Effect of Uneven Distribution of Material Property on Buckling Behavior of Strip during Hot Finishing Rolling

Abstract

Unlike traditional austenitic rolling, there is phase transformation during hot finishing rolling for thin gauge and high strength steel or non-oriented electrical steel. The transverse differences of temperature and asynchronous phase transformation result in uneven distribution of material property of rolled strip, further change the buckling behavior of strip during hot finishing rolling. By replacing the elastic modulus constant in the traditional buckling model with the distribution function of tangent modulus obtained by multiphase compression experiments and multifield coupling simulation, the effect of uneven distribution of material property on the critical buckling stress and buckling wave length are analyzed. The results show that for the global longitudinal wave, the critical buckling stress at the exit of stand is greater than that at the entry. But the opposite is true for the local longitudinal wave. Under the effect of uneven distribution of material property, the critical buckling stress and buckling wavelength of global and local center waves change little. The critical buckling stress of global edge wave is almost unchanged, but the buckling wavelength decreases slightly. The critical buckling stress and buckling wavelength of local edge wave are reduced obviously, and the buckling wavelength is decreased by about 10%. It means that the existence of soft ferrite at the strip edge easily makes the wave mode develop to “fragmented edge wave”, which is consistent with the actual phenomenon.

1. Introduction

In the manufacturing process, the thermal gradients or uneven deformation of workpiece will lead to the generation of residual stress, which will cause unwanted distortion and deteriorate dimensional accuracy.^1,2,3,4) As an important form of material processing, rolling is also troubled by residual stress.^5,6,7) For the hot rolling of strip, the flatness problem will come out when the internal residual stress exceeds the critical value.^8,9,10,11) The determination of the critical stress depends on the buckling model.^{12,13,14,15,16,17)} Due to the high-temperature phase transformation characteristics of some steel grades, like thin gauge and high strength steel or non-oriented electrical steel,¹⁸⁾ the multiphase rolling caused by uneven temperature distribution along the strip width is inevitable during hot finishing rolling as shown in Fig. 1. In this case, the buckling behavior is not only affected by geometric size and external load, but also by the uneven distribution of material property as a result of temperature difference and asynchronous phase transformation along strip width. The existing online shape control system does not consider this point, so there are often abnormal edge waves during the multiphase rolling.

Fig. 1.

Multiphase rolling process and the observed abnormal edge waves. (Online version in color.)

In the recent years, the effect of uneven distribution of material property on bucking behavior becomes one of the most actively research, especially for FGM (Functionally Graded Materials) plate composed of different strength materials along the thickness direction.^19,20,21,22) The influence of material inhomogeneity on buckling has been proved that it can not be ignored. However, it has not been paid much attention in the theoretical research field of strip rolling process, which does occur in the actual production and bring abnormal wave shape between mill stands.^23,24) At present the commonly used buckling model in actual strip production is a simple but effective instability criterion proposed by Shohet²⁵⁾ through rolling experiments of stainless steel and aluminum, which was closely related to the geometric size. In order to make the model more accurate, Yang²⁶⁾ abstracted the functional forms of internal stress and wave from the actual wave shape and established the buckling model based on the sheet stability theory to calculate the critical buckling stress under arbitrary tension stress. Fischer²⁷⁾ assumed the cosine distribution and polynomial distribution to describe the residual stress field and applied the Ritz approach to determine the critical load, wavelength and shape parameters. Kpogan²⁸⁾ presented a simplified numerical method assuming harmonic buckling mode along the sheet length which can be used to predict efficiently the response of long thin plates under effect of residual stress induced by production process such as rolling. Tenenbaum²⁹⁾ proposed a new analytical solution used for finding the buckling loads of rectangular plates with rotationally restrained edges, which fits all the combinations of possible boundary conditions, of the deflection, slope, shear force and bending moment along the edges of the plate. Yang³⁰⁾ considered the additional stress caused by uneven temperature and tension, and the effect of geometrical nonlinearities due to pressure load on the thermal buckling and dynamic characteristics of plates were investigated. It can be seen from the above researches that the effects of geometrical size, external boundary conditions and internal stress on buckling behavior are the focus of the previous researches, and the effect of transverse material inhomogeneity due to temperature difference and asynchronous phase transformation is seldom considered in the rolling process, which exactly makes the buckling behavior in the dual-phase region different from that in the single-phase region. Our previous research has proved that the transverse differences of temperature and asynchronous phase transformation during the hot finishing rolling resulted in uneven distribution of material property^31,32,33,34) and further affected the post-buckling behavior of strip, such as wave height.³⁵⁾ However, the critical buckling stress and buckling wavelength have not been determined, which is more important for online shape control in practical production to draw up the flatness dead zone and explain the cause of wave shape. This requires us to adopt the small deformation theory to establish the buckling model, which is different from the large deformation theory used in the previous study of post buckling.

In order to study the buckling behavior of strip in the dual-phase region, the elastic modulus constant in the traditional buckling model is replaced with the distribution function of tangent modulus obtained by hot compression experiments and multi-field coupling simulation. The effect of uneven distribution of material property caused by temperature difference and asynchronous phase transformation on the critical buckling stress and buckling wave length are analyzed.

2. Transverse Distribution of Tangent Modulus

As a typical steel with dual-phase characteristics, the non-oriented electrical steel is selected as research object. The chemical composition of the test specimens is listed in Table 1. To characterize the non-uniform distribution of deformation resistance along the width of strip due to transverse temperature difference and asynchronous phase transformation, the transverse distribution functions of tangent modulus for dual-phase strip are obtained by hot compression experiments and multi-field coupling simulation. As we have done in our previous research (see reference 35), hot compression experiments were carried out in austenite region (1000–1100°C) and ferrite region (800–900°C), respectively. The true stress-true strain curves during the plastic hardening stage at different temperatures in different phase regions are obtained, as shown in Fig. 2. The tangent modulus is corresponding to the slope of each fitting line. The relationships of tangent modulus in austenite region and ferrite region varying with temperature can be obtained, as shown in Fig. 3. Based on the the transverse distribution of temperature and phase structure at stand F4 (see Fig. 1) in dual-phase region calculated by FEM model,³¹⁾ the transverse distributions of tangent modulus with and without the transverse differences of temperature and phase transformation are calculated, as shown in Eqs. (1) and (2).

Table 1. Chemical composition of non-oriented electrical steel tested (mass/%).

C	Si	Mn	P	S
0.0031	0.77	0.25	0.02	0.005

Fig. 2.

True stress-true strain curves during the plastic hardening stage at different temperatures in different phase regions.³⁵⁾ (Online version in color.)

Fig. 3.

Curve of tangent modulus versus temperature.³⁵⁾

(1) Ignoring the uneven distribution of temperature and phase structure:   

$$E\left(x\right)=898.124$$(1)

(2) Considering the uneven distribution of temperature and phase structure:   

$$\begin{array}{l}E\left(x\right)=905.1-260.8\times {\left(\frac{x}{b}\right)}^2+1\mathrm{\hspace{0.17em} }366\times {\left(\frac{x}{b}\right)}^4\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-1\mathrm{\hspace{0.17em} }539\times {\left(\frac{x}{b}\right)}^6-108\times {\left(\frac{x}{b}\right)}^8\end{array}$$(2)

where E(x) is the tangent modulus at the position x along the strip width, MPa.

3. Buckling Model Considering Uneven Distribution of Material Property

Based on the sheet stability theory, the buckling model is established. The transverse distribution function of tangent modulus is used to replace the elastic modulus constant in traditional buckling theory. The typical functions of stress and wave for global and local longitudinal buckling are determined. The critical buckling stress and buckling wavelength are calculated by the energy principle under the condition of inhomogeneous material.

3.1. Basic Equations

The static equilibrium diagram of sheet under micro-bending condition is shown in Fig. 4.

Fig. 4.

Static equilibrium diagram of sheet under Micro-bending condition.

3.1.1. Physical Equation

According to the assumptions of sheet stability theory, the normal stress σ_z is ignored. The relationship between stress and strain can be obtained from Hooke’s law.   

$$\{\begin{array}{l}{\sigma }_x=\frac{E\left(x\right)}{1-{\mu }^2}\left({\epsilon }_x+\mu {\epsilon }_y\right)\\ {\sigma }_y=\frac{E\left(x\right)}{1-{\mu }^2}\left({\epsilon }_y+\mu {\epsilon }_x\right)\\ {\tau }_{xy}=\frac{E\left(x\right)}{2\left(1+\mu \right)}{\gamma }_{xy}\end{array}$$(3)

where E(x) is the transverse distribution function of tangent modulus which is used to replace the elastic modulus constant in traditional buckling theory.

3.1.2. Geometric Equation

According to the assumptions of sheet stability theory, the strain can be approximated to the derivative of the corresponding displacement.   

$$\{\begin{array}{l}{\epsilon }_x=\frac{\partial u}{\partial x}=-z\frac{{\partial }^{2}w}{\partial x^2}\\ {\epsilon }_y=\frac{\partial v}{\partial y}=-z\frac{{\partial }^{2}w}{\partial y^2}\\ {\gamma }_{xy}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}=-2z\frac{{\partial }^{2}w}{\partial x\partial y}\end{array}$$(4)

where u, v, w are the displacements in x, y, z direction, respectively.

The relationship between stress and displacement can be obtained from Eqs. (3) and (4).   

$$\{\begin{array}{l}{\sigma }_x=-\frac{E\left(x\right)z}{1-{\mu }^2}\left(\frac{{\partial }^{2}w}{\partial x^2}+\mu \frac{{\partial }^{2}w}{\partial y^2}\right)\\ {\sigma }_y=-\frac{E\left(x\right)z}{1-{\mu }^2}\left(\frac{{\partial }^{2}w}{\partial y^2}+\mu \frac{{\partial }^{2}w}{\partial x^2}\right)\\ {\tau }_{xy}=-\frac{E\left(x\right)z}{1+\mu }\frac{{\partial }^{2}w}{\partial x\partial y}\end{array}$$(5)

In addition, the relationship between bending moment and stress is as follows:   

$$M_{ij}={\int }_{-h/2}^{h/2}z{\sigma }_{ij}\text{d}z$$(6)

where h is sheet thickness.

The relationship between bending moment and displacement can be obtained from Eqs. (5) and (6).   

$$\{\begin{array}{l}{M}_x=-\frac{E\left(x\right)h^3}{12(1-{\mu }^2)}\left(\frac{{\partial }^{2}w}{\partial x^2}+\mu \frac{{\partial }^{2}w}{\partial y^2}\right)\\ M_y=-\frac{E\left(x\right)h^3}{12(1-{\mu }^2)}\left(\frac{{\partial }^{2}w}{\partial y^2}+\mu \frac{{\partial }^{2}w}{\partial x^2}\right)\\ M_{xy}=-\frac{E\left(x\right)h^3}{12(1-{\mu }^2)}\left(1-\mu \right)\frac{{\partial }^{2}w}{\partial x\partial y}\end{array}$$(7)

3.1.3. Equilibrium Equation

The equilibrium equation under micro-bending condition is as follows:   

$$\begin{array}{l}\frac{{\partial }^2{M}_x}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}}{\partial x\partial y} +\frac{{\partial }^2{M}_y}{\partial y^2}+N_x\frac{{\partial }^{2}w}{\partial x^2}\\ +2N_{xy}\frac{{\partial }^{2}w}{\partial x\partial y}+N_y\frac{{\partial }^{2}w}{\partial y^2}=0\end{array}$$(8)

where mid-plane forces N_ij=σ_ijh

3.1.4. Buckling Differential Equation

By substituting Eq. (7) for Eq. (8), the buckling differential equation of sheet can be obtained.   

$$\begin{array}{l}\frac{E\left(x\right)h^3}{12\left(1-{\mu }^2\right)}\left(\frac{{\partial }^{4}w}{\partial x^4}+2\frac{{\partial }^{4}w}{\partial x^2\partial y^2} +\frac{{\partial }^{4}w}{\partial y^4}\right)\\ =N_x\frac{{\partial }^{2}w}{\partial x^2}+2N_{xy}\frac{{\partial }^{2}w}{\partial x\partial y}+N_y\frac{{\partial }^{2}w}{\partial y^2}\end{array}$$(9)

3.2. Energy Principle

The energy principle is used to solve the buckling differential equation. According to the small deformation assumption, only the bending stress can produce strain energy under the micro-bending condition. Therefore, the strain energy of sheet is expressed as:   

$$U=\frac{1}{2}\underset{\mathrm{\Omega }}{\iint }\left({\sigma }_x{\epsilon }_x+{\sigma }_y{\epsilon }_y+{\tau }_{xy}{\gamma }_{xy}\right)\mathrm{d}x\mathrm{d}y\mathrm{d}z$$(10)

The physical equation and geometric equation are substituted into Eq. (10) and integrated in z direction. The strain energy can be derived as follows:   

$$\begin{array}{l}U=\frac{h^3}{24(1-{\mu }^2)}\underset{\mathrm{\Omega }}{\iint }E\left(x\right)[{\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}^2+{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}^2\\ \hspace{1em}+2\mu \frac{{\partial }^{2}w}{\partial x^2}\frac{{\partial }^{2}w}{\partial y^2}+2\left(1-\mu \right){\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2]\mathrm{d}x\mathrm{d}y\end{array}$$(11)

The external potential energy can be considered as the sum of the work done by the mid-plane forces N_x, N_y and N_xy in the buckling process.   

$$T=\frac{1}{2}\underset{\mathrm{\Omega }}{\iint }\left[N_x{\left(\frac{\partial w}{\partial x}\right)}^2+N_y{\left(\frac{\partial w}{\partial y}\right)}^2+2N_{xy}\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\right]\mathrm{d}x\mathrm{d}y$$(12)

According to the assumptions, the transverse force component and the shear force component are neglected, i.e. N_x=N_xy=0. The longitudinal force component is only considered, i.e. N_y=σ_yh. The longitudinal stress σ_y is composed of two parts. One is the average tensile stress σ_f, and the other is the residual stress σ_r. The residual stress σ_r is equal to the product of critical buckling stress σ₀ and distribution function, which is related to uneven plastic deformation, temperature drop and phase transformation in hot rolling.   

$${\sigma }_y\left(x\right)={\sigma }_f+{\sigma }_r\left(x\right)$$(13)

Therefore, the total potential energy of the sheet is as follows:   

$$\begin{array}{l}\mathrm{\Pi }=U+T\\ \hspace{1em}=\frac{h^3}{24(1-{\mu }^2)}\underset{\mathrm{\Omega }}{\iint }E\left(x\right)[{\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}^2+{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}^2\\ \hspace{1em}+2\mu \frac{{\partial }^{2}w}{\partial x^2}\frac{{\partial }^{2}w}{\partial y^2}+2\left(1-\mu \right){\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2]\mathrm{d}x\mathrm{d}y\\ \hspace{1em}+\frac{h}{2}\underset{\mathrm{\Omega }}{\iint }\left[{\sigma }_f+{\sigma }_r\left(x\right)\right]{\left(\frac{\partial w}{\partial y}\right)}^2\mathrm{d}x\mathrm{d}y\end{array}$$(14)

According to the sheet stability theory, the critical buckling state of sheet can be reached when the external potential energy equals the strain energy. According to the energy stationary principle, the buckling critical stress can be obtained by finding the stationary value of total potential energy.

The second-order variation of total potential energy is as follows:   

$$\begin{array}{l}{\delta }^2\left(\mathrm{\Pi }\right)=\frac{h^3}{24(1-{\mu }^2)}\underset{\mathrm{\Omega }}{\iint }E\left(x\right)[{\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}^2+{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}+2\mu \frac{{\partial }^{2}w}{\partial x^2}\frac{{\partial }^{2}w}{\partial y^2}+2\left(1-\mu \right){\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2]\mathrm{d}x\mathrm{d}y\\ \hspace{1em}\hspace{1em}\hspace{1em}+h\underset{\mathrm{\Omega }}{\iint }\left[{\sigma }_f+{\sigma }_r\left(x\right)\right]{\left(\frac{\partial w}{\partial y}\right)}^2\mathrm{d}x\mathrm{d}y\end{array}$$(15)

To make the second-order variation of total potential energy have a stationary value, the following condition needs to be met:   

$$\partial \left[{\sigma }^2\right(\mathrm{\Pi }\left)\right]/\partial r=0$$(16)

where r is the deflection amplitude.

The critical buckling stress σ₀(l) can be obtained from the above equation. Let ∂ σ₀(l)/∂l = 0, the buckling wavelength l_cr is calculated. Then substitute l_cr into σ₀(l), and the critical buckling stress σ₀ is finally obtained.

3.3. Buckling Model of Global Longitudinal Wave

The global longitudinal waves mainly include the global center wave and the global edge wave as shown in Fig. 5, which can be described by high-order polynomial. One wave is selected as the research unit. The wavelength in the rolling direction is l, and the strip width is 2b.

Fig. 5.

Global center wave and global edge wave.

The distribution of residual stress can be written as:   

$${\sigma }_r\left(x\right)={\sigma }_0\left[e_0+e_2{\left(\frac{x}{b}\right)}^2+e_4{\left(\frac{x}{b}\right)}^4+e_6{\left(\frac{x}{b}\right)}^6+e_8{\left(\frac{x}{b}\right)}^8\right]$$(17)

where σ₀ is the critical buckling stress. e_i are the stress coefficients, of which values correspond to different buckling forms.

According to the self-equilibrium of internal stress, the coefficient e₀ can be calculated as follows:   

$$e_0=-\left(\frac{e_2}{3}+\frac{e_4}{5}+\frac{e_6}{7}+\frac{e_8}{9}\right)$$(18)

The transverse distribution function of wave can be expressed as follows:   

$$w_x=r_x\left[r_0+r_2{\left(\frac{x}{b}\right)}^2+r_4{\left(\frac{x}{b}\right)}^4+r_6{\left(\frac{x}{b}\right)}^6+r_8{\left(\frac{x}{b}\right)}^8\right]$$(19)

where r_x is the deflection amplitude along strip width. r_i is the wave coefficient.

The deflection of strip along rolling direction varies periodically, so it can be described by sinusoidal function.   

$$w_y=r_{y}sin\left(\frac{\pi y}{l}\right)$$(20)

where l is buckling wavelength. r_y is the deflection amplitude along rolling direction.

Then the total wave can be represented as we adopted in our previous research (see reference 35):   

$$\begin{array}{l}w=w_x\times w_y\\ =r_w\left[r_0+r_2{\left(\frac{x}{b}\right)}^2+r_4{\left(\frac{x}{b}\right)}^4+r_6{\left(\frac{x}{b}\right)}^6+r_8{\left(\frac{x}{b}\right)}^8\right]\times sin\left(\frac{\pi y}{l}\right)\end{array}$$(21)

The distribution functions of residual stress and total wave are substituted into the expression of total potential energy, and then the corresponding critical buckling stress σ₀ and buckling wavelength l_cr are obtained by making the second-order variation of total potential energy have a stationary value.

3.4. Buckling Model of Local Longitudinal Wave

The local longitudinal waves mainly include the local center wave and the local edge wave as shown in Fig. 6. The critical buckling stress σ₀ and buckling wavelength l_cr are calculated also by stationary principle of potential energy.

Fig. 6.

Local center wave and local edge wave.

(1) For local center wave, the residual stress is expressed by high-order polynomial and the wave function is piecewise as follows:   

$$\begin{array}{l}{\sigma }_r\left(x\right)={\sigma }_0[e_0\left(1-\frac{b_w}{b}\right)+e_2\left({\left(\frac{x}{b_w}\right)}^2-\frac{b_w}{3b}\right)+e_4\left({\left(\frac{x}{b_w}\right)}^4-\frac{b_w}{5b}\right)\\ \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }+e_6\left({\left(\frac{x}{b_w}\right)}^6-\frac{b_w}{7b}\right)+e_8\left({\left(\frac{x}{b_w}\right)}^8-\frac{b_w}{9b}\right)]\end{array}$$(22)

  

$$\{\begin{array}{l}w\left(x,y\right)=r_w\left[r_0+r_2{\left(\frac{x}{b_w}\right)}^2+r_4{\left(\frac{x}{b_w}\right)}^4+r_6{\left(\frac{x}{b_w}\right)}^6+r_8{\left(\frac{x}{b_w}\right)}^8\right]\times sin\left(\frac{\pi y}{l}\right)\\ \hfill \left(-b_w\le x\le b_w\right)\\ \\ w\left(x,y\right)=0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(-b\le x<-b_w,b_w<x\le b\right)\end{array}$$(23)

where b_w is the half width of buckling part in the middle of strip.

(2) For local edge wave, only one side is taken to study due to the symmetry. The residual stress and wave are expressed by power function.   

$${\sigma }_r\left(x\right)={\sigma }_0\left[{\left(\frac{x-b+b_w}{b_w}\right)}^{n_E}-\frac{b_w}{\left(n_E+1\right)b}\right]$$(24)

  

$$\{\begin{array}{l}w\left(x,y\right)=r_w{\left(\frac{x-b+b_w}{b_w}\right)}^{n_w}sin\left(\frac{\pi y}{l}\right)\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(b-b_w\le x\le b\right)\\ w\left(x,y\right)=0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(0\le x<b-b_w\right)\end{array}$$(25)

where b_w is the side width of buckling part.

4. Results and Discussion

Buckling occurs at the entry and exit of stand will change the back tension and front tension of the rolling strip respectively, which further affects the stability of rolling. Compared to the strip at the exit of stand, the strip thickness at the entry of stand is greater and the tension is smaller. As we know, the greater the thickness, the greater the critical buckling stress. Meanwhile, the smaller the tension, the smaller the critical buckling stress. It can be seen that the effects of thickness and tension on the critical buckling stress are opposite. Therefore, the combined effect of thickness and tension on the critical buckling stress are needed to be compared whether at the entry or the exit of stand. At the same time, the strip at the entry and exit of F4 stand are both located in the dual-phase region due to the time in roll gap is very short. The uneven distribution of material properties caused by the transverse differences of temperature and phase transformation not only affects the critical buckling stress, but also changes the buckling wavelength.

Based on the above analysis, the effects of temperature difference and asynchronous phase transformation should be considered, and the strips at the entry and exit of dual-phase stand F4 are selected for comparative analysis. The buckling critical stress and buckling wavelength of global longitudinal wave (global center wave and global edge wave) and local longitudinal wave (local center wave and local edge wave) are calculated and analyzed. The geometric size and mechanical parameters at dual-phase stand F4 are shown in Table 2.

Table 2. Geometric size and mechanical parameters at dual-phase stand F4.

Half strip width/mm	Poisson ratio	Entry thickness/mm	Exit thickness/mm	Front tension/MPa	Back tension/MPa
625	0.3	10.66	6.6	12	8

4.1. Effect of Uneven Distribution of Material Property on Buckling of Global Center Wave

According to the typical form of global center wave, the corresponding wave coefficients²⁶⁾ are shown in Table 3.

Table 3. Wave coefficients of global center wave.

r0	r2	r4	r6	r8
1	−2.467	2.023	−0.645	0.0892

According to the self-equilibrium of internal stress, the stress coefficients of global center wave²⁶⁾ are shown in Table 4.

Table 4. Stress coefficients of global center wave.

e2	e4	e6	e8
2.468	−2.031	0.658	−0.095

The transverse distributions of tangent modulus E(x) under two conditions, whether considering the effect of uneven distribution of material property or not, are substituted into the buckling model of global center wave to calculate the critical buckling stress and buckling wavelength. It can be seen from Tables 5 and 6 that the critical buckling stress of the global center wave at the exit of F4 stand is obviously greater than that at the entry of the stand. It indicates that for the global center wave, although the critical buckling stress theoretically decreases with the reduction of strip thickness from the entry to the exit of the stand, it actually increases due to the increase of tension. That is to say, the effect of tension on the critical buckling stress of global center wave is more significant than that of strip thickness. At the same time, the temperature drop and phase transformation at the strip edge have small effect on the critical buckling stress and buckling wavelength of global center wave.

Table 5. Effect of uneven distribution of material property on the critical buckling stress of global center wave.

Calculation cases	Critical buckling stress at the entry/MPa	Critical buckling stress at the exit/MPa
Considering the effects	25.03	36.12
No considering the effects	25	36.11

Table 6. Effect of uneven distribution of material property on the buckling wavelength of global center wave.

Calculation cases	Buckling wavelength at the entry/mm	Buckling wavelength at the exit/mm
Considering the effects	822.59	822.59
No considering the effects	841.37	841.37

4.2. Effect of Uneven Distribution of Material Property on Buckling of Global Edge Wave

According to the typical form of global edge wave, the corresponding wave coefficients²⁶⁾ are shown in Table 7.

Table 7. Wave coefficients of global edge wave.

r0	r2	r4	r6	r8
0	1	0	0	0

According to the self-equilibrium of internal stress, the stress coefficients of global edge wave²⁶⁾ are shown in Table 8.

Table 8. Stress coefficients of global edge wave.

e2	e 4	e6	e 8
−1	0	0	0

The transverse distributions of tangent modulus E(x) under two conditions, whether considering the effect of uneven distribution of material property or not, are substituted into the buckling model of global edge wave to calculate the critical buckling stress and buckling wavelength. It can be seen from Tables 9 and 10 that the critical buckling stress of the global edge wave at the exit of F4 stand is obviously greater than that at the entry of stand, which is consistent with the law of the global center wave. That is to say, the effect of tension on the critical buckling stress of global edge wave is more significant than that of strip thickness. At the same time, the temperature drop and phase transformation at the strip edge have small effect on the critical buckling stress of global edge wave, and the buckling wavelength is reduced by about 4%.

Table 9. Effect of uneven distribution of material property on the critical buckling stress of global edge wave.

Calculation cases	Critical buckling stress at the entry/MPa	Critical buckling stress at the exit/MPa
Considering the effects	22.02	31.89
No considering the effects	21.9	31.85

Table 10. Effect of uneven distribution of material property on the buckling wavelength of global edge wave.

Calculation cases	Buckling wavelength at the entry/mm	Buckling wavelength at the exit/mm
Considering the effects	928.48	928.48
No considering the effects	893.35	893.35

4.3. Effect of Uneven Distribution of Material Property on Buckling of Local Center Wave

According to the typical form of local center wave, the half width of buckling part in the middle of strip b_w is set as 100 mm. The transverse distributions of tangent modulus E(x) under two conditions, whether considering the effect of uneven distribution of material property or not, are substituted into the buckling model of local center wave to calculate the critical buckling stress and buckling wavelength. It can be seen from Tables 11 and 12 that the critical buckling stress of the local center wave at the exit of F4 stand is obviously less than that at the entry of stand, which is contrary to the law of global longitudinal waves. That is to say, the effect of strip thickness on the critical buckling stress of local center wave is more significant than that of tension. At the same time, the temperature drop and phase transformation at the strip edge have small effect on the critical buckling stress and buckling wavelength of local center wave.

Table 11. Effect of uneven distribution of material property on the critical buckling stress of local center wave.

Calculation cases	Critical buckling stress at the entry/MPa	Critical buckling stress at the exit/MPa
Considering the effects	32.9	24.46
No considering the effects	33.04	24.52

Table 12. Effect of uneven distribution of material property on the buckling wavelength of local center wave.

Calculation cases	Buckling wavelength at the entry/mm	Buckling wavelength at the exit/mm
Considering the effects	131.61	131.61
No considering the effects	131.68	131.68

4.4. Effect of Uneven Distribution of Material Property on Buckling of Local Edge Wave

According to the typical form of local edge wave,²⁶⁾ the width of wave on one side b_w is set as 100 mm, the stress exponent n_E is set as 2.11, and the wave exponent n_w is set as 1.73.²⁶⁾ The transverse distributions of tangent modulus E(x) under two conditions, whether considering the effect of uneven distribution of material property or not, are substituted into the buckling model of local edge wave to calculate the critical buckling stress and buckling wavelength. It can be seen from Tables 13 and 14 that the critical buckling stress of the local edge wave at the exit of F4 stand is obviously less than that at the entry of stand, which is consistent with the law of the local center wave. It indicates that for the local bilateral wave, although the critical buckling stress theoretically increases with the increase of tension from the entry to the exit of the stand, but actually decreases due to the decrease of strip thickness. That is to say, the effect of strip thickness on the critical buckling stress of local edge wave is more significant than that of tension. At the same time, the temperature drop and phase transformation at the strip edge reduce the critical buckling stress and buckling wavelength of local edge wave, and the buckling wavelength is decreased by about 10%. So the existence of soft ferrite at the strip edge makes the wave mode develop to “fragmented edge wave”.

Table 13. Effect of uneven distribution of material property on the critical buckling stress of local edge wave.

Calculation cases	Critical buckling stress at the entry/MPa	Critical buckling stress at the exit/MPa
Considering the effects	33.74	27.17
No considering the effects	26.2	24.28

Table 14. Effect of uneven distribution of material property on the buckling wavelength of local edge wave.

Calculation cases	Buckling wavelength at the entry/mm	Buckling wavelength at the exit/mm
Considering the effects	158.42	158.42
No considering the effects	143.77	143.77

5. Model Application

The established buckling model considering the effect of uneven distribution of material property caused by temperature difference and asynchronous phase transformation has been applied to the online shape preset model. The chemical composition, temperature and reduction obtained from FSU (Finishing Set Up) module are used to determine the start and end temperatures of phase transition. The phase transition fraction is calculated by the phase transition kinetic equation. Subsequently, the buckling model established in this paper is used to precisely determine the flatness dead band as shown in Fig. 7. After model application, there is almost no abnormal wave shape between stands during hot finishing rolling of non-oriented electrical steel and the flatness hit rate increases from 92.3% to 97.5%. With the improvement of strip flatness, the rolling stability and the steel production have increased significantly.

Fig. 7.

Application of established buckling model in online shape preset model. (Online version in color.)

6. Conclusion

Based on the modified sheet stability theory, considering the transverse difference of temperature and phase transformation in hot finishing rolling of strip, the transverse distribution function of tangent modulus obtained by experimental regression is used to replace the elastic modulus constant in traditional buckling theory. The effects of thickness, tension, temperature and phase transformation on the buckling behavior of strip at the entry and exit of dual-phase stand F4 are compared and analyzed, which improve the traditional buckling theory. The established buckling model is finally applied in online shape preset model and reduce the occurrence of wave shape between stands.

(1) For global longitudinal waves, the critical buckling stress at the exit of dual-phase stand F4 is obviously greater than that at the entry of the stand. That is to say, the effect of tension on the critical buckling stress of global longitudinal wave is more significant than that of strip thickness.

(2) For local longitudinal waves, the critical buckling stress at the exit of dual-phase stand F4 is obviously less than that at the entry of stand. That is to say, the effect of strip thickness on the critical buckling stress of local center wave is more significant than that of tension, which is contrary to the law of global longitudinal waves.

(3) For center waves, the temperature drop and phase transformation at the strip edge have small effect on the critical buckling stress and buckling wavelength.

(4) For edge waves, the temperature drop and phase transformation at the strip edge have small effect on the critical buckling stress of global edge wave, and the buckling wavelength of global edge wave is reduced by about 4%. In addition, they reduce the critical buckling stress and buckling wavelength of local edge wave, and the buckling wavelength is decreased by about 10%. So the existence of soft ferrite at the strip edge makes the wave mode develop to “fragmented edge wave”, which is consistent with the actual phenomenon.

Funding Information

The work is financially supported by the National Natural Science Foundation of China [grant number 51674028 and 52004029] and the Fundamental Research Funds for the Central Universities [FRF-TT-20-06 and FRF-AT-20-06]. The authors gratefully acknowledge these supports.

Author Contributions

Seven authors have made irreplaceable work on this paper. Chao Liu: conceptualization, methodology, investigation, writing – original draft, and funding acquisition. Anrui He: conceptualization, writing – review and editing, and funding acquisition. Fengwei Jing: industrial validation, and funding acquisition. Wenquan Sun: industrial validation. Jian Shao: industrial validation. Chihuan Yao: formal analysis, and visualization. Hairui Wu: data curation. All authors have discussed and agreed to the published version of the manuscript.

Conflict of Interest

Authors state no conflict of interest.

Data Availability Statement

The raw data related to this manuscript will be made available on request.

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