Mathematical Problems in Engineering

Volume 2015, Article ID 612840, 5 pages

http://dx.doi.org/10.1155/2015/612840

## The Influence of Bauschinger Effect in Straightening Process

^{1}Heavy Industry Engineering Center of China Ministry of Education, Taiyuan 030024, China^{2}Material Science & Engineering Science College, Taiyuan University of Science and Technology, Taiyuan 030024, China^{3}Technology Center, Taiyuan Heavy Industry Co., Ltd., Taiyuan 030024, China

Received 22 October 2014; Revised 6 January 2015; Accepted 6 January 2015

Academic Editor: Yannis Dimakopoulos

Copyright © 2015 Hai-Lian Gui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The yield stress changes during the straightening process, because of the Bauschinger effect. This effect leads to a different reduction rate in each straightening roller. To accurately estimate the straightening effect of the rolled piece, the Bauschinger effect must be considered in calculating the reduction rate. In this paper, the straightening model is described by the fast multipole boundary element method. The Bauschinger effect model is discussed in the elastic loading region and elastic plastic loading region. Young’s modulus reduction and the reverse yield stress reduction are obtained for the straightening force model. The straightening force formula including the Bauschinger effect is determined by analyzing the tension and compression processes. This formula reflects the changes of the yield stress and the straightening force in the tension and compression processes. It is concluded that the Bauschinger effect is a very important factor for the precise estimation of the straightening force.

#### 1. Introduction

The Bauschinger [1] effect was proposed by Bauschinger in 1881. He noted that after plastic deformation the elastic limit of a metal was lower for subsequent loading in the reverse direction than for loading in the same direction. In recent years, this phenomenon has been a subject of numerous studies that showed that this effect is more general and complex than it had been thought originally. Much of this study has been summarized in recent review papers. Liu et al. [2] have experimentally studied the Bauschinger effect in sheet metal. A device to avoid the bending of the specimen in compression progress was designed and the rounding of the reverse flow curve was obtained, yielding at low reverse stresses. Ghorbanpour et al. [3] studied an independent kinematic hardening material model with the reverse yielding point defined by the Bauschinger effect factor (BEF).

The Bauschinger phenomenon can cause reverse yielding to occur at the surface of the straightening pieces. Reverse yielding caused by highly compressive residual stresses can affect the performance characteristics and product quality. Therefore, the Bauschinger effect is a very important factor in the straightening process. Many studies have discussed the influence of various factors, such as the roller distance, roll gap of up roll and down roll, straightening, and the metal material, on the straightening force in the framework of the traditional straightening theory. Zhou et al. [4] obtained the reverse bend curvature and work hardening modulus using the ANSYS/LS-DYNA analysis. Sun and Chen [5] analyzed the frame of prestress straightening machine based on the ANSYS Workbench. Li et al. [6] optimized the parameter by simulation of the straightening process using ANSYS/LS-DYNA. The relationship between the straightening force and the relative strain on the column neutral layer is obtained by using the polynomial fit. However, in the traditional straightening analysis, the Bauschinger effect had been ignored. This may lead to a systematic error for the work curve in straightening process that can reach up to 40%.

Unlike the traditional methods, the straightening model used in this work is described by the fast multipole boundary element method (FM-BEM). The Bauschinger effect is considered in this model and the work curve for the straightening process is obtained using the analysis of the FM-BEM results. The stress of the cross section in the loading process and unloading process is then discussed. It is concluded that the Bauschinger effect is important for the precise determination of the deformation resistance and improving straightening effect.

#### 2. The Straightening Model Based on FM-BEM

The straightening process is considered as an inverse problem of the plate bending. Using the fundamental solution of plate bending, the boundary integral equation of the straightening process can be written as [7] where denotes the vertical force; is the boundary coefficient; and denote the normal angle and the tangential angle at an arbitrary point on the boundary, respectively; and are the known values of and ; and denote the bending moment and the torsional moment in a cross section of the plate, respectively; and are the known values of and ; is the shear force; and is the known value of .

is the fundamental solution of the plate bending problem and can be written as where , , and is the bending rigidity.

#### 3. Bauschinger Effect Model

First, the initial stress and strain relationship are given in Cartesian coordinates for the loading process and the unloading process.(1)Loading process is as follows: in elastic state in strain hardening state (2)Unloading process is as follows: in elastic state in strain hardening state where and , respectively, denote the stress in the loading process and the unloading process, and denote the strain in the loading process and the unloading process, and denote the elastic modulus in the loading process and the unloading process, , , , denote the strain hardening coefficients, and and denote loading the yield strain in the loading process and the unloading process.

In the straightening process, the straightening piece deformation is considered as a plane strain problem. It follows the Von Mises yield law: , where is an equivalent stress and is the yield limit:

When , the neutral layer is offset downward, while, for , the neutral layer is offset upward and means that the neutral layer has a zero offset; namely, the geometric central layer and the stress neutral layer coincide. The stress distribution of the rolled piece cross section is shown in Figure 1, where is the rolled piece thickness, is the roller distance, and and are, respectively, the thickness of the elastic deformation in the tensile zone and in the compression region.