1 Introduction

In the framework of increasingly stricter pollutant emission regulations and the need of energy efficiency, Lutsey [1] pointed out as the automotive industry leads the technology development aimed to reduce energy consumption of vehicles. Graf [2] highlighted how such reduction can be achieved, for instance, by designing lighter cars through the use of materials with improved strength-to-weight ratios like aluminum alloys in substitution of steel. Hakamada et al. [3] concluded that, although the use of parts made by magnesium and aluminum alloys requires more energy than steel at the production stage, the vehicle mass reduction increases fuel efficiency sufficiently to reduce the overall energy use. Mayyas et al. [4] claimed that another viable solution is redesigning and optimizing vehicle components/assemblies, or, where steel is needed for strength reasons, “tailoring” the whole structure such that specific material properties are located exactly where they are needed. As described by Merklein et al. [5], sheets with different chemical compositions, thicknesses, or coating can be joined together to form, as it is called, a tailor welded blank (TWB), a semi-finished product subjected, after joining, to a stamping process and designed to satisfy specified performances in specified locations. This technique allows the use of thicker or stronger materials in the critical regions of a component to reach structural and safety requirements, while thinner or lighter materials can be used in other regions to reduce the weight. Zadpoor et al. [6] reported that TWBs are commonly welded by laser welding from different sheets in the butt joint configuration prior to forming. The laser welding process is used for its well-known advantages such as high strength of the joint, deep penetration, localized heat affected zone, small distortions, and high processing speed which results in high productivity. Hong and Shin [7] explained that the reasons why laser welding is a more attractive joining method compared to older ones are to be found in the abovementioned benefits. A coherent laser beam is focused on the surface of the piece to a small spot producing high power densities that quickly melt and partially vaporize the metal, generating a characteristic cavity known as “keyhole,” prevented from collapsing by the vapor pressure. Yan and Takayama [8] showed accurately how the laser-material interaction is a very complex phenomenon involving phase transformations and changes in chemical composition. However, the development of numerical models that are able to predict the weld pool geometry, thermal cycles, or the effects of the welding, such as mechanical distortions, is useful to set and optimize process parameters. For these purposes, and other engineering problems, the complexity of the laser welding phenomena can be reduced by using an equivalent moving heat source calibrated with the experimental melt pool geometries, removing the fluid flow modeling difficulties without losing the capability to compute relevant data and to investigate some aspects of the process. Such an approach does not take into account several phenomena involved in the laser-matter interaction like those associated with the dynamics in the weld pool, metal evaporation, plasma formation, absorption mechanism in the keyhole, and interaction between laser beam and plasma plume [9]. At the same time, like other finite-element-based simulation approaches, allows studying many aspects related to the interaction between thermal and stress fields experienced by the material during the welding process [10]. The high-temperature region (i.e., the weld pool) is approximated by a meshed solid and a given heat input is applied to each element according to the relative motion of the laser beam. There are several methods to simulate the heat input of the laser beam. Casalino et al. [11] modeled the moving source by associating an internal heat generation to specific elements in the weld zone, but, as reported by Aghaee Attar et al. [12], the most widespread methods are based on a moving volumetric heat source. Several volumetric heat sources have been proposed in the last decades. Goldak et al. [13] proposed a non-axisymmetric three-dimensional heat source in a double ellipsoid configuration. Although this model has been extensively used in FEM simulations, it is not applicable to deep penetration high density welding processes, such as keyhole processes. Du et al. [14] developed a heat source comprising a plane heat source on the top surface of the blank and a cylindrical heat source along the laser beam axis, which takes into account the plasma effect and the keyhole absorption. Wu et al. [15] proposed a three-dimensional conical heat source, where the diameter of the heat intensity distribution region, characterized by a Gaussian form at any plane perpendicular to the laser beam axis, is linearly decreased along the thickness of the workpiece. However, neither the cylindrical shape nor the conical one is able to predict the root widening phenomenon of the weld pool associated with a deep penetration laser beam welding. Thus, new types of welding heat source were proposed. Farrokhi et al. [16] presented a modified version of the three-dimensional conical heat source developed by Wu and coworkers, consisting in two conical moving heat sources, in which the heat density distribution region is first linearly decreased along the thickness and then it follows a reverse cone configuration to obtain an hourglass shape distribution. The same approach has been used by Zhan et al. [17]. To better calibrate the weld pool dimensions, a variable peak value of thermal flow along the thickness was used by Rong et al. [18]. In particular, the peak flux in the center of each plane perpendicular to the laser beam axis was expressed as a function of the distance from the middle plane and of an adjustment coefficient. Depending on the value of this coefficient, the peak flux is increased or decreased along the thickness direction. Because of the axial symmetry of the volumetric heat sources presented so far, welding between blanks of the same thickness can only be modeled. However, TWBs are often characterized by welding of blanks with different thicknesses, in which the fused zone shape is asymmetric and it can be represented as a combination of two half hourglass-like shapes, each with specific dimensions depending on thickness. Koo et al. [19] studied the phenomena involved during laser welding that promote the formation of an hourglass shaped fusion zone in TWB. However, their numerical results are presented in an iso-thickness configuration. Li et al. [20] investigated by experiments and numerical simulations the effect of process parameters on the shape of the welding pool during laser welding of dissimilar thicknesses. In their study, they demonstrated that the difference in thickness in the butt joint has a large impact on the weld deviation since it creates different cooling rates. In spite of this, in their experiments the depth of the fusion zone was identical in the two welded thicknesses. This is not the case in a common industrial TWB.

The aim of this work is to develop a new volumetric heat source to be implemented in a FEM simulation based on thermal conduction heat transfer and capable to predict the dimensions of a double half hourglass weld pool shape of a full penetration laser welding performed on TWB composed of sheets with different thicknesses. The parameters of a variable peak flux heat source with a double conical shape were found by the energy balance differentiating them from the two different thicknesses. The heat source was calibrated first using two iso-thickness welding configurations and then on a different thickness welding configuration.

2 Material and methods

2.1 Experimental procedure

Laser welding experiments have been conducted by Baosteel Tailored Blanks Italy on Al-Si coated 22MnB5 steel sheets. The sheet thicknesses were 1.2 and 1.5 mm, whereas the coating thickness was nominally 30 μm. Steel sheets were laser-welded by a TRUMPF TruDisk 6006 solid-state disk laser (specifications listed in Table 1) and a weld system for linear and non-linear welding.

Table 1 Specifications of laser welding system
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Experiments were run in three different configurations. Sheets of the same thickness were joined in two configurations (1.2-1.2 mm and 1.5-1.5 mm), whereas the third one concerned the welding of different thicknesses (1.5-1.2 mm). The same process parameters were adopted for all the configurations described. The welding speed was 5 m/min; the laser power was 5.4 kW. The focal plane was positioned at 1.5 mm over the clamping table, and the beam axis was positioned at 0.08 mm from the weld line (in the different thickness configurations, the beam axis always hit the thicker sheet). The shielding gas was argon, blown on the upper surface of the sheets (the nozzle was inside the laser head), with a flow rate of 17 l/min. For any configuration, the steel sheets were welded in the as-received condition without any filler material, so the Al-Si layer was not removed. After welding, the specimens were prepared for the metallographic analysis in the as-welded condition. No heat treatment was performed on the welded joints in order to preserve a good visual differentiation of fused zone. The cross-sections of the welded joints were cut with the precision cutting machine Struers Secotom-50, mounted with the mounting press Struers CitoPress-30, and then ground and polished with the automatic polisher Struers Tegramin-25. Finally, specimens were etched with a 2% nital solution and observed by optical microscopy.

2.2 Heat source model

In order to accurately describe the weld geometry between two sheets of different thickness, the peak index increment-double cone (PII-DC) heat source, proposed by Rong et al. [18], has been appropriately modified. As shown in Fig. 1, the heat source used is composed of two double conical sources, one for each sheet, combined with two further three-dimensional elements called “conical hoofs” capable of reproducing the bevel generated by welding of sheets of different thickness. Each “conical hoof” represents one of the two parts obtained by cutting, with an inclined plane, a half truncated cone.

Fig. 1
figure 1

Schematic diagram of the proposed heat source model

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Assuming that the heat flux ql, in each cross-section, is characterized by a Gaussian distribution and that its peak value qm varies along the thickness direction (z-direction), the heat source can be mathematically expressed using the following formulation

$${q}_l\left(r,z\right)={q}_m(z)\bullet \exp \left(-\frac{3{r}^2}{r_0^2}\right)$$
(1)

where r0 is the distribution parameter (linearly variable along the thickness) and r is the radial coordinate. For a moving laser heat source in y direction with a welding speed v, we have

$$r=\sqrt{{\left(y- vt\right)}^2+{x}^2}$$
(2)

With reference to Fig. 2a, for each zone of the heat source, the distribution parameter r0 can be expressed as

$${r}_{01}(z)={r}_i-\left({r}_i-{r}_b\right)\frac{z_i-z}{z_i-{z}_b}\kern2.5em {z}_b\le z<{z}_i$$
(3)
$${r}_{02}(z)={r}_t-\left({r}_t-{r}_i\right)\frac{z_t-z}{z_t-{z}_i}\kern2.75em {z}_i\le z\le {z}_a$$
(4)
$${r}_{01}^{\ast }(z)={r}_i^{\ast }-\left({r}_i^{\ast }-{r}_b^{\ast}\right)\frac{z_i^{\ast }-z}{z_i^{\ast }-{z}_b^{\ast }}\kern1.75em {z}_b^{\ast}\le z<{z}_i^{\ast }$$
(5)
$${r}_{02}^{\ast }(z)={r}_t^{\ast }-\left({r}_t^{\ast }-{r}_i^{\ast}\right)\frac{z_t^{\ast }-z}{z_t^{\ast }-{z}_i^{\ast }}\kern2em {z}_i^{\ast}\le z\le {z}_t^{\ast }$$
(6)

where rb, ri, rt, zb, zi, zt, za, rb*, ri*, rt*, zb*, zi*, and zt* are 13 defined parameters to describe the shape of the laser weld; the star apex indicates the quantities relating to the thicker sheet while the subscripts 1 and 2 indicate, respectively, the lower and upper zone of the heat source.

Fig. 2
figure 2

a 2D schematic of the heat source and heat intensity distribution at b zt plane, c zi* plane, d zi plane, and e the bottom surface. Heat source parameters: P = 5.4 kW, ηl = 0.28044, α = 2, zt = 1.2 mm, rt = 0.471 mm, zt = 1.2 mm, rt = 0.471 mm, zt = 1.2 mm, rt = 0.471 mm, za = 1.324 mm, rt = 0.471 mm, zt = 1.2 mm, ri = 0.314 mm, zi = 0.45 mm, rb = 0.494 mm, rt* = 0.517 mm, zt* = 1.5 mm, ri* = 0.378 mm, zi* = 0.629 mm, rb* = 0.743 mm

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The value of the thermal flux at the center of the heat source qm (z) is expressed as a function of a peak index λ using the following equations.

$${q}_{m1}(z)={q}_0\exp \left(\ln {\lambda}_1\frac{z_i-z}{z_i-{z}_b}\right)\kern2.25em {z}_b\le z<{z}_i$$
(7)
$${q}_{m2}(z)={q}_0\exp \left(\ln {\lambda}_2\frac{z-{z}_i}{z_a-{z}_i}\right)\kern2.25em {z}_i\le z\le {z}_a$$
(8)
$${q}_{m1}^{\ast }(z)={q}_0\exp \left(\ln {\lambda}_1^{\ast}\frac{z_i^{\ast }-z}{z_i^{\ast }-{z}_b^{\ast }}\right)\kern1.75em {z}_b^{\ast}\le z<{z}_i^{\ast }$$
(9)
$${q}_{m2}^{\ast }(z)={q}_0\exp \left(\ln {\lambda}_2^{\ast}\frac{z-{z}_i^{\ast }}{z_t^{\ast }-{z}_i^{\ast }}\right)\kern1.75em {z}_i^{\ast}\le z\le {z}_t^{\ast }$$
(10)

where q0 is the peak value at the zi and zi* sections

$${q}_m\left({z}_i\right)={q}_m^{\ast}\left({z}_i^{\ast}\right)={q}_0$$
(11)

Substituting Eqs. 7, 8, 9, and 10 into Eq. 1, the heat flow can be written as

$${q}_{l1}\left(r,z\right)={q}_0\exp \left(\ln {\lambda}_1\frac{z_i-z}{z_i-{z}_b}\right)\exp \left(-\frac{3{r}^2}{{r_{01}}^2}\right)\kern2.25em {z}_b\le z<{z}_i$$
(12)
$${q}_{l2}\left(r,z\right)={q}_0\exp \left(\ln {\lambda}_2\frac{z-{z}_i}{z_a-{z}_i}\right)\exp \left(-\frac{3{r}^2}{{r_{02}}^2}\right)\kern2.25em {z}_i\le z\le {z}_a$$
(13)
$${q}_{l1}^{\ast}\left(r,z\right)={q}_0\exp \left(\ln {\lambda}_1^{\ast}\frac{z_i^{\ast }-z}{z_i^{\ast }-{z}_b^{\ast }}\right)\exp \left(-\frac{3{r}^2}{{r_{01}^{\ast}}^2}\right)\kern1.75em {z}_b^{\ast}\le z<{z}_i^{\ast }$$
(14)
$${q}_{l2}^{\ast}\left(r,z\right)={q}_0\exp \left(\ln {\lambda}_2^{\ast}\frac{z-{z}_i^{\ast }}{z_t^{\ast }-{z}_i^{\ast }}\right)\exp \left(-\frac{3{r}^2}{{r_{02}^{\ast}}^2}\right)\kern1.75em {z}_i^{\ast}\le z\le {z}_t^{\ast }$$
(15)

The adjustment coefficients (λ1, λ2, λ1*, and λ2*) for the upper and lower zones of the heat source, respectively for thin and thick sheet, are expressed using the following equations.

$${\lambda}_1=\alpha \bullet \frac{r_b}{r_i}$$
(16)
$${\lambda}_2=\alpha \bullet \frac{r_t}{r_i}$$
(17)
$${\lambda}_1^{\ast }=\alpha \bullet \frac{r_b^{\ast }}{r_i^{\ast }}$$
(18)
$${\lambda}_2^{\ast }=\alpha \bullet \frac{r_t^{\ast }}{r_i^{\ast }}$$
(19)

where α is a coefficient introduced to synchronously regulate λ1, λ2, λ1*, and λ2*.

The resulting distributions of the heat flux on the zt plane, on the zi* plane, on the zi plane, and on the bottom surface are reported in Fig. 2b–e.

The key problem is how to determine the parameter q0. According to thermal energy conservation law, indicating with Pl the laser power and with ηl the energy transfer efficiency (ETE) of laser, an equilibrium equation was developed

$${\eta}_l{P}_l={Q}_l={\sum}_k{Q}_{lk}={\sum}_k\left({\iiint}_{V_k}{q}_l\left(r,z\right)\ rdrd\theta dz\right)\kern3.5em k=1,2,3,{1}^{\ast },{2}^{\ast },{3}^{\ast }$$
(20)

Into Eq. 20, Ql represents the effective absorbed laser power distributed within the heat source in different quotas Qlk, proportional to the volume Vk of each part that composes it.

For truncated cone parts, Qlk can be expressed as

$${Q}_{l1}={\int}_{z_b}^{z_i}{\int}_0^{\pi }{\int}_0^{r_{01}}{q}_{l1}\left(r,z\right)\ rdrd\theta dz$$
(21)
$${Q}_{l2}={\int}_{z_i}^{z_t}{\int}_0^{\pi }{\int}_0^{r_{02}}{q}_{l2}\left(r,z\right)\ rdrd\theta dz$$
(22)
$${Q}_{l{1}^{\ast }}={\int}_{z_b^{\ast}}^{z_i^{\ast }}{\int}_0^{\pi }{\int}_0^{r_{01}^{\ast }}{q}_{l1}^{\ast}\left(r,z\right)\ rdrd\theta dz$$
(23)
$${Q}_{l{2}^{\ast }}={\int}_{z_i^{\ast}}^{z_a}{\int}_0^{\pi }{\int}_0^{r_{02}^{\ast }}{q}_{l2}^{\ast}\left(r,z\right)\ rdrd\theta dz$$
(24)

Due to the symmetry of the heat source with respect to the x-z plane, Ql3 and Ql3*, quantities referring to the “conical hoofs” (see Appendix 1), can be written as

$${Q}_{l3}=2\bullet {\int}_{z_t}^{z_a}\left({\int}_0^{\varphi }{\int}_0^{r_3}{q}_{l2}\left(r,z\right)\ rdrd\theta +{\int}_{\varphi}^{\pi /2}{\int}_0^{r_{02}}{q}_{l2}\left(r,z\right)\ rdrd\theta \right) dz$$
(25)
$${Q}_{l{3}^{\ast }}=2\bullet {\int}_{z_a}^{z_t^{\ast }}{\int}_0^{\varphi^{\ast }}{\int}_{r_3^{\ast}}^{r_{02}^{\ast }}{q}_{l2}^{\ast}\left(r,z\right) rdrd\theta dz$$
(26)

As shown in Eq. 27, isolating the initial peak value of thermal flow q0 from Eqs. 21, 22, 23, 24, 25, and 26, it can be expressed as

$${q}_0=\frac{\eta_l{P}_l}{I_1+{I}_2+{I}_3+{I_1}^{\ast }+{I_2}^{\ast }+{I_3}^{\ast }}$$
(27)

where I1, I2, I3, I1*, I2*, and I3* are the values of the integrals reported in the Appendix 2.

2.3 Finite element modeling

After the definition of the heat source model, the weld pool geometry and the temperature profile were determined by FEM. For the simulations, the commercially available software ABAQUS\CAE was used.

In absence of data of some of the thermo-physical properties of the material, such as latent heat of fusion and vaporization, liquidus and solidus temperature, the values referred to chemically similar steels and to pure iron were picked from Wang et al. [21], while density, specific heat capacity, and thermal conductivity were extracted from the work of Kaars et al. [22]. Thermo-physical properties of the investigated steel are summarized in Table 2.

Table 2 Thermo-physical properties of the material implemented in the numerical simulation
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Three case studies were considered. The first two cases employed two sheets of thicknesses 1.2 mm and 1.5 mm, respectively, to determine the energy transfer efficiency, which can be very low for thin plates, as demonstrated by Swift-Hook and Gick [23] and dependent on the thickness of the sheet as can be seen in the work of Sharma and Molian [24]. Secondly, an average value of the ETE was imposed for the simulation of the TWB obtained from the welding of the sheets with the abovementioned initial thicknesses of 1.2 and 1.5 mm. To save computational time in the first two cases, the full size of the specimen was reduced to a symmetric sheet of size 12.5 mm × 25 mm. The model is shown in Fig. 3. For the same reason, this portion of the sheet was divided into three regions with different mesh refinement, as can be seen in Fig. 4, with the finer one constructed by structured hexahedral elements of type DC3D8, an 8-node linear heat transfer brick of dimensions 0.1 mm × 0.1 mm.

Fig. 3
figure 3

Symmetric plate of size 12.5 mm × 20 mm

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Fig. 4
figure 4

Finite element mesh with three levels of mesh refinement and detail view of the finest one

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On the top surface of the sheet and on the bottom surface of the welding bead region, a constant convection heat transfer coefficient of h = 10 W/m2K was imposed, together with an emissivity of 0.5 applied to the whole part. The effect of the shielding gas was neglected since an insignificant influence in the temperature distribution of a much higher value of h was found. A transient heat transfer procedure was set, with no mechanical boundary conditions and a welding time period of 0.2 s. The new volumetric heat source was implemented by an external subroutine called DFLUX, available in the ABAQUS library, defining a non-uniform distributed flux as a function of position and time. The same assumptions, except for the symmetry, have been made for the TWB. In this case, the geometry of the weld bead was constructed from the acquired macrographs, taking into account the peculiar bevel feature.

3 Results and discussion

3.1 Welding results

Figure 5 shows the cross-sections of welded joints in the as-welded conditions. A full-penetrated weld with an hourglass shape was achieved in all the considered combinations, so the aforementioned double conical heat source model was considered suitable for all the configurations. In Table 3, the cross-section measurements according to the geometrical parameters of the double conical heat source model defined in Fig. 2a are reported.

Fig. 5
figure 5

Cross-sections of welded joints in as-welded conditions: a 1.5-1.5 mm; b 1.2-1.2 mm; c 1.5-1.2 mm

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Table 3 Geometrical parameters of the welding experiments in the three different configurations
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3.2 Numerical results

The first step was to calibrate the model in the iso-thickness configurations (1.2 mm and 1.5 mm) through the corresponding experimental profile. In order to reduce the number of calibration parameters, the double conical heat source model developed by Farrokhi et al. [16], with the effective laser power distributed in the lower and upper cone, as shown in the work by Thasanaraphan [25], was applied. In that formulation, the only unknown (and uncertain) parameter affecting the dimension of the weld bead was the ETE. Thus, a variable ETE in the range between 0.1 and 0.9 was implemented in the simulations. The experimental and the simulated values of the area of the fused zone in the cross-section of the weld bead were compared and the difference between the two was minimized by linear interpolation as shown in Fig. 6. The values of the ETE in the two cases were found to be very low compared to values that can be as high as 90% or more for keyhole processes, as pointed out by Swift-Hook and Gick [23]. However, these results are consistent with those found by Sharma and Molian [24] and by Swift-Hook and Gick [23] for thin plates. In welding thin sheets with large power levels like in the investigated cases, a through thickness keyhole is usually formed, letting a large portion of the laser light be transmitted beyond the sheets. For the thinner sheet (1.2 mm), a value of 0.244 was found, while for the thicker one (1.5 mm) a value of 0.317 minimized the abovementioned difference. Results with the corresponding ETE values are depicted in Fig. 7. An average ETE value of 0.2805 between these two was used for the TWBs subsequent simulations.

Fig. 6
figure 6

Numerical results with different values of the energy transfer efficiency for the iso-thickness configurations

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Fig. 7
figure 7

Cross-section of the melt pool geometries of the two sheets: a 1.2 mm, b 1.5 mm. The grey area represents the fused zone

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To calibrate the fused zone in the TWB, a variable α (1, 2, 4, 6, 10) was imposed, with fixed ETE and fixed geometrical dimensions (rb, rt, ri, rb*, rt*, ri*). Results, in terms of values of the peak of the heat flux (q0) calculated solving Eq. 27, are reported in Table 4. Deviations, in terms of percentage errors between experimental and numerical geometrical parameters, are shown in Fig. 8. A value of α = 6 was found to minimize the overall error between experimental and simulation values. Numerical results for α = 6, including the body heat flux, are shown in Fig. 9.

Table 4 Peak value of the heat flux calculated for different α values
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Fig. 8
figure 8

Deviations between experimental and numerical values of the fused zone dimensions

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Fig. 9
figure 9

Numerical results on the different thickness configuration with α = 6. a Temperature distribution on the TWB; b volume in which the body heat flux is applied with related values; c volume of the material where the temperature is above the solidus

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As it can be seen in Fig. 8, while an acceptable average error was achieved, especially on characteristics radii dimensions, a significant deviation persists in the prediction of the neck height both in the best alpha configuration and the other simulated ones. No such large errors were found neither by Farrokhi et al. [16] nor by Rong et al. [18] in the iso-thickness configuration. Thus, one possible reason for these deviations could be the different heat diffusion phenomena along the sheets that take place in welds with different thicknesses together with consequent thermal fluxes in the thickness direction.

In order to have an additional physical dimension for the numerical–experimental comparison of the thermal fields achieved on the material in the welding process, the dimensions of the heat affected zone (HAZ) were measured and compared. The experimental HAZ was evaluated as the distance of the margin of the microstructurally altered zone from the center axis and an averaged value (among three different values across the sheet thickness) of 0.87 mm and 0.94 mm was measured for the thinner and the thicker sheets, respectively. The numerical extension of the HAZ was evaluated measuring the distance from the center axis where the material reached a maximum temperature of 1068 K (Ac1 for the investigated steel according to [22]) during the welding process. Values of 0.75 mm for the thinner sheet and 0.85 mm for the thicker one were found denoting an error of 13.8% and 9.6%, respectively.

While few experimental observations of the molten pool are available in previous studies like the one by Artinov et al. [26], several published numerical results indicate that the width of the molten pool is usually larger (transversal direction), as shown by Zhan et al. [17], and longer (longitudinal direction), as shown by Artinov et al. [27], on the weld top rather on the bottom, contrary to what was found in this work (Fig. 9c). It should be recalled that published works, dealing with both experimental and numerical results, are related to iso-thickness configurations and heat diffusion in the joint is strongly influenced by the thickness of the welded sheet. It is thought that the upper part of the weld pool is likely to be more rapidly cooled down by the greater heat losses in the upper part of the joint since the different thicknesses are welded with the lower edge at the same z level. A different weighting of the geometrical parameters or a different distribution of the heat flux along the thickness and/or between the two different thicknesses, e.g., including additional boundary conditions to the energy balance, could be applied to avoid or reduce thermal fluxes in the thickness direction trying to reach a better correlation also in the neck height prediction. Those modifications are beyond the scope of this work and could be considered as future investigations for improving the achieved results.

4 Conclusions

An original approach for applying a volumetric heat source to a laser beam welding configuration where a butt joint involves different thicknesses was developed and assessed. The energy balance was applied to find the parameters of the peak index increment-double cone heat source combined with two further three-dimensional elements called “conical hoofs” capable of reproducing the bevel generated by welding of sheets of different thickness with an hourglass shaped fusion zone. Two iso-thickness configurations were used to find a suitable energy transfer efficiency for the different thickness configurations. Numerical results in the dissimilar thickness configuration show a fairly good prediction capability of the model in terms of dimensions of the fused zone into the weld. Larger discrepancies were found in the prediction of the height of the hourglass neck. While the approach could be considered as generally feasible in simulating the laser welding process, the parameters found in this work are related to a specific welding configuration. Further research efforts are needed to find a suitable relation between process parameters and geometrical ones in order to use the developed approach also in different welding configurations.