Development of Mathematical Models for Dissimilar Welding Pool Geometries

Abstract

This study presents the design of experiment methodology (DOE) for the development of mathematical models for predicting and optimizing fusion zone dimensions of aluminum/titanium welded components by using CO₂ laser welding (LW) process. The implemented DOE methodology used the Taguchi approach and orthogonal array to mathematical models to predict and an optimize fusion zone dimensions. These dimensions are the welding penetration (L ₁ ), welding width at the surface (W ₁ ) and cross section area of melted zone (A ₁ ) in the aluminum plate. The laser power, welding speed, and defocusing distance ranges were experimentally determined with the objective of producing a welded joint with adequate penetration, minimum fusion zone size and acceptable welding profile. The fusion zone area and the shape of weld profile of the dissimilar aluminum/titanium were evaluated based on the selected laser welding parameters. Taguchi approach was used as a statistical design of experiment (DOE) technique for optimizing the selected welding parameters in terms of the minimizing area of melted zone (A ₁ )_, welding width at the surface (W ₁ ) and maximizing the penetration (L ₁ ). The results showed that, for the dissimilar titanium–aluminum overlap welds, the prediction of fusion depth and width that were obtained by the developed models are in good agreement with the results obtained experimentally. Since this methodology does not require complicated or excessive computation, it is especially useful for the actual welding process applications. It is also provided a robust approach to adaptive welding as well as to stabilize weld quality.

1 Introduction

Welding quality is strongly characterized by the weld penetration and the weld pool geometry [1]. Due to that the weld pool geometry plays an important role in determining the mechanical properties of the welded joints. Therefore, the selection of the welding process parameters is essential for obtaining optimal weld pool geometries. The important problem to be solved in welding engineering is to develop a model for determining the optimal process parameters.

A mathematical model for weld heat sources based on a Gaussian distribution of power density in space was presented by Goldak et al. [2]. In particular a double ellipsoidal geometry was proposed so that the size and shape of the heat source could be easily changed to model both the shallow penetration arc welding processes and the deeper penetration laser and electron beam processes.

Hsu and Rubinsky [3] have investigated a about a two-dimensional, quasi-stationary finite element numerical model to study the fluid flow and the heat transfer phenomena which occur during constant travel speed, keyhole plasma arc welding of metal plates.

Based on the computation model for quasi-steady heat transfer problems of welding with the boundary element method, Hang and Okada [4] had developed a computer program that used for the computation of thermal cycles at heat affected zones with gas shielded metal arc welding “GMAW” on medium thickness plates.

In a study by Zhang et al. [5], a polar coordinate model was proposed to characterize the weld pool geometrically. The identification of its parameters involves complicated non-linear optimization which cannot be done in real time using conventional algorithms. A neural network ANN was therefore proposed to identify the parameters in real time. Weld pool geometry was computed via numerical solution of a boundary integral equation used as a model for the autogenous full penetration welding of pure materials by Yeh and Brush [6]. Wahab and Painter [7] measured the full 3-dimensional weld pool shape for the GMAW process, and to study the use of this information within numerical models.

A computational modelling of welding phenomena within a versatile numerical framework was presented by Taylor et al. [8]. Fusion zone area and shape were evaluated by Gunaraj and Murugan [9] as a function of the selected submerged arc welding (SAW) parameters. Response surface methodology (RSM) was used as statistical design of experiment (DOE) technique for optimizing the selected welding parameters in terms of minimizing the fusion zone. In this study the modified Taguchi method was adopted to solve the optimal weld bead geometry with four the-smaller-the-better quality characteristics.

2 Experimental Procedure

Two sheets of Al 6082 T6 and Ti G2 with dimensions of 160 × 80 × 1 mm were selected to be joined by CO₂ laser welding.

2.1 Materials Specifications

The chemical and mechanical properties of materials selected are presented in Tables 1 and 2.

Table 1 Chemical composition of the Ti G2 and the Al 6082 (wt%)

Table 2 Mechanical properties of the welded materials

2.2 Sample Preparation

A small sample was cut from each welded plate perpendicular to the welding line to study the weld pool shape and geometry. The titanium side of the weldment was etched in reagent consisting of (10 ml HF and 5 ml HNO₃ in 85 ml of water), and the rest of the regions of the weldment were etched with Keller’s reagent (1 % HF, 1.5 % HCl, 2.5 % HNO₃ and H₂O solution).

2.3 Welding Parameters Selection and Experiment Design

The selected welding parameters for these dissimilar materials are: Laser power, welding speed and focus point position. Table 3 shows the welding input variables and experiment design levels. The welding experiments were carried out in the Mechanical School workshop following the Taguchi designed matrix in random order generated by the Design Expert software, as presented in Table 4. The welding pool geometry, mechanical destructive tests (tensile shear strength) and cost per meter welded calculations were carried out in the joined specimens and the results are presented in Table 4. Each presented result in Table 4 in each column is an average of at least of three readings.

Table 3 Process parameters and design levels used

Table 4 Welding input variables, experiment design levels and the welding pool geometry calculations

The measured results of welding pool area for each sample are presented in Table 4. Figure 1a–d shows the effect of the welding parameters and the variation on the total weld pool (fusion area) ‘A ₁ ’ at aluminum plate only, welding widths at the specimen surface of aluminium ‘W ₁ ’ and welding widths at the penetration of welding depth in aluminum plate ‘L ₁ ’ of some experiments selected from Table 4.

Fig. 1

a Welding pool specimen 1, b Welding pool specimen 4, c Welding pool specimen 8 and d Welding pool specimen 14. Effect of the welding parameters and the variation on the total weld pool dimensions

3 Results and Discussion

The fusion zone dimensions in the Al plate were measured by using the transverse sectioned specimens, optical microscope and image analysis software. The measured responses are listed in the same Table 4. Design Expert 7 software was used for analysing the responses. The fit summary output indicates that the quadratic models which are developed are statistically significant for the prediction of the responses (W ₁ and L ₁ ); therefore, they will be used for further analysis. It has been seen from the achieved results that the welding pool geometry, shape and penetration are controlled by the rate of heat input, which is a function of laser power and welding speed. The focusing position has also a strong effect on the responses.

3.1 Analysis of Variance

The test for significance of the regression model and the test for significance on individual model coefficients were performed using the Design Expert 7 software. The backward elimination regression method was applied and exhibited in ANOVA Tables 5 and 6 for the suggested reduced quadratic models. Tables 5 and 6 summarize the analysis of variances of the responses and show the significant models. The same tables also show the other adequacy measures R ², adjusted R ² and adequacy precisions. All adequacy measures were close to 1, which is reasonable and indicates an adequate model. The adequate precision compares the range of the predicted value at the design points to the average predicted error. In this study the values of adequate precision for the W ₁ and L ₁ are significantly greater than 4. The adequate precision ratio above 4 indicates adequate model discrimination. The developed quadratic models in terms of coded factors and actual values are exhibited in Eqs. 1–4.

Table 5 ANOVA for response ‘W ₁ ’

Table 6 ANOVA for response ‘L ₁’

Final Equation in Terms of Coded Factors:

$$W_{1} = 3 8 1. 50 + 5. 9 2P + 4 6. 3 5S - 2 2. 9 5F + 200. 8 1P^{ 2} - 7 5. 9 4F^{ 2}$$

(1)

Final Equation in Terms of Actual Factors:

$$W_{1} = 2591.09 - 5441.71P + 0.09S - 349.65F + 2914.29P^{2} - 303.75F^{2}$$

(2)

Final Equation in Terms of Coded Factors:

$$L_{1} = 279.16 + 60.30P + 8.32S - 21.60F + 44.13PS + 62.72F^{2}$$

(3)

Final Equation in Terms of Actual Factors:

$$L_{1} = 851.34 - 610.87P - 0.30S + 207.68F + 0.34PS + 250.88F^{2}$$

(4)

For the response W ₁ of the developed model, the analysis of variance indicates that the welding speed ‘S’ and focus position ‘F’ are the stronger welding parameters affecting the responses. The focus position ‘F’ has a greater affect on the response W ₁ than the laser power ‘P’. The W ₁ model indicates that the studied parameters (S, F) significantly affect the response. For the response L ₁ of the developed model, the analysis of variance indicates that the laser power ‘P’ and focus position ‘F’ are the stronger welding parameters affecting the responses. Focus position ‘F’ has a greater affect on the response W ₁ than welding speed ‘S’. The L ₁ model indicates that the studied parameters (P, F) significantly affect the response. The L ₁ model indicates that the welding parameters have interactions between P and S exhibited in Fig. 2. The figure exhibits the interaction of the welding speed with the laser power at focus position F = −0.5 mm.

Fig. 2

Interactions between the welding parameters (P, S) with respect to the depth of penetration response at F = −0.5 mm

3.2 Model Validation

The aim of this step is to predict and verify the improvement of the response using the optimal levels of the welding process parameters. Figures 3 and 4 show the relationship between the actual and predicted values of W ₁ and L ₁ , respectively. These figures indicate that the developed models are adequate because the residuals in prediction of each response are negligible, since the residuals tend to be close to the diagonal line.

Fig. 3

The predicted values of the L ₁ versus actual measured values

Fig. 4

The predicted values of W ₁ versus actual penetration depth of measured values

Furthermore, to verify the satisfactoriness of the developed models, three confirmations experiments were carried out using new test conditions at different parameters conditions, obtained using the software and the developed mathematical models. The values of W ₁ and L ₁ for validation experiments were calculated using the software. Table 7 summarizes the experimental conditions, the actual experimental values, the predicted values and the percentages of absolute errors. It could be concluded that the models developed can predict the responses with a very small errors. W ₁ and L ₁ were greatly improved through this optimization.

Table 7 Confirmation experiments of the responses (W ₁ and L ₁ )

3.3 Effect of the Parameters on Responses

The reason for predicting the welding pool geometry is to develop a model which would include the optimization step.

• Welding Pool Width at the Work Piece Surface (W ₁).

The results and the model obtained for the response indicate that the S and F are the most important factors affecting the W ₁ value. An increase in S leads to a decrease in W ₁ and the increase of F leads to an increase in W ₁ . This is due to the fact that the laser beam is traveling at high speed over the welding line when S is increased. Therefore, the heat input decreases leading to less volume of the base metal being melted, consequently the width of the welded zone decreases. Moreover, a defocused beam, which is a wider laser beam, results in spreading the laser power over a wide area. Therefore, a wide area of the base metal will be melted leading to an increase in W ₁ or vice versa. The result shows also that P contributes a secondary effect in the response width dimensions. Increasing P results in a slight increase in W ₁ , due to the increase in the power density. Figure 5 shows contour plots for the effect of the parameters on the W ₁ width. Figure 5 illustrates the relationship between S and P with their impact on the welding pool width at the surface of the aluminum plate (W ₁ ) at F = −0.5 mm.

Fig. 5

Contour graphs of the effect of P, S parameters at F = −0.5 mm on the response L ₁

• Welding Pool Width at the Middle of the Work Piece (L ₁ )

The results and the model obtained for the response indicate that the P and F are the most important factors affecting the W ₁ value. An increase in P leads to an increase in L ₁ and the increase in F leads to increase in L ₁ . This is due to the fact that an increase in the amount of laser power P that is transfer to the work piece leads to an increase in the response. Therefore, the heat input increases leading to an increase in the amount that the base metal melts, consequently the penetration of the welded zone increases. Moreover, a defocused beam, which is in a wider laser beam, results in spreading the laser power over a wide area. Therefore, a wide area of the base metal will be melted leading to an increase in L ₁ or vice versa. The result also shows that S contributes in a secondary effect in the response width dimensions. S is inversely proportional to the heat input. Increasing S results in a slight decrease in L ₁; this is due to the decrease in the heat input. Figure 6 shows contour plots for the effect of the process parameters on the L ₁ width. Figure 6 illustrates the relationship between S and P with their impact on the welding pool depth of the aluminum plate (L ₁ ) at F = −0.5 mm.

Fig. 6

Contour graphs exhibiting the effect of P, S parameters at F = −0.5 mm on the response W ₁

4 Conclusions

Laser welding of nonferrous with nonferrous dissimilar material has been studied experimentally and analyzed statistically and the following points are concluded:

• 1. The dissimilar joint between aluminum alloy Al 6082 and titanium G2 alloys were successfully welded by CO₂ laser welding with a single pass and without filler material using the overlap joint design.

• 2. Laser welding is a very successful method for joining dissimilar nonferrous metals.

• 3. The models developed can satisfactorily predict the responses within the studied domain.

• 4. Appling of a DOE inspired by the Taguchi technique, best operating parameters were achieved and then develop models to control the welding parameters.