## Abstract

The grain structure of a weld seam influences its susceptibility to hot cracking during the welding process. The previously derived explicit analytical expressions allow for the accurate prediction of both the morphology of the grain structure and the grain size in a wide range of processing parameters. This model is now combined with the pressure balance model of Rappaz, which describes the formation of hot cracks by the balance between solidification shrinkage and thermomechanical deformation. The combination of the two models allows for the description of the impact of the welding parameters on the strain rate limit that a laser welded seam can withstand without the formation of hot cracks. It reveals that the absorbed line energy per depth is the key parameter to influence the value of this limit. The model was validated for the case of laser beam welding of the technical aluminum alloy AA6016. The calculated critical strain rates agree well with the experimentally determined critical strain rates measured by means of digital image correlation.

## Introduction

During laser beam welding, oriented dendrites grow from the side of the weld towards the centerline. In case of laser beam welding of aluminum alloys, when the value of the power per unit depth \( P_{\text{Depth}} = \frac{P}{s} \), where *P* is the laser power and *s* the welding depth, or the Peclet criterion \( Pe_{\text{Ryk}} = \frac{{w_{\text{weld}} \cdot v}}{4 \cdot \kappa } \) as defined by Rykalin,[1] where *w*_{wled} is the width of the welded seam, *v* the feeding rate, and *κ* the thermal diffusivity, exceeds an alloy-specific threshold, an equiaxed dendritic grain structure forms in the center area of the weld.[2]

Centerline hot cracks are a fatal welding defect and may propagate along the grain boundaries between the equiaxed dendritic grains in this central area of the weld. The grain structure of the weld has a major influence on the susceptibility of the welds to hot cracking.[3,4] The horizontal section in Figure 1 shows a centerline crack within an equiaxed dendritic grain structure of a weld seam in AA6016 alloy. The section was etched according to Barker[5] and illuminated by polarized light.

According to the RDG criterion of Rappaz,[6] the solidification shrinkage and the thermomechanical deformation of the work piece lead to a pressure drop in the melt between the growing grains during the solidification process. A hot crack starts to form when this pressure drop exceeds the value of what was termed a critical depression.[6] Rappaz derived a critical strain rate limit from this criterion to quantify the hot cracking susceptibility (HCS) of aluminum alloys. The major influence of strains and strain rates on hot crack formation has been proven in recent work by means of optical measurement techniques.[7,8,9,10]

Figure 2 sketches the typical processing zone that is present during laser beam welding. The origin of the coordinate system was chosen to be located in the centre of the vapor capillary, with (positive) *x* pointing in the direction of the moving laser beam. The solidification occurs at the trailing end of the melt pool as indicated by the large white arrow. The zone between the liquidus isotherm *T*_{Liquidus} and the solidus isotherm *T*_{Solidus} is referred to as the solidification zone. The solidification starts at a solid fraction of *f*_{s} = 0 (all the material is liquid) and is completed at a solid fraction of *f*_{S} = 1 (all the material is solid).

The liquidus and solidus isotherms cross the centerline of the weld (*y* = 0) at *x*_{L} and *x*_{S}, respectively. The condition of coherency, where *f*_{S} = *f*_{S,coh}, is defined to be the solidification condition at which the growing grains build a coherent network. At the centerline this condition is fulfilled at the position *x*_{coh}. Between this state and the complete solidification, the flow of the liquid melt between the grains is impeded and volume changes cannot be compensated anymore by backfilling with melt.[11] Such volume changes may result from the density change (shrinkage) during solidification and from the thermomechanical deformation of the heated material next to the solidification zone. These thermomechanical deformations affect the solidification zone by transverse strains and strain rates,[7] as sketched by the gray arrows in Figure 2.

It is well known that the grain structure of welds influences the hot cracking susceptibility.[3] Pellini[12,13] explained this influence by the distribution of the strain, which affects the solidification zone. He suggested a decrease of the HCS by the even distribution of the strain over several grain boundaries of a solidifying microstructure. Coniglio and Cross[4,14] stated that there is a linear relation between the critical deformation rate, which is required for the formation of hot cracks, and the number of grain boundaries, which are affected by this load. This linear relation was already confirmed experimentally[15] by the determination of the critical strain rate for the formation of hot cracks in a self-restraint hot cracking test.[16] An increase of the number of grain boundaries, which are affected by the transverse thermomechanical load, results from an increase of the width *w*_{eqx} of the zone containing an equiaxed dendritic grain structure, subsequently referred to as the equiaxed zone, and from the decrease of the size *d*_{eqx} of the equiaxed dendritic grains,[15] as defined in Figure 2.

The grain structure model[2] (GSM) allows for the explicit description of the grain structure that is formed by laser beam welding under the assumption of an ideal two-dimensional heat conduction.

Usually the susceptibility of welds to hot cracking is reduced by optimizing the material properties, *e.g*., with the addition of filler wire[17] with a high content of silicon or by the addition of nucleators such as titanium.[15] The influence of the welding parameters on the susceptibility to hot cracking is only described by empirical studies[18,19,20,21] or numerical calculations of the thermomechanics.[22,23,24]

The present work shows that the combination of the grain-structure model[2] (GSM) with the RDG criterion[6] allows for the explicit analytical description of the influence of the welding parameters on the hot cracking susceptibility of laser beam welds in aluminum alloys, quantified by the critical strain rates for the formation of hot cracks.

## Model

According to the RDG criterion,[6] hot cracks form when the pressure drop Δ*p*_{d}^{Footnote 1} in the melt between the solidifying grains exceeds the critical depression Δ*p*_{c},

where the pressure drop Δ*p*_{d} is the sum of the two contributions Δ*p*_{sh} and Δ*p*_{ε}. The pressure drop

results from the solidification shrinkage. It is determined by the local velocity of the isotherms \( v_{\rm T} \)[6] (which for welding equals the local solidification rate), solidification shrinkage coefficient *β*, viscosity *μ*, and spatial distributions of both the liquid fraction \( f_{\text{L}} (x) = 1 - f_{\text{S}} (x) \) and permeability *K*(*x*) along the centerline at *y* = 0 between *x*_{coh} and *x*_{L}.[6,25] The contribution resulting from the thermomechanical deformation of the workpiece amounts to

where \( \dot{\varepsilon } \) is the strain rate, which is assumed to not depend on *x*.[6,25]

Deviating from the original introduction of the RDG criterion,[6] the lower limit of the integrals in Eqs. [2] and [3] is determined by the condition of coherency, as described in the introduction with Figure 2. According to,[25,26] the temperature \( T_{\text{coh}} \) of the solid fraction of coherency *f*_{S,coh} and hence the position \( x_{\text{coh}} \) have to be chosen for the lower limit to obtain a more realistic representation of the hot cracking phenomena.

Inserting Eqs. [2] through [4] in Eq. [1] and after solving for \( \dot{\varepsilon } \), it is found that the solidifying material cannot withstand strain rates exceeding the critical value

and a hot crack will form.

For the description of centerline cracking, one needs to consider only the solidification processes right at *y* = 0 where the velocity \( v_{\text{T}} (y = 0) \) equals the welding velocity \( v \)[27] as soon as a quasi-stationary temperature field has developed.

The permeability *K*(*x*) of the equiaxed dendritic grain structure at the centerline of a weld can be approximated by

following the Kozeny–Carman equation[28,29] and the assumption of spherical grains[3,30] with the diameter *d*_{eqx}. This average grain diameter of the equiaxed dendritic grains in the center area of a laser-welded seam (see Figure 2) can be approximated by the linear fit

to empirical data published in,[2] where \( \eta_{\text{abs}} \) is the absorptance of the laser beam in the sample and

is the depth-specific line energy.

Note that for all equations that result from the GSM,[2] an ideal two-dimensional heat conduction was assumed. The origin of the *x*–*y*-coordinates is the line source, *i.e*., the position of the capillary (see Figure 2). The equations derived from the GSM are only valid for *x* < 0 and where solidification takes place.[2]

The upper boundary of the integrals in Eqs. [2] and [3] is given by the location *x*_{L} at which the solidification starts[6] (*i.e*., where the temperature equals *T*_{Liquidus}) and is given by Reference 2

where \( \lambda_{\text{th}} \) is the heat conductivity, *T*_{amb} = 20 °C the ambient temperature, and \( \kappa \) the thermal diffusivity.

The solidification is completed at the temperature *T*_{Solidus} where a solid fraction of *f*_{S} = 1 is reached at the centerline of the weld at the *x*-position[2]

The solid fraction of coherency *f*_{S,coh} yields the lower boundary of the integrals[25,26] in Eqs. [2] and [3] and the corresponding *x*-position[2]

remarks the position at which the solidifying grains reach coherency at the centerline of the weld (see Figure 2). The coherency temperature *T*_{coh} *=* *T*(*f*_{S} *=* *f*_{S,coh}) can be derived from the solidification path *f*_{S}(*T*), which depends on the composition of the alloy and is calculated according to Scheil.[31]

With the known solidification path *f*_{S}(*T*), the corresponding spatial function *f*_{S}(*x*), as required for Eqs. [2] and [3], is obtained by using the temperature distribution

at the centerline of the weld, as derived in Reference 2.

As mentioned in the introduction, the strain rate, which affects the solidification zone, distributes over more than one grain boundary[12,13] in case of welds with an equiaxed dendritic grain structure (see Figure 2).

This number of grain boundaries is given by

where \( w_{\text{eqx}} \) is the width of the zone with an equiaxed grain structure and \( d_{\text{eqx}} \) is the average grain diameter,[4,15] as given by Eq. [7]. From the GSM,[2] the width

of the zone containing equiaxed dendritic grains is found to be a function of *E*_{Depth}, where the thermal efficiency \( \eta_{\text{T}} \) is a function of the Peclet value according to Rykalin[1] as defined in the introduction and \( w_{\text{eqx,norm}} = \frac{{w_{\text{eqx}} (E_{\text{Depth}} )}}{{w_{\text{weld}} (E_{\text{Depth}} )}} \) is the share of equiaxed dendritic grains.

To take into account the linear dependence of the critical strain rate \( \dot{\varepsilon }_{\text{crit}} \), at which hot cracking starts to occur, on the number of grain boundaries \( N_{\text{GB}}, \)[4,15] which are affected by the thermomechanical load, we multiply Eq. [5] by \( \frac{1}{{N_{\text{GB}} }} \). This results in the critical strain rate limit

which should not be exceeded to avoid the formation of a centerline crack in a laser beam weld, as a function of *E*_{Depth} and *v*.

According to Eqs. [5], [6], [7], [9], [11], [13], and [14], all variables, which are not material properties but influence the result of Eq. [15], depend on *E*_{Depth}.

Note that this model is only valid for welds with an equiaxed dendritic grain structure, *i.e*., for welds welded with a depth-specific power \( P_{\text{Depth}} = v \cdot E_{\text{Depth}} > P_{\text{Depth,eqx}} \) above the alloy-specific threshold \( P_{\text{Depth,eqx}} \) for the formation of an equiaxed dendritic grain structure.[2]

## Application of the Model to AA6016

Figure 3 shows the spatial distribution of the solid fraction *f*_{S} and the temperature *T* along the centerline (*y* = 0) for the example of welding a *s* = 2.4-mm-thick sheet of AA6016 with a velocity of \( v \) = 6 m/min and a laser power of *P* = 4.3 kW, *i.e*., with a depth-specific line energy of *E*_{Depth} = 17.92 J/mm^{2}. The solidification path *f*_{S}(*T*) of the AA6016 alloy was calculated according to Scheil[31] with Thermocalc2016b assuming a ternary system with a silicon content of 1.07 wt pct and a magnesium content of 0.41 wt pct according to Reference 32. Its spatial evolution *f*_{S}(*T*(*x*)) is represented by the blue curve with the left vertical axis. The temperature was calculated by Eq. [12] using the material properties listed in Table I.

The temperatures *T*_{Liquidus} = 652 °C and *T*_{Solidus} = 534 °C, shown by the dashed orange lines in Figure 3, were derived from the solidification path *f*_{S}(*T*) of the AA6016 alloy. From this, *x*_{S} and *x*_{L} were calculated according to Eqs. [9] and [10], as represented by the blue dashed lines.

In case of welding AA6016, the depth-specific power of \( P_{\text{Depth}} > P_{\text{Depth,eqx}} = 677_{ - 50}^{ + 65} \,{\text{W}}/{\text{mm}} \) must be exceeded for the formation of an equiaxed dendritic grain structure.[2]

To consider the uncertainties of the values of the influencing variables, the model was solved for a large range of possible values of the respective quantities, as listed in Table II.

The absorptance *η*_{abs} = 0.8 was calculated according to Gouffé[35] under the assumption of a cylindrical capillary with an aspect ratio of 10 in liquid aluminum with an absorptivity of 12 pct at a wavelength of 1030 nm. The maximum value *η*_{abs} = 1 is the maximum possible value.

The range of the heat efficiency *η*_{T} is a function of the Peclet criterion \( Pe_{\text{Ryk}} = \frac{{w_{\text{weld}} \cdot v}}{4 \cdot \kappa } \).[1] The highest possible heat efficiency in two-dimensional heat conduction is *η*_{T} = 0.48.[1] Due to the criterion \( Pe_{\text{Ryk}} > Pe_{\text{Ryk,eqx}} = 0.287_{ - 0.046}^{ + 0.036} \) for the formation of an equiaxed dendritic grain structure,[2] the minimum heat efficiency is *η*_{T} = 0.29 according to Rykalin’s calculations.[1]

The range of the solid fraction of coherency, which determines the coherency temperature by means of the solidification path, was chosen according to the range of different values used in the literature.[3,25]

The share of equiaxed grains in the weld \( w_{\text{eqx,norm}} = \frac{{w_{\text{eqx}} (E_{\text{Depth}} )}}{{w_{\text{weld}} (E_{\text{Depth}} )}} = 0.45_{ - 0.15}^{ + 0.15} \) is chosen according to the experimental results presented in Reference 2 which range from 0.3 up to 0.6.

The critical cavitation depression *∆p*_{c} = 2000 Pa results from the assumption of Rappaz.[6] To respect possible deviations, it was chosen to range from 0 Pa up to 4000 Pa in the following calculations.

The minimum welding velocity results from the validity criterion \( v > \frac{{P_{\text{Depth,eqx}} }}{{E_{\text{Depth}} }} \) as introduced in Eq. [15]. The maximum welding velocity is chosen to be 100 times higher than the minimum velocity, which is required for the corresponding depth-specific line energy *E*_{Depth}.

Figure 4 shows the number of grain boundaries *N*_{GB} with the blue curve, the average grain size *d*_{eqx} with the green curve and the width of the equiaxed zone *w*_{eqx} with the orange curve calculated according to Eqs. [7], [13], and [14]. The variables were chosen as listed in Tables I and II.

Figure 4 shows that the number of grain boundaries increases with increasing depth-specific line energy.

The black curve in Figure 5 represents the critical strain rate \( \dot{\varepsilon }_{\text{crit}} \) as a function of *E*_{Depth} in case of welding an AA6016 alloy as calculated with Eq. [15] with the material properties listed in Table I and the typical values listed in Table II. The gray scatter band represents the range of possible critical strain rates \( \dot{\varepsilon }_{\text{crit}} \) within the range of the lowest and highest considered values listed in Table II.

The calculated strain rates are compared with the experimental results published in References 15 and 16 (hatched symbols) and new measurements (fully filled symbols) resulting from the same experimental setup.[16]

This self-restraint hot cracking test follows the concept of Kutsuna[36] and the SEP 1220[37]: A centerline crack initiates at the beginning of the weld and the thermomechanical load, which affects the weld, decreases during welding until the propagation of the centerline crack ends. Additionally, the change of the distance between two points on opposite sides of the weld is measured by means of digital image correlation.[16] The ratio between this change and the initial distance of 5 mm between the points expresses the strain, which affects the weld.[7] The critical strain rate is defined as the temporal derivation of this strain, which affects the end of the melt pool at the position where the propagation of the centerline crack ends.[16]

All investigated sheets were alloys within the AA6016 standard. The color of the symbol indicates the welding velocity and the shape of the symbols the diameter of the laser beam on the surface of the welded sample. The measured values represent the average of five measurements. The length of the vertical error bars represents the range between the minimum and maximum measured values. The horizontal error bars represent the uncertainty of the absorbed depth-specific line energy \( \eta_{\text{abs}} \cdot E_{\text{Depth}} \) resulting from the variation of the absorptance \( \eta_{\text{abs}} = 0.8_{ - 0.1}^{ + 0.2} \), as stated in Table II.

Figure 5 shows that the experimental results agree well with the results of the presented model. Despite the wide ranges of the values of the constants and of the variable welding velocity *v* (see Table II), the scatter band is comparatively narrow. This shows that if the validity criterion \( v > \frac{{P_{\text{Depth,eqx}} }}{{E_{\text{Depth}} }} \) is fulfilled, \( v \) has a negligible influence on the critical strain rate and proves the major influence of *E*_{Depth} on the critical strain rate.

Positive values of \( \dot{\varepsilon } \) indicate a tensile strain rate, and negative values indicate a compressive strain rate. Both the calculated curve and the experimental results prove that the solidification zone of the weld has to be compressed to balance the solidification shrinkage in case of welding AA6016 alloys. If this compression is not sufficient, *i.e*., the compressive strain rate is too low (small negative values of \( \dot{\varepsilon } \)), a hot crack will form. The region of hot crack formation, *i.e*., the region above the calculated curve of the critical strain rate, is hatched in red. The parameter range, which will not lead to the formation of a centerline crack in the laser beam weld, is beneath the calculated curve of the critical strain rate (large negative values of \( \dot{\varepsilon } \)) and hatched in green.

According to Eq. [15], the critical strain rates are compressive for all welding parameters in case of the materials properties of AA6016 (Table I). However, the curve of the critical strain rate shows that the compressive critical strain rate of a laser beam weld, *i.e*., the hot cracking sensitivity, can be reduced by an increase of the depth-specific line energy *E*_{Depth}.

Note that an increase of the depth-specific line energy *E*_{Depth} may lead to an increase of the thermomechanical load, which results from the clamping of the component or from the weld configuration itself.[4] This is not considered in the model.

For a further optimization of the hot cracking sensitivity of laser beam welds up to positive critical strain rates, the material properties have to be changed, *e.g*., with the addition of filler wire[17] or by means of alloys, which are especially designed for laser beam remote welding.[15,38]

## Conclusion

The explicit analytical model for the prediction of the grain structure of laser beam welds, derived in previous work of the authors, was combined with the RDG-criterion of Rappaz, which describes the hot crack formation. This combined model allows for the first time the explicit analytical description of the influence of the welding parameters on the critical strain rates that determine the limit at which the formation of hot crack starts to occur.

The calculated critical strain rates agree well with the experimental results measured by means of digital image correlation.

From the analytical expressions, the depth-specific line energy \( E_{\text{Depth}} = \frac{P}{v \cdot s} \) was now identified to be the key parameter to influence the critical strain rate, which determines the hot cracking sensitivity of a laser beam weld.

The novel calculated results and the experimental results show that *E*_{Depth} has to be increased to reduce the sensitivity of a laser beam weld to hot cracking.

## Change history

### 05 November 2019

In the original article the indefinite integral sign of the solidification path is missing in Eq. [5].

## Notes

- 1.
The values of Δ

*p*_{d}, Δ*p*_{c}, Δ*p*_{sh}, and Δ*p*_{ε}are positive in case of a pressure drop.

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## Acknowledgments

This work was partly supported by Constellium and funded in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—389369540.

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Hagenlocher, C., Weller, D., Weber, R. *et al.* Analytical Description of the Influence of the Welding Parameters on the Hot Cracking Susceptibility of Laser Beam Welds in Aluminum Alloys.
*Metall Mater Trans A* **50, **5174–5180 (2019). https://doi.org/10.1007/s11661-019-05430-7

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