Predicting the weld zones size in FSSW of 304L stainless steel plates by mathematical model based on RSM

2.1. Material specification

The chemical composition of 304L SS is given in Table 1 and its temperature-dependent physical and mechanical properties are shown in Table 2.

2.2. Response Surface Methodology (RSM)

The main aim of DOE is to identify the points/conditions where the experiments should be carried out or evaluated. After collecting a large set of experimental data, several techniques can be used to fit these data; and then, they may also be used in the numerical solutions. The RSM is a mathematical and statistical technique for DOE. An important feature of the RSM is the design of experiments. The aim of this technique is to optimize the test results that are affected by several independent variables (input variables). In fact, RSM assists in the selection of the appropriate experimental design and the definition of the tests. The application of the RSM also covers the mathematic-statistical treatment of the test data by fitting the polynomial function to them. In this line, Box et al. developed the RSM in early 50 s (Khuri and Mukhopadhyay, 2010Khuri, A.I., Mukhopadhyay, S. (2010). Response surface methodology. Wiley Interdiscip. Rev. Comput. Stat. 2 (2), 128 -149. https://doi.org/10.1002/wics.73.). This method uses the fit of predefined models to the experimental results. For this purpose, linear or square polynomial relationships are implemented to explain the system studied and, subsequently, to identify (modeling and displacing) experimental parameters to obtain an optimum condition (Khuri and Mukhopadhyay, 2010Khuri, A.I., Mukhopadhyay, S. (2010). Response surface methodology. Wiley Interdiscip. Rev. Comput. Stat. 2 (2), 128 -149. https://doi.org/10.1002/wics.73.). Usually a second-order model together with the two-level factorial design are used, but this method may fail if extra effects, such as second-order effects, are significant. In this case, a central point in two-level factorial designs is implemented to evaluate the relationship. The next step is to obtain the additional terms that explain the interaction between the different test parameters. Using the above techniques, the following polynomial known as Box-Behnken (Khuri and Mukhopadhyay, 2010Khuri, A.I., Mukhopadhyay, S. (2010). Response surface methodology. Wiley Interdiscip. Rev. Comput. Stat. 2 (2), 128 -149. https://doi.org/10.1002/wics.73.) can be suggested:

where k is the number of variables, β₀ is the constant term, β_i are the polynomial coefficients, x_i refers to the variables, β_ij introduce the interaction between these variables and ε is the residual associated to the error in test results.

2.3. Numerical method

To capture the thermo-mechanical response under the given system and process parameters, coupled temperature and displacement numerical formulations are used along with the heat generation factor. The frictional heat generation formulation involves the calculation of the heat flux at the interface elements located on the contacting parts (tool and work-piece). The heat flux is applied as a thermal load to the volume of elements on each part. Thermal distributions to each surface are given in Equations (2) and (3):

where, q_tool and q_WP are the thermal loads, the heat flux to the tool and the work-piece respectively, q_g is the total heat generated by the interface element due to the friction, q_r is the heat flux due to the radiation, q_k is the heat flux due to the conduction, and f_tool and f_WP stand for the fraction of the total generated heat flux (q_g) to the tool and the work-piece (q_tool+ q_WP= 1).

In order to incorporate the hardening effect due to the plastic deformation (as a function of applied temperature), Johnson Cook material constitutive model is employed. This model is mathematically given in Eq. (4), which is extensively used for extrusion, forging and impact analyses (Al-Moussawi et al., 2017Al-Moussawi, M., Smith, A.J., Faraji, M. (2017). Friction stir welding of EH46 steel grade at dwell stage: Microstructure evolution. Metallogr. Microstruct. Anal. 6, 489-501. https://doi.org/10.1007/s13632-017-0390-5.; Zhu and Chao, 2004Zhu, X.K., Chao, Y.J. (2004). Numerical simulation of transient temperature and residual stresses in friction stir welding of 304L stainless steel. J. Mater. Process. Technol. 146 (2), 263-272. https://doi.org/10.1016/j.jmatprotec.2003.10.025 ; Khuri and Mukhopadhyay, 2010Khuri, A.I., Mukhopadhyay, S. (2010). Response surface methodology. Wiley Interdiscip. Rev. Comput. Stat. 2 (2), 128 -149. https://doi.org/10.1002/wics.73.; Johnson and Cook, 1983Johnson, G.R., Cook, J.R. (1983). Constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. Proceedings of 7th Symposium on Ballistics. (Netherlands), pp. 541-547.):

where $\stackrel{-}{\sigma }$ is the effective yield strength, $\frac{.}{\epsilon }pl$ is the equivalent plastic strain rate, ${\epsilon }_0^{.}$ is a coefficient to normalize the strain rate, A is the yield stress constant, B is the strain hardening constant, n is the strain hardening exponent, C is the strain rate hardening constant and m is the temperature dependent coefficient. Also, $\widehat{\theta }$ presents the non-dimensional temperature, which is given as below (Al-Moussawi et al., 2017Al-Moussawi, M., Smith, A.J., Faraji, M. (2017). Friction stir welding of EH46 steel grade at dwell stage: Microstructure evolution. Metallogr. Microstruct. Anal. 6, 489-501. https://doi.org/10.1007/s13632-017-0390-5.; Zhu and Chao, 2004Zhu, X.K., Chao, Y.J. (2004). Numerical simulation of transient temperature and residual stresses in friction stir welding of 304L stainless steel. J. Mater. Process. Technol. 146 (2), 263-272. https://doi.org/10.1016/j.jmatprotec.2003.10.025 ; Khuri and Mukhopadhyay, 2010Khuri, A.I., Mukhopadhyay, S. (2010). Response surface methodology. Wiley Interdiscip. Rev. Comput. Stat. 2 (2), 128 -149. https://doi.org/10.1002/wics.73.; Johnson and Cook, 1983Johnson, G.R., Cook, J.R. (1983). Constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. Proceedings of 7th Symposium on Ballistics. (Netherlands), pp. 541-547.):

where θ, θ_ref, θ_melt are the current, reference and melting temperatures. Johnson Cook material constitutive parameters for 304L SS are presented in Table 3 (Maurel-Pantel et al., 2012Maurel-Pantel, A., Fontaine, M., Thibaud, S., Gelin, J.C. (2012). 3D FEM simulations of shoulder milling operations on a 304L stainless steel. Simul. Model. Pract. Theory 22, 13-27. https://doi.org/10.1016/j.simpat.2011.10.009 ).

By employing the arbitrary lagrangian-eulerian (ALE) formulation, large deformations may be simulated by re-meshing the FE model in ABAQUS by “8-node C3D8RT” element Abaqus Version 6.14 (2014)Abaqus Version 6.14 (2014). User’s Manual, Dassault Systemes, Simulia Corp.; Providence, RI, USA.. The main point is to control the distortion of the elements by improving the aspect ratio of the distorted elements. This is done by the inter-step conversion of the results at the integration points and re-adjustment of the nodal positions as close as possible to the original coordinates. As compared to the pure Eulerian formulations, the advantage in using ALE formulation is that in the ALE approach the “free” surfaces have Lagrange properties in the normal direction, in a way that the surface tracking and partially filled elements are avoided. Hence, the position of the surface of the domain is found directly by solving the governing equation and the iterations are not required (Al-Moussawi et al., 2017Al-Moussawi, M., Smith, A.J., Faraji, M. (2017). Friction stir welding of EH46 steel grade at dwell stage: Microstructure evolution. Metallogr. Microstruct. Anal. 6, 489-501. https://doi.org/10.1007/s13632-017-0390-5.). The FSSW geometry consists of three components including a rigid tool and two 304L SS plates. The dimensions of the plates and the tool location are given in Fig. 1.

According to Fig. 2, boundary conditions are applied in the modeling as follows: upper and lower edges of the work-pieces are restrained and fixed in the horizontal direction; the welding tool has transitional and rotational movement. The convection coefficient of 30 W/m^{2 o}C at 25 °C is applied to all surfaces that are exposed to the surrounding air (Awang, 2007Awang, M. (2007). Simulation of friction stir spot welding (FSSW) process: Study of friction phenomena. ProQuest, Graduate Theses, Dissertations, and Problem Reports. 8426, West Virginia University. https://researchrepository.wvu.edu/etd/8426.; Jiji, 2006Jiji, L.M. (2006). Heat Convection. Springer-Verlag Berlin Heidelberg. https://doi.org/10.1007/978-3-540-30694-8.).

2.4. Experimental method

Two similar 304L SS plates are joined by FSSW process and the temperature levels of the selected points around the weld zone are measured during the welding process, and then, the microstructure, hardness and thickness of the weld zones are determined. The FSSW tool, made of Tungsten Carbide, and its dimensions are shown in Fig. 3a and Fig. 3b, respectively.

For the welding process, a Universal Milling Machine shown in Fig. 4a is used and Fig. 4b shows the plates are fixed by a set of fixtures during the welding. Fig. 4a and 4b show that the welding tool is installed on the spindle and rotates around the axis perpendicular to the surface of the plate. Welded plates are shown in Fig. 4c.

An Axial force exerted by the milling machine head is applied to the material and welding is being possible by the extreme plastic deformation in the solid phase that includes recrystallization of the base material (BM), and eventually, a strong metallurgical connection is created. In this way, the stir zone (SZ), thermo-mechanical affected zone (TMAZ) and heat affected zone (HAZ) are formed around the weld spot. In order to study the microstructure of the welded specimens, macrographs are taken from the mid-plane perpendicular cross section of the weld zone according to ASTM E3-01 (2001)ASTM E3-01 (2001). Standard Practice for Preparation of Metallographic Specimens. ASTM International, West Conshohocken, PA, USA. https://doi.org/10.1520/E0003-01 . The marble solution with 50 ml hydrochloric acid, 50 ml distilled water and 10 g copper sulfate is used to etch the specimen’s surface and the macro-graphic images of the specimens are taken using an optical microscope. Then, the surface hardness of the specimens in different points of the weld cross section are determined based on ASTM E384 (2017)ASTM E384 (2017). Standard Test Method for Microindentation Hardness of Materials. ASTM International, West Conshohocken, PA, USA. https://doi.org/10.1520/E0384-17. using Micro Vickers method.

Temperature of the points have been measured by a K type thermocouple during the FSSW process. These thermocouples are linked to a PC equipped with a data acquisition system through the digital thermometers (TM-747D) or analogue to digital converters indicating the temperature change/history during the process.

2.5. Design of experiment conditions by response surface method

Rotational speed of the tool, plunging depth and dwell time are selected as process variables in this research identifying the major parameters affecting the nugget of the FSSW and the structure of the weld zones. In order to study the effect of these parameters on the size of the weld zones, the tests are conducted based on DOE method and RSM has been used for proposing a model to predict the zone sizes. The mentioned parameters are considered as input data and three levels of them (given in Table 4) are selected.

These levels are selected based on the previous experiences that produce appropriate joints and the reported values for the similar material (Lakshminarayanan et al., 2015Lakshminarayanan, A.K., Annamalai, V.E., Elangovan, K. (2015). Identification of optimum friction stir spot welding process parameters controlling the properties of low carbon automotive steel joints. J. Mater. Res. Technol. 4 (3), 262-272. https://doi.org/10.1016/j.jmrt.2015.01.001.; Ahmed et al., 2016Ahmed, M.M.Z., Ahmed, E., Hamada, A.S., Khodir, S.A., Seleman, M.E.S., Wynne, B.P. (2016). Microstructure and mechanical properties evolution of friction stir spot welded high-Mn twinning-induced plasticity steel. Mater. Des. 91, 378-387. https://doi.org/10.1016/j.matdes.2015.12.001.). This set of input data is the foundation of the numerical and experimental analyses in this research.

3.1. Microstructures

The cross section images of the FSSW for two 304L SS plates with magnification factor 50 (Mag. =50x) are shown in Fig. 5a. In this case, the tool rotational speed is 750 rpm, the plunging depth is 0.1 mm and the dwell time is 4 s. For better resolution, macrographs with Mag. = 200x are shown in Fig. 5b. Considering the structural characteristics, three weld zones: stir zone (SZ), thermo-mechanical affected zone (TMAZ) and heat affected zone (HAZ), can be seen in this figure. The procedure is repeated for all other case studies with different welding parameters listed in Table 4.

3.2. Hardness

In the present research, Vickers Hardness for different points on the plate surface along a radial line starting from the outer surface of the weld nugget has been investigated. Considering that the process of changes in the hardness-distance diagram is almost the same for all case studies, for example in Fig. 6, this process shown for case study with tool rotational speed: 750 rpm, dwell time: 4 s, and plunging depth: 0.1 mm. The vertical axis of this figure is located at the outer surface of the weld nugget. As shown in this figure, the SZ hardness is maximum (250 Vickers) and it decreases by moving away from this area. In other words, SZ has the highest amount of hardness. The reason for that is the microstructure of its fine grains, which can be clearly seen in this figure. It can also be caused by the plastic deformation as a result of pressure by the tool pin and work hardening phenomenon. By entering to the TMAZ, the amount of hardness is reduced due to the lower amount of plastic deformation or stir. In other words, the reason for which was low is the mixing and reduction of the effect of welding fineness in this zone. In the HAZ, the decreasing trend of the hardness continues to reach the base metal hardness, which is equal to 192 Vickers. The SZ hardness numbers for three case studies are presented in Table 5.

As it is shown in this figure, the SZ hardness is maximum (250 Vickers) and it decreases by moving away from this area. In other words, SZ has the highest hardness because of its microstructure. By entering in the TMAZ, the hardness is reduced due to the lower amount of plastic deformation or stir. In the HAZ, the decreasing trend of the hardness continues to reach the base metal hardness, which is equal to 192 Vickers. The SZ hardness numbers for three case studies are presented in Table 5.

3.3. Temperature distribution

Figure 7 shows the temperature distribution for the various weld zones in t=6 s. It can be seen that SZ has the maximum temperature level (almost 1200 ºC) due to the heat generated from the plastic stirring and friction. Temperature level history for three points (A, B and C) on the section mid line and in the different zones that are obtained using the FEM solutions are shown in Fig. 8. In FEM simulation for this case study, the rotational speed of the tool, the plunging depth and the dwell time are 750 rpm, 0.1 mm and 4 s respectively. The difference in temperature level also affects the microstructure of the zones. Also, this figure shows the maximum temperatures difference of the various regions is approximately 200 °C.

For validating the FEM results, temperature levels of point “D” (see Fig. 7) have been measured. Fig. 9 shows the temperature history of both FE solution and experimental measurements for selected point (“D”).

In addition, the experimentally measured temperature levels are compared with those obtained using the FE solutions in Table 6. As it can be seen, there is a good agreement between the experimental data and the numerical results with maximum deviation of 40 °C (3.39%)

3.4. Developing the mathematical model

As it was mentioned before, for studying the impact of the rotational speed (R), plunging depth (P) and dwell time (D) on the size of the weld zones, DOE conditions are determined by the RSM. Based on this method, fifteen tests are defined and conducted and their conditions are presented in Table 7.

Since carrying out these tests is expensive, the numerical model that has been validated in Section 3.3, is used to obtain the temperature distribution and the weld zone size for each case and their results are employed to develop a mathematical model by the RSM. The size of each zone is approximated based on their temperature level: between 1200 °C and 1000 °C is known as SZ+TMAZ and between 1000 °C and 850 °C for HAZ. Based on these data an order- two polynomial have been developed using the RSM to predict the weld zone size for FSSW of 304 SS plates according to the selected operating conditions as below:

3.5. Validation of the mathematical model

In Fig. 10 and Fig. 11 the results obtained from the finite element model and the mathematical model for the size of HAZ and SZ+TMAZ are compared. According to the R² values (>95%), it can be seen that there is an acceptable agreement between the results.

The experimental data are used to validate the results of the mathematical model achieved by the RSM. For this purpose, three different cases are randomly selected in the range of experiments, and the results of the mathematical model are compared with those obtained by the measurements. The comparisons for SZ+TMAZ and HAZ sizes are presented in Table 8 and Table 9, respectively. As it can be seen from the error values given in these tables, the maximum error is 0.1 mm (9%) which indicates a good agreement between the results.

3.6. Effect of the welding parameters on the size of the weld zones

Figure 12 shows the direct impact of the welding parameters: rotational speed, dwell time, and plunging depth on the size of SZ+TMAZ and HAZ. As it can be seen in Fig. 12a, the size of SZ+TMAZ increases by increasing the rotational speed, plunging depth and dwell time. By increasing these parameters, the amount of input heat to the work-piece increases and the stirring effect around the pin intensifies and leads to a larger size of SZ+TMAZ. It is clear that the impact of the rotational speed on the size of SZ+TMAZ at very low revolutions (from 500 rpm to 550 rpm) is insignificant. This figure also shows that the increase of the SZ+TMAZ size by increasing the plunging depth is lower than the increase of the size of this area by increasing other parameters. In Fig. 12b it is evident that by increasing the rotational speed, the size of HAZ increases to a maximum value and then decreases. By increasing the rotational speed, the temperature increases and that causes the HAZ size to enlarge, but with further increase of this parameter, HAZ size decreases because the amount of stirring increase and SZ+TMAZ moves outward and replaces the HAZ.

The impact of the dwell time on the HAZ size is similar to the impact of the rotational speed. Initially, the size of HAZ rises to a maximum amount and then is reduced due to the enlargement of the SZ+TMAZ. The trend of change of the HAZ size due to the increase in the plunging depth is similar to the other parameters.

In addition to the direct effect of parameters on the size of different zones, the interference effect of parameters is also very important. Figure 13 shows the interference effect of the parameters on SZ+TMAZ size. According to Fig. 13 (a) , at low rotational speeds, due to the increase in the plunging depth, the size of SZ+TMAZ rises. The increase rate of SZ+TMAZ size is reduced by raising the rotational speed. However, by reaching a speed of 750 rpm, the increase rate of size for this zone rises. Actually, at speeds greater than 750 rpm, the effect of increasing the plunging depth is greater and causes a faster growth in this zone. Fig. 13 (b-c) show the interaction between the dwell time and the rotational speed. As it is seen, by increasing of these two parameters at the same time, the size of SZ+TMAZ increases. This rate, at speeds lower than 750 rpm, is low and at speeds greater than 850 rpm is high. Actually, the simultaneous effect of these two parameters is much greater in higher values. Since the rotational speed and the dwell time have the same direct effect on the size of SZ+TMAZ, the interference effect of the dwell time and the plunging depth is similar to the interference effect of the rotational speed and the plunging depth.

Figure 14a shows the interference effect of the plunging depth and the rotational speed parameters on HAZ size. The maximum value of this zone occurs when the rotation speed is between 750 and 1000 rpm and the plunging depth is between 6.0 and 14.0 mm. At lower values than these ranges, the HAZ size is reduced due to the lower input heat. Besides, at higher values than these ranges, the HAZ size is reduced because of SZ expansion. Figure 14 (b-c) show that the highest value of the HAZ size occurs when the rotational speed is between 750 and 1000 rpm and the dwell time is between 3.5 and 7 s due to the interference effect between the rotation speed and the dwell time. Due to the same direct effect of the rotating speed and the dwell time, the interference effect of the dwell time and the plunging depth is similar to the interfering effect of the rotating speed and the plunging depth on the size of the HAZ zone.