Fatigue assessment of welded joints and crack growth considering residual stress

There are different approaches to investigate the fatigue life of welded joints, however, less has been done considering residual stresses. The heat and residual stress distribution of a butt weld joint are determined using ABAQUS software and the results are validated against experimental and numerical studies in the literature. The software FE-SAFE is used to study the fatigue life of a GMAW welded butt joint under high cycle loading considering the residual stress. The corresponding S-N curve is determined

The S-N method basically relies on experiment, and this is why most of the numerical approaches are based on fracture mechanics method.In fracture mechanics approach the Stress Intensity Factor (SIF) is calculated at the existing crack tip and the fatigue life is determined using the well-known Paris law [4].Many research works have been devoted to SIF analysis in different geometrical, material and loading conditions.Rahbar Ranji et al [5] studied the effect of pitting corrosion on SIF in a surface crack.Lee et al [6] simulated the growth of a crack based on SIF by regenerating the crack tip mesh as the crack grows.Akeel et al [7] used FEM to analyze SIF of a crack in wheel with contact to railhead.Maitireyimu et al [8] studied SIF in surface cracks in round bars under rotary bending types of loading.Fageehi [9] used computer codes ANSYS and FRANC2D/L to study fatigue crack growth by estimation of SIF.
Despite the many benefits of welding, the detrimental effects of the induced residual stresses are inevitable.Many research works have done on measuring and controlling residual stress using recent advanced measurement techniques such as deep hole drilling [10], high energy synchrotron x-ray diffraction [11], and neutron diffraction [12].Kurdyavtsev and Kleiman [13] used ultrasonic wave method to measure residual weld stress.Zhenga et al [14] used ultrasonic impact treatment to predict weld residual stress distribution of butt-and T-weld joints.Peng et al [15] combined digital image correlation with the hole-drilling method to measure residual stress.Some researchers used numerical methods to simulate residual stress distribution in welding joints.Almeida et al [16] used thermal Finite Element Analysis (FEA) to calculate the residual stresses and deformations introduced by welding process.DeWald and Hill [17] extended the contour method to allow for the measurement of spatially varying multi-axial residual stresses in prismatic, continuously processed bodies.Xiong et al [18] developed a method to use an artificial neural network in the FE model updating technique to obtain the residual stress distribution.
The induced residual stress of welding is due to non-uniform heating and cooling.Some researchers have used numerical methods to analyze heat distribution in welded joints for estimating residual stress distribution.In these studies, modelling the heat source due to welding is a challenging task.Goldak et al [19] defined a double symmetric ellipse for heat source model, which is used by many researchers for residual stress analysis.Mok et al [20] used Goldak's heat source model to analyze welding residual stresses in a T-joint of steel specimen using ABAQUS software.Deng and Murakawa [21] determined heat distribution in multi pass welds in stainless steel pipes using Goldak's double ellipsoidal model and measured the induced residual stress.Mokrov et al [22] modified the Goldak's model and Winczek [23] proposed an analytical model based on volumetric model of the heat source and a semi-infinite body model for describing the temperature field of multi-pass arc weld.
High tensile residual stress in front of weld could lead to cracking and consequently reducing the fatigue life.To study fatigue life of a welded structure, it is necessary to have accurate distribution of stresses and strains.Berg et al [24] studied the effect of residual stresses on fatigue crack propagation rate using extended finite element method (XFEM).Fustar et al [25] reviewed the most important methods for design and fatigue life assessment of welded steel structures.Hobbacher [26] has reviewed the impact of imperfections on fatigue life of welded joints.Backstrom and Marquis [27] reviewed and analyzed the results of 233 experiments on welded joints that are prone to fatigue.Nguyen and Wahab [28] developed a model to predict the combined effect of weld toe undercut, residual stress and misalignment on fatigue strength of butt weld joints.He et al [29] numerically studied the effect of weld joint geometrical parameters on fatigue life using ABAQUS/FE-SAFE softwares.Lopez-Jauregi et al [30] developed a procedure to estimate the fatigue life of welded joints considering residual stress.Neto et al [31] studied numerically the effect of residual stress on fatigue crack growth.Esnaola et al [32] determined temperature and residual stress distribution in a multi pass butt weld using new method to define heat source energy.
It is the main aim of the present work to investigate the fatigue life and crack propagation in a butt weld joint numerically.The ABAQUS software is used for thermomechanical analysis to determine temperature and residual stress distributions in a butt weld.The results compared with available experimental and numerical results in the literature.Fatigue life of butt welds are determined and validated using FE-SAFE software.XFEM software used to study the crack propagation path in a butt weld joint.

Numerical analysis
Welding analysis needs interaction of two engineering fields, i.e., thermal, and structural/mechanical analyses.In FEM, this interaction can be considered either directly, called coupled thermomechanical analysis or indirectly, decoupled thermal and mechanical analyses.The main assumption in decoupled thermomechanical analysis is that thermal field influences structural field, however structural field has negligible influence in the thermal field.This is why it is also called one-way coupled thermo-structural problem.Two separate one-time analyses are performed without any convergence study.The equations governing the temperature and the displacement of the structure are solved separately with two distinct solvers.
The decoupled thermomechanical solution started by determination of the temperature distribution from transient thermal analysis.An accurate estimation of the temperature field in a weld is of paramount important in this solution, and most of the studies have used Goldak's double symmetry ellipsoidal model [19].In the thermal field analysis finite second order elements that have temperature as the only degree of freedom in their nodes are employed.The main inputs are heat transfer equation and boundary conditions.Solving heat transfer equation, the temperature at each node is determined which is used as input data in mechanical/structural analysis.Expansion and shrinkage due to the changes of the temperature and volume variation of elements impose residual stress.
In mechanical/structural analysis the stiffness matrix is determined using basic plasticity equation.Thermoelasto-plastic behavior of material with temperature dependent properties of material including thermal expansion coefficient, elastic modulus, yields stress and Poisson's ratio are considered.Temperature at each node together with structural boundary conditions are applied and stress and strains of each node are calculated.As a result, the residual stresses and deformations created by heat cycle are known.
The heat due to welding varies mechanical and thermal properties of material due to microstructure changes.Volumetric changes due to phase transformation induces thermal stress and strain.In return, the plastic deformation of material creates heat.Thus, the assumption that structural field has no effect on thermal field is inaccurate, and welding is two-way phenomena, in which heat creates stresses and deformations induce heat.Thus, coupled analysis should be used in which governing equations of heat and displacement of the structure are solved simultaneously with a single solver and all degree-of-freedoms including thermal and structural degree of freedoms are applied.The heat transfer problem is solved first, with the temperature history being saved for each node.Then, structural nodal loads are calculated using the thermal expansion coefficient in static conditions through the nodal temperature history [33].The heat source can be simulated either inducing heat to specified volume of the weld zone, or modeling input heat source as body flux.

Validation of FEA for residual stress analysis due to welding
In the present study, the coupled (two-way) method is employed for residual stress analysis of welding, and heat source is modelled as body flux on each weld component.To evaluate the accuracy of FEA for residual stress analysis a butt weld of Gas metal arc welding (GMAW) between two plates with dimensions of 200 × 10 mm is taken from the research work of Esnaola et al [32], which is shown in figure 1.
The heat source is modelled stepwise, which is the simplest form for considering a movable body flux.The heat flux is applied at the starting point of weld, and then after a proper time step corresponding to the speed of the heat source, is shifted to a new position.The thermal stress in each weld pass is calculated from heat distribution.This procedure is employed along the whole welding line.The steel used is designated as S275JR according to standard EN10025-2 [34], a non-alloy structural steel which can be welded without restrictions.The considered electrode is selected according to AWS/ASME SFA 5.18 standard [35] as PRAXAIR M-86 with diameter of 1.2 mm.Both materials have the same ultimate stress and strain; however, the yield stress of electrode is 45% higher than steel.The Stargon 82 with 8% Co2 is used as shield gas.The chemical composition of the S275JR steel is given in table 1 and the mechanical properties of both S275JR and weld material are depicted in table 2. The type of weld is three pass butt welds, and every pass of weld is modelled using 40 solid elements of C3D8R in ABAQUS software with the length of 5 mm along the weld line and coarser size of element further from weld line (figure 2).
Esnaola et al [32] determined heat distribution using thermocouple and thermographic camera and the result compared with FEM. Figure 3 shows the heat distribution determined in this study using FEM.Comparing against FE and experiment results of Esnaola et al [32], it can concluded that distribution and the maximum heat determined in the current study are close to FE result of Esnaola et al [32].
Esnaola et al [32] determined residual stress distributions using hole drilling method and FEM.Comparing the transverse residual stress distribution obtained by FEM (figure 4) with Esnaola et al [32], it can be concluded that the maximum tensile residual stress calculated is 208.3MPa, against 241.3MPa, of Esnaola et al [32], and maximum compressive residual stress is 372.4MPa, and in Esnaola et al [32], is 373.8MPa.

Validation of FEM for fatigue analysis
The software, FE-SAFE is one of the earliest software which has been developed for fatigue analysis based on FEM.This software is used for fatigue analysis of a dog-bone specimen (figure 5) which is validated against numerical and experimental results of Aldeeb and Abduelmula [1].
The considered model is built according to the ASTM E466-07 standard [36] by Mazak CNC tool at the Northumbria University.The material is S275 steel that its chemical compositions and mechanical properties are given in tables 3 and 4.
The cyclic load with frequency of 4 Hz, load ratio of −1, and Goodman model for the determination of the fatigue life are applied for eight different stress amplitudes.Table 5 shows the results of the current study which are compared against experimental and numerical results [1].As seen, the results of the present study are lower than both numerical and experimental results, however, the difference is as low as 1% when applied load is high, say 1.9 kN (low cycle fatigue), and reaches to 10% when applied load is low, say 1.2 kN (high cycle fatigue) which still is within the accepted limit.
Figure 6 depicts the von-Mises stress distribution in the dog-bone sample corresponding to the applied load of 1.9kN determined in the current study.As seen, stress distribution in both studies is the same and maximum von Mises stress in current study is 264.4MPa against 266.93 MPa in Aldeeb and Abduelmula [1].

Fatigue analysis of a butt weld joint
Lopez-Jauregi et al [30] carried out experiment and FEA to determine fatigue life of butt and fillet weld joints in S275JR steel.The considered weld joint in this work has limited length.To study the effect of the weld length on fatigue strength of the butt joint, the butt weld between two plates shown in figure 1 is chosen and fatigue life is determined using FE-SAFE software.Five cyclic loadings with stress amplitudes of 80, 90, 100, 115, 125 MPa, frequency of 5 Hz, and loading ratio of −1 are applied.Figure 7 shows the von-Mises stress distribution at different stress amplitudes in butt weld shown in figure 1. Figure 8 depicts the S-N curve of this butt weld determined using FE-SAFE software which is compared against base steel and butt weld used in Lopez-Jauregi et al [30].As seen, the fatigue life of steel is reduced due to welding, and the results of the present study with longer weld length is lower than Lopez-Jauregi et al [30].Thus, one can conclude that the longer the length of welds, the higher residual stress and lower fatigue life.

Fatigue crack initiation and propagation
Using XFEM software of ABAQUS, crack nucleation and growth are determined in the butt weld shown in figure 1.First the odb file of the thermomechanical analysis of welding is used as predefined field and then in the interaction modulus, the geometry of the crack is defined.The position of crack nucleation is not defined in order it be determined by software.In the property modulus, the Maxps Damage criteria is selected as damage for separation law.The fracture toughness of steel S275JR is taken from Brnic et al [37].To determine crack  nucleation and propagation, a static step is defined for nucleation and then by defining direct cyclic step, propagation of crack is generated.Due to limitation in ABAQUS, only one cyclic load with amplitude of 125 MPa is considered.Figure 9 shows different steps of crack nucleation and propagation.The crack nucleation position in the XFEM software is the weakest element based on fatigue damage criteria in the FE-SAFE software.In this analysis two simultaneous cracks have started from two opposite edges which have maximum restraint due to the applied boundary condition, and growth simultaneously towards the center of plate.Comparing the length of cracks in different loading steps, it is clear that at the early steps (short cracks) the rate of increase of crack length is low, while for long cracks, this rate is higher.

Conclusion
The software ABAQUS is used for thermomechanical analysis of a butt weld and temperature and residual stress distributions are determined and compared with experiment and numerical results available in the literature.The software FE-SAFE is used for fatigue analysis considering residual stress distribution.The method is validated by analyzing the fatigue life of a butt weld in a dog-bone specimen and the results compared against experiment and numerical results.The validated method is employed to study the fatigue life of a butt weld and the results are compared with an experiment carried out on the same butt weld but with smaller size.The result shows that though the butt weld reduces the fatigue life of base metal, however the longer the length of welds, the higher residual stress and lower fatigue life.The crack nucleation and propagation steps in the butt weld under cyclic load are determined using software XFEM.The result shows that the crack would start from the point with the highest restriction, and the rate of increase of crack length is higher for longer cracks than short cracks.

Figure 2 .
Figure 2. Finite element model of three pass butt weld.

Figure 4 .
Figure 4. Transverse residual stress distribution in three pass butt weld.

Figure 5 .
Figure 5.The geometrical model considered for fatigue analysis.

Figure 7 .
Figure 7. Von-Mises stress distribution in the butt weld joint (figure 1) at the different stress amplitudes in fatigue analysis.

Figure 8 .
Figure 8.The S-N curve of base steel S275JR and butt weld joints.

Figure 9 .
Figure 9. Steps of crack nucleation and propagation in butt weld shown in figure 1 at stress.

Table 1 .
Chemical composition of the steel S275JR.

Table 2 .
Mechanical properties of steel S275JR and electrode PRAXAIR M-86.

Table 5 .
The fatigue life of the dog-bone specimen shown in figure5.