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Mathematical model for vertical rolling deformation based on energy method

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Abstract

An approach to analyzing three-dimensional vertical rolling is presented on the basis of energy method. The double parabola function model is applied to describe dog bone shape in the deformed region between the vertical rolls. The DSF (dual stream function) method is utilized to obtain three-dimensional velocity and strain rate fields. The values of dog bone shape dimensions and roll force are obtained when the total power functional achieves minimum, which is received according to double parabola model, velocity field, and the first variational principle. The validity of the proposed approach is discussed by contradistinguishing the present predictions with other models’ results and measured data in a hot strip plant in miscellaneous rolling conditions. Moreover, the impacts of different rolling conditions on the dog bone shape and stress state coefficient are researched, respectively.

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Abbreviations

W0, WE :

Half of the initial and final slab width at entrance and exit

ΔW :

Half of the reduction, ΔW = W0−WE

W x :

Half of the width in deformation zone.

ΔWx :

Half of the width reduction in deformation zone, ΔWx = W0−Wx

h 0 :

Half of the initial slab thickness at entrance

hI, hII, hIII :

Half of slab thickness in zone I, II and III

h bx :

Peak height of dog bone, hbx = h0 + 2βh0ΔWx/Ax

h rx :

Edge height of dog bone, hrx = h0 + βh0ΔWx/Ax

R :

Radius of work roll

l :

Projected length of roll slab contact arc

v 0 :

Inlet velocity of slab

v R :

Roll speed

θ :

Bite angle, θ = sin−1(l/R)

α :

Contact angle

A,β :

Undetermined parameters

A x :

Width parameter

U :

Flow volume per second

ϕ, ψ :

Stream functions

vx, vy, vz :

Components of velocity vector

U :

Flow volume per second, U = 3v0h0A0

J ∗ :

Total power

W i :

Internal plastic deformation power

W f :

Friction power

W s :

Shear power

σs :

Material yield stress

k :

Yield shear stress, \( k={\upsigma}_s/\sqrt{3} \)

m :

Friction factor

J * min :

Minimum value of total power

M :

Roll torque

F :

Roll force

n σ :

Stress state coefficient

χ :

Arm factor

x, y, z :

The directions of length, thickness, and width

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Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan, Shanxi, 030024, People’s Republic of China

    Y. M. Liu, P. J. Hao, T. Wang & Z. K. Ren

  2. State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang, Liaoning, 110819, People’s Republic of China

    J. Sun & D. H. Zhang

  3. Engineering Research Center of Advanced Metal Composites Forming Technology and Equipment, Ministry of Education, Taiyuan, Shanxi, 030024, People’s Republic of China

    Y. M. Liu, P. J. Hao, T. Wang & Z. K. Ren

  4. TYUT-UOW Joint Research Centre, Taiyuan, Shanxi, 030024, People’s Republic of China

    Y. M. Liu, P. J. Hao, T. Wang & Z. K. Ren

  5. Shagang School of Iron and Steel, Soochow University, Suzhou, 215021, People’s Republic of China

    S. H. Zhang

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  1. Y. M. Liu
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  7. S. H. Zhang
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Liu, Y.M., Hao, P.J., Wang, T. et al. Mathematical model for vertical rolling deformation based on energy method. Int J Adv Manuf Technol 107, 875–883 (2020). https://doi.org/10.1007/s00170-020-05094-3

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