Analysis of the Influence of Residual Stress on Fatigue Life of Welded Joints

This work presents a new computational approach to estimate the fatigue life in welded joints, taking into account three-dimensional, non-proportional, and out-of-phase stresses; the thermomechanical properties of steel; parameters of the welding process; and the effects of residual stresses. We use the mechanical equations of continuous and cumulative fatigue damage found from Multiaxial RainFlow, the Wang and Brown and critical plane methods, and the Findley criteria approach. The numerical finite element method, available with software, is used to simulate the formation of residual stress and quantify its influence by solving equations. Based on the analysis of numerical results, we were able to prevision the crack location and angles of propagation of a high cycle fatigue crack with totally random loads in a metallic structure. We conclude that the result of the computational method is valid and can be used to estimate and life in fatigue in welded joints.


INTRODUCTION
In many structural applications of steel or composite steel-concrete (QIAN et al., 2014) and (QIAN et al., 2014), for example in bridges, off-shore structures, cars, cargo vehicles, and railway equipment among others, welded joints are usually subjected to three-dimensional, cyclic, non-proportional loads (ESMAEILI et al., 2015) for long periods.According to (HUO;WANG;ZHANG, 2005), welded joint failure represents 80% of all fatigue failures.
High temperatures developed in the welding process can produce significant metallurgical and mechanical transformations in the fused zone (FZ) and the Heat-Affected Zone (HAZ), reducing reliability.This functional degradation can be modeled using mathematical equations (BUI et al., 1994) and (KOHANDEHGHAN;SERAJZADEH, 2011).According to (HE et al., 2014), the resistance to multiaxial fatigue in the HAZ and the weld metal (WM) are, respectively, 55% and 60% lower than that of the base metal (MB).
In the last twenty years, the properties of the MB have been extensively studied.However, the fatigue behavior of welded joints has rarely been investigated, especially in the high cycle fatigue regime (HE et al., 2014).
In the welded joint, the residual stress (RS) appears and remains in the absence of external loads or changes in temperature.It arises because metallic materials have different thermomechanical coefficients and elastic/plastic strain rates as a function of temperature (SILVA et al., 2016).
During the welding process, significant metallurgical changes occur in the MB, HAZ, and FZ (KOHANDEHGHAN; SERAJZADEH, 2011) and (LI et al., 2017), in addition to the evident changes in the mechanical properties between the MB areas and HAZ, where RS occurs.
According to (GAN et al., 2016), in the last decades, the RS from the welding process has been the object of study by engineers and designers because it is related to both manufacturing welding process and fatigue life of the component.
Several methodologies are proposed for the study of fatigue in welded joints (KHURSHID et al., 2016): hot spots, structural stress, local stresses,, and modified Wöhler and von Mises (AUSIELLO et al., 2011) curves.Some of these are based on the critical plane model, effective integral method, or an equivalent voltage hypothesis.
As computer simulation tools for structural design and manufacturing processes evolve, there is an inherent need to optimize and automate the analytical or computational tools to estimate fatigue life in various types of welded joints.
The geometric relationship of the welded joint to the direction of the structural stress means that the multiaxial fatigue strength can be optimized for compressive RS or reduced by tractive RS (KREBS; KASSNER, 2007).For these reasons, three-dimensional analysis is necessary for each region of the HAZ under different load conditions.
According to BHATTI et al. (2015) and EISAZADEH et al. (2015), the RS can influence the performance of the multiaxial fatigue of the structure.In this sense, understanding the formation of the RS in the weld and its effects is extremely important for mechanical engineers and designers working in metal industries.The interpretation of the effects of RS on dynamic performance is a challenge, but the effects have been optimized in recent decades (JAMES et al., 2007).
While external loads acting on components are clearly important, other factors, like microstructure and RS, also contribute to failure.In cases where failure cannot be tolerated, engineers must be able to predict the behavior of welded joints, whether via measurements or computer simulations.Knowledge of the RS intensity and directions are essential in such situations (WITHERS, 2007), (SAVARIA;BRIDIER;BOCHER, 2016), and(GADALLAH et al., 2018).
The first objective of this work is to carry out the 3D heat transfer simulation of the welding process according to Goldak's equation (GOLDAK; CHAKRAVARTI; BIBBY, 1984) and the formation of the RS in the flat welded joint, especially in the HAZ/MB.Next, we perform the numerical modeling of multiaxial stresses from three-dimensional random loads using Comsol Multiphysics (COMSOL, 2019), applying the methods of Wang & Brown (WANG;BROWN, 1996), Multiaxial RainFlow, and Findley criteria (FINDLEY, 1959).Finally, we predict the fatigue life and determine the crack propagation angle.
The present study represents an advancement of the research into the relationship between the welding process and the fatigue life.According to (GADALLAH et al., 2018), the integrity of the welded structures represents a problem that requires a rigorous analysis, especially for structures exposed to cyclic loads, but also for metallurgical transformations due to transient thermal changes, and for non-uniform heat distribution in the weld region, where adjacent regions have different temperature peaks (LINDGREN, 2006) and(JIA et al., 2014).In these regions, the observed maximum values of temperatures tend to be reduced because of their distance from the displacement line of the heat source.The transient thermal changes are the metallurgical changes observed mainly due to the cooling rate in the temperature range between 1073 K to 773 K in each of the sub-regions of a HAZ or even in different HAZs in the same component.

Findley model (1959)
The fatigue failure process consists of three phases: crack initiation, crack propagation, and rupture of the remaining section due to overload (HOBBACHER et al., 2016).As a rule, the first two stages correspond to 99% of the component's fatigue life.There are criteria for estimating life in fatigue based on the analysis of stresses, deformations, and energy.The analysis of multiaxial stresses is suitable for high cycle fatigue, the use of deformation is suitable for low cycle fatigue and the methods based on energy analysis can be applied to either stress or deformation analysis (HE et al., 2014).
Some fatigue life prediction models, in the multiaxial case, are based on the set of methodologies cited in (CROSSLAND, 1956) and (SINES, 1959).
According to (PAPADOPOULOS, 2001), the sets of methodologies give more reliable predictions when the stresses appear in phase, as opposed to random, non-proportional, or out of phase tensions.
The critical plane is the method that determines the planes for which deformations and stresses, whether shear   or normal   , define the progress of the nucleation and crack propagation stages.
Several models conceived in recent years employ the approach of the critical plane, among which (BROWN; MILLER, 1973), (FATEMI;SOCIE, 1988) and (MCDIARMID, 1991) stand out.(MAMIYA; ARAÚJO; CASTRO, 2009) concentrated their efforts on developing the methods more suitable for out-of-phase and non-proportional loads.
Methods based on stress analysis that employ the concept of the critical plane (PEDERSEN, 2016), (FATEMI;SOCIE, 1988) and (FINDLEY, 1959) can determine the propagation plane of the crack mathematically (KLUGER; ŁAGODA, 2016).Life in fatigue depends on a combination of stresses acting on that critical plane, which allows the global analysis of stresses and microstructure.
To calculate the damage in fatigue of a high non-proportional and out of phase multiaxial cycle, it is necessary to determine the octahedral   and   stresses acting in the critical plane, as shown in Figure 1.The stresses   and σ ij (x, y, z) are decomposed into  ′′ and  ′′ (x',y',z') as a function of the angles of the plane, to help in later determination of the planes in which the values of the stresses   and σ n are maximized (MAMIYA; ARAÚJO; CASTRO, 2009) and (PEDERSEN, 2016).
Figure 1 shows an element on the surface in the cartesian coordinate system O-X-Y-Z, in which the values of τ i′j′ (x',y',z') and σ i′j′ (x',y',z') in the O-X'-Y'-plane Z are obtained.
Latin American Journal of Solids and Structures, 2020, 17(3), e262 4/26 According to (EUROCODE 3, 2005) and (KHURSHID et al., 2016), the best method for analyzing life in multiaxial fatigue, in particular in the high-cycle regime of non-proportional, out of phase, and cyclic loading of welded joints, is based on a combination of stress checking   and   .The Findley model (FINDLEY, 1959) is widely used for this purpose in winLiFE® software (WINLIFE -STZ-VERKEHR, 2019) and Comsol Multiphysics ®(COMSOL, 2019).
The Findley criterion was the first to use the concept of cumulative damage at the critical level to determine the angles at which the values on the left side of Equation ( 1) are maximized (ESMAEILI et al., 2015) and (FINDLEY, 1959).This allows the analysis of all of the planes in the multiaxial fatigue cycle.It should be noted that for each fatigue time -t1(s) there is a pair of equidistant planes in which the values of  ′′ (x',y',z') and  ′′ (x',y',z') are maximized, but for analysis purposes, only one plane is used in the critical plane method (BAPTISTA et al., 2016).
In Equation 1,   * is the fatigue strength coefficient, b is the high cycle fatigue strength exponent   is the sensitivity factor,  ,á is the maximum value of the normal stress and   is the amplitude of the shear stress in the cycle.
The parameter   * (2)  , which is, according to (SOCIE, D. F. & MARQUIS, 2000), identical to fatigue limit -βF, carries within itself the limiting case of the SN fatigue curve, which is why the stresses that result in values lower than βF, in the predictions of life, tend to infinity in steels.
In the current norms or codes, there are no methodologies or forms of fatigue life predictions applicable in multiaxial loadings with or without proportionality, or for loads that are completely out of phase.The lack of regulations clearly attributes the allocation of eventual types of joints to the designer; these are not evidently listed in the standards, into lower, upper, or extra classes when possible (TREMARIN; PRAVIA, 2017).The vast majority of studies in the last 70 years, focused their efforts on the analysis of proportional, phased, and uniaxial stress (TCHOUPOU, 2015).
In this study, the virtually generated stresses do not exceed 50% of the flow stress of the material at any pair of angles, because there is a possibility of rebalancing the RS in the event of plastic deformation, even if only locally.

Goldak equation
One of the best known three-dimensional and non-linear models used in computer simulation of heat conduction for welding processes using a mobile thermal source was presented by (ROSENTHAL, 1946) and explored by (EISAZADEH et al., 2015).This method describes a volume of material in a transient liquid state (Figure 2), which forms the weld bead when moving along the metal part.The volume must be adjusted to be as similar as possible to the volume of the fusion pool (DARMADI; KIET-TIEU; NORRISH, 2014).The model is described by Equation (3).
Latin American Journal of Solids and Structures, 2020, 17(3), e262 5/26 Precise determination of the values of (a, b, cf, and cr), which correspond to the size of the melt pool (JOSHI et al., 1970) (Figure 2), is essential for finding the position and the size of the two main zones, FZ and HAZ, and the position of the maximum temperatures, as well as the respective distributions of the isotherms in the HAZ (BHATTI et al., 2015).In Equation (3), v corresponds to the travel speed on the x-axis.
Figure 2 shows schematically the two ellipses that model the volume of the solid as a function on the axes z<0 and z> 0, x <0 and x> 0 and y<0 and y> 0 and the intensity of the incident heat flow described in Equation (3).
In Equation ( 2), n represents the efficiency of the process, 60% and 98%, depending exclusively on the welding process, I is the welding current (amp), and V is the welding voltage (volts).Source: Adapted (ROSENTHAL, 1946).
For Yadaiah (YADAIAH; BAG, 2014), Goldak's mathematical model achieved a considerably more accurate temperature distribution in the melting zones compared to the other available models, for example the Gauss method, in the welding process.Also, according to (YADAIAH; BAG, 2014), unlike the Gauss model, the (ROSENTHAL, 1946) model penetrates the solid.This characteristic is more suitable for mobile sources (DARMADI; KIET-TIEU; NORRISH, 2014).Especially interesting is the work of (MIKAMI; NAKAMURA; MOCHIZUKI, 2016), which carried out two simulations.In the first, the thermal source moved only on the locomotion axis (x).In the second simulation, a weaving movement (cyclic displacement perpendicular or main axis, similar to the sinusoidal curve) was inserted in addition to the displacement, in order to represent with greater fidelity the movement of the torch or electrode in the welding operation.The weaving movement did not cause significant changes to the results obtained in the first simulation.

Thermal-mechanical properties
Unlike the linear form of heat transfer, characteristic of some manufacturing processes, for the welding process we need to consider the evolution of the microstructures and thermomechanical properties of the materials, depending on the instantaneous temperature or the microstructural change.For the proper definition of the isotherms and the elastic and plastic deformations arising from the dilations in the HAZ and MB, it is essential to define the thermomechanical properties as a function of temperature.
Latin American Journal of Solids and Structures, 2020, 17(3), e262 6/26 Figure 3 shows the temperature dependence of thermomechanical properties in the range from 273 K to 2500 K.It should be noted that some properties have zero value above the temperature of 1749 K as a result of the melting of the material.
Simulation is considered by countless researchers as the least linear description of the welding process, having the greatest complexity in the analysis of the phenomena suitable for computer modeling (LINDGREN, 2006).

Residual Stresses
According to (ROSSINI et al., 2012), RS is generated in several manufacturing processes that involve at least one of the following: material deformation, heat treatment, heating, machining (HUA et al., 2005), or processing operations that allow the microstructural transformation of the material.RS originates from numerous sources and can be present in semi-industrialized raw or unprocessed material, in components with final geometry, or introduced during the last stage of manufacture.
RS is a complex quantity that is difficult to evaluate.According to (EUROCODE 3, 2005) and (BS, 2005), most fatigue life estimates do not consider the RS or overestimate it, proposing a range of values strongly linked to the component's geometry.
The RS is the result of different properties of the material in the HAZ/MB/FZ regions at high temperatures and different rates of elastic and plastic deformation.It appears as the material changes its structure to balance the forces acting on volume, resulting in micro-displacements in the solid interior.
It is therefore important to predict the effects of the thermal cycle on the metallic material during the welding process, in particular the peak temperatures in each region, the timing welding time -t2(s) of each peak in the thermal cycle in each region (FZ, HAZ, MB and sub-regions), and the subsequent cooling rates.Finding intensity and direction of the RS in the fatigue life predictions makes it possible to quantify the performance degradation and to characterize and relate the microstructural changes with their thermomechanical properties (NI; WAHAB, 2017).
According to (HUANG; LIU; XIE, 2013) RS consists of a set of stresses within the solid, generated during the manufacturing processes, which remain in equilibrium.These stresses are caused by non-uniform elastic or plastic deformations (EISAZADEH et al., 2015), applied stresses, or temperature variations.In the process of welding, the source of RS is the high localized heat input.
The properties of materials used in structural constructions are profoundly altered, as they can suffer distortions, reduced dimensional stability, corrosion, or even be considerably influenced by the existence of RS (ROSSINI et al., 2012).
RS plays a significant role in the durability of welded joints (LOPEZ-JAUREGI et al., 2015), (SONSINO, 2009) (GOEDEL et al., 2018), and can change the direction and speed of propagation of cracks induced by welding.The RS can be divided into elastic, plastic, thermal and transformation stress (EISAZADEH et al., 2015).
Latin American Journal of Solids and Structures, 2020, 17(3), e262 7/26 There are many computational models of elastoplastic deformations, such as isotropic hardening, kinetic isotropic hardening, mixed isotropic hardening, tangent modulus, von Mises, Tresca, and Mohr-Coulomb, among others.In this work, we shall use the isotropic hardening model.
Equation (4) (CAPRACE et al., 2017) represents one possible mathematical model of isotropic hardening suitable for computer simulation, where σys0 corresponds to the yield stress and σh(ε) is the plastic deformation function.
σ(ε) =  0 +  ℎ (), (4) In Figure 4, the material behaves elastically up to the deformation of  0 = 0.02.However, from the point  0 to the point   the material is deformed plastically according to different curves as a function of temperature.We interpolated the values in the curves as a function of temperature because heat transfer is a continuous process.In comparison with (HILDEBRAND; SOLTANZADEH, 2014), the present work contains a very low percentage of alloy elements, which prevents the formation of martensite.Due to the low percentage of carbon, RS from phase transformations can be disregarded in this case (CHO; KIM, 2002) and (MIKIHITO; YOSHITO, 2016) and (CAPRACE et al., 2017).However, ferrite-austenite phase transformations are implicit in the evolution of thermomechanical properties, as shown in Figure 3.
In many cases, the best approach to solving problems in engineering is to prioritize the main phenomena and to overlook the complementary analyses or insignificant factors.As mentioned in (DENG et al., 2016), the influence of phase transformations in the formation of RS in low carbon steel or low carbon alloy, or even in high strength steels, can be neglected.
In the comparative study in (CAPRACE et al., 2017), finite element method in the Sysweld® (SYSWELD, 2019) and ANSYS® (ANSYS, 2019) software was used to determine the RS with or without coupling the influence of phase transformations and displacement restrictions during the welding processes.This study found that the biggest sources of errors are in the calibration of the model as a function of the welding parameters and thermomechanical properties, highlighting the coupling and phase transformations.Still, according to (LIU et al., 2018), computer simulations of the welding process and formation of RS using ANSYS® (ANSYS, 2019) were satisfactory, while the wrong estimates of the thermomechanical properties of the base material and process parameters of welding were again important sources of errors.
According to (JAVADI; HASANI; SADEGHI, 2015), the intensity of the RS may increase up to 16% because of displacement restrictions, which prevent the deformation of components in the welding process.This is in agreement with the previous results of (JAMES et al., 2007), who also found that the restrictions cause amplification in the formation Latin American Journal of Solids and Structures, 2020, 17(3), e262 8/26 of RS, and of (KOHANDEHGHAN; SERAJZADEH, 2011), who carried out component experiments on the aluminum metallic material.
The effects of RS on the fatigue phenomenon, which may increase or reduce the average stress in the fatigue cycle, are important for quantifying the influence using the Gerber or Goodman Relations models in the case of uniaxial loads.In exceptional cases, RS can even cause early fractures during the first cycles of fatigue.
Therefore, a full understanding of how the RS acts on the surface or interior of the material is an advantage from the engineering point of view, with a greater possibility of gains in fatigue phenomena controlled by stresses typical for high fatigue cycles (WITHERS; BHADESHIA, 2001).
The RS in the component, in addition to being strongly influenced by the restrictions, is closely linked to the distance from the restrictions, as shown in Figure 5.
Figure 5 can be interpreted as the RS balance since, in both XX and YY planes, the intensity of the stresses has zero-sum, as well as between the planes, which is, in this case, considered as the ZZ plane.Thus, in addition to displacement restrictions, the position measurement of the RS is an important factor to be considered for the purpose of resolving different stress components.

METHODOLOGY
In order to determine the angles (Φ, ϴ) (Figure 1) of the crack propagation plane, and determine the fatigue life prevision without the influence of RS, a generic component was subjected to a set of three-dimensional loads, with completely random non-proportional traction F (N), moment M (Nm) and shear force V (N), represented in Figure 6b) and c), respectively.The loads are applied to the opposite face, which is the movement face of the fusion pool, in order to represent the action of any load and its effects on any point in the structure.We used the thermomechanical properties at 298 K shown in Figure 3 and considered only the elastic deformations.The stresses   and   in each t1(s), at points strategically positioned in the HAZ and MB, are determined from three-dimensional stresses in a computer simulation using the finite element method in the software Comsol Multiphysic, version 5.2®.
The evolution of the three-dimensional stresses σ x, σ y, σ z, τ xy, τ xz and τ yz obtained from the finite element method at any point is illustrated in Figure 6d).To maximize the Equation (1) by determining the best angles (Φ, ϴ), we transform the function of angles (Φ, ϴ) in the three-dimensional stresses in σ x′ σ y ′ σ z ′ τ xy′ τ xz′ and τ yz′ determining the values of the octahedral stress τ n and σ n .
The statistical tool in RainFlow's second uniaxial flow cannot be applied at this point to any case of uniaxial stress (BOSCO JÚNIOR, 2007) and (ASTM E 1049(ASTM E -85, 1985)).However, we can group all the values of the change in a single set of values at time t1, according to the method of Wang & Brown (WANG; BROWN, 1996) illustrated in Figure 7, steps 1 and 2. We obtain the strain values   ,   ,   ,   ,   ,   for each set of stresses    ,  ,  ,      and then use the adjustment Equation (5) (WANG; BROWN, 1996), to calculate the equivalent deformation in Equation ( 6), . (6) Finally, we use Multiaxial RainFlow, as illustrated in Figure 6e) and Figure 7 step 3.In Equation ( 6), the symbol ∆ is the variation of the value of the chosen strain element relative to its highest value calculated in Equation ( 5), and v is the Poisson's ratio.
We select only the pairs of strains forming the peaks and valleys of the deformations, and their respective stresses, grouped in multiples of half a cycle as in Figure 6e) and Figure 7 step 4.
The stresses corresponding to peaks and valleys of deformations are transformed in other planes until the sum of damages is maximized, as shown in Figure 6g), h), and i).The last operation performed in the system is represented by the algorithm between steps f), g), h), and i) in Figure 6 (only the arrows in red color), which illustrates the execution of the solver function in any calculation spreadsheet.The solver function varies Φ from 0 to 90 º and ϴ from 0 to 90º simultaneously, looking for the combination that maximizes the damage calculated by Equation (1) and Palmgren-Miner Equation, Figure 6i).For each pair of angles, it calculates and adds the damages, finding the shortest fatigue life using only the properties of the materials under multiaxial fatigue test, shown in Table 1.
The second sequence, illustrated in Figure 6a), simulates the movement of the thermal source, starting with the blue arrow in Figure 6, obtaining the evolution of temperature and RS at each time t2(s), with thermomechanical properties (Figure 3  The events described in Figure 8 are details Figure 6a).We imposed the displacement restriction on the finite element mesh nodes on the face perpendicular to the displacement face of the fusion pool, with the restriction on the deformations arising from welding, as shown in Figure 10.
Figure 8 schematically shows the information and correlation flow during the computer simulation under the boundary conditions represented in Figure 9 and Figure 10a) and b).
It is important to note that the successive points in Figure 9 are separated in the Y direction by 2.5 mm and in the Z direction by 1.25 mm.This difference is the result of the study's focus on the thermal and mechanical influence on the HAZ and MB.
Latin American Journal of Solids and Structures, 2020, 17(3), e262 11/26 Figure 10a), b) shows the thermal and mechanical boundary conditions, respectively.Doing several simulations based in results convergence authors decided choice the tetragonal element, for the computer simulation of the welding process, with dimensions varying between 1.25 and 2.5 mm, without chamfer, illustrated in Figure 6 a) and Figure 10a), b).In Figure 6, there are two complete simulation flows in which the values of RS and thermal cycles obtained in step a) occur or do not occur.In the flow described by the blue arrow in Figure 6, we would only be analyzing the influence of RS, completely excluding all other possible variations.
This method can be applied in any position in the HAZ / MB / FZ, mainly on the surface of the solid.In addition to discovering the angles at which crack propagation occurs, it will be possible to determine the location of the first crack, and the reduction in fatigue life.
Finally, all steps are repeated to determine the influence of welding and the source movement on the RS   ,   ,   ,  ,      at t(2).These stresses are subsequently aggregated according to (AHLSTRÖM, 2016) and (SRIVASTAVA et al., 2017) to obtain the stresses    ,  ,        , and redoing all calculations determines new angles and new projections of life in fatigue.There is no consensus in the literature regarding the exact time t2(s) of the computer simulation for the acquisition of RS.In this work, we only consider the values of RS obtained in time t2(s) = 86400 seconds.This time represents 24 h after the execution of the welding process.To reduce processing time, we opted to carry out all simulations with half of the component as a mirrored image of the other half, without losing the representativeness of the data, as shown in Figure 9, Figure 10 and Figure 11.
We performed only a computer simulation with 85686 tetrahedral elements, where the domain was segmented into two subdomains, as shown in Figure 11.The first subdomain had 74141 elements, and the second subdomain had 11545 elements, so that the system contained in total 379582 degrees of freedom and 6603108 different equations.The simulation of structural analysis is performed simultaneously with thermal analysis.Both simulations are non-linear, generating isotherms, micro deformations, and RS, mainly in the first domain.Both simulations were performed using the Comsol Multiphysics software, version 5.2, with the aid of the MUMPS -Multifrontal Massively Parallel Solver method to solve non-linear equations during 64 hours of simulation of the welding process in 86400 seconds.A surface, positioned perpendicular to the displacement of the thermal source, with pre-positioned points, separated by 2.5 mm from each other in the y-direction and 1.25 mm in the z-direction, as shown in Figure 9, is virtually inserted in the solid.Figure 11 shows the refinement of the mesh at the place of insertion of the perpendicular plane (Figure 9).This refinement is necessary since in this place temperature and RTs will be calculated, thus requiring the construction of a more refined mesh.

RESULTS
Figure 12 presents the results of the computer simulation of the welding process.The sequence of figures represents the evolution of temperature in time (t2).The isotherms due to the mobile, continuous, three-dimensional thermal source show the evolution during times t2 (s) of 0, 1, 5, 35, 200, and 400 s.At time t2(s) = 5 s, the thermal source is completely inside the solid, and the temperature reaches 2450 K, in the center of the melt pool, a result very similar to other studies.At time t2(s) = 35 s, the thermal source is not completely inside the solid any longer, and there is a significant reduction in temperature, again agreeing with the results from other studies, for example (SHEN et al., 2017) and (DARMADI; KIET-TIEU; NORRISH, 2014).
Figure 13 shows the temperature profile projected in the virtual internal plane (YZ) perpendicular to the movement of the thermal source positioned in the center of the component similarly to Figure 9, at times t2 (s) equal to 2, 4, 6, and 10 s.
Latin American Journal of Solids and Structures, 2020, 17(3), e262 14/26 In Figure 13, at the time t2(s) = 2s after reaching the plate, that the temperature reached approximately 2600 K.After the thermal source t2(s) increased to 10 s, temperature decreased, reaching approximately 800 K. Figure 14 shows thermal cycles of successive virtual points, spaced on the surface of the solid 2.5 mm apart in the direction of the Y-axis and 1.25 mm in the direction of the Z-axis, starting from the center of the fusion pool.In Figure 14, it is clear that points 0 and 1 (at 0 mm) and 10 and 11 to (at 2.5 mm) in the center of the melting pool reached the melting temperature 1749 K of the material -solidus temperature.However, the distant points, which are the main objects of our study, did not reach the temperature of fusion.Thus, all points between 5 mm and 22.5 mm from the center of the fusion pool did not undergo significant metallurgical changes characteristic of the HAZ.Nevertheless, we will analyze all points for the RS, because RS can present effects in regions relatively distant from the classic HAZ.In addition to the significant difference in the maximum values of temperature reached by each analyzed point, we found that different points had different cooling rates, with the points closest to FZ having higher cooling rates.Such variation explains the differences in mechanical properties between different points next to the HAZ since in Figure 3 and Figure 4, differences in thermomechanical properties are considerable.
Figure 15 shows the 3D projection of the simultaneous maximum temperature in several planes (YZ) perpendicular to the displacement of the thermal source.Note that, like in Figure 14, at the points on the surface (this figure is under the Y-axis), there is a significant variation of temperature demarcating the regions of FZ, HAZ, and MB.In Figure 15, the instantaneous maximum value of the thermal cycle (at t2(s)) isn't important, only the maximum value of temperature.The graph clearly delimits the FZ and the HAZ, since only the points with simultaneous displacements along Y and Z axis of less than approximately 5 mm are in the FZ, while the points positioned simultaneously between approximately 5 mm and 7.5 mm are in the classic HAZ.The points with coordinates greater than 7.5 mm, despite not being in the HAZ, are influenced by the RS formed in the vicinity.Figure 16 shows the RS at the time t2(s) = 86400 s on the x, y, z and xy, xz, and zy axes, for the consecutive spacing of 2.5 mm on the solid surface exactly aligned with the middle of the component (as illustrated in Figure 10).Knowledge of this dependence allows the maximization of the expression for the RS, as explained in Figure 5.A careful analysis of the behavior of the RS curves in the various vectors in Figure 16 reveals that at the points in the FZ from 0 to 2.5 mm, there are significant changes in the stress, in some cases even with a change of direction.The points located between 2.5 and 5 mm now maintain the trend observed in the points between 0 and 2.5 mm, and stabilize the variation vectors x, y, yz, or reverse the trends for the vectors z, xz, xy and z.
Notably, any changes in position, whether along the x-axis or z-axis, will result in significant changes that cannot be disregarded, as shown in Figure 17, in which we present, in a three-dimensional format, the change of RS along the Z vector due to the displacement along the Z and Y axes.
Again it is important to note that the measurement of the position of the RS in relation to the component geometry is extremely important due to the imposed restrictions and the balance of the RS in 3D space.In this particular case, all measurements of RS were performed in the center of a parallelepiped component, as shown in Figure 10.
In Figure 17, the evolution of RS is shown only in the Z plane for the points referenced in Figure 9.We note that the vast majority of the points have values very close to the value of 100 MPa, with the exception of the points in the FZ and HAZ, which have positive (traction) and negative (compression) values.
In Figure 17 and Figure 18 shows the evolution of RS in the Y plane.The vast majority of points do not present any values of RS, except in FZ and HAZ, where the values are positive.The Figure 19 shows similar behavior value of RS in FZ and HAZ is immensely larger than elsewhere on the X plane.
When comparing Figure 17, Figure 18 and Figure 19 with Figure 16 it is extremely important to point out that in Figure 16 only the values of the surface points are presented, whereas in the 3D graphs, points inside the solid are predominant.In a systemic analysis of 3D images, it is agreed that the oscillations in the X, Y, or Z planes are in the FZ and HAZ and tend to decrease due to the distance.The solutions developed in previous studies considered only the effects of constants , b, and  * in Table 1.We emphasize that the results shown here can undergo profound changes due to the above constants, as indicated in literature, but also depending on the position of the collection of RS and multiaxial stress values.
In addition to the above, we performed a set of 10000 simulations (t1), from which we obtained approximately 3340 oscillations.The cycles (peaks and valleys), grouped by the Multiaxial Rainflow method, were all multiplied by a fixed value representing the consecutive application of a set of 10000 stress fluctuations.
Figure 21 shows the angles (Φ, ϴ) varying from 0º to 90º.Note that there is little variation in the angles ϴ, which oscillates around the value of 85º.Note that only the points of the HAZ and MB and the solid surface are represented by points 20,30,40,50,60,70,80,90 in Figure 9.The variation of the crack propagation angles on the surface, shown in Figure 21, is only a variation as a function of position, so the reduced amplitude is in accordance with the theory, as there are no significant variations in the stress between various points.
In Figure 22, we illustrate in a schematic way the fatigue life predictions without the influence of the RS at the same points as in Figure 21.We note that there is a significant change in the fatigue life predictions in the points at a distance less than 17.5 mm from the line of locomotion of the fusion pool although there are no significant changes in the propagation angles of that crack.In a comparative analysis of Figure 24 and Figure 23, it can be seen that the existence of RS significantly changes the prediction of life in fatigue, with possible reduction by up to 50% of the expectation.In addition, there is a change in the trend observed in Figure 22, in which the values were of the order of magnitude of a hundred years for the reference points greater than 70.Such trend or the order of magnitude is not observed in Figure 24.
In the point-to-point comparative analysis of Figure 22 and Figure 24, it can be seen that the intensity of the RS at the point at 5 mm (reference point 20) reduced the fatigue life expectancy prevision.
In Figure 25, the evolution of fatigue life predictions in the points as a function of the angles (Φ, ϴ) is shown in a two-dimensional form, clearly demonstrating the maximization of damage for the pairs of angles (10º, 85º) in the situation without TR.

CONCLUSIONS
This work brought a new level of understanding of relationship between the welding process and its effects on the fatigue phenomenon.Although there is specialized software for computer simulation via FEM of the welding process with the possibility of calculating the residual stresses (RS), it is rare for software to connect the RS with the countless multiaxial fatigue prediction models, as was achieved in this set of comprehensive connected simulations.
Existing computer simulation methods for welding, RS and fatigue are available in open code, versatile and suitable for immediate application in different geometries, steel families, process variables, with low cost, plausible adjustments, and possible additions or deletions of variables, which are fully coupled with the formation of RS.
There are countless works of fatigue life prediction in the base metal, but there are few and limited works that focus on understanding the consequences of the formation of the HAZ and the relationship with cyclical mechanical loads, especially in the phenomenon of high cycle fatigue.Thus, this work adds another piece of knowledge in understanding the relationship between welding/fatigue, as seen in the bibliographic analysis.
The methodology of this research was designed from the beginning foreseeing the possibility of fatigue life prediction, crack emergence position, and crack emergence angles.Special focus was on the HAZ region in the presence or absence of RS, a challenge which we have solved at the computational level.
We developed a method, in partially open code, to prevision the crack location and propagation in steel structure, welded or not.In addition, we were able to predict the angles of propagation of a high cycle fatigue crack with totally random loads, something that only a few models are able to deal with statistically, making this work applicable in the majority of everyday situations.
Finally, the current technical standards are solely and exclusively focused on the definition of methodologies for the phenomenon of uniaxial fatigue.Our intention is to propose the addition of a constant adjustment to the Findley models, only applicable to welded materials.This objective clearly represents an advancement in the state of the art picture.
This article reported a new approach to a more accurate estimate of the fatigue limit under non-proportional and out of phase multiaxial loads.
Latin American Journal of Solids and Structures, 2020, 17(3), e262 22/26 The Fiedley criterion was carefully explored with preference for the Wang & Brown and Multiaxial RainFlow methods.We observed that in the analysis without influences of RS, it produced good prevision of life in fatigue, but also allowed the addition of the specific influences of the RS.

Figure 1 -
Figure1-The X, Y, Z coordinate system, and the X', Y', Z 'auxiliary system, showing a possible critical plane.

Figure 3 -
Figure 3 -Evolution of thermomechanical properties as a function of temperature for steel SAE 1020.

Figure 5 -
Figure 5-Representation of longitudinal    and transversal stresses    in segments with displacement restrictions on the upper, lower, and parallel faces of the thermal source displacement.

Figure 6 - 26 Figure 7 -
Figure 6 -Flow of advanced steps for determination of the critical plane with RS.
, Figure 4) taken into account.The boundary conditions of Figure 8 were obtained in the same virtual positions as the extraction of three-dimensional stresses.The positions were strategically allocated at predetermined points in the HAZ and MB, as shown in Figure 9, with the direction of displacement of the thermal source parallel to the x-axis, similar to Figure 2, to obtain the values   ,   ,   ,  ,      .

Figure 8 -
Figure 8 -Flow chart of the simulation of residual stress formation.Source: Adapted (SRIVASTAVA et al., 2017)

Figure 9 -
Figure 9-Reference points and point A, broken in both directions from position 0.

Figure 10 -
Figure 10 -Boundary conditions thermomechanical simulation of the welding process.a) thermal boundary conditions mirrored by symmetry, b) mechanical boundary conditions mirrored by symmetry.Source: Adapted (SERINDAG et al., 2017)

Figure 11 -
Figure 11 -Representative image of virtual elements, with necessary refinement in subdomain 1, referring to the position of the virtual board, Figure 9.

Figure 12 -
Figure12-The evolution of the isotherms due to the thermal source in movement.The times t2 (s) are 0, 1, 5, 35, 200, and 400 s.On the right is the Kelvin temperature bar.

Figure 13 -
Figure 13 -Illustration of a plane (Figure 9) perpendicular to the displacement trajectory of the thermal source positioned in the center of the fusion pool component through the plate, at time t2 (s) equal to 2, 4, 6 and 10 s.On the right is the Kelvin temperature bar.

Figure 14 -
Figure 14 -Evolution of the virtual Kelvin temperature from the center of the fusion pool, at successive points spaced 2.5 mm apart, referred to in Figure 9.

Figure 15 -
Figure 15-3D plot simultaneously correlating the maximum temperature K, and the position of the respective displacements on the Y-axis (surface of the solid) and on the Z-axis (into the solid).

26 Figure 17 -
Figure 17 -3D representation of RS (Pa) in the Z plane.The time of simulation t2 is 86400 s.The colored bar on right is provided for comparison purposes.

Figure 18 - 26 Figure 19 -
Figure 18 -3D representation of the RS (Pa) in the Y plane.The time of simulation t2 is 86400 s.On right, the colored bar on the right is provided for comparison purposes.

Figure 20 -
Figure 20 -Main stresses  1 ,  2 and  3 as a function of the distance from fusion pool.

Figure 21 -
Figure 21 -Variation of crack propagation angles (Φ, ϴ) as a function of position relative to the melt pool displacement line, without the influence of RS on the surface layer, as shown in Figure 9.

Figure 22 - 26 Figure 23
Figure 22 -Fatigue life prediction as a function of position in relation to the melt pool displacement line, without the influence of TR on the surface layer, as shown in Figure 9.

Figure 23 -
Figure 23-Variation of crack propagation angles(Φ, ϴ)as a function of position relative to the melt pool displacement line, with the influence of the TR of the superficial layer, as shown in Figure 9.In Figure24, representing the fatigue life prevision, we note that there is a significant variation starting from the maximum value of 55 years for (Φ, ϴ) around the value of 45 0 on the surface of the solid represented by the points20,  30, 40, 50, 60, 70, 80, 90 in  Figure 9.

Figure 24 -
Figure 24-Fatigue life prevision in years, as a function of the position in relation to the displacement line of the fusion pool, without the influence of TR in the superficial layer, as shown in Figure 9.

Figure 25 -
Figure 25-Change in the prediction of fatigue life as a function of the pair of angles (Φ, ϴ) without RS.

Table 1 -
Process characteristics, dimensions of the melting pool and material properties, steel SAE 1020.