Issue 
Mechanics & Industry
Volume 21, Number 5, 2020



Article Number  505  
Number of page(s)  10  
DOI  https://doi.org/10.1051/meca/2020061  
Published online  05 August 2020 
Regular Article
Comparative study of the effective parameters on residual stress relaxation in welded aluminum plates under cyclic loading
^{1}
Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, 16758136 Tehran, Iran
^{2}
Faculty of Mechanical Engineering, K.N. Toosi University of Technology, 167653381 Tehran, Iran
^{*} email: ali_pourkamali@srttu.edu
Received:
27
March
2019
Accepted:
23
June
2020
The prediction of residual stress relaxation is essential to assess the safety of welded components. This paper aims to study the influence of various effective parameters on residual stress relaxation under cyclic loading. In this regard, a 3D finite element modeling is performed to determine the residual stress in welded aluminum plates. The accuracy of this analysis is verified through experiment. To study the plasticity effect on stress relaxation, two plasticity models are implemented: perfect plasticity and combined isotropickinematic hardening. Hence, cyclic plasticity characterization of the material is specified by low cycle fatigue tests. It is found that the perfect plasticity leads to greater stress relaxation. In order to propose an accurate model to compute the residual stress relaxation, the Taguchi L18 array with four 3level factors and one 6level is employed. Using statistical analysis, the order of factors based on their effect on stress relaxation is determined as mean stress, stress amplitude, initial residual stress, and number of cycles. In addition, the stress relaxation increases with an increase in mean stress and stress amplitude.
Key words: Taguchi / stress relaxation / plasticity / cyclic load / aluminum
© AFM, EDP Sciences 2020
1 Introduction
Welding has been widely used in production and repair of components. Due to highlocalized temperature during welding and subsequent rapid cooling, the tensile residual stress is developed in and near to the weld area, which can significantly reduces the life expectancy of structures [1–3]. To extend the service life of engineering structures, there are different methods to reduce the harmful tensile residual stress, such as the heat treatment [4,5], shot peening [6–8], and mechanical loading [9–23]. However, the welded structures tolerate the residual stress as well as the stress resulted by inservice mechanical loads. Therefore, to assess the safety of structures, it is necessary to take into account the residual stress redistribution. It is understood that when the total stress exceeds than the local yield stress of material, the residual stress relaxation occurs as a result of plastic deformation. To predict the plasticity, different constitutive models from simple elasticperfectly plastic to more complicated rules have been developed. Some studies used the analytical method for modeling the residual stress relaxation [9–11]. Such methods typically make some simplifying assumptions and, as a result, were unable to accurately model the complexities involved with the 3D welding residual stress. Therefore, numerical method based on the finite element (FE) analysis has been developed. Yi and Lee [12] carried out the FE analysis to evaluate the residual stress relaxation in the steel welded plates under cyclic load. They considered the stress relaxation only on the axial direction. Moreover, they did not taken into account the material hardening. Barsoum [13], and Katsuyama and Onizawa [14] also studied the welding residual stress relaxation without cyclic plasticity modeling. They performed FE analysis based on the elasticperfectly plastic behavior. Since these studies did not consider the cyclic plasticity hardening, the residual stress state was not accurately predicted. Katsuyama et al. [15] considered a bilinear stress–strain equation with linear kinematic hardening to study the welding residual stress relaxation under repeated loading. They observed that the higher loading caused the greater relaxation. Dattoma et al. [16] assumed the isotropic hardening behavior to predict the stress relaxation. They concluded that the stress relaxation occurs only in the first cycle. Their results revealed that the stress relaxation depends not only on the load amplitude but also on the load direction. Qian et al. [17] showed that the different types of steel exhibited very different residual stress relaxation behavior. Gannon et al. [18] used the constitutive model combined of perfectly plastic behavior and the kinematic hardening model. They concluded that for shipbuilding steel, the kinematic hardening has a little influence on stress relaxation. They also showed that the residual stress reduction occurs entirely during the first cycle if the magnitude of the cyclic load is not subsequently exceeded. Cho and Lee [19] studied the interaction between the residual stress and cyclic loading in a welded steel pipe. They proposed a kinematic hardening model [20] to take into account the cyclic plastic strain. Their simulation indicated that the stress relaxation depends on the direction and magnitude of external load. Xie et al. [21] proposed a model to predict the residual stress relaxation in the welded plates based on the experimental and FE results. They used the combined isotropic and kinematic hardening in the constitutive model. Farajian et al. [22,23] measured the initial and relaxed residual stresses in the welded plates under cyclic load by means of Xray method. They concluded that the stress relaxation occurs at the first loading cycle when the von Mises stress exceeds than the monotonic yield strength.
It is clear that the welding residual stress relaxation is a complex phenomenon, which depends upon a wide range of variables. Further, experimental design is a useful method to identify the effective parameters and their impacts on response variable. In this study, experimental design is done to study the characteristics of welding residual stress relaxation under cyclic load. First, a 3D FE model is developed to predict the initial residual stress. Moreover, to validate the model, the predicted residual stress was compared with the experimental result. Subsequently, the analysis is developed to apply the cyclic load, with various amplitudes and cycle numbers. The stress relaxation is highly dependent on the plasticity, particularly in metals that show strain hardening. To study the plasticity behavior on stress relaxation, two different models are used: (1) perfect plasticity, and (2) combined nonlinear isotropickinematic hardening. Hence, cyclic plasticity characterization of the material is specified by low cycle fatigue tests. In order to provide a simple analytical model for prediction the stress relaxation, the Taguchi L18 array with one 6level factor and four 3level factors is used. The specification of cyclic load and the initial welding residual stress were chosen as the design factors. Variance analysis is also performed to determine the impact and contribution of each factor on the residual stress relaxation.
2 Experimental procedure
The experimental procedure consists of two steps: material properties and specimen production, and residual stress measurement, as described below.
2.1 Material properties and specimen production
The experimental specimen was prepared through singlepass buttwelding of the 5000series aluminum plates with length 350 mm, width 250 mm, and thickness 2 mm. The GTAW process was done with welding parameters given in Table 1. The thermal specifications [24] and the mechanical properties of the material versus to the temperature are plotted in Figure 1. Due to the high impact of yield strength on the welding residual stress, the mechanical properties were determined by hightemperature tensile test, based on the ASTM E 21 standard [25]. The specimen geometry, which was used in this test, is shown in Figure 2.
Parameters of the welding process.
Fig. 1 Temperaturedependent properties of material. 
2.2 Residual stress measurement
During the past years, various techniques have been developed to measure the residual stress in welded structures. Among these methods, the holedrilling is a widely used technique. It offers advantages of good accuracy and reliability, standardized test procedure, and nonsignificant damage [26]. In this study, in order to verify the FE analysis, the welding residual stresses were measured through the incremental holedrilling method. In this method, a small hole is drilled in an incremental number of steps and the corresponding strains are measured on the surface. The residual stresses are calculated with these measured strains. Hence, five strain gages of type rosette were bonded to the specimen surface as shown in Figure 3. The blindholes with depth 1 mm were drilled precisely at the center of the rosette using an airturbine at rotation speed 300,000 rpm, feed rate 0.2 mm/min, depth increment 0.05 mm, and delay time 5 seconds. The cutting tool was a tungsten carbide inverted cone end mill with 1.6 mm in diameter and six cutting edges.
The depth of each hole was 1 mm, being drilled in 20 steps 2.3 of 50 µm each.
Fig. 3 Residual stress measurement (a) strain gauge position (b) drilling process. 
3 Welding simulation of the buttjointed plates
3.1 General consideration
The thermoelastoplastic analysis was performed based on the couple method between the thermal and mechanical solutions. The analysis was done in three stages as seen in Table 2. The 3D FE model was constructed and meshed using ABAQUS software, as shown in Figure 4. For the boundary condition, the nodes at distance 16 mm from the weld centerline were fixed due to the clamps of welding machine, and then released due to removing the clamps. The thermal gradient is highest at the welded zone, so the finer mesh size was used near the weld. The identical geometry and mesh configurations were utilized in the thermal and mechanical analyses. Whereas, the element types DC3D20 and C3D20R were used in the thermal and mechanical analyses, respectively.
Stages of the welding simulation.
Fig. 4 FE model of the welding simulation. 
3.2 Thermal analysis
For the heat source simulation, the most popular model, proposed by Goldak et al. [27] was used. In this model, the heat distribution for the front and rear ellipsoids (Fig. 5) were calculated separately as follows [27]:(1) (2)where, a_{f} , a_{r} , b, and c are the dimensional constants of the Goldak's model, Q = VI is the total heat input, η is the thermal efficiency which is assumed to be 60% for the GTAW process [28], and f_{f} and f_{r} are the heat fraction coefficients in the front and rear ellipsoids, respectively, f_{f} + f_{r} = 2 [27]. In this study, a_{f} , a_{r} , b, and c are considered 4, 12, 4, and 2 mm, respectively and f_{f} and f_{r} are 1.5 and 0.5.
3.3 Mechanical analysis
Total strain rate during welding is decomposed into the four components [29]:(3)where , , , and are the elastic, plastic, thermal, and solidstate phase transformation strain rates, respectively. During welding of the 5000 seriesaluminum, solidstate phase transformation does not occur, so is ignored.
3.4 Result of welding simulation
Figure 6 shows the predicted and the measured residual stresses. The stresses were extracted from the midsection perpendicular to the weld line. As seen, the maximum longitudinal residual stress (160 MPa) equals to the yield strength, approximately. Whereas, the maximum transverse residual stress (34 MPa) is much lower than the yield strength. Hence, a good agreement was found between the test and the FE analysis.
Fig. 6 (a) Longitudinal and (b) transverse residual stress distributions. 
4 Simulation of the welding residual stress relaxation under cyclic loading
4.1 Plastic constitutive model
By using the von Mises criterion, the yield function can be expressed as [30]:(4)where, σ _{ D } is the deviatoric part of the stress tensor, X _{ D } is the deviatoric part of the back stress, σ _{0} is the initial yield stress, and R is the isotropic hardening variable.
4.1.1 Isotropic hardening
The isotropic hardening (IH) assumes the uniform expansion without translation of the yield surface within the stress space, which can be explained by the following equation [30]:(5)where, p is the equivalent plastic strain, b determines the rate of isotropic hardening change, and q is the maximum change in the size of the yield surface.
4.1.2 Nonlinear kinematic hardening
The kinematic hardening assumes that the yield surface translates in the stress space without any expansion. The nonlinear kinematic hardening model (NLKH) can be expressed as [31]:(6)where, C and γ are the material parameters describing the nonlinear kinematic hardening. ABAQUS software provides a method to directly implement the combined isotropic and kinematic hardening model using empirical data of cyclic straincontrolled test and static tensile test [32].
4.2 Cyclic plasticity response
The low cycle fatigue (LCF) tests were performed based on the ASTM E606 standard [33] to characterize the cyclic hardening properties. The straincontrolled tests were carried out on flat specimens by a servohydraulic testing machine with 100 KN capacity (Fig. 7), sinusoidal wave shape, and frequency of 1 Hz. The cyclic stress–strain curves were determined using the companion test method. This method requires a series of test specimens, which each specimen is subjected to a constant strain amplitude [34]. The tests were carried out on each specimen until it fractured in two parts. Figure 8 shows the variation of stress amplitude with the number of cycles for strain ranges of 1.1%, 1.2%, and 1.9%. As seen, the material presents significant hardening due to cyclic loading.
Fig. 8 Variation of stress amplitude with number of cycles. 
4.3 FE analysis with cyclic loading
To study the plasticity behavior on the residual stress relaxation, two different plasticity constitutive models were implemented into the FE analysis: (1) perfect plasticity without hardening (NH), and (2) combined isotropic hardening with nonlinear kinematic hardening (IH&NLKH). In this analysis, the same FE model of the welding simulation was used, except that the boundary condition and the loading were modified. The residual stress field determined in previous section was used as the initial load. The fully reversed cyclic loading, parallel to the weld line, was applied to the one end of the plate and the other end was fixed.
Figures 9 and 10 show the longitudinal and transverse residual stress distributions after different number of cycles with the load amplitude Sa = 100 MPa for NH and IH&NLKH models, respectively. As can be seen, for both plasticity models the tensile residual stress in the longitudinal and the transverse directions will be decreased by the cyclic loading. However, the relaxation is limited to the initial cycle, which the residual stress stabilized after that, i.e. the subsequent cycles do not contribute to the relaxation significantly. In addition, the stress relaxation along the longitudinal direction is greater than that along the transverse direction.
Figures 11 and 12 show the stabilized longitudinal residual stresses at different load amplitudes for NH and IH&NLKH models, respectively. As seen, the external cyclic load causes redistribution in the welding residual stress. It can be concluded that the amount of relaxation depends on the level of applied cyclic load, i.e. a higher load amplitude results in a greater residual stress change.
Figure 13 shows the percentage reduction of the longitudinal residual stress due to relaxation versus to the load amplitude (data is extracted at a distance of 16 mm from weld centerline). This figure indicates that the NH model causes a greater stress relaxation than the IH&NLKH model. The reason is that the magnitude of the stress relaxation depends on the accumulated plastic strain, which is stronger in the perfectly plastic model. Experiments have shown that the 5000series aluminums exhibit cyclic strain hardening with the Bauschinger effect [35]. However, the perfectly plastic model assumes no strain hardening in tension and compression; the initial yield function is unchanged in the stress space during plastic deformation. Therefore, to more accurate assessment of the welding residual stress relaxation in aluminumwelded plates, the IH&NLKH model should be used, which will be utilized in the subsequent section.
Fig. 9 (a) Longitudinal, (b) transverse residual stress distributions after cyclic loading with Sa = 100 MPa for NH model. 
Fig. 10 (a) Longitudinal, (b) transverse residual stress distributions after cyclic loading with Sa = 100 MPa for IH&NLKH model. 
Fig. 11 Stabilized longitudinal residual stress at different load amplitude for NH model. 
Fig. 12 Stabilized longitudinal residual stress at different load amplitude for IH&NLKH model. 
Fig. 13 Percentage reduction of longitudinal residual stress versus to load amplitude. 
5 Design of experiments
Since there are many effective parameters on the welding residual stress relaxation, design of experiments based on the Taguchi method [36] was done to study the impact of main parameters. The characteristics of the cyclic load consist of stress amplitude (S _{ a }), mean stress (S _{ m }), and number of cycles (N) were selected as the loading parameters, and the maximum von Mises of the initial welding residual stress (σ _{ Irs }) was selected as the technological parameter. Based on the number of design factors and their levels, the Taguchi L18 array was used in the experimental design, as shown in Table 3. The maximum von Mises of the residual stresses after applying the cyclic loading, namely relaxed residual stress, (σ _{ Rrs }) was selected as the response variable. Using the FE analysis, the response variable values were determined for all experiments, as plotted in Figure 14.
Design factors and their levels.
Fig. 14 Relaxed residual stresses for all experiments (MPa). 
5.1 Taguchi analysis
The results of the Taguchi analysis are given in Table 4. This table presents the averages of the relaxed residual stresses for every level of design factors. Delta is the maximum difference of response variable between the levels of each factor, and Rank specifies the rank of each factor based on the Delta values. As seen, the order of factors based on their effects on the relaxed residual stress is determined as follows: mean stress, stress amplitude, initial residual stress, and finally number of cycle. This can be also observed in the plot of design factor impacts shown in Figure 15. As seen, the mean stress and the stress amplitude affect the relaxed residual stress significantly; an increase in the mean stress and the stress amplitude causes the relaxed residual stress to reduce, while the effect of the number of cycles is rather insignificant.
Average values of relaxed residual stresses in every level of each factor (MPa).
Fig. 15 Design factor effects on the relaxed residual stress. 
5.2 Regression analysis
The regression analysis was performed to derive the relationship between the design factors and the response variable. By this analysis, the relaxed residual stress was estimated through the following first order equation,(7)
In equation (7), each numerical coefficient reveals the importance of the corresponding parameter. In addition, the negative sign of coefficients declares reduction of the relaxed residual stress. In this study, the high value of R ^{2} (variation percentage of the response) indicates that the fit of the experimental data to the model is satisfactory.
5.3 Analysis of variance
Analysis of variance (ANOVA) was then carried out to interpret the results of the experiments and detect the amount of contribution of each variable. The ANOVA result for the first order regression is shown in Table 5. The sum of squares indicates the relative importance of each factor; the factor with the biggest sum of squares has the greatest impact. The pvalue also shows the significance level of each factor; a smaller pvalue corresponds to a greater significance. Table 5 indicates that mean stress (p = 0), stress amplitude (p = 0), initial residual stress (p = 0.004), and number of cycles (p = 0.56) in this order are the most significant factors affecting the relaxed residual stress. Moreover, mean stress with 68.5% contribution has the most impact on the relaxed residual stress. Notice that these conclusions are consistent with the Taguchi analysis results.
Figure 16 shows the residual plots for the proposed first order equation. This figure can be used to asses if the model satisfies the assumptions of the analysis. A residual is the difference between an observed value and its corresponding fitted value. The normal probability plot approximately follows a straight line. This means that the data are normally distributed. Histogram of the residuals also indicates that the data is not skewed and normally disturbed. In addition, the residual versus order plot displays no pattern in distribution of data, which indicates that the residual values are independent of each other.
In order to verify the regression analysis, the results of the FE method were compared with the values derived from equation (7) as shown in Table 6. As seen, the maximum relative difference is 12%. This is compatible with the residual plots.
ANOVA results for the first order regression.
Fig. 16 Residual plots for the first order regression. 
Comparison between the results of the FEM and the model (MPa).
6 Conclusions
This paper studied the influence of the main parameters on the residual stress relaxation of aluminum weldedplates under cyclic loading. The induced residual stress for the welded plates was predicted using FE analysis. The experimental residual stress measurements are in good agreement with the predicted stresses. To study the plasticity effect on the stress relaxation, two different models were considered: perfect plasticity, and combined nonlinear isotropickinematic hardening. In addition, the effect of various loading and technological parameters on the stress relaxation was studied through the Taguchi method. In summary, the following conclusions can be drawn from this research:

The residual stress relaxation is highly dependent on the plasticity behavior. The NH causes to a greater relaxation compared to IH&NLKH.

The IH&NLKH model successfully predicts the redistribution of welding residual stress in a strain hardening aluminum due to its ability to precisely determine the accumulated plastic strain.

The residual stress is stabilized after the first cycle and the amount of stress relaxation occurred in subsequent cycles is insignificant.

Increase in the mean stress and the stress amplitude cause the stress relaxation to increase, while the effect of number of cycles is rather insignificant

The most significant factors on welding residual stress relaxation in the order of significance are the following: the mean stress, the stress amplitude, the initial residual stress, and the number of cycles.
Nomenclature
a_{f}, a_{r}, b, c : Dimensional constants of the Goldak's model (mm)
b : Rapidity of isotropic hardening change
C : Nonlinear kinematic hardening parameter (MPa)
f : Von Mises yield function (MPa)
f_{f}, f_{r} : Heat fraction in front and rear ellipsoids of the Goldak's model
q : Saturated value of isotropic hardening (MPa)
Q : Total heat power of welding source (W)
R : Isotropic hardening value (MPa)
S_{a},S_{m} : Stress amplitude and mean stress (MPa)
X_{D} : Deviatoric part of back stress tensor (MPa)
γ : Nonlinear kinematic hardening parameter
, , : Elastic, plastic, thermal, and phase transformation strain rates
η : Thermal efficiency of welding heat source
σ _{0} : Initial yield stress (MPa)
σ _{ D } : Deviatoric part of the stress tensor (MPa)
σ _{ Irs }, σ _{ Rrs } : Von Mises values of initial and relaxed welding residual stresses (MPa)
Conflict of interest
The authors declare that they have no conflict of interest.
Funding
This research did not receive any specific grant.
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Cite this article as: Y. Ghaderi Dehkordi, A. Pourkamali Anaraki, A.R. Shahani, Comparative study of the effective parameters on residual stress relaxation in welded aluminum plates under cyclic loading, Mechanics & Industry 21, 505 (2020)
All Tables
All Figures
Fig. 1 Temperaturedependent properties of material. 

In the text 
Fig. 2 Specimen geometry of high temperature tensile test [25]. 

In the text 
Fig. 3 Residual stress measurement (a) strain gauge position (b) drilling process. 

In the text 
Fig. 4 FE model of the welding simulation. 

In the text 
Fig. 5 Doubleellipsoidal heat source model [27]. 

In the text 
Fig. 6 (a) Longitudinal and (b) transverse residual stress distributions. 

In the text 
Fig. 7 LCF tests (a) experimental setup (b) specimen geometry [33]. 

In the text 
Fig. 8 Variation of stress amplitude with number of cycles. 

In the text 
Fig. 9 (a) Longitudinal, (b) transverse residual stress distributions after cyclic loading with Sa = 100 MPa for NH model. 

In the text 
Fig. 10 (a) Longitudinal, (b) transverse residual stress distributions after cyclic loading with Sa = 100 MPa for IH&NLKH model. 

In the text 
Fig. 11 Stabilized longitudinal residual stress at different load amplitude for NH model. 

In the text 
Fig. 12 Stabilized longitudinal residual stress at different load amplitude for IH&NLKH model. 

In the text 
Fig. 13 Percentage reduction of longitudinal residual stress versus to load amplitude. 

In the text 
Fig. 14 Relaxed residual stresses for all experiments (MPa). 

In the text 
Fig. 15 Design factor effects on the relaxed residual stress. 

In the text 
Fig. 16 Residual plots for the first order regression. 

In the text 
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