light scattering of laser-transparent part has a significant effect on the T-NPFD of the laser beam.
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In the unreinforced laser-transparent parts, the crystalline phase causes light
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scattering to occur and consequently causes a significant widening of the transmitted
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laser beam and a lowering of the peak height relatively to the unscattered laser beam. And with the increase of the thickness of unreinforced laser-transparent parts (2 mm PA66 and 2.5 mm PA66), the difference of T-NPFD is more evident. Furthermore,
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the glass fiber reinforced laser-transparent part (2 mm PA66GF30) has greater effect on light scattering than the unreinforced laser-transparent part. And after transmission through the glass fiber reinforced laser-transparent material, the laser intensity shows a wider distribution and a lower peak height. That means the presence of glass fiber has greater effect on the laser intensity distribution than the polymer, which can be also seen from Table 3. Therefore, the laser-transparent part’s scattering effect cannot be ignored in the numerical simulation of LTW, especially when the laser-transparent material contains the glass fiber reinforcing agent. In order to make the simulated results more accurate and reliable, it is necessary to model the laser heat source considering light scattering.
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Fig. 7. Scattered T-NPFD along y-axis after transmission through 2mm PA66, 2.5mmPA66 and 2mm PA66GF30
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The proposed energy transformation algorithm can be used to determine the
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point energy intensity of the laser beam on the basis of the T-NPFD, which is obtained after transmission through the laser-transparent part. In the transformation
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algorithm, the line energy intensity i can be transformed into the point energy intensity j . Then the normalized intensity profile (following Gaussian distribution)
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can be fitted based on the point energy intensity ( j ( j 1, 2, , n )). After transmission through 2mm PA66, 2.5mm PA66 and 2mm PA66GF30, the normalized laser intensity distribution considering light scattering can be modeled, which are shown in Fig. 8, 9 and 10 respectively.
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Fig. 8. Fitted results of intensity profile considering light scattering – transmission through 2mm PA66
(a) fitted Gaussian curve along Y–Z plane; (b) normalized intensity profile of scattered laser light
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Fig. 9. Fitted results of intensity profile considering light scattering – transmission through 2.5 mm PA66
(a) fitted Gaussian curve along Y–Z plane; (b) normalized intensity profile of scattered laser light
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Fig. 10. Fitted results of intensity profile considering light scattering – transmission through 2mm PA66GF30 (a) fitted Gaussian curve along Y–Z plane; (b) normalized intensity profile of scattered laser light
It can be seen that the intensity profiles of the laser beam are following the Gaussian distribution very well after transmission through the laser-transparent parts. Table 4 shows the fitted values of the Gaussian parameters after transmission through 2 mm PA66, 2.5mm PA66 and 2mm PA66GF30 respectively. Compared with the original laser intensity distribution (energy profile of unscattered light which is not considering light scattering), the scattered intensity profiles show wider distribution and lower peak height. And when the laser-transparent part contains the glass fiber 21
ACCEPTED MANUSCRIPT reinforcing agent, the difference of intensity profile is bigger. This indicates that the change of laser intensity distribution occurs during the process of laser beam transmission through the laser-transparent part. The effect of light scattering on the
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laser intensity cannot be ignored when we describe the heat source at the weld interface.
a
2mm PA66
0.4608
2.5mm PA66
0.4111
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Laser-transparent material
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Table 4 Fitted values of Gaussian parameters after transmission through laser-transparent parts
2mm PA66GF30
0.3164
b
R-square
0.9295
0.9915
1.025
0.9921
1.5
0.9914
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For different laser-transparent parts (2mm PA66, 2.5mm PA66 and 2mm PA66GF30), the scattered heat source ( I1,2,3 ) within the scope of laser spot
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( x 2 y 2 R ) can be respectively modeled in terms of the laser intensity for any laser power setting PE , which are given by: x2 y 2 0.92952
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I1 0.3841PE e
I 2 0.3425PE e
x y
(5)
2
1.0242
(6)
x2 y 2
1.49232
(7)
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I 3 0.2957 PE e
2
where x and y are the coordinate values of each point within the scope of laser spot, which are relative to the center of laser spot. Compared with the heat source without considering light scattering (see Eq. (3)), some laser energy is absorbed and scattered during the process of transmission through the laser-transparent part. This causes the difference of heat source after transmission through the laser-transparent part. And for the glass fiber reinforced laser-transparent part, the effect of light scattering on the intensity profile of heat source is more apparent. This indicates that the light scattering needs to be considered when the laser heat source is described at the weld interface. The proposed method for modeling the heat source considering light scattering will have a great significance on 22
ACCEPTED MANUSCRIPT improving the accuracy and reliability of numerical simulation in LTW.
5. Conclusions
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In the present study, the knife-edge and the non-contact line scanning experiments were presented to describe the T-NPFD of the laser beam. Subsequently, the heat
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source considering light scattering was modeled based on the T-NPFD and the energy transformation algorithm. The results showed that:
(1) The line energy intensity was determined by the knife-edge and the
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non-contact line scanning experiments. And the results of knife-edge experiment indicated that the laser intensity within the beam cross-section followed the Gaussian
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distribution.
(2) The direct-scattered model was served to validate that the scattered T-NPFD
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profile of the laser beam followed the Gaussian distribution.
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(3) The energy transformation algorithm was proposed to transform the line energy intensity into the point energy intensity. And the Gaussian fitting theory was
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used to describe the distribution of laser intensity within the scope of laser spot. (4) Compared with the unscattered intensity profile, the light scattering had a significant influence on the intensity distribution of heat source at the weld interface.
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Especially for the glass fiber reinforced laser-transparent part, the scattered intensity profile showed a wider distribution and a lower peak height. This indicates that the proposed method for modeling the heat source considering light scattering will have a great significance on the further investigations of light scattering and numerical simulation in LTW.
Acknowledgments The authors acknowledge the National Natural Science Foundation of China (Grant No. 51275219), College Students' Innovation Practice Fund of Industry center in Jiangsu University (ZXJG201592).
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ACCEPTED MANUSCRIPT References [1] E. Ghorbel, G. Casalino, S. Abed, Laser diode transmission welding of
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distribution measurement in laser transmission welding of thermoplastics, Journal of Materials Processing Technology, 210–1 (2010) 24–31. [18] M. Aden, A. Roesner, A. Olowinsky, Optical characterization of polycarbonate: Influence of additives on optical properties, Journal of Polymer Science Part B: Polymer Physics, 48–4 (2010) 451–455. [19] M. Chen, G. Zak, P.J. Bates, Description of transmitted energy during laser transmission welding of polymers, Welding in the World, 57–2 (2013) 171–178. [20] X.F. Xu, A. Parkinson, P. J. Bates, et al., Effect of part thickness, glass fiber and crystallinity on light
scattering during laser transmission welding of
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ACCEPTED MANUSCRIPT & Technology, 51–1 (2007) 73–79. [22] E. Azhikannickal, P.J. Bates, G. Zak, Laser Light Transmission Through Thermoplastics as a Function of Thickness and Laser Incidence Angle:
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[24] A.C.A. Asséko, B. Cosson, M. Deleglise, et al., Analytical and numerical modeling of light scattering in composite transmission laser welding process,
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[27] S.S. Gajapathi, S.K. Mitra, P.F. Mendez, Part I: Development of new heat source model applicable to micro electron beam welding, Science and Technology of Welding and Joining, 17–6 (2012) 429–434. [28] X. Wang, H. Chen, H.X. Liu, et al., Simulation and optimization of continuous laser transmission welding between PET and titanium through FEM, RSM, GA and experiments, Optics and Lasers in Engineering, 51–11 (2013) 1245–1254. [29] S.S.
Gajapathi,
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ACCEPTED MANUSCRIPT [31] J. Chen, C.S. Wu, M.A. Chen, Improvement of welding heat source models for TIG-MIG hybrid welding process, Journal of Manufacturing Processes, 16–4 (2014) 485–493.
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[32] M. Kubiak, W. Piekarska, S. Stano, Modelling of laser beam heat source based on experimental research of Yb: YAG laser power distribution, International Journal
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of Heat and Mass Transfer, 83 (2015) 679–689.
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Design, 60 (2014) 94–101.
[34] Z. Peng, S.M. El–Sayed, On positive definite solution of a nonlinear matrix
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equation, Numerical Linear Algebra with Applications, 14–2 (2007) 99–113. [35] F.A. Shah, M.A. Noor, Some numerical methods for solving nonlinear equations
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mathematics and computation, 167–1 (2005) 286–298. [37] H. Guo, A simple algorithm for fitting a Gaussian function, IEEE Signal Process. Mag, 28–5 (2011) 134–137.
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[38] K.L. Ho, L. Greengard, A fast semidirect least squares algorithm for hierarchically block separable matrices, SIAM Journal on Matrix Analysis and Applications, 35–2 (2014) 725–748. [39] D. Jukić, R. Scitovski, Solution of the least-squares problem for logistic function, Journal of computational and applied mathematics, 156–1 (2003) 159–177. [40] F. Zhang, M. Wei, Y. Li, et al., Special least squares solutions of the quaternion matrix equation AX=B with applications, Applied Mathematics and Computation, 270 (2015) 425–433.
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Appendix A: Energy transformation algorithm On the basis of the inversion algorithm [12], an energy transformation algorithm
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is proposed to transform the 1-D line energy intensity (i.e. T-NPFD) into the 2-D point energy intensity, which can be used to model the laser heat source. According to the symmetry of laser energy profile, it only needs to obtain half of the point energy
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intensity distribution along y-axis direction. The energy transformation algorithm is related to the geometry depicted in Fig. 11. With a sufficiently small increment yi ,
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the point energy intensity at the position rj is assumed to be a constant j (2-D
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point energy intensity with coordinate ( rj ,0)) within the area Aij . Subsequently, j
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can be transformed and calculated based on the line energy intensity i . The
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transformation formula between i and j can be shown as [12]: n
i yi 2 Aij j j i
(A.1)
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where each i yi is a weighted sum of j , and the weighting factors are given by the area Aij . Therefore, the energy transformation algorithm (transforming i into
j ) can be defined as a linear equation:
j
1 Aji1 i yi 2 i
(A.2)
From Fig. 11, it is clear that the area Aii and Aij can be calculated by a two-dimensional integration respectively. And the formulas of the two-dimensional integration are given by [12]:
Aii
yi 2 y yi i 2 yi
0
( yi
yi 2 2 ) y 2
dxdy
(A.3)
and 28
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yi 2 y yi i 2 yi
yi 2 2 ) y 2 yi 2 2 ( yi ) y 2 (yi
(A.4)
dxdy
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respectively.
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Fig. 11. Geometric illustration of energy transformation algorithm [12]
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Appendix B: Direct-scattered model In this analytical model, the incident laser beam can be discretized into a
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sequence of narrow beams with power Pi and width Wi of the ith beam, as shown in Fig. 12(a). The power Pi from the ith beam is attenuated and scattered after transmission through the laser-transparent part. Here, the power Pi (h) ( h is the thickness of laser-transparent part) at the interface includes the scattered power
Psi (h) and the unscattered power Pdi (h) , which is shown in Fig. 12(b).
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Fig. 12. Schematic of the direct-scattered model [19,20] (a) the ith laser point beam discretized from a laser beam profile; (b) the ith laser point beam
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transmission and scattering in laser-transparent PA66
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In the direct-scattered model, is defined as the scattering coefficient of the scattered laser power Psi (h) divided by the total laser power Pi (h) at the interface,
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and is the scattering standard deviation. The total transmitted power flux at the interface is the sum of all the discretized point laser beams. Then the complete
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T-NPFD of the laser beam at the weld interface ( ( y, h) ) can be described by using a series of related parameters, as given by [19]: n
( y, h) (1 ) ( y, 0) Wi ( yi , 0) i 1
1 e 2
w ( k yi )2 2 2 2
(B.1)
If parameters and are known, ( y, h) can be described accurately using Eq. (B.1). In Eq. (B.1), let y wk 2 , then an equation (Eq. (B.2)) can be obtained when we equate Eq. (2) and Eq. (B.1):
n Pk 1 wk (1 ) ( , 0) ( yi , 0)Wi P0 * (0) 2 i 1
1 e 2
(
wk yi )2 2 2 2
1
(B.2)
A multivariable fitting technique can be used to obtain the scattering parameters 30
ACCEPTED MANUSCRIPT ( , , and * (0) ) that best fits Eq. (B.2). And in the fitting program, Eq. (B.2) can be expressed as the following implicit equation: (B.3)
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w w 1 f , , * F P0 , Pk , k , yi , ( yi , 0), ( k , 0) (0) 2 2
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where parameters ( yi ,0) and ( wk 2,0) can be calculated using the knife-edge approach. Parameters P0 , Pk , yi and wk 2 are given by the non-contact line
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scanning experiment. Then the scattering parameters , and * (0) can be determined using a program algorithm in MATLAB, and the algorithm is shown as
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follows:
Step 1: The preparation of experimental data ( P0 , Pk , wk 2 , yi , (yi ) ,
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( wk 2) ).
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Step 2: The values of formula (
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calculated.
wk yi )2 / 2 for each point beam are 2
Step 3: A large amount of continuous values of , and * (0) are
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presented, and the values of Wi
w 1 ( k yi )2 and 22 2 can be obtained for each 2 e
scanning line ( k ) and each point beam ( i ). Step 4: Based on the values of Wi
w 1 ( k yi )2 and 22 2 , the power ratio of 2 e
w ( k yi )2 2 n wk 1 1 2 2 (1 ) ( ) (y ) W e numerical calculation i i * can be (0) 2 2 i 1
obtained, which is defined as r1 .
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ACCEPTED MANUSCRIPT Step 5: The power ratio ( P0 Pk ) of the line scanning experiment is defined as r2 , n
2 and the least value of S ( S (r1 r2 ) ) can be used to determine the scattering
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i 1
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parameters , and * (0) .
Appendix C: Gaussian fitting theory
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In the process of curve fitting, an analytical expression is used to describe the dependency between x and y, and the analytical function is given as [34,35]: (C.1)
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y f ( x, c)
where c is the undetermined coefficient ( c (c1 , c2 , , cn ) ). When the analytical
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nonlinear analytical function.
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function (Eq. (C.1)) is nonlinear, the Gaussian curve can be fitted based on this
Generally, the Gaussian function can be expressed as [36-38]:
f ( x; a, b, t) a exp((( x t ) b)2 )
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(C.2)
where a , b and t are the Gaussian parameters. It can be seen from Fig. 13 that
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the Gaussian function can be graphed with a symmetrical bell-shaped curve centered at the position x t , with a representing the peak height and
2b 2 controlling
the width of Gaussian curve. On both sides of the peak, the tails (low-amplitude) of the curve quickly fall off and approach the x-axis.
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Fig. 13. Gaussian curve [37,38]
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Taken the natural logarithm of the Gaussian function in Eq. (C.2), a new quadratic function can be given by: (C.3)
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Y Ax2 Bx C
where
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Y ln( f ( x)) 2 A 1 b 2 B 2t b C t 2 b 2 ln(a )
According to the least squares criterion [39,40], parameters A , B and C can be calculated from x2 3 x x 4
x x2 x3
1 A Y x B = xY 2 x 2 C x Y
(C.4)
where
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(m 0,1, 2)
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Simultaneous solution of Eq. (C.2), (C.3) and (C.4), the Gaussian parameters a ,
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b and t can be obtained as:
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a exp( B 2 4 A C ) b 1 A t B 2 A
(C.5)
Once the Gaussian parameters ( a , b and t ) are knowing, the Gaussian function
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(Eq. (C.2)) can be obtained. Then the laser intensity can be modeled by using the
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Gaussian fitting results.
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Graphical abstract
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ACCEPTED MANUSCRIPT Highlights: 1) The knife-edge and non-contact line scanning methods are served to obtain the
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line energy intensity distribution.
intensity into 2-D point energy intensity.
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2) An energy transformation algorithm is proposed to transform 1-D line energy
3) The heat source considering light scattering is modeled at the weld interface.
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4) The effect of thickness and reinforcement on light scattering and laser intensity
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is investigated.
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S0264-1275(16)30327-6 doi: 10.1016/j.matdes.2016.03.052 JMADE 1530
To appear in: Received date: Revised date: Accepted date:
22 January 2016 29 February 2016 10 March 2016
Please cite this article as: Huixia Liu, Wei Liu, Xuejiao Zhong, Baoguang Liu, Dehui Guo, Xiao Wang, Modeling of laser heat source considering light scattering during laser transmission welding, (2016), doi: 10.1016/j.matdes.2016.03.052
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ACCEPTED MANUSCRIPT Modeling of laser heat source considering light scattering during laser transmission welding
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Huixia Liu*, Wei Liu, Xuejiao Zhong, Baoguang Liu, Dehui Guo, Xiao Wang
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School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China Abstract: The scattering effect of laser-transparent part has significant influence on the intensity profile of heat source during laser transmission welding (LTW). The
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purpose of the present study is to propose a method for modeling the laser heat source considering light scattering. The knife-edge experiment is served to obtain the
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normalized power flux distribution (NPFD) without considering light scattering, and the non-contact line scanning experiment is used to describe the NPFD considering light scattering. Subsequently, the energy transformation algorithm is presented to
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transform the normalized line energy intensity into the normalized point energy
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intensity. Then the heat source considering light scattering can be modeled based on the distribution of point energy intensity. Compared with the heat source without
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considering light scattering, the intensity profile of heat source considering light scattering shows a wider width and a lower peak height. Especially when the laser-transparent part contains reinforcements, the difference of intensity profile is
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more evident than the unreinforced laser-transparent part. This indicates that light scattering has a significant influence on the laser intensity at the weld interface. The proposed method for modeling the heat source, considering light scattering, can optimize the laser source in the numerical simulation of LTW. Keywords: Laser transmission welding; Light scattering; Laser beam profile; Energy transformation algorithm; Heat source considering light scattering
1. Introduction Laser transmission welding (LTW) is a technique increasingly being used for joining laser-transparent and laser-absorbent thermoplastic materials [1–3]. Along
*
Corresponding author, E-mail address: [email protected] (H. Liu). Tel.: +86 0511 88797998; fax: +86 0511 88780276. 1
ACCEPTED MANUSCRIPT with the deepening investigations of LTW, the numerical simulation of LTW plays a more and more important role in predicting the conditions for the onset of welding, the temperature contours or even the start of thermal degradation [4–8]. During the
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process of laser beam transmission through the laser-transparent part, the laser light is scattered by crystal structures, reinforcing agent, pigments, filler, etc. This causes the
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difference of laser energy distribution before and after transmission through laser-transparent part. Therefore, light scattering has significant effect on the laser intensity distribution at the weld interface.
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Many investigations have involved in the measurements of transmission, reflection and light scattering. The main research methods for light scattering include
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pinhole approach, knife-edge technique, non-contact scanning method and analytical model, etc. Van Gemert et al. [9] used the diffusion approximation in one-dimension
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to solve the radiative transport equation in a slab. The absorption and scattering
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coefficients were obtained from the measurements of the diffuse transmission, collimated transmission and diffuse reflection. Alexander-Katz [10] presented an
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analytical expression for gloss and an experimental method to achieve the correlation length with a glossmeter was proposed. Tsilingiris [11] developed an analysis for the comparative evaluation of the total infrared transmission of various polymer material
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films. And the results indicated that the total transmission of a polymer film is determined by the specific spectral transmission characteristics of the materials. Plass et al. [12] presented an approach to the high-resolution knife-edge laser beam profile, and the inversion algorithm used to obtain the beam profile from knife-edge data was presented. Roundy [13] described the general state of the art of laser beam profile analysis, and many quantitative measurements (such as non-electronic tools, electronic measurement method, etc.) were made on the laser beam profile. Becker and Potente [14] used a pinhole and a power meter to assess the measurement of power flux (power per unit area) distribution at the weld interface after transmission through the unreinforced polypropylene.
Ishii et al. [15] estimated the
keV-submicron-ion-beam width using a knife-edge approach, and the beam width measurement system was constructed based on the flat-top model. Mayboudi et al. [16] 2
ACCEPTED MANUSCRIPT used the knife-edge and pinhole methods to measure the laser beam power profiles. Zak et al. [17] proposed a novel experimental technique to measure the scattering of the laser beam caused by the laser-transparent part in LTW. And the transverse
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normalized power flux distribution (T-NPFD) (also described as the transverse distribution of power flux within the laser beam cross-section) can be defined as the
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power per unit width normalized. Aden et al. [18] investigated the effect of additives on the weld seam properties, and the optical properties (scattering coefficient, absorption coefficient, and anisotropy factor) of the weldments were obtained. Chen
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et al. [19] presented an analytical model to describe the energy transmission in LTW of light scattering polymers. And the normalized power flux distribution model of
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the transmitted power from the discretized input beam was created. Xu et al. [20] studied the effect of thickness, glass fiber and crystallinity on the distribution of laser
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light. And an experimental technique was used to assess the transmitted energy
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distribution.
The above studies have explored the measurements of transmission and
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reflection, the phenomenon of light scattering when the laser beam propagates through the laser-transparent part. These investigations complement and perfect the studies on optical properties of polymer, and it has a significant influence on the
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temperature simulation for LTW incorporating scattering effect. Generally, the laser heat source is described as the absorbed light energy (absorbed by welding material) within the scope of laser spot. And the light energy can be defined by using the laser intensity, heat flux, or heat generation rate, etc. The heat source causes the materials temperature rise, and leads to the material melting and joining at the interface. Some investigations have involved in the laser intensity or heat source incorporating light scattering effect. Ilie et al. [21] proposed a method to quantify the scattering phenomena in the semi-transparent polymers by connecting the optical properties and the laser intensity spatial distribution into the medium. Then the numerical simulation of LTW was carried out using the intensity distribution (served to define the heat source), and the simulated results were validated by infrared thermography. Azhikannickal et al. [22] employed a transmission measurement method refraction, 3
ACCEPTED MANUSCRIPT absorption and reflection of the laser light. A model, presented to predict transmission as a function of thickness and laser incident angle, was used to validate the accuracy of experimental results. Aden et al. [23] reported the laser welding experiments and
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measurements of optical polymer properties (transmittance, reflectance and collimated transmittance) for thermoplastic polypropylene PP and Polyamide PA66 as
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well as for thermoplastic elastomer PEBAX. The results were characterized by measuring the heat affected zone, and the scattering of laser radiation influenced the welding results considerably. Asséko et al. [24] presented the analytical and
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numerical models for laser beam scattering in the thermoplastics composites. Also, the laser intensity at the weld interface was described and optimized for good
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weldability. These studies have investigated the energy distribution at the weld interface (considering light scattering), and the scattering effect is incorporated in the
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temperature simulation of LTW. However, the laser intensity distribution has not been
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modeled using the T-NPFD of the laser beam. Thus the purpose of present investigation is to propose a method to model the laser intensity of the heat source
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based on the T-NPFD. And since the T-NPFD can be used to describe the 1-D line energy intensity distribution along the weld width direction, an energy transformation algorithm is proposed to transform the 1-D line energy intensity into the 2-D point
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energy intensity. Then the laser intensity of the heat source can be modeled in terms of the point energy intensity. Furthermore, the effect of light scattering on the laser intensity is investigated and the heat source considering light scattering is modeled. In the present study, the purpose is to propose a method for modeling the heat source considering light scattering. The knife-edge experiment is used to describe the unscattered (without considering light scattering) T-NPFD within the laser beam cross-section. The non-contact line scanning experiment is presented to determine the scattered (considering light scattering) T-NPFD after transmission through three different laser-transparent parts (2 mm PA66, 2.5 mm PA66 and 2 mm PA66GF30). And the direct-scattered model, an analytical model that describes the energy transmission in LTW of light scattering polymers, is served to validate whether the scattered energy profile (intensity distribution of laser source considering light 4
ACCEPTED MANUSCRIPT scattering) follows Gaussian distribution. Since the T-NPFD can be used to describe the 1-D line energy intensity distribution along the weld width direction [17], an energy transformation algorithm is proposed to transform the 1-D line energy
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intensity (power per unit width of the laser beam normalized, 1/mm) into the 2-D
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point energy intensity (power per unit area of the laser beam normalized, 1/mm2).
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And the Gaussian fitting theory is served to determine the intensity distribution of the complete laser beam. Since the heat source is defined as the absorbed light energy, we cannot describe it only using the modeled laser intensity. The laser and material
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properties (such as laser efficiency, transmittance, reflectivity, etc.) are also considered when we model the heat source based on the laser intensity. Moreover, the
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heat source considering light scattering is also modeled in terms of the laser intensity incorporating light scattering effects. Compared with the unscattered energy profile
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(intensity distribution of laser source without considering light scattering), the effect
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of light scattering on the heat source is analyzed and discussed. And for the glass fiber reinforced laser-transparent part, the effect of glass fiber on the light scattering and
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the heat source is also investigated. The proposed method for modeling the heat source considering light scattering will have great significance on improving the
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accuracy and reliability of numerical simulation in LTW.
2. Experimental equipment and materials 2.1. Experimental equipment The power meter used in the knife-edge approach is manufactured by GENTEC-EO. The power meter mainly consists of a power meter probe (UP19K-30H-H5) and a power indicator (MAESTRO). The DILAS Compact 130/140 semiconductor continuous laser at 980±10nm wavelength is used in the present study. The maximum power of the laser is 130W, and the optical fiber transmission is used. A three-axis motion system is used for the welding experiment and its measurement range is W300× L300× H200 mm. The pneumatic clamping device is used to ensure the enough clamping force, which can be calculated and transformed from the 5
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2.2. Experimental materials
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The materials used in the present study are DuPont Zytel® 101L NC010
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polyamide 66 (PA66) manufactured by injection molding. The milky white and pure PA66 is used as the laser-transparent part. The black PA66 is used as the laser-absorbent part, where 0.2 wt.% carbon black (CB) is served as absorbent. All the
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specimens are cleaned using KQ3200E ultrasonic cleaning machine before welding, and then the specimens are placed in the drying oven for about 36h. The dimensions
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of the experimental materials are as follows
(1) Milky white PA66: 20×50×2 mm, 20×50×2.5 mm. (2) Milky white PA66GF30 (PA66 with 30% by weight of glass fiber
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reinforcements): 20×50×2 mm.
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(3) Black PA66 (laser-absorbent part): 20×50×2.5 mm.
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In order to investigate the effect of light scattering on the heat source, the transmittance of K9 glass ( Tg ), the transmittance of transparent PA66 ( Tr ) and the reflectivity of opaque PA66 ( Ro ) are measured by Cary 5000 UV Vis NIR absorption
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spectrometer manufactured by Varian. Each optical parameter is measured by means of three measurements taken on average. Note that, the reflectivity represents the total reflectivity which includes surface reflection and body reflection. Thus the reflectivity increases with the increase of material thickness. The average measurements for optical parameters at the wavelength = 980 nm are shown in Table 1. Table 1 Optical parameters of experimental materials at 980 nm wavelength Polymer name
Thickness (mm)
Milky white PA66
2
Milky white PA66
2.5
Transmittance (%) 82.944 81.345 82.071 70.893 72.146 72.061
Mean value 82.12 Mean value 71.70 6
Reflectivity (%) 7.158 7.370 7.462 14.982 17.541 16.137
Mean value 7.33 Mean value 16.22
Absorptivity (%) 10.789 9.837 11.024 11.629 12.301 12.310
Mean value 10.55 Mean value 12.08
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35.690 34.718 35.972
Mean value 35.46
19.539 20.473 20.738
Mean value 20.25
43.678 44.821 44.371
Mean value 44.29
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Milky white PA66GF30
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3. Experiments and method
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In order to investigate the effect of light scattering on the laser intensity, the knife-edge and non-contact line scanning experiments are carried out respectively. The T-NPFD without considering light scattering is described by the knife-edge
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approach; the T-NPFD considering light scattering is obtained by using the non-contact line scanning method. After knowing the T-NPFD of the laser beam, an
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energy transformation algorithm (discussed in appendix A) is proposed to transform the 1-D line energy intensity into the 2-D point energy intensity. And then the laser
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heat source can be modeled in terms of the point energy intensity. During the
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modeling process, the following hypothesis should be made: 1) The intensity distribution of the laser beam follows the symmetrical Gaussian
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distribution after transmission through the laser-transparent part. 2) The heat source is modeled as the surface heat flux per unit area (W/mm2) on the top surface of laser-absorbent part.
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3) The heat source considering light scattering can be defined on the basis of the modeled laser intensity in the joining zone.
4) Isotropic material properties.
3.1. Knife-edge experiment The knife-edge approach is simplicity, ability to be carried out with relatively inexpensive equipment, and tolerance to high laser power. The knife-edge experiment is most commonly used to measure the energy profile of the laser beam. The knife-edge approach presented in this paper is used to determine the T-NPFD profile in the absence of light scattering [12,16]. In order to obtain the accurate results, the knife-edge experiment is carried out at the same position of the top surface of laser-absorbent part. Therefore, the working distance is given at 117mm in the 7
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knife-edge experiment, which is showing in Fig.1.
Fig. 1. Schematic of experimental setup for knife-edge experiment
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During the process of knife-edge experiment, the knife-edge incrementally
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moves a sharp linear edge underneath a laser beam and recording the power by means
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of power meter. The knife-edge is moved along y-axis in d steps, and this movement happens from the point where the laser beam is completely obscured to the point where it is completely revealed while recording the power by means of a power
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meter [16]. The recorded power increases from zero to full power as the knife-edge moving along y-axis. Here, the power of each position can be assumed to be Pi , and the power reading increases from the position yi to yi d . The T-NPFD (in unit of 1/length) can be described as the quantity of interest NPFD ( x, y) (in unit of 1/area) perpendicular to the weld line direction (i.e., along y-axis). And in the present knife-edge approach, the T-NPFD along y-axis can be defined as [16]: ( y)
Pi PE d
(1)
where ( y) is the T-NPFD of the laser output (without considering light scattering);
PE is the laser power. For this experiment, a steel sheet placed underneath the 8
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3.2. Non-contact line scanning experiment
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The knife-edge approach is mainly served to measure the power profile of the
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laser beam, and the knife-edge moves perpendicular to the direction of propagation of the laser beam in this measurement. If the laser beam is not the direct light, the measured results of knife-edge experiment will be incorrect. After transmission
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through the laser-transparent part, the laser light is scattered, and the scattered laser beam will not be perpendicular to the moving direction of the knife-edge. Therefore,
through the laser-transparent part.
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we cannot use this method to measure the beam power profile after transmission
The non-contact line scanning technique uses the idea of keeping the two parts
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slightly separated by thin shims in order to avoid their joining during the weld in
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order to facilitate quick examination of the joint interface [17]. This technique relies
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on the measurement of line widths for a sequence of scanning lines by a laser at progressively increasing power. Therefore, the non-contact line scanning experiment can be used to measure the scattered laser light after transmission through the
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laser-transparent part. And the T-NPFD considering light scattering can be described using the non-contact line scanning method. During the non-contact line scanning experiment, the laser-transparent part is separated from the laser-absorbent part (used as the sensor polymer) by parallel 0.5 mm thick metal shims to avoid contact between the parts after heating. The schematic of non-contact line scanning experiment is shown in Fig. 2. A sequence of parallel laser line scans at a constant speed ( v ) are made over the surface of laser-transparent PA66. The laser line scan starts incrementally with the lowest laser power ( Plow ), when no melt line appears, until the first (thinnest) melt line is generated at a threshold laser power P0 . The thinnest melt width observed is w0 , and the laser power ( P ) will be increased up to the highest power possible without significant 9
ACCEPTED MANUSCRIPT degradation of PA66. Then, a series of laser power Pk (k 1, 2, , n) for each scan line will be presented to obtain a sequence of scan lines, and the melt width wk
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( k 1, 2, , n ) on the top surface of laser-absorbent PA66 for each laser power level
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( Pk ) is recorded.
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Fig. 2. Schematic of non-contact line scanning experiment [17,19]
This line scanning experiment allows one to measure a series of melt widths on the basis of the different laser scanning powers. For a symmetrical laser beam (symmetrical about y=0), P0 is the critical power at a specific speed, when the laser-absorbent PA66 just begins to melt. Once the laser power increases from P0 to
Pk , the line width is increased from w0 to wk correspondingly. The energy intensity at P0 is equated with that at the edge of the weld line width ( y wk 2 ) at power Pk , as given by [17]:
* (
wk P * (0) ) 0 2 Pk
(2) 10
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scaling factor defined by the requirement that the integral of * ( y) along y direction must be equal to 1.
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In order to investigate the laser-transparent part’s scattering effect and the laser intensity distribution at the weld interface, milky white PA66 with a range of thickness and glass fiber contents are used as the laser-transparent parts in the
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non-contact line scanning experiment. The unreinforced PA66 with 0.2 wt.% CB is used as the laser-absorbent part. The working distance (117mm) is consistent with
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that of the knife-edge approach. The process parameters of the non-contact line scanning experiment are given in Table 2.
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material 2mm PA66 2.5mm PA66 2mm PA66GF30
Power range
Laser scanning speed
Line energy range
(mm/s) 25 25 10
(J/mm)
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Laser-transparent
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Table 2 Process parameters of the non-contact line scanning experiment
(W)
12～43 17～53 28～60
0.48～1.72 0.68～2.12 2.8～6
For the purpose of obtaining accurate results, the non-contact scanning
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experiments are performed three times for each set of process parameters. And the final measured line widths are the average value of these three experiments ( wk (wk1 wk2 wk3 ) 3 , k 1, 2, , n ). Fig. 3 shows a typical image of the laser-absorbent PA66 after a sequence of scanning lines transmission through 2 mm laser-transparent PA66 at a speed of 25 mm/s.
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Fig. 3. Laser scanning lines produced on laser-absorbent PA66 after transmission through 2 mm
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transparent PA66 at a speed of 25 mm/s (a) original image; (b) zoomed in
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Although the scattered T-NPFD at any position can be calculated from Eq. (2), the ratio of P0 Pk at high and low position of y is difficult to obtain using the non-contact line scanning experiment. Thus we cannot conclude that the energy
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intensity profile of laser beam follows the Gaussian distribution. In order to validate whether the scattered T-NPFD follows the Gaussian distribution, the direct-scattered model is used in this paper. And for this analytical model, the process of modeling and solution is shown in appendix B. On the basis of the direct-scattered model, the fitted values of the scattering parameters , and * (0) are given in Table 3. It can be seen that the scattering coefficient ( ) of the three different laser-transparent materials are 0.65, 0.72 and 0.91 respectively. This indicates that the scattering coefficient of laser-transparent part will be increased with the material thickness. And moreover, the presence of glass fiber reinforcing agent apparently increases the scattering coefficient of the laser-transparent part. 12
ACCEPTED MANUSCRIPT Table 3 Light scattering parameters of different laser-transparent materials
Laser-transparent material
1 (0)
R-square
2mm PA66 2.5mm PA66 2mm PA66GF30
0.65 0.72 0.91
0.93 1.098 1.26
1.6578 1.8248 2.23
0.99 0.998 0.9977
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3.3. Method for modeling heat source
*
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According to the knife-edge and non-contact line scanning experiments, the T-NPFD without considering light scattering is obtained by the knife-edge approach,
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and the T-NPFD considering light scattering is described using the non-contact line scanning method. As we have discussed in the introduction, the T-NPFD can be also
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defined as the 1-D line energy intensity distribution along the weld width direction.
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Thus an energy transformation algorithm is proposed to transform the 1-D line energy intensity (power per unit width of the laser beam normalized, 1/mm) into the 2-D
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point energy intensity (power per unit area of the laser beam normalized, 1/mm2). The description and discussion of the energy transformation algorithm is given in Appendix A. Then the point energy intensity, transformed from the line energy
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intensity, can be used as the original data for modeling the heat source. The previous investigations for temperature simulation showed that the laser intensity can be generally assumed to be following the symmetrical Gaussian distribution [25-33]. Also, we have assumed that the laser intensity of the scattered light follows the Gaussian distribution. Therefore, the laser intensity can be fitted in terms of the Gaussian fitting theory, which is discussed in appendix C. For the reason that the laser energy distribution is assumed to be symmetrical about the center of laser spot, the coordinate of the symmetrical center ( t ) can be simplified as zero ( t 0 ) when we model the laser heat source. According to the Gaussian fitting theory and the above simplification, the normalized laser intensity, which is shown in Fig. 4, can be described using the point energy intensity. Then the laser heat source can be modeled and the laser-transparent 13
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part’s scattering effect on the heat source is analyzed and discussed.
Fig. 4. Modeling process of the normalized laser intensity
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4. Results and discussion
4.1. Heat source without considering light scattering
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According to the computational formula of T-NPFD without considering light
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scattering (see Eq. (1)), the T-NPFD profile along the Y direction can be obtained in
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the absence of the laser-transparent part’s scattering effect, which is shown in Fig. 5. It can be seen that the T-NPFD profile follows the Gaussian distribution. And we can assume that the complete energy profile of the laser beam follows the Gaussian
spot.
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distribution, and the laser intensity profile is symmetrical about the center of laser
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Fig. 5. Y-axis unscattered T-NPFD profile from knife-edge approach
Knowing the unscattered T-NPFD, the energy transformation algorithm can be
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used to transform the line energy intensity i into the point energy intensity j . Then the normalized intensity profile can be obtained on the basis of j
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( j 1, 2, , n ) and the Gaussian fitting theory. The fitted results are shown in Fig. 6. It can be seen that the goodness of fit is very good, and the Gaussian distribution can
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be served to accurately describe the laser intensity distribution. During the process of modeling the heat source, we assume that the radius of laser spot can be defined as the effective heating radius of laser source. The area within the laser spot will be loaded heat flux, which is following the distribution of laser intensity (modeled in Fig. 6). And the area outside the laser spot will not be given heat flux. In the absence of light scattering, for any laser power setting PE , the laser heat source ( I 0 ) within the scope of the laser spot ( x 2 y 2 R ) can be modeled on the basis of the laser intensity (described in Fig. 6), which is given by:
I 0 0.6615Pa e
x2 y 2 0.64692
(3)
where x and y are the coordinates of each position within the scope of the laser spot, which are relative to the center of laser spot; Pa is the laser power on the top surface 15
ACCEPTED MANUSCRIPT of laser-absorbent PA66, and the computational formula of Pa is as follows:
Pa PETrgTr (1 r )
(4)
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where is defined as the laser efficiency; Trg is defined as the transmittance of K9;
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Tr is defined as the transmittance of laser-transparent PA66; r is defined as the
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reflectivity of laser-absorbent PA66.
Fig. 6. Fitted results of intensity profile without considering light scattering (a) fitted Gaussian curve along Y–Z plane; (b) normalized intensity profile of unscattered laser light 16
ACCEPTED MANUSCRIPT 4.2. Heat source considering light scattering The T-NPFD of the laser beam considering light scattering can be described after
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transmission through the laser-transparent part. After transmission through 2 mm
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PA66, 2.5 mm PA66 and 2 mm PA66GF30 respectively, the experimental and
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modeled T-NPFD profiles are obtained by the non-contact line scanning method and the direct-scattered model, which are shown in Fig. 7. It can be seen that the agreement between the experimental and modeled T-NPFD is very well, and the
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Gaussian distribution can be served to describe the laser energy distribution at the weld interface. Furthermore, the scattered T-NPFDs are compared to the unscattered
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(without considering light scattering) T-NPFD obtained by the knife-edge approach. From the comparison, it can be seen that the light scattering of laser-transparent part has a significant effect on the T-NPFD of the laser beam.
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In the unreinforced laser-transparent parts, the crystalline phase causes light
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scattering to occur and consequently causes a significant widening of the transmitted
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laser beam and a lowering of the peak height relatively to the unscattered laser beam. And with the increase of the thickness of unreinforced laser-transparent parts (2 mm PA66 and 2.5 mm PA66), the difference of T-NPFD is more evident. Furthermore,
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the glass fiber reinforced laser-transparent part (2 mm PA66GF30) has greater effect on light scattering than the unreinforced laser-transparent part. And after transmission through the glass fiber reinforced laser-transparent material, the laser intensity shows a wider distribution and a lower peak height. That means the presence of glass fiber has greater effect on the laser intensity distribution than the polymer, which can be also seen from Table 3. Therefore, the laser-transparent part’s scattering effect cannot be ignored in the numerical simulation of LTW, especially when the laser-transparent material contains the glass fiber reinforcing agent. In order to make the simulated results more accurate and reliable, it is necessary to model the laser heat source considering light scattering.
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Fig. 7. Scattered T-NPFD along y-axis after transmission through 2mm PA66, 2.5mmPA66 and 2mm PA66GF30
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The proposed energy transformation algorithm can be used to determine the
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point energy intensity of the laser beam on the basis of the T-NPFD, which is obtained after transmission through the laser-transparent part. In the transformation
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algorithm, the line energy intensity i can be transformed into the point energy intensity j . Then the normalized intensity profile (following Gaussian distribution)
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can be fitted based on the point energy intensity ( j ( j 1, 2, , n )). After transmission through 2mm PA66, 2.5mm PA66 and 2mm PA66GF30, the normalized laser intensity distribution considering light scattering can be modeled, which are shown in Fig. 8, 9 and 10 respectively.
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Fig. 8. Fitted results of intensity profile considering light scattering – transmission through 2mm PA66
(a) fitted Gaussian curve along Y–Z plane; (b) normalized intensity profile of scattered laser light
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Fig. 9. Fitted results of intensity profile considering light scattering – transmission through 2.5 mm PA66
(a) fitted Gaussian curve along Y–Z plane; (b) normalized intensity profile of scattered laser light
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Fig. 10. Fitted results of intensity profile considering light scattering – transmission through 2mm PA66GF30 (a) fitted Gaussian curve along Y–Z plane; (b) normalized intensity profile of scattered laser light
It can be seen that the intensity profiles of the laser beam are following the Gaussian distribution very well after transmission through the laser-transparent parts. Table 4 shows the fitted values of the Gaussian parameters after transmission through 2 mm PA66, 2.5mm PA66 and 2mm PA66GF30 respectively. Compared with the original laser intensity distribution (energy profile of unscattered light which is not considering light scattering), the scattered intensity profiles show wider distribution and lower peak height. And when the laser-transparent part contains the glass fiber 21
ACCEPTED MANUSCRIPT reinforcing agent, the difference of intensity profile is bigger. This indicates that the change of laser intensity distribution occurs during the process of laser beam transmission through the laser-transparent part. The effect of light scattering on the
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laser intensity cannot be ignored when we describe the heat source at the weld interface.
a
2mm PA66
0.4608
2.5mm PA66
0.4111
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Laser-transparent material
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Table 4 Fitted values of Gaussian parameters after transmission through laser-transparent parts
2mm PA66GF30
0.3164
b
R-square
0.9295
0.9915
1.025
0.9921
1.5
0.9914
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For different laser-transparent parts (2mm PA66, 2.5mm PA66 and 2mm PA66GF30), the scattered heat source ( I1,2,3 ) within the scope of laser spot
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( x 2 y 2 R ) can be respectively modeled in terms of the laser intensity for any laser power setting PE , which are given by: x2 y 2 0.92952
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I1 0.3841PE e
I 2 0.3425PE e
x y
(5)
2
1.0242
(6)
x2 y 2
1.49232
(7)
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I 3 0.2957 PE e
2
where x and y are the coordinate values of each point within the scope of laser spot, which are relative to the center of laser spot. Compared with the heat source without considering light scattering (see Eq. (3)), some laser energy is absorbed and scattered during the process of transmission through the laser-transparent part. This causes the difference of heat source after transmission through the laser-transparent part. And for the glass fiber reinforced laser-transparent part, the effect of light scattering on the intensity profile of heat source is more apparent. This indicates that the light scattering needs to be considered when the laser heat source is described at the weld interface. The proposed method for modeling the heat source considering light scattering will have a great significance on 22
ACCEPTED MANUSCRIPT improving the accuracy and reliability of numerical simulation in LTW.
5. Conclusions
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In the present study, the knife-edge and the non-contact line scanning experiments were presented to describe the T-NPFD of the laser beam. Subsequently, the heat
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source considering light scattering was modeled based on the T-NPFD and the energy transformation algorithm. The results showed that:
(1) The line energy intensity was determined by the knife-edge and the
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non-contact line scanning experiments. And the results of knife-edge experiment indicated that the laser intensity within the beam cross-section followed the Gaussian
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distribution.
(2) The direct-scattered model was served to validate that the scattered T-NPFD
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profile of the laser beam followed the Gaussian distribution.
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(3) The energy transformation algorithm was proposed to transform the line energy intensity into the point energy intensity. And the Gaussian fitting theory was
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used to describe the distribution of laser intensity within the scope of laser spot. (4) Compared with the unscattered intensity profile, the light scattering had a significant influence on the intensity distribution of heat source at the weld interface.
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Especially for the glass fiber reinforced laser-transparent part, the scattered intensity profile showed a wider distribution and a lower peak height. This indicates that the proposed method for modeling the heat source considering light scattering will have a great significance on the further investigations of light scattering and numerical simulation in LTW.
Acknowledgments The authors acknowledge the National Natural Science Foundation of China (Grant No. 51275219), College Students' Innovation Practice Fund of Industry center in Jiangsu University (ZXJG201592).
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Kanaya–Okayama heat source in modelling micro electron beam welding, Science and Technology of Welding and Joining, 17–6 (2012) 435–440. [30] N. Ma, Z. Cai, H. Huang, et al., Investigation of welding residual stress in flash-butt joint of U71Mn rail steel by numerical simulation and experiment, Materials and Design, 88 (2015) 1296–1309. 26
ACCEPTED MANUSCRIPT [31] J. Chen, C.S. Wu, M.A. Chen, Improvement of welding heat source models for TIG-MIG hybrid welding process, Journal of Manufacturing Processes, 16–4 (2014) 485–493.
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[32] M. Kubiak, W. Piekarska, S. Stano, Modelling of laser beam heat source based on experimental research of Yb: YAG laser power distribution, International Journal
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of Heat and Mass Transfer, 83 (2015) 679–689.
[33] J. Zhang, Y. Shen, B. Li, et al., Numerical simulation and experimental investigation on friction stir welding of 6061-T6 aluminum alloy, Materials and
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Design, 60 (2014) 94–101.
[34] Z. Peng, S.M. El–Sayed, On positive definite solution of a nonlinear matrix
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equation, Numerical Linear Algebra with Applications, 14–2 (2007) 99–113. [35] F.A. Shah, M.A. Noor, Some numerical methods for solving nonlinear equations
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by using decomposition technique, Applied Mathematics and Computation, 251
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(2015) 378–386.
[36] D. Jukić, R. Scitovski, Least squares fitting Gaussian type curve, Applied
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mathematics and computation, 167–1 (2005) 286–298. [37] H. Guo, A simple algorithm for fitting a Gaussian function, IEEE Signal Process. Mag, 28–5 (2011) 134–137.
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[38] K.L. Ho, L. Greengard, A fast semidirect least squares algorithm for hierarchically block separable matrices, SIAM Journal on Matrix Analysis and Applications, 35–2 (2014) 725–748. [39] D. Jukić, R. Scitovski, Solution of the least-squares problem for logistic function, Journal of computational and applied mathematics, 156–1 (2003) 159–177. [40] F. Zhang, M. Wei, Y. Li, et al., Special least squares solutions of the quaternion matrix equation AX=B with applications, Applied Mathematics and Computation, 270 (2015) 425–433.
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Appendix A: Energy transformation algorithm On the basis of the inversion algorithm [12], an energy transformation algorithm
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is proposed to transform the 1-D line energy intensity (i.e. T-NPFD) into the 2-D point energy intensity, which can be used to model the laser heat source. According to the symmetry of laser energy profile, it only needs to obtain half of the point energy
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intensity distribution along y-axis direction. The energy transformation algorithm is related to the geometry depicted in Fig. 11. With a sufficiently small increment yi ,
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the point energy intensity at the position rj is assumed to be a constant j (2-D
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point energy intensity with coordinate ( rj ,0)) within the area Aij . Subsequently, j
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can be transformed and calculated based on the line energy intensity i . The
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transformation formula between i and j can be shown as [12]: n
i yi 2 Aij j j i
(A.1)
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where each i yi is a weighted sum of j , and the weighting factors are given by the area Aij . Therefore, the energy transformation algorithm (transforming i into
j ) can be defined as a linear equation:
j
1 Aji1 i yi 2 i
(A.2)
From Fig. 11, it is clear that the area Aii and Aij can be calculated by a two-dimensional integration respectively. And the formulas of the two-dimensional integration are given by [12]:
Aii
yi 2 y yi i 2 yi
0
( yi
yi 2 2 ) y 2
dxdy
(A.3)
and 28
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yi 2 y yi i 2 yi
yi 2 2 ) y 2 yi 2 2 ( yi ) y 2 (yi
(A.4)
dxdy
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respectively.
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Fig. 11. Geometric illustration of energy transformation algorithm [12]
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Appendix B: Direct-scattered model In this analytical model, the incident laser beam can be discretized into a
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sequence of narrow beams with power Pi and width Wi of the ith beam, as shown in Fig. 12(a). The power Pi from the ith beam is attenuated and scattered after transmission through the laser-transparent part. Here, the power Pi (h) ( h is the thickness of laser-transparent part) at the interface includes the scattered power
Psi (h) and the unscattered power Pdi (h) , which is shown in Fig. 12(b).
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Fig. 12. Schematic of the direct-scattered model [19,20] (a) the ith laser point beam discretized from a laser beam profile; (b) the ith laser point beam
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transmission and scattering in laser-transparent PA66
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In the direct-scattered model, is defined as the scattering coefficient of the scattered laser power Psi (h) divided by the total laser power Pi (h) at the interface,
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and is the scattering standard deviation. The total transmitted power flux at the interface is the sum of all the discretized point laser beams. Then the complete
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T-NPFD of the laser beam at the weld interface ( ( y, h) ) can be described by using a series of related parameters, as given by [19]: n
( y, h) (1 ) ( y, 0) Wi ( yi , 0) i 1
1 e 2
w ( k yi )2 2 2 2
(B.1)
If parameters and are known, ( y, h) can be described accurately using Eq. (B.1). In Eq. (B.1), let y wk 2 , then an equation (Eq. (B.2)) can be obtained when we equate Eq. (2) and Eq. (B.1):
n Pk 1 wk (1 ) ( , 0) ( yi , 0)Wi P0 * (0) 2 i 1
1 e 2
(
wk yi )2 2 2 2
1
(B.2)
A multivariable fitting technique can be used to obtain the scattering parameters 30
ACCEPTED MANUSCRIPT ( , , and * (0) ) that best fits Eq. (B.2). And in the fitting program, Eq. (B.2) can be expressed as the following implicit equation: (B.3)
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w w 1 f , , * F P0 , Pk , k , yi , ( yi , 0), ( k , 0) (0) 2 2
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where parameters ( yi ,0) and ( wk 2,0) can be calculated using the knife-edge approach. Parameters P0 , Pk , yi and wk 2 are given by the non-contact line
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scanning experiment. Then the scattering parameters , and * (0) can be determined using a program algorithm in MATLAB, and the algorithm is shown as
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follows:
Step 1: The preparation of experimental data ( P0 , Pk , wk 2 , yi , (yi ) ,
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( wk 2) ).
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Step 2: The values of formula (
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calculated.
wk yi )2 / 2 for each point beam are 2
Step 3: A large amount of continuous values of , and * (0) are
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presented, and the values of Wi
w 1 ( k yi )2 and 22 2 can be obtained for each 2 e
scanning line ( k ) and each point beam ( i ). Step 4: Based on the values of Wi
w 1 ( k yi )2 and 22 2 , the power ratio of 2 e
w ( k yi )2 2 n wk 1 1 2 2 (1 ) ( ) (y ) W e numerical calculation i i * can be (0) 2 2 i 1
obtained, which is defined as r1 .
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2 and the least value of S ( S (r1 r2 ) ) can be used to determine the scattering
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parameters , and * (0) .
Appendix C: Gaussian fitting theory
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In the process of curve fitting, an analytical expression is used to describe the dependency between x and y, and the analytical function is given as [34,35]: (C.1)
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y f ( x, c)
where c is the undetermined coefficient ( c (c1 , c2 , , cn ) ). When the analytical
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nonlinear analytical function.
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function (Eq. (C.1)) is nonlinear, the Gaussian curve can be fitted based on this
Generally, the Gaussian function can be expressed as [36-38]:
f ( x; a, b, t) a exp((( x t ) b)2 )
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(C.2)
where a , b and t are the Gaussian parameters. It can be seen from Fig. 13 that
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the Gaussian function can be graphed with a symmetrical bell-shaped curve centered at the position x t , with a representing the peak height and
2b 2 controlling
the width of Gaussian curve. On both sides of the peak, the tails (low-amplitude) of the curve quickly fall off and approach the x-axis.
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Fig. 13. Gaussian curve [37,38]
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Taken the natural logarithm of the Gaussian function in Eq. (C.2), a new quadratic function can be given by: (C.3)
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Y Ax2 Bx C
where
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Y ln( f ( x)) 2 A 1 b 2 B 2t b C t 2 b 2 ln(a )
According to the least squares criterion [39,40], parameters A , B and C can be calculated from x2 3 x x 4
x x2 x3
1 A Y x B = xY 2 x 2 C x Y
(C.4)
where
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(m 0,1, 2)
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Simultaneous solution of Eq. (C.2), (C.3) and (C.4), the Gaussian parameters a ,
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b and t can be obtained as:
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a exp( B 2 4 A C ) b 1 A t B 2 A
(C.5)
Once the Gaussian parameters ( a , b and t ) are knowing, the Gaussian function
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(Eq. (C.2)) can be obtained. Then the laser intensity can be modeled by using the
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Gaussian fitting results.
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Graphical abstract
35
ACCEPTED MANUSCRIPT Highlights: 1) The knife-edge and non-contact line scanning methods are served to obtain the
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line energy intensity distribution.
intensity into 2-D point energy intensity.
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2) An energy transformation algorithm is proposed to transform 1-D line energy
3) The heat source considering light scattering is modeled at the weld interface.
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4) The effect of thickness and reinforcement on light scattering and laser intensity
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is investigated.
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