An Analytical Model for Prediction of Solidification Cracking Susceptibility in Aluminum Alloys Taking into Account the Effect of Solidification Rate

Abstract

Solidification cracking in pulsed laser welding of aluminum alloys has been the subject of a number of researches. The purpose of this study is to analyze the possible effects of high solidification rates involved in pulsed laser welding on the solidification path of aluminum alloys and solidification cracking sensitivity. For the analysis, conditions at the interface are not necessarily considered to be at equilibrium. Accordingly, the solidification path is modified taking into account the effects of solidification rate on the solidus, liquidus, and eutectic temperatures, the liquidus slope, and the interface parameters, including the partition coefficient, the cell/dendrite tip radius, and the liquid composition in the cellular dendrite tip. In line with the previous researches, cracking susceptibility is assumed to be proportional to the range of temperature squared for solidification crack formation ($\Delta T^{2}$). The results of calculations for hypo-eutectic Al-Cu alloys show that cracking susceptibility increases and shifts to a higher copper composition with the increasing solidification rate. The given calculations are evaluated against the reported experimental results.

Appendices

Appendix A

$m_{\text{v}}$, solidification rate dependent on the liquidus slope is related to $k_{\text{v}}$ as follows[7]:

$$ m_{\text{v}} =m_{0} \left( {\frac{{1 - k_{\text{v}} + k_{\text{v}} \ln (k_{\text{v}} /k_{0} )}}{{1 - k_{0} }}} \right) $$

(A1)

$k_{0}$ and $k_{\text{v}}$ are the equilibrium partition coefficient and the solidification rate dependent on the partition coefficient, respectively. Assuming the local equilibrium at the interface, the equilibrium partition coefficient is defined as the ratio of the concentration of the solute element in solid ($C_{\text{S}}$) to that in the liquid ($C_{\text{L}}$) at the interface.[24]

Appendix B

As the solidification rate is increased, since the equilibrium partition coefficient has no longer enough time to draw back the atoms of the solute element into the liquid of the interface tip to maintain the local equilibrium,[8,9] in this case, the equilibrium partition coefficient is not convincing; so the partition coefficient becomes a function of the growth rate (v) and is called $K_{\text{v}}$. As the growth rate is increased, the partition coefficient is increased from its equilibrium value, $k_{0}$, to 1.[8] Aziz’s model predicts the solidification rate-dependent partition coefficient as follows[4,5]:

$$ K_{\text{v}} =\left[ {K_{0} + V/V_{\text{Di}} } \right]/\left[ {1 + V/V_{\text{Di}} } \right] $$

(B1)

$V_{\text{Di}}$ is the diffusive speed of the solute element across the interface, and is a kinetic rate parameter for the distribution of the solute element in the interface and its trapping phenomenon.[40,41,42] Galenko and Sobolev developed the Aziz model of solute trapping by introducing V_Db, the diffusive speed of the solute element in the liquid bulk, which is a diffusive parameter for the diffusivity of the solute element in the liquid bulk under local nonequilibrium conditions and not dependent on the kinetic interface.[9,43] Sobolev presented a local nonequilibrium diffusional model (LNDM) for the rapid solidification of binary alloys, according to which in the interface rate of V=$V_{\text{Db}}$, the complete trapping of the solute element occurs.[43] In this way, in 2013, the Aziz relation (Eq. [B1]) was changed by Sobolev to Eq. [B2].

$$ K_{\text{v}} =\frac{{K_{0} \left( {1 - V^{2} /V_{\text{Db}}^{2} } \right) + V/V_{\text{Di}} }}{{\left( {1 - V^{2} /V_{\text{Db}}^{2} } \right) + V/V_{\text{Di}} }}, $$

(B2)

where v is the solidification rate (the rate of interface movement), $v_{\text{Db}}$, is the diffusive speed of the solute element in the liquid bulk.[7] This relation indicates that the partition of the solute element depends on both $V_{\text{Db}}$ and the $V_{\text{Di}}$.[44] In LNDM model, $K_{\text{v}}$ takes into account both interfacial kinetic effects corresponding to the CGM model (effect of V_Di) and local nonequilibrium diffusion effects (effect of V_Db).[7]

Appendix C

$C_{\text{L}}^{*}$, is obtained from the following equation[20,24]:

$$ C_{\text{L}}^{*} =\frac{{C_{0} }}{{\left[ {1 - \left( {1 - K_{\text{v}} } \right){\text{Iv}}\left( {P_{\text{c}} } \right)} \right]}} $$

(C1)

${\text{Iv}}\left( {P_{\text{c}} } \right)$ is Ivantsov function expressed as[24]

$$ {\text{Iv}}\left( {P_{\text{c}} } \right)=P_{\text{c}} \exp \left( {P_{\text{c}} } \right)E_{1} \left( {P_{\text{c}} } \right), $$

(C2)

where $P_{\text{c}}$ is the peclet number of the solute diffusion, and $E_{1}$ is an exponential integral derived from the following relationships[24]:

$$ P_{\text{c}} =r_{\text{t}} v/2D_{\text{L}} $$

(C3)

$${{E}_{1}}({{P}_{\text{C}}})=\int_{{{P}_{\text{C}}}}^{\infty }{\frac{\exp (-x)}{x}\text{d}x}, $$

(C4)

and $r_{\text{t}}$ is the radius of the cell/dendrite tip. $D_{\text{L}}$, the diffusion coefficient in the bulk liquid, is also determined as follows[25]:

$$ D_{\text{L}}=D^{0}_{{\text{L}}} {{\exp}}( - {Q_{\text{L}}} /RT), $$

(C5)

where R is the universal constant of gases, $Q_{\text{L}}$ is the activation energy for diffusion, and $D_{\text{L}}^{0}$ is the diffusion constant.[25,42] It can be said that $P_{\text{c}}$ is the dimensionless growth rate, and ${\text{Iv}}\left( {P_{c} } \right)$ is the dimensionless undercooling.[3] The Ivantsov function is approximated for the peclet number of $1 \le p \le \infty$ as follows[24]:

$$ {\text{Iv}}\left( {P_{\text{c}} } \right) =\frac{{P_{\text{c}}^{4} + \left( {8.57P_{\text{c}}^{3} } \right) + \left( {18.06P_{\text{c}}^{2} } \right) + \left( {8.63P_{\text{c}} } \right) + 0.268}}{{P_{\text{c}}^{4} + \left( {9.57P_{\text{c}}^{3} } \right) + \left( {25.6P_{\text{c}}^{2} } \right) + \left( {21.1P_{\text{c}} } \right) + 3.96}} $$

(C6)

The above equation is a good approximation for the exact solution of the Ivantsov function in a wide range of undercooling conditions.[24]

The cell/dendrite tip radius,$r_{\text{t}}$, is an important dimensional parameter in the cellular/dendritic solidification.[24,42] In addition to the solidification rate, $r_{\text{t}}$ depends on the composition of alloys.[24,42] In pulsed laser welding, metallographic investigations indicate that a cellular structure is created instead of a dendritic structure.[16] Under these conditions, a macroscopic thermal flux is obtained from the superheated liquid into solid, creating a positive temperature gradient at the cell tips. Using the general equation of stability in the cellular growth model and assuming that $r_{\text{t}}$ is equal to the critical wavelength for the zero expansion instability, $r_{\text{t}}$ is obtained from the following equation[24,42,44]:

$$ r_{\text{t}} =\left\{ {\left( {1/\sigma^{*} } \right)\left[ {\frac{\varGamma }{{\left( {m_{\text{v}} G_{\text{c}} \xi_{\text{c}} - G} \right)}}} \right]} \right\}^{1/2} ,\; (G > 0) $$

(C7)

It is assumed that the thermal conductivity (k) and the thermal diffusivity (α) parameters related to the solid and liquid phases $(K_{\text{s}} =K_{\text{l}} =K,\; \alpha_{\text{s}} =\alpha_{\text{l}} =\alpha )$ are equal and for the sake of simplicity, the effect of the latent heat is ignored.[24] As a result, the temperature gradients in both phases set equal to the gradient at the tip. That is, $G_{\text{s}} =G_{\text{l}} =G$, in which G, is the applied temperature gradient at the interface. In the above equation, $\sigma^{*} =0.2533$ is a stable constant.[24,42] ${{\varGamma }}$ is the Gibbs–Thompson constant. G_c, the compositional gradient, is determined from the equilibrium conditions of the solute element at the cell tip as follows[42]:

$$ G_{\text{c}} =- vC_{\text{L}}^{*} \left( {1 - k_{\text{v}} } \right)/D_{\text{L}} $$

(C8)

${{\xi }}_{\text{c}}$, peclet function, is a function of the corresponding peclet number of the solute diffusion ($P_{\text{c}}$), reaching 1 in the small peclet numbers. ${{\xi }}_{\text{c}}$ is obtained in high peclet numbers, generated at high solidification rates by the following equation[24,44]:

$$ \xi_{\text{c}} =1 - \frac{{2k_{v} }}{{\left[ {1 + (2\pi /P_{\text{c}} )^{2} } \right]^{1/2} - 1 + 2k_{v} }} $$

(C9)

$r_{\text{t}}$ is a function of v, i.e., the solidification rate, and C₀, i.e., the alloy composition. In other words, there is a specific $r_{\text{t}}$ for each specific solidification rate and alloy composition.

Taking all three Eqs. [C3], [C8], and [C9] into account, it can be said that $r_{\text{t}}$, $P_{\text{c}}$ and V are the interdependent variables.

Appendix D

Based on TMK model, the eutectic temperature drop, $\Delta T_{\text{Eutectic}}$, is determined by the Eq. [D1], which depends on the value of λ; the value of λ can be obtained by virtue of the principles of the minimum undercooling. According to Eq. [D2], it is related to v.[22,25] Obtaining λ from Eqs. [D2] through [D7] and inserting it in Eq. [D1], $\Delta T_{\text{Eutectic}}$ can be derived.

$$ \lambda \Delta T_{\text{Eutectic}} =ma^{\text{L}} \left[ {1 + \frac{P}{P + \lambda (\partial P/\partial \lambda )}} \right] $$

(D1)

$$ \lambda^{2} v=a^{\text{L}} /Q^{\text{L}}, $$

(D2)

where

$$ a^{\text{L}} =2\left[ {\frac{{a_{\alpha }^{\text{L}} }}{{fm_{\alpha } }} + \frac{{a_{\beta }^{\text{L}} }}{{\left( {1 - f} \right)m_{\beta } }}} \right] $$

(D3)

and

$$ Q^{\text{L}} =\left( {{\raise0.7ex\hbox{${Q_{0} }$} \!\mathord{\left/ {\vphantom {{Q_{0} } {D_{\text{L}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${D_{\text{L}} }$}}} \right)\left[ {P + \lambda \left( {{\raise0.7ex\hbox{${\partial P}$} \!\mathord{\left/ {\vphantom {{\partial P} {\partial \lambda }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\partial \lambda }$}}} \right)} \right] $$

(D4)

$$ Q_{0} =\frac{{c_{0} }}{{f\left( {1 - f} \right)}}, $$

(D5)

where $m_{\alpha }$ and $m_{\beta }$ are the liquidus slopes consistent with the phases α and β (both are considered positive), respectively. $a_{\alpha }^{\text{L}}$ and $a_{\beta }^{\text{L}}$ are the capillarity constants (given in Table I). $c_{0}$ is the Length of the eutectic tie-line, and f is the volume fraction of α phase. As stated in Eq. [C5], $D_{\text{L}}$ is the diffusion coefficient of the solute element in the liquid.[22] For the second type of metastable phase diagrams, $k_{\alpha } =k_{\beta } ={\text{constant}}$, the values of P and $P + \lambda \left( {{\raise0.7ex\hbox{${\partial P}$} \!\mathord{\left/ {\vphantom {{\partial P} {\partial \lambda }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\partial \lambda }$}}} \right)$ series in Eqs. [D6] and [D7] are given as functions of the volume fraction of the phase α (f), the peclet number at the eutectic temperature (p), and the partition coefficient (k) obtained from the following series[22]:

$$ P=\mathop \sum \limits_{n=1}^{\infty } \left( {\frac{1}{n\pi }} \right)^{3} \left[ {{ \sin }\left( {n\pi f} \right)} \right]^{2} \times \frac{{p_{n} }}{{\sqrt {1 + p_{n}^{2} } - 1 + 2k}} $$

(D6)

$$ P + \lambda \left( {{\raise0.7ex\hbox{${\partial P}$} \!\mathord{\left/ {\vphantom {{\partial P} {\partial \lambda }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\partial \lambda }$}}} \right)=\mathop \sum \limits_{n=1}^{\infty } \left( {\frac{1}{n\pi }} \right)^{3} \left[ {{ \sin }\left( {n\pi f} \right)} \right]^{2} \times \left[ {\frac{{p_{n} }}{{\sqrt {1 + p_{n}^{2} } - 1 + 2k}}} \right]^{2} \frac{{p_{n} }}{{\sqrt {1 + p_{n}^{2} } }}, $$

(D7)

where $p_{n} =2n\pi /p$ and $p=v\lambda /2D_{\text{L}}$.[23,24]