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Computers and Structures 80 (2002) 967–976 www.elsevier.com/locate/compstruc Effects of temperature-dependent material properties on welding simulation X.K. Zhu, Y.J. Chao * Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA Received 22 August 2001; accepted 11 January 2002 Abstract Detailed three-dimensional nonlinear thermal and thermo-mechanical analyses are carried out using the finite element welding simulation code––WELDSIM [Chao et al., In: Advances in Computational & Engineering Science, vol. II, Tech Science Press: Paledale, USA: 2000. pp. 1206–1211]. The objective is to investigate the effect of each temperature-dependent material property on the transient temperature, residual stress and distortion in computational simulation of welding process. Welding of an aluminum plate using three sets of material properties, namely, properties that are functions of temperature, room temperature values, and average values over the entire temperature history in welding, are considered in the simulation. Results show that (a) the thermal conductivity has certain effect on the distribution of transient temperature fields during welding, (b) the yield stress and Young’s Modulus have significant and small effects, respectively, on the residual stress and distortion, after welding, and (c) except for the yield stress, using material properties at the room temperature gives reasonable predictions for the transient temperature fields, residual stress and distortion. Since high temperature material properties are either difficult to obtain or do not exist for many materials, an engineering approach is proposed based on the results in this study. The engineering approach suggests using simplified properties constituted by a piecewise linear function with temperature for the yield stress and constant room-temperature values of all other properties for computational weld simulation. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Welding simulation; Material property; Residual stress; Welding distortion; Thermal modeling; Three dimension; Finite element analysis 1. Introduction Many metallic structures in industry are assembled through some kind of welding process which is composed of heating, melting and solidification using a heat source such as arc, laser, torch or electron beam. The highly localized transient heat and strongly nonlinear temperature fields in both heating and cooling processes cause nonuniform thermal expansion and contraction, and thus result in plastic deformation in the weld and surrounding areas. As a result, residual stress, strain and * Corresponding author. Fax: +1-803-777-0106. E-mail address: [email protected] (Y.J. Chao). distortion are permanently produced in the welded structures. High tensile residual stresses are known to promote fracture and fatigue, while compressive residual stresses may induce undesired, and often unpredictable, global or local buckling during or after the welding. It is particularly evident with large and thin panels, as used in the construction of automobile bodies and ships. These adversely affect the fabrication, assembly, and service life of the structures. Therefore, prediction and control of residual stresses and distortion from the welding process are extremely important in the shipbuilding and automotive industry. Over the past 20 years, research has been conducted enabling the use of advanced analytical procedures to more accurately simulate the welding process. Due to 0045-7949/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 0 4 0 - 8 968 X.K. Zhu, Y.J. Chao / Computers and Structures 80 (2002) 967–976 the complexity of the physical processes involved in welding, however, simple mathematical solutions cannot address the practical manufacturing processes. Furthermore, it is also impossible for any experimental technique to obtain a complete mapping of the residual stress and distortion distribution in a general welded structure. Computational simulation thus plays an indispensable role in the integrity analysis of such welded structures. Hibbitt and Marcal [8] marked the first step in applying a two-dimensional (2D) finite element analysis (FEA) to predict residual stresses in a weldment. Due to computational and cost limitations, FEA simulation efforts during the 70s and 80s were focused on simplified 2D geometries [9,14]. In reality, the thermal and stress–strain responses of all weldments are three-dimensional (3D). With recent advancements in computational power, FEA simulation of transient temperature and residual stresses in welding has gradually become feasible. Recently, 3D welding simulation was conducted using commercial FEA software, as reported by Tekriwal and [18,19], Brown and Song [1], Michaleris and DeBiccari [15], Dong et al. [6], and Chao et al. [4]. In a computational simulation of welding process, material modeling is one of the key problems. Recently, Lindgren [10] gives a detailed review on this topic, which includes the development of material constitutive relationships, material microstructures and material properties as functions of temperature. Most publications in welding simulation adapted material properties that are dependent on temperature. However, in practice complete temperature-dependent material property data required for welding simulation are difficult to obtain, especially at high temperatures. To circumvent this problem, assumptions and simplifications are often proposed for welding simulation. Stanley and Chau [17] first explored effects of the temperature-dependent properties on thermal stresses in cylinders. Free and Goff [7] assumed all material properties (except the yield stress) remaining as the room temperature values to model welding. Canas et al. [2] studies the influence of strain hardening and temperature-dependent properties on the residual stresses. Ma et al. [13] simulated residual stresses using constant material density, Poisson ratio and thermal expansion coefficient. Little and Kamtekar [12] and Little et al. [11] presented the effect of various thermal properties and weld efficiency on transient temperature during welding. Chen et al. [5] and Shi et al. [16] numerically studied the effect of material properties on the welding simulation using extrapolated material properties at high temperature and using different fictitious data of material properties at high temperature covering a bandwidth which includes the extrapolated unknown data, respectively. These authors then concluded that the unavailable material property data at high temperature have almost no effects on the residual stresses and distortion. To reduce computational time, many numerical analyses often used a cut-off temperature above which no changes in the mechanical properties are accounted for. Tekriwal and Mazumder [19] showed that the residual stresses from FEA have small changes for carbon steels when the cut-off temperature varied from 600 to 900 °C, but the computational time is significantly different. Although various assumptions and simplifications were applied in welding simulation, as cited in the previous paragraph, the potential errors from these assumptions and simplifications to the transient temperature fields, residual stress and distortion are not assessed and are uncertain [16]. To this end, the present paper systematically investigates the errors associated with each of the material properties in a welding simulation. We performed detailed 3D thermal and thermomechanical welding simulations for a 5052-H32 aluminum alloy by using three sets of material properties, i.e. those at the room temperature, averaged over the temperature history in the welding process, and as functions of temperature. WELDSIM, a 3D FEA software developed at the University of South Carolina by the authors, is used [4] in the simulation. By comparing the results for the three cases, the effect of each temperaturedependent material property on transient temperature, residual stress and distortion is investigated. Since high temperature material properties are either difficult to obtain or do not exist for many materials, an engineering approach is proposed based on the results in this study. The engineering approach suggests using simplified properties constituted by a piece-wise linear function with temperature for the yield stress and constant room-temperature values of all other properties for computational weld simulation. It is shown that this approach yields transient temperature fields, residual stress and distortion with sufficient accuracy, yet circumvents the difficulty in finding and using exact high temperature material properties. 2. Geometry and computational simulation 2.1. Geometry configuration For convenience, the classical gas metal arc welding (GMAW) experiment by Masubuchi [14] is modeled in the present welding simulation. Fig. 1 shows the geometry and welding configuration of the problem under consideration. As shown in the figure, the moving torch is applied on the longitudinal upper edge of the plate of 1220 mm long, 152.4 mm wide and 12.5 mm thick. The plate is simply supported at both ends. Such a geometry and arrangement are chosen to produce a complete set of experimental data that include transient temperature, X.K. Zhu, Y.J. Chao / Computers and Structures 80 (2002) 967–976 Fig. 1. Configuration of the welding plate (dimensions are in millimeter). residual stress, and distortion at various locations in the plate. As such, it provides excellent information to validate results from any computer simulation. The material used in the welding was 5052-H32 aluminum alloy. Thermal and mechanical properties of the material are shown in Fig. 2(a) and (b), respectively. Detailed experimental setup and procedures can be found in Masubuchi [14]. Six single-direction strain gages were used for measuring longitudinal strains. Four thermocouples were also mounted on the plate at dis- 969 tances of 12.7, 38.1, 76.2 and 144.8 mm from the top, as shown in Fig. 1, to record the temperature history. A dial gage was used to measure the transient deflection of the beam at the lower, midpoint of the beam during welding. After welding was completed and the plate cooled to the room temperature, the longitudinal residual stresses along the middle section were determined using rosette strain gages by sectioning the plate. The arc traveling speed is 7.34 mm/s which yields a 166 s total time of welding. The welding current is 260 A, and the arc voltage is 23 V. A Gaussian distribution with an effective arc radius of 6 mm and thermal efficiency of 64.3% are used in the current computer simulation for the heat flux applied to the plate. 2.2. Computational simulation A half model in the thickness direction was used in the computer simulation due to symmetry of the plate. One layer of element is used in the FEA model, which consists of 900 eight-node brick solid elements and 2020 nodes in both the heat transfer and the thermomechanical analyses. The smallest element used in the modeling is in the area of the weld and has the dimensions of 12:2 mm 6 mm 6:25 mm. The finite element mesh is the same as that shown in Fig. 2 of Chao et al. [4] and yields results with sufficient accuracy. It should be mentioned that in a mesh sensitivity study [4] a finer FEA mesh with 1800 elements and 3030 nodes leads to almost identical results as the present FEA mesh, but the computational time increased significantly. An uncoupled thermal and thermo-mechanical analysis is adapted in this calculation. The thermal analysis was performed first and the transient temperature outputs from this analysis are saved for the subsequent thermo-mechanical analysis. In the thermal analysis, the transient temperature field T is a function of time t and the spatial coordinates ðx; y; zÞ, and is determined by the 3D nonlinear heat transfer equation: kT;ii þ Qint ¼ cqT_ ð1Þ where k is the conductivity, Qint is the internal heat source rate, c is the specific heat and q is the density of materials. The comma (,) denotes partial differentiation with respect to a spatial coordinate, the dot (Þ denotes differentiation with respect to time t. Heat flux to the system is input by a moving source on the boundary. To consider heat convection and radiation on the plate surfaces, the heat flux loss is evaluated by qs ¼ bðT T0 Þ þ eBðT 4 T04 Þ Fig. 2. Variation of material properties with temperature of 5052-H32 aluminum alloy (a) thermal properties, (b) mechanical properties. ð2Þ where T0 is the room temperature, b is the convection coefficient, e is the emissivity of the plate surfaces and 970 X.K. Zhu, Y.J. Chao / Computers and Structures 80 (2002) 967–976 B ¼ 5:67 1012 W=cm2 °C is the Stefan–Boltzmann constant. In this calculation for the aluminum alloy, b ¼ 30 W=m2 °C and e ¼ 0:03. In addition, a latent heat of fusion is taken as 400 J/g to consider the phase transformation of the aluminum. In the thermo-mechanical analysis, the plastic deformation of materials is assumed to obey the Mises yield criterion and the associated flow rule. The rate relationship between thermal stresses, rij , and strains, eij , is described by e_ij ¼ 1þm m oa r_ ij r_ kk dij þ ksij þ a þ ðT T0 Þ T_ E E oT ð3Þ where E is Young’s modulus, m is Poisson ratio, a is the thermal expansion coefficient. sij ¼ rij ð1=3rkk dij Þ are the components of deviatoric stresses and k is the plastic flow factor. k ¼ 0 for elastic deformation or re < rs , and k > 0 for plastic deformation or re P rs , here rs is the yield stress and re ¼ ð3=2sij sij Þ1=2 is the Mises effective stress. A cut-off temperature of 400 °C (i.e. about 2=3 of the aluminum melting temperature 607 °C) is used in the numerical calculations to reduce unnecessary computational time. All FEA computations are performed on PCs by using WELDSIM [4]. To save computational time without loss accuracy of results, different time steps are used for the two analyses. The time step used in this study is 0.1 s for the heat transfer analysis, and 1.0 s for the stress-deformation analysis. As such, the CPU time in a typical welding simulation is about 1.2 min for the temperature computation, and 81.8 min for the stress and deformation computation. 2.3. WELDSIM––a WELDing SIMulation code WELDSIM is a computer simulation software, developed at the University of South Carolina. The overall structure of this software can be found in Chao et al. [4] and Chao and Qi [3]. WELDSIM is a 3D, nonlinear finite element computer code for the determination of transient, as well as residual, temperature, stress, strain and distortion of welded structures in a welding process. Advanced features in the FEA code make WELDSIM robust and computationally efficient. Its efficiency has been verified through comparisons of computed results with experimental data (see [3,4]). In general, in WELDSIM, a heat transfer analysis and a subsequent thermo-mechanical analysis can be run sequentially or in a sequentially coupled manner. A moving heat source simulating the torch is modeled first to generate the temperature fields in the structure at various time steps during the welding process, and then the temperature history is used for the calculation of thermal stress and displacement fields in the structure during and after the welding process. The finite element formulation and solver in WELDSIM follow the standard procedures in computational mechanics. Details can be found in Chao and Qi [3]. Unique features in welding implemented into the code include: moving heat source model, complex heat input models for various types of heat sources, latent heat for simulating melting, dummy elements for simulating multi-pass welding, cutoff temperature to increase the computation speed, effective weight factor for elastic to plastic transition, flexibility in convergence criterion for both the heat transfer and stress analyses, both small and large deformation options for predicting distortion, and effect of welding fixture to the residual stress and distortion. 3. Effect of thermal properties on temperature simulation This section reports the numerical results of transient temperature field for the three cases using the thermal physical properties at room temperature, averaged over temperature history and as functions of temperature. All FEA results are compared with the experimental data obtained by Masubuchi [14]. Accordingly, the effect of various thermal properties on temperature simulation is revealed and discussed. 3.1. Thermal modeling The heat transfer equation (1) clearly indicates that the material density q, specific heat c and thermal conductivity k are three primary thermal physical properties in the thermal analysis. To avoid interaction of these parameters in the study, each parameter is taken as the room temperature (RT) value, average value over temperature history and the value as functions of temperature, respectively, in the FEA computation. For comparison, eight test cases as follows are considered in our thermal modeling: T1––all three parameters q, c, and k are functions of temperature, as shown in Fig. 2(a). T2––density q at RT, q ¼ q0 ¼ 2:69 g/cm3 , c and k are functions of temperature. T3––averaged density q ¼ q ¼ 2:64 g/cm3 , c and k are functions of temperature. T4––specific heat c at RT, c ¼ c0 ¼ 0:94 J=g °C, q and k are functions of temperature. T5––averaged specific heat c ¼ c ¼ 1:07 J=g °C, q and k are functions of temperature. T6––conductivity k at RT, k ¼ k0 ¼ 1:67 W=cm °C, q and c are functions of temperature. T7––averaged conductivity k ¼ k ¼ 2:06 W=cm °C, q and c are functions of temperature. T8––all three parameters are taken as the RT values, i.e. q ¼ q0 , c ¼ c0 and k ¼ k0 . X.K. Zhu, Y.J. Chao / Computers and Structures 80 (2002) 967–976 971 Since all thermal properties are functions of temperature in Case T1, the FEA result of temperature is referred to as the baseline or standard numerical solution. All other FEA results of temperature for Cases T2– T8 will be compared to that from Case T1, as well as the test data, to evaluate the accuracy and error of simulation results for each case. 3.2. Numerical results for temperature Fig. 3 shows the transient temperature history for various densities at four test points for the three cases T1, T2 and T3. The four points are the locations of the thermocouples at distances of 12.7, 38.1, 76.2 and 144.8 mm from the top of the beam, as shown in Fig. 1, and named as #1, #2, #3 and #4, respectively, in Fig. 3. The experimentally measured temperature values at these points obtained by Masubuchi [14] are also plotted in the figure. Fig. 3 shows that the FEA results for Cases T1, T2 and T3 are identical to each other, and match well with the test data. Therefore, it is concluded that a constant material density value at room temperature can be used in numerical simulation to obtain good temperature results. Fig. 4 shows the influence of various specific heat on the distribution of temperature at the four tested points for cases T1, T4 and T5. It is seen that the FEA results for Cases T1, T4 and T5 are close to each other with very small differences, and match well with the test data. Therefore, a constant specific heat value at room temperature can be used in numerical simulation to obtain good temperature results. Fig. 5 shows the distribution of temperature at the four tested points for various thermal conductivities for cases T1, T6 and T7. The data in Fig. 5 indicate that (a) Fig. 4. Effect of specific heat on the transient temperature at four points. Using specific heat at the room temperature (Case T4), average value (Case T5) and function of temperature (Case T1). Fig. 5. Effect of thermal conductivity on the transient temperature at four points. Using thermal conductivity at the room temperature (Case T6), average value (Case T7) and function of temperature (Case T1). Fig. 3. Effect of density on the transient temperature at four points. Using density at the room temperature (Case T2), average value (Case T3), and function of temperature (Case T1). the FEA results for the RT conductivity (Case T6) overestimated the temperature relative to those from the standard Case T1, with the maximum error less than 10%, and (b) the numerical results from using the averaged conductivity (Case T7) are nearly the same as those obtained from the standard Case T1. Moreover, all FEA results for Cases T1, T6 and T7 agree well with the test data. Therefore, two conclusions can be made here, (a) the averaged conductivity can be used to obtain as good numerical prediction as using temperaturedependent conductivity, and (b) using a constant conductivity at room temperature can also give reasonable approximation of transient temperature field. 972 X.K. Zhu, Y.J. Chao / Computers and Structures 80 (2002) 967–976 In addition, when all three thermal properties are taken as the room temperature values, i.e. Case T8, the FEA results of the transient temperature fields at the four test points are almost identical to those for Case T6, as shown in Fig. 5. Hence, they are not reproduced here. Comparing to the standard Case T1, the results from Case T8 only show a less than 10% difference. In conclusion, from our studies in thermal modeling of welding all thermal properties can be simply taken as the room temperature values to yield the numerical temperature results with sufficient accuracy. This validate the assumption of constant thermal properties used by Free and Goff [7] and Canas et al. [2] in their welding simulations. 4. Effect of mechanical properties on deformation simulation This section presents the numerical results of residual stress and deformation fields for three sets of mechanical properties, i.e. property values at the room temperature, the average values over the temperature history and the values as functions of temperature. All FEA results are also compared with the experimental data obtained by Masubuchi [14]. As such, the effect of various mechanical properties on the deformation simulation can be evaluated. D3––E ¼ EðT Þ, rs ¼ rs ðT Þ and a ¼ aðT Þ. Temperature field is obtained from Case T8, i.e. all thermal properties are taken as the room temperature values. D4––Young’s modulus at RT, E ¼ E0 ¼ 69:3 GPa, rs ¼ rs ðT Þ and a ¼ aðT Þ. D5––averaged Young’s modulus, E ¼ E ¼ 44:5 GPa, rs ¼ rs ðT Þ and a ¼ aðT Þ. D6––yield stress at RT, rs ¼ rs0 ¼ 193 MPa, E ¼ EðT Þ and a ¼ aðT Þ. D7––averaged yield stress, rs ¼ rs ¼ 77:6 MPa, E ¼ EðT Þ and a ¼ aðT Þ. D8––thermal expansion at RT, a ¼ a0 ¼ 22:73 lm= m °C, E ¼ EðT Þ and rs ¼ rs ðT Þ. D9––averaged thermal expansion, a ¼ a ¼ 25:35 lm=m °C, E ¼ EðT Þ and rs ¼ rs ðT Þ. D10––RT values for E and a, E ¼ E0 , a ¼ a0 , and rs ¼ rs ðT Þ. Since all thermal and mechanical properties are functions of temperature in Case D1, the FEA result of residual stress and deformation is referred to as a standard numerical solution in the deformation simulation. All other FEA results of residual stress and deformation for Cases D2–D8 will be compared to that for the standard Case D1 in order to evaluate the accuracy of simulation results for each case. 4.2. Effect of thermal properties on the residual stress and distortion 4.1. Deformation modeling From the elastic–plastic constitutive Eq. (3), it is seen that Young’s modulus E, the yield stress rs and thermal expansion coefficient a are three primary mechanical properties in the thermo-mechanical analysis. The other two parameters, i.e. the Poisson ratio m and plastic hardening modulus Et have been proved no effects on deformation modeling, respectively, by Tekrival and Mazumder [19] and Canas et al. [2]. Therefore, in our current calculation, m ¼ 0:33 and Et ¼ 0 are assumed. Parameters E, rs and a are taken as the room temperature (RT) value, average value over temperature history and the value as functions of temperature in FEA modeling. For comparison, 10 cases of thermal properties are considered in our thermal modeling as follows (notice that Cases D4–D10 use the temperature field from Case T1) D1––E ¼ EðT Þ, rs ¼ rs ðT Þ and a ¼ aðT Þ are functions of temperature, as shown in Fig. 2(b). Temperature field is obtained from Case T1, i.e. all thermal properties are functions of temperature. D2––E ¼ EðT Þ, rs ¼ rs ðT Þ and a ¼ aðT Þ. Temperature field is obtained from Case T7, i.e. the average value of conductivity. Fig. 6(a) shows the distributions of residual stress in the longitudinal direction, rxx , along the middle section of the beam for Cases D1, D2 and D3. Fig. 6(b) shows the transient deflection of the lower midpoint of the beam for the three cases. The experimental data of the residual stress and transient deflection measured by Masubuchi [14] are also plotted in the two figures. It should be noted that all mechanical properties used in these three cases are functions of temperature. The objective of the comparison shown in Fig. 6 is to investigate the change of thermal properties on the deformation simulation. Again, the FEA results of the residual stress and transient deflection predicted by using the averaged conductivity (i.e. Case D2) are almost the same as those from the standard Case D1. Except in the area close to the weld line, the residual stress distribution in other areas predicted by using thermal properties at room temperature (i.e. Case D3) is in good agreement with those of Case D1, and the maximum error is <10%. The transient deflections predicted by Case D3 also agree with those of Case D1, and the maximum error is about 10%. All FEA results for the three cases match well with the test data for both the residual stress and the transient deflection. It may be concluded that the various thermal X.K. Zhu, Y.J. Chao / Computers and Structures 80 (2002) 967–976 Fig. 6. Effect of thermal properties on deformation simulation. Using averaged conductivity (Case D2), the room temperature properties (Case D3) and functions of temperature for all thermal properties (Case D1). (a) Residual stress along the middle section of the beam, (b) transient deflection at the lower mid-point of the beam. properties studied have no significant effects on deformation simulation. 4.3. Effect of mechanical properties on the residual stress and distortion The distribution of the residual stress along the middle section and the transient deflection at the lower midpoint of the beam for Cases D1, D4 and D5 are shown in Fig. 7(a) and (b), respectively. Both numerical results are compared with the test data. The comparison in Fig. 7 indicates that (a) using the room temperature value of Young’s modulus (i.e. Case D4) yields numerical results almost as good as the standard Case D1 for both the residual stress and the transient deflection, and (b) the FEA results match well with the test data. The difference for the residual stress is only in the weld region, and the maximum error is about 13%. On the other hand, using the averaged Young’s modulus (i.e. Case D5) predicts the deflection after welding, i.e. after 500 s, 973 Fig. 7. Effect of Young’s modulus on deformation simulation. Using room temperature value of Young’s modulus (Case D4), average value (Case D5) and function of temperature (Case D1). (a) Residual stress along the middle section of the beam, (b) transient deflection at the lower mid-point of the beam. with an error about 11% relative to the standard Case D1, while the FEA residual stress from Case D5 significantly deviates from that of the standard Case D1. The maximum error of residual stress for Cases D5 compared to the standard Case D1 is more than 40%. In conclusion, Fig. 7 shows that the predicted results using the room temperature value of Young’s modulus in a welding simulation are acceptable. Fig. 8(a) and (b) shows the distribution of the residual stress along the mid-section of the beam and the transient deflection at the lower midpoint of the beam, respectively, for Cases D1, D6 and D7. It is interesting to see from Fig. 8(a) that using the room temperature value of the yield stress, rs0 (i.e. Case D6), in the simulation predicts ZERO value of the residual stress as well as the permanent deflection after welding. It is well known that the residual stress in weldment is caused by plastic strains in regions where high temperature occurs during welding. However, the constant rs0 overestimates the material yield ability, which actually reduces with 974 X.K. Zhu, Y.J. Chao / Computers and Structures 80 (2002) 967–976 Fig. 8. Effect of yield stress on deformation simulation. Using room temperature value of yield stress (Case D6), average value (Case D7) and function of temperature (Case D1). (a) Residual stress along the middle section of the beam, (b) transient deflection at the lower mid-point of the beam. increasing temperature in the high temperature region, so that only elastic strains, but no plastic strains, occur in the weldment. This is clearly demonstrated by the variation of the transient deflection for Case D6 as shown in Fig. 8(b). The deflection has the same tendency as the test data during the welding, but restores to zero value after the welding is completed and the room temperature is reached. When the yield stress is changed to the average value, the numerical results for Case D7, as shown in Fig. 8, have the same tendency as the test data for both the residual stress and deflection. But there exhibits a large difference between Cases D7 and D1. Accordingly, the yield stress has to be considered as a function of temperature to predict correct deformation in welding simulation. Fig. 9(a) and (b) compares the variations of the residual stress and deflection for Cases D1, D8 and D9. There is nearly no difference between the three cases, and all FEA results match well with the test data for both the residual stress and transient deflection. It can be concluded that the thermal expansion coefficient can Fig. 9. Effect of thermal expansion coefficient on deformation simulation. Using room temperature value of thermal expansion coefficient (Case D8), average value (Case D9) and function of temperature (Case D1). (a) Residual stress along the middle section of the beam, (b) transient deflection at the lower mid-point of the beam. be simply taken as the room temperature value in welding simulation. 4.4. An engineering approach using simplified properties The analyses and comparison above revealed that all thermal and mechanical properties, except for the yield stress, can be simply taken as the room temperature values in welding simulation to give the residual stress and distortion with sufficient accuracy. The yield stress has to be considered as a function of temperature in the thermo-mechanical simulation. Since the yield stress at high temperature for engineering alloy is often not readily available for welding simulation, an engineering approximation to this particular material property at high temperatures is investigated and proposed here. It is assumed that the yield stress for the material under consideration takes the room temperature value when X.K. Zhu, Y.J. Chao / Computers and Structures 80 (2002) 967–976 0 6 T 6 100 °C, 5% of the room temperature value when T P T1 ¼ 2=3 of the melting temperature of the material, and a linear function of temperature in between, i.e. 100 °C < T < T1 . Note that T1 is essentially the cut-off temperature used widely in numerical simulation for welding to account for the close-to-zero yield stress as the melting temperature of the material is approached. Five percent (other than zero) is chosen arbitrarily to avoid any numerical difficulties. We have performed numerical tests and it was found that the selection of five percent as well as a precise cut-off temperature is not sensitive to the final results. For the 5052-H32 aluminum alloy, the melting temperature is 607 °C and thus T1 ¼ 400 °C. The assumed temperature-dependent yield stress for the 5052-H32 aluminum alloy can then be expressed mathematically by the following piece-wise linear function 8 < rs0 ¼ 193 MPa; rs ¼ 5%rs0 þ 400T 95%rs0 ; 300 : 5%rs0 10 MPa; 975 0 6 T 6 100 °C 100 < T < 400 °C T P 400 °C ð4Þ The variation of the yield stress in (4) with temperature is plotted in Fig. 2(b) in dashed line. The simplified yield stress in (4) and the room temperature values for all other properties constitute an engineering approximation for material properties in welding simulation. The residual stress and deflection from FEA using this simplification are depicted in Fig. 10. It is observed that the residual stresses are in very good agreement with the test data as well as the standard FEA results of Case D1, except perhaps for the weld area. The maximum error of the peak value of the residual stress is about 13% in the weld area. The computed transient and permanent deflections also match well with both the test data and the standard FEA results. The maximum difference in deflection between Case D1 and the engineering approach is <10%. These results indicate that the proposed engineering approach using simplified material properties is a simple, yet effective, way in welding simulation for the aluminum alloy. 5. Concluding remarks Fig. 10. Comparison of the engineering approach for deformation simulation. Using temperature-dependent yield stress and room temperature properties for others (Case D10); all properties are functions of temperature (Case D1); and the engineering approach using approximate rs as in Eq. (4). (a) Residual stress along the middle section of the beam, (b) transient deflection at the lower mid-point of the beam. To investigate the effects of temperature-dependent material properties on transient temperature, residual stress and distortion in computational simulation of a welding process detailed 3D nonlinear thermal and thermo-mechanical analyses are performed using FEA method. Three sets of material property for the 5052H32 aluminum alloy are investigated, i.e. values at the room temperature, average values over temperature history, and functions of temperature. Numerical results are compared between the three sets, as well as compared with the test data. The main results and conclusions are summarized as follows: (1) The thermal conductivity has some effect on the distribution of transient temperature field during welding. The material density and specific heat have negligible effect on the temperature field. Although using the average values over the temperature history is better, adopting room temperature values of all thermal properties in the welding simulation can predict reasonable results for the transient temperature distribution. (2) The yield stress is the key mechanical property in welding simulation. Its value has significant effect on the residual stress and distortion. If the room temperature value of the yield stress is taken, the FEA computation predicts zero residual stress and no permanent distortion because only elastic strain, and no plastic strain, occurs under this circumstance. The temperature dependency of the yield stress must be considered in a welding process simulation to obtain correct results. 976 X.K. Zhu, Y.J. Chao / Computers and Structures 80 (2002) 967–976 (3) Young’s modulus and the thermal expansion coefficient have small effects on the residual stress and distortion, respectively, in welding deformation simulation. It is found that the numerical results obtained by using the room temperature value of Young’s modulus are much better than those using its average value over temperature history. (4) An engineering approach using simplified properties, constituted by a piece-wise linear function with temperature of the yield stress and the constant room temperature values of all other properties, is proposed and validated for computational welding simulations of aluminums. Similar approach can be extended to welding simulation for steels. The proposed simplification methodology for material properties is most useful for welding simulation when detailed material properties at high temperatures are unavailable. Acknowledgements The authors wish to thank the support of this work by the National Science Foundation through Grant CMS0116238, program director Dr. K.P. Chong. References [1] Brown S, Song H. Finite element simulation of welding of large structures. J Eng Ind 1992;114:441–51. 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