A general autofrettage model of a thick-walled ...

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A general autofrettage model of a thick-walled cylinder based on tensile–compressive stress–strain curve of a material X P Huang School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China. email: [email protected] The manuscript was received on 21 October 2004 and was accepted after revision for publication on 24 March 2005. DOI: 10.1243/030932405X16070

Abstract: The basic autofrettage theory assumes elastic–perfectly plastic behaviour. Because of the Bauschinger effect and strain-hardening, most materials do not display elastic–perfectly plastic properties and consequently various autofrettage models are based on different simplified material strain-hardening models, which assume linear strain-hardening or power strain-hardening or a combination of these strain-hardening models. This approach gives a more accurate prediction than the elastic–perfectly plastic model and is suitable for different strain-hardening materials. In this paper, a general autofrettage model that incorporates the material strain-hardening relationship and the Bauschinger effect, based upon the actual tensile–compressive stress–strain curve of a material is proposed. The model incorporates the von Mises yield criterion, an incompressible material, and the plane strain condition. Analytic expressions for the residual stress distribution have been derived. Experimental results show that the present model has a stronger curve-fitting ability and gives a more accurate prediction. Several other models are shown to be special cases of the general model presented in this paper. The parameters needed in the model are determined by fitting the actual tensile–compressive curve of the material, and the maximum strain of this curve should closely represent the maximum equivalent strain at the inner surface of the cylinder under maximum autofrettage pressure. Keywords: thick-walled cylinder, von Mises yield criterion, Bauschinger effect, strain-hardening, autofrettage model

1

INTRODUCTION

The autofrettage process is a practical method for increasing the elastic-carrying capacity and the fatigue life of a thick-walled cylinder such as a cannon or a high-pressure tubular reactor. The essence of the autofrettage technique is the introduction and utilization of residual stresses. These residual stresses are generated after pressurizing to cause yielding partway through the cylinder wall. The reliable prediction of the influence of residual stresses on the elastic-carrying capacity, fatigue crack growth, and fracture in a thickwalled cylinder requires accurate estimation of the residual stress field [1]. Residual stress distributions can be determined by experiment or calculation. The calculation procedures usually involve making simplifying assumptions about the material behaviour, which JSA85 # IMechE 2005

may limit their accuracies [2]. The basic autofrettage model proposed by Hill [3] is an elastic–perfectly plastic model, shown in Fig. 1(a). 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 strainhardening characteristics, have been proposed [4]. These are an unloading linear strain-hardening model [5, 6], a bilinear strain-hardening model [7–9], a loading elastic–perfectly plastic and unloading power strainhardening model [9, 10], a loading and unloading power strain-hardening model [11], and a loading linear and unloading power strain-hardening model [12], shown in Fig. 1. These models give more accurate solutions than the elastic–perfectly plastic model and each of them suits different strain-hardening materials. J. Strain Analysis Vol. 40 No. 6

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X P Huang

Fig. 2 General material tensile–compressive stress–strain curve

(a) Loading phase O–A–B. In the Cartesian coordinate system ð"O
Þ, shown in Fig. 2, there is an initial tensile loading regime, O–A, during which the steel behaves elastically up to the yield point
s ð"s Þ; the elastic modulus over this range is E . The material then behaves plastically (A–B). This phase may involve significant non-linearity. The relationship of stress–strain can be expressed as Fig. 1 Four typical autofrettage models: (a) elastic–perfectly plastic model; (b) linear strain-hardening model; (c) unloading power function strain-hardening model; (d) power function strain-hardening model

Linear elastic regime O–A


¼ E"

ð" 4 "s Þ

Strain-hardening regime A–B


¼ A1 þ A2 "B1 Numerical analysis procedures are required when material behaviour is so complex that an analytical model cannot properly represent the material or boundary conditions. Such procedures have been employed by Parker and his co-workers [13, 14]. In this paper, a general autofrettage model considering the material strain-hardening relationship and the effect of Bauschinger, based on the actual tensile– compressive curve of the material and the von Mises yield criterion, is proposed.

THEORETICAL ANALYSIS AND FORMULA DERIVATION

2.1 2.1.1

Material stress–strain relationship and fundamental assumptions Description of the material stress–strain relationship

A general material tensile–compressive stress–strain curve is shown in Fig. 2. The curve can be divided into four segments, O–A, A–B, B–D, and D–E, and can be expressed by four equations. J. Strain Analysis Vol. 40 No. 6

ð" 5 "s Þ

ð2Þ

(b) Unloading phase B–D–E. In the Cartesian coordinate system ("
B

), shown in Fig. 2, there is an unloading elastic regime, B–D, during which the steel behaves elastically up to the yield point
D ð"
s Þ; the elastic modulus over this range is E 0 . The material then behaves plastically (D–E). This phase exhibits significant non-linearity. The relationship of stress– strain can be expressed as Elastic regime B–D



¼ E 0 "
2

ð1Þ

ð"
4 "
s Þ

ð3Þ

Strain hardening regime D–E



¼ A3 þ A4 ð"
ÞB2 2.1.2

ð"
5 "
s Þ

ð4Þ

Fundamental assumptions

(a) Unique curve assumption. The relationship between the equivalent strain (strain intensity) "i and the equivalent stress (stress intensity)
i under complex stress states is the same as the strain–stress relationship under uniaxial tensile–compressive loading, i.e. equations (1) to (4) remain valid when JSA85 # IMechE 2005

A general autofrettage model of a thick-walled cylinder

"‚
ð"
‚

Þ is replaced by "i ‚
i ð"
i ‚

i Þ. Stress intensity
i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i ¼ 12 ½ð
 
r Þ2 þ ð
r 
z Þ2 þ ð
z 
 Þ2  Strain intensity "i pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð"  "r Þ2 þ ð"r  "z Þ2 þ ðz " Þ2 "i ¼ 3

and the corresponding strains "r and " , are given in terms of the radial displacement, u, by

ð5Þ

ð6Þ

(b) Incompressible volume assumption

" þ "r þ "z ¼ 0

2.2

ð8Þ

Yield criterion

2.3

ð9Þ

and the deformation theory constitutive equations are valid [15].

"r ¼

3"i ð

m Þ 2
i r

" ¼

3"i ð

m Þ 2
i 

"z ¼

3"i ð

m Þ 2
i z

ð12Þ


m ¼ 13ð
r þ
 þ
z Þ

ð13Þ

Substituting "z ¼ 0 into equations (12) and (13) gives


z ¼ 12ð
 þ
r Þ

Loading stress analysis

Basic equations

Assuming small strain and no body forces in the axisymmetric state of generalized plane strain, the radial and tangential stresses,
r and
 , must satisfy the equilibrium equation

d
r
r 
 þ ¼0 dr r

ð14Þ

Substituting equation (14) into equation (5) gives

The loading stress analysis is based on the Cartesian coordinate system "O
, shown in Fig. 2. The radii of elastic–plastic zones in the cylinder wall are shown in Fig. 3. 2.3.1

ð11Þ

where

The yield criterion adopted here is the von Mises criterion, i.e.


i ¼
s

du dr u " ¼ r "r ¼

ð7Þ

(c) Plane strain assumption

"z ¼ 0

601

ð10Þ

2
 ¼ pffiffiffi
i þ
r 3 1
z ¼ pffiffiffi
i þ
r 3

ð15Þ

The radial stress
r can be obtained from equations (10) and (15) ð 2

r ¼ pffiffiffi i dr ð16Þ 3 r From equations (6), (7), and (11)

2 c "i ¼ pffiffiffi 2 3r

ð17Þ

where c is an integration constant. 2.3.2

Stress analysis

(a) Loading elastic zone (rc 4 r 4 ro ). Substituting equations (1) and (17) into equation (16) gives


r ¼ 

2Ece þ c01 3r2

ð18Þ

(b) Loading plastic zone (ri 4 r 4 rc ). Substituting equations (2) and (17) into equation (16) gives  B1 2 A2 2 1 pffiffiffi ðcp ÞB1 2B þ c02
r ¼ pffiffiffi A1 ln r  pffiffiffi r 1 3 3 3B1 Fig. 3 Radii of elastic–plastic zones

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ð19Þ J. Strain Analysis Vol. 40 No. 6

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X P Huang

where ce , cp , c01 , and c02 are integration constants, which can be determined from boundary conditions. 2.3.3

Boundary conditions

The autofrettaged cylinder is subjected to internal pressure pa and the external pressure is zero. The following boundary conditions exist under autofrettage pressure: 1. Both of the stress intensities in the elastic zone ð
i Þe and plastic zone ð
i Þp at the elastoplastic radius reach the yield strength, i.e.

ð
i Þe jr ¼ rc ¼
s ð
i Þp jr ¼ rc ¼
s

ð20Þ

2. The radial stress
r at the inner surface and outer surface are


r jr ¼ ri ¼ pa
r jr ¼ ro ¼ 0

ð21Þ

The following integration constants are determined from equations (15), (18), (19), and boundary condition equations (20) and (21) pffiffiffi 3
s 2 1 r2 rc ‚ ce ¼ c01 ¼ pffiffiffi
s 2c 2 E 3 ro pffiffiffi 1=B1 3
s  A1 cp ¼ r2c A1 2   2
s  A1 rc 2B1 ffiffiffi c02 ¼ pa  pffiffiffi A1 ln ri þ p 3 3 B1 r i 2.3.4

Loading stress distribution

(a) Elastic zone (rc 4 r 4 ro )  
1 1
r ¼ psffiffiffi r2c 2  2 ro r 3  
1 1
 ¼ psffiffiffi r2c 2 þ 2 ro r 3


1
z ¼ psffiffiffi r2c 2 3 ro

ð22Þ

(b) Plastic zone (ri 4 r 4 rc )     2 r
s  A1 2B1 1 1
r ¼ pffiffiffi A1 ln rc  2B  pa þ pffiffiffi 1 ri r 1 3 3B1 r2B i     2 r þ 1 A1
 ¼ pffiffiffi ln ri 3  
s  A1 2B1 1 1 ffiffiffi  pa þ p rc þ ð2B  1Þ 1 1 r2B1 3B1 r2B i J. Strain Analysis Vol. 40 No. 6

    1 r þ 1 A1
z ¼ pffiffiffi 2 ln ri 3  
s  A1 2B1 1 1 þ pffiffiffi rc þ ðB1  1Þ 2B  pa 1 r 1 3B1 r2B i ð23Þ

2.4

Unloading stress analysis

Unloading stress analysis is based on the Cartesian coordinate system "
B

, shown in Fig. 2. The radii of elastic–plastic zones in the cylinder wall are shown in Fig. 3. The procedure of unloading stress analysis is the same as that for loading stress analysis. Elastic unloading and the elastic–plastic unloading are each discussed below. 2.4.1

Yielding on unloading

For sufficiently thick cylinders 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 in magnitude to that in tension, reverse yielding will not happen until k ¼ ro =ri exceeds a value of about 2.2 [3]. It can take place at lower k values if yielding occurs in compression at a lower stress than in tension due to the Bauschinger effect. This effect can be taken into account by letting the yield stress in compression be bef
s , where bef is the Bauschinger effect factor [2]

bef ¼

j
 s j
s

ð24Þ

In general, bef is found to be material dependent and sensitive to the amount of previous plastic strain. Typical values of bef in the range 0.3–1.0 have been measured [16]. The greater the previous plastic strain, the smaller is the Bauschinger effect factor. The lower the bef , the earlier is the onset of reverse yielding and the greater is the impact upon residual stress distribution. In this paper, the Bauschinger effect is treated as constant and incorporated via parameter
D. Parameter
D is also dependent on the prior plastic strain. From Fig. 2, bef can be expressed as

bef ¼

1
D ð"max Þ  A1  A2 "Bmax
s

ð25Þ

It is very important to determine the maximum strain "max of the stress–strain curve of a material; this will be discussed in section 4. 2.4.2

Elastic unloading

When the autofrettage pressure is totally removed, the unloading pressure p
is equal to pa. If there is JSA85 # IMechE 2005

A general autofrettage model of a thick-walled cylinder

no reverse yielding, the unloading is perfectly elastic. In this case, the unloading stresses in the whole wall (ri 4 r 4 ro ) follow the Lame´ equations   pa r2o
2
r ¼ 2 ri 1  2 ro  r2i r   p r2

 ¼ 2 a 2 r2i 1 þ o2 ro  ri r p

z ¼ 2 a 2 r2i ro  ri

ð26Þ 2.4.3

Elastic–plastic unloading (reverse yielding taking place)

(a) Unloading elastic zone (rd 4 r 4 ro )  
1 1

r ¼ pDffiffiffi r2d 2  2 ro r 3  
1 1

 ¼ pDffiffiffi r2d 2 þ 2 ro r 3


1

z ¼ pDffiffiffi r2d 2 3 ro ð27Þ (b) Unloading plastic zone (ri 4 r 4 rd )     2 r
D  A3 2B2 1 1
p ffiffi ffi p ffiffi ffi þ A3 ln
r ¼ rd  2B  pa 2 ri r 2 3 3B2 r2B i     2 r þ 1 A3

 ¼ pffiffiffi ln ri 3  
D  A3 2B2 1 1 þ pffiffiffi rd þ ð2B2  1Þ 2B  pa 2 r 2 3 B2 r2B i     1 r þ 1 A3

z ¼ pffiffiffi 2 ln ri 3  
 A3 2B2 1 1 pffiffiffi  pa þ D rd þ ðB  1Þ 2 2 r2B2 3 B2 r2B i ð28Þ 2.5

Residual stress distribution

The residual stress distribution can be determined by subtracting the unloading stress from the corresponding loading stress, i.e.
R ¼


. Because ri 4 rd 4 rc 4 ro , the residual stress distribution should be discussed for two different cases and in three zones. 2.5.1

Elastic unloading (no reverse yielding)

(a) Elastic loading and elastic unloading zone (rc 4 r 4 ro ) JSA85 # IMechE 2005

603

   
s 2 1 1 pa r2o 2 p ffiffi ffi  1 
R ¼  r r r c i r2o r2 r2o  r2i r2 3    
s 2 1 1 pa r2o R 2
 ¼ pffiffiffi rc 2 þ 2  2 ri 1 þ 2 ro  r2i r ro r 3
s 2 1 pa
R r2i z ¼ pffiffiffi rc 2  2 3 ro ro  r2i

ð29Þ

(b) Plastic loading and elastic unloading zone (ri 4 r 4 rc )     2 r
s  A1 2B1 1 1 R þ pffiffiffi
r ¼ pffiffiffi A1 ln rc  2B 1 ri r 1 3 3B1 r2B i   pa r2i 2  2 ro 1  2 ro  r2i r     2 r
s  A1 2B1 ffiffiffi þ 1 A1 þ p rc
R  ¼ pffiffiffi ln ri 3 3B1     1 1 p r2  2B þ ð2B1  1Þ 2B  2 a 2 r2o 1 þ i2 ro  ri r r 1 ri 1     1 r
s  A1 2B1 ffiffiffi þ 1 A1 þ p
R rc z ¼ pffiffiffi 2 ln r 3 B1 3 i   1 1 p  2B þ ðB1  1Þ 2B  2 a 2 r2o 1 1 r ro  ri ri ð30Þ 2.5.2

Elastic–plastic unloading (reverse yielding taking place)

(a) Elastic loading zone and elastic unloading zone (rc 4 r 4 ro )   1 1 1 2 2 p ffiffi ffi ð

R ¼ r 
r Þ  s c D d r r2o r2 3   1 1 1 2 2 p ffiffi ffi
R ð
¼ r 
r Þ þ  s c D d r2o r2 3

1 2 2 1
R z ¼ pffiffiffið
s rc 
D rd Þ 2 ro 3

ð31Þ

(b) Plastic loading zone and elastic unloading zone (rd 4 r 4 rc )     2 r
s  A1 2B1 1 1 R þ pffiffiffi
r ¼ pffiffiffi A1 ln rc  2B 1 ri r 1 3 3B1 r2B i  
1 1  pDffiffiffi r2d 2  2  pa ro r 3     2 r
s  A1 2B1 R ffiffiffi þ 1 A1 þ p
 ¼ pffiffiffi ln rc ri 3 3B1     1 1
D 2 1 1  2B þ ð2B1  1Þ 2B  pffiffiffi rd 2 þ 2  pa r 1 ro r 3 ri 1 J. Strain Analysis Vol. 40 No. 6

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X P Huang


R z

    1 r
s  A1 2B1 ffiffiffi þ 1 A1 þ p ¼ pffiffiffi 2 ln rc ri 3 3B1   1 1
1  2B þ ðB1  1Þ 2B  pDffiffiffi r2d 2  pa r 1 3 ro ri 1

3.2

ð32Þ (c) Plastic loading zone and plastic unloading zone (ri 4 r 4 rd )   2 r R
r ¼ pffiffiffi ðA1  A3 Þ ln r 3 i  
s  A1 2B1 1 1 þ pffiffiffi rc  2B 1 r 1 3B1 r2B i  
D  A3 2B2 1 1 p ffiffi ffi rd  2B  2 r 2 3B2 r2B i     2 r p ffiffi ffi þ 1 ðA1  A3 Þ
R ln ¼  ri 3  
s  A1 2B1 1 1 ffiffiffi þ p rc þ ð2B  1Þ 1 1 r2B1 3B1 r2B i  
D  A3 2B2 1 1 ffiffiffi rd þ ð2B  1Þ  p 2 2 r2B2 3B2 r2B i     1 r R þ 1 ðA1  A3 Þ
z ¼ pffiffiffi 2 ln ri 3  
s  A1 2B1 1 1 þ pffiffiffi rc þ ðB1  1Þ 2B 1 r 1 3B1 r2B i  
D  A3 2B2 1 1 ffiffiffi  p rd þ ðB  1Þ 2 2 r2B2 3B2 r2B i ð33Þ

3

AUTOFRETTAGE PRESSURE

3.1

The relationship of pa  rc

The autofrettage pressure pa can be derived from

ð
r Þe jr ¼ rc ¼ ð
r Þp jr ¼ rc     2 rc
s  A1 rc 2B1 þ pffiffiffi pa ¼ pffiffiffi A1 ln ri 3B1 ri 3  2
r ð1  B1 Þ
s  A1 pffiffiffi  psffiffiffi c  3 ro 3B1 ð34Þ

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When the autofrettage pressure pa is greater than the critical autofrettage pressure pacr , reverse yielding will take place near the inner surface of the autofrettaged cylinder. The reverse yielding elastoplastic radius rd can be determined by     2 r
 A3 rd 2B2 pffiffiffi pa ¼ pffiffiffi A3 ln d þ D ri 3B2 ri 3  2
r ð1  B2 Þ
D  A3 pffiffiffi  pDffiffiffi d  ð35Þ r 3 o 3B2 3.3

Critical autofrettage pressure pacr

The critical autofrettage pressure is defined as the autofrettage pressure when reverse yielding just takes place at the inner surface of the autofrettaged cylinder. Because of the Bauschinger effect, the reverse yield will take place easily. In the present model, the Bauschinger effect can be considered using parameter
D. From equation (35) and letting rd ¼ ri , the critical autofrettage pressure is   2 
r ð36Þ pacr ¼ pDffiffiffi 1  i ro 3 Reverse yielding will take place when pa > pacr . The reverse yielding radius can be determined from equation (35), where it can be seen that parameter
D is important and can be determined by fitting the stress–strain curve of the material. The unloading stress–strain curve is defined by the maximum tensile strain in the uniaxial tension–compression test of the material. Determining the maximum strain of the tensile–compressive stress–strain curve of a material is very important.

4

The elastoplastic radius rc is a basic design parameter in autofrettage processes of a thick-walled cylinder. The autofrettage pressure pa depends upon the parameter rc.

The relationship of rd  pa

ACCURATE DETERMINATION OF THE EXPERIMENTAL TENSILE–COMPRESSIVE CURVE

It is essential to obtain an appropriate experimental uniaxial tension–compression curve to define equivalent stress during autofrettage loading and unloading. This curve should match the behaviour of the material at the bore of the tube, in particular its maximum equivalent strain value. If the desired autofrettage radius is rc , then for elastic–perfectly plastic behaviour   p E ref ð37Þ rc ¼ ri exp
s Et where Et and E are Young’s modulus of the material under operation temperature and room temperature respectively and p is operation pressure. JSA85 # IMechE 2005

A general autofrettage model of a thick-walled cylinder

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Table 1 Calculation parameters of 30CrNiMo8
s (MPa)

E (MPa)

A1 (MPa)

A2 (MPa)

B1


D (MPa)

E 0 (MPa)

A3 (MPa)

A4 (MPa)

B2

960.7

207 000

918.2

9157

1.0

1420

201 000

5.0

10 850

0.47

The maximum strain "max is then determined by  ref 2 r "max ¼ "s c ð38Þ ri This provides a good, initial estimate of the maximum plastic strain for the non-ideal case. An initial uniaxial test is performed to a strain level somewhat greater than "max . In practice it is advisable to run this test to a strain level slightly greater than that envisaged for any likely tube geometry to avoid the need for re-testing. This initial behaviour is then used to obtain values for parameters E , A1 , A2 , and B1 . These are then employed in selecting an appropriate autofrettage radius and pressure for the loading phase. Some iteration may be necessary to ensure that the maximum plastic strain is properly fitted by the various parameters since it is pointless to fit behaviour beyond the calculated maximum strain. Once a specific pressure has been selected, the precise maximum plastic strain at the peak of the loading cycle "
max , may be calculated. A second uniaxial test is then performed to a level of "
max in tension, followed by load reversal to define the unloading behaviour. Once again, to avoid the need for retesting, it is advisable to continue the load reversal beyond the likely equivalent strain value at the bore of the tube.

5

EXPERIMENTAL VALIDATIONS

The experimental material of the specimen is 30CrNiMo8, made in Germany. The tensile–compressive stress–strain curve of the material in this experiment is shown in Fig. 4. The parameters needed in the calculation were determined by fitting the tensile–compressive stress–strain curve using equations (1) to (4) and are listed in Table 1. The dots shown in Fig. 4 are determined by equations (1) to (4) using the data in Table 1. This shows that equations (1) to (4) can fit the stress–strain curve well. Note that there is a small difference at the two Table 2 ri (mm)

ro (mm)

19.3

43.7


c rc

¼ calculation value;

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rm c

Fig. 4

Stress–strain curve of 30CrNiMo8

junctions of the elastic–plastic curves between the actual curve and the fitting curve. To overcome this, parameters
s and
D should correspond to the values of intersection of equations (1) and (2), and equations (3) and (4) 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. 5. The calculated elastoplastic radius is smaller than the measured value and the predicted residual stresses are in good agreement with test data. This difference can be improved by adopting modified yield criterion (see reference [17]).

6

COMPARISON OF AUTOFRETTAGE MODELS

There are several autofrettage models based on different simplified material strain-hardening characteristics mentioned in section 1. They are the elastic–perfectly plastic model, the unloading

Radii and autofrettage pressure of the cylinder pa (MPa)

pacr (MPa)

rcc (mm)



rm c (mm)

rd (mm)

740

658.3

28.0

30.2

20.6

¼ experimental value.

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X P Huang

Fig. 5 Residual stress distribution

Fig. 6 Stress–strain relationship assumptions of the three models

linear strain-hardening model, the bilinear strainhardening model, the loading elastic–perfectly plastic and unloading power strain-hardening model, the loading and unloading power strain-hardening model, and the loading linear and unloading power strain-hardening model. The present model does not need the prior assumption of the material stress–strain curve type, such as the elastic–perfectly plastic or linear hardening, but depends on the tensile–compressive curve of the material. Comparing Figs 1 and 2, it can be seen that the present model is a general form of those models. It has a very strong curve-fitting ability, which can give better simulation of the material stress–strain relationship than other models, such as the elastic– perfectly plastic model and the bilinear strainhardening model, shown in Fig. 6. These models can be drawn from the present model by letting some parameters attain certain values. The

relationships between these models and the present model are listed in Table 3. The present model is a general model for it can replace the other models.

Table 3

7

CONCLUSIONS

A general autofrettage model that incorporates the material strain-hardening relationship and the Bauschinger effect, based upon the actual tensile– compressive stress–strain curve of a material, has been proposed. The model incorporates the von Mises yield criterion, an incompressible material, and plane strain. Experimental results show that the present model has a strong curve-fitting ability and the predicted residual stresses are in good agreement with test data. The required parameters are determined by fitting the actual tensile–compressive curve of the material using equations (1) to (4).

The relationship between the past models and the present model

Case

A1

A2

A3

A4

B1

B2

Comment

1 2


s
s

0 0

2
s
D  A4 "
s

0 6 0 ¼

1 1

1 1

3 4


s  A2 "s
s

6¼ 0 0


D  A4 "
s 0

6¼ 0 6¼ 0

1 1

1 6¼ 1

5

0

6¼ 0

0

6¼ 0

6¼ 1

6¼ 1

6


s  A2 "s

6¼ 0

0

6¼ 0

1

6¼ 1

Elastic–perfectly plastic model [3] Loading elastic–perfectly plastic and unloading linear strain-hardening model [4] Bilinear strain hardening model [8] Loading elastic–perfectly plastic and unloading power strain-hardening model [10] Loading and unloading power strain-hardening model [11] Loading linear and unloading power strainhardening model [12]

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A general autofrettage model of a thick-walled cylinder

Reverse yielding will take place when the autofrettage pressure pa is higher than the critical pressure pacr . Some models as special cases of the present model can be drawn by letting some parameters attain certain values. The Bauschinger effect is a function of the prior plastic strain and is incorporated via parameter
D. In order to represent properly the Bauschinger effect, the maximum tensile strain of the tensile–compressive stress–strain curve should be approximately equal to the maximum equivalent strain at the inner surface of the cylinder under autofrettage pressure. For the equations to remain valid throughout, parameters B1 ¼ 6 0 and B2 6¼ 0. ACKNOWLEDGEMENT The author greatly appreciates the help of Professor A. P. Parker who supplied many references and gave some useful suggestions and comments.

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11 Su, H. J. and Huang, X. P. Autofrettage technology research (II) (in Chinese). J. Daqing Petroleum Inst., 1995, 19(2), 78–82. 12 Huang, X. P. and Cui, W. C. Autofrettage analysis of thick-walled cylinder based on tensile–compressive curve of material. Key Engng Mater., 2004, 274–276, 1035–1040. 13 Parker, A. P. Autofrettage of open-end tubes – pressures, stresses, strains and code comparisons. Trans. ASME, J. Pressure Vessel Technol., 2001, 123, 271–281. 14 Parker, A. P., Troiano, E., Underwood, J. H., and Mossey, C. Characterization of steels using a revised kinematic hardening model incorporating Bauschinger effect. Trans. ASME, J. Pressure Vessel Technol., 2003, 125, 277–281. 15 Jia, N. W. Plastic mechanics (in Chinese), 1992 (Chongqi University Press, China). 16 Milligan, R. V., Koo, W. H., and Davidson, T. E. The Bauschinger effect on a high-strength steel. J. Basic Engng, 1966, 88, 480–488. 17 Huang, X. P. and Cui, W. Effect of Bauschinger effect and yield criterion on residual stress distribution of autofrettaged tube under different end conditions. Accepted by International Conference of Gun Tubes, Keble College Oxford, UK, 2005.

REFERENCES 1 Stacey, A. and Webster, G. A. Fatigue crack growth in autofrettaged thick-walled high pressure tube material. In High pressure in science and technology (eds C. Homan, R. K. MacCrone, and E. Walley), 1984, pp. 215–219 (Elsevier, New York). 2 Stacey, A. and Webster, G. A. Determination of residual stress distributions in autofrettaged tubing. Int. J. Pressure Vessels and Piping, 1988, 31, 205–220. 3 Hill, R. The mathematical theory of plasticity, 1950 (Oxford University Press, New York). 4 Zhang, Y. H., Huang, X. P., and Pan, B. Z. Fracture and fatigue control design in pressure vessels (in Chinese), 1977 (Petroleum Industry Press, Beijing). 5 Chen, P. C. T. Generalized plane-strain problems in an elastic–plastic thick-walled cylinder. In Transcripts of 26th Conference of Army Mathematicians, 1980, pp. 265–275. 6 Chen, P. C. T. The Bauschinger and hardening effects on residual stresses in an autofrettaged thick-walled cylinder. Trans. ASME, J. Pressure Vessel Technol., 1986, 108, 108–112. 7 Lazzarin, P. and Livieri, P. Different solution for stress and strain fields in autofrettaged thick-walled cylinders. Int. J. Pressure Vessels and Piping, 1997, 31, 231–238. 8 Xue, Q. L. The theory of autofrettage in thick-walled and its experiments study (in Chinese). China Pressure Vessel Technol., 2001, 18(3), 1–6. 9 Livieri, P. and Lazzarin, P. Autofrettaged cylindrical vessels and Bauschinger effect: an analytical frame for evaluating residual stress distributions. Trans. ASME, J. Pressure Vessel Technol., 2002, 124, 38–45. 10 Pan, B. Z., Zhu, R. D., and Su, H. J. Autofrettage theory and experimental research (I) (in Chinese). J. Daqing Petroleum Inst., 1990, 12(1), 14–16.

JSA85 # IMechE 2005

APPENDIX Notation

A1 ‚ A3 A2 ‚ A4 B1 ‚ B2 bef E1 ‚ E2 pa pacr r ri ro rc rd

constants in equations (2) and (4) coefficients in equations (2) and (4) exponential in equations (2) and (4) Bauschinger effect factor loading and unloading Young’s modulus autofrettage pressure critical autofrettage pressure variable radius bore radius outside radius elasto–plastic radius reverse yielding radius

 " "s "
s

E
R
s

coefficient of yield criterion strain loading yield strain unloading yield strain stress unloading yield stress residual stress loading yield stress

Subscripts  tangential directions r radial directions
unloading

J. Strain Analysis Vol. 40 No. 6