IAS-2008-66-546ST Finite element simulation of residual stresses in laser heating G. H. Farrahi 1, M. Sistaninia 2, H. Moeinoddini 3 1,2-School of Mechanical Engineering, Sharif University of Technology, Tehran, I. R. of Iran 2-Department of Mechanical Engineering, Iran University science and Technology, Tehran, Iran Abstract The laser heating involves the use of high intensity laser radiation to rapidly heat the surface of metals to forming or hardening metals. During a laser heating event, pressures well above the dynamic yield strength of the material are imparted on the target. This will lead to distortion because both elastic and plastic deformation will be involved. After the laser beam has dispersed, the deformation remains then the compressive stresses convert to tensile. Because the flow stress and thermal expansion of a material are temperature dependent, this finally leads to distortion of the sample with internal stresses within the workpiece. 3 Dimension Finite element analysis techniques have been applied to predict the residual stress in a semi infinite induced from laser heating. The analytical development, including the loading history and the elastic plastic constitutive model, is discussed. Analysis results are presented. The three-dimensional thermal results of the FEM model presented in this paper compared with the results of an analytical thermal modeling. Keywords: Laser heating- Residual stresses- FEM - Thermal modeling. Introduction Theoretical and experimental investigation of laser heating of materials, the resulting thermal stress fields and temperature field, date back to the 1960 s. Laser heating, although possible for a number of years, is a technology which is still in its infancy. Until recently, the widespread use of lasers for materials processing has been hindered by the size, complexity and high investment cost of the laser systems [1]. In recent years, considerable attention has been paid to the numerical analysis of laser heating of materials,[2-6]. The process involves the use of high intensity laser radiation to rapidly heat the surface of steel. In this process residual stresses are imparted at and near the surface of the target material. Laser processing generates different internal stress situations depending on the control temperature, feed rate and composition of the steels, especially its carbon content and austenite-martensite phase transformation. In this article only the residual stress due to elastic and plastic deformation was modeled. Laser heating analysis procedure The heat produced due to plastic deformation is very small and can be ignored [7]. Thermo-mechanical problem can therefore be decoupled and solved sequentially in two steps. A transient thermal analysis was first carried out to determine temperature distribution which was then input as a body load for the stress analysis to determine the residual stresses and distortions. Ansys/Explicit could be used to model both the initial load (Thermal load), and to determine the final residual stress field, and ANSYS APDL language was used for incorporating the laser beam motion scheme, for scans and subsequent application of thermal loads in the stress analysis. The convergence toward the residual stress state is extremely slow. A number of techniques have been proposed to facilitate rapid convergence, but none have been found to be sufficiently robust and/or reliable. The thermal results of the FEM model compared with the results of an analytical model. Thermal modeling The transient thermal field in laser heat treatment of medium carbon steel was analyzed by employing both threedimensional analytical model and finite element model. In finite element model the laser beam was considered as a moving plane heat flux to establish the temperature rise distribution in the workpiece. While in analytical model laser beam was considered as an internal generation of heat. Analytical model An analytical model, based on the well-known transient heat conduction equation, was adopted to describe the time and space temperature distribution T(r,t) in the material under the action of a laser beam. 1 T 2 f(,) rt T = (1) α t K 1 Professor 2 M.S. Student(corresponding author) 3 M.S. Student
where α = K/ ρ c p, and K denote thermal conductivity of the material. The heat source term f(r,t), at the right side of Eq. (1), is identified as the energy distribution of the laser beam. In the case of three-dimensional transient, nonhomogeneous heat conduction problem given by equation (1), the solution for T(r,t) is expressed in terms of the three-dimensional Green s function, as[8], α t T(,) r t = d G(, t ', ) f ( ', ) d ' K τ τ τ ν τ = 0 r r r R + G( r, t r', τ) 0 F ( ') d ' (2) R τ = r ν where F ( r ') is the initial temperature distribution. The three-dimensional Green s function can product of the three one-dimensional Green s function [8]. Gxyzt (,,, x', y', z', τ ) = G1( xt, x', τ ) G2( yt, y', τ ) G3( zt, z', τ ) (3) where each of the one-dimensional Green s functions G1, G2 and G3 depends on the extent of the region (i.e., finite, semi infinite, or infinite). However, the three-dimensional Green s function, for z ' = 0, obtain as, ' ' ' [ ] 3/2 Gxyzt (,,, x, y, z = 0, τ ) = 2 4 πα ( t τ ) ' 2 ' 2 2 ( x x ) + ( y y ) + z exp 4 α( t τ) (4) And, α T( xyzt,,, ) = K f( x', y', ) 2[4 ( t )] ( y y') + ( x x') + z'. exp d y ' dx 'd τ 4 α( t τ) (5) t 1.5 τ πα τ 0 2 2 In this equation, f( x', y', τ ) is the beam intensity distribution, which for a beam with Gaussian distribution that move along a straight line is, f( x', y', τ ) = 2 2 3 Q ( x' ντ ) + ( y') (6) exp 3 2 2 π r o r o Finite element model The basic heat-transfer equations considered are F = ( KT ) (7-a) Which relates heat flux F to thermal gradient? T ( K T) ρcp = G (7-b) t In finite element model the laser beam consider as a moving plane heat flux to establish the temperature rise distribution in the workpiece. In this model we divide the time during laser beam continuous irradiation to n time increment Δt, during every increment Δt we assume that laser beam don t move and the program determine thermal load for every node on the heated surface, that depend on the laser beam position, heat intensity distribution and the node position. Laser beam position depend on the moving pattern, speed and time thus the value of load for every node on the heated surface is a function of time and place. Care must be taken, however, in the determination of heat flux for every node if the incident intensity is discontinuous, as would be the case of the uniform disc beam mode intensity distribution 2 2 2 Q/ π r (, ) o x + y r Ixy = o 0 otherwise. (8) In every time increment finite element code ANSYS is used to compute the solution to the heat transfer equation (7), and the result of every time increment become the initial condition of next time increment [9]. For small Δt we can assume that the laser move continuously. The flow chart of thermal finite element simulation is shown in Fig. 1. In the model, it is necessary to make a decision about the element size and shape, time increment and number of step n* for every time increment. Since these quantities are not independent, the following relation can be written. 2 x Fo = * α t * t t = * n (9) The Fourier number Fo, which includes material thermo-diffused efficiencyα, time step Δt* and node spacing x should be below 2. This decision has been made with the help of the analytical model. Stress numerical simulation As mentioned above, the stress analysis is performed separately from the thermal analysis. However, because of the variation of nodal temperatures with time, the mechanical stresses and strains also vary with time. This indicates that the structural analysis is time-dependent although the inertia and damping forces need not be included, that is so-called quasistatic analysis. Loading For every loadstep the result of thermal modeling of the same loadstep become the thermal load for that loadstep. Thus the number of loadstep for stress analysis is as many as the number of loadstep for thermal analysis. The flow chart of the finite element simulation is shown in Fig. 2. Constitutive modeling Laser heating generates stresses exceeding yield stress within the target material. This behavior is modeled analytically using a bilinear kinematic stress strain curve (Fig. 3) with strain-rate independence of the yield strength. Rate-independent
plasticity is characterized by the irreversible straining that occurs in a material once a certain level of stress is reached. Model of calculation The specimen is discretized by means of an 8-nod brick element for mechanical analysis to attain high precision and the meshes are as the meshes for thermal analysis. Generally, the finite element model extends about 4 times the spot radius (R) along both the heating surface and through the depth. The analyses which follow employ a square mesh with an average element edge length on the order of 7% of the spot radius. The material investigated in this study is AISI 304L (stainless steel). A cubic of 30 mm along the direction of laser moving and 16 mm vertical the direction of laser moving and 8 mm thickness was used for the analysis. Temperature dependent physical and mechanical properties were used for the model (Table 1) [10 11]. The material was assumed to be homogenous and isotropic. The ambient temperature was set at 25 o C. The following are the processing variables chosen for the heat treatment of the specimen: laser power 250 J/s ; scan velocity, 40 mm/s, the beam diameter of 3mm for uniform and 4mm for Gaussian distribution with the same power on a circular disc beam heat source. Results Discussion Position of the global coordinate system with reference to the specimen for Gaussian distribution is shown in Fig. 4. Figure 5 shows the temperature change with time at various points on the workpiece surface for the Gaussian and uniform beam modes. In the Gaussian beam mode as shown in Fig. 5, the temperature history at any selected point can be described by a continuous smooth curve because the heat flux on the sliced solution domain increases and decreases continuously. In contrast the temperature history in the uniform beam mode appears as a discontinuous curve because of its suddenly decreasing characteristics of the heat flux as shown in Fig. 5. The longitudinal thermal stress distributions on the heated surface during heating and cooling are shown in Fig. 6. The simulation conditions of Figs. 5 and 6 were selected to be the same (Gaussian distribution) and the simulation conditions of Figs 5 and 6 also were selected to be the same (uniform distribution). In Fig.7 variation of Longitudinal stress distribution from heated surface into the bulk at y=0 during heating and cooling along z-dir. for Gaussian beam mode and uniform beam mode are shown. As you seen in Fig. 6 and Fig. 7 upon the initial heat-up the localization of sever temperature gradients in the immediate vicinity of the beam line produces the compressive yielding in this region. During the loading, the material under the spot deforms. This material wants to expand radially, but is constrained by the surrounding material. This constraint results in compressive radial stresses. As you see, the stresses exceeding yield stress within the target material. Thus both elastic and plastic deformation will be involved. After that the laser has dispersed the plastic deformation remains and the state of thermal stress changes to the tensile residual stress during cooling. Fig. 8 shows the variation of Sy distribution and Fig. 8 shows the variation of Sz distribution from heated surface into the bulk at y=0 for Gaussian beam mode during heating and cooling along z-dir. The radial stresses (S x, S y ) are particularly important in the Laser heating simulation because as you see in Fig. 6 and 7 they ultimately become the residual stresses. The axial stress Sz are insignificant once the initial heating-up through the target. The variation of Sy stress distribution on the heated surface during heating and cooling also are shown in Fig. 9. Concluding Remarks By using the three-dimensional finite element model the transient thermal stress and the residual stress in laser heat treatment of a stainless steel were analysed. By using the proposed model, the thermal and residual stresses in the laser heat treatment were successively calculated. The simulation results revealed that upon the initial heat-up the localization of sever temperature gradients in the immediate vicinity of the beam line produces the compressive yielding in this region. However after that the laser has dispersed the state of thermal stress changes to the tensile residual stress. Table(1) Material Properties [10-11] Temperature 200 300 400 Thermal Conductivity (W/mK) 12.6 14.9 16.6 Specific heat C p (J/kgK) 402 477 515 Density ρ (kg/m 3 ) 7854.01 Thermal expansion α (1/K) 1.7e-05 1.7e-05 1.8e-05 Poisson's ratio ν 0.29 0.29 0.2925 Elastic modulus E(Gpa) 193 191 183 Yield stress (Mpa) 410 331 265 Tangent modulus (Mpa) 29370 27791 23435 600 800 1000 1200 1500 19.8 22.6 25.4 28 31.7 557 582 611 640 682 7854.01 1.9e-05 1.95e-05 2e-05 ----- ----- 0.296 0.308 0.33 0.36 0.39 168 148 128 110 ----- 214 112 66 ------- ------- 21761 18413 16739 ------- ------
References [1] A review of the use of high power diode lasers in surface heating E. Kennedy1, G. Byrne1, D. N. Collins2 [2] Hong Shen *, Yongjun Shi, Zhenqiang Yao, Jun Hu, An analytical model for estimating deformation in laser forming, Computational Materials Science 37 (2006) 593 598. [3] Hong Shen *, Yongjun Shi, Zhenqiang Yao, Numerical simulation of the laser forming of plates using two simultaneous scans, Computational Materials Science 37 (2006) 239 245 [4] Shakeel Safdar, Lin Li, M.A. Sheikh, Zhu Liu, Finite element simulation of laser tube bending: Effect of scanning schemes on bending angle, distortions and stress distribution, Optics & Laser Technology 39 (2007) 1101 1110. [5] J.Domes, DMuller,and H.W.Bergann, 3rd Eurpean conference on Residual Stress, Frankfurt, DGM, 1993 [6] F.R. Liu, K.C. Chan, C.Y. Tang, Numerical simulation of laser forming of aluminum matrix composites with different volume fractions of reinforcement, Materials Science and Engineering A 458 (2007) 48 57. [7] Structure with material non linearities and coupled field analysis guide. In: ANSYS theory manual. ANSYS, Inc., 2003. [8] M.N. Özisik, Heat conduction, second edition, 1993. [9] ASYS Help System, Theory Reference, ver 10. [10] Engineering Properties of Steel: American Society for Metals. [11] Rasmussen Jr. K. Full-range stress strain curves for stainless steel alloys 2001; 2005. Figures and Drawings Yes START Define element type, material properties Build geometric model, generate meshes Define temperature for every node Discover the laser position Find thermal load for every node on the heated and Apply thermal boundary conditions surface Formulate the conductivity and specific heat matrices Calculate the temperature Distribution at time t Lode step end? END No Let time t n+1 =t n + t Fig. 1 Flow chart of thermal modeling during laser hardening using a finite element method.
Change element to structural analysis Input temperature field result and boundary Calculate the non-linear structural equation Lode step end? Let time t n+1 =t n + t Analyze the result END Fig. 2 Flow chart of thermal modeling during laser hardening using a finite element method. Fig. 5. Temperature histories of Gaussian beam mode (4mm beam diameter) uniform beam mode (3mm beam diameter) Fig. 3 Bilinear kinematic stress strain curve Fig. 4 Schematic diagram of Gaussian beam mode Fig. 6 Longitudinal stress distribution on the heated surface during heating and cooling along y-dir. Gaussian beam mode uniform beam mode
Fig. 9 Variation of vertical stress (Sy) distribution on the heated surface y-dir. for Gaussian beam mode Fig. 7 Variation of Longitudinal stress distribution from heated surface into the bulk at y=0 during heating and cooling along z-dir. Gaussian beam mode uniform beam mode Fig. 8 Variation of stress distribution from heated surface into the bulk at y=0 for Gaussian beam mode during heating and cooling along z-dir. Sy (Mpa) Sz (Mpa)