Effect of laser shock processing on residual stress evolution in martensitic stainless steel multi-pass butt-welded joints

Laser shock processing (LSP) is an innovative approach, which effectively improves the mechanical behavior of metallic structures by introducing compressive residual stress. To evaluate the residual stress evolution in low-carbon 13Cr4Ni martensitic stainless steel multi-pass butt-welded joints induced by LSP, a two-step numerical simulation including welding analysis, at first, followed by LSP calculation with the simulated welding stress results being taken into account, was performed based on ABAQUS software. Effects of LSP parameters such as power density, spot size, overlapping rate and numbers of laser shock on the residual stress variations, were systematically investigated. To validate the reliability and accuracy of the numerical simulation, experiments of welding and LSP were conducted in sequence. The residual stress after welding and LSP were investigated by x-ray diffraction method. Results demonstrate that the simulated results show a good agreement with the experimental datas. The welding residual stress distribution is uneven. Larger tensile stresses appear on the weld surface and its adjacent heat-affected zone, which could be converted into high-level compressive stress after LSP. Furthermore, an ideal residual stress field can be obtained after two successive laser shocks with an overlap rate of 75% when the power density, spot diameter, and pulse width are 7.6 GW cm−2, 4 mm, and 25 ns, respectively.


Introduction
13Cr4Ni type low-carbon martensitic stainless steels are an economical option for the requirement of superior mechanical properties and higher corrosion resistance, and their welded components and structures, such as the end-stage steam turbine blades and hot-oil pump shafts, have been extensively used in high stress and corrosive environments [1,2]. However, higher welding residual stresses inevitably occur in the weld due to the uneven heating and subsequent rapid cooling during the welding process. Such welding stress (generally tensile stress) may affect the in-service properties of welded structures, cause crack initiation, accelerate crack propagation, and finally, even lead to disastrous consequences under the combined action of service load and environment [3]. This is unfavorable for the service performance and life of the welded components. Therefore, it is of great importance to mitigate or control the welding residual stress in the welded structures.
Compared with other techniques to eliminate residual stress, as a promising surface modification technique, LSP is expected to improve mechanical properties of the weldments in welding manufacturing, owing to its unique advantages of contact-free, good adaptability and flexibility in various conditions, small distortion and damage, clean and sustainable application [4,5]. During LSP, a short-pulse (dozens of nanoseconds (ns)) laser beam with high-power density (in GW сm −2 ) irradiates the material surface as shown in figure 1. The treated material will vaporize and form plasma after absorbing high laser energy. Because of the confining effect from the constraint layer (usually water), the rapidly expanding plasma is trapped resulting in the induced shockwave pressure produced on the material surface and continuously propagating into the material. When the induced shockwave pressure (up to several GPa) exceeds the dynamic yield strength of the treated material, plastic deformation is generated in the material surface and subsurface within a very short time (ns level). In this case, favorable compressive stresses will be generated and contribute to the performance improvement of material.
Laser shock parameters are of crucial importance in generating the desired compressive stresses during LSP. However, for want of the precise control about material behaviour and laser impact conditions during LSP procedure, setting parameters suitable for the required performances quantitatively and accurately, is difficult experimentally. Meanwhile, high cost, time-consumption, and relatively high reliability and stability requirements of laser systems has also limited the wider application in industries. As a particularly interesting tool, numerical simulation of LSP, which can efficiently predict and evaluate residual stress induced by LSP, help to further understand LSP process, reduce time and cost of experiments, and conveniently optimize the laser parameters for various LSP conditions, has attracted more attention of researchers. Fabbro, R et al [6] investigated the transmission of breakdown plasma generated in water during laser shock processing experiments theoretically and developed a numerical model based on a rate equation formalism has been to calculate the characteristics (peak irradiance and duration) of the laser pulses. Zhang et al [7] researched the residual stress hole on the surface of 304 stainless steel thin plate subjected to two sided laser shock processing by numerical simulation and experiment method. Luo et al [8] investigated the effects of surface curvature on tridirectional (radial, axial, and depth) residual stress distributions in the LSPed 316 L stainless steel specimens experimentally and theoretically and compared using finite element analysis. Keller et al [9] studied the influence of prestress on laser shock peening-induced residual stresses in the aluminium alloy AA2024 by applying the defined elastic prestress through experiment and simulation methodology to determine the prestress-residual stress relationship. Kaufman et al [10] studied the effect of laser parameters on residual stresses and corrosion fatigue of AA5083 both with (LSP) and without protective coating (LPwC) at multiple pulse densities. Ma et al [11] investigated the effects of constituent distribution and microstructure change on high-temperature creep properties of the aluminized layer in 321 stainless steel treated by laser shock processing (LSP). However, existing research about LSP are mainly focused on the laser-target material interaction mechanisms [6,12], effects of laser parameters on residual stress variation [8][9][10], and improvements in the mechanical performance of lasershock treated material [10,11,13,14]. There is little discussion on the LSP procedure analysis of the welded structures with the initial stress conditions (especially the welding residual stress) taken into consideration. Also, little effort has been devoted to simulating the role of LSP on the residual stress variation in the welded joints and structures during LSP. Hence, the residual stress evolution in the welded joints subjected to LSP is worth studying with the welding stress states being considered.
In this study, taking welding residual stress as the initial stress conditions, a novel three-dimensional (3D) numerical analysis of residual stress evolution in low-carbon 13Cr4Ni martensitic stainless steel welded joints subjected to LSP was carried out. Effects of laser parameters on residual stress evolution were investigated in detail. Based on the simulated results, experiments of welding and LSP were performed in sequence under the same condition as those of simulation. The residual stress after welding and LSP were measured by x-ray diffraction method. The simulation results of welding and LSP were verified and compared with the experiments. An optimum laser parameters can be obtained, and some reliability optimization measures for the LSPed welded joints will be carried out in future based on the presented results.

Numerical simulation procedures
The numerical simulation, including welding and LSP analysis, was shown in figure 2. The welding simulation was performed firstly through a sequentially uncoupled thermo-mechanical analysis. Based on the calculated welding results, LSP simulation was subsequently carried out. To accurately capture the elasto-plastic response of the material induced by laser shock, the simulation of dynamic laser shock procedure was conducted by using the ABAQUS/Explicit program. The static analysis of LSP was completed through the ABAQUS/Standard program with the obtained transient results input so as to release the strain energy, from the dynamic shock, stored in the materials.

Finite element model and prediction of the welding procedure
The multi-pass butt-welded joints with the dimensions of 60 mm × 40 mm × 6 mm were composed of two 13Cr4Ni martensitic stainless steel plates as shown in figure 3. Figure 3(a) shows the model dimensions and welding sequences. Figure 3(b) is the FE model, in which the welding heat source is assumed to move along the Z -direction on the weld centerline. In welding simulation, the element types used for temperature and stress calculation were DC3D8 and C3D8R, respectively. To acquire more precise results of temperature histories and welding residual stress, a finer mesh (the element size 0.1 mm in minimum) was employed in the weld zone, while a coarser meshes (the element edge length 0.2-2.0 mm) was used for the adjacent heat affected zone and the other parts of the model. There are 56,750 elements and 63,301 nodes in the 3D model.  To facilitate stress analysis, it is assumed that the simulated residual stress along welding direction (i.e. Z -direction) to be the longitudinal residual stress, the residual stress perpendicular to welding direction (i.e. X -direction) is the the transverse residual stress and the stresses along Y -direction is the through-thickness stress. The material properties of the weld metal were assumed to be same with the base metal. Considering the effect of transient temperature fields on the welding residual stress, the temperature-dependent thermo-physical and thermo-mechanical properties were employed, as shown in figure 4.

Thermal analysis
To acquire the transient temperature distributions during the welding simulation the thermal analysis was executed firstly. Each weld was sequentially deposited with the movement of the welding heat source through the element birth technique. According to Goldak et al [16] and Kong et al [17], the double ellipsoidal model shown in figure 5 was used and modeled by a volumetric heat source through the self-developed subprogram.The front and rear heat flux density distributions are expressed as equations (1) and (2), respectively.   In addition, considering heat losses, both the heat convection and radiation are adopted as the boundary conditions according to Newton's law of cooling. Heat losses due to the combined effects of the convection and radiation are considered and the total temperature-dependent heat transfer coefficient h is expressed as follows [18]: where T is the local temperature of the weld surface.

Mechanical analysis
The mechanical analysis was sequentially conducted with the temperature field results as the input data. The same FE meshes associated with stress element and thermo-mechanical properties were applied. To prevent the displacements of the FE model the boundary constrains shown in figure 3(b) were adopted. In order to introduce of the welding residual stress more conveniently and accurately, the same geometric model and mesh sizes used in welding analysis were also adopted, and the corresponding stress element type DC3D8 was employed in LSP simulation. As the initial stress conditions, the welding stress results obtained from welding simulation were introduced the LSP model in advance. Then LSP simulation was carried out at one-side on the weld surface of the welded joints.

Laser shock loading
Laser shock wave loading is a complicated nonlinear dynamic problem Generally, it was assumed that the pressure distribution over the area covered by laser pulse was uniformly. According to previous research [19], the laser-induced shock wave pressure can be simplified and modeled as a transient pressure with a specific spatial profile, and the correlative characteristics of which can be determined depending on the given laser parameters. A possible simple triangular approximations can be employed to describe the pressure-time history of the laser-induced shock wave pressure in the simulation. The temporal variation of the shock wave pressure can be expressed as shown in figure 6. The pressure rises rapidly to the peak pressure for the first few nanoseconds and then decays gradually to zero within the following rest time. The duration of the laser-induced shock wave is roughly three times the full width at half maximum of the laser pulse or even longer. In current LSP simulation, according to Fabbro's model [20], the peak pressure P 0 of the pulsed laserinduced shock wave could be evaluated as: where P 0 is the peak pressure (in GPa); a is the fraction of the internal energy devoted to the thermal energy (generally 0.1 0.25, a =~here 0.2 is used) [21]. I 0 is the incident laser power density (in GW cm −2 ), which can be determined by the given laser shock parameters: where E (in J), r 0 ( in mm) and t (in ns) are the laser output energy, the radius of the laser spot and the pulse duration of the laser shock wave, respectively. Generally, the duration τ of the laser pulse employed in the LSP simulation is in the range of 10-50 ns. Z (in / g cm s 2 ( ) ) is the reduced shock impedance between the confining layer and the ablative layer material , which could be estimated by: here Z 1 for the confining water is 0.165 10 6 / g cm s , 2 ( ) Z 2 for the ablative layer of the stainless steel is 3.96 10 6 / g cm s 2 ( ) [22].

Material constitutive model
During LSP procedure, the laser treated material underwent an extremely high strain rate exceeding / 10 s. 6 To accurately simulate the materials' dynamic response to high velocity shock waves, the appropriate material constitutive model is needed. The well-known Johnson-Cook strain sensitive plasticity model was selected to simulate the high strain rate behavior of the material [21,23]. The flow stress σ according to the Johnson-Cook model is expressed as follows: where e represents the equivalent plastic strain, A is the initial yield strength at room temperature, B is the strain hardening modulus, and n is the strain hardening coefficient. C represents the strain rate sensitivity, e  is the strain rate, 0 e  is the reference strain rate, T* is the homologous temperature expressed as with T r the reference temperature and T m the melting temperature, while m denotes the thermal softening coefficient. In the case of LSP, the thermal effect of the Johnson-Cook equation was not considered due to the shielding effect from the ablative layer, and that the local temperature increment induced by the shock waves is very small [22]. The Johnson-Cook model material parameters of the 13Cr-4Ni martensitic stainless steel used in this simulation are given in table 2.

Experimental procedures
A commercial 13Cr4Ni type low-carbon martensitic stainless steel plate (purchased from Honghuida Commercial and Trading Co. Ltd, Dalian, China) was chosen as the base metal. The filler metal is the flux-cored wires of 410NiMo martensitic alloy with diameter of 1.2 mm. The nominal composition of the base metal and filler metal are listed in table 3. Two as-received 13Cr4Ni martensitic stainless steel plates were into the rectangular samples by wire-cut electrical discharge machining. Prior to welding, the plates were mechanically polished and cleaned. Figure 7 shows the configuration dimensions and the clamping conditions of the buttwelded joints, in which a single V-groove of 60°was used. Two plates were welded by manual GTAW under the condition of the same welding sequences and parameters as those chosen in welding simulation. The welding parameters of welding voltage, current and velocity, as well as the preheat and inter-pass temperatures are shown in table 4. LSP experiments of the welded joints were performed by using a Q-switched Nd:YAG laser system operating at a repetition rate of 1 Hz with the wavelength of 1064 nm. To ensure the desired coverage of the treated area through successive overlapped laser impacts, specimens were vertically fixed to a self-designed clamp devices and driven to move up and down, left and right. Samples were classified into different groups according to the laser parameters, and compared with the simulation results subjected to the same laser shock condition. Here, laser scanning speed ν is about 1.6 mm s −1 , the step distance s between two adjacent laser spots is about 0.8 mm.
During LSP experiments, a black paint as the opaque overlay was painted to the surface of shocked zone of the specimens. The flowing water with a thickness of about 2.5 mm was chosen to be the confining overlay and fully covered the specimen surface waiting to be treated. The comparison of macroscopic appearance of the specimens with and without LSP is shown in figure 8.
The residual stresses after welding and laser shock were measured on an x-ray diffractometer (Proto, Taylor, MI, USA) using the sin2 y method with CrKα (in 2.2897 nm) radiation at 2 139 , o q = respectively. The principle and computational formulas of x-ray residual stress measurement can be found in the literature [2]. To ensure the accuracy in operation of the stress measurement, the specimens surface irradiated by x-rays should keep a plane stress state during the measurement. The measurement locations are shown in figure 9. The   measurements were carried out at the weld surface along and perpendicular to the weld centerline as well the mid-section along thickness-direction. To measure the residual stresses distributed along the thicknessdirection, the layers of specimens were progressively removed by the electro polisher with CH 3 OH:H 2 SO 4 electrolytic solution. The obtained results were considered to be the longitudinal residual stress, transverse residual stress and the residual stress in thickness-direction, respectively.

Results and discussion
4.1. Welding simulation results and validation Figure 10 shows the simulated welding residual stresses. Obviously, the distribution of welding residual stress is uneven. Larger longitudinal tensile stresses shown in figure 10(a) occur on the weld surface and its adjacent fusion zone between the upper welded passes and the base metal. Larger transverse tensile stresses shown in figure 10(b) appear in the fusion zones between the middle welded passes and the base metal. In the case of the residual stresses distributed through thickness direction, as shown in figure 10(c), the larger tensile stresses generate in the weld centre zone. Through the analysis and contrast of these residual stresses results, it can be found that the residual stress in thickness-direction is smaller than longitudinal and transverse residual stresses.  To verify the validity of welding simulation, the computed residual stress were compared with the measurements. The comparison between them are presented in figure 11. It can be seen that the simulated results are in good agreement with the experimental values. This indicates that the predicted welding residual stresses are reasonable, and the presented calculation of the welding analysis is reliable.

Verification of mesh-sensitivity
Meshing generation is an important factor which affects the simulation results. Therefore, to ensure the efficiency and accuracy the suitable mesh density was necessary. To determine a suitable mesh size for LSP simulation, without considering the initial residual stress state, the mesh convergence analysis was carried out. Six different mesh sizes of 0.5 mm, 0.2 mm, 0.1 mm, 0.05 mm, 0.02 mm and 0.01 mm were considered in the laser peened region for mesh refinement, respectively. However, in the case of 0.01 mm, the calculation is not convergent and the results are failed to be obtained. The comparison of residual stress after LSP with different mesh sizes is shown figure 12. It can be seen that LSP simulation was sensitive to mesh density. The contrast shows that the element size of 0.1 mm is seemed sufficient to obtain accurate results without huge computational costs. Therefore, to capture accurate simulated results and save calculation cost, the mesh size of 0.1 mm is an ideal option for both welding and LSP simulation.

Validation of LSP FE model
To verify the reliability of the FE model used in LSP simulation, without considering the initial residual stress, LSP simulation was performed and validated by the experimental results presented in the literature [24]. Experiments of laser shock peening was performed on the root part of a steam turbine blade (made of DINX10CrNiMoV1222 martensitic stainless steel) by Sundar R et al [24] with an indigenously developed flash lamp pumped electro-optically (E-O) Q-switched Nd:YAG laser system. The same laser parameters as Sundar R. used in experiment are also adopted. The comparison of the computed results with those of Sundar R were shown figure 13. It can be seen that both the longitudinal residual stress and residual stress along thickness-  direction obtained from the presented LSP simulation are in agreement with the experimental results. Therefore, the FE model used for LSP simulation is reliable.

Influence of laser parameters on residual stress evolution
To evaluate the influence of peak pressure P 0 on residual stress variation conveniently, laser power intensity I 0 was introduced during LSP simulation. Here, five I 0 of 3.2 GW cm −2 , 5.3 GW cm −2 , 7.6 GW cm −2 , 13.6 GW cm −2 , and 21.2 GW cm −2 were used to predict the residual stress variation after single laser shock at the center of the weld. Meanwhile, the same laser parameters such as pulse width of 25 ns, spot radius of 2 mm normally used in existing experiments [25] were employed. Figure 14 shows the transverse residual stresses distribution over the welded joints surface marked in red under different I . 0 For comparison, the transverse welding residual stress is also added.
From figure 14(a) it can be seen that the transverse residual stress distribution over the laser-treated regions has changed after single laser shock. Due to the existence of larger welding tensile stress, the compressive stresses can be generated until I 0 exceeds 3.2 GW cm −2 . The maximum compressive stresses obtained in the shocked region increase with increasing I . 0 When I 0 is continuously increased to 21.6 GW cm −2 , 'residual stress hole' (compressive stress obtained in the impacted region center is smaller than that of the surrounding areas) appears on the laser treated surface. As shown in figure 14(b), when I 0 increased from 5.3 GW cm −2 , to 7.6 GW cm −2 , and 13.6 GW cm −2 , the maximum residual stresses were −30.39 MPa, −93.69 MPa, and −118.47 MPa, respectively. When I 0 closed to 21.6 GW cm −2 , the corresponding residual stress obtained in the treated region was −116.76 MPa. Obviously, the magnitude of the compressive residual stresses increased gradually to saturation with I 0 increasing to its threshold. This phenomenon may relate to the work hardening and strain hardening behavior of the welded joints during LSP treatment.   Figure 15 is the residual stress distribution over the mid-section and stress variation through thickness direction after single LSP with different I . 0 As shown in figure 15(a), the compressive stress obtained along thickness-direction shows a pronounced increasing tendency with the increase of I 0 . Figure 15(b) shows the amplitude and depth of the compressive stresses on the mid-section along thickness-direction. When I 0 increased from 5.3 GW cm −2 to 7.6 GW cm −2 and 13.6 GW cm −2 , the corresponding compressive stress increased by 208% (from −30.39 MPa to −93.69 MPa) and 26.5% (from −93.69 MPa to −118.47 MPa), respectively. The depths of compressive stress along Y -axis increased by 81.25% (from 0.48 mm to 0.87 mm) and 34.48% (from 0.87 mm to 1.17 mm), respectively. When I 0 increased to 21.6 GW cm −2 , the maximum and depth of compressive stress decreased by 1.4% (from −118.47 MPa to −116.76 MPa), and 18.8% (from 1.17 mm to 1.39 mm). Apparently, both the amplitude and depth of the compressive residual stress increased and tended to saturation when I 0 increased to its threshold. The similar phenomenon has already been reported by Fang et al [26]. A distinct improvement of residual stress can be acquired when I 0 increased to 7.6 GW cm −2 . This means that the favorable compressive residual stress can be obtained by single laser shock with I 0 of 7.6 GW cm −2 .
As mentioned above, the compressive residual stress can be obtained in the LSP-treated region after single LSP. However, since the impacted region is small, the stress distribution induced by single LSP was non-uniform and little improved in the less affected area. Therefore, overlapping of laser shock may be efficient to produce more uniform residual stress distribution in relatively larger range of the welded joints. The overlapping rate f can be expressed as: / R 2 100% ( ) f w =´where w is the coincidence length of two adjacent laser spots, R is the laser spot radius. Figure 16(a) shows the schematic of overlapping loading on the laser treated surface. To investigate the effects of overlapping LSP on residual stress, five overlapping rates of 0%, 25%, 50%, 75% and 85% (shown in figure 16(b)) were selected, with the laser power density of 7.6 GW cm −2 , pulse width of 25 ns, and spot diameter of 4 mm. Meanwhile, it was assumed that the material properties of the welded joints remain the same during the overlapping LSP. Actually, work hardening will inevitably occur in some overlapping regions during repeated laser impacts. The advantages and also the disadvantages of the Johnson-Cook mode is accounting for strain hardening, strain rate effects, and thermal effects (here neglected) on the flow stress in isolation, which may result in the calculation error of the LSP simulation results. Furthermore, the fluctuation ratio was also chosen to evaluate the effects of overlapping rate on the residual stress distribution. Here, the  -´in which max l and min l are the maximum and minimum residual stress, respectively. It can be seen that the smaller the fluctuation ratio y is, the more uniform the resultant stress distribution. Figure 17(a) depicts the surface longitudinal residual stress distributions on the marked area of the welded joints with five .
j Compared with the as-welded joints, the tensile residual stresses are entirely absent, and a relatively uniform distribution of compressive stresses occurs in the overlap surface of the treated region of the weld joints after LSP. Figure 17(b) shows that when j increases, the longitudinal stress curve tends to be smooth. The maximum compressive stresses with f changing from 0% to 25%, 50%, 75% and 85% were −150.03 MPa, −210.67 MPa, −252.72 MPa, −285.02 MPa, and −290.28 MPa. As f increased, the magnitude of the surface longitudinal compressive stress gradually increased. This phenomenon can be attributed to the fact that the previously shocked zone in the overlapping region may be treated again by the laser-induce shock wave from subsequent laser shocks. That is to say, the overlapping region of the weld surface suffered repeated LSP treatments compared with the single LSP case. In this case, a higher compressive residual stress can be obtained. Meanwhile, the corresponding y were 85.1%, 65.4%, 38.9%, 10.08% and 10.71%, respectively. Apparently, when f increases, y of the compressive residual stresses continuously decreases to saturation. This further proves that overlapping LSP can improve the residual stress distribution of the welded joints. Figure 18(a) shows the longitudinal residual stress distribution on the mid-section of the welded joints induced by five .
f As can be seen that the compressive stress distribution gradually tends to be uniform.
Moreover, the higher , f the more even the compressive stress distribution in the LSP treated region of the welded joints. As is shown in figure 18( [27]. This result may be related to work hardening of the material from the  repeated laser impact. Additionally, it can be seen that high-level compressive residual stresses fields, with good uniformity and larger depth, can be acquired when the overlapping LSP rate is 75%. Compared with the single LSP case, the magnitude and depth as well the distribution of the compressive stress induced by overlapping LSP have been significantly improved. As the overlapping rate gets closer to a certain limit, saturation occurs and the magnitude and depth of the compressive stress can no longer be increased even increasing the overlapping rate. In such case, applying successive LSP shocks maybe an option to overcome this limitation. To investigate the effect of multiple LSP shots on the residual stress evolution, five different successive LSP shots were employed with the overlapping rate of 75%, laser power density of 7.6 GW cm −2 , pulse width of 25 ns, and spot diameter of 4 mm. Figure 19 shows the surface longitudinal residual stress distribution after multiple LSP shots. As shown in figure 19(a), the distribution of surface longitudinal compressive stress tends to be more even and stable as the number of multiple LSP shots increased. The corresponding magnitude of compressive stress (shown in figure 19(b)) increased evidently compared with that of the single shock. The average compressive stress was increased by 20.67% from −204.23 MPa to −246.45 MPa, and 9.13% from −246.45 MPa to −268.94 MPa after the second and third laser shock, respectively. The increments of the average compressive stresses were 6.54% (from −268.94 MPa to −286.53 MPa), and 3% (from −286.53 MPa to −295.13 MPa) after the forth and fifth LSP shot. It can be seen that the improvement of compressive stress is almost steady after two or three successive LSP impacts with the shock number continually increasing. This may relate to the work hardening in the surface layer of the weld from the previous shocks during the multiple LSP process. Figure 20 shows the longitudinal residual stress distribution over the mid-section of the welded joints after multiple LSP shots with an overlapping rate of 75%. From figure 20(a), the compressive stress gradually closed to saturation with the shock number rising. As shown in figure 20(b), after five successive laser impacts, the maximum compressive stress were increased by 12.2% (from −234.16 MPa to −262.73 MPa), 7.83% (from  −262.73 MPa to −282.11 MPa), 3.47% (from −282.11 MPa to −291.9 MPa), and 3.16% (from −291.9 MPa to −301.12 MPa), respectively. The corresponding depths increased by 13.58% (from 2.003 mm to 2.275 mm), 10.68% (from 2.275 mm to 2.518 mm), 6.91% (from 2.518 mm to 2.692 mm), and 2.38% (from 2.692 mm to 2.756 mm), respectively. It can be seen that both the amplitude and depth of the compressive stress can be improved by multiple LSP shots. Multiple LSP treatment not only increases the magnitude of compressive residual stress, but also drives the compressive stress deeper into the weld surface. According to the contrastive analysis of five successive laser impacts, the improvements of the magnitude and depth progressively declined and reached saturation with the increase of the shock number. The improvements after the second shot are significantly higher than those of the third, fourth, and fifth shot. This may be due to the work hardening from the previous shocks on the shocked region in the weld surface, which decreases the attenuation of the subsequent laser induced shockwave and prompts a higher peak pressure to permeate deeper into the weld surface during the successive laser shocks.

Validation of LSP parameters
To validate the effects of these optimum laser parameters obtained from LSP simulation on residual stress variation, LSP experiment were conducted under the same conditions mentioned above. Laser parameters such as a laser power density of 7.6 GW cm −2 , pulse width of 25 ns and spot diameter of 4 mm, two successive laser impacts with an overlapping rate of 75% were employed in the experiment. Comparison of the simulation and experiment results is shown in figure 21. Figure 21 shows the distribution of surface longitudinal residual stress and the stress profiles along thickness-direction on the mid-section. It can be found that both the level and distribution of longitudinal residual stress and stress along the thickness-direction from simulation and experiment present the similar tendency. However, it should be noted that there is little differences between the simulation and experiment results. The simulated residual stress was lower than those of the experiments. The causes responsible for this phenomenon may be the variation of material properties. The material properties of the welded joints was assumed to be constant during LSP simulation. However, work hardening will inevitably occur, especially in the repeated laser impact regions. Johnson-Cook mode used in LSP simulation is mainly accounting for strain hardening, strain rate effects and thermal effects (here neglected) on the flow stress in isolation, which may also result in the calculation error between the LSP simulation and experiment results. Furthermore, it's too hard to accurately describe the constitutive relation, considering the potential changes of work hardening, grain refining and so on, which happened in the laser treated region as those might occur in experiments. So, at this point the reasonable error was allowed to live.

Conclusions
In this study, the residual stress evolution in Cr13Ni4 martensitic stainless steel multi-pass butt-welded joints subjected to LSP were systematically investigated. Based on a same FE model, simulations of welding and LSP considering welding residual stress were carried out sequentially. Experiments of welding and LSP are conducted under the same parameter conditions as simulation. The FE model, simulated results as well the optimal parameters were all validated experimentally.
(1)Results of numerical simulation and experiments indicated that the simulated residual stress showed a good agreement with the experiment data. The distribution of welding residual stress is uneven, and larger tensile stresses can be converted into compressive residual stress after LSP treatment.
(2)Effect of laser parameters such as laser power density, overlapping rates, and multiple LSP shots on the residual stress variation was remarkable. Higher laser power density is beneficial to a higher amplitude, deeper compressive residual stress. Compared with single LSP, overlapping LSP is good for a high-level of compressive stress with good uniformity. Multiple LSP impacts with a certain overlapping rate may be a favorable choice to achieve a more uniform distribution and larger affected depth of compressive stress.
(3)The desired compressive stress field can be achieved by an optimum LSP parameters constitution, which can effectively avoid the 'residual stress hole' during LSP. Results obtained in this study will provide evidence in theory for further experiment and production of laser shock stainless steel welded joint, and can be considered as a crucial reference for laser shock manufacturing.