简介概要

Planar anisotropy of commercially pure titanium sheets

来源期刊：Rare Metals2013年第1期

论文作者：Qing-Dong Zhang Qiang Cao

文章页码：1 - 4

摘要：Commercially pure titanium （CP Ti） sheets show typical planar anisotropy due to the inherently crys-tallographic texture and manufacturing process. To characterize the planar anisotropic behaviors of CP Ti sheets in the forming process, uniaxial tensile tests of TA0 sheets were performed along rolling, transverse, and diagonal directions at room temperature; corresponding stress-strain curves and Lankford coefficients were obtained. Based on Hill’48 and Barlat’89 yield functions, the planar anisotropy of TA0 sheets was investigated. In order to verify the accuracy of two models, we compared the experimental and predicted values of yield stress and Lankford coefficients. It reveals that Barlat’89 criterion with M=10 is good agreement with experimental data, and the obtained function can be used in simulation of forming process.

Rare Metals 2013,32(01),1-4

Planar anisotropy of commercially pure titanium sheets

Qing-Dong Zhang Qiang Cao

School of Mechanical Engineering, University of Science and Technology Beijing

摘要：

Commercially pure titanium (CP Ti) sheets show typical planar anisotropy due to the inherently crys-tallographic texture and manufacturing process. To characterize the planar anisotropic behaviors of CP Ti sheets in the forming process, uniaxial tensile tests of TA0 sheets were performed along rolling, transverse, and diagonal directions at room temperature; corresponding stress-strain curves and Lankford coefficients were obtained. Based on Hill'48 and Barlat'89 yield functions, the planar anisotropy of TA0 sheets was investigated. In order to verify the accuracy of two models, we compared the experimental and predicted values of yield stress and Lankford coefficients. It reveals that Barlat'89 criterion with M=10 is good agreement with experimental data, and the obtained function can be used in simulation of forming process.

作者简介：Qiang Cao e-mail: 444032673@qq.com; me818@me.ustb.edu.cn;

收稿日期：27 June 2012

基金：financially supported by the National Natural Science Foundation of China (No. 51075031,50831008);

Planar anisotropy of commercially pure titanium sheets

Abstract：

Commercially pure titanium (CP Ti) sheets show typical planar anisotropy due to the inherently crys-tallographic texture and manufacturing process. To characterize the planar anisotropic behaviors of CP Ti sheets in the forming process, uniaxial tensile tests of TA0 sheets were performed along rolling, transverse, and diagonal directions at room temperature; corresponding stress-strain curves and Lankford coefficients were obtained. Based on Hill’48 and Barlat’89 yield functions, the planar anisotropy of TA0 sheets was investigated. In order to verify the accuracy of two models, we compared the experimental and predicted values of yield stress and Lankford coefficients. It reveals that Barlat’89 criterion with M=10 is good agreement with experimental data, and the obtained function can be used in simulation of forming process.

Keyword：

Commercially pure titanium; Planar anisotropy; Yield criterion; Barlat’89 model;

Received： 27 June 2012

1 Introduction

Titanium and its alloys are widely used in aerospace,defense,biomedical,and energy industry due to their high specific weight,corrosion resistance,biocompatibility,and heat resistance[1,2].As a category of titanium and its alloys,commercially pure titanium(CP Ti)has been a potential material for structural components in electronic industry recently[3].

Owing to the inherently crystallographic texture and manufacturing process,cold-rolled CP Ti sheets get strong plastic anisotropy in rolling and transverse directions,respectively[4].Moreover,this planar anisotropy has a great influence on the strain distribution during the forming process.To simulate the forming process more realistically,it is important to establish accurate anisotropic yield criteria of metal sheets.

The previous research in the area of anisotropic yield criteria can be pided into microcosmic and macroscopic model.The first macroscopic yield function was introduced by Hill[5]in 1948 as an extension of the isotropic von Mises yield criterion.Thereafter,several yield functions based on the results of polycrystalline calculations were proposed by Hosford[6],Bassani[7],Gotoh[8],Logan and Hosford[9],and Budiansky[10].Unfortunately,these criteria are not enough to explain the material’s macroscopic plastic anisotropy and are not suitable for finite element numerical simulation.Recently,Barlat et al.have made a significant contribution to studying on anisotropic yield criteria.The Barlat’89,Barlat’91,Barlat’93,Barlat’97,and Barlat’2003[1,11–15]yield criteria were proposed one by one.Since its expression format is simple and its anisotropy parameters are obtained easily,Barlat’89yield criterion is used more widely than the other criteria in numerical simulation.Hence,Hill’48 and Barlat’89 yield criteria are chosen to describe the planar anisotropy of CP Ti sheets in this paper.

For the research of plastic anisotropy in CP Ti sheets,Wang and Song[16]studied the effects of electropulsing on anisotropy of cold-rolled CP Ti sheet.Nasiri-Abarbekoh et al.[17]focused on the effects of rolling reduction on mechanical properties anisotropy of CP Ti.Serebryany et al.[18]considered the influence of texture on the normal plastic anisotropy parameter R for example of CP Ti sheet samples.Chen and Chiu[3]investigated the anisotropic mechanical behavior of CP Ti sheets at various temperatures.Although influencing factors of anisotropic behavior were analyzed above,the anisotropic yield criterion of CP Ti sheets was not acquired.Port et al.[19]described anisotropic mechanical behavior of CP Ti according to an elastic–plastic model based on the quadratic Hill’s criterion.However,Barlat’89 yield criterion was not considered.It is necessary to compare the results of Hill’48 yield criterion to Barlat’89 one and choose the better one to characterize the anisotropic behavior of CP Ti sheets.

The present study is focused on how to describe the initial planar anisotropy of TA0 sheets precisely.Two phenomenological yield functions,Hill’48 and Barlat’89,were chosen.First,the mechanical properties of CP Ti sheets were acquired with various uniaxial tensile tests performed along rolling,transverse,and diagonal directions.Then,the parameters of Hill’48 and Barlat’89anisotropic yield criteria were obtained,respectively.To identify the material properties in a comprehensive way,the yield stress and Lankford coefficients(r value)directionalities were investigated by Hill’48 and Barlat’89 yield models.The values are compared with experimental results to obtain the most accurate model,which can be used in the simulation of forming process.

2 Experimental

In this paper,for a typical type of cold-rolled CP Ti sheets,TA0 sheets were considered.The components are shown in Table 1.Uniaxial tensile tests were carried out to determine yield strength,ultimate tensile strength,r values,and stress–strain curves using the specimens taken along 0°,45°,and 90°from the rolling direction at a constant strain rate of 0.001 s^-1.The initial thickness of TA0 sheet is1.5 mm.The gage length of TA0 sheets is 50 mm,and the width is 12.5 mm.

Figure 1 shows the true stress–strain curves of TA0sheets in the directions of rolling(0°),transverse(90°),and diagonal(45°).It shows that TA0 sheets display obvious anisotropy.The breaking elongation rate and ultimate tensile strength of TA0 sheets in rolling direction are maximal following diagonal and transverse directions.On the contrary,the order of yield strength is TD[DD[RD.

The three parameters of the Hill’48 and Barlat’89 yield criteria are obtained from the r values in three directions,respectively.The values of yield stress and Lankford coefficients in three directions are listed in Table 2.

Fig.1 True stress–strain curves of TA0 sheets in the directions of rolling(0°),transverse(90°),and diagonal(45°)

3 Yield functions for anisotropic metal sheets

As mentioned above,there are many yield criteria describing the initial planar anisotropy of the sheet metals.The expression of the yield criterion is known to have a significant effect on the calculated strain and stress distribution[20].In this study,two phenomenological yield criteria are considered.

The Hill’48 yield criterion[5]is a generalization of von Mises yield criterion.The form presenting planar anisotropy is taken in this paper as follows:

where F,H,and Q are anisotropic coefficients,and"r is effective stress.The parameters F,H,and Q can be determined from experimental results as follows:

Another phenomenological yield criterion,Barlat’89[11]is also called Barlat-Lian yield criterion.Yield condition for plane stress is indicated in this model.The yield function is:

Table 1 Chemical compositions(wt%) 下载原图

Table 1 Chemical compositions(wt%)

Table 2 Experimental values of material parameters for TA0 sheets 下载原图

Table 2 Experimental values of material parameters for TA0 sheets

Table 3 Anisotropic coefficients for Hill’48 yield function 下载原图

Table 3 Anisotropic coefficients for Hill’48 yield function

where a,c,h,and p are material constants.The exponent M is related to the crystallographic structure of the material and is set to 6 for BCC metal and 8 for FCC metal according to the work by Logan and Hosford[9].For HCP metal,such as CP Ti,the value of M is unknown.It is recommended for 8–12 by Guo et al.[21].

The parameters a,c,and h can be determined from experiments:

Although the parameter p cannot be calculated explicitly,the relationship of r₄₅and p exists for uniaxial tensile tests.Moreover,the value of parameter p can be obtained by iteration according to Eq.6:

where g(p)is a new function of p

4 Material characterization using yield models

Based on r values along 0°,45°,and 90°from the rolling direction,the input data for the yield functions and calculated anisotropic coefficients are listed in Tables 3 and 4,respectively.For Barlat’89 yield function,the parameter p is different with various values of M.

Fig.2 Normalized yield surfaces for two different yield criteria

Fig.3 Comparison of experimental and predicted yield stress distribution

In Fig.2,the yield loci for two yield criteria are displayed in a normalized stress space for TA0 sheets when S?r_xy="r?0.The black circle dot stands for the experimental result.It reveals that the calculated yield loci using both Hill’48 and Barlat’89 models can represent the initial yield behavior of TA0 in rolling and transverse directions roughly.

Figures 3 and 4 show the predictions of the normalized yield stress and r value anisotropy,respectively.As shown in Fig.3,the normalized yield stress anisotropy is better described by the Barlat’89 criterion than Hill’48.Furthermore,the predicted Barlat’89 yield criterion with M=10fits the experimental results best.In the case of r value anisotropy,the results predicted by both Hill’48 yield model and Barlat’89 yield model with M=10 show good agreements with experiments,as shown in Fig.4.

Table 4 Anisotropic coefficients for Barlat’89 yield function 下载原图

Table 4 Anisotropic coefficients for Barlat’89 yield function

Fig.4 Comparison of experimental and predicted r values distribution

In conclusion,the result from Barlat’89 criterion with M=10 exhibits good agreement with experimental data.This yield model can express the planar anisotropic behaviors of TA0 sheets accurately.The expression of yield criterion is shown as follows:

5 Conclusion

In this paper,plastic,planar,and anisotropic behavior of TA0 sheets was investigated by two phenomenological yield functions of the Hill’48 and Barlat’89.By comparing the experimental values of yield stress and r to predicted ones,Barlat’89 model is more accurate than the Hill’48model.When M=10,the model possesses the highest precision.The obtained Barlat’89 yield criterion can be used in the simulation of forming process.The initial yield function of TA0 sheet is shown as follows: