1 Introduction

Manufacturing of parts as a single unit is some time impossible since the part may be made up of so many intricate shapes. In such situations, different methods are used in the manufacturing system to join the parts to obtain the complete unit. For producing that product one of the most desired demands is joining. Laser welding is one of the most important modern and advanced joining process which has the ability to weld the structural components having macro to micro level thickness of various materials like steel, high temperature alloy, plastic etc. The main advantages of laser welding over conventional welding processes are: narrower weld zone, high depth of penetration into the material, lower heat affected zone (HAZ), high precision and accuracy and the requirement of single sided access to the workpiece only. Potential benefits realized by the application of laser welding include reduced flange widths, increased structural strength and high speed automated processing. In this present study CO2 laser welding is used to join thin AISI 304 Steel sheet.

From published literatures, it has been observed that very few researchers were studied on CO2 laser welding process. Mei et al. (2013) studied on late-model fiber laser and CO2 laser welding of auto body galvanized steel and common cold rolled steel. They studied on effect of process parameters on surface formation, bead shape and mechanical properties of similar and dissimilar metal welded joints. They found that the fiber laser welding is better for tensile-shear capacity and hardness as compared to CO2 laser welding. Zhang et al. (2014) performed the welding of 12 mm thick stainless steel plates by using 10 kW high power fiber laser to investigate microstructure and mechanical properties. They observed that the focal position is a main parameter in high-power fiber laser welding of thick plates. They also observed that for achieving good full penetration joint there is a critical range of welding speed. Lakshminarayanan and Balasubramaniam (2012) investigated the microstructure and mechanical properties AISI 409 M stainless steel welded plates by using 3.5 kW CO2 laser power source. They observed that the coarse grain structure in base metal and fine dendritic grains structure in welded zone which resulted in more tensile strength of welded joints as compared with base metal. Sathiya and AbdulJaleel (2010) developed Artificial Neural Networks (ANN) programming in MatLab software to study the effect of input parameters such as laser power, scanning speed and focal position on weld bead geometry of laser welded AISI 904L super austenitic stainless steel plates. Suresh Kumar (2014) investigated temperature distribution in various zone of laser welded material. He observed that the size of the heat affected zone is directly proportional to the heat input and the type of grain structure depends on cooling rate. He also observed that the cooling rate is directly proportional to the laser scanning speed, thickness, thermal conductivity of the material and inversely proportional with the heat input.

Piekarska and Kubiak (2011) developed mathematical and Finite Element (FE) models on laser–arc hybrid welding of S355 steel. They studied the effect of welding parameters such as distance between laser beam and arc, laser scanning speed and heat sources power distributions on size and shape of weld geometry. Tu et al. (2003) performed the 20 kW CO2 laser welding on mild steel. They observed that the plasma absorption has an important effect in certain welding conditions. The plasma absorption gradually decreases from the top portion of key-hole to the bottom portion of key-hole which widened the weld bead shape at top portion. Moraitis and Labeas (2008) developed a FE model based on the keyhole theory in order to investigate the stress, strain and distortion fields over the welded aluminum 6061-T6 plates. They used 4 kW CO2 laser with 2 mm focused radius and 50 mm/s welding speed on lap-joint of aluminum specimens. They observed that the strains distribution was non-uniform and the distortion was maximum at the adjacent of the laser entry. Tsirkas et al. (2003) studied on 3D finite element model (FEM) for predict the distortions of laser welded butt joint of AH36 shipbuilding steel plates. They observed that large temperature gradient developed near to heat source. Abdulateef (2009) studied the effect of laser spot welding process parameters on residual stresses and temperature distribution over the welded mild steel plates based on analytical and finite element analyses. They found that the maximum temperature developed at weld line and inversely decreases away from the weld line. They also studied on the effect of laser welding parameters on residual stresses. Armentani et al. (2007) developed a FE model using element “birth and death” technique to estimate the effect of temperature dependent material properties and weld efficiency on transient temperature distribution and residual stresses in butt weld joints of Inconel Alloy 600. They found that the tensile longitudinal residual stresses decreases with the increases of thermal conductivity. The temperature distribution over the welded plates is also highly affected by the thermal conductivity of the material. Mohammed (2011) developed a three-dimensional FE model for the CO2 laser welding on Al-6061-T6 alloy. She used a Gaussian heat source for the finite element modeling. Ronda and Siwek (2011) developed numerical model using computational fluid dynamics (CFD) software, Fluent for determining the interaction between key-hole and welding pool during laser welding. They observed that shape and depth of melted zone depends on the surface tension, the sulphur content and the surface temperature at conduction welding mode.

Deng and Murakawa (2008) used FE method to simulate welding distortion in a low carbon steel butt welded joint with 1 mm thickness. They observed that there was no temperature gradient through the thickness of the 1 mm thin welded plate. Jiang et al. (2012) carried out the three-dimensional finite element analysis by applying a heat source of Goldak double ellipsoidal distribution. They have investigated that the temperature field and residual stress in the repair weld of a stainless steel clad plate. They observed that the repair length has effect on yield strength, longitudinal stress and the transverse stress in the weld and HAZ.

So, from the above literature review it is observed that there are very few published literatures on FEM simulation of CO2 laser welding process. But there is no literature found on 3D FE simulation using volumetric heat source of CO2 laser welding of AISI 304 stainless steel thin plate. In this present study both 3D FE and experimental investigations of CO2 laser butt welding on thin AISI 304 Steel sheet were carried out for determining the transient thermal history, residual stresses and residual deformation for butt welded plates. In this model temperature dependent thermo-mechanical properties of AISI 304 steel, element birth and death technique and moving volumetric heat source were used. The weld bead geometry was also incorporated in this present study. The numerical results were validated by experimentally obtained results.

2 Experimental Details

In the present study, the single-sided butt welding was performed by 2.5 kW continuous wave CO2 laser machine (LVD, model: Orion 3015) which is shown in Fig. 1b. In this experiment AISI 304 stainless steel sheets were used as study material. The dimensions of each plate are \(100 \,{\text{mm }} \times 100 \,{\text{mm }} \times 1\, {\text{mm}}\) as shown in Fig. 3. Several experiments trail runs were performed for different operating parameters to determine the acceptable range of parameters. From the trail runs the welding parameters were taken as laser scanning speed (300, 400 and 500 mm/min) and laser power (1300, 1500 and 1700 W). The tacking method performed by TIG welding for good alignment of the edges.

Fig. 1
figure 1

Experimental set-up: a temperature measuring set-up b experimental set-up c deformation measuring set-up

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The sample were fixed with a fixture and put on the laser machine table with the help of clamp as shown in the Fig. 1b. The temperature distribution over the plate was measured by K type thermocouple with the help of Agilent data accusation system. After competition of welding process the welded samples were cooled to the ambient temperature. The angular deformation of the sample was measured with the help of coordinate measuring machine (ZeissTM, model: Vista, Oberkochen, Germany) which is shown in Fig. 1c.

3 Finite Element Analysis Details

The transient temperature distribution, residual stresses and distortion was investigated by means of coupled thermo-mechanical Finite Element Method (FEM) for CO2 laser but welding of AISI 304 austenitic steel. The temperature dependent thermo-mechanical properties was incorporated in 3D FEM model and the surrounding temperature of 26 °C to be considered as ambient temperature. The value of convection coefficient was taken as 20 W/m2 °C. After the welding phase was finished, the time step was progressively increased up to 450 s to cool down the plates to ambient temperature. Time and distance travelled by laser source was used to calculate the welding speed. To calculate the results, two steps were considered: (1) temperature history from transient thermal analysis (2) for the calculation of residual stresses and distortion, temperature history taken as input of thermal load. Here the effect of welding parameters i.e. laser welding speed and laser power on thermal history, residual stresses and distortion were studied.

The thermo-mechanical analysis was carried out in the present work for laser welding is represented in a block diagram as shown in Fig. 2.

Fig. 2
figure 2

Thermo-mechanical analysis of laser butt welding

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3.1 Thermal Analysis

The governing differential equation for heat conduction for a homogenous, isotropic solid without heat generation in the rectangular coordinate system (x, y, z) can be expressed as:

$$\tfrac{\partial }{\partial x}[K\tfrac{\partial T}{\partial x}] + \tfrac{\partial }{\partial y}[K\tfrac{\partial T}{\partial y}] + \tfrac{\partial }{\partial z}[K\tfrac{\partial T}{\partial z}] = \rho c\tfrac{\partial T}{\partial t}$$
(1)

where \(K\) = thermal conductivity, \(T\) = temperature, \(\rho\) = density of the material, \(c\) = specific heat, t = time.

The following boundary conditions were considered.

  1. (i)

    Initial condition

A specified initial temperature for the laser welding that covers all the elements of the specimen:

$$T = T_{{\infty }} \quad {\text{for }}t \, = \, 0$$
(2)

where \(T_{{\infty }}\) is the ambient temperature.

To develop second boundary condition, we consider energy balance at the work surface as:

Heat supply = Heat loss.

  1. (ii)

    First boundary condition

A specific heat flows acting over surface S1 (as shown in Fig. 3):

$${\text{q}}_{n} = q_{\sup }$$
(3)

The quantity \(q_{n}\) represents the component of the conduction heat flux vector normal to the work surface. The quantity \(q_{\sup }\) represents the heat flux supplied to the work surface in \(\frac{W}{{m^{2} }}\), from an external laser heat source arc.

$$q_{n} = \left\{ q \right\}^{T} \left\{ n \right\}$$
(4)

where \(\left\{ n \right\}\) = unit outward normal vector.

Fig. 3
figure 3

Schematic diagram of laser welded butt joint

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  1. (iii)

    Second boundary condition:

Considering heat loss due to convection over surface S2 (Newton’s law of cooling)

$${\text{Q}}_{\text{conv}} {\text{ = h }}\left( {{\text{ T}} - {\text{ T}}_{{\infty }} } \right)$$
(5)

where h = convective heat transfer coefficient (W/m2K), T = plate temperature, T ͚ = ambient temperature.

3.2 Mechanical Analysis

Welding structural distortion is the result of thermal stresses. However, an accurate prediction of distortion is extremely difficult because thermal and mechanical properties of welding include local high temperatures, temperature dependence of material properties and a moving heat source. The structural analysis of welding is involved with thermal stress and material non-linearity. The formulations for stress–strain evaluation (Fanous et al. 2003) used in the present study are presented below.

Stress is related to the strains by:

$$\left\{ \sigma \right\} = \left[ D \right]\left\{ {\varepsilon^{e} } \right\}$$
(6)

where \(\left[ D \right]\) = stress–strain matrix, \(\left\{ {\varepsilon^{e} } \right\}\,\,\, = \,\,\left\{ {\varepsilon^{{}} } \right\} - \,\left\{ {\varepsilon^{t} } \right\}\,\) = elastic strain vector, \(\left\{ {\varepsilon^{t} } \right\}\,\) = thermal strain vector, \(\left\{ \sigma \right\}\) = stress vector = \(\left[ {\sigma_{X} \,\sigma_{Y\,} \,\sigma_{Z\,} \sigma_{XY\,} \,\sigma_{YZ} \,\sigma_{ZX} } \right]^{T}\) and \(\left\{ \varepsilon \right\}\) = total strain vector = \(\left[ {\varepsilon_{X} \,\varepsilon_{Y} \,\varepsilon_{Z} \,\varepsilon_{XY} \,\varepsilon_{YZ} \,\varepsilon_{XZ} } \right]^{T}\).

Equation (6) may also be represented to:

$$\left\{ \varepsilon \right\}\,\, = \,\,\,\left\{ {\varepsilon^{t} } \right\}\, + \,\left[ D \right]^{ - 1} \left\{ \sigma \right\}$$
(7)

The welding structural analysis involved large displacements (strain) and a rate-independent thermo–elasto-plastic material model with temperature dependent material properties incorporated into the finite element modelling. Kinematic work hardening together with von Mises yield criterion and associative flow rules were assumed in the analysis.

3.2.1 Work Hardening Rule

The hardening rule describes the changing of the yield surface with progressive yielding, so that the conditions for subsequent yielding may be established. Hardening is described by a combination of two specific types of hardening, namely isotropic hardening and kinematic hardening. In case of steel the rate independent plasticity, kinematic hardening and bilinear von Mises yield criteria (Cheng 2005; Alberg 2005; Tsai and Eagar 1983) are applicable.

Isotropic hardening rule The Isotropic hardening rule presumes that the yield surface remains centered about its centerline in stress space and expands with develop plastic strains. The yielding occurs when (Jiang et al. 2015):

$$f = \sigma_{eq} - K - \sigma_{y} = 0$$
(8)

where the \(\sigma_{eq}\) is the von Mises equivalent stress, the \(\sigma_{y}\) is the initial yield stress, and K is the degree of strain hardening.

Kinematic hardening rule The Kinematic hardening rule which is used in present analysis assumes that the yield surface remains constant in size and the surface translates in stress space with progressive yielding. The yielding occurs when (Jiang et al. 2015):

$$f = f\left( {\sigma - \alpha } \right) - \sigma_{y} = 0$$
(9)

where σ is the stress, and α is a back stress tensor.

3.2.2 Yield Criterion

Yield criterion is a hypothesis defining the limit of elasticity in a material and the onset of plastic deformation under any possible combination of stresses. For multi-component stresses, this is represented as a function of the individual components, f ({σ}), which can be interpreted as an equivalent stress (Kong et al. 2011):

$$\sigma_{e} = f\left( {\left\{ \sigma \right\}} \right)$$
(10)

where

{σ} = stress vector

When the equivalent stress is equal to a material yield stress σy,

$$f\left( {\left\{ \sigma \right\}} \right) = \sigma_{y}$$
(11)

The material will develop plastic strains. If σe is less than σy, the material is elastic in nature and the stresses will be developing according to the elastic stress–strain relations. It is to be noted that the equivalent stress can never exceed the material yield since in this case plastic strains would develop instantaneously, thereby reducing the stress to the material yield.

3.3 Modeling Details

The eight- node brick elements with linear shape functions were used in meshing the FE model. The SOLID70 thermal element was used in transient thermal analysis due to its three-dimensional thermal conduction capability. The transient thermal history obtained from the 3-D thermal analysis was used as an input to perform the non-linear elasto-plastic thermo-mechanical analysis. For structural analysis, SOLID70 thermal element was replaced with comparable structural element i.e. SOLID185 to perform the structural analysis The schematic diagram and finite element model geometry of the but-weld joint is shown in Figs. 3 and 4.

Fig. 4
figure 4

Finite element model for butt weld

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3.3.1 Mesh Model

A rigorous mesh sensitivity analysis was carried out to determine the optimum meshing parameters like element size, number of elements and spacing ratio. The uniform fine mesh was used along welding region for obtaining the accurate temperature field in high temperature gradient regions of fusion zone and heat affected zone and the outer region was discretized with coarse mesh to reduce the computational time. In the fusion zone, elements of size 0.5 mm × 0.5 mm were taken and in the outer region 40 elements were taken with a spacing ratio of 0.05. Four equidistant elements were taken along the thickness direction. Total number of elements after meshing was 34,400. The mesh model of the worksheet is shown in Fig. 5.

Fig. 5
figure 5

Mesh of FE model

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3.4 Heat Source Model

The heat source is playing an important role in welding operation to get overall thermal history of welded specimen. The geometry of weld bead was taken from the experimental result. The numerical simulation of the welding process was performed by applying a uniform heat generation on weld bead. The moving co-ordinate volumetric heat source model was developed by considering element “birth and death” technique. The element “birth and death” technique was applied to only thermal analysis part. Initially, all the elements located at the weld zone are killed, and an empty weld zone is obtained. Subsequently, the elements located at the weld zone are activated bead by bead with time to simulate the weld bead formation by the volumetric heat source. The heat generation per unit volume was calculated by using the following equation (i.e. Eq. 6).

$$Q = \frac{\eta W}{v}$$
(12)

where \(\eta\) is the absorption efficiency of the welded material and W is the laser power and v is the volume of the active weld bead element as shown in Fig. 4. The experimental weld bead geometry is shown in Fig. 6.

Fig. 6
figure 6

The experimentally obtained cross-sectional views of fusion zone and heat affected zone in the weld by CO2 laser welding

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The accuracy of thermal analysis was verified by comparing the weld bead shape predicted in FEM and cross-sectional view of weld bead obtained by microscope under the operating parameters laser power 1500 W and laser scanning speed 400 mm/min. Figure 6 shows the cross-sectional view of the weld.

4 Material Properties

The chemical composition of AISI 304 steel (Berrettaa et al. 2007) is shown in Table 1.

Table 1 Weight% of elements
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Mechanical and thermal properties of AISI 304 steel (Vakili-Tahami 2013) are shown in Fig. 7a, b.

Fig. 7
figure 7

Mechanical properties of AISI 304 stainless steel a thermal conductivity, specific heat, density, convection coefficient b Young modulus, yield stress, thermal expansion coefficient

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5 Result and Discussion

In this work, a thermo-mechanical analysis method was used to simulate for autogenous laser welding. In this model, the analysis was directed in two steps. First, transient-thermal analysis was conducted to determine the temperature distribution over the plates. Second step, the transient- temperature distribution was used as thermal load in the FEM model to conduct the mechanical elastic–plastic calculation analysis.

5.1 FE Thermal Analysis Results

Absorption coefficient 21.52% was determined by back substitution analysis in heat source model (Eq. 12) to obtain the good agreement between experimentally obtained temperature distribution and the simulation results at laser power 1500 W and laser scanning speed 400 mm/min. The FE thermal analysis results are shown in Figs. 8, 9, 10, 11, 12, 13, 14 and 15. Figure 8 show the moving heat source’s transient temperature contour over the top surface and Fig. 9 shows the temperature contour over the top surface at a time of 150 s during cooling phase.

Fig. 8
figure 8

Temperature distribution during welding process at 4.5 s

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Fig. 9
figure 9

Temperature distribution during cooling phases at time 150 s

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Fig. 10
figure 10

Peak temperature distribution for varying laser power away from weld centre

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Fig. 11
figure 11

Peak temperature distribution for varying laser welding speed away from weld centre line

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Fig. 12
figure 12

Fusion zone and Heat affected zone in butt weld

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Fig. 13
figure 13

Size of fusion zone and heat affected zone for varying a laser power b laser welding speed

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Fig. 14
figure 14

Comparison of numerical and experimental thermal profiles at 7, 14, 22 and 32 mm away from weld centre line

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Fig. 15
figure 15

Overall numerical and experimental temperature contour

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Figures 10 and 11 represent the peak temperature distribution on top surface from the weld line towards the transverse direction with the varying laser power and laser welding speed respectively. Figure 12 shows the size of fusion zone and heat affected zone in weld for laser power 1500 W and laser welding speed 400 mm/min. Figure 13a and b shows the size of fusion zone and heat affected zone for varying laser power and varying laser welding speed respectively.

From Figs. 10 and 11, it can be observed that the peak temperatures increase with the increase of the laser power and decrease of laser welding speed respectively. In Fig. 12, line drawn corresponding to the temperature 1440 °C, represent the liquid weld pool boundary, which extended 1.508 mm from the weld centre line. The boundaries calculated between the temperatures 1440–950 °C represent the locations where phase transformations would occur under equilibrium conditions (Kim et al. 2010). From Fig. 13a, it is observed that the welding fusion zone and heat affected zone increases with increase of laser power. From Fig. 13b it is also observed that the welding fusion zone and heat affected zone increases with decrease of laser welding speed.

5.2 Experimental Validation of Thermal Model

The temperature distribution was measured by K type thermocouple which was connected to Agilent data logger to record the transient thermal history. The thermocouples locations are shown in Fig. 14.

Figure 14 represents the experimental and numerical comparison of transient thermal history at different locations. From Fig. 14, it is observed that the temperature distribution obtained from experimental observation matched fairly well with that of simulation results. Figure 15 shows the comparison of different zones of thermal contour obtained both from experiment and FE analysis.

The numerical predicted a cross-sectional view shows that the different temperature field by specifying the colours i.e. the weld zone temperature varies from 1440 to 1712.51 °C, heat affected zone temperature varies from 950 to 1440 °C and temperature of the base metal is to be below 950 °C (Kim et al. 2010). It is observed that the maximum temperature reaches at weld zone when the welding process was done. The temperature value decreases when further moving away from the centre line of the weldment. Similarly, maximum temperature values decrease as the distance from centre line increases.

5.3 Structural Analysis Results

The results obtained from FE transient thermal analysis were used as a load for the structural analysis to get the welding residual stresses and distortions. The boundary conditions of the mechanical analysis comprise clamping of the one edge parallel to the weld line shown in Fig. 16.

Fig. 16
figure 16

Details of the worksheet geometry and boundary conditions

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The results of structural analysis are shown in Figs. 17, 18, 19, 20, 21and 22. The structural analysis results are discussed in two subsections i.e. residual stresses and distortions.

Fig. 17
figure 17

Longitudinal stress distribution in butt welded plates

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Fig. 18
figure 18

Transverse stress distribution in butt welded plates

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Fig. 19
figure 19

von Mises stress distribution in butt welded plates

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Fig. 20
figure 20

Longitudinal, transversal and von Mises residual stress distributions perpendicular to the weld line on top surface

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Fig. 21
figure 21

Longitudinal, transversal and von Mises residual stress distributions perpendicular to the weld line on bottom surface

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Fig. 22
figure 22

Longitudinal and transversal residual stress distributions along weld line

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5.3.1 Residual Stresses

Figures 17, 18 and 19 represent the longitudinal, transverse and von Misses residual stresses distribution contour in which the laser power and laser scanning speed were considered as 1500 W and 400 mm/min respectively.

It is observed that the longitudinal and transverse residual stresses are tensile in nature along and near the weld line whereas these are compressive in nature away from the weld zone It is observed that the maximum magnitude of longitudinal residual stress is close to the yield strength of the base metal and larger than the transverse residual stress, which cause the plastic deformation to be retained in welded specimen, which results residual stress remain in the cooling stage. The residual stresses in the plastic region exceed the room temperature yield strength because of material strain hardening, which is such that the initial short-time compressive yielding produces an expansion of the yield surface followed, during cool-down, by yielding in tension at a stress level higher than the mono tonic yield strength. This type of behaviour is found in published literatures (Kong et al. 2011; Kim et al. 2009; Denga and Kiyoshimab 2010; Sun et al. 2014).

The von Mises residual stress distribution is tensile in nature and the peak value of von Mises stress is almost equal to that of the yield strength of the base metal.

Figures 20 and 21 represent the longitudinal, transverse and von Mises residual stresses distribution perpendicular to the weld line on the top and bottom surface respectively.

Figures 20 and 21 represent the longitudinal and transverse residual stresses distribution which is tensile in nature along and near the weld zone and it becomes compressive in nature further away from the weld zone. The maximum magnitude of longitudinal residual stress is much higher than that of transverse residual stress. The peak value of longitudinal stress is \(302.11 \,{\text{MPa}}\) which is much larger than the transverse tensile stress of \(99.953 \,{\text{MPa}}\) in welded plates as shown in Fig. 20. It is also observed from Figs. 20 and 21 that the stresses distribution pattern over the top surface of the plate is almost similar which is tensile in nature throughout the entire welded plate. Because the residual stress magnitude highly depends on temperature gradient through the thickness of the welded plate. The thickness of the workpiece used in this study was relatively thin, which leads to generation of negligible temperature gradient through the thickness of the welded plate.

The maximum magnitude of von Misses stress is observed near the weld zone and its gradually decreases away from the weld zone. Figure 22 represent that the longitudinal, transversal and von Mises residual stresses distribution along the weld line on top surface of the welded plate.

From Fig. 22, it is clear that the distribution of the longitudinal residual stress is almost uniform in nature along the weld line and its magnitude is very small near the starting and end position of the welded joint. It is also observed that the longitudinal residual distribution pattern is similar to the transversal residual stress and it is almost uniform along weld line except at the starting and end position. But at beginning and end of the weld line the transverse residual stress and longitudinal residual stress distributions are compressive in nature where the magnitude of compressive transverse residual stress is much higher than the longitudinal residual stress. The heat input has great effect on the transverse stress than longitudinal stress. The increase in heat input increase the welding temperature which leads to increase the deformation. The increase in deformation can release more residual stresses and therefore the transverse stress is decreased (Jiang et al. 2011).

5.3.2 Residual Deformations

Two different boundary conditions were used to analyse the weld induced residual deformations i.e. (1) one side clamped and (2) both sides clamped. The results of distortion are shown in Figs. 23, 24 and 25. Distortions occur in the welded plate due to non-uniform expansion and contraction during heating and cooling cycle of the welding process. The distortion contour for one sided clamped condition is shown in Fig. 23.

Fig. 23
figure 23

Distortion pattern of the welded plates

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Fig. 24
figure 24

Distortion perpendicular to weld line: a both edges parallel to the weld line fixed b one edge parallel to the weld line fixed

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Fig. 25
figure 25

Comparison of numerical and experimental distortions: a along X-axis b along Y-axis

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Figure 24a, b represent the angular distortions along Y-axis i.e. perpendicular to the weld line for two different cases which are both edges parallel to the weld line fixed and one edge parallel to the weld line fixed condition respectively.

In case of both edges fixed condition, the distortion increases gradually towards the weld line and its maximum magnitude is \(0.039514 \,{\text{mm}}\) as shown in Fig. 24a. In case of one edge fixed condition, the maximum deflection (i.e. \(0.19 \,{\text{mm}}\) occur at free edge of the plate and minimum deflection near the welding region as shown in Fig. 24b.

The angular deformation of the sample was measured with the help of coordinate measuring machine for individual set of experiments before and after welding. The comparison between numerical and experimental measured distortions is shown in Fig. 25.

Figure 25a, b represent that there is a close agreement between the simulation and experimentally measured distortion.

6 Conclusions

From the above investigation the following conclusions can be made:

  • A heat source model for selecting the geometry of the weld bead was successfully designed using a comparison of the FE simulation results and experimental data for AISI 304 thin stainless steel sheet. The simulation of the welding process is followed by a number of loading steps simulating the cooling stage, which lasts for about 450 s. The global coupled thermo-mechanical LBW model predicts reliably the transient temperature, residual stress, and residual distortion fields for different varying process parameters, i.e. laser power, laser welding speed and boundary conditions.

  • Numerically predicted various zones and temperature distribution with the designed heat source was verified experimentally. The transient thermal history obtained from FE analysis and those obtained from experimental measurements compared fairly well with a maximum variation of \(3.74 \%\) with the peak temperatures. This means the present model methodology can be used to any type of weld bead geometry for better accuracy.

  • The peak temperature, fusion zone and heat affected zone increase with increasing of laser power and decreasing of laser welding speed. Because the melting temperature zone and recrystallization temperature zone increases with increase of laser power and decrease of laser welding speed.

  • The longitudinal and transverse residual stresses distribution is tensile in nature along and near the weld zone and it becomes compressive in nature further away from the weld zone. The maximum magnitude of longitudinal residual stress is much higher than that of transverse residual stress. But at beginning and end of the weld line the transverse residual stress and longitudinal residual stress distributions are compressive in nature where the magnitude of compressive transverse residual stress is much higher than the longitudinal residual stress.

  • The residual deformations obtained through the FE analyses and those obtained from experimental measurements compared fairly well with a maximum variation of \(5.071\%\) only.

  • It was observed that boundary conditions have significant effect in weld induced distortions. When both edges parallel to weld line were constrained then the maximum angular distortion was \(0.0395 \,{\text{mm}}.\) When one edge parallel to weld line was clamped and other edges were free then the maximum angular distortion was \(0.19157 \,{\text{mm}}\) which is much higher than previous case.