Analytical and experimental investigation on the distribution of solidification crack initiation sites throughout a laser spot weld

Abstract

The objective of this this study is to investigate the distribution of crack initiation sites in the unsteady solidification conditions of pulsed laser welding for an AA2024 aluminum alloy, using both analytical and experimental methods. The employed analytical model considers the competition between volume change rate and liquid flow rate during the final stages of solidification. The former has a direct relationship with solidification rate (r), the latter has an inverse relationship with the length of the crack-vulnerable zone (l), and the risk of cracking decreases with lower values of r × l. According to this model, the distribution of crack initiation sites is more strongly influenced by the solidification rate profile, specifically the solidification rate at the fusion line (r_FL) and at the weld center (r_WC). The crack initiation site will move toward the weld center as (r_WC-r_FL)/r_FL increases. The model states that in square-wave pulse welded samples, a decrease in r_FL caused by preheating causes crack initiation sites to move from the FL toward the WC and crack severity to decrease. Moreover, it was shown that ramp down pulse shaping was more effective at reducing cracks, as both r and l can be controlled and, in turn, r × l can be reduced enough to prevent crack initiation. The model’s reliability was assessed using experimental pulsed laser welding tests that considered the influence of base metal preheating and temporal ramp down pulse shaping.

Appendices

Appendix A

Gill et al. [28] reported that the solidification rate dependent liquidus slope (m_r) in Eq. (12) can be related to solidification rate dependent distribution coefficient (k_r) as follows:

$${m}_{r}={m}_{0}\left[\frac{1-{k}_{r}+{k}_{r}ln\left(\frac{{k}_{r}}{{k}_{0}}\right)}{1-{k}_{0}}\right]$$

(23)

And similarly Sobolev [29] reported that ${C}_{L}^{*}$ can also be stated as a function of k_r by:

$${C}_{L}^{*}=\frac{{C}_{0}}{1-\left(1-{k}_{r}\right)Iv\left({P}_{C}\right)}$$

(24)

Iv (P_C) is Ivantsov function and P_C is the Peclet number of the solute diffusion at dendrite/cell tip. Peclet number was related to tip radius (r_t), the solute element diffusion coefficient in the liquid (${D}_{L}$), and solidification rate (v) as follows:

$${P}_{C}=\frac{{r}_{t}v}{2{D}_{L}}$$

(25)

It was proposed a local non-equilibrium diffusional model (LNDM) developed for the binary alloys rapid solidification conditions [29,30,31]. Based on this model, k_r can be calculated as a function of solidification rate (r), the solute element diffusive speed in the liquid (v_Db), and diffusive speed of the solute element across the interface (v_Di) by following equation.

$${k}_{v}=\frac{{k}_{0}\left(1-\frac{{r}^{2}}{{v}_{Db}^{2}}\right)+\frac{r}{{v}_{Di}}}{\left(1-\frac{{r}^{2}}{{v}_{Db}^{2}}\right)+\frac{r}{{v}_{Di}}}$$

(26)

Reasonably based on this model when solidification rate reach to the speed of diffusive solute element in the liquid (r = v_Db) then k_r = 1, i.e., solute trapping at the interface occurs completely.

Appendix B

$\Delta {T}_{E}$ Can be computed using a model called TMK model [23]. TMK model showed that the parameters of $\Delta {T}_{E}$ at high solidification rates are dependent on r, as follows:

$$\Delta {T}_{E}=\frac{{m}_{\alpha }+{m}_{\theta }}{2}{\left[\left[Z+\lambda \left(\left. ^{\displaystyle \partial Z}\!\!\right/ \!\!_{\displaystyle \partial \lambda }\right)\right]\times \frac{2{C}_{0}v}{f\left(1-f\right){D}_{L}}\times \left(\frac{{a}_{\alpha }^{L}}{{fm}_{\alpha }}+\frac{{a}_{\theta }^{L}}{\left(1-f\right){m}_{\beta }}\right)\right]}^{\displaystyle \left. ^{1}\!\right/ \!_{2}}\left(1+\frac{Z}{P+\lambda \left(\left. ^{\displaystyle \partial Z}\!\!\right/ \!\!_{\displaystyle \partial \lambda }\right)}\right)$$

(27)

Here, ${m}_{\theta }$ and ${m}_{\alpha }$ are the slopes of liquidus for θ and α phases. ${a}_{\alpha }^{L}$ and ${a}_{\theta }^{L}$ are the capillarity constants. ${C}_{0}$ is the eutectic tie-line length, f is the α phase volume fraction, and $\lambda$ is the distance between the eutectic layer. P and $P+\lambda \left({~}^{\partial P}\!\left/ \!{~}_{\partial \lambda }\right.\right)$ are two series that can be related to the distribution coefficient at α/liquid interface (k_α), the Peclet number at the eutectic temperature (p), and f as follows:

$$Z=\sum\limits_{n=1}^\infty\left(\frac1{n\pi}\right)^3\sin^2\left(n\pi f\right)\times\frac{p_n}{\sqrt{1+p_n^2}-1+2k_\alpha}$$

(28)

and

$$Z+\lambda \left(\left. ^{\displaystyle \partial Z}\!\!\right/ \!\!_{\displaystyle \partial \lambda }\right)=\sum_{n=1}^{\infty }{\left(\frac{1}{n\pi }\right)}^{3}{sin}^{2}\left(n\pi f\right)\times {\left[\frac{{p}_{n}}{\sqrt{1+{p}_{n}^{2}}-1+2{k}_{\alpha }}\right]}^{2}\frac{{p}_{n}}{\sqrt{1+{p}_{n}^{2}}}$$

(29)

where

$${p}_{n}=\frac{2n\pi }{p}$$

(30)

p can be stated as a function of the distance between eutectic layers (λ) [24] and the solute element diffusion coefficient in the liquid (D_L) by:

$$p={~}^{r\lambda }\!\left/ \!{~}_{2{D}_{L}}\right.$$

(31)

Appendix C

The physical properties and constants for Al-Cu alloys that were used in this investigation are given in Table 6.

Table 6 Physical properties of Al (α)-Al₂Cu (θ) system required for calculations [24]