Significance of Finite Element Models and Solid-State Phase Transformation on the Evaluation of Weld Induced Residual Stresses

Abstract

In the present study, different finite element (FE) models were prepared to investigate weld induced residual stresses in thick multi-pass butt welded joint of SA516 Gr. 70 plates. Both 3D and 2D full geometry models and their axisymmetric half models were taken into consideration. The competence of these FE models on the accuracy of predicting residual stress distribution across the weld cross-section was investigated by comparing it with the experimental results. Blind hole drilling technique and deep hole drilling technique were employed to evaluate the surface and through-thickness residual stress distributions, respectively. In addition, the change in volume and yield strength of weld material due to austenitic phase transformation was also incorporated in the material modeling to observe the effect of solid-state phase transformation (SSPT) on the evaluation of residual stresses. Computed residual stresses obtained from different FE models indicate that the 3D FE models procured the best accuracy compared with the experimental results. On the other hand, 2D models can save a significant amount of computational time with reasonable accuracy. Incorporation of SSPT in the 3D FE full model exhibited a better agreement of predicted results with the experimental measurements.

Graphic Abstract

1 Introduction

During welding, residual stresses get induced in components due to excessive localized heating causing non-uniform heat distribution following steep temperature gradients. Moreover, during the heating and solidification process, the phase transformation of the subjected material can significantly influence the evolution of residual stresses [1]. The presence of locked in residual stresses makes the joint prone to intergranular stress corrosion cracking, fatigue failure, and brittle fracture [2,3,4]. So, to avoid failure of components under loading conditions, the magnitude and distribution of residual stresses in welded joints should be taken into consideration [5]. Finite element (FE) modeling of welded joints has become a widespread practice alongside experimental techniques to evaluate induced residual stresses. Numerical modeling of welded joints emerged in the early 1970s with limited computational competency [6]. However, over the past decades, with technologically advanced computers, remarkable improvements have been seen in FE modeling in terms of accuracy and efficiency. In the course of time, several FE models have been developed to study the thermomechanical behavior of welded joints. Nevertheless, accurate modeling of temperature-dependent material properties has been carried out to investigate the thermo-metallurgical behavior. Dean Deng investigated the effect of weld bead deposition sequence on the residual stress distribution in a J groove weld of tube block made of austenitic stainless steel based on both the FE modeling and experimental approach [7]. Lee et al. studied the effect of preheating the base material and welding direction on the magnitude of residual stresses using 3D sequential thermal elastic–plastic FE modeling and verified the simulation with test results [8]. Chen et al. investigated residual stresses induced in a thin steel plate using numerical modeling and validated it with the experimental results exhibited by the X-ray diffraction technique [9]. Residual stress distribution through the thickness of thicker components was studied by Jiang et al. using 2-dimensional FE modeling and neutron diffraction technique [10]. Farahani et al. presented a residual stress prediction model using 3-dimensional FE modeling in medium carbon steel weld incorporating solid-state phase transformation and post-weld heat treatment (PWHT) [11].

The numerical modeling of welded joints is commonly performed as a sequentially coupled analysis. In the first step, the thermal analysis is done where due to the welding heat input, nodal transient temperature history is generated. Then, the temperature history is applied to a similar structural model as the thermal loads [12]. Efficient modeling of the welded joints involves meticulous interaction of the model with boundary conditions and moving welding heat source with continuous deposition of filler material following temperature-dependent material properties. Recently, the implementation of 3D FE models is being emphasized over 2D models with significant improvement in computer hardware [13,14,15]. However, from the past literatures, it has been seen that the 2D models are also comparatively efficient in predicting weld induced residual stresses [16,17,18,19]. In 2D models, a single plane of the weld cross-section is modeled normal to the travel path of the moving heat source considering plane strain condition. Here the deformation in the longitudinal direction is not taken into consideration, which plays as the major drawback of the 2D models. So, to get a better view of welding residual stresses, efforts were made to develop 3D weld models with moving heat source. But, with increasing number of elements in 3D models, the computational expense also increases. The total number of elements used in the numerical model directly attributes to the time required for the solver to run the simulation. In that case, 2D and axisymmetric models with a lesser number of elements require lower computational time. Li et al. developed 3D FE models of P92 steel welds and meshed with around 75,900 cubic elements to investigate the residual stress distribution [20]. Yaghi et al. also studied residual stress fields on P92 steel weld employing a 2D numerical model comprising only 2919 elements [21].

With substantial progress in FE modeling of welded joints, few potential aspects of FE models still limit its proficiency. As an example, the effect of microstructural changes associated with the heating and cooling cycle of welding is challenging to integrate into the numerical model. In the case of high strength carbon steel welds, it is suggested that the effect of solid-state phase transformation (SSPT) of the weld material on the evolution of residual stresses should be incorporated [11]. SSPT has a substantial impact on the physical and mechanical properties of the weld material, which may induce volumetric changes in the weld and its adjacent region. The changes in volume and yield strength of the weld material due to SSPT have been emphasized in FE models by many researchers [22].

In the present study, the competence of various FE models to evaluate weld induced residual stresses has been investigated. Two 3D and two 2D models, each having one axisymmetric half and one full geometry models, were prepared for a thick multipass butt welded joint of SA516 Gr. 70 ferritic steel plates. Blind hole drilling and deep hole drilling technique were performed to evaluate the surface and through-thickness residual stress fields, respectively. Residual stresses evaluated from the numerical models were compared with the experimental results to find out the most efficient model. Finally, the significance of SSPT on the weld induced residual stresses were observed by taking volume and weld material yield strength change into account.

2 Experimental Details

ASME SA516 Gr. 70 material was used as the base material for the present investigation. The length, width, and thickness of the plates are 300, 110, and 25 mm. V type groove geometry was chosen for the butt configuration, as shown in Fig. 1. The gas metal arc welding process was employed with a root gap of 1.5 mm. ER70S6 was selected as the filler material of 1.2 mm in diameter. The chemical compositions of the base material and filler material are given in Table 1. One directional welding sequence was maintained for all the passes. It took total 10 number of weld passes, excluding the root pass, to fill the groove. The parameters opted for the welding procedure are given in Table 2. After welding, the weld reinforcement was removed at the measurement location to prepare the weld sample for experimental measurements.

Fig. 1

Weld specimen and groove geometry (all dimensions are in mm)

Table 1 Chemical compositions of SA516 Gr.70 and ER70S6

Table 2 Welding parameters

Experimentally residual stress measurements were done using two semi destructive methods, popularly known as the blind hole drilling (BHD) method and deep hole drilling (DHD) technique. BHD method is widely used for the purpose of surface residual stress evaluation. This is a standardized practice that is well known for its accuracy and reliability. The BHD method comprises minor damage made on the specimen in terms of a small drilled hole, which is mostly bearable or easily repairable. Here, after material removal, the hole diameter changes due to elastic unloading, and consequently, the surrounding material gets deformed associated with the amount of strain energy released. With the help of strain gauges, the surface deformation of the surrounding material is quantified, and the resultant residual stresses are evaluated. In the present investigation, ASTM standard, E837-13a, was followed using K-RY61-1.5/120R prewired strain gauge rosettes (center hole diameter-1.8 mm) for the BHD measurements. Figure 2 shows the locations of BHD measurement and the corresponding relative distance between the installed strain gauges.

Fig. 2

Locations of residual stress measurements

DHD technique was used for the purpose of through-thickness residual stress evaluation. DHD was performed at the weld zone and heat-affected zone (HAZ), where previously BHD measurements were carried out (Fig. 2). This technique was initially studied by Zhdanov and Gonchar [23], Beaney [24], Jesensky, and Vargova [25]. Zhdanov and Gonchar applied the DHD technique on a steel weld specimen, where they trepanned out a 40 mm diameter core with an 8 mm coaxial inner hole [23]. Beaney had introduced the electro-chemical machining process for the trepanning operation to minimize plasticity error due to machining operation. A reference hole of 3.175 mm diameter was made using a gun drill to achieve a comparatively steady and straight reference hole. Two beams equipped together with strain gauges were used to measure the reference hole diameter [24]. Procter and Beaney had improved the procedure by using non contacting capacitance gauges to measure the inner hole diameter instead of using strain gauged beams [26]. Jesensky and Vargova placed strain gauges inside the reference hole and to the faces of the hole, followed by a 32 mm core removal [25]. Smith et al. measured the reference hole diameter using an air probe in a 6 mm gun drilled hole, and electro-discharge machining (EDM) process was utilized for the trepanning operation [27]. Mahmoudi et al. proposed a modified DHD technique to eliminate the issue of plasticity surrounding the hole during trepanning. It was also well validated by experimenting on a quenched cylinder of steel, and the observations were found to be in a reasonably close match with neutron diffraction measurements [28]. Hossain et al. successfully validated the over coring DHD process with a simulation of a quenched solid cylinder and a welded pipe [29].

DHD technique comprises four necessary steps [30, 31], as shown in Fig. 3. In the very first step, a reference hole of 5 mm in diameter was drilled through the thickness at the measurement location in a CNC milling machine. In the next step, the diameter $(d(\theta ))$ of the reference hole was measured using an air probe at three different angles (0°, 45°, 90°) in each step following 1 mm of interval through the thickness. The annular measurement of the hole can be taken in more number of angular directions $(\theta = \theta_{1} ,\theta_{2} ,\theta_{3} \ldots \theta_{N} )$ for better results. In the third step, the reference hole was trepanned (inner diameter 15 mm and outer diameter 25 mm) coaxially through the thickness up to 23 mm to relax the locked-in strains in the core column. In the final step, after trepanning, the inner diameter $d^{\prime}(\theta )$ of the hole was remeasured at the same previous locations and angular directions. The diametral deformation $(\Delta d(\theta ) = d^{\prime}(\theta ) - d(\theta ))$ of the reference hole corresponding to different depth intervals were then used to calculate the in-plane residual stress magnitudes. The normalized strain $\varepsilon ({\uptheta })$ associated with the $\sigma_{x }$, $\sigma_{y }$, $\sigma_{xy }$ and $\sigma_{z }$ residual stresses can be evaluated following the below equation,

$$\varepsilon (\theta ) = \frac{\Delta d(\theta )}{{d(\theta )}} = - \frac{1}{E} \left[ { \sigma_{x } f(\theta ) + \sigma_{y } g(\theta ) + \sigma_{xy } h(\theta ) - \mu \sigma_{z } } \right]$$

(1)

where $f\left( \theta \right) = 1 + 2{\text{cos}}2\theta$, $g\left( \theta \right) = 1 - 2{\text{cos}}2\theta$, $h\left( \theta \right) = 4{\text{sin}}2\theta$, $E$ is young’s modulus, and $\mu$ is poisons ratio [28]. In the above equation, the x-axis lies with the $\theta = 0^\circ$ angular direction. The stress magnitude $\sigma_{z }$ was assumed to be zero incorporating plane stress condition for the present investigation. Equation 1 can be expressed in a matrix form for N number of angular diametral measurements as below,

$${\overline{\mathbf{u}}} = - \frac{1}{E}{\mathbf{M}} \times \sigma$$

(2)

where

$${\overline{\mathbf{u}}} = \left[ {\begin{array}{*{20}c} {\overline{u}_{r} \left( {\theta_{1} } \right)} \\ \vdots \\ {\overline{u}_{r} \left( {\theta_{i} } \right)} \\ \vdots \\ {\overline{u}_{r} \left( {\theta_{N} } \right)} \\ \end{array} } \right],\quad {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {f(\theta_{1} )} & {g(\theta_{1} )} & {h(\theta_{1} )} \\ \vdots & \vdots & \vdots \\ {f(\theta_{{\varvec{i}}} )} & {g(\theta_{{\varvec{i}}} )} & {h(\theta_{{\varvec{i}}} )} \\ \vdots & \vdots & \vdots \\ {f(\theta_{N} )} & {g(\theta_{N} )} & {h(\theta_{N} )} \\ \end{array} } \right]\;{\text{and}}\;\sigma = \left[ {\begin{array}{*{20}c} {\sigma_{x} } \\ {\sigma_{y} } \\ {\sigma_{xy} } \\ \end{array} } \right]$$

Fig. 3

Schematic representation of deep hole drilling technique

Lastly, the unidentified magnitude of residual stresses can be evaluated as,

$$\hat{\sigma } = - E{\mathbf{M}}^{*} \times {\overline{\mathbf{u}}}$$

(3)

where

$${\mathbf{M}}^{\user2{*}} = \left( {{\mathbf{M}}^{{\mathbf{T}}} \times {\varvec{M}}} \right){ }^{ - 1} \times {\mathbf{M}}^{{\mathbf{T}}}$$

(4)

Here, $\hat{\sigma }$ is the most favourable stress vector that best fits the annular deformation and ${\mathbf{M}}^{\user2{*}}$ is the pseudo inverse of matrix ${\mathbf{M}}$ [32].

3 Finite element analysis

Total four 3D and 2D thermo elastic–plastic FE models, along with their axisymmetric half models, were prepared to predict the weld induced residual stresses in a 25 mm thick SA516 Gr. 70 butt welded joint. Abaqus code was used for numerical modeling. The simulations were conducted on a computer with a 3.30 GHz CPU and a 32 GB memory. Directly coupled computational models were employed to analyze the thermomechanical behavior of the welded joint. By the application of coupled temperature displacement elements, the thermal and mechanical analyses were performed simultaneously in a single solution. The dimension of the model and weld passes were approximately kept similar to the actual weld specimen following the weld macrograph. In the present study, at first, a 3D model was developed, and following the same dimensions, a 3D axisymmetric half model was prepared. Similarly, 2D and 2D axisymmetric models were also prepared, keeping the same weld geometry as of 3D models. The FE models with their weld pass sequence are shown in Fig. 4. In order to save total computational time, finer meshing grids were assigned to the weld zone, and biased meshing order was employed for the base material. Here, the mesh density gradually becomes coarser away from the weld fusion boundary. Figure 5a, b shows the meshing grids of 3D models. Hexahedral mesh type with C3D8T and CPE4T coupled temperature displacement elements were used for the 3D and 2D models, respectively. The total number of elements and nodes employed in different FE models are given in Table 3. The sequential weld bead deposition was achieved by the application of model change option following element birth and death technique.

Fig. 4

Finite element models a 3D full, b 2D full, c 3D axisymmetric (half), d 2D axisymmetric (half)

Fig. 5

Meshed finite element models a 3D full geometry and b 3D axisymmetric

Table 3 Simulation cases

3.1 Thermal Analysis

Transient nonlinear heat transfer of the welding thermal cycles was achieved from the moving heat source along the weld torch path. The governing equation behind the transient thermal history can be expressed as below,

$$\rho c\frac{\partial T}{{\partial t}}\left( {x,y,z,t} \right) = - \nabla \vec{q}\left( {x,y,z,t} \right) + Q\left( {x,y,z,t} \right)$$

(5)

where $\rho$ is the material density (g/mm³), $c$ is specific heat (J/(g °C)), $T$ is the instantaneous temperature (°C), $t$ is time, $\vec{q}$ is the heat flux vector (W/mm²), $Q$ is the rate of internal heat generation (W/mm³), $x,y,$ and $z$ are local coordinates and $\nabla$ is spatial gradient operator. The standard Fourier’s law of heat conduction was taken into consideration as:

$$\vec{q} = - k\nabla T$$

(6)

where $k$ is the temperature-dependent thermal conductivity of the material, which is assumed to be isotropic in nature.

The accuracy of the transient temperature history obtained from the thermal analysis closely depends upon the chosen heat source model. In the present analysis for the 3D models, a volumetric heat source was applied with separately defined heat inputs under the double ellipsoidal local coordinate system suggested by Goldak et al. [33, 34]. The moving heat source was employed using a Fortran based user subroutine linked with the Abaqus code. The total heat input is divided into two parts, one for the front portion from the eye of the arc and one for the rear portion. It is expressed as,

for the front,

$$Q(x,y,z) = \frac{{6\sqrt 3 f_{f} Q_{w} }}{{abc_{f} \pi \sqrt \pi }} \cdot e^{{ - 3\frac{{x^{2} }}{{a^{2} }}}} \cdot e^{{ - 3\frac{{y^{2} }}{{b^{2} }}}} \cdot e^{{ - 3\frac{{z^{2} }}{{c_{f}^{2} }}}}$$

(7)

for the rear-end,

$$Q(x,y,z) = \frac{{6\sqrt 3 f_{r} Q_{w} }}{{abc_{r} \pi \sqrt \pi }} \cdot e^{{ - 3\frac{{x^{2} }}{{a^{2} }}}} \cdot e^{{ - 3\frac{{y^{2} }}{{b^{2} }}}} \cdot e^{{ - 3\frac{{z^{2} }}{{c_{r}^{2} }}}}$$

(8)

where $Q$ is the amount of heat found instantaneously at any location along the weld path at time $t$ in the reference system $Q = Q\left( {x,y,z} \right)$ as a function of three-dimensional local coordinates $x, y,$ and $z$. $Q_{w}$ is the welding heat input estimated from the input current and voltage parameters. $f_{f}$ and $f_{r}$ denotes the fractions of heat present in the front and rear parts of the heat source. It should be noted that $f_{f} + f_{r} = 2$ and in the present analysis, the values of $f_{f}$ and $f_{r}$ were assumed as 0.6 and 1.4, respectively [11]. The shape parameters of the heat source were taken as $a = 6$, $b = 7$, $c_{f} = 6$ and $c_{r} = 4 \times c_{f} = 24$ [34]. Arc efficiency was assumed as 80% [35]. The shape parameters were chosen in a way that it produces a proper molten weld pool [9].

The 2D models were developed considering plane strain condition. Constant distributed volumetric heat flux $DFLUX$ was considered for the 2Dmodels. The volumetric heat flux was applied to each weld bead after the activation step of corresponding elements. The magnitude of the heat flux depends upon the welding input parameters; it can be expressed as [36],

$$DFLUX = \frac{U.I.\eta }{{V_{p} }}$$

(9)

where $U$ is voltage, $I$ is current, $\eta$ is arc efficiency, and $V_{p}$ is the volume of corresponding weld bead. The heat flux was applied following an amplitude. In 2D models, a single plane of weld cross-section was taken into consideration from the middle portion. The required time for the heat source to pass through a single plane was noted, and in the amplitude, the heat flux was activated for that particular span of time. The total time duration of the amplitude was kept as a summation of the time required for the electrode to complete one single pass and engaged cooling time between two successive welds. The relationship between welding heat input parameters and line energy is represented as,

$$Q = \frac{U.I.\eta }{\nu }$$

(10)

where $\nu$ is weld torch travel speed.

The heat loss from the model was considered in terms of convection and radiation. Convection was kept limited to the base metal, and radiation was assigned only for the weld metal. Newton’s law was followed for the heat loss by means of convection as,

$$Q_{c} = - h_{c} \left( {T_{s} - T_{i} } \right)$$

(11)

where $h_{c}$ is the heat transfer coefficient assumed as 15 × 10^–6 W/(mm² °C) [37], $T_{s}$ is the surface temperature of the specimen and $T_{i}$ is the ambient room temperature taken as 21 °C. radiation was considered following Stefan–Boltzmann law as represented below,

$$Q_{r} = - \epsilon \sigma \left\{ {\left( {T_{s} + 273.15} \right)^{4} - \left( {T_{i} + 273.15} \right)^{4} } \right\}$$

(12)

where $\epsilon$ is emissivity assumed as 0.8, and $\sigma$ is Stefan–Boltzmann constant.

Temperature-dependent material properties were taken into consideration except for density (7900 kg/m³), which was assumed to be constant throughout the analysis. Properties of general structural steel were used in modeling due to the lack of availability of temperature-dependent material properties of the corresponding material. Relevant temperature-dependent thermal and mechanical properties are given in Fig. 6 [38,39,40]. An inter-pass temperature of 100–180 °C was maintained between two successive weld passes [19].

Fig. 6

Temperature-dependent material properties a thermal properties and b mechanical properties [39,40,41]

3.2 Structural Analysis and Solid-State Phase Transformation

The structural analysis of the FE models was computed alongside the thermal analysis simultaneously by the application of coupled temperature and displacement elements. The material properties of the weld zone and the heat-affected zone (HAZ) were assumed to be similar. Displacement boundary conditions were employed at the edges of the models to restrict the rigid body motion. In the case of axisymmetric models, the planes of symmetry were constrained in the direction normal to the weld axis.

In the present study, the effect of solid-state phase transformation (SSPT) and change in yield strength due to phase transformation on the evaluation of residual stresses were also investigated. When high strength carbon steels are heated beyond the A₁ temperature limit, the base material starts getting transformed into austenite from pearlite and ferrite. These steels get completely transformed into austenite after crossing the A₃ temperature limit. The pearlite–ferrite phase holds a BCC structure, and during heating, it endures a reduction in volume after getting fully transformed into the austenitic phase consisting of FCC structure. During cooling, the decomposition of the austenitic phase depends upon the rate of cooling. Hansen studied the effect of SSPT on the evaluation of residual stresses incorporating phase transformation from ferrite to austenite in terms of the temperature-dependent coefficient of thermal expansion. Based on surface residual stress distribution, no significant effect of SSPT was noticed [41]. However, in a study by Taraphdar et al., it was observed that in FE modeling, change in mechanical material properties significantly alters the through-thickness residual stress distribution but may not affect the surface residual stress magnitudes [30]. In the present analysis, the significance of SSPT from ferrite to austenite phase was emphasized in the 3D full geometry FE model to observe its effects on the residual stress fields.

Considering SSPT during welding, the total strain rate can be expressed as the summation of individual strain rate components as below,

$$\dot{\varepsilon } = \mathop {\varepsilon^{E} }\limits + \mathop {\varepsilon^{P} }\limits + \mathop {\varepsilon^{T} }\limits + \mathop {\varepsilon^{\Delta Vol} }\limits$$

(13)

where $\mathop {\varepsilon^{E} }\limits$, $\mathop {\varepsilon^{P} }\limits$, $\mathop {\varepsilon^{T} }\limits$, and $\mathop {\varepsilon^{\Delta Vol} }\limits$ represent the strain rate components of elastic loading, plastic loading, thermal loading, and volumetric change due to metallurgical effects, respectively. The above equation can be presented in the form of increments for the purpose of numerical simulation, as in Eq. 14,

$$\Delta \varepsilon_{total} = \Delta \varepsilon^{E} + \Delta \varepsilon^{P} + \Delta \varepsilon^{T} + \Delta \varepsilon^{Vol}$$

(14)

where $\Delta \varepsilon^{E}$ is the strain increment due to elastic loading computed by isotropic Hook’s law and temperature-dependent material properties like Young’s modulus and Poisson’s ratio. $\Delta \varepsilon^{P}$ is the strain increment due to plastic loading which follows von Mises yield criterion, isotropic hardening model and mechanical properties. The thermal strain increment $(\Delta \varepsilon^{T} )$ and volumetric strain increment $(\Delta \varepsilon^{Vol} )$ are calculated combinedly, considering it as a function of temperature-dependent thermal expansion coefficient ($\alpha )$. Here the original trend of the coefficient of thermal expansion was modified at the phase change temperature range to include the volumetric changes. The combined effect of thermal loading and volumetric change on the strain increment due to austenitic transformation can be found as [42],

$$\Delta \varepsilon^{T} + \Delta \varepsilon^{Vol} = \alpha \Delta T + \frac{{ - 2.288 \times 10^{ - 3} \Delta T}}{{A_{3} - A_{1} }}$$

(15)

where $\Delta T$ is temperature increment. The detailed steps of analytical calculation attributed to SSPT can be found elsewhere [42, 43]. The variation in yield strength during different phases was also incorporated by altering the magnitudes of yield strength in the phase transformation temperature range.

4 Results and Discussion

Four numerical models were prepared, and the predicted residual stresses along the surface and through the thickness were compared with the experimental results. The simulation cases studied in the present work are shown in Table 3.

4.1 Evaluation of Microstructures

Figure 7 represents the macrostructure of the weld joint and microstructure of various zones across the weld cross-section. As shown in Fig. 7b, the base region of the weld consisted of coarse ferrite (F) and pearlite (P) grains. Equiaxed ferrite and acicular ferrite (AF) grains with grain boundary ferrites (GBF) were observed in the weld zone (Fig. 7c). The region of HAZ shows the variation in microstructure as move away from the fusion line, and it depends on the temperature experienced by the region. On the basis of temperature experienced, a narrow HAZ is subdivided into three regions, namely: coarse-grained HAZ (CGHAZ), fine-grained HAZ (FGHAZ), and the inter-critical HAZ (ICHAZ) as indicated in Fig. 8a. The corresponding predicted thermal histories of these regions are shown in Fig. 8b. The nodes representing peak temperature (CGHAZ-1300 °C, FGHAZ-950 °C, ICHAZ-750 °C) of CGHAZ, FGHAZ, and ICHAZ are 0.66, 1.89, and 2.92 mm away from the fusion boundary, respectively. The CGHAZ were found to be in a combination of upper bainite (UB), grain boundary ferrite (GBF), and Widmanstätten ferrite (WF) grains, as depicted in Fig. 7d. The FGHAZ had an appearance comprising pearlites in a polygonal ferrite matrix, as observed in Fig. 7e. Accumulation of carbides with irregular grain boundaries and ferrite grains were observed in ICHAZ, as shown in Fig. 7f.

Fig. 7

a Macrostructure, b parent material, c weld zone, d coarse-grained heat-affected zone (CGHAZ), e fine-grained heat-affected zone (FGHAZ), f inter-critical heat-affected zone (ICHAZ)

Fig. 8

a Subdivisions of the HAZ, b predicted thermal histories experienced at the HAZ

4.2 Temperature Contours

For the purpose of developing a reliable FE model, it is required to attain an appropriate fusion zone which attributes the shape and size of the weld pool and the HAZ. The heat source parameters were adjusted accordingly to obtain melting temperature at the fusion boundary. The temperature contours shown in Fig. 9 were captured along the planes of weld cross-section in the middle of the weld axis of different FE models. The represented contours were acquired when the corresponding nodes of those particular planes were enduring peak temperature. It can be observed from the temperature contours of the root, first and final passes that the fusion boundary attained reasonable melting temperature, confirming the desired fusion of the weld metal. In a comparison of temperature distribution in different FE models, a significantly close match was observed among themselves.

Fig. 9

Temperature contours of different finite element models

4.3 Comparison of Surface Residual Stresses

Three paths were defined to compare the evaluated residual stresses from different FE models, as shown in Fig. 5. Line 1, 2, and 3 were used to present the residual stress distribution at the surface, through the thickness at the weld zone, and the HAZ, respectively, for both the 3D and 2D models.

The comparison of residual stresses evaluated from different FE models is shown in Fig. 10 by means of stress contours of the weld cross-section. From the stress contours, it can be observed that the intensity of tensile longitudinal residual stress was found to be comparatively higher than the transverse residual stress at the upper layer of the weld zone. The tensile nature of longitudinal residual stress gradually decreased through the thickness and went into a mild compressive zone at the bottom. Whereas the transverse residual stress appeared to be compressive in the middle of the weld cross-section and transformed into tensile in nature at the weld top and weld root. It is evident from the stress contours that the maximum transverse residual stress appeared at the HAZ and few layers beneath the top surface of the weld zone. However, maximum longitudinal residual stress was found at the weld center. It can also be observed that 2D models exhibited a comparatively lower magnitude of residual stresses but revealed similar zones of stresses as of the 3D models. This may happen because the 3D models take weld start and end effects into account, but the 2D models fail to capture the end effects [16]. At the locations of weld start and end, zones of higher compressive transverse residual stresses are seen [44], and to balance the equilibrium state of induced residual stresses higher magnitude of tensile residual stresses is observed at the weld center in the 3D models in comparison to 2D models.

Fig. 10

Longitudinal and transverse residual stress contours of different finite element models

For a better understanding of the surface residual stress distribution across the weld cross-section, residual stresses evaluated by the FE models and experiments (BHD) have been plotted following Line 1, as shown in Fig. 11. The predicted results were found to be in good agreement with the experimental results. However, in careful observation, some mismatches between the predicted and experimental results can be observed. This may occur due to the initial presence of residual stresses in the plates which underwent different thermal exposures during production. The interaction of these initial stresses with the weld induced residual stresses can influence the final distribution of residual stresses across the weldment. It should be noted that the initial presence of residual stresses was not incorporated in the FE models. Among all the cases of simulations, the predicted longitudinal residual stresses from cases A, B, and C show almost no difference at the weld zone, Fig. 11a. Surprisingly the 2D axisymmetric model (case D) predicted the surface residual stress magnitude with a very close match with the BHD results at the weld zone and the HAZ. The peak magnitude of longitudinal residual stresses at the surface of the weld zone evaluated by the simulation cases A, B, C, D, and the experiment are 259, 255, 262, 204, and 209 MPa, respectively.

Fig. 11

Surface residual stress distribution a longitudinal stress and b transverse stress

Figure 11b represents the surface transverse residual stress distribution evaluated from the FE models and experiments (BHD). The predicted transverse residual stresses along the line 1 from the simulation cases A and B (3D models) are in a close match. Whereas, the surface transverse residual stresses predicted by the 2D models follow the same trend as of the 3D models but differ in magnitude. Overall, the predicted transverse residual stresses along the line 1 from the cases A, B, and C produced a reasonably close match with the experimental results. Unlike longitudinal residual stress, here, the 2D axisymmetric model underestimated peak transverse residual stress at the surface of the weld zone. The peak magnitude of transverse residual stresses at the surface of the weld zone evaluated by the simulation cases A, B, C, D, and the experiment are 59, 58, 41, 22, and 46 MPa, respectively. It is interesting to observe that the 2D full geometry model closely predicted the transverse residual stresses at the weld zone and the HAZ. So, from Fig. 11a, b, it is clear that the predicted longitudinal and transverse residual stress distribution at the surface is clearly sensitive to the kind of FE model is being employed.

It is of interest that the 3D and 2D axisymmetric half models did not encounter a similar number of weld thermal cycles as of the full models. From Fig. 4, it can be seen that the total number of weld passes is 10 (excluding root pass), whereas, in the case of axisymmetric half models, only 7 number of weld passes were taken into account in the analysis. Thus, the half models endured a comparatively lesser number of weld thermal cycles. However, the predicted results from these models are impressively in good agreement with the experimental results. So, the deviation of predicted results between the 3D or 2D full models and their axisymmetric half models is because of the lesser number of weld thermal cycles experienced by the half models.

4.4 Comparison of Through-Thickness Residual Stresses

Figure 12a, b represents the through-thickness longitudinal and transverse residual stress distribution along line 2, respectively. Here the predicted through-thickness residual stress fields from different simulation cases have been compared with the DHD test results. The major fact observed from the through-thickness residual stress trends is that the peak magnitudes of residual stresses were found at around 6 mm below the top surface of the weld zone. The peak magnitude of longitudinal and transverse residual stresses exhibited from the DHD test results are 309 and 176 MPa, respectively. In comparison to the surface residual stress magnitudes, the peak magnitudes of the deep interior stresses were found to be significantly higher in both the longitudinal and transverse directions. The through-thickness residual stress trends obtained from most of the simulation cases were found to be in good agreement with the experimental results. However, the 2D axisymmetric model significantly under predicted the through-thickness residual stress distribution. The 3D full geometry model comparatively exhibited a better match with the experimental results than the rest of the models. 3D axisymmetric and 2D full geometry FE models also impressively predicted the through-thickness residual trends. Overall, simulation cases A, B, and C procured the through-thickness residual stresses with reasonably close matches with the experimental results.

Fig. 12

Through thickness residual stress distribution at the weld zone a longitudinal stress and b transverse stress

In a similar way, the through-thickness residual stresses were also evaluated at the HAZ along line 3 using the DHD technique and FE models, as shown in Fig. 13a, b. The peak magnitude of longitudinal and transverse residual stress through the thickness at the HAZ was obtained as 145 and 135 MPa, respectively. Same as of weld zone, the 3D full geometry FE model produced best-predicted results of through-thickness residual stresses at the HAZ. Simulation cases B, C, and D also predicted a similar trend of both the longitudinal and transverse residual stresses with acceptable match with the experimental results.

Fig. 13

Through thickness residual stress distribution at the HAZ a longitudinal stress and b transverse stress

Among all the cases, 3D models were found to be most efficient in predicting residual stresses. These results indicate that dimensional differences in FE modeling of any individual geometry work as an influential factor in predicting both the surface and through-thickness residual stress distribution.

4.5 Comparison of Residual Stresses Incorporating SSPT

In this section, the effect of SSPT on the evaluation of residual stresses using the 3D full geometry FE model has been explained. Two simulation cases, namely case E and case F have been studied where the change in volume fraction and yield strength of the weld material due to austenitic phase transformation have been incorporated in the numerical simulations. In Fig. 14, the comparison of the predicted longitudinal and transverse residual stress contours of the weld cross-section procured from the above mentioned two simulation cases have been represented. In a closer observation, it can be seen that the size of the zone having tensile residual stresses at the upper half of the cross-section got reduced in dimension and appeared as in a concentrated form compared to what was observed in case A. Figure 15 represents the comparison of surface residual stress distribution evaluated from the simulation cases E and F with the experimental results (BHD). From Fig. 15a significant effect of SSPT on the longitudinal residual stress distribution can be observed. Due to the integration of SSPT in the material modeling, cases E and F produced a better match of surface longitudinal residual stress distribution with the experimental test results. Almost similar trends from cases E and F were obtained. However, a substantial difference between cases A and E/F can be noticed. Similarly, in the transverse direction between simulation cases E and F, negligible differences were observed in the trend along the weld cross-section except at the HAZ, as shown in Fig. 15b. It is also evident that the FE models (Case E and F) incorporating SSPT under predicted the peak surface transverse residual stress magnitude at the weld zone but provided a better match at the HAZ and base region. However, it is interesting that the 3D full geometry model predicted the surface residual stress distribution more accurately after incorporating SSPT in material modeling.

Fig. 14

Longitudinal and transverse residual stress contours incorporating SSPT

Fig. 15

Surface residual stress distribution incorporating SSPT a longitudinal stress and b transverse stress

In the case of through-thickness residual stress fields at the weld zone, cases E and F produced similar trends as of case A with some minor differences. Nevertheless, the trends attained from cases E and F produced a better match with the DHD test results throughout the thickness in the longitudinal direction, as shown in Fig. 16a. Cases E and F under predicted the through-thickness transverse residual stresses up to a depth of 6 mm from the top surface, and after that, it provided a reasonably close match with experimental (DHD) results, as shown in Fig. 16b. No significant differences were observed between the cases E and F in their prediction of through-thickness residual stress distribution at the weld zone.

Fig. 16

Through thickness residual stress distribution at the weld zone incorporating SSPT a longitudinal stress and b transverse stress

Figure 17a, b represents the comparison of evaluated through-thickness residual stress fields of cases E and F with the experimental and case A results at the HAZ. Here, cases E and F produced a significantly better match with the experimental results in both the longitudinal and transverse directions. Whereas case A over predicted the through-thickness residual stress values at the upper layer of the HAZ.

Fig. 17

Through thickness residual stress distribution at the HAZ incorporating SSPT a longitudinal stress and b transverse stress

Overall, it was observed that incorporating SSPT in the 3D full geometry FE model produced the best match of both the surface and through-thickness residual stress distribution with the experimental results. From the comparison of different cases of FE models, it can be concluded that though the change in yield strength has a significant influence on the evaluation of residual stresses, but only the incorporation of volumetric change can be sufficient in the material modeling. The present computational model can be further improved by introducing more accurate weld bead shape and size along with the incorporation of the initial stresses present in the base metal.

4.6 Comparison of Computational Time

Figure 18 represents the comparison of the total computational time required for each FE model to complete the analysis. The total time consumed by the case A was assumed as the reference, and completion time percentages for other FE models were then calculated relative to case A. According to Fig. 18, the 3D axisymmetric model (case B) took around 37% of the total time required for the 3D full geometry to finish the analysis. Moreover, the 3D axisymmetric model impressively predicted both the surface and through-thickness residual stress distribution with reasonable accuracy. This indicates that 3D axisymmetric models of multipass butt joints can be equally competent for weld induced residual stress analysis.

Fig. 18

Comparison of computational time required for different FE models

Despite the fact that 2D FE models (cases C and D) under predicted the residual stress values in few cases, both the 2D models satisfactorily predicted the surface residual stress distribution considering its total computational time consumed during the analysis. From Fig. 18, one can easily figure out how much computational time can be saved by the application of 2D models. So, if the main purpose of the analysis is to find the surface residual stress distribution, then 2D models can be a good option with minimum computing time.

5 Conclusions

In the present study, different 3D and 2D FE models were prepared to observe their competence in predicting both the surface and through-thickness residual stress distribution in an SA516 Gr. 70 multipass butt welded joint. The predicted residual stress fields from these FE models were compared with the experimental results to understand better their accuracy and total required computational time to complete the analysis. The effect of SSPT on the predicted residual stress distribution was also studied. The following conclusions can be drawn from the observations.

• The peak magnitudes of both the longitudinal and transverse residual stresses were observed at a depth of 6 mm below the top surface of the weld zone.

• In comparison to 2D models, the 3D models were found out to be in the best match with the experimental results. These models can provide a better overview of residual stresses distributed across the weld cross-section.

• Considering the total computational time required for the 2D models to complete the analysis, the 2D FE models can be a good option over the 3D models with competitive accuracy. It can be beneficial where a large number of weld passes are subjected to be taken into account for thick multi-pass welded joints.

• The change in volume due to the austenitic transformation incorporated into the 3D FE full model enhanced the prediction accuracy of the model.

• The change in yield strength of material due to SSPT has some specific effects on the evaluation of residual stresses but were find out to be not so important to account in the FE model.