Effect of Bauschinger Effect and Yield Criterion on ...

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Nov 10, 2005 - Keble College Oxford, hosted by Cranfield University, RMCS, Shrivenham, SN6. 8LA. 212 / Vol. 128, MAY 2006. Copyright © 2006 by ASME.
X. P Huang School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, China e-mail: [email protected]

W. C. Cui China Ship Scientific Research Center, P.O. Box 116, Wuxi, Jiangsu, 214082, China e-mail: [email protected]

Effect of Bauschinger Effect and Yield Criterion on Residual Stress Distribution of Autofrettaged Tube Many analytical and numerical solutions for determining the residual stress distribution in autofrettaged tube have been reported. The significance of the choice of yield criterion, the Bauschinger effect, strain hardening, and the end conditions on the predicted residual stress distribution has been discussed by many authors. There are some different autofrettage models based on different simplified material strain-hardening behaviors, such as a linear strain-hardening model, power strain-hardening model, etc. Those models give more accurate predictions than that of elastic–perfectly plastic model, and each of them suits different strain-hardening materials. In this paper, an autofrettage model considering the material strain-hardening relationship and the Bauschinger effect, based on the actual tensile-compressive stress-strain curve of material, plane-strain, and modified yield criterion, has been proposed. The predicted residual stress distributions of autofrettaged tubes from the present model are compared to the numerical results and the experimental data. The predicted residual stresses are in good agreement with the experimental data and numerical predictions. The effect of Bauschinger effect and yield criterion on residual stress is discussed based on the present model. To predict residual stress distribution accurately, it is necessary to properly model yield criterion, Bauschinger effect, and appropriate end conditions. 关DOI: 10.1115/1.2172621兴 Keywords: autofrettage, residual stress distribution, Bauschinger effect, yield criterion

Introduction The autofrettage process is a practical method for increasing the elastic-carrying capacity and the fatigue life of a thick-walled tube, such as a cannon or a high-pressure tubular reactor, etc. The essence of the autofrettage technique is the introduction and utilization of residual stresses. These residual stresses are generated after pressurization causes yielding partway through the tube wall. The reliable prediction of the influence of residual stresses on the elastic-carrying capacity, fatigue crack growth, and fracture in a thick-walled tube requires accurate estimation of the residual stress field 关1兴. Residual stress distributions can be determined by experiments or calculations. The calculation procedures usually involve making simplifying assumptions about the material behavior which may limit their accuracies 关2兴. The basic autofrettage model proposed by Hill 关3兴 is elastic perfectly plastic. Because of the Bauschinger effect and strain hardening, most materials do not satisfy the elastic–perfectly plastic assumption, and consequently, alternative autofrettage models, based on various simplified material strain-hardening characteristics, have been proposed 关4兴. These are the unloading linear strain-hardening 关5兴, bilinear strain-hardening 关6,7兴, loading elastic–perfectly plastic and unloading power strain-hardening 关7,8兴, loading and unloading power strain-hardening 关9兴, and loading linear and unloading power strain-hardening 关10兴 models. These models give more accurate solutions than the elastic–perfectly plastic model, and each of them suits different strain-hardening materials. Kendall 关11兴 proposed a quadratic fit to the Bauschinger-unloading profile. This fit is a function of prior plastic strain and is based on the work of Milligan et al. 关12兴. It was also the basis of 关13兴 in which several Tresca, plane stress solutions are presented. In later extensive exContributed by the Pressure Vessels and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received November 10, 2005; final manuscript received November 22, 2005. Review conducted by Anthony Parker. Paper presented at the Gun Tubes Conference 2005, April 10–14, 2005, Keble College Oxford, hosted by Cranfield University, RMCS, Shrivenham, SN6 8LA.

212 / Vol. 128, MAY 2006

perimental work and associated curve-fitting 关14,15兴, a nonlinear kinematic hardening fit to both loading and unloading profiles is proposed; the latter also represents unloading as a function of prior plastic strain. This work shows that Kendall’s fit 关11兴 is appropriate to A723-type steels, but that other candidate pressure vessel steels exhibit significantly different profiles. Associated work 关16兴 indicates the crucial importance of end conditions 共e.g., plane stress, plane-strain, and open-end conditions兲 in analyzing the autofrettage process. In this paper, a general autofrettage model considering the material strain-hardening relationship and the effect of Bauschinger effect, based on the actual tensile-compressive curve of material and the modified yield criterion and plane strain, incompressible material, is proposed. Based on this model, the effect of Bauschinger effect and yield criteria on residual stress distribution are discussed.

Theoretical Analysis Material Stress-Strain Relationship. A general material tensile-compressive stress-strain curve is shown in Fig. 1. The curve can be divided into four segments, O-A, A-B, B-D, and DE, and be expressed by four equations. 1. Loading phase O-A-B: in the Cartesian coordinate system 共␧O␴兲, shown in Fig. 1. An initial tensile loading regime, O-A, during which the steel behaves elastically up to the yield point ␴s共␧s兲, the elastic modulus over this range is E1. The material then behaves plastically, A-B. This phase may involve significant nonlinearity. The relationship of stress and strain can be expressed as Linear elastic regime O-A

␴ = E 1␧

共␧ 艋 ␧s兲

共1兲

Strain-hardening regime A-B

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␴ = A 1 + A 2␧ B1

共␧ 艌 ␧s兲

共2兲

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Fig. 2 Radii of elastic plastic zones

␧␪ + ␧r + ␧z = 0

共7兲

3. Plane-strain assumption 共with closed ends兲 ␧z = 0 Fig. 1 General material tensile-compressive stress-strain curve

2. Unloading phase B-D-E: in the Cartesian coordinate system 共␧*B␴*兲, shown in Fig. 1, unloading elastic regime, B-D, during which the steel behaves elastically up to the yield point ␴E共␧s*兲, the elastic modulus over this range is E2. The material then behaves plastically, D-E. This phase behaves significant nonlinearity. The relationship of stress and strain can be expressed as Elastic regime B-D

␴ = E 2␧ *

*

共␧ 艋 *

␧s*兲

共3兲

Yield Criterion. The experiments have shown that von Mises yield criterion and Tresca yield criterion are more suitable for the elastic-plastic analysis of thick-walled tubes than other yield criteria 关5兴. Stacey and Webster 关2兴 found that close agreement with experiment is achieved when the unloading stress-strain behavior of the material is modeled accurately and the average of the Tresca and von Mises yield criteria is used 关1兴. Some researchers have suggested that adopting von Mises yield criteria will give a more accurate solution than that of Tresca. The yield criterion can be rewritten as the unified form as

␴i =

共␧* 艌 ␧s*兲

共4兲

1. Unique curve assumption The relationship between equivalent strain 共strain intensity兲 ␧i and equivalent stress 共stress intensity兲 ␴i under complex stress states is the same as the strain-stress relationship under uniaxial tensile-compressive loading, i.e., Eqs. 共1兲–共4兲 remain valid when ␧ , ␴共␧* , ␴*兲 is replaced by ␧i , ␴i共␧* , ␴*兲. Stress intensity ␴i

␴i = 冑 21 关共␴␪ − ␴r兲2 + 共␴r − ␴z兲2 + 共␴z − ␴␪兲2兴 冑2 3

Residual Stress Distribution. Loading and unloading stress analysis is performed in the Cartesian coordinate system ␧O␴ and ␧*B␴* shown in Fig. 1, respectively. The radii of elastic plastic zones in the tube wall were shown in Fig. 2. The residual stress distribution can be determined by using loading stress minus corresponding unloading stress, i.e., ␴R = ␴ − ␴*. The residual stress calculation should be expressed in two different cases and three zones, respectively. Ideal Elastic Unloading. Loading elastic zone 共rc 艋 r 艋 ro兲



共5兲

冑共␧␪ − ␧r兲2 + 共␧r − ␧z兲2 + 共␧z − ␧␪兲2

共6兲



␴␪R

␴rR = ␴␪R

冉 冉

冉 冊 冉 冊

冊 冊

r2o pa ␣␴s 2 1 1 rc 2 − 2 − 2 2 ri2 1 − 2 2 r ro r ro − ri

r2o pa ␣␴s 2 1 1 = rc 2 + 2 − 2 2 ri2 1 + 2 2 r ro r ro − ri

Loading plastic zone 共ri 艋 r 艋 rc兲

2. Incompressible material assumption

␴rR =

共9兲

where ␣ = 1, ␣ = 2 / 冑3, and 1 ⬍ ␣ ⬍ 2 / 冑3, for Tresca, von Mises,

Strain intensity ␧i ␧i =

冑3 2 ␣␴s

and modified yield criterion, will give.

Strain hardening regime D-E

␴* = A3 + A4共␧*兲B2 Fundamental Assumptions

共8兲

冋 冉冊 再冋 冉 冊 册



1 ␴s − A1 2B1 1 ␣ r 2A1 ln + r − 2 ri B1 c ri2B1 r2B1



冊册



pa r2o



r2o ri2

冉 冊

1 ␴s − A1 2B1 1 ␣ r = + 共2B1 − 1兲 2B 2 ln + 1 A1 + r 2 ri B1 c ri2B1 r 1

1−

册冎

ri2 r2 −

pa r2o −

r2o ri2



ri2 1+ 2 r

冊冧



共10兲

共11兲

Elastic Plastic Unloading. Loading elastic zone and unloading elastic zone 共rc 艋 r 艋 ro兲



冉 冉

1 1 ␣ 共␴sr2c − ␴Er2d兲 2 − 2 2 ro r 1 1 ␣ ␴␪R = 共␴sr2c − ␴Er2d兲 2 + 2 2 ro r

␴rR =

冊 冊



共12兲

Loading plastic zone and unloading elastic zone 共rd 艋 r 艋 rc兲 Journal of Pressure Vessel Technology

MAY 2006, Vol. 128 / 213

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冋 冉冊 再冋 冉 冊 册





Loading plastic zone and unloading plastic zone 共ri 艋 r 艋 rd兲



冊册





冉冊



册冎





冊册





␴s − A1 2B1 1 ␴E − A3 2B2 1 1 1 ␣ r 2共A1 − A3兲ln + r − − rd − 2 ri B1 c ri2B1 r2B1 B2 ri2B2 r2B2 1 1 ␴s − A1 2B1 1 ␴E − A3 2B2 1 ␣ r 2 ln + 1 共A1 − A3兲 + r + 共2B1 − 1兲 2B − rd + 共2B2 − 1兲 2B ␴␪R = 2 ri B1 c ri2B1 r 1 B2 r 2 ri2B2

␴rR =

再冋 冉 冊 册



Autofrettage Pressure The relationship of pa ⬃ rc

再 冉冊

␴s − A1 ␣ rc pa = 2A1 ln + 2 ri B1

再 冉冊

␴E − A3 ␣ rd 2A1 ln + 2 ri B2





共13兲

册冎 冧

共14兲

Discussion

冋冉 冊 册 冋 冉 冊 册冎 rc ri

2B1

rc − 1 − ␴S 1 − ro

2

共15兲

The relationship of pa ⬃ rd pa =

冉 冊

␴s − A1 2B1 1 1 r ␣␴E 2 1 ri2 ␣ 2A1 ln + rc − − − pa r 2B1 − 2B1 2 ri B1 r 2 d r2o r2 ri ␴s − A1 2B1 1 1 ␣ r ␣␴E 2 1 1 + 共2B1 − 1兲 2B ␴␪R = 2 ln + 1 A1 + r − + − pa r 2 ri B1 c ri2B1 r 1 2 d r2o r2

␴rR =

冋冉 冊 册 冋 冉 冊 册冎 rd ri

2B2

− 1 − ␴E 1 −

rd ro

2

共16兲

Critical autofrettage pressure The critical autofrettage pressure is defined as the autofrettage pressure when the reverse yield just takes place at the inner surface of the autofrettaged tube. Replace rd with ri in Eq. 共16兲, the critical autofrettage pressure is

冋 冉 冊册

␣␴E ri 1− pacr = 2 ro

2

共17兲

Validations

Effect of Bauschinger Effect on Residual Stress. For sufficiently thick tubes and depths of yielding during the autofrettage process, reverse yielding may take place adjacent to the inner surface when the internal pressure is removed. For a yield stress in compression equal to that in tension, reyielding occurs when ro / ri ⬎ 2.22. It can take place at lower k values due to the Bauschinger effect. In general, Bauschinger effect coefficient bef is found to be material dependent and sensitive to the amount of prior plastic strain. Typical values of bef in the range 0.3–1.0 have been measured 关12兴. The greater the previous plastic strain the smaller the Bauschinger effect coefficient is. The smaller Bauschinger effect coefficient causes the reverse yielding to take place more easily and affects the residual stress distribution. In the present model, the effect of Bauschinger effect is considered by parameter ␴E.

␴E = A1 + A2␧b1 + bef␴s

共18兲

The effect of Bauschinger effect on residual stress is illustrated in Fig. 6. The smaller the bef, the larger the reversed yielding radius is and the less compressive the hoop residual stress is near the bore. Plastic strain through the wall thickness is not constant; thus, in reality bef varies with radius. In this analysis, bef was set pragmatically and the parameter ␴E was determined by stress-strain

Experimental Validation. The experimental material of the specimen is 30CrNiMo8. The tensile-compressive stress-strain curve of the material is shown in Fig. 3. The parameters needed in the present model were determined by fitting the tensilecompressive stress-strain curve using Eqs. 共1兲–共4兲 and are listed in Table 1. The dots shown in Fig. 3 are determined by Eqs. 共1兲–共4兲 using the data in Table 1. It shows that Eqs. 共1兲–共4兲 fit the strssstrain curve well. There is a small difference at the two knots of the elasticity and plasticity of the stress-strain curve. To eliminate the difference, parameters ␴s and ␴E should be correspond to the values of intersection of Eqs. 共1兲 and 共2兲 and Eqs. 共3兲 and 共4兲, respectively. The internal and external radii, autofrettage pressure, and some important results are listed in Table 2. The predicted residual stress distributions and the experimental data measured by Sach’s boring method are shown in Fig. 4. This figure shows that the calculated elasto-plastic radius is a little smaller than the measured value when ␣ = 1.11, and the predicted residual stresses are in good agreement with test data. Numerical Validation. The residual stress distributions predicted by the present model for 30CrNiMo8 may be compared to some numerical results of Parker 关17兴 that are shown in Fig. 5. Parker’s results are for A723 steel of the same yield strength and include Bauschinger effect that varies as a function of plastic strain and, hence, radius as defined in Ref. 关14兴. There is some difference between the various results near the bore. To understand these differences now consider some important parameters affecting residual stress distribution. 214 / Vol. 128, MAY 2006

Fig. 3 Stress-strain curve of 30CrNiMo8

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Table 1 Calculation parameters of 30CrNiMo8

␴s 共MPa兲

E1 共MPa兲

A1 共MPa兲

A2 共MPa兲

B1

␴E 共MPa兲

E2 共MPa兲

A3 共MPa兲

A4 共Mpa兲

B2

bef

960.7

207000

928.1

7026

1.0

1420

201000

−5.0

10850

0.405

0.47

curve fitting in this model. For simulating the stress-strain relationship of the tube under autofrettage pressure with the uniaxial tensile-compressive curve of the material, the maximum strain of the curve should be approximately equal to or less than the von Mises equivalent strain at the inner surface of the tube under autofrettage pressure. Table 2 Radii and autofrettage pressure of the tube: rcc-calculation value; rcm-experimental measuring value ri 共mm兲

ro 共mm兲



pa 共MPa兲

pacr 共MPa兲

rcc 共mm兲

rcm 共mm兲

rd 共mm兲

19.3

43.7

1.11

740

601.9

29.43

30.2

21.16

Effect of Yield Criterion on Residual Stress. The effect of yield criterion on residual stress under same Pa is shown in Fig. 7. The yield radius using Tresca yield criterion is larger than that using the von Mises yield criterion, but both criteria give a similar value of residual stress at the bore under the same autofrettage pressure. The effect of yield criterion on residual stress under same rc is shown in Fig. 8. Larger residual stresses are given by using von Mises criterion under same rc. Some experimental results indicate that a proper value of the yield criterion parameter ␣ will properly model different conditions. In the present paper, ␣ = 1.11 is proper for giving more accurate prediction of residual stress.

Conclusions A general autofrettage model considering the material strainhardening relationship and Bauschinger effect, based on actual tensile-compressive curve of material, modified yield criterion, and plane-strain, incompressible conditions, has been proposed. Experimental results show that the present model has strong curve-fitting ability, and the predicted residual stresses are in good

Fig. 6 Effect of Bauschinger effect on residual stress Fig. 4 Comparison of predicted results with test data

Fig. 5 Comparison of predicted results with numerical simulation data

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Fig. 7 Effect of yield criterion on residual stress under same Pa

MAY 2006, Vol. 128 / 215

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References

Fig. 8 Effect of yield criterion on residual stress under same rc

agreement with test data and numerical simulation data. Those parameters needed in the present model are determined by fitting the actual tensile-compressive curve of the material using Eqs. 共2兲–共4兲. Because the Bauschinger effect is material dependent and sensitive to the amount of prior plastic strain, the parameter of the Bauschinger effect should be a function of radius. In this analysis, bef was set pragmatically. The maximum tensile strain of the tensile-compressive curve should be approximately equal to or less than the equivalent strain at the inner surface of the tube under autofrettage pressure. The yield criterion will influence the distribution of residual stress, and an appropriate choice of parameter ␣ will suit different conditions.

Acknowledgment The authors greatly appreciate Professor A. P. Parker, who supplied many references and provided some useful suggestions and comments to this work.

216 / Vol. 128, MAY 2006

关1兴 Stacey, A., and Webster, G. A., 1984, “Fatigue Crack Growth in Autofrettaged Thick-Walled High Pressure Tube Material,” High Pressure in Science and Technology, C. Homan R. K. MacCrone, and E. Walley, eds., Elsevier, New York, pp. 215–219. 关2兴 Stacey, A., and Webster, G. A., 1988, “Determination of Residual Stress Distributions in Autofrettaged Tubing,” Int. J. Pressure Vessels Piping, 31, pp. 205–220. 关3兴 Hill, R., 1950, The Mathematical Theory of Plasticity, Oxford University Press, London. 关4兴 Zhang, Y. H., Huang, X. P., and Pan, B. Z., 1997, Fracture and Fatigue Control Design in Pressure Vessels 共in Chinese兲, Press of Petroleum Industry, Beijing, China. 关5兴 Chen, P. C. T., 1980, “Generalized Plane-Strain Problems in an Elastic-Plastic Thick-Walled Cylinder,” Trans. 26th Conference of Army Mathematicians, pp. 265–275. 关6兴 Lazzarin, P., and Livieri, P., 1997, “Different Solution for Stress and Strain Fields in Autofrettaged Thick-Walled Cylinders,” Int. J. Pressure Vessels Piping, 31, pp. 231–238. 关7兴 Livieri, P., and Lazzarin, P., 2002, “Autofrettaged Cylindrical Vessels and Bausching Effect: An Analytical Frame for Evaluating Residual Stress Distributions,” ASME J. Pressure Vessel Technol., 124, pp. 38–45. 关8兴 Pan, B. Z., Zhu, R. D., and Su, H. J., 1990, “Autofrettage Theory and Experimental Research 共I兲” 共in Chinese兲, J. Daqing Pet. Inst., 12共1兲, pp. 14–16. 关9兴 Su, H. J., and Huang X. P., 1995, “Autofrettage Technology Research 共II兲” 共in Chinese兲, J. Daqing Pet. Inst., 19共2兲, pp. 78–82. 关10兴 Huang, X. P., and Cui, W. C., 2004, “Autofrettage Analysis of Thick-Walled Cylinder Based on Tensile-Compressive Curve of Material,” Key Eng. Mater., 274–276, pp. 1035–1040. 关11兴 Kendall, D. P., 1998, “Unpublished discussion of a technical report ‘The Bauschinger Effect in Autofrettaged Tubes—A Comparison of Models Including the ASME Code’ ” by A. P. Parker, and J. H. Underwood, Technical report ARCCB-TR-98010, US Army ARDEC, Watervliet, New York. 关12兴 Milligan, R. V., Koo, W. H., and Davidson, T. E., 1966, “The Bauschinger Effect in a High Strength Steel,” J. Basic Eng., 88, pp. 480–488. 关13兴 Parker, A. P., Underwood, J. H., and Kendall, D. P., 1999, “Bauschinger Effect Design Procedures for Autofrettaged Tubes Including Material Removal and Sachs’ Method,” ASME J. Pressure Vessel Technol., 121, pp. 430–437. 关14兴 Parker, A. P., Troiano, E., Underwood, J. H., and Mossey, C., 2003, “Characterization of Steels Using a Revised Kinematic Hardening Model Incorporating Bauschinger Effect,” ASME J. Pressure Vessel Technol., 125, pp. 277– 281. 关15兴 Troiano, E., Parker, A. P., Underwood, J. H., and Mossey, C., 2003, “Experimental Data, Numerical Fit and Fatigue Life Calculations Relating to Bauschinger Effect in High Strength Armament Steels,” ASME J. Pressure Vessel Technol., 125, pp. 330–334. 关16兴 Parker, A. P., 2001, “Autofrettage of Open End Tubes—Pressures, Stresses, Strains and Code Comparisons,” ASME J. Pressure Vessel Technol., 123, pp. 271–281. 关17兴 Parker, A. P., private communication.

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