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Collaborative Simulation of Nugget Growth and Process Signals for Resistance Spot Welding


In this work, an electrical–thermal–mechanical coupled finite element (FE) model for resistance spot welding (RSW) process is developed to achieve a simultaneous simulation of not only nugget growth but also process signals including dynamic resistance and electrode displacement. The model entails a multi-objective optimization problem of hard-to-measure physical quantities. During the optimization process, the limitation of adjusting only the interface contact parameters is found and corrected by introducing an enhanced thermal conductivity due to molten metal flow. Experimental validation confirmed simulation accuracy and adaptability to sheet metal thickness and welding process parameter variation. The model can directly generate process signal data with accurate quality indexes, which supports solving the problem of insufficient labeled data in model training for RSW quality prediction.

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The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (Grant Nos. 52205396 and 52025058), the Postdoctoral Science Foundation of China (Grant No. 2022T150409), and the State Key Laboratory of Advanced Brazing Filler Metals and Technology (Grant No. SKLABFMT202101).

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Correspondence to Yong-Bing Li.

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Recommended for publication by Commission III Resistance Welding, Solid State Welding, and Allied Joining Process.

This paper was presented at the IIW 2022 International Conference on Innovative Welding and Joining Technologies to achieve Carbon Neutrality and promote Sustainable Development held in Tokyo, Japan.


Appendix 1



measured nugget diameter (mm)


electrode tip diameter (mm)


measured indentation diameter (mm)


simulated nugget diameter at moment t (mm)

eD N

relative calculation error of nugget diameter (%)

eP N

relative calculation error of weld penetration (%)


relative calculation error of dynamic resistance (%)


relative calculation error of electrode displacement (%)


electrical contact resistance of A/B interface at temperature T and pressure P (Ωm2); A, B ∈ {S, E}, where S stands for sheet, E stands for electrode

h a

convective heat transfer coefficients of air (W/(m2·℃))

h w

convective heat transfer coefficients of cooling water (W/(m2·℃))

h S

sheet thickness (mm)


factor related to the oxide layer or coating layer on the material surface


Lorentz constant (WΩ/℃2)


pressure (MPa)


measured nugget penetration (mm)


simulated weld penetration at moment t (mm)


measured dynamic resistance signal at moment t (μΩ)


simulated dynamic resistance at moment t (μΩ)


resistance of A/B interface (μΩ); A, B ∈ {S, E, P}, where S stands for sheet, E stands for electrode, P stands for probe


bulk resistances of one electrode (μΩ)


bulk resistances of one sheet (μΩ)

R ti

resistance measurement at condition i (μΩ); i = 1 ~ 3


measured electrode displacement signal at moment t (μm)


simulated electrode displacement at moment t (μm)


time (ms)

t w

heating time (ms)


temperature (°C)

T 0

room temperature (°C)

T m

melting point of sheet (°C)


thermal contact resistance of A/B interface at temperature T and pressure P (℃m2/W); A, B ∈ {S, E}, where S stands for sheet, E stands for electrode


displacement (m)

α A/B

adjustable correction coefficient for electrical contact resistance at A/B interface; A, B ∈ {S, E}, where S stands for sheet, E stands for electrode

β A/B

adjustable correction coefficient for thermal contact resistance at A/B interface; A, B ∈ {S, E}, where S stands for sheet, E stands for electrode

λ m

thermal conductivity above the melting point


yield strength of the softer of materials A and B at temperature T; A, B ∈ {S, E}, where S stands for sheet, E stands for electrode


potential (V)


the resistivity of material A at temperature T (μΩm); A ∈ {S, E}, where S stands for sheet, E stands for electrode

Appendix 2

2.1 Control equations

  1. 1)

    Stress equilibrium equation

    $$\left\{\begin{array}{l}\frac{1}{r}\frac{\partial }{\partial r}\left(r{\sigma }_{rr}\right)-\frac{{\sigma }_{\theta \theta }}{r}+\frac{\partial {\sigma }_{zr}}{\partial z}=0\\ \frac{1}{r}\frac{\partial }{\partial r}\left(r{\sigma }_{zr}\right)+\frac{\partial {\sigma }_{zz}}{\partial z}=0\end{array}\right.$$

    where (r, θ, z) describes a cylindrical coordinate system. σrr, σθθ, and σzz represent radial, circumferential, and axial normal stress components, respectively. σzr is the shear stress component. The stress does not change along the circumference with the axial symmetry condition. Meanwhile, the influence of inertial force and gravity is ignored.

  1. 2)

    Elastic–plastic constitutive equation

    $$\left\{\begin{array}{l}\left\{d\varepsilon \right\}=\left\{d{\varepsilon }_{e}\right\}+\left\{d{\varepsilon }_{P}\right\}+\left\{d{\varepsilon }_{T}\right\}\\ \left\{d{\varepsilon }_{e}\right\}={\left[{D}_{e}\right]}^{-1}\left\{d\sigma \right\}+\frac{\partial {\left[{D}_{e}\right]}^{-1}}{\partial T}\left\{\sigma \right\}dT\\ \left\{d{\varepsilon }_{P}\right\}=d\lambda \left\{\frac{\partial Y}{\partial \sigma }\right\}\\ \left\{d{\varepsilon }_{T}\right\}=\left\{\alpha \right\}dT\end{array}\right.$$

    where {σ} and{dσ}, respectively, represent stress and stress increment tensors. {dε}, {dεe}, {dεP}, and {dεT} stand for the tensors of total strain increment, plastic strain increment, and thermal strain increment, respectively. [De] is the elastic stiffness matrix, [α] is the linear expansion coefficient matrix, and T stands for temperature. In addition, Y is the subsequent yield function of the material and obeys Von Mises yield criterion. dλ is a parameter related to the hardening rule and obeys the isotropic strengthening criterion.

  1. 3)

    Current continuity equation

    $$\left\{\begin{array}{l}j=-\frac{1}{\rho }\nabla \varphi \\ \nabla \cdot j=\frac{1}{r}\frac{\partial }{\partial r}\left(\frac{r}{\rho }\frac{\partial \varphi }{\partial r}\right)+\frac{\partial }{\partial z}\left(\frac{1}{\rho }\frac{\partial \varphi }{\partial z}\right)=0\end{array}\right.$$

    where (r, θ, z) describes a cylindrical coordinate system. j stands for current density, φ is potential, and ρ is the temperature-dependent resistivity of the material. The above equation is obtained from the current continuity equation for the constant electric field. In MFDC RSW process with constant current mode, the magnitude and frequency of current variation are low. Thus, the electric field within each time step can be approximated as quasi-static.

  1. 4)

    Heat conduction equation

    $$\left\{\begin{array}{l}\frac{\partial {H}_{v}}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}\left(\lambda r\frac{\partial T}{\partial r}\right)+\frac{\partial }{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right)+{q}_{v}\\ {q}_{v}=\rho {j}^{2}=\frac{1}{\rho }{\left(\nabla \varphi \right)}^{2}=\frac{1}{\rho }\left[{\left(\frac{\partial \varphi }{\partial r}\right)}^{2}+{\left(\frac{\partial \varphi }{\partial z}\right)}^{2}\right]\end{array}\right.$$

    where (r, θ, z) describes a cylindrical coordinate system. T stands for temperature, φ is potential, t refers to time, Hv represents the volume enthalpy of the material, and λ is the thermal conductivity of the material. The internal heat source qv represents the power of joule heat generated by the welding current passing through the unit volume of material.

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Xia, YJ., Lv, TL., Ghassemi-Armaki, H. et al. Collaborative Simulation of Nugget Growth and Process Signals for Resistance Spot Welding. Weld World 67, 1377–1392 (2023).

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