Numerical evaluation of surface welding residual stress behavior under multiaxial mechanical loading and experimental validations

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Numerical Evaluation of Surface Welding Residual Stress Behaviour under Multiaxial Mechanical Loading and Experimental Validations Kimiya Hemmesi , Pierre Mallet , Majid Farajian PII: DOI: Reference:

S0020-7403(18)33196-5 https://doi.org/10.1016/j.ijmecsci.2019.105127 MS 105127

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International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

28 September 2018 1 September 2019 2 September 2019

Please cite this article as: Kimiya Hemmesi , Pierre Mallet , Majid Farajian , Numerical Evaluation of Surface Welding Residual Stress Behaviour under Multiaxial Mechanical Loading and Experimental Validations, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105127

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Highlights 

The relaxation/or redistribution of welding residual stresses under uniaxial and multiaxial mechanical loading conditions is studied by means of numerical analysis.



Advanced x-ray diffraction technique is used to validate the simulation results.



The influence of the applied hardening model on the accuracy of relaxation analyses is investigated.

Numerical Evaluation of Surface Welding Residual Stress Behaviour under Multiaxial Mechanical Loading and Experimental Validations Kimiya Hemmesi1*, Pierre Mallet2, Majid Farajian2 1

Institute of Computational Material Science, IAM CMS, KIT, Straße am Forum 7, 76131, Karlsruhe, Germany 2 Institute for Mechanics of Materials IWM, Wöhler str. 11, 79108, Freiburg, Germany *Corresponding author E-mail: [email protected]

Keywords: Welding residual stresses, Relaxation/redistribution, Tubular joints, Finite element analysis, x-ray diffraction.

Abstract Welding residual stresses have been always seen as a threat to the integrity of structures and components during service loadings. However residual stresses could partially or even totally relax under thermal or mechanical external loads if the sum of residual stresses and load stresses exceeds the material yield strength. If the stable portion of residual stresses after relaxation is still significant, care must be taken that residual stresses, similar to mean stresses, could shift the applied stress range. Accordingly, for a reliable structural integrity assessment an accurate insight into the stable residual stress field is essential. A three-dimensional FE model was implemented in software ABAQUS in order to study the relaxation and redistribution of welding residual stresses in a bead on tube weld under multiaxial mechanical loading. The predefined field of residual stress was initially calculated through a welding simulation in software SYSWELD. To verify the numerical simulations, a couple of tubular samples made of steel S355J2H were TIG welded and then mechanically loaded at different applied load levels. The surface residual stresses were measured by means of x-ray diffraction (XRD) in the as-welded condition and also after applying the mechanical loads for further comparisons. Pure torsion and multiaxial tension-torsion loading conditions in the form of quasi-static loading-unloading were investigated experimentally. The predicted welding residual stress relaxation conforms well the measurement results. The accordance between the numerical and experimental relaxation results confirmed the validity of the FE model. The validated model was then used to evaluate numerically the behavior of residual stresses under cyclic torsion loading with regard to the influence of the used material model. Based on the results, for such a material, the major relaxation occurs during the first cycle due to the monotonic yielding but further relaxation may take place in the upcoming cycles due to the accumulation of plastic strains.

1. Introduction and background The fatigue behavior of welded components could be significantly influenced by the existing welding residual stresses. Tensile residual stresses may assist to open the fatigue cracks during loading and therefore accelerate the crack growth rate. In contrast, compressive residual stresses due to their crack closure effect could reduce the rate of fatigue crack growth [1-10]. In this regard not only the initial magnitude and distribution of residual stress field, but also its stable state after relaxation and redistribution during the operation is essential for the accurate component design and the respective lifetime assessments. Depending on the process condition, monotonic or cyclic loading as well as the material nature, the relaxation pattern could differ. In case of ignoring service load induced relaxation, one can hardly judge about the capability of the available reliability assessment models. In fact knowing the real state of residual stress in the material could help create weight-efficient and cost-efficient structures. In this work, residual stress refers to type I or macro residual stress. Relaxation means the partial or total decay of residual stresses due to the transformation of elastic residual strains to plastic strains under the external loadings. Relaxation phenomenon could be divided into quasi-static (due to yielding), cyclic, and crack-induced relaxation. Quasi-static relaxation occurs due to macroscopic yielding during the first few cycles when the sum of residual stresses and load stresses exceeds the monotonic yield strength of the

material. However during cyclic loading even if the mentioned summation remains below the yield strength, microscopic plastic deformation in the microstructure level may cause further relaxation depending on the material microstructure [11]. This is related to the movement of dislocations which depends on the density of dislocations and microstructural barriers such as grain boundaries and precipitates [12, 13]. Accordingly type of the initial material as well as the weld induced microstructural changes are determining factors. In the final stages of loading, the rate of relaxation may increase due to the activation of another mechanism which is crack initiation and propagation. Even at room temperatures, due to the abundance of influencing parameters such as the complexity of the initial field of residual stress, load type, load amplitude and material property it is difficult to accurately determine the relaxation of welding residual stresses [14]. One way is to measure residual stresses at different stages of loading which is very expensive and time consuming and even sometimes inapplicable. Beside measurements, there are two common methods for the prediction of residual stress relaxation, namely the empirical [15-17] and numerical methods; the first method is easier for use but less capable in terms of considering all the influencing parameters but the latter has shown to be of a high potential in delivering accurate results [11, 14, 17]. In the context of empirical methods, by equating the residual stresses with mean stresses, the proposed mean stress relaxation equations have been also applied for the case of residual stress. In such relationships the deterioration of mean or residual stress is expressed as a function of number of cycles; for instance as a linear logarithmic function [18, 19]: ( )

( )

( )

( ) is the mean stress at Nth cycle and where N is the number of cycles, is the mean stress at the first cycle, A and M are constants which incorporate the influence of material yield strength, stress and strain amplitudes. It has been shown that such relationships are applicable only after the first cycle. Beside the logarithmic function, the relationship between the mean stress relaxation and number of cycles could be of the form of power function [20, 21]: ( )

( )

where r is the stress relaxation exponent which depends on the applied strain amplitude. As noted by Landgraf and Chernenkoff [21] the rate of relaxation increases with the increase of the strain amplitude: (

)

( )

where is the total strain range and is the threshold strain range required for the occurrence of relaxation which has been proposed to be determined according to the material Brinell hardness: [

(

)]

( )

It should be noted that in none of these models the stress ratio effect is included. Furthermore the mentioned models are not applicable for predicting the drastic quasi-static relaxation of residual stress in the early cycles. To overcome this problem Lindgren and Lepistö [22] proposed that the stress relaxation exponent can be divided into quasi-static and cyclic relaxation terms; the latter that defines the rate of cycle dependent relaxation, is independent of the mean stress, while the former depends on the mean stress since quasi-static relaxation normally occurs when the strain amplitude is high enough for yielding: ( ) On this basis if the strain amplitude is lower than the threshold no mean stress relaxation will occur. The cycle dependent mean stress relaxation will begin when the strain amplitude goes beyond the threshold. In this case if the strain amplitude is high enough for yielding, plastic deformation and consequently quasi-static relaxation will occur during the first cycles and the remaining mean stress will be subjected afterward to the cyclic relaxation. Finally, in the case of significant yielding quasi-static relaxation might happen alone without further cyclic relaxation. In terms of time-dependent stress-strain evolution due to the motion of dislocations under mechanical loading, the nature of residual stress relaxation is very close to the creep process [23]. On this basis Qian et al. [24] proposed a phenomenological model for the prediction of residual stress relaxation by treating the dislocations motion during cyclic loading similar to the creep process. The model exhibited good agreement with the experimental results for AISI 1008 and AISI 4140 steels at different applied stresses and load cycles. Nowadays, besides empirical models which are basically developed based on the interpolation or extrapolation of experimental residual stress relaxation results, attention is widely drawn to the finite element techniques hoping that

more influencing factors could be included in the prediction of residual stress relaxation. However the focus in most research works is dedicated to the relaxation of surface treatment induced residual stresses which have been intentionally applied to the surfaces of metallic components in order to improve the fatigue resistance [25-27]. In the context of welding residual stresses, Dattoma et al. [28] studied numerically the relaxation of welding residual stresses under cyclic mechanical loading in AISI 316 specimen using a bilinear isotropic hardening model. Based on their FE model the predicted relaxation was almost limited to the first cycle. But this observation requires to be verified experimentally. The combined isotropic-nonlinear kinematic hardening model proposed by Chaboche [29, 30] was used by Lee et al. [31] in an attempt to predict the relaxation of welding residual stresses by means of FEM in butt welds made of mild carbon steel under cyclic mechanical loading. Though the FE method developed in this study seems to predict effectively both quasi-static and cyclic residual stress relaxation, however the results are not supported experimentally. Similarly, Cho and Lee [32] developed a cyclic plasticity constitutive model to describe the real state of welding residual stress under cyclic mechanical loading in a girth weld made of super duplex stainless steel. A gradual decrease of residual stress during the first few cycles was predicted through the application of this model. Xie et al. [14] studied the weld residual stress relaxation under cyclic loading in a 316L stainless steel weld joint through a numerical simulation incorporating a proposed cyclic plasticity constitutive model. The accuracy of the constitutive model was examined experimentally by means of x-ray diffraction technique. Similarly in this study it was shown that the significant portion of relaxation occurs during the first few cycles. Particularly about 45-60% of the maximum residual stresses were released after the first cycle due to the quasi-static relaxation. Zhuang and Halford [15] employed the Chaboche isotropic-nonlinear kinematic hardening model for an FE model in order to follow the relaxation behavior of the cold-work induced residual stresses. A physics-based analytical model was then proposed in this study based on the degree of cold-work on one hand and the accurate material cyclic stress-strain response on the other hand. Barsoum [33] presented a 3D finite element model to study the relaxation of residual stresses in tubular welded joints under torsional fatigue loading. The initial field of residual stress was calculated through a sequentially coupled thermo-mechanical multi-pass weld simulation following by experimental validation. For further relaxation analysis, rate-independent isotropic bilinear hardening model was used. Under the applied load which is in the high cycle fatigue range, slight relaxation of residual stresses was predicated. Li et al. [34] proposed on the basis of 3D elastic-plastic FE analyses, a formula for the prediction of stable residual stress in the hot-spot of ship-shaped structures under variable amplitude cyclic loading as a function of initial residual stress and applied load. Summing up the FE based residual stress relaxation studies, it turns out that in most cases, the majority of residual stress relaxation was predicted to be in the first cycle or during the first few cycles, depending on the used material model. Nevertheless, there is still a need for considering more complex loading conditions together with more advanced experimental techniques to validate the simulation procedures. This study aims to combine the numerical and experimental methods to describe the behavior of welding residual stresses under multiaxial mechanical loading. Indeed the objective of this work is to understand and quantify the theory behind this phenomenon. A bead on tube weld made of structural steel S355J2H was chosen for the evaluations. The surface residual stresses in the as-welded condition had been initially determined using the x-ray technique together with the finite element method (FEM). The welding process was simulated by means of software SYSWELD 12.5 through a 3D thermal-mechanical analysis and the respective results including temperature history and residual stress field were calibrated with regard to the experimental results. After the as-welded samples were carefully examined, they were subjected to pure torsion and tension-torsion quasi-static loads at different stress levels. The surface residual stresses were measured again after each unloading using the x-ray method. Parallel to experimental measurements a 3D elastic-plastic FE model incorporating the nonlinear isotropic hardening model [35] was developed in ABAQUS 6.14 to follow numerically the behavior of welding residual stresses under mechanical loading. The model in relaxation analyses was completely the same as the one from welding simulation and the residual stress field calculated in SYSWELD was applied as a predefined field to the model for relaxation. The relaxation of welding residual stresses was then investigated by comparing the experimental and numerical results. Having a calibrated FE model, the relaxation of residual stresses under cyclic loading was then studied numerically. In this regard, the cyclic relaxations were examined in the light of the use of a proper hardening model. Based on the results, it can be concluded that finite element approach is a powerful tool for predicting the relaxation of welding residual stresses under different mechanical loading conditions. It was also observed that for the respective material in this study (S355J2H) the static relaxation mechanism which occurs after the first load cycle is predominant. This happens when the superposition of the virgin residual stresses and the load stresses exceeds the yield strength of the material. In such condition, Bauschinger effect and consequently the kinematic hardening model have minor effect on the accuracy of predicted relaxation. Nevertheless relaxation may take place in the upcoming cycles due to the accumulation of plastic strains depending on the loading condition.

2. Experimental work Two tubular samples made of seamless cold-drawn post heat-treated S355J2H were used in this work to study the welding residual stresses and their relaxation behavior under mechanical loading. For each of the loading cases; a) pure torsional and b) tension-torsional, one specimen was investigated separately; sample-1 and sample-2 respectively. The raw material had an outside diameter of 25 mm and a thickness of 5.6 mm. This material which is widely used in the production of engineering components and structures has a ferritic-pearlitic microstructure with an average hardness of 180 HV 10. The material proof and ultimate stresses are 400 and 700 MPa respectively.

2.1.

Welding the specimens

As illustrated in Fig.1, welded samples were produced by means of Tungsten Inert Gas (TIG) technique in the form of single bead on tube dummy welds [36]. The tubular specimens were machined out of the raw material in order to feature the final dimensions. In this way the outside diameter and thickness of the samples in the gage section were 21 mm and 3.6 mm respectively (Fig. 1). After machining and before welding, the specimens were heat treated at 600 °C under shielding gas for 30 minutes in order to release any machining induced residual stresses. In the next step, 5 mm-wide TIG weld beads were produced in the middle of the specimens by setting the welding voltage to 12 V, the current to 130 A and the welding speed to 30 cm/min. A filler material of the same type as the base material was used for the welding. The cross-sectional macrograph of the weld is shown in Fig. 2. During welding, S355J2H undergoes metallurgical phase transformations due to consecutive heating and cooling. As a result, a significant portion of the ferritic-pearlitic microstructure transforms into martensite and bainite in the heat affected zone (HAZ) and weld metal (WM) [37]. Therefore the material hardness is higher in these areas so that it even reaches 340 HV in the weld area [36].

Fig. 1 Single bead on tube dummy TIG weld and the weld specimen dimensions.

Fig. 2 Weld cross-section macrograph and different state of microstructure in the Weld Metal (WM), Heat Affected Zone (HAZ) and Base Material (BM).

2.2.

Measurement of residual stresses

The welding residual stresses were determined experimentally by means of diffraction technique. In the as-welded condition, x-ray (XRD) was used to measure the surface welding residual stresses namely the initial residual stress profiles along four longitudinal lines passing over the weld start point and the other three quadrants, Q1 to Q4 (Fig. 1). Details of the measurements and the respective results are given in Ref. [36, 38]. Another set of XRD measurements were conducted in order to evaluate the behavior of welding residual stresses under uniaxial and multiaxial mechanical loadings. It should be noted that all relaxation studies were conducted at Q3 quadrant which is the 180° location from the weld start point. As given by detail in Ref. [36] the relaxation study of „surface‟ residual stresses by means of XRD was performed for a few steps of quasi-static loading-unloading (5 steps for any types of loading). Pure Torsion and multiaxial tension-torsion loading conditions were considered for these studies. It is clear that the specimens being studied in this phase are those for which the initial field of residual stress had been determined earlier. Under pure torsion, sample-1 was gradually loaded and unloaded at five different nominal shear stress levels (100, 175, 261, 278.5 and 283 MPa). The XRD measurement was conducted after each load-unload cycle at a given level. As the next case study, the behavior of residual stresses was studied under multiaxial loading. For this purpose a 100 MPa nominal shear stress was combined with five different nominal tensile stress levels (203, 254, 305, 356 and 452 MPa) and applied to sample-2. Similar to the pure torsion case, the sample was removed from the testing machine after each load-unload cycle to investigate the changes in the residual stresses. The loading histories in both case-studies are given in Fig. 3. In this figure, „T-RSM‟ and „TT-RSM‟ terms stand for the intervals of „Residual Stress Measurement‟ on the samples under pure Torsion (sample-1) and multiaxial Tension-Torsion (sample-1) loadings respectively (RSM0 refers to the as-welded condition).

Fig. 3 History of the applied loads for the residual stress relaxation tests to sample-1 under Pure torsion (left) and sample-2 under multiaxial tension-torsion (right); RSM refers to the Residual Strain/Stress Measurement intervals.

Parallel to surface residual stress measurements, the evolution of surface strain field during loading was investigated using strain gauges. As shown in Fig. 4, a set of three stain gauges was applied on the surface of the samples at the weld area (DMS1), HAZ (DMS2) and the base material (DMS3). The goal was to investigate the evolution of the strain field under loading in different material areas. This way it might be possible to find out the relationship between the elastic and/or plastic behavior of the material and the relaxation of residual stresses. Moreover the strain gauge responses could be treated as an extra criterion in the numerical simulations for the validation of the used material models.

DMS2 DMS3

DMS1 Fig. 4 Arrangement of the strain gauges over the weld (DMS1), HAZ (DMS2) and base material (DMS3).

3. FE simulation of initial welding residual stress field and experimental calibration The numerical approach in this study is composed of two main steps. In the first step the exact field of welding residual stresses is to be determined through finite element simulation of the welding process. Details of the simulation procedure are given previously in Refs. [37] and [38]. Indeed the prerequisite for an exact residual stress relaxation study is to have a relatively precise initial residual stress field. According to the results of the previous works FE method could be used as a powerful tool for the prediction of welding residual stresses. Despite this fact, there are still ambiguities in the way towards the accurate simulation of a complex process such as weld. In the next step, the behavior of welding residual stresses under uniaxial and multiaxial mechanical loadings in terms of relaxation or redistribution will be investigated and results will be given later in Section 4. A three-dimensional FE model was developed for the simulation of the welding process in SYSWELD 8.5. In the thermal analysis a moving heat source of double-ellipsoid type proposed by Goldak et al. [39] was used to compute the temperature history. The respective continuous cooling transformation diagram was implemented to the model to calculate the temperature dependent metallurgical phase proportions. Structural steel S355J2H undergoes both diffusion and non-diffusion types of solid state phase transformations during cooling. Depending on the cooling rate at any material point in the weld and HAZ, martensite, bainite or ferrite may be produced [37]. Following the verified thermal simulation, a mechanical analysis was conducted decoupled from the thermal one in order to calculate the welding residual stresses. In the mechanical simulation the total strain increment (dεtot) is composed of the elastic strain increment (dεe), plastic strain increment (dεp), thermal strain increment (dεth) and phase transformation induced strain increment (dεtr): dεtot= dεe+dεp+dεth+dεtr

(6)

The first two terms in the above equation were incorporated into the simulation through the isotropic Hooke‟s law for the elastic term and the Von Mises yield function together with the isotropic hardening model for the plastic term. As shown previously in Ref. [37], the choice of the isotropic hardening is adequate for the simulation of welding process in the desired material S355J2H. All required material properties were defined as a function of temperature. The thermal strain increment was computed on the basis of the defined temperature-dependent thermal expansion coefficient. Finally the phase transformation induced strain increment incorporates into the model the volumetric changes and the transformation induced plastic deformations. The residual stress results from the welding simulation are given in Refs. [38] and [40]. Despite the good overall agreement between the measurement and calculation results, there remained still some discrepancies in the weld area and its immediate vicinity. Understanding the reason behind such a discrepancy requires further effort particularly for such a material which undergoes phase transformations and for such a highly constrained geometry. As the prerequisite for an exact residual stress relaxation analysis is to have a precise initial welding residual stress field, a MATLAB script was developed in order to redistribute the calculated residual stresses in the weld area so that the mentioned discrepancy is minimized. The modified residual stress field was then applied for the further relaxation assessments. Fig. 5 illustrates the comparison between x-ray measurements and modified finite element results of surface residual stresses. As can be seen in Fig. 5 the agreement between the measured and calculated residual stresses (after modification) on the specimen surface is very good. 500

EXP FEM-MOD

300

Hoop residual stress (MPa)

Axial Residual stress (MPa)

500

100

-100 -300 -500 -8

-6

-4

-2

0

2

4

Distance from center-line (mm)

6

8

EXP FEM-MOD

300 100 -100 -300 -500 -8

-6

-4

-2

0

2

4

6

8

Distance from center-line (mm)

Fig. 5 Measured (XRD) [36] and predicted (and modified) surface welding residual stress profiles in the as-welded condition.

4. Numerical analysis of welding residual stress behavior under mechanical loading The finite element analysis was used later to evaluate the behavior of residual stresses under mechanical loading condition. The FE analysis was carried out in ABAQUS 6.14-3 using a three-dimensional model same as the one

used for the welding simulation. Isoperimetric 8-nodes solid elements with reduced integration (C3D8R) were used for the analyses. The applied FE mesh whose quality is proven through a mesh sensitivity analysis is shown in Fig. 6. In the next step, the modified version of the calculated residual stress field from SYSWELD was initially applied to the model in ABAQUS as a predefined stress field. The mapped residual stress field is equilibrated (balanced) by defining a zero external load condition in ABAQUS (Fig. 6). After these primary steps, the main elastic-plastic simulation was established to study the relaxation of welding residual stresses under different mechanical load cases. The mechanical properties over the whole FE model was defined uniformly and equal to the properties of the HAZ, because the hot spots of the plastic deformation and residual stress redistribution are located mostly in the HAZ. Gleeble simulation was performed to reproduce the HAZ material for subsequent mechanical properties testing [41]. The respective material flow curve is given in Table 1. Von Mises yield criterion together with the nonlinear isotropic hardening model was used for the quasi-static load cases described in section 2.2. The classical yield surface function is: (

)

(7)

where σij and αij are stress and back-stress tensors respectively, σ0 is the size of the yield surface and f (σij - αij) is the equivalent von Mises stress as below: (

)

√ (

) (

)

(8)

where and are the deviatoric parts of the stress and back-stress tensors respectively. The associated plastic flow rule which defines the direction of plastic strain increment after yielding is described as below: (9) where is a nonnegative scalar plastic multiplier and is equal to the equivalent plastic strain increment is obtained by: ̅

√

̅ which

̅

(10) (11)

Fig. 6 Predefined field of weld residual stress from SYSWELD to ABAQUS; axial and hoop components. The given stress field is balanced after a self- equilibrating analysis.

Table 1 Flow stress of S355J2H - HAZ.

Plastic strain

0.0

0.003

0.01

0.03

0.05

0.1

0.3

0.5

0.8

True stress (MPa)

390

507.7

575.5

639.0

685.4

719.3

765.8

791.2

812.4

The behavior of the material under cyclic loading was described by the combined isotropic-kinematic hardening model proposed by Chaboche [29, 30]. The Chaboche model is able to incorporate into the analysis different phenomena such as Bauschinger effect, cyclic hardening, ratcheting and relaxation of mean stress. With the application of this model, the residual stress relaxation under cyclic loading was then investigated through a couple of pure numerical analyses. The cyclic behavior of the material was obtained based on the strain-controlled low cycle fatigue (LCF) testing of the Gleeble specimens with the strain ratio of -1. The combined isotropic-kinematic hardening model is composed of two components namely the isotropic term which defines the expansion of the yield surface in the stress space and the nonlinear kinematic term which describes the translation of the yield surface through the definition of the back-stress tensor. The evolution of the back-stress components in the nonlinear kinematic hardening rule for the isothermal and monophasic condition is written as follow: ̇

∑ [

(

) ̇̅

̇̅ ]

(12)

Ci is the initial linear kinematic hardening coefficient and γi is the recall term which defines the rate at which the linear kinematic hardening degrades as the plastic strain increases. γi actually introduces the respective nonlinearity to the model. Index i defines the number of C and γ pairs which means that the overall back-stress is composed of multiple back-stress components in order to improve the accuracy of the results. The ̇̅ term in the above equation describes the equivalent plastic strain rate. In order to define the kinematic hardening parameters the model could be fit either to the monotonic stress-strain curve or to the initial or stable stress-strain hysteresis loops. The isotropic hardening term which defines the size of the yield surface through σ0 is defined as below: (

̅

)

(13)

where is the yield stress at zero plastic strain and and b are material isotropic hardening parameters which must be calibrated according to the cyclic hardening material data obtained from the first few stress-strain hysteresis loops. The Chaboche hardening model parameters for the present study are given in Table 2. Table 2 Chaboche hardening model parameters for S355J2H - HAZ.

Yield stress (MPa) 390

150.0

b

C1

γ1

C2

γ2

C3

γ3

40.0

17000

1000

17000

2200

17000

10000

The capability of the numerical model in predicting the relaxation of residual stresses was primarily controlled according to the experimental results. Using the validated FE model, as mentioned earlier a series of additional analyses were conducted in a pure numerical way in order to study the influence of applied material hardening model on the relaxation results under cyclic loading condition.

5. Results and discussions After defining the realistic field of welding residual stress by means of numerical and experimental approaches, the behavior of residual stresses under mechanical loading would be the next concern. Residual stresses may be released partially or completely under static or cyclic mechanical loading due to the occurrence of plasticity. In case of cyclic loading, cyclic plasticity may also cause more relaxation during the first few cycles until the residual stress state reaches a stable condition. The accuracy in defining the stable state of stress is very important for the further life time assessments. Indeed the relaxation is associated with the movement of dislocations which can be seen from two sights, macro- and micro plasticity. Most studies including the current research are based on the macro scale analyses. In order to have more precise predictions under cyclic loading, consideration of the micro plasticity is highly recommended. However as far as the macro scale is concern, a robust numerical model may still lead to a relatively accurate estimation of residual stress relaxation depending on the used material model and constitutive equation. The relaxation or redistribution of residual stresses in the material may differ depending on the loading condition on one hand e.g. loading amplitude, loading path and multiaxiality, and static or cyclic material property on the other hand. As mentioned in section 1, in many cases the majority of residual stress relaxation may occur during the first cycle. This happens when the superposition of load stresses and residual stresses exceeds the yield point of the material. Further relaxation is associated with the cyclic yield strength.

It should be noted that most of the research works in this regard are limited to the relaxation of those residual stresses which are produced by the processes other than welding e.g. mechanical surface treatments. In the present work, after initial residual stress determination, in order to study the behavior of surface residual stresses, one specimen was further studied under pure torsion and the other under tension-torsion loading. All studies regarding the behavior of residual stresses under mechanical loading were performed along an axial line on the surface at 180º from the weld start point.

5.1. Quasi-static pure torsion loading As explained in section 2.2, the first specimen (sample-1) subjected to gradually increasing torsion load was being unloaded in order to measure the residual stresses after applying every specific shear stress level and before reloading up to the next level (T-RSM1 to T-RSM5). Fig. 7 and Fig. 8 shows the variation of welding residual stresses in the axial and hoop directions obtained from measurements in comparison with the predicted results under pure torsion loading condition. As can be seen, the experimental and numerical results agree well with each other. It seems that the classical failure criteria used in plasticity could describe the relaxation phenomenon as a consequence of local plastic deformations. On this basis, though the load stress is of shear type, once the von Mises equivalent stress as a function of loads stresses and residual stresses exceeds at some locations the tensile yield strength of the local material, the residual stresses start to vary at the respective area. According to Fig. 7, by increasing in steps the applied nominal shear stress the first considerable variation of the residual stress is observed after a nominal shear stress of 261 MPa is applied (T-RSM3). In this case the axial compressive residual stresses at the weld toe and in the HAZ start to relax. In the weld bead centerline where high tensile residual stresses are present no changes could be observed. By increasing the torque and thus applied nominal shear stress, the relaxation continues at the weld toe and its vicinity in the HAZ, while the tensile residual stresses in the weld bead remain unchanged. In the hoop direction as shown in Fig. 8 some variations in the residual stress profiles are observed but these changes are not as pronounced as the relaxation in the axial direction.

Fig. 7 Measured - XRD [36] (a) and predicted (b) surface welding residual stress relaxation in the axial directions under pure torsion loading (Sample-1).

Fig. 8 Measured - XRD [36] (a) and predicted (b) surface welding residual stress relaxation in the hoop directions under pure torsion loading (Sample-1).

As can be seen at higher load levels the amount of predicted relaxation in the axial direction is slightly lower than the experimental one. The question arises here if the material property on one hand and the material hardening model on the other hand are accurate enough. In order to obtain a sufficiently accurate material model or properties, the strain field in the material at different load levels was investigated by means of strain gauges. As explained in section 2.2 measurements were conducted using three strain gauges which have been applied to the weld bead, HAZ and the base metal. The obtained results in terms of the elastic and plastic strains at each load level are compared with the calculated ones as shown in Fig. 9. In this figure, patterned and solid colors indicate the elastic and plastic strains respectively. Meanwhile the blue color represents the measured results while the red color indicates the results calculated by FEM. „c‟ and „b‟ and „a‟ symbols represent the strain components in the axial, 45° and hoop directions. The results are shown in terms of micro-strains. As can be seen in Fig. 9 at lower load levels (100 MPa and 175 MPa) the material deforms purely elastic in the „b‟ direction (45°) and the level of deformation obtained from measurements and calculations are conforming very well. Since the applied load is pure torsion, the strains in the other directions („a‟ and „c‟) are insignificant. At higher load levels, the material in the weld bead remains more or less elastic due to a larger geometrical cross-section. In the HAZ and base material plastic deformation appears to some extent which explains the relaxation of residual stresses but as can be seen both elastic and plastic strains are over predicted. Apart from the possible measurement errors, it could be related to the use of the cyclic vs monotonic stress-strain curve in the material model. This will be explained with more details in section 5.3. As explained in section 4 the material property in the base material was defined same as in the HAZ which could also influence the precision of the results.

Fig. 9 Comparison between the calculated elastic and plastic load strains and the strain gauge responses on the specimen surface at the weld bead, HAZ and base material under pure torsion; vertical axes represent the micro-strains. ‘c’ and ‘b’ and ‘a’ directions represent the strain components in the axial, 45° and hoop directions.

5.2. Quasi-static multiaxial loading In another experiment the behavior of the welding residual stresses under multiaxial loading was studied. Since in the previous experiment a pure nominal shear stress of up to 175 MPa did not lead to relaxation, it was aimed here to investigate the influence of the combination of 100 MPa nominal shear stress with gradually increasing tensile stresses on the residual stress field. Here the sample (sample-2) was kept under the same torsion loading during all loading sessions which induced a nominal shear stress of 100 MPa. However the tensile load level was increased

step by step at each loading session. After each tension-torsion loading and unloading session sample-2 was removed from the testing machine and the changes in the residual stress were investigated as shown in Fig. 3 (TTRSM1 to TT-RSM5). The comparison between the predicted and measured residual stress relaxation results in the axial and hoop directions under tension-torsion loading is shown in Fig. 10 and Fig. 11. The agreement between the measurements and calculations is reasonably good. Again same as under the pure torsion condition the amount of relaxation is slightly under estimated using the FEM. But anyhow, it seems that even for such a complicated multiaxial loading condition, the assumed yielding criterion is valid. It is observed that increasing the applied tensile stress combined with the applied torsion stress does not induce plastic deformation in the weld and its vicinity. So the residual stresses in both axial and hoop direction show a high level of stability until the applied tensile stress reaches the yield strength of the material i.e. 450 MPa (TT-RSM5). At this level of applied tensile load stress the compressive residual stresses at the weld toe and HAZ are almost eliminated. In the weld bead again no considerable changes are observed. As can be seen in Fig. 10 there are some discrepancies between the measurement and predicted results at the tensile stress level of 452 MPa (TT-RSM5). Indeed based on the XRD measurements the compressive residual stresses in the HAZ not only eliminate but also turn into tensile stresses to some extent. In the meantime the relaxation in the weld area is almost zero. This is while according to the simulation results the residual stresses in the HAZ still tend to be compressive even after relaxation. Moreover a slight relaxation (about 100 MPa) was predicted in the weld area. It is possible that the initial simplification in the definition of the material property has led to this discrepancy. As mentioned in section 4, for the simulation, the mechanical properties of weld metal, HAZ and base material have been set equal to those of the HAZ. In reality, the weld metal is slightly harder and the base metal is softer than the HAZ. Therefore it is expected to have an overestimation of the relaxation in the weld metal and an underestimation in the base metal in the FEM results. Another potential reason for such a discrepancy could be related to the defined plasticity model. This item will be discussed with details in section 5.3.

Fig. 10 Measured - XRD [36] (a) and predicted (b) surface welding residual stress relaxation in the axial directions under tension-torsion loading (Constant Nominal shear stress = 100 MPa) (Sample-2).

Fig. 11 Measured - XRD [36] (a) and predicted (b) surface welding residual stress relaxation in the hoop directions under tension-torsion loading (Constant Nominal shear stress = 100 MPa) (Sample-2).

To better track the source of discrepancy, strain measurements by means of strain gauges were performed similarly as explained for the pure torsion and results are given in Fig. 12. Unfortunately the strain gauge responses failed at the first and the last axial stress levels (203 MPa and 452 MPa) due to technical problems. As can be seen for the three remaining stress levels the material condition remains elastic which describes the zero residual stress relaxation at these levels. It is clear that the predicted elastic strains agree pretty well the measured responses.

Fig. 12 Comparison between the calculated elastic and plastic load strains and the strain gauge responses on the specimen surface under tension-torsion; vertical axes represent the micro-strains. ‘c’ and ‘b’ and ‘a’ directions represent the strain components in the axial, 45° and hoop directions. The strain gauges responses were failed for the first and the last axial stress levels 203 MPa and 452 MPa.

Comparing the relaxation results after every steps of loading (as shown in Fig. 13) between the pure torsion and tension-torsion loading conditions reveals that the relaxation of different components of residual stress is very sensitive to the direction of applied stresses [42]; it means that under different loading conditions, even if the equivalent stress states are similar (e.g. Step-4 for the weld and Step-5 for the HAZ), the residual stress components which are parallel to the applied stress amplitude tend to relax more drastically. Besides, the load sequence may also affect the relaxation behavior; indeed the gradual relaxation during the previous load-steps could affect significantly the level of remaining residual stresses after a specific load-session.

Fig. 13 Von-Mises stress history in the weld (left) and HAZ (right) under pure torsion (T) and tension-torsion (TT) loading.

5.3. Cyclic pure torsion loading As explained in section 4, isotropic hardening model was used preliminarily for the residual stress relaxation analyses under uniaxial and multiaxial quasi-static loading conditions. The validity of the model was investigated through a wide range of experimental measurements. The calculated relaxation results were conforming well with the experiments. Having this validated FE model, in this section the relaxation behavior under cyclic pure torsion loading with different loading levels is investigated numerically by focusing on the influence of applied material hardening model for the case of cyclic loading. For this purpose the FE model was loaded for few cycles under the following loading conditions: 1234-

Asymmetric torque-controlled (Loading-unloading) Asymmetric rotation-controlled Symmetric torque-controlled (Fully reversing load) Symmetric rotation-controlled

All calculations were conducted twice using isotropic and Chaboche combined isotropic-kinematic [28, 29] hardening models. The combined hardening parameters were defined by fitting the model to the cyclic test data in terms of the stable stress-strain hysteresis (Table 2) [37]. Monotonic tensile test data were used for the case of isotropic hardening model (Table 1). The relaxation behavior of axial residual stresses at point A (in the HAZ) and the respective material stress-strain response under loading are shown in Fig. 14 and Fig. 15. As shown in these figures, point A is located on the surface of the model and 5 mm away from the weld center line at Q3 (180° from the weld start point). As shown in Fig. 14 (a-1), under torque-controlled loading-unloading using both isotropic and combined hardening models the relaxation of residual stresses at every load level is limited to the first cycle. It means that the plastic deformation occurs only during the first cycle when the summation of load stresses and residual stresses exceeds the material yield strength. For the subsequent cycles the material cyclic behavior turns into an elastic shake down with no more cyclic plasticity. In fact the superposition of remaining residual stresses and load stresses does not lead to any new plastic deformation, further no plastic deformation occurs in the opposite direction as a result of biased stress-strain state. Since the proportion of isotropic hardening of S355J2H is dominant [37], ratcheting phenomenon did not occur in this case. As can be seen in Fig. 14 (a-1) the amount of relaxation at lower load levels is similar for both isotropic and combined hardening cases. At higher load levels the isotropic hardening model leads to a higher residual stress relaxation but this is not related to the characteristics of the hardening model. Indeed the respective difference in the results is only the matter of fitting choice. As mentioned the combined hardening model can be fitted to monotonic or cyclic material stress-strain data in different ways. In this case the stable stress-strain hysteresis loop was chosen for fitting which causes that the initial monotonic stressstrain response does not match perfectly the monotonic axial test data. As the torque-controlled condition is closer to the stress-controlled condition therefore a slight difference in the definition of monotonic stress-strain curve causes significantly different plastic strains at a certain stress level (Fig. 14 (a-2)). This effect is less pronounced in the Asymmetric rotation controlled condition which is closer to the strain-controlled condition as illustrated in Fig. 14 (b-2). This figure shows the material response at point A for the first three load levels shown in Fig. 14 (b-1). According to Fig. 14 (b-1), both isotropic and combined hardening models lead to relatively similar results in predicting the residual stress relaxation at every load level. As can be seen, in this loading condition at lower strains relaxation occurs only after the first cycle using both isotropic and combined hardening models. At higher load levels some slight cycle-dependent relaxation appears due to limited cyclic plasticity. According to Fig. 15, for the symmetric loading conditions the relaxation results based on both isotropic and combined hardening models are relatively the same. For these loading conditions, cyclic relaxation occurs at higher stress/or strain amplitudes due to the cyclic deformation. The stress-strain hysteresis loops for both torque-controlled and rotation-controlled conditions and for the third load level is shown in Fig. 15 (a-2) and Fig. 15 (b-2). As can be seen, for these loading conditions cyclic plastic deformation occurs during a few cycles. It can be concluded that the relaxation does not occur necessarily only after the first cycle. It depends actually on the load ratio and amplitude. But this cyclic relaxation is usually limited to the first few cycles and once the cyclic stress-strain response reaches its stable condition no more relaxation will appear. Though the applied stress is constant, the superimposed stress and consequently the plastic strain will be decreasing with the increase of cycle number. This happens indeed due to the gradual decrease of residual stress. Therefore residual stress relaxation depends on the initial condition of the material namely the material yield strength and the initial field of residual stresses on one hand, and on the applied loading condition including the stress/or strain amplitude and ratio as well as the number of cycles on the other hand.

Fig. 14 The relaxation behavior of axial residual stresses at point A and the respective material stress-strain response under (a) Asymmetric Torque-controlled (Loading-unloading) and (b) Asymmetric Rotation-controlled loadings.

Fig. 15 The relaxation behavior of axial residual stresses at point A and the respective material stress-strain response under (a) Symmetric Torque-controlled (Fully reversing load) and (b) Symmetric Rotation-controlled loadings.

Based on the results in this section it can be concluded that the use of combined hardening model does not influence that much the relaxation of residual stresses in the studied material (SS355J2H) because the fraction of isotropic hardening is dominant for this material. The significant difference due to the use of combined hardening model for the loading-unloading condition is related to the accuracy of the monotonic material response.

6. Conclusions The relaxation behavior of welding residual stresses under uniaxial and multiaxial loading conditions was studied numerically by means FEM. The validity of numerical investigations of surface residual stress relaxation was controlled according to the previously done x-ray diffraction residual stress measurements. The initial field of residual stress in the as-welded condition was determined by means of diffraction technique and FE simulations prior to relaxation. The whole experimental work was conducted in the quasi-static loading condition. The comparison between the relaxation results from measurements and FE calculations revealed very good agreement. In the meantime, the behavior of residual stresses under cyclic loading condition was studied using a purely numerical approach. Finally the influence of the applied hardening model on the accuracy of relaxation analyses was investigated. The following conclusions can be made out of the results in this study: 

Based on the observations and calculation, it was shown that The FE method can predict very well the relaxation behavior of welding residual stresses under uniaxial and multiaxial loading conditions.



The mechanisms of residual stress relaxation depend on the characteristics of the material therefore use of a proper constitutive model with proper material properties in the FE simulations would significantly increase the validity of predictions.



The von Mises yield criterion together with the local monotonic yield strength of the material has proven to be capable of describing the relaxation phenomenon as a consequence of local plastic deformation.



Under similar applied equivalent stresses, depending on the loading direction on one hand, and the loading sequence on the other hand, the remaining residual stresses might be different.



For the studied material S355J2H, in the HAZ and under cyclic torsion loading, use of a combined isotropic kinematic hardening model does not bring any significant advantage over an isotropic one (particularly under symmetric loading conditions).



In asymmetric loading conditions particularly the torque-controlled one, though isotropic and combined hardening models led to different relaxation results, this does not have any physical background and is only due to the difference in the model parameters fitting approaches.



The major relaxation occurs during the first cycle due to the monotonic yielding. Further relaxation may take place in the upcoming cycles due to the accumulation of plastic strains.



Occurrence of cyclic relaxation besides material mechanical behavior depends strongly on the loading conditions e.g. the load amplitude and load ratio; in this study for asymmetric loading conditions the material experienced elastic shake down after the first cycle which does not lead to any further cyclic relaxation. Since the isotropic proportion in the hardening behavior of S355J2H is dominant, therefore ratcheting phenomenon did not occur in these cases. In contrast at symmetric loading conditions some cyclic relaxation occurred at higher strain amplitudes due to the cyclic plasticity.

Acknowledgment The paper was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft - DFG) as part of the project “Numerical description of the behavior of welding residual stress field under multiaxial mechanical loading.” The authors would like to thank the DFG for its support.

Data availability All data generated or analyzed during this study are included in this published article.

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Graphical abstract