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Three-dimensional analytical model for residual stress in the weld of thick plates by friction stir welding

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Abstract

The residual stress is non-uniform in the thickness direction for thick plates jointed by friction stir welding (FSW). An analytical method by using the material flow in the welding process is proposed to solve the non-uniform residual stress of thick plates. The streamline model for the material flow is established in terms of the relationship between stream function and velocity fields in the weld. The cumulative plastic strain is computed by integrating the strain rate along each streamline. The residual stress field is obtained based on characteristic strain theories. Experiments of FSW are conducted for thick plates of aluminum alloy 2024-T3. The residual stresses in the welded region are measured by both X-ray diffraction and the contour method. The relative error of the predicted residual stress in the top surface of the weld is 4.8%. The accuracy of the analytical model can be verified. The analytical and experimental results show that the distribution of residual stress is asymmetric with a conventional “M” shape, and the distance of the two peak stresses is 40.63 mm in the top surface of the weldment. The flow mechanism of material and residual stress in different regions of thick plates by FSW are revealed. The proposed model may be used for the prediction and analysis of the non-uniform distribution of residual stress in thick plates jointed by FSW.

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Abbreviations

\({\xi }_{\mathrm{tp}}\) :

Dimensionless variable, \({\xi }_{\mathrm{tp}}=z/H\)

H :

Thickness of workpiece

\({v}_{\mathrm{weld}}\) :

Welding speed

\({r}_{\mathrm{pin}}\) :

Radius of pin

r :

Distance between reference point and welding center

\(\omega\) :

Angular speed of rotating pin

Re:

Reynolds number

\({v}_{x,F}\), \({v}_{y,F}\), \({v}_{z,F}\) :

Velocity components in flow arm zone

\({v}_{x,N}\), \({v}_{y,N}\), \({v}_{z,N}\) :

Velocity components in nugget zone

\(\psi\) :

Stream function

\(\Omega\) :

Vorticity function

\(\theta\) :

Angel from reference point to welding center

\({\mathrm{v}}\) :

Vector of material flow velocity in the weld

A :

Vector potential function

\({\chi }_{c}\) , \({\psi }_{c}\) , \({\phi }_{c}\) :

Clebsch scalar potentials

F, \(\dot{{\varvec{F}}}\) :

Deformation gradient tensor and the derivative to time

u :

Displacement tensor

E :

Green strain tensor

L :

Velocity gradient tensor

D :

Symmetrical deformational rate

W :

Asymmetrical rotational rate

\(\dot{{\varvec{E}}}\) :

Strain rate tensor

u, v, w, \(\dot{u}\), \(\dot{v}\), \(\dot{w}\) :

Displacement components and the derivative to time

\({{\varvec{\varepsilon}}}_{ij}\) :

Total strain tensor

\({{\varvec{\varepsilon}}}_{ij}^{*}\) :

Characteristic strain tensor

\(\nabla\) :

Gradient operator

\(\Delta {u}_{s}\) :

Displacement increment in the streamline

\({v}_{\mathrm{ave}}\) :

Average speed of material flow

\({e}_{ij}\) :

Elastic strain tensor

\({\sigma }_{ij}\) :

Stress tensor

\({C}_{ijkl}\) :

Elastic modulus tensor

\({f}_{i}\) :

Body force

\({n}_{j}\) :

Unit normal vector to the boundary

\({e}_{ijk}\) :

Permutation symbol

\({G}_{ij}\) :

Green’s function

\({\mathrm{N}}(\xi )\), \({\mathrm{D}}_{g}(\xi )\) :

Material parameters depend on the structural stiffness

\(\xi\) :

Instantaneous coordinate

\(\lambda\), \(\mu\) :

Lame constants

v :

Poisson ratio

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Funding

This work presented received the financial support from the National Key Research and Development Program of China (2019YFA0709001) and the National Natural Science Foundation of China (No. 52275502).

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Authors and Affiliations

  1. Shanghai Key Laboratory of Digital Manufacture for Thin-Walled Structures, Shanghai Jiao Tong University, Shanghai, 200240, China

    Haidong Yu, Chang Gao, Ke Yuan & Wei Chen

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  1. Haidong Yu
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  2. Chang Gao
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Contributions

Haidong Yu conceived and designed the study. Material preparation, methodology, and analysis were performed by Chang Gao and Ke Yuan. Experiments were performed by Wei Chen. The draft of the manuscript was written by Haidong Yu and Chang Gao. All authors read and approved the final manuscript.

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Correspondence to Haidong Yu.

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Yu, H., Gao, C., Yuan, K. et al. Three-dimensional analytical model for residual stress in the weld of thick plates by friction stir welding. Int J Adv Manuf Technol 129, 4369–4381 (2023). https://doi.org/10.1007/s00170-023-12600-w

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