98 Al-Khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) Heating and Melting Model Induced by Laser Beam in Solid Material Faiz F. Mustafa Department of Manufacturing Operations Engineering /Al-Khwarizmi College of Engineering University of Baghdad (Received 2 March 2008; accepted 15 June 2008) Abstract An analytical method and a two-dimensional finite element model for treating the problem of laser heating and melting has been applied to aluminum 2519T87and stainless steel 304. The time needed to melt and vaporize and the effects of laser power density on the melt depth for two metals are also obtained. In addition, the depth profile and time evolution of the temperature before melting and after melting are given, in which a discontinuity in the temperature gradient is obviously observed due to the latent heat of fusion and the increment in thermal conductivity in solid phase. The analytical results that induced by m6.10 laser irradiation is in good agreement with numerical results. Keywords: Laser; Heating; Melting; Finite element method; Non-linear heat transfer 1. Introduction High power density, such as laser beam, has been increasingly utilized in industrial manufacturing. The process of laser beam in solid material offers a great potential for the new product design, for example, welding, cutting of metals, drilling of holes, laser shock hardening and laser glazing. When a high power laser irradiates a material surface, a part of the laser energy is absorbed and conducted into the interior of the material. If the absorbed energy is high enough, the material surface will melt and even vaporize. The study of laser induced heating and melting has attracted great interest [1–7] and the results obtained are of great importance for achieving high quality materials processing with lasers. To simplify the mathematical problem, it is necessary to assume that the process of laser heating and melting is a linear process, that is to say, the physical parameters of the material, including density, thermal conductivity, thermal capacity, optical absorptivity, etc., are independent of the temperature. In this study, a one-dimensional heat conduction problem is solved approximately in the solid and liquid regions by assuming that the thermo-physical properties of the material are independent of the temperature. Numerical simulation of laser heating and melting with two dimensional finite element methods has been used to evaluate temperature distribution and other variables. A nonlinear transient thermal analysis was performed using temperature dependent material properties used to evaluate temperature distribution The computations of the depth profile and time evolution of the temperature before melting as well as after melting are carried out for the two materials, the variation of the melt depth with time, the effects of the laser power density on the melt depth and the irradiation time on melting and vaporization are calculated. 2. Mathematical Model The geometry of laser irradiation and the resulting liquid and solid regions are shown in Fig. 1. The diameter of the laser beam is broad enough compared to the molten region and the thickness of the material is much greater than the Faiz F. Mustafa Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) 99 thermal penetration depth, so that the problem can be solved in one dimension and a semi-infinite model may be accepted. The analytical solution applied to aluminum and stainless steel, the thermo-physical parameters in solid and liquid phases are shown in Table 1 were taken from [8-9]. Accordingly the surface of the material reaches the fusion point or not, the whole process of laser induced heating and melting in the material is divided into two steps: before melting and after melting. Fig. 1. The Geometry of Laser Irradiation. Table 1 Thermophysical Parameters and Absorptivity for Materials [8-9]. _______________________________________ Material Aluminum Stainless-steel properties (2519T87) (304) ______________________________________ )/( 3 mkg s  2823 7860 )/( 3 mkg L  2485 6980 )/( kgKJC s 896 465 )/( kgKJC L 1080 691 )/( mKWK s 100 53 )/( mKWK L 238 120 )(KT m 933 1811 )(KT V 2793 3134 )10( 15  kgL 3.88 3.65 s A 0.0588 0.386 L A 0.064 0.346 ______________________________________ 2.1. Step A. Before Melting The thermal conduction model can be described by the following equations before the temperature of the surface reaches the fusion point: ,0 ),(1),( 2 2       t tyT y tyT S S S  ,0  y …(1) , ),( IA y tyT K S S S     ,0y …(2) ,),( TtyTS  ,y …(3) ,),( TtyTS  ,0t …(4) where SSS KT ,, and S A are the temperature, thermal conductivity, thermal diffusivity and absorptivity of the solid phase respectively. T is the ambient temperature and I is the power density of laser beam. We first assume a temperature profile, which satisfies the boundary condition (3): ,)(),( )(/ ty WS etTtyT   …(5) where )(tT W represents the temperature of the surface and )(t is a temporal function representing the temperature penetration depth in the solid. Substituting expression (5) into Eqs. (2) and (1), we get the following relations: ),( )( )( 2 tT tdt tdT W SW    …(6) ),()( t K IA tT S S W  …(7) According to Eqs. (6) and (7), one gets ,)2()( 2 1 t KC IA TtT SSS S W    …(8) .2)( tt S   …(9) 2.2 Step B. After Melting The thermal conduction equations in liquid and solid regions can be described as ,0 ),(1),( 2 2       t tyT y tyT L L L  ),(0 tSy  …(10) ,0 ),(1),( 2 2       t tyT y tyT S S S   ytS )( …(11) with boundary conditions , ),( IA y tyT K L L L     ,0y …(12) Faiz F. Mustafa Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) 100 , ),(),( mLS TtyTtyT  ),(tSy  …(13) , ),(),()( y tyT K y tyT K dt tdS L L L S SS       …(14) ,),( TtyTS  y …(15) and initial condition ,0)( tS , m tt  … (16) where ii KT , and iiii CK  / are temperature, thermal conductivity and thermal diffusivity of the i th phase respectively ( si  (solid phase) or l (liquid phase)), i and iC are the density and heat capacity of the i th phase. L A is the absorptivity of liquid phase, m T is the melting point and L is the fusion latent heat of the irradiated material, )(tS is the position of the interface between solid phase and liquid phase. The temperature profiles in liquid region and solid region are assumed as )(/ )(),( ty mL letTtyT   …(17) )(/)( ),( ttSy mS SeTtyT   … (18) which satisfy Eqs. (13) and (15), )(t L  and )(t S  are two temporal functions representing the temperature penetration depth in liquid and solid regions. Satisfying Eq. (10) at 0y by using expression (17) and Eq. (11) at )(tSy  by using expression (13), we obtain ,0 )( )()( 2  t tT dt tdT L WLW   …(19) . )( )( tdt tdS S S    …(20) Substituting expression (17) into Eq. (12), we get . )( )( IA t tTK L L WL   …(21) Substituting expressions (17) and (18) into Eq. (14) and by using Eq. (13), we obtain ). )()( ( )( )( t K t K L T td tdS S S L L S m   …(22) According to Eqs. (19)–(22), the following relations are obtained: )(tT W , 2 2/1 2 22        Ct K IA L LL  …(23) , 2 )( 2/1 2 22        Ct K IA IA K t L LL L L L   …(24) , 2 )()( 2/1 2 22        Ct K IA L KT ITA L t L LLSm S mL S S      …(25)   , /2 ln 2 )( 2/1222 2/1 2 22 m LLL L LL L L T CKtIA Ct K IA IA K tS            where .)( 2 22 22 2  TT AK AK TC m SLS LSL m    3. Finite Element Modeling During the laser irradiation many mechanisms are taking place in the basin material; a very narrow zone under the laser beam is suddenly heated, vaporized and locally fused. A two- dimensional finite element model was used to simulate the laser processes on the depth of the solid material using the commercial code ANSYS (9). The geometry of the structure, shown in Fig 2-a, was modeled using a two dimensional solid 8-nod element to simulate the laser processes on the depth of the solid material. The width of the material is broad enough compared to the molten region and the depth of the material is much greater to than the thermal penetration depth. The accuracy of the FE method depends upon the density of the mesh used in the analysis. The laser spot temperature is higher than the melting point of the material, and it drops sharply in regions away from the laser spot. Therefore in order to obtain the correct temperature field in the region of high temperature gradients it was necessary to have a more refined mesh close to the laser spot, while in regions located away a more coarse mesh was used as shown in Fig. 2-b. The final mesh was the result of compromise between computing time and accuracy. A transient heat transfer analysis was performed with an appropriate time-stepping scheme to achieve fast convergence of the solution and reasonable accuracy. The thermal analysis was conducted using temperature dependent thermal material properties the values of these properties for aluminum 2519 T87and stainless steel 304 are shown in Table 2 were taken from [10-11]. The governing partial differential equation for the transient heat conduction is                                    t T TCTQ y T x T TK P )()()( 2 2 2 2  …(27) where yx, are the Cartesian coordinates,  Q the internal heat generation,  the density, K the thermal conductivity and P C the specific heat are function of temperature T . The temperature dependent material properties were inserted in FE code in the table form. …(26) Faiz F. Mustafa Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) 101 (a) (b) Fig. 2. (a) 2-D Solid 8-nod Element. (b) 2-D Finite Element Geometry and Mesh. 3.1 Heat Input The heat input is generally calculated from the energy supplied, during laser irradiates at the solid material. The geometry considered in this dissertation is a finite, rectangular work-piece irradiated by a laser beam impinging on its surface and subjected to convection and radiation heat losses. A laser power P is assumed to have a uniformly distributed over a circular area of radius ω on the surface of a work-piece, so that a laser beam is assumed to have a uniform intensity distribution I defined as 2 /PI  …(28) 3.2 Boundary and Initial Conditions During the laser irradiates, the heat was supplied to the specimen surface by the beam laser. This heat is transferred to the metal by the conduction and convection. A part of this heat energy is lost by free convection and radiation. The heat loss by free convection follows Newton’s law, where the coefficient of convective heat transfer was assumed to vary with both temperature and orientation of boundary. )( TT L KNu q c  …(29) where K is the thermal conductivity of the material, L the characteristic length of the plate (or surface), T the ambient temperature, and Nu Nusselt number defined by 3/13/1 Pr67.5 GrNu  …(30) where Pr is the Prandtl number and Gr is the Grashof number, both of them being functions of ambient air properties and temperature differences between the surface and the environment. Heat loss due to thermal radiation between the spacemen and environment are important when the temperature difference is high. This radiation was modeled by the standard Stefan-Boltzman relation )( 44 TTqr   …(31) where  is the heat emissivity and  the Stefan- Boltzman constant. Radiation is assumed from the surface to the surroundings. The material is assumed to be at room temperature. 4. Results and Discussions We apply the above analytical solutions to aluminum and stainless steel. Table 2 shows their thermophysical parameters in solid and liquid phases and their absorptivity at wave length m6.10 laser irradiation. Faiz F. Mustafa Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) 102 Table 2 Variation of Thermal Properties with Temperature [10-11]. _________________________________________________________ Temp k (W/m/c) Cp (J/Kg/c)  (Kg/m3) (°C) Thermal conductivity Specific heat Density S.S AL S.S AL S.S AL ________________________________________________________ 0 53 100 465 896 7860 2823 100 52 140 485 915 7721 2754 200 50 160 494 950 7648 2705 300 46 155 509 952 7603 2654 400 42 145 529 1080 7670 2613 500 39 145 562 = 7635 2559 600 36 180 582 = 7405 2500 700 32 238 638 = 7365 2485 800 25 238 691 = 7330 2485 900 27 - 690 - 7303 - 1000 28 - 688 - 7272 - 1100 29 - = - 7235 - 1200 30 - = - 7210 - 1300 31 - = - 7170 - 1400 32 - = - 7131 - 1500 70 - = - 7103 - 1600 120 - = - 7042 - 1700 120 - = - 6980 - ________________________________________________________ 4.1. Melt Depth The propagation of the solid–liquid interface expressed by Eq. (26) shows that the melt depth increases rapidly at the beginning of laser irradiation and then slowly after a certain time. Such a trend is also observed in experimental studies for the laser drilling of aluminum [12]. The Mathematical and finite element calculated melt depth evolution of aluminum induced by m6.10 laser irradiation compared to the experimental data [12] is shown in Fig. 3. This figure shows a good agreement between the theory and the experiment for irradiation times less than about 4sec, but some deviations appear as the time increases. In fact, vapor and plasma will occur at large irradiation times; which block the incident laser light and absorb a portion of laser energy, but such effects are ignored in this work. As a result, the theoretical results seem to overestimate the melt depth for large irradiation times. Therefore, the model expressed by Eq. (26) is accurate when the vapor or plasma is not very strong, which is usually the case at the beginning of the irradiation process. The Mathematical and finite element variations of melt depth with m6.10 laser irradiation time for aluminum and stainless steel are shown in Fig. 4. At the beginning, the melting velocity is high and then decreases to a low value. The trend is the same as that observed in experimental studies. Some deviations appear between mathematical and finite element especially at the time increases because the temperature independences thermal material properties of the mathematical model. 0 0.005 0.01 0.015 0.02 0.025 0 1 2 3 4 5 6 7 8 9 10 Time (sec) M e lt d e p th ( m ) Mathematical Finite element Experimantal Fig. 3. Melt Depth Evaluation of Aluminum. Faiz F. Mustafa Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) 103 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6 7 8 9 10 Time (sec) M e lt d e p th ( m ) S.S-Mathematical S.S-Finite element AL-Mathematical AL-Finite element Fig. 4. Melt Depth Evaluation of Stainless-Steel and Aluminum I= 108 W/m2. 4.2. Temperature Profile and Evolution The mathematical and finite element temperature fields as the functions of depth for aluminum at different irradiation times are plotted in Fig. 5. (a) and (b). The curves 1 and 2 represent the temperature distribution in the solid before melting, while the curves 3 and 4 represent that after melting. For both models the temperature within the liquid phase decreases rapidly from the surface temperature to the melting temperature. Beyond the solid–liquid interface in the solid phase region, the temperature decreases to the ambient temperature with a relatively gradual gradient. Such a discontinuity in the temperature gradient is obviously observed due to the latent heat of fusion and the increment in thermal conductivity in the solid phase. It is also seen from the figure that the evolution of the surface temperature after melting is much faster than before melting, which results from the higher absorptivity and lower thermal conductivity in liquid phase. Fig. 6. (a) and (b), gives the finite element temperature profile for aluminum and stainless steel at the same power density I= 10 8 W/m 2 for different time irradiate, we can see with the different time the melt depth of aluminum are less than that of stainless steel due to high thermal conductivity and low absorptivity. 0 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 3 4 5 6 7 8 9 10 Distance (mm) T e m p e r a tu r e ( k ) 1 2 3 4 T=Tm 0 200 400 600 800 1000 1200 1400 1600 0 1 2 3 4 5 6 7 8 9 10 Distance (mm) T e m p e r a tu r e ( k ) 1 2 3 4 T=Tm Fig. 5. Temperature Distribution of Aluminum at Different Irradiation Time I= 7*10 8 W/m 2 . (a) Mathematical, (b) Finite Element. 4.3. The Effects of Power Density on Melt Depth The variations of melt depth versus incident power density for a given laser irradiation time for aluminum and stainless steel are plotted in Fig. 7. In the two cases (mathematical and finite element), the melt depth increases rapidly with increasing power density when the power density is low, and increases slowly at higher power densities. It is shown that the melt depths of aluminum are less than that of stainless steel at low power density, while greater than that at higher power densities. Aluminum has high thermal conductivity and low absorptivity, so that they are difficult to melt compared to stainless steel at low laser power density. With the increment of incident power density, more laser energy can be conducted to the solid–liquid interfaces of aluminum due to their higher thermal conductivity so that more laser energy needs to be used to achieve melting. Therefore, the melt depths of aluminum will exceed these of stainless steel at higher power density. (a) (b) Faiz F. Mustafa Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) 104 (a) (b) Fig. 6. Temperature Distribution During Laser Irradiation at Different Times, I=10 8 W/m 2 . (a) Aluminum, (b) Stainless Steel. t=2 sec t=4 sec t=6 sec t=8 sec Faiz F. Mustafa Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) 105 0 20 40 60 80 100 120 140 1.00E+08 1.00E+09 1.00E+10 I (W/m^2) M e lt d e p th ( m m ) S.S-Mathematical S.S-Finite element AL-Mathematical AL-Finite element Fig. 7. Influence of Power Density on Melt Depth at Time= 5 sec. 4.4. The Time for Surface to Melt and Vaporization Let )(tT W in Eq. (8) equal the fusion temperature m T . The time for the surface reaching the melting point m t is 2 2 )(2 )( IA KCTT t S SSSm m   …(32) Let )(tT W in Eq. (23) equal the vaporization temperature V T .The time for the surface reaching the vaporization point v t is LL LV v IA KCT t  2 22 )(2 )(   …(33) The results of two metals are shown in Table 3. It is shown that stainless-steel needs least time to reach fusion temperature and vaporization temperature then aluminum due to its low thermal conductivity although it has relatively high fusion and vaporization temperatures. Table 3 The Time Needed to Melt and Vaporize for Materials at Different Power Densities. ________________________________________________________ )/( 2 mWI 108 109 1010 ________________________________________________________ Melting Al 1.175 1.175*10^-2 1.170*10^-4 time(s) S.S 0.148 0.148*10^-2 0.148*10^-4 Vaporizing Al 55.51 55.51*10^-2 56.01*10^-4 time(s) S.S 1.730 1.730*10^-2 1.730*10^-4 _______________________________________________________ 5. Conclusions An analytical method for treating the problem of the laser heating and melting is considered in this paper by suggesting a simple and reasonable temperature profile and compared with a two dimensional finite element model. We apply the two methods to aluminum 2519T87 and stainless steel 403. The temperature profile and evolution of aluminum before melting as well as after melting is described. A discontinuity in the temperature gradient is obviously observed due to the latent heat of fusion and the increment in thermal conductivity in solid phase. The calculated melt depth evolution of aluminum is in good agreement with the experimental results. The effects of laser power density on the melt depths for two metals are also obtained. It can also be concluded that stainless steel needs least time to reach fusion temperature and vaporization temperature than aluminum due to its low thermal conductivity although it has relatively high fusion and vaporization temperatures. References [1] El-Adawi MK, El-Shehawey EF. Heating a slab induced by a time-dependent laser irradiation—An exact solution. J. Appl Phys 1986; 60 (7): 2250–5. [2] Hassan AF, El-Nicklawy MM, El-Adawi MK. A general problem of pulse laser heating of a slab. Opt Laser Technol 1993; 25 (3): 155–62. [3] Rantala TT, Levoska J. A numerical simulation method for the laser-induced Faiz F. Mustafa Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) 106 temperature distribution. J. Appl. Phys. 1989; 65 (12): 4475–9. [4] Armon E, Zvirin Y, Laufer G. Metal drilling with a CO2 laser beam. I. Theory. J Appl. Phys 1989; 65(12): 4995–5002. [5] Kar A, Mazumder J. Two-dimensional model for material damage due to melting and vaporization during laser irradiation. J. Appl. Phys. 1990; 68: 3884–91. [6] V. A. Karkhin, V. A. Lopota, and N. O. Palova. Effect of phase transformations on residual stresses in laser welding. Welding International 2003 17 (8) 645–649 [7] Han GuoMing, Zhao Jian, Li JianQang. Dynamic simulation of the temperature field of stainless steel laser welding. School of Material Science and Engineering, Tianjin University, China, 8 June 2005. [8] S.A. Tsirkas, P. Papanikos, Th. Kermanidis. Numerical simulation of the laser welding process in butt-joint specimens. J Materials Processing Technology 2003; 134:59-69. [9] Justin D. Francis. Welding Simulations of Aluminum Alloy Joints by Finite Element Analysis. Master of Science in Aerospace Engineering April 2002 Blacksburg, Virginia [10] A. Anca, A. Cardona, and J.M. Risso. 3D- Thermo-Mecanical Simulation of Welding Processes. Centro Internacional de M´etodos Computacionales en Ingenier´ıa (CIMEC), Bariloche, Argentina, November 2004. [11] D. Berglund, L.E. Lindgren and A. Lundbäck. Three-Dimensional Finite Element Simulation of Laser Welded Stainless Steel Plate. Computer Aided Design, Luleå University of Technology, 97187 Luleå, Sweden, 2004. [12] Armon E, Zvirin Y, Laufer G. Metal drilling with a CO2 laser beam. II. Experiment. J. Appl. Phys. 1989; 65 (12): 5003–8. Faiz F. Mustafa Al-Khwarizmi Engineering Journal, Vol. 4, No. 3, PP 98-107 (2008) 107 النمىذج الحراري واالنصهاري الناتج من الحسمة الليسرية على المىاد الصلبة فائس فىزي مصطفى ظايعح تغذاد/ كهٛح ُْذسح انخٕاسصيٙ/ قسى ُْذسح عًهٛاخ انرصُٛع الخالصة ذى اسرخذاو ًَٕرض , نٕصف ظاْشج اسذفاع انحشاسج ٔانٕصٕل انٗ دسظح االَصٓاس انُاذط يٍ حضيح نٛضسٚح يسهطح عهٗ ظسى صهة ٔانحذٚذ انًقأو (2519T87) سٚاضٙ احاد٘ انثعذ تاالضافّ انٗ اسرخذاو طشٚقح انعُاصش انًحذدِ شُائٛح انثعذ نكم يٍ يادج االنًُٕٛو َٕع دسظح االَصٓاس ٔانرثخش ٔذأشٛش كصافح طاقح حضيح انهٛضس عهٗ عًق االَصٓاس ٖذى حساب انٕقد انالصو نهٕصٕل ال. (304) نهصذاء َٕع حٛس , ْزا تاالضافّ انٗ انعًق انعاَثٙ يع ذغٛش دسظاخ انحشاسج نًخرهف االصياٌ قثم ٔتعذ حذٔز عًهٛح االَصٓاس. انحاصم نكال انًادذٍٛ نٕحع عذو اسرًشاسٚح اَسٛاتٛح ذذسض دسظاخ انحشاسج تشكم ٔاضح عُذ االَرقال يٍ انطٕس انسائم انٗ انطٕس انصهة ٔرنك تسثة انحشاسج نقذ أضحد انُرائط انشٚاضّٛ . انكايُح نالَصٓاس ٔانضٚادج انحاصهح فٙ يعايم انرٕصٛم انحشاس٘ نهطٕس انصهة عًا ْٕ عهّٛ فٙ انطٕس انسائم .m6.10.ذٕافق ظٛذ يع انُرائط انعذدٚح تاسرخذاو طشٚقح انعُاصش انًحذدج انُاذعح يٍ نهحضيح انهٛضسٚح تطٕل يٕظٙ