A Weighting Grade-Based Optimization Method for Determining Refill Friction Stir Spot Welding Process Parameters

Friction stir welding is a solid-state joining technique that is finding greater use in aircraft applications for manufacturing thin-walled aircraft components, especially stiffened panels. The FSW process is capable of welding some previously unweldable precipitation-strengthened aluminum alloy sheets. In a classical FSW process, a rotating cylindrical tool composed of a probe and a shoulder is plunged into the materials to be joined. Mechanical deformation of the materials is possible via the heat generated through friction (Ref 1, 2). FSW joints have higher strengths than riveted joints and much lower residual stresses than a typical fusion welded joint, like resistance spot welding (RSW) (Ref 3).

Refill friction stir spot welding (RFSSW) is a modern solid-state friction welding process which was developed by Helmholtz-Zentrum Geesthacht (Ref 4) for welding two or more similar or dissimilar sheets in a lap configuration. RFSSW is a solid-state joining technology which connects two similar or dissimilar materials together with minimal heat input or distortion. In the aircraft industry, friction welding is replacing other methods of joining such as riveting, screwing and adhesive bonding because (i) RFSSW does not require additional elements, (ii) there is no need to drill the materials to be joined, and (iii) the welds do not protrude above the surface of the joined elements.

The influence of the welding process parameters on the load capacity of joints and the weld microstructure has become the focus of much research investigation (Ref 5-7). Oberembdt et al. (Ref 5) concluded that the most effective variable for controlling joint strength is plunge depth. Shen et al. (Ref 6) concluded that the overlap shear strength of single-lap joints increases with the increase in weld time and plunge depth due to increasing nugget diameter. Li et al. (Ref 8) discussed the effect of tool rotational speed on weld formation, microstructure and the mechanical properties of the RFSSW welds. They concluded that increasing the tool rotational speed decreases the lap shear load of the weld. Tier et al. (Ref 9) concluded that the plunge depth and rotational speed are the most significant variables during RFSSW.

Optimization of the FSW and RFSSW process parameters depends upon the ability to measure and control the process variables (Ref 10). To select the appropriate welding parameters, several methods can be used based on the welder’s own experience and analytical and numerical methods. Some of these methods are time-consuming and expensive. An economical way to select the optimal process parameters is to conduct experiments using the design of experiments (DOE) technique (Ref 7, 10, 11).

The friction welding of Alclad aluminum alloy sheets requires special attention and precise welding procedures, different to the procedures found in the FSW process. Firstly, Alclad within the weld nugget causes heterogeneity in the structure of the weld, especially in the area of the nugget axis, because it cannot be stirred together with the base materials to an adequate extent. Secondly, the Alclad layer is characterized by a significantly higher value of thermal conductivity than the 7075-T6 aluminum alloy. So, the heat generated in the welding process is non-uniformly distributed in the vicinity of the weld area due to the rapid dissipation of heat through the Alclad layer.

Optimization methods are applied to reduce the number of experiments and determine the maximum value of the regression function, usually taking into account only one criterion, the load capacity of the joint. The many defects which can arise in the RFSSW process are correlated with many process parameters and physical phenomena. Therefore, the aim of this paper is to present an alternative method for optimizing the parameters of the RFSSW process for joining 7075-T6 aluminum alloy sheets based on scalarization of the objective function using the weighting grades method in order to maximize the load capacity of the joint and minimize the standard deviation of the results.

The results obtained from tests on the influence of tool rotational speed and duration of welding on the load capacity of joints are shown in Fig. 4(a) and 5. An increase in tool rotational speed causes an increase in load capacity of the joint in the case of welding durations of 1.5 and 2.5 s. For a welding duration of 1.5 s, an increase in tool rotational speed causes an increase in load capacity of 17.9%, while for a welding duration of 2.5 s the increase is 11.5% (Fig. 4a). In the case of welds performed with a welding duration x₃ = 3.5 s, it should be noted that an increase in tool rotational speed from 2000 to 2400 rpm enabled the load capacity of the joint to increase by 8.3%. A further increase in the tool rotational speed caused a reduction in load capacity of the joint by 7.25%. An increase in tool rotational speed causes an increase in the weld temperature leading to better plasticization of the material and increasing strength of the joint. However, a long duration of welding causes an excessive rise in temperature and overheating of the material. A high temperature in the welding zone also causes the plasticized material to stick to the rotating tool surfaces. The sticking material also made it difficult to obtain the assumed plunge depth of the tool, and as a result a much larger dispersion of the test results was observed than is the case with joints fabricated at lower tool rotational speeds (Fig. 4b). The standard deviation of the test results was determined on the basis of five-element specimens. A smaller value of the standard deviation is associated with greater repeatability of the process, which is reflected in the reliability of the welded structure of the aircraft. An evaluation of the repeatability of the process was carried out using the Hartley test. In order to verify the hypothesis about the repeatability of variance, the statistic of the F_max test was determined:where \(s^{2} \left( y \right)_{i\hbox{max} }\) and \(s^{2} \left( y \right)_{i\hbox{min} }\) are the maximum and minimum values from the set of all variances, respectively.

The critical values of the Hartley test f_{max(a, k, v)} for the number of degrees of freedom k = m = 36, v = r-1 = 5-1 = 4 for the significance level α = 0.05 were read out from the statistical tables and compared with the calculated value of F_max. An analysis of the test showed that the results obtained at different values of the welding process parameters do not have the same variance (F_max > f_{max(a, k, v)}).

The curves of the effect of tool plunge depth on the load capacity of the joint have the same trend (Fig. 4b). The increase in tool plunge depth from 1.3 to 1.5 mm produces an increase in the load capacity of the joint. The highest value of load capacity of the joint is observed for a tool plunge depth of 1.5 mm. After exceeding this value, all characteristics show a trend to lowering of the strength of the joint as tool plunge depth increases. In the case of a curve corresponding to the highest rotational speed used, an initial significant increase in weld strength was achieved, reaching a maximum value of 8100 N, after which the trend reverses as the tool plunge depth increases and the joint strength decreases. The reduction in the load capacity of the joint with increasing tool plunge depth is caused by a weakening of the bottom part of the weld by the operation of the sleeve in the second stage of welding (Fig. 2b). Analysis of the test results also indicated some dependence regarding the values of variance and standard deviation. Apart from the assumed tool rotational speed, the highest standard deviation value was observed for a tool plunge depth of 1.5 mm, i.e., at the plunge depth ensuring the highest load capacity of the joint (Fig. 4b).

A cross section of the spot weld reveals four metallurgical zones: the base material (BM), heat-affected zone (HAZ), thermomechanically affected zone (TMAZ) and stir zone (SZ). In the cross section of an RFSSW joint, four principal geometric and metallurgical features can be found: partial bonding (PB), hooks (Hs), lack of mixing (LM) and bonding ligaments (BLs) (Fig. 6).

These features have a significant effect on joint failure strength under tensile/shear loading by affecting the fracture mechanism. Partial bonding is a transition region formed when the plasticized material is pushed back to its original position by the pin in the refill stage of the welding (Fig. 2c). The connection between sheets in this region is not very strong. The hook is formed during deformation of the lower sheet during the last stage of welding, when the pin pushes the material, leading to bending of the sheet interface. A bonding ligament is composed of pure aluminum, whose properties are different in relation to the BM (7075-T6 aluminum alloy). The thickness of the bonding ligament decreases with increasing duration of welding and tool rotational speed. Voids (Vs) and incomplete refilling (IR) can be attributed to insufficient flow of the SZ material at the refilling stage and a poor metallurgical bonding effect and residual heat stress after welding (Ref 13).

For tensile/shear tested specimens, three main types of failure mechanisms were revealed: interfacial failure (IF), plug failure (PF) and U-shape or L-shape pullout failure, denoted as U-type POF and L-type POF, respectively. Table 2 summarizes the fracture modes of the welds investigated. The first mechanism is characterized by failure via crack propagation through the weld (Fig. 7a). In the second case, joints will fail as a PF, a plug is characterized by the weld zone tearing clear of one of the sheets leaving a visible protrusion (Fig. 7b). Pullout failure occurs via complete withdrawal of the weld nugget from the sheet (Fig. 7c and d).

Interfacial failure is the main mechanism of failure which was observed in the case of the welds fabricated at a sleeve plunge depth lower than that of the upper sheet thickness. However, pullout failure was observed in the case of welds fabricated at sleeve plunge depths of 1.3 or 1.5 mm and the highest values of tool rotational speed (2800 rpm) and duration of welding (3.5 s). Both high welding time and high tool rotational speed led to a recrystallization process caused by high temperature. So, a fine-grained structure with higher mechanical strength than BM is observed in the weld. The tear plug fracture with a tear on the lower sheet is the preferred failure mode due to its higher associated plastic deformation and energy absorption.

Increasing the sleeve plunge depth to a value higher than the thickness of the upper sheet causes the existence of a clear structural notch on the perimeter of the weld where the edge of the sleeve cuts the lower sheet. In this region, the BM material is not sufficiently plasticized and mixed in the weld region. Complete or partial plug failure is then observed in the area of the lower sheet. A lack of sufficient mixing of material between both stir and thermomechanically affected zones due to low tool rotational speed or duration of welding is the principal cause of plug fracture mode.

Figure 8, 9, 10 summarizes the selected fractures of the welds destroyed by the three main types of failure mechanism. The notations of the weld fractures correspond to the parameters listed in Table 2. The surfaces of the welds destroyed by interfacial failure are characterized by ductile fracture (Fig. 8). In the case of the specimens fabricated at the sleeve plunge depth of 1.5 mm (Fig. 8d), the fracture surfaces are characterized by the presence of Alclad layer mixed in BM. The center area of the fracture surface is also contaminated by randomly distributed small fragments of oxides which originally were located on the surface of the Alclad layer.

The fracture surface of the specimens destroyed by the mechanism of plug failure is characterized by dimples (Fig. 9) which are located around the perimeter of the weld. The size and shape of the dimples depend on the direction of the stress interaction. The dimples are typical for the ductile type of fracture and are mainly located in the TMAZ of the weld.

The crack initiation of the specimens destroyed by pullout failure was located in the vicinity of the weld periphery in which the fine-grained structure is observed. The material with fine-grained structure is characterized by higher mechanical strength than BM, and the combined brittle-ductile fracture type can be revealed in the micrographs (Fig. 10). The inhomogeneity of the weld structure around its periphery is a result of the presence of aluminum oxides which are a source of crack nucleation.

The selection of the optimal parameters of the RFSSW process requires the determination of an adequate mathematical model in the form of a regression function W(x). Regression analysis has been carried out using the least squares method. The criterion for assessing the quality of the approximation takes the form:where the value of function R is a certain measure of the deviation of the approximation function W(x) and approximated function f(x), i = 1, …, N number of experiments.

For the needs of statistical analysis, the following designation was adopted: tool rotational speed x₁, plunge depth x₂ and duration of welding x₃.

During the approximation, the most frequently selected basic functions are monomials according to Weierstrass’s theorem. This theorem defines that for every function f(x) specified and continuous on a closed and limited interval [a, b] exists a polynomial W = b₀ + b₁x₁ + b_mx^m that approximates monotonously the function f(x) on the interval [a, b]. During the analysis, however, it was not possible to obtain such a polynomial with a rational m level, which could be considered as adequate. Therefore, the m-degree algebraic polynomial was adopted for the definition of the interactions between the RFSSW process parameters:In Eq 3, there exists L-number of unknown coefficients \(b_{0} , b_{i}^{\left( 1 \right)} , b_{ij}^{\left( 1 \right)} , b_{ij \ldots \ln }^{\left( 1 \right)} , b_{ij}^{\left( 2 \right)} , b_{ii \ldots m}^{\left( m \right)}\) while i, j,…, n = 1,…, S variables of a polynomial (3).

The Fisher-Snedecor test was used to assess the adequacy of the regression equation with the test results. At the first stage of the analysis, the adequacy variance was determined, according to the following formula:where \(\bar{y}_{i}\) is average value of measurement results in the i-th experiment, \(\overline{\overline{{y_{1} }}}\) is value calculated from the regression equation for the levels of input and output factors in i-th experiment, k is number of terms in regression equation (without a free term) after rejection of insignificant terms and N the total number of experiments.

Then the value determined for the test coefficient F:was compared with the critical value determined from the Fisher-Snedecor distribution table. This allowed an adequate regression equation to be obtained describing the influence of the RFSSW process parameters on the load capacity of joints (W_F(x)) and standard deviation of the load capacity (W_σ(x)), with the following form:

The maximum error value of the regression function did not exceed 6.11%, while the mean square error between the experimental results and the results of mathematical modeling was 2.55%.

In order to replace classic resistance welding by RFSSW technique, thus ensuring a better quality of welds, it is necessary to select optimal welding parameters. This selection must guarantee not only a high load capacity of the joints produced, but also high process stability, characterized by the lowest possible value of the variance of the results obtained. This requires multicriteria optimization of the process and finding a compromise solution that meets the aforementioned mathematical model.

For the writing of the multicriteria problem, the following designations have been adopted:

The problem of multicriteria optimization of the selection of process parameters can be written in the form of:

The one-criteria problem:is an i-th partial problem, where the vector z^io ∈ D in which the i-th objective function reaches the extremum searched.

The function describing the load capacity of the joint W_F(x) reaches a maximum value of 7759.54 N at a tool rotational speed x₁ = 2800 rpm, tool plunge depth x₂ = 1.55 mm and duration of welding x₃ = 1.58 s. In contrast, the minimum load capacity of 5456.25 N is observed at x₁ = 2000 rpm, x₂ = 1.9 mm and x₃ = 1.5 s. The maximum and minimum values of the W_F(x) function correspond to the extreme values of the RFSSW process parameters at which the tests were carried out. The second function describing the standard deviation of the results of the load capacity W_σ(x) of the joint takes the highest value of 374.71 N at a tool rotational speed x₁ = 2800 rpm, tool plunge depth x₂ = 1.47 mm and duration of welding x₃ = 3.5 s, while the minimum value of 16.67 N is found to be at x₁ = 2241.35 rpm, x₂ = 1.9 mm and x₃ = 1.87 s.

Usually the ideal solution of function (9) is not available, which means that in the set of permissible solutions D, there is no vector z^o for which all objective functions reach the extreme searched for. Therefore, a search was made for effective solutions during the solution of the problem being examined. The aim of the methodology presented in the article was to select one optimal solution from the set of effective solutions. Goal programming was applied to solve the problem being studied. In goal programming, the main control parameters are aspiration levels treated as goals to be achieved for individual assessment functions. An aspiration level a_i was assumed for each assessment function f_i(z). For each permissible solution z ∈ D, the deviations downwards and upwards of the assessment function f_i(z) from the aspiration levels are defined as:

An optimal solution of the problem being studied is the permissible solution which minimizes the deviations from the aspiration levels. As a goal function, a weighted sum of deviations was adopted:where u_i are non-negative weights corresponding to individual deviations.

Deviations were introduced into the goal programming model by means of an additional system of goal equations:where \(d_{i}^{ - } ,\;d_{i}^{ + }\) are non-negative state variables expressing, respectively, the upper and lower deviation of the current value of the i-th assessment function from the appropriate level of aspiration a_i.

In order to find an optimal (compromise) solution, the method of weighting of the goal variable was applied. So the set of equations of the problem takes the form:The form of the solution (12) depends on the weight value u_i and the levels of aspiration a_i adopted. During the calculations, it was assumed that the load capacity of the RFSSW joint is more important than the variance of the welding process (u₁ = 0.6, u₂ = 0.4). In the first stage of the calculations, the level of aspiration regarding the load capacity of the joint was assumed to be a₁ = 7300 N and the variance of the process a₂ = 100 N. A solution slightly deviating from the required levels was obtained, which provided a joint load capacity of 7299 N with a standard deviation of 112.31 N (Table 3). Decreasing the level of aspiration regarding the load capacity of the joint to 7200 N provides a solution (x₁ = 2182 rpm, x₂ = 1.7 mm, x₃ = 3.5 s) much closer to the requirements (Fig. 11). In the case considered here, the load capacity of the joint deviates from the assumed one by 0.0027 N, and the standard deviation by 0.149 N.

A change in the aspiration levels to a₁ = 7400 N and a₂ = 140 N moves the optimal solution to a point corresponding to a tool rotational speed x₁ = 2216 rpm, x₂ = 1.55 mm and x₃ = 3.46 s (point P4 in Fig. 12). By applying the parameter settings for the welding process given above, it is possible to obtain a joint load capacity of 7400 N with a standard deviation of the tensile/shear test results of 152.46 N. To validate the results obtained by the goal programming, six experiments have been carried out with the process parameters given above. The average results of tensile/shear tests were: maximum load capacity of the joint 7115 N and standard deviation 102.3 N. The load capacity is overestimated by the analytical model; however, the standard deviation value is lower than the numerical ones.

The problem presented in the paper was considered from the point of view of a technologist whose task is to determine the process parameters that ensure the load capacity of joint required by the constructor while ensuring the required welding process capability. In the case considered, it was assumed that the required load capacity of the joint is 7400 N with a standard deviation of 140 N. During the calculations one can also assume much higher requirements of load capacity of the joint (from those accepted by the constructor), which allows one to obtain solution results similar to the values obtained during the experiment. Such a solution is not only burdened with a larger standard deviation of the results but also it results in a higher labor demand in the welding process because it requires a much higher rotational speed which increases the amount of heat emission. This process also causes significant plasticization of the material which tends to stick to the working surfaces of the tool and requires frequent cleaning of the tool, which increases the labor demand of the RFSSW process and reduces its durability. So, in the case of the parameters considered, the maximum load capacity obtained experimentally was not the most advantageous solution.

The article presents the results of research aimed at optimizing RFSSW process parameters in order to provide the highest possible load capacity of the joint and at the same time the smallest standard deviation of the tensile/shear test results. The analyses carried out allow one to draw the following conclusions: