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Metallurgical and Materials Transactions B

, Volume 50, Issue 6, pp 3126–3126 | Cite as

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

  • Anshan Yu
  • Xiangjie YangEmail author
  • HongMin Guo
  • Kun Yu
  • Xiuyuan Sun
  • Zixin Li
Correction
  • 207 Downloads

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.

Notes

Copyright information

© The Minerals, Metals & Materials Society and ASM International 2019

Authors and Affiliations

  • Anshan Yu
    • 1
    • 2
  • Xiangjie Yang
    • 1
    • 2
    Email author
  • HongMin Guo
    • 2
    • 3
  • Kun Yu
    • 1
    • 2
  • Xiuyuan Sun
    • 1
    • 2
  • Zixin Li
    • 1
    • 2
  1. 1.School of Mechanical and Electrical EngineeringNanchang UniversityNanchangP.R. China
  2. 2.Key Laboratory of Near Net Forming in Jiangxi ProvinceNanchangP.R. China
  3. 3.Department of Materials Science and EngineeringNanchang UniversityNanchangP.R. China

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