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The curvature mathematical modeling of the double-layer metal clad plate by the asynchronous rolling

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Abstract

The asymmetric phenomenon of the rolling process will cause rolling clad plate bending, and the inability to quantify accurately the bending curvature will reduce production efficiency and increase production costs in industrial production. Therefore, this paper establishes a curvature theoretical model of double-layer metal on the basis of the stream function method, which has high accuracy and can be used for predicting the plate curvature after asynchronous rolling. The stream function field, velocity field, and strain rate field model are established based on the geometric relationship of the rolling deformation zone. The total power model is represented by the five-node Gauss quadrature algorithm according to the complex forming process, and then, the real kinematic parameters are obtained by optimizing the total power. The theoretical model of plate curvature caused by linear strain difference and shear strain difference is calculated based on the kinematic parameters optimized from the total power in the final. This paper creatively proposes a method of integrating the positive strain rate and shear strain rate with time to obtain the strain after rolling, which avoids the error caused by the slab method linearizing the non-uniform distribution of shear stress along the vertical, and the nonlinear problems caused by yield criterion in solving the compressive stress. The article analyzes the accuracy of the velocity model and obtains the variation law of total power with reduction ratio and the variation law of curvature with velocity ratio and layer thickness ratio, and some methods to reduce outlet curvature are also proposed. The maximum, minimum, and average relative errors are 12.02%, 0.66%, and 4.95% by comparing theoretical results with experiments, which verifies the accuracy of the analytic model. The calculating time is less than 5 s, which is of great significance for the setting and optimization of rolling process parameters and online prediction.

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Abbreviations

\(R\) :

Top and bottom work roll radius

\(r\) :

Bonding surface radius

\({v}_{1},{v}_{2}\) :

Top and bottom work roll linear velocity

\({t}_{i},{t}_{o}\) :

Total thickness before and after rolling

\({t}_{ia},{t}_{oa}\) :

Cladding layer thickness before and after rolling

\({t}_{ib},{t}_{ob}\) :

Substrate layer thickness before and after rolling

\({\Gamma }_{1},{\Gamma }_{2},{\Gamma }_{3}\) :

Contact boundary function

\({\Gamma }_{i},{\Gamma }_{o}\) :

Inlet and outlet velocity discontinuity line

\(\Delta h\) :

The rolling reduction

\({\psi }_{a}^{0},{\psi }_{b}^{0}\) :

Basic stream function

\({\psi }_{a}{\prime},{\psi }_{b}{\prime}\) :

Additional stream function

\({\psi }_{a},{\psi }_{b}\) :

Complete stream function

\({\varphi }_{a},{\varphi }_{b}\) :

Volume flow of the layers a and b

\({a}_{1},{a}_{2},{b}_{1},{b}_{2}\) :

Parameters of the velocity gradient function

\({U}_{xa},{U}_{ya}\) :

Horizontal and vertical velocity of the layer a

\({U}_{xb},{U}_{yb}\) :

Horizontal and vertical velocity of the layer b

\({\stackrel{\cdot }{\varepsilon }}_{xa},{\stackrel{\cdot }{\varepsilon }}_{ya}\) :

Horizontal and vertical strain rate of the layer a

\({\stackrel{\cdot }{\varepsilon }}_{xb},{\stackrel{\cdot }{\varepsilon }}_{yb}\) :

Horizontal and vertical strain rate of the layer b

\({\stackrel{\cdot }{\varepsilon }}_{xya},{\stackrel{\cdot }{\varepsilon }}_{xyb}\) :

Shear strain rate of the layers a and b

\({\stackrel{\cdot }{\varepsilon }}_{ea},{\stackrel{\cdot }{\varepsilon }}_{eb}\) :

Equivalent strain rate of the layers a and b

\({\varepsilon }_{u},{\varepsilon }_{l}\) :

The linear strain on the top and bottom surfaces of the clad plate

\({\stackrel{\cdot }{E}}_{a}({A}_{a}),{\stackrel{\cdot }{E}}_{b}({A}_{b})\) :

Deformation power

\(\stackrel{\cdot }{E}({\Gamma }_{i}),\stackrel{\cdot }{E}({\Gamma }_{o})\) :

Shear power

\({\stackrel{\cdot }{E}}_{a}({\Gamma }_{1}),\stackrel{\cdot }{E}({\Gamma }_{2})\) :

Frictional power

\({A}_{a},{A}_{b}\) :

Area of layers a and b in the deformation zone

\({\tau }_{f1},{\tau }_{f{2}}\) :

Shear stress

\(t\) :

The consumption time of the strain

\(\Delta {v}_{fa}\) :

Velocity difference between the top work roll and layer a

\(\Delta {v}_{fb}\) :

Velocity difference between the bottom work roll and layer b

\(\Delta {v}_{ai},\Delta {v}_{bi}\) :

The velocity difference between the inlet velocity and velocity discontinuity line Γi of the cladding and substrate layer

\(\Delta {v}_{ao},\Delta {v}_{bo}\) :

The velocity difference between the outlet velocity and velocity discontinuity line Γo of the cladding and substrate layer

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Funding

The authors gratefully appreciate the financial support by the National Natural Science Foundation of China under Grant 52175354, the Shandong Provincial Natural Science Foundation of China under ZR2022ME082, and the Central Leading Local Science and Technology Development Fund Project under Grant YDZJSX2022A052.

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Authors and Affiliations

  1. School of Mechanical Engineering, Taiyuan University of Science and Technology, Taiyuan, 030024, People’s Republic of China

    Lian-Yun Jiang, Gui-Wen Liu, Qi-Qi Ma, Jia-Yu Song & Jian-Hui Shi

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  2. Gui-Wen Liu
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Contributions

Lian-Yun Jiang: conceptualization; writing—original draft preparation; visualization; and project administration.

Gui-Wen Liu: writing—original draft preparation and visualization.

Qi-Qi Ma and Jia-Yu Song: review and editing, and supervision.

Jian-Hui Shi: writing—review and editing, and supervision.

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Correspondence to Gui-Wen Liu.

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Jiang, LY., Liu, GW., Ma, QQ. et al. The curvature mathematical modeling of the double-layer metal clad plate by the asynchronous rolling. Int J Adv Manuf Technol 129, 5457–5471 (2023). https://doi.org/10.1007/s00170-023-12674-6

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