Metallurgical and Materials Transactions A

, Volume 50, Issue 5, pp 2246–2258 | Cite as

A CFD-FEM Model of Residual Stress for Electron Beam Welding Including the Weld Imperfection Effect

  • Lvjie Liang
  • Renzhi Hu
  • Jingsheng Wang
  • Manlelan Luo
  • Anguo Huang
  • Bing Wu
  • Shengyong PangEmail author


Residual stress of electron beam welding (EBW) is associated with the heat transfer, fluid flow, and keyhole dynamics occurred in the process. In engineering, it was commonly simulated by pure heat conduction-based thermomechanical modeling with an experimentally calibrated heat source model. This method, however, is difficult to be used to consider the effect of weld imperfections such as the spiking and porosity defect. In this study, we proposed a novel combined computational fluid dynamics and finite element method (CFD-FEM) model to predicate the residual stresses of EBW. This model allows us to include the integral effect of heat transfer, fluid flow, and keyhole dynamics as well as process defects on residual stresses for the first time. A phenomenological heat source model dependent on the dynamic keyhole was considered in the CFD model part. A mapping algorithm was developed to build a new heat source model of FEM modeling part from the CFD simulations. Parametrical simulations and experiments of EBW of a 10-mm-thickness Ti-6-Al-4-V alloy plate were carried out and good agreements were obtained by the comparisons. It was shown that for deep penetration EBW the keyhole oscillates in radial and depth directions and the spiking defect occurs, particularly under large heat input parameters. Interestingly, the equivalent and principle residual stresses exhibits strong oscillations in the bottom part of welds along the welding direction, in which the amplitudes are around 9 pct (73.69 MPa) of the equivalent stress and 24 pct (121.84 MPa) of the principle stress in penetration direction, respectively, when the power of electron beam is 1.5 kW. As the power of electron beam increases, the effect of spiking on the residual stress enhances. When the welding power increases to 2.25 kW, the amplitudes of equivalent residual stress increase to 12 pct (119.2 MPa). Our results provide a quantitative method connecting the weld pool dynamics and process defects with the residual stress of EBW, which previously could be understood only in a qualitative way, and demonstrate the possibility of residual stress predications only from the information of materials and process parameters before welding, without the knowledge of post-weld profiles.


\( U \)


\( \vec{U} \)

Movement speed of the free surface interface

\( T \)


\( T_{\text{ref}} \)

Reference temperature

\( T_{\infty } \)

Ambient temperature

\( T_{i} \)

Temperature of the CFD element

\( p \)


\( p_{\text{f}} \)

Free surface

\( p_{\text{r}} \)

Recoil pressure

\( g \)

Gravitational vector

\( c \)

Specific heat

\( C\rho \)

Heat capacity

\( K \)

Heat conductivity

\( k_{i} \) (i = x, y, z)

Thermal conductivity in the x, y, and z direction

\( Q \)

Volumetric energy generation

\( Q_{\text{total}} \)

Total energy input

\( Q_{\text{d}} \)

Dynamical keyhole-based heat source term

\( Q_{i} \)

Heat flux of each element of FEM model

\( \vec{n} \)

Unit normal vector

\( \overrightarrow {{t_{1} }} \), \( \overrightarrow {{t_{2} }} \)

Unit vector orthogonal cutting, \( \overrightarrow {{t_{1} }} \) is perpendicular to \( \overrightarrow {{t_{2} }} \)

\( V_{i} \)

Volume of the element of FEM model

\( h_{\text{r}} \)

Radiant heat transfer coefficient

\( D^{P} \)

Plastic matrix

\( D^{e} \)

Elastic matrix


Young’s modulus

Greek symbols

\( \mu \)

Kinetic viscous coefficient

\( \rho \)


\( \beta \)

Thermal expansion coefficient

\( \phi \)

Distant to the surface

\( \sigma \)

Surface tension coefficient

\( \kappa \)


\( \varepsilon_{\text{r}} \)

Surface emissivity

\( \sigma_{\text{s}} \)

Stefan–Boltzmann constant

\( \Omega \)

Heat source zone

\( d\sigma \)

Stress increment

\( d\varepsilon \)

Strain increment

\( \alpha \)

Coefficient of liner expansion



This research was supported by the Key R&D Projects (National Key projects) (Nos. 2017YFE0100100, 2018YFB1105300), the National Natural Science Foundation of China (NSFC) (No. 51675202), and the National Basic Research Program of China (973 Program, No. 2014CB046703).


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Copyright information

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

Authors and Affiliations

  • Lvjie Liang
    • 1
  • Renzhi Hu
    • 2
  • Jingsheng Wang
    • 2
  • Manlelan Luo
    • 2
  • Anguo Huang
    • 2
  • Bing Wu
    • 3
  • Shengyong Pang
    • 2
    Email author
  1. 1.State Key Laboratory of Digital Manufacturing Equipment and TechnologyHuazhong University of Science and Technology(HUST)WuhanP.R. China
  2. 2.State Key Laboratory of Material Processing and Die & Mould TechnologyHuazhong University of Science and Technology(HUST)WuhanP.R. China
  3. 3.Science and Technology on Power Beam Processes LaboratoryAVIC Manufacturing Technology InstituteBeijingP.R. China

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