Effects of Temperature and Phase Transformation on Post-buckling Behavior of Non-oriented Electric Steel during Hot Finishing Rolling

Abstract

The traditional buckling model is based on the assumption of homogeneous material. However, for non-oriented electrical steel with high-temperature phase transformation, the transverse differences of temperature and phase transformation during the hot finishing rolling result in uneven distribution of material properties in the dual-phase region. In order to study the effect of inhomogeneous material on the post-buckling behavior of strip, the relationships between tangent modulus and temperature in the austenite region and ferrite region are firstly obtained by hot compression experiments. Secondly, the transverse distribution function of tangent modulus is calculated according to the distributions of temperature and phase structure in the dual-phase region. Finally, the large deflection theory of thin plate is modified, and the elastic modulus constant is replaced by the distribution function of tangent modulus. The post-buckling model considering inhomogeneous material is established to analyze the effect of temperature and phase transformation on the wave height. The results show that strip thickness and tension have great effect on the post-buckling deformation of global longitudinal wave, but little effect on local longitudinal wave. The temperature drop and phase transformation at the strip edge have no significant effect on the wave heights of global and local center waves, but they reduce the wave heights of global and local edge waves by 6% and 20%, respectively.

1. Introduction

The essence of the flatness problem occurring during the production of strip is the buckling deformation.^1,2,3,4) When the internal residual stress exceeds the critical buckling stress, the observable wave will come out. The quantification of the buckling deformation depends on the post-buckling model.^5,6,7,8) Unlike traditional austenitic rolling, there is high-temperature phase transformation during hot finishing rolling of non-oriented electrical steel.^{9,10,11,12,13)} The post-buckling behavior is not only affected by internal stress, but also by uneven temperature drop and asynchronous phase transformation along strip width, which result in uneven distribution of material properties, thus affecting the buckling wave height. These increase the difficulty of tension control between stands.

In the study of post-buckling, Yang¹⁴⁾ abstracted the functional forms of internal stress and wave from the actual wave shape and obtained the relationship of internal stress and wave height based on the large deflection theory of thin plate. Bian et al.¹⁵⁾ constructed a general wave function with spline curve to replace infinite degree of freedom with finite degree of freedom, and solved the post-buckling deformation of strip under longitudinal residual stress by using the large deflection theory of thin plate and the energy principle. Qing¹⁶⁾ used commercial finite element software to simulate the post-buckling behaviors of global and local longitudinal wave of rectangular thin plate by the arc-length incremental loading method. Su¹⁷⁾ established the post-buckling deformation model based on the large deflection theory of thin plate and obtained the wave heights under the conditions of small tension and large tension. It can be seen from the above that the existing researches mainly focus on the effects of internal stress on the post-buckling behavior without considering the effect of inhomogeneous material. But for hot finishing rolling of non-oriented electrical steel, the uneven distribution of material properties along strip width makes its post-buckling behavior different from that of plain carbon steel.

In order to study the effect of inhomogeneous material on the post-buckling behavior of strip in the dual-phase region, the true stress-true strain curves at different temperatures in austenite region and ferrite region are firstly obtained by hot compression experiments. The relationship between tangent modulus and temperature in each single-phase region are obtained by regression. Secondly, the dual-phase stand F4 is selected and the transverse distribution function of tangent modulus is calculated according to the distributions of temperature and phase structure. Finally, the large deflection theory of thin plate is modified, and the elastic modulus constant is replaced by the distribution function of tangent modulus. The post-buckling model considering inhomogeneous material is established to analyze the effect of temperature and phase transformation on the buckling wave height.

2. Transverse Distribution of Tangent Modulus

The transverse differences of temperature and phase structure existing in hot finishing rolling of non-oriented electrical steel result in the non-uniform distribution of material properties along the strip width direction. In the well-known Shohet instability criterion theory,¹⁸⁾ the tangent modulus in the plastic stage, not the elastic modulus in the elastic stage, is properly selected due to plastic deformation is easy to occur under high temperature and internal stress. In this paper, the tangent modulus is also used to characterize the resistance to instability of material. The transverse distribution of tangent modulus under the effect of temperature drop and phase transformation is quantified by experimental regression method.

According to the critical temperature range of phase transformation of non-oriented electrical steel,¹⁹⁾ hot compression experiments were carried out at three temperatures in austenite region and ferrite region, respectively. The true stress-true strain curves during the plastic hardening stage at different temperatures in different phase regions are obtained. The corresponding expressions of stress-strain relationship are obtained by linear fitting method as shown in Fig. 1. The slope of each fitting line is the desired tangent modulus as shown in Table 1.

Fig. 1.

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

Table 1. Tangent modulus at different temperatures.

Temperature/°C	800	850	900	1000	1050	1100
Tangent modulus/MPa	564.29	459.91	365.33	826.42	678.91	575.96

It can be seen from the curve of tangent modulus versus temperature as shown in Fig. 2 that tangent modulus varies almost linearly with temperature in ferrite region, and also in austenite region. Therefore, the expressions of tangent modulus in austenite region and ferrite region varying with temperature can be obtained as follows by linear fitting method.   

$$\{\begin{array}{l}E\left(T\right)=-2.51T+3\mathrm{\hspace{0.17em} }323.59\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{In}\mathrm{\hspace{0.17em} }\text{austenite}\mathrm{\hspace{0.17em} }\text{region}\\ E\left(T\right)=-1.99T+2\mathrm{\hspace{0.17em} }154.34\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{In}\mathrm{\hspace{0.17em} }\text{ferrite}\mathrm{\hspace{0.17em} }\text{region}\end{array}$$(1)

where E(T) is the tangent modulus varying with temperature, MPa.

Fig. 2.

Curve of tangent modulus versus temperature.

The transverse distribution of temperature and phase structure at stand F4 in dual-phase region are taken from previous work.²⁰⁾ The transverse distributions of tangent modulus with and without the transverse differences of temperature and phase transformation are calculated by Eq. (1) and the mixing rule^21,22) as shown in Fig. 3.

Fig. 3.

Transverse distribution of tangent modulus. (Online version in color.)

The distribution functions of tangent modulus under two conditions are obtained as follows by polynomial fitting. In the formula, the unit of tangent modulus is MPa.

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

$$E\left(x\right)=898.124$$(2)

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

$$\begin{array}{c}E\left(x\right)=905.1-260.8\times {\left({\displaystyle \frac{x}{b}}\right)}^2+1\mathrm{\hspace{0.17em} }366\times {\left({\displaystyle \frac{x}{b}}\right)}^4\\ -1\mathrm{\hspace{0.17em} }539\times {\left({\displaystyle \frac{x}{b}}\right)}^6-108\times {\left({\displaystyle \frac{x}{b}}\right)}^8\end{array}$$(3)

3. Post-buckling Model Considering Inhomogeneous Material

If the internal stress of strip is greater than the critical buckling stress, the post-buckling stage will begin. The buckling deformation of strip can be calculated based on the large deflection theory of thin plate.^23,24,25)

3.1. Basic Equations

3.1.1. Physical Equation

According to Hooke’s law, the relationship between mid-plane stress and mid-plane strain is as follow:   

$$\{\begin{array}{l}{\sigma }_x={\displaystyle \frac{E\left(x\right)}{1-{\mu }^2}}\left({\epsilon }_x+\mu {\epsilon }_y\right)\\ {\sigma }_y={\displaystyle \frac{E\left(x\right)}{1-{\mu }^2}}\left({\epsilon }_y+\mu {\epsilon }_x\right)\\ {\tau }_{xy}={\displaystyle \frac{E\left(x\right)}{2\left(1+\mu \right)}}{\gamma }_{xy}\end{array}$$(4)

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

3.1.2. Equilibrium Differential Equation

  

$$\{\begin{array}{l}{\displaystyle \frac{\partial N_x}{\partial x}}+{\displaystyle \frac{\partial N_{xy}}{\partial y}}=0\\ {\displaystyle \frac{\partial N_y}{\partial x}}+{\displaystyle \frac{\partial N_{xy}}{\partial y}}=0\\ {\displaystyle \frac{E\left(x\right)h^3}{12(1-{\mu }^2)}}\left({\displaystyle \frac{{\partial }^{4}w}{\partial x^4}}+2{\displaystyle \frac{{\partial }^{4}w}{\partial x^2\partial y^2}} +{\displaystyle \frac{{\partial }^{4}w}{\partial y^4}}\right)=N_x{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+2N_{xy}{\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}+N_y{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\end{array}$$(5)

where the mid-plane forces N_x=hσ_x, Ny=hσ_y, Nxy=hτ_xy.

3.1.3. Geometric Equation

The mid-plane strain can be derived by the mid-plane displacement:   

$$\{\begin{array}{l}{\epsilon }_x={\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial w}{\partial x}}\right)}^2\\ {\epsilon }_y={\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial w}{\partial y}}\right)}^2\\ {\gamma }_{xy}={\displaystyle \frac{\partial v}{\partial x}}+{\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial w}{\partial x}}{\displaystyle \frac{\partial w}{\partial y}}\end{array}$$(6)

3.1.4. Deformation Compatibility Equation

The deformation compatibility equation can be obtained by sorting Eq. (6):   

$$\frac{{\partial }^2{\epsilon }_x}{\partial y^2}+\frac{{\partial }^2{\epsilon }_y}{\partial x^2}-\frac{{\partial }^2{\gamma }_{xy}}{\partial x\partial y}={\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2-\frac{{\partial }^{2}w}{\partial x^2}\frac{{\partial }^{2}w}{\partial y^2}$$(7)

3.2. Post-buckling Model of Global Longitudinal Wave

The distribution function of residual strain is expressed by high-order polynomial:   

$${\epsilon }_p\left(x\right)=\mathrm{\Delta }{\epsilon }_p\sum _{k=0}^8{e}_k\left[{\left(\frac{x}{b}\right)}^k-\frac{1}{\left(k+1\right)}\right]$$(8)

where Δε_p is the critical buckling strain which is calculated by the pre-buckling model, e_k are strain coefficients.

The distribution function that characterizes the wave is:   

$$w\left(x,y\right)=r_w{\sum _{k=0}^8{r}_k\left(\frac{x}{b}\right)}^{k}sin\left(\frac{\pi y}{l_w}\right)$$(9)

where r_w is the wave height, r_k is the wave coefficients, l_w is the wavelength which is calculated by the pre-buckling model.

The bending deformation energy U_b and mid-plane strain energy U_m are as follows:   

$$\begin{array}{l}{U}_b={\displaystyle \frac{1}{2}}{\int }_0^{l_w}{\int }_{-b}^b{\displaystyle \frac{E\left(x\right)h^3}{12(1-{\mu }^2)}}[{\left({\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right)}^2+{\left({\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\right)}^2\\ \hspace{1em}+2\mu {\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}+2\left(1-\mu \right){\left({\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}\right)}^2]\text{d}x\text{d}y\end{array}$$(10)

  

$$U_m=\frac{h}{2\left(1-{\mu }^2\right)}{\int }_0^{l_w}{\int }_{-b}^{b}E\left(x\right)\left({\epsilon }_x^2+{\epsilon }_y^2+2\mu {\epsilon }_x{\epsilon }_y+\frac{1-\mu }{2}{\gamma }_{xy}^2\right) \text{d}x\text{d}y$$(11)

Because the mid-plane is free in the width direction, so the transverse stress σ_x and the shear stress τ_xy are ignored. Therefore, the strains are derived as follows:   

$${\epsilon }_x=-\mu {\epsilon }_y,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{\gamma }_{xy}=0$$(12)

The mid-plane strain energy can be obtained by taking Eq. (12) into Eq. (11).   

$$U_m=\frac{h}{2}{\int }_0^{l_w}{\int }_{-b}^{b}E\left(x\right){\epsilon }_y^2\text{d}x\text{d}y$$(13)

Due to the tension between stands is small, the post-buckling behavior under small tension is solved only. The displacement boundary conditions are as follows:   

$${\left(v\right)}_{y=0}=0,\begin{array}{cc}& \end{array} {\left(v\right)}_{y=l_w}={\epsilon }_{0}y$$(14)

where ε₀ is the strain at the boundary.

Assuming that the compressive strain is positive, the total longitudinal strain can be expressed as follow:   

$${\epsilon }_y(x)=-{\epsilon }_p\left(x\right)-{\epsilon }_f-\frac{\partial v}{\partial y}-\frac{1}{2}{\left(\frac{\partial w}{\partial y}\right)}^2$$(15)

where ε_f is the tensile strain.

Equation (16) is derived by differentiating for Eq. (15).   

$$\frac{{\partial }^{2}v}{\partial y^2}+\frac{\partial w}{\partial y}\frac{{\partial }^{2}w}{\partial y^2}=0$$(16)

Equation (9) is substituted into Eq. (16), and integrated over y twice combining with Eq. (14). The displacement in y direction is as follow:   

$$v=-\frac{\pi r_w{\hspace{0.17em}}^2}{8l_w}{\left[{\sum _{k=0}^8{r}_k\left(\frac{x}{b}\right)}^k\right]}^{2}sin\left(\frac{2\pi y}{l_w}\right)+{\epsilon }_{0}y$$(17)

Equations (17) and (9) are substituted to Eq. (15), and the total longitudinal strain is calculated as follow:   

$${\epsilon }_y(x)=-{\epsilon }_p\left(x\right)-{\epsilon }_f+{\epsilon }_0-\frac{{\pi }^2{r}_w{}^2}{4l_w^2}{\left[{\sum _{k=0}^8{r}_k\left(\frac{x}{b_w}\right)}^k\right]}^2$$(18)

The total potential energy is obtained by substituting Eq. (9) to Eq. (10) and substituting Eq. (18) to Eq. (13). The following equations according to the minimum potential energy principle²⁶⁾ are solved concurrently, and the wave height r_w is obtained.   

$$\{\begin{array}{l}\partial \left(U_b+U_m\right)/\partial {\epsilon }_0=0\\ \partial \left(U_b+U_m\right)/\partial r_w=0\end{array}$$(19)

3.3. Post-buckling Model of Local Longitudinal Wave

The local longitudinal waves mainly include the local center wave and the local edge wave, of which strain function and wave function are all different.

(1) For local center wave, the residual strain is expressed by high-order polynomial, and the wave function is piecewise. The total potential energy can be obtained. According to the principle of minimum potential energy, the wave height r_w can be obtained.   

$$\begin{array}{l}{\epsilon }_p\left(x\right)=\mathrm{\Delta }{\epsilon }_p[e_0\left(1-{\displaystyle \frac{b_w}{b}}\right)+e_2\left({\left({\displaystyle \frac{x}{b_w}}\right)}^2-{\displaystyle \frac{b_w}{3b}}\right)+e_4\left({\left({\displaystyle \frac{x}{b_w}}\right)}^4-{\displaystyle \frac{b_w}{5b}}\right)\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+e_6\left({\left({\displaystyle \frac{x}{b_w}}\right)}^6-{\displaystyle \frac{b_w}{7b}}\right)+e_8\left({\left({\displaystyle \frac{x}{b_w}}\right)}^8-{\displaystyle \frac{b_w}{9b}}\right)]\end{array}$$(20)

  

$$\{\begin{array}{l}w\left(x,y\right)=r_w\left[r_0+r_2{\left({\displaystyle \frac{x}{b_w}}\right)}^2+r_4{\left({\displaystyle \frac{x}{b_w}}\right)}^4+r_6{\left({\displaystyle \frac{x}{b_w}}\right)}^6+r_8{\left({\displaystyle \frac{x}{b_w}}\right)}^8\right]\times sin\left({\displaystyle \frac{\pi y}{l}}\right)\\ \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(-b_w\le x\le b_w\right)\\ \\ w\left(x,y\right)=0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(-b\le x<-b_w,b_w<x\le b\right)\end{array}$$(21)

where b_w is the half width of local buckling part.   

$$\begin{array}{l}{U}_b+U_m={\displaystyle \frac{1}{2}}{\int }_0^{l_w}{\int }_{-b_w}^{b_w}{\displaystyle \frac{E\left(x\right)h^3}{12(1-{\mu }^2)}}{\left({\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right)}^2+{\left({\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\right)}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+2\mu {\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+2\left(1-\mu \right){\left({\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}\right)}^2\text{d}x\text{d}y\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }+{\displaystyle \frac{h}{2}}{\int }_0^{l_w}{\int }_{-b_w}^{b_w}E\left(x\right){\left[-{\epsilon }_p(x)-{\epsilon }_f+{\epsilon }_0-{\displaystyle \frac{{\pi }^2{r}_w{}^2}{4l_w^2}}{\left[\sum _{k=0}^8{r}_k{\left({\displaystyle \frac{x}{b_w}}\right)}^k\right]}^2\right]}^2\text{d}x\text{d}y\end{array}$$(22)

(2) For local edge wave, only one side is taken to study due to the symmetry. The residual strain and wave are expressed by power function. The total potential energy can be obtained. According to the principle of minimum potential energy, the wave height r_w can be obtained.   

$${\epsilon }_p\left(x\right)=\mathrm{\Delta }{\epsilon }_p\left[{\left(\frac{x-b+b_w}{b_w}\right)}^{n_E}-\frac{b_w}{\left(n_E+1\right)b}\right]$$(23)

  

$$\{\begin{array}{l}w\left(x,y\right)=r_w{\left({\displaystyle \frac{x-b+b_w}{b_w}}\right)}^{n_w}sin\left({\displaystyle \frac{\pi y}{l}}\right)\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(b-b_w\le x\le b\right)\\ w\left(x,y\right)=0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(0\le x<b-b_w\right)\end{array}$$(24)

where b_w is the side width of local buckling part.   

$$\begin{array}{l}{U}_b+U_m={\displaystyle \frac{1}{2}}{\int }_0^{l_w}{\int }_{b-b_w}^b{\displaystyle \frac{E\left(x\right)h^3}{12(1-{\mu }^2)}}{\left({\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right)}^2+{\left({\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\right)}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+2\mu {\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+2\left(1-\mu \right){\left({\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}\right)}^2\text{d}x\text{d}y\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }+{\displaystyle \frac{h}{2}}{\int }_0^{l_w}{\int }_{-b_w}^{b_w}E\left(x\right){\left[-{\epsilon }_p(x)-{\epsilon }_f+{\epsilon }_0-{\displaystyle \frac{{\pi }^2{r}_w{}^2}{4l_w^2}}{\left({\displaystyle \frac{x-b+b_w}{b_w}}\right)}^{2n_w}\right]}^2\text{d}x\text{d}y\end{array}$$(25)

4. Result and Discussion

Under the effects of temperature and phase transformation, the strip shows different material properties along the width direction, and then affect the post-buckling deformation of global and local longitudinal wave. At the same time, due to the strip thickness and tension at the entry and exit of stand are different, the post-buckling deformation will also be affected. Therefore, the strips at the entry and exit of dual-phase stand F4 are selected for comparative analysis. The geometric size and mechanical parameters of strip are shown in Table 2.

Table 2. Geometric size and mechanical parameters at 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 Temperature and Phase Transformation on Post-buckling of Global Center Wave

According to the typical form of global center wave, the corresponding wave coefficients^14,17) 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 strain coefficients of global center wave^14,17) are shown in Table 4.

Table 4. Strain coefficients of global center wave.

e2	e4	e6	e8
2.468	−2.031	0.658	−0.095

The wave coefficients and strain coefficients are substituted into the post-buckling model to calculate the wave height. It can be seen from Table 5 that the temperature drop and phase transformation at the edge have no significant effect on the wave height of global center wave. In addition, the wave height of global center wave at the exit of stand is obviously higher than that at the entry, which indicates that the changes of strip thickness and tension have great influence on the post-buckling deformation of global center wave.

Table 5. Effect of temperature and phase transformation on the wave height of global center wave.

Calculation cases	Wave height at the entry/mm	Wave height at the exit/mm
Considering the effects	70.83	85.9
No considering the effects	72.15	87.52

4.2. Effect of Temperature and Phase Transformation on Post-buckling of Global Edge Wave

According to the typical form of global edge wave, the corresponding wave coefficients are shown in Table 6.

Table 6. 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 strain coefficients of global edge wave are shown in Table 7.

Table 7. Strain coefficients of global edge wave.

e2	e4	e6	e8
−1	0	0	0

The wave coefficients and strain coefficients are substituted into the post-buckling model to calculate the wave height. As shown in Table 8, the wave height of global edge wave decreases by about 6% due to the existence of soft ferrite at the strip edge. In addition, the wave height of global edge wave at the exit of stand is obviously higher than that at the entry, which indicates that the changes of strip thickness and tension have a great influence on the post-buckling deformation of global edge wave.

Table 8. Effect of temperature and phase transformation on the wave height of global edge wave.

Calculation cases	Wave height at the entry/mm	Wave height at the exit/mm
Considering the effects	79.85	96.93
No considering the effects	74.19	90.2

4.3. Effect of Temperature and Phase Transformation on Post-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 wave coefficients and strain coefficients are the same as those of global center wave. The wave coefficients and strain coefficients are substituted into the post-buckling model to calculate the wave height. It can be seen from Table 9 that the temperature drop and phase transformation at the edge have no significant effect on the wave height of the local center wave. In addition, the wave heights at the entry and exit of stand are basically the same, which indicates that the strip thickness and tension have little effect on the post-buckling deformation of the local center wave.

Table 9. Effect of temperature and phase transformation on the wave height of local center wave.

Calculation cases	Wave height at the entry/mm	Wave height at the exit/mm
Considering the effects	16.01	15.42
No considering the effects	15.99	15.39

4.4. Effect of Temperature and Phase Transformation on Post-buckling of Local Edge Wave

According to the typical form of local edge wave, the width of wave on one side b_w is set as 100 mm, the strain exponent n_E is set as 2.11, and the wave exponent n_w is set as 1.73.^14,17) The wave coefficients and strain coefficients are substituted into the post-buckling model to calculate the wave height. As shown in Table 10, the wave height of local edge wave significantly decreases by about 20% due to the presence of soft ferrite at the edge. In addition, the wave heights at the entry and exit of stand are basically the same, indicating that the strip thickness and tension have little effect on the post-buckling deformation of the local edge wave.

Table 10. Effect of temperature and phase transformation on the wave height of local edge wave.

Calculation cases	Wave height at the entry/mm	Wave height at the exit/mm
Considering the effects	18.17	18.13
No considering the effects	14.69	15.57

5. Conclusion

Based on the modified large deflection theory of thin plate, considering the transverse difference of temperature and phase transformation in hot finishing rolling of non-oriented electrical steel, 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 post-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 main conclusions are as follows:

(1) By comparing the post-buckling wave heights of strip at the entry and exit of stand F4, it is found that the strip thickness and tension has greater influence on the post-buckling deformation of global center wave and global edge wave, but less influence on the post-buckling deformation of local center wave and local edge wave.

(2) The temperature drop and phase transformation have no obvious effect on the post-buckling wave heights of global and local center waves, but reduce the wave heights of global and local edge waves by 6% and 20%, respectively.

Acknowledgements

The authors would like to thank the National Natural Science Foundation of China (Nos.51674028), the Innovation Method Fund of China (2016IM010300) and the Fundamental Research Funds for the Central Universities (FRF-TP-18-105A1) for supporting this research.

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