Correction to: Numerical Simulation and Experimental Validation of Nondendritic Structure Formation in Magnesium Alloy Under Oscillation and Ultrasonic Vibration

The Original Article is available

Correction to: Metallurgical and Materials Transactions B https://doi.org/10.1007/s11663-019-01654-5

The equation 1 “1 − ϕ” should be “1 − ϕ2”.

The equation 2 last “∂u/∂t” should be “∂ϕ/∂t” .

The corrected equations are

$$ \begin{aligned} &\tau \left( \varvec{n} \right)\left[ {1 + (1 - k)u} \right]\frac{\partial \varphi^{2}}{\partial t} \\ &\quad = \varphi (1 - \varphi ^{2}) - \lambda \left( {1 - \varphi^{2} } \right)^{2} \left( {u + \theta_{\text{sys}} } \right) + \nabla \cdot \left\{ {\left[ {W(\varvec{n})} \right]^{2} \nabla \varphi } \right\} \\ & \qquad - \,\frac{\partial }{\partial x}\left[ {W(\varvec{n})W^{\prime}(\varvec{n})\frac{\partial \varphi }{\partial y}} \right] + \frac{\partial }{\partial y}\left[ {W(\varvec{n})W^{\prime}(\varvec{n})\frac{\partial \varphi }{\partial x}} \right] \\ \end{aligned} $$
(1)
$$ \left( {\frac{1 + k}{2} - \frac{1 - k}{2}\varphi } \right)\left( {\frac{\partial u}{\partial t} + U \cdot \nabla u} \right) = \nabla \left( {D\frac{1 - \varphi }{2}\nabla u + J_{AT} } \right) + \frac{1}{2}\frac{\partial \varphi}{\partial t}\left[ {1 + \left( {1 - k} \right)u} \right] $$
(2)

The caption of Figure 21 should say, “Microstructure of solidified AZ91D at a fixed oscillation frequency (1 Hz), ultrasonic power (1000 W), and oscillation amplitude (4π/3) for inclined angles of (a) 25 deg, (b) 20 deg, and (c) 15 deg.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xiangjie Yang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yu, A., Yang, X., Guo, H. et al. Correction to: Numerical Simulation and Experimental Validation of Nondendritic Structure Formation in Magnesium Alloy Under Oscillation and Ultrasonic Vibration. Metall Mater Trans B 50, 3126 (2019). https://doi.org/10.1007/s11663-019-01694-x

Download citation