Theoretical and finite-element modeling of autofrettage process in strain-hardening thick-walled...

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International Journal of Pressure Vessels and Piping 84 (2007) 310–319

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Theoretical and finite-element modeling of autofrettageprocess in strain-hardening thick-walled cylinders

M.H. Hojjati�, A. Hassani

Faculty of Mechanical Engineering, University of Mazandaran, Babol, Iran

Received 21 May 2006; accepted 11 October 2006

Abstract

The optimum autofrettage pressure and the optimum radius of the elastic–plastic boundary of strain-hardening cylinders in plane

strain and plane stress have been studied theoretically and by finite-element modeling. Equivalent von-Mises stress is used as yield

criterion. Comparison of the results of the two methods shows good agreement. Although there is no explicit expression for the optimum

autofrettage pressure in plane stress, the equation for plane strain can be used with good accuracy. It has also been observed that the

optimum autofrettage pressure is not a constant value but depends on working pressure.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Autofrettage; Thick-walled cylinder; Strain-hardening material; Plane strain; Plane stress

1. Introduction

Pressure vessels are widely used in the nuclear, chemical,and military industries and also for fluid transmission andstorage applications [1]. In recent decades, various methodshave been proposed for strengthening the vessels. Theautofrettage process is possibly the most well knownmethod. Autofrettage is a process in which the cylinder issubjected to a certain amount of pre-internal pressure sothat its wall becomes partially plastic. The pressure is thenreleased and the residual stresses lead to a decrease in themaximum von-Mises stress in the working loading stage.This means an increase in the pressure capacity of thecylinder. A key problem in the analysis of autofrettageprocess is to determine the optimum autofrettage pressureand the corresponding radius of the elastic–plastic bound-ary where the maximum equivalent von-Mises stress in thecylinder becomes a minimum.

Brownell and Young [2] proposed an iterative calculationmethod to determine the optimum radius of the elastic–plasticboundary, which was tedious to use. Franklin and Morrison[3] performed an analysis of residual stresses and deformation

e front matter r 2006 Elsevier Ltd. All rights reserved.

vp.2006.10.004

ing author. Fax: +98 111 3234201.

ess: [email protected] (M.H. Hojjati).

of autofrettaged cylinders. Blazinski [4] studied the auto-frettage of cylinders made of elastic–perfectly plastic material.Chen [5] also studied the stress analysis and deformation ofautofrettaged cylinders. Kong [6] applied a graphical methodto determine the optimum radius of the elastic–plasticboundary which was inaccurate and tedious. Gao [7]obtained the elasto-plastic solution of a strain-hardeningcylinder subjected to internal pressure in plane stress byassuming an isotropic hardening material model (with noBauschinger effect). Avitzur [8] studied the stress distributionafter depressurization of a thick-walled cylinder made ofelastic–perfectly plastic material based on the von-Mises andTresca yield criteria in plane stress and plane strain. Zhu andYang [9] determined the optimum radius of the elastic–plasticboundary and the optimum autofrettage pressure of acylinder made of elastic–perfectly plastic material in planestrain. Gao [10] obtained the stress, strain and displacementcomponents of a strain-hardening cylinder subjected topressures greater than the elastic-limit pressure in planestrain. Majzoobi et al. [1] investigated the autofrettageprocess for strain-hardening cylinders in plane stress using acommercial finite element code. They showed that thenumber of autofrettage stages has no influence on optimiza-tion of the stress distribution [1]. Majzoobi et al. [11] by usingfinite-element modeling also showed that the effect of the

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ARTICLE IN PRESSM.H. Hojjati, A. Hassani / International Journal of Pressure Vessels and Piping 84 (2007) 310–319 311

autofrettage process on optimization of compound cylindersis negligible. Chen [12] studied influence of Bauschingereffects and hardening on residual stresses of an autofrettagedcylinder. Parker and Underwood [13] investigated influenceof Bauschinger effect on stresses and fatigue lifetimes inautofrettaged cylinders.

In this paper, the optimum radius of elastic–plasticboundary and the optimum autofrettage pressure of astrain-hardening cylinder based on the von-Mises yieldcriterion in plane-stress and plane-strain conditions aredetermined by assuming an isotropic material model.Finite-element analysis is performed to model the cases.The results of theoretical calculations are compared withfinite-element modeling.

2. Theoretical analysis of autofrettage process

Since the autofrettage pressure is greater than the elastic-limit pressure (pressure at which yielding commences atinner surface), this pressure causes plastic deformation inthe wall of cylinder up to a certain radius. So, thegoverning equations for material behavior in the elasticand plastic zones are needed. Gao [7,10] obtained the stresscomponents in a strain-hardening cylinder in plane-stressand plane-strain conditions by assuming a homogeneousand isotropic hardening model which are summarized here.

2.1. Stress distribution in plane-strain and plane-stress

conditions

The stress–strain relationship of a strain-hardeningmaterial shown in Fig. 1, is assumed as

si ¼E�i sipsy;

A�ni si4sy;

((1)

Fig. 1. The stress–strain relation [7].

where si and ei are the equivalent stress and strain,respectively; A is a material constant equal to A ¼ sy=�n

y;and n is the strain-hardening exponent of the material(0pno1).This material model is for loading; therefore unloading

and reloading is treated elastically. Radial and hoopcomponents of stress in plane strain and plane stress forelastic response can be calculated from the followingequations also, known as Lame’s equations [8]:

srðrÞ ¼ �ðb=rÞ2 � 1� �

pi

ðb=aÞ2 � 1; syðrÞ ¼

ðb=rÞ2 þ 1� �

pi

ðb=aÞ2 � 1. (2)

It can be assumed that the elastic zone of vessel is acylinder of inner radius r and outer radius b which issubjected to internal pressure pr; therefore, using Eq. (2),the stress components in the elastic zone of a cylinder inplane strain and plane stress are [7,10]

sy ¼prr

2

b2� r2

1þb2

r2

� �; sr ¼

prr2

b2� r2

1�b2

r2

� �, (3)

where pr in plane-strain and plane-stress conditions isobtained from Eqs. (4) and (5), respectively, [7,10]:

pr ¼syffiffiffi3p 1�

r2

b2

� �, (4)

pr ¼sy b2

� r2� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3b4þ r4

q . (5)

The relation between internal pressure and the radius ofthe elastic–plastic boundary in plane-strain and plane-stress conditions is determined by Eqs. (6) and (7),respectively, [7,10]:

pi ¼syffiffiffi3p 1�

r2

b2

� �þ

1

n

r2n

a2n� 1

� �� , (6)

pisy¼ 2ffiffi

3p

cosðfrþfnÞ

cosðfaþfnÞ

ð3n2þnÞ=ð3n2þ1Þ

� expffiffi3p

nðn�1Þ3n2þ1

ðfa � frÞ

h icos fa;

r ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisinðfaþp=6Þsinðfrþp=6Þ

qcosðfrþfnÞ

cosðfaþfnÞ

2n=ð3n2þ1Þ

� expffiffi3p

21�n2

3n2þ1ðfr � faÞ

h i;

fr ¼ cos�1ffiffi3p

2ðb2�r2Þffiffiffiffiffiffiffiffiffiffiffiffiffið3b4þr4Þp ;

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

(7)

where fn is defined as cos�1ðffiffiffi3p

n=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3n2 þ 1p

Þ [7].The three unknowns, fa, fr and r in Eq. (7) are

calculated in terms of the internal pressure pi. By

Table 1

Composition of aluminum alloy (wt%) [1]

Al Zn Cu Si Fe Ni Mn S

81.72 6.63 3.66 4.82 2.46 0.21 0.35 0.15

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Table 2

Characteristics of geometry, mechanical properties and boundary conditions

Model Inner radius a

(mm)

Outer radius b

(mm)

Elastic

modulus E

(MPa)

Poisson’ s

ratio uYield sress sy(MPa)

Strain

hardening

exponent n

Boundary

conditions

PV1 6.5 14.3 75 000 0.33 120 0.24 Plane strain

PV2 6.5 14.3 75 000 0.33 120 0.24 Plane stress

PV3 6.5 14.3 75 000 0.33 120 0 Plane strain

PV4 6.5 14.3 75 000 0.33 120 0 Plane stress

Table 3

The results of theoretical analysis (PV1, PV2, PV3 and PV4 models)

Model pw (MPa) pe (MPa) py (MPa) ropt (mm) paopt

(MPa)

PV1 48 54.97 132.8 9.816 99.8

PV2 48 54.58 128.6 9.94 101

PV3 48 54.97 109.25 9.191 88.66

PV4 48 54.58 105.09 9.17 88

Fig. 2. Maximum von-Mises stress versus autofrettage pressure for a

working pressure of 48MPa (PV2 model).

Fig. 3. Maximum von-Mises stress versus radius of elastic–plastic for a


Fig. 4. Maximum von-Mises stress versus autofrettage pressure for a


M.H. Hojjati, A. Hassani / International Journal of Pressure Vessels and Piping 84 (2007) 310–319312

substituting a and b, in turn for the radius of theelastic–plastic border, r, the elastic-limit pressure pe(pressure at which yielding commences at inner surface)and the plastic-limit pressure py (pressure at whichplasticity has spread throughout the cylinder) are obtained,respectively. Eqs. (8) and (9) present these two limitpressures in plane strain while Eqs. (10) and (11) give thelimit pressures in plane stress:

pe ¼syffiffiffi3p 1�

a2

b2

� �, (8)

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Fig. 5. Maximum von-Mises stress versus radius of elastic–plastic for a


Table 4

The results of finite-element optimization (PV1 model)

Autofrettage pressure

(MPa)

Max. von-

Mises stress

(MPa)

Stage 1 60 96.289

Stage 2 100.99 69.894

Stage 3* 99.609 69.83

Stage 4 102.72 69.843

Fig. 6. Maximum von-Mises stress versus

M.H. Hojjati, A. Hassani / International Journal of Pressure Vessels and Piping 84 (2007) 310–319 313

py ¼syffiffiffi3p

1

n

b2n

a2n� 1

� �, (9)

pe

sy¼

k2� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3k4þ 1

� �q , (10)

py ¼ sy2ffiffiffi3p

cosððp=2Þ þ fnÞ

cosðfa þ fnÞ

ð3n2þnÞ=ð3n2þ1Þ

� exp

ffiffiffi3p

nðn� 1Þ

3n2 þ 1fa �

p2

�� cos fa, ð11Þ

where fa is the root of the following equation:ffiffiffi3p

2k2¼ sinðfa þ p=6Þ

cosððp=2Þ þ fnÞ

cosðfa þ fnÞ

4n=ð3n2þ1Þ

� expffiffiffi3p 1� n2

3n2 þ 1

p2� fa

�� . ð12Þ

2.2. Optimum radius of elastic–plastic boundary and

optimum autofrettage pressure

The optimum radius of the elastic/plastic interface andthe corresponding autofrettage pressure are defined withthe condition of minimizing the maximum equivalent von-Mises stress. The position of maximum von Mises stress isat the elastic–plastic boundary [9]; therefore, the optimumradius of the elastic–plastic interface should be calculatedwith the condition of minimizing von Mises stress at r ¼ r.Thus, in order to determine stress components resultingform the autofrettage pressure at the elastic/plastic

autofrettage pressure in PV1 model.

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Fig. 7. Variation of von-Mises stress versus the radius of cylinder (pi ¼ 99.609MPa), PV1 model.

Table 5

The results of finite-element optimization (PV2 model)

Autofrettage pressure

(MPa)

Max. von-

Mises stress

(MPa)

Stage 1 60 96.377

Stage 2 101 70.022

Stage 3 99.231 69.982

Stage 4* 100.74 69.926

Fig. 8. Maximum von-Mises stress versus autofrettage pre


boundary ðr ¼ rÞ, one can apply Eq. (3) which is correctonly in elastic zone of autofrettaged cylinder.The residual stresses are added to the corresponding

working stresses to give the total stresses in order toestablish the von-Mises stress. The residual stresses areobtained by subtracting the elastic recovery stresses fromthe stresses resulting from the autofrettage pressure [8].In plane strain, the stresses resulting from the auto-

frettage pressure at r ¼ r are determined by substitutingpi ¼ pa and r ¼ r into Eq. (3). Elastic recovery stress

ssure for a working pressure of 48MPa in PV2 model.

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Table 6

Comparison of finite-element optimization for PV1, PV2, PV3 and PV4

models

Model Working

pressure

(MPa)

Optimum radius

of elastic–plastic

(mm)

Optimum

autofrettage

pressure

(MPa)

Max.

von-

Mises

stress

(MPa)

PV1 48 9.7663 99.609 69.83

PV2 48 10.06 100.74 69.926

PV3 48 — 89.202 75.476

PV4 48 — 88.6 76


components at r ¼ r are determined by substituting pi ¼ pa

and r ¼ r into Eq. (2); therefore, the residual stresses atr ¼ r are

sresr ðrÞ ¼syffiffiffi3p�b2þ r2

� �r2n� a2n� a2 r=a

� �2nþ a2

�r2n b2

� a2� � ,

(13)

sresy ðrÞ ¼syffiffiffi3p

b2þ r2

� �r2n� a2n� a2 r=a

� �2nþ a2

�r2n b2

� a2� � .

(14)

The components of radial stress spwr and hoop stress spwy(from the working pressure) are determined by substitutingpi ¼ pw into the Eq. (2). Radial and hoop components oftotal stress are then

stotr jr¼r ¼ sresr þ spwr ¼1

3

�b2þ r2

� �r2 b2

� a2� �

n

�ffiffiffi3p

syr2n�ffiffiffi3p

sya2n�ffiffiffi3p

sya2 ra

�2n�

þffiffiffi3p

sya2 þ 3a2npw

�, ð15Þ

stoty jr¼r ¼ sresy þ spwy ¼1

3

b2þ r2

� �r2 b2

� a2� �

n

�ffiffiffi3p

syr2n�ffiffiffi3p

sya2n�ffiffiffi3p

sya2 ra

�2n�

þffiffiffi3p

sya2 þ 3a2npw

�. ð16Þ

The von-Mises stress isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2sr � syð Þ

2þ sy � szð Þ

2þ sz � srð Þ

2� �r

¼ svoneq . (17)

By substituting the axial stress in plane strain ðsz ¼12ðsy þ srÞÞ into Eq. (17), the von-Mises stress is obtained as

svoneq ¼

ffiffiffi3p

2sy � srð Þ. (18)

Thus, the von-Mises stress at r ¼ r is

svoneq jr¼r ¼

ffiffiffi3p

syn r2 � a2� �

þffiffiffi3p

sya2 1� r=a� �2n

�þ 3a2npw

�b2ffiffiffi

3p

r2n b2 � a2� � .

(19)

The optimum radius of the elastic–plastic junction is theradius where the von-Mises stress is a minimum:

dsvoneq

dr¼ 0. (20)

Substituting Eq. (19) into Eq. (20) and solving for theoptimum radius of the elastic–plastic interface leads to

rvonopt ¼ a 1þ

ffiffiffi3p

npw

sy 1� nð Þ

� �1=2n

. (21)

Substituting Eq. (21) into Eq. (6), gives the optimumautofrettage pressure:

pvona opt ¼

ffiffiffi3p

sy �na2 1�ffiffi3p

npwsy n�1ð Þ

�1=n

þ b2n �ffiffi3p

pwsy n�1ð Þ

þ 1 ��

3b2n.

(22)

Eq. (22) can be normalized as the following:

pvona opt

sy¼

ffiffiffi3p

3�

a

b

�21�

ffiffiffi3p

n

n� 1ð Þ

pw

sy

� �� 1=n

�þ 1�

ffiffiffi3p

n� 1ð Þ

pw

sy

� �� . ð23Þ

Eq. (23) shows the dependency of the optimum auto-frettage pressure on the ratio of radii and also strain-hardening exponent of material, i.e. n.In plane stress, to calculate the radius of the elastic–

plastic junction at various autofrettage pressures, the set ofEq. (7) is available. Due to the lack of an explicitexpression between elastic/plastic interface radius andautofrettage pressure, Eq. (7) have to be solved withiterative methods. The value of autofrettage pressure isvaried incrementally from the elastic-limit pressure to theplastic-limit pressure in Eq. (7) from which the correspond-ing elastic/plastic interface radius is obtained numerically.Then, using the calculated radii, the equivalent von-Misesstresses are established. In order to determine the optimumradius of the elastic–plastic interface and the optimumautofrettage pressure, the maximum von-Mises stress atr ¼ r are plotted versus r or pa. The position of theminimum von-Mises stress on the curves indicates theoptimum autofrettage pressure/radius of elastic–plasticinterface.In all theoretical calculations, the material and geometry

specifications of an aluminum alloy (Table 1) cylindricalvessel as summarized in Table 2 are used.

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Fig. 9. Maximum von-Mises stress versus autofrettage pressure for a working pressure of 48MPa in PV3 model.

Fig. 10. Maximum von-Mises stress versus autofrettage pressure for a working pressure of 48MPa in PV4 model.


3. Finite-element modeling

Commercially available software has been used for finite-element modeling of the autofrettaged vessel [14]. Theelement PLANE42 with the capability of elastic and plasticmaterial modeling has been used. To get reasonable accuracy,110 elements are used in the radial direction in both plane-strain and plane-stress cases. In plane strain, the models PV1and PV3 are used (see Table 2). To create the plane-strain

case, the upper and lower surfaces of the cylinder areconstrained in the Y direction. In plane stress, the modelsPV2 and PV4 are used (see Table 2). To create plane stress,the upper and lower surfaces should be stress free.

4. Results and discussion

All the theoretical and finite-element results reportedhere are based on using the material properties summarized

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Fig. 11. Nondimensional optimum autofrettage pressure versus strain-hardening exponent for different Ks, in PV1 model, pw ¼ 48MPa.


in Table 2. A working pressure of 48MPa is used in bothplane-strain and plane-stress cases. The geometry of thecylinder is as recorded in Table 2. Models PV1 and PV3represent the plane-strain condition while models PV2 andPV4 represent plane stress.

Table 3 records the theoretical results for elastic pressurelimit, plastic pressure limit, optimum radius of elastic–plasticinterface and optimum autofrettage pressure for plane-strainand plane-stress cases at a working pressure of 48MPa. Inplane stress, Figs. 2 and 3 show the maximum von-Misesstress versus the autofrettage pressure and the radius of theelastic–plastic interface, respectively, for model PV2. Figs. 4and 5 show the same results for PV4 model. In plane strain,according to Table 3, the elastic-limit pressure for PV1 modelequals 54.96MPa and the plastic-limit pressure equals132MPa; therefore, the autofrettage pressure should be variedfrom 54.96 to 132MPa. However, in order to study the effectof autofrettage pressures less than pe, the autofrettage pressurehas been varied from 35 to 132MPa in finite-elementmodeling. The stages of the optimization process are givenin Table 4. The optimum autofrettage pressure of 99.609MPais extracted from this table. The von-Mises stress is thenplotted versus autofrettage pressure in Fig. 6.

To obtain the radius of the elastic–plastic boun-dary, which corresponds to the optimum autofrettagepressure of 99.609MPa, the cylinder subjected to aninternal pressure of 99.609MPa has been analyzed andthe radius at which the von-Mises stress is 120MPa (i.e. theyield strength) is found. The variation of von-Mises stressversus the radius of the cylinder is shown in Fig. 7. Thisfigure shows that the radius of 9.7663mm corresponds to avon-Mises stress of 120MPa. This is the optimum radius ofthe elastic–plastic interface by finite-element modeling forthis case.

In plane stress, the stages of the finite element optimiza-tion process for PV2 model are presented in Table 5. Theanalysis is repeated at different autofrettage pressures anda working pressure of 48MPa. The maximum von-Misesstress is then plotted versus autofrettage pressure as shownin Fig. 8 and the optimum value of autofrettage pressure isfound to be 100.74MPa. To obtain the radius of theelastic–plastic interface of PV2 model which corresponds tothis optimum autofrettage pressure, a cylinder which issubjected to internal pressures of 100.74MPa is analyzedand then the radius at which the von Mises stress is120MPa is found to be 10.06mm. The results of the finite-element simulations in plane strain and plane stress arecompared in Table 6.In order to show the influence of strain-hardening

exponent of material (n) on the optimum autofrettagepressure in plane-strain and plane-stress conditions, themaximum von-Mises stress results from finite-elementmodeling versus the autofrettage pressure in PV3 andPV4 models are plotted in Figs. 9 and 10, respectively.Fig. 11 shows the simultaneous influence of strain-hard-ening exponent (n) and the ratio of outer to inner radii K

on the nondimensional optimum autofrettage pressure (i.e.Pa opt/sy in Eq. (23)) at a working pressure of 48MPa. Thisfigure clearly demonstrates that increasing the strain hard-ening exponent n and the ratio of outer to inner radii K leadsto an increase in the optimum autofrettage pressure.

5. Comparison of theoretical and finite-element results

Tables 7–10 compare the results of theoretical analysiswith those of finite-element analysis in PV1, PV2, PV3 andPV4 models, respectively. These tables clearly show very

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Table 7

Comparison of theoretical and finite-element results in plane-strain

condition (PV1 model)

ropt (mm) popt (MPa)

Theoretical analysis 9.8158 99.8

FEA 9.7663 99.609

Difference 0.5 0.20%

Table 8

Comparison of theoretical and finite-element results in plane-stress


ropt (mm) popt (MPa)

Theoretical through a computer code 9.94 101

FEA 10.06 100.74

Difference 1.2 0.3%

Table 9

Comparison of theoretical and finite-element results in plane-strain


popt (MPa)

Theoretical analysis 88.66

FEA 89.202

Difference 0.6%

Table 10

Comparison of theoretical and finite-element results in plane-stress


popt (MPa)

Theoretical analysis 88

FEA 88.6

Difference 0.7%

Table 11

Comparison of optimum autofrettage pressure in plane-strain and plane-

stress cases for PV1 and PV2 models ðn ¼ 0:24Þ

pw (MPa) paopt in plane

stress (MPa)

paopt in plane

strain (MPa)

Difference

(%)

10 65 65.2 0.3

20 76 75 1.3

30 85 84.34 0.8

40 94 93.15 0.9

48 101 99.8 1.2

50 103 101.4 1.58

55 106 105.3 0.6

Table 12

Comparison of optimum autofrettage pressure in plane-strain and plane-

stress cases for PV3 and PV4 models ðn ¼ 0:0Þ

pw (MPa) paopt in plane

stress (MPa)

paopt in plane

strain (MPa)

Difference

(%)

10 62 62.74 1.19

20 70 70.17 0.24

30 77 77.21 0.27

40 83 83.78 0.94

48 88 88.66 0.75

50 89 89.82 0.92

55 92 92.62 0.67


good agreement between the two for all cases with amaximum difference of 1.2%.

Table 11 (for n ¼ 0:24) and Table 12 (for n ¼ 0:0)compare the optimum autofrettage pressure in plane strainand plane stress at various working pressures. These tablesshow that the maximum difference between optimumautofrettage pressure in plane-stress and plane-strain casesis about 1.58%. This indicates that although there is noexplicit functional relationship for popt in plane stress, theequation of optimum autofrettage pressure for plane straincan be used for this case with good accuracy.

6. Conclusions

Autofrettage of a cylinder made of strain-hardeningmaterial based on the von-Mises yield criterion has beenstudied theoretically for plane-strain and plane-stressconditions and corresponding equations for optimumautofrettage pressure and optimum elastic–plastic bound-ary have been developed. The process has also been

simulated by a commercial finite-element code. The resultsof finite-element modeling are in good agreement withthose of the theoretical analysis. Comparison of optimumautofrettage pressure in plane-stress and plane-strain casesfor various working pressures reveals that the difference ofoptimum autofrettage pressure in these two differentconditions is negligible. This indicates that existing explicitrelationships for calculating the optimum autofrettagepressure for plane-strain condition can be used for planestress with good accuracy. It has also been observed thatincreasing the strain hardening exponent n and the ratio ofouter to inner radii K leads to an increase in the optimumautofrettage pressure.

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Theoretical and finite-element modeling of autofrettage process in strain-hardening thick-walled...

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