Accounting for shear deformation in fast models for plate rolling

Abstract

For on-line prediction of roll force and torque, fast models have been available for a long time, which are mainly based on the slab method or other solution methods that allow for computing times in the range of seconds. Such fast models typically treat the rolling process as a plane strain problem and neglect shear deformation, which is always present in real processes. The shear deformation results in inhomogeneous strain profiles and thus might lead to inhomogeneity in microstructure over plate thickness. In this paper, a novel method is presented that allows for superposition of shear strain onto the strain state obtained from the slab method. The shear strains are interpolated from an extensive finite element (FE) parameter study of rolling processes that covers the entire parameter range of today’s plate rolling. For the regimes of very thick and thin plates different interpolation functions are introduced. It is shown that when the proposed shear strain model is combined with the slab method, similar results are obtained as with a full-scale FE calculation of the rolling problem but with a calculation time in the range of seconds.

Appendix

The proposed analytical functions for the hypothetical shear strain $\gamma_{xy}^{\prime }$ operate within certain validity boundaries. In terms of process parameters these are given by the chosen parameters as follows: h ₀  = 5–900 mm and ε _h  = 1–30 %. Due to their analytical nature the domains of the functions have to be analyzed in order to avoid possible discontinuities in the operation of the model.

1.1 Regime I

If the base of the denominator equals zero the function becomes undefined. Consequently the following restriction can be derived:

$$0\mathop < \limits^{!} 0.7807 + k - \frac{{0.8003 \cdot h_{1} }}{{h_{0} }} \Leftrightarrow h_{1} \mathop < \limits^{!} \frac{0.7807 + k}{0.8003} \cdot h_{0}$$

Since naturally h ₁ < h ₀, it follows that k $\mathop > \limits^{!}$ 0.0194. For k ∈ [0, 0.02], we propose to use $\gamma_{xy}^{\prime }$ values of k = 0.02, which leads to mathematical consistency and no relevant deviation in the results, as can be seen in Fig. 7.

1.2 Regime II

As the denominator approaches zero, the equation approaches infinity. Substituting $h_{0} = h_{1} \; \cdot \;\exp \left( { - \frac{\sqrt 3 }{2}\bar{\varepsilon }} \right)$, the following restriction can be formulated:

$$0\mathop < \limits^{!} h_{1}^{2} + 7.251 \cdot k \cdot h_{1} - 6.564 \cdot k \cdot h_{1} \; \cdot \;\exp \left( { - \frac{\sqrt 3 }{2}\bar{\varepsilon }} \right)$$

Solving for $\bar{\varepsilon }$, by applying h ₁  > 0 and k ≤ 1 yields a criterion for the maximal deformation as a function of the initial height h ₁ .

$$\bar{\varepsilon }\mathop < \limits^{!} \frac{2}{\sqrt 3 }\; \cdot \;\ln \left( {\frac{{h_{1} + 7.251}}{6.564}} \right)$$

Extensive analyses of industrial data substantiate, that all process parameters of heavy plate rolling (h ₀  > 3 mm) satisfy the above conditions in any case.