4. Inherent Deformation Theory
Aiming to obtain the welding distortion of large welding structure, many studies were done on the simplified numerical simulation method of welding [
26,
27,
28,
29,
30]. This originated the inherent deformation theory. Although this method cannot simulate every step of the welding process, it can solve complex engineering problems and easily predict welding distortion.
While the uniform movement of the heat source, in a certain distance since the start to the edge of the plate is given, elastic, plastic, thermal, and creep strains are generated by heat input and are to be expected to be uniform along the welded line. The elastic component of these strains is reversible and is removed after cooling; however, plastic, creep, and thermal parts of the strain remain, and their sum is defined as the inherent strain [
31], which is expressed by Equation (4):
In a welded component, the creep and thermal strains are too small and usually are ignored [
32]. Therefore, in welding, the amount of inherent strain in each direction is basically the amount of plastic strain in the same direction, so Equation (4) turns into Equation (5):
These deformations can be calculated for a small part of the structure by its simulation, and then, it can be propagated through the entire structure [
33]. However, because it is difficult to apply inherent strains on the model, it is possible to assume that there is an inherent amount of deformation for each section perpendicular to the welded line.
Lu et al. [
34] compared the inherent strains theory with the use of FEM thermo–elastic–plastic analysis, using a T-joint part constituted by a web with 60 mm × 60 mm × 6 mm and a flange with 100 mm × 60 mm × 6 mm.
The welding distortions were computed using the two methods: FEM and inherent deformation theory. Based on thermo–elastic–plastic finite element method, the welding process was simulated by life and death element, moving the heat source and transient state thermal field. It was concluded from the comparison (
Table 3) with the simulation results by using the two methods consistently. Therefore, the inherent strains method can be suitably applied to predict structural welding distortion in large and complex structures.
Farajpour and Ranjbarnodeh [
35] studied a dissimilar welded structure that was simulated with 3D solid and shell elements in ANSYS 11.0 finite element software (Ansys, Inc., Canonsburg, PA, USA), to obtain the deformations. Also, the weldment was simulated using an inherent deformation method. Both results were compared to the experimental ones. This comparison showed that it was feasible to simulate a dissimilar welded joint using both methods. The conventional simulation had 9-node solid elements, which needs a large memory as well as a time-consuming analysis. On the other hand, simulation with the inherent deformations uses 4-nodes shell elements, reducing the required memory, consequently saving time and costs of the analysis.
The following results were drawn from this study [
35]:
The experimental weld pool width was 5.6 mm, while the one from the thermal simulation result was 5 mm, showing an 11% error rate;
Empirical results showed a distortion of 5.85 mm, whereas the analysis from the inherent strains model showed a result of 5 mm (17% error rate);
This method reduces analysis time by 70%.
Wei et al. [
36] proposed a simple and efficient method, to estimate inherent deformation of typical weld joints by inverse analysis. In this method, the inherent deformation was introduced into the elastic FEM as the initial strain, however, the values of the inherent deformations for all welded joints included in the structure, must be known previously.
Through this work, it was concluded that a good agreement between the predicted (inherent strains theory) and the measured deformations was achieved, which demonstrates that this method can be employed with inverse analysis, in an effective way to predict welding distortions of large welded plate structures.
6. FEM Simulation vs. Experimental Results
Stamenkovíc and Vasovíc [
38] studied manual metal arc welding (MMAW) of carbon steel plates. The finite element analysis of residual stresses in butt welding of two similar plates was performed with ANSYS.
The welding simulation considered a sequential coupled thermomechanical analysis and the element birth and death method was applied. The values achieved through the software, were compared with the experimental ones, and a good agreement between them was shown (
Figure 4).
Ding et al. [
39] studied the residual stress of the wide butt welds through an 8-experiment test program. The model was developed by SYSWELD, to simulate the residual stress produced by the wide butt welding. The conclusions of this study were the following:
The FE model underestimates the residual stress in the transverse and longitudinal direction to the welding seam by 7% and 2%, respectively.
This study demonstrated the existence of high residual stress during the welding process of wide butt-welding seams (
Table 6).
While the pattern was similar across different welding widths, the strength of the residual stress increased when the width of the weld increased.
Zubairuddin et al. [
40] investigated the residual stress and distortion induced in 3 mm thick modified 9Cr-1Mo steel plates, using gas tungsten arc welding (GTAW) as a joining process. SYSWELD software was performed for the thermomechanical analysis.
Distortion of the weld joint was measured using vertical electronic height gauge and the finite element analysis of distortion of the weld joint was carried out by applying both large and small distortion theories. Comparison of experimental and numerical results showed better accuracy if large distortion theory was applied.
It can also be concluded that:
Kik and Górka [
41] investigated numerical simulations based on the real experiments of S700MC steel T-joint laser and hybrid welding. The simulation was carried out with SYSWELD. Some variables such as the distribution of temperature fields, thermal cycles, distributions of individual metallurgical phases and hardness, strains, and plastic deformations were calculated for one selected joint from both mentioned methods. The main objective was to determine the differences in the stress distributions, and their minimal and maximal values. The results of the experiment showed that after the calibration of heat source models, it was possible to obtain results of thermometallurgical analyses with a good agreement with the experimental test results. It was also possible to have other information such as:
Metallurgical phases and cooling rates were reflected in the hardness distributions;
The concentration of martensite influenced the von Mises stress values;
The values in the laser welding were higher 50 MPa, than the hybrid welding case;
The maximum stress values were concentrated in the HAZ, for both welding processes;
Areas subjected to strains were influenced by the thermal cycles.
Pavani et al. [
42] studied manual metal arc welding (MMAW) process of carbon steel plates. The finite element analysis was performed by ANSYS software. The welding simulation considered a sequential coupled thermostructural analysis and the element birth and death method were applied. It was observed that the stress, in the direction of the width of the test plate, has the highest influence on the formation of cold cracks. The instantaneous stress on the weld surface was 800–1000 MPa, and below the weld was 500–600 MPa.
Dean et al. [
43] investigated the effects of solid-state phase transformation on residual stresses and deformations in low and medium carbon steels, welded by tungsten inert gas (TIG) arc welding process. In this study, continuous cooling transformation (CCT) diagrams were used to predict the fractions of martensite in the HAZ. The analysis of low carbon steel revealed that the residual stresses and distortions did not seem to be affected by phase transformation during cooling. However, for medium carbon steel, the residual stresses and deformation were significantly affected by the low-temperature phase transformation.
Seles et al. [
44] presented a paper about a finite element procedure for the prediction of welding-induced residual stresses and distortions in large structures, using an arranged temperature approach considered in ABAQUS software. To validate the results obtained from this method, two numerical samples were tested: one was a butt-welding of two plates and the other a T-joint fillet. The technique of birth and death was applied, and a comparison with experimental results was done, demonstrating a good agreement between the heat generation rate approach as well as the experimental measurements.
Lei et al. [
45] studied the characteristics of residual stresses, using a finite element method, on a dissimilar welded pipe with the following type of steels: T92 and S30432. It was also studied in this paper, the effects of heat input, layer number, and groove shape on the residual stress distribution to find the approach to reduce these stresses. The numerical results demonstrated that the hoop and axial stress in HAZ of T92 steel side, of the dissimilar welded joint, had high-pitched gradients. Other results can be seen in
Table 8.
Dhinakaran et al. [
46] studied autogenous plasma arc welding of thin titanium alloy with 2 mm thickness, by numerical and experimental tests. The finite element code COMSOL multiphysics (COMSOL Inc., Stockholm, Sweden) was applied to perform non-linear instable heat transfer analysis using parabolic Gaussian heat source. Some variables such as thermal conductivity, density, and specific heat were used to improve the efficiency of the simulation process. The experimental tests were directed by varying the welding speed and current using Fronius plasma arc. The simulation and experimental results were in good agreement between them.
Belitzki et al. [
47], in his experience, developed a method to minimize distortions caused by laser beam welding. This method consisted of using a meta-model by means of an artificial neural network to predict local distortions in a complex structure, depending on several welding parameters. In this model, a genetic algorithm was used to find the suitable parameters of laser welding in this specific structure. A thermomechanical simulation was performed by SYSWELD software, whereby it was concluded that this method was reliable in identifying optimized welding parameters and predicting distortions.
Chukkan et al. [
48] developed a methodology to verify the state of residual stress in a butt-welded plate by using neutron diffraction for measurements. The measured residual stress was used to estimate the stress distribution, which was mapped by FEM on a weld plate. It was shown that precise residual stress field reconstruction was possible, in and around the area of measurement. It was concluded that the mapping method was a simple and computationally efficient technique for the prediction of the global residual stress field, especially for fracture and fatigue analysis.
Zhang et al. [
49] studied the prediction of distortions using a full-size element model, performed by ABAQUS in a 1/8 VV, which is a vacuum vessel highly used in China. To study the distortions, three different tungsten inert gas (TIG) welding sequences were simulated on this vessel. Through this paper, it was possible to verify that in three different welding sequences, the maximum distortion occurred on the shells near the transition structures. Since sequence 1 resulted in the minimum welding stress and the lowest distortion among all three, this should be the improved welding sequence to diminish distortions in this material.
Liu et al. [
50] studied the simulations and experiments regarding fatigue behavior of the RAFM (reduced activation martensitic/ferritic) steel, which is an important material for future fusion reactor blankets. The simulation behavior of TIG and electron beam welding (EBW), were carried out by using ANSYS software, using the same gradient load. The experimental results were studied to analyze the impact of fatigue resistance on RAFM steel, and the results indicate that the EB welding was stable under increasing stress.
Rathod et al. [
51] studied the influence of the choice of welding processes, and how they can impact the life performance of critical nuclear components such as reactor pressure vessels and steam generators. Welding joints at a thickness of 130 mm were welded in three different processes: narrow-gap gas tungsten arc (NG-GTAW), narrow-gap submerged arc (NG-SAW), and reduced pressure electron beam welding (RPEBW). Through this experience, it was possible to conclude the following:
EB welding resulted in the largest HAZ;
The level of butterfly distortion was highest for the GTAW joint 3.65°, followed by the SAW joint (1.87°), and the EB weld with the least value (0.08°);
Since it was noted a much higher weld heat input during the filling passes for GTAW, when compared to SAW, there was a bigger distortion;
Concerning maintaining weld quality, NG-GTAW was the most challenging.
Joshi et al. [
52] presented a paper about the numerical measurements of welding-induced distortions in a 4-lacing dragline cluster built in a workshop, performed with GMAW process. It was concluded that welding-induced distortions produced very little dimensional inaccuracies. The utmost deviation seemed to be in the second lacing (4 mm), which would result in reducing the strength and overall load-bearing ability of the joint. To eliminate this inaccuracy, a jig should be constructed.
Kobayashi et al. [
53] presented a paper about studying the distortions applied in a complicated structure such as a compressor impeller. This analysis and experiment results have revealed that it is possible to evaluate welding distortion with high accuracy in an impeller by using numerical analysis (heat source model and phase transformations). This study showed that there is an agreement between the numerical and experimental results, revealing that the simulation was accurate. It was possible to see that while radial-direction distortions were not so large, the following distortions displacement of cover height and displacement of discharge width, were.
Bonnaud et al. [
54] performed full 3D simulations in studying the start/stop in partial repair effects. The start and stop events have been simulated in 3D, and comparison, with 2D results, indicate a significant increase in weld residual stresses. These events are harmful to reliability since the extra thermal and mechanical loading increases the stress height, which can influence the crack depth or the initiation of failure.
Pasternak et al. [
55] presented a study in which a welded I-girder from two structural steel grades (S355 and S690) was submitted to a numerical simulation in order to verify residual welding stresses. Given the results shown in this paper, simplified models are still in need. Despite its reliability in stress results, and the fact that they are suitable for load-carrying calculation, it needs further studies in matters of civil engineering.
Yuewei et al. [
56] investigated the characteristics of welding fiber laser keyhole, using a 3D numerical simulation model and a Gaussian heat source model. Welding parameters under different process conditions were analyzed through numerical simulation. The validity of this model was confirmed with experimental results.
Derakshshan et al. [
57] studied the prediction of residual stresses and distortions in welding thin plate structures, using a 3D thermometallurgical-mechanical model. Three different laser-based welding processes were tested: autogenous laser welding (ALW), cold wire assisted laser welding (CWLAW), and hybrid laser arc welding (HLAW). These processes were compared with submerged arc welding (SAW) in software and experimental results. For this paper, SYSWELD software was used, in which deformation theories were applied to predict residual stresses and deformations. It was concluded that lower heat input can influence distortion values, which supports the theory that laser welding processes should substitute the arc welding method.
Balram et al. [
58] developed an investigation about residual stress in dissimilar TIG weldments of AISI 304 and Monel 400. For this work, a finite element model was created to predict temperature fields and residual stress distribution. For this FEM model, a coupled sequentially thermomechanical transient analysis was applied in ANSYS software. One of the conclusions was that the finite element 3D heat source model was an efficient method to predict with accuracy residual stresses in the weldments. The predicted results agreed with the experimental ones with a 5% error.
Lee et al. [
59] developed a simulation model considering both in-plane and out-of-plane distortions. This model was validated with case study analysis and the results demonstrated good agreement in predicting and diagnosing the in-plane variation.
Li et al. [
60] performed a finite element analysis to understand multi-layer rotating arc narrow gap MAG welding for medium steel plate. Temperature field was solved and analyzed in multi- layer rotating arc welding based on element birth and death technique. The simulation results were in good agreement with the experimental data, 1.5 mm of difference between them. Residual stress and deformations were calculated based on temperature fields in four welding conditions. This method is useful for microstructural and welding analysis.
D’Ostuni et al. [
61] performed a study in a dissimilar welding butt joint (titanium and aluminum), using a fiber laser welding method. 2D and 3D Gaussian heat source were used to study the thermal analysis of this welding process. The experimental fusion zone of the joint was compared with the numerical one. During the welding cycle, the actual temperature was registered and was validated by the numerical model. To calculate fusion zone’s dimension, the 2D model demonstrated better accuracy than the 3D. Although, the 3D heat source resented better results in the matter of welding pool and cooling rate simulation.
Casalino et al. [
62] studied the effects of Yb fiber laser welding method in a 2 mm thickness AA5754 and Ti6Al4V butt joints. This Yb fiber laser operated on the upper surface of the Ti sheet. To confirm experimental results, a FEM analysis using ANSYS code was performed to support the results from the experimental work. It was possible to verify that the numerical model was accurate, as well as, the thermal behavior and pool shape geometry.