CHEN DTIC -LE- -,-E · SUPPLEMENTARY NOTES Presented at the Fifth Army Conference on Applied Mathematics and Computing, U.S. Military Academy, West Point, New York, 15-18 June 1987.

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TECHNICAL REPORT ARCCB-TR-88030

A SIMPLE ANALYSIS OF THE,.

SWAGE AUTOFRETTAGE PROCESS'ii

PETER C. T. CHEN

DTIC-LE- -,-E

AUG 1 9 1988 0

JULY 1988

US ARMY ARMAMENT RESEARCH,DEVELOPMENT AND ENGINEERING CENTER

CLOSE COMBAT ARMAMENTS CENTERBEN]T LABORATORIES

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A SIMPLE ANALYSIS OF THE SWAGE AUTOFRETTAGE FPROCESSFinal

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Peter C. T. Chen

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IS. SUPPLEMENTARY NOTES

Presented at the Fifth Army Conference on Applied Mathematics and Computing,U.S. Military Academy, West Point, New York, 15-18 June 1987.

Published in Proceedings of the Conference.

19. KEY WORDS (Continue on reveree aide If neceesezy and Identify by block number)

Gur: Tube SwagingAutofrettage PlasticityResidual Stress Bauschinger Effect . ,,.

Hardening

2 ABST"RACT (lCou 1aue e wreso aiv ff ne.weary ad idenjlify by block number)

Many solutions have been reported for the hydraulic autofrettage process. In

this report a simple analysis of the swage autofrettage process is pres2nted.The contact pressure at different locations is determined as a function ofinterference. The deformation and stress distribution during autofrettaoe is

obtained. At the end of the autofrettage process, the permanent bore enlarge-ment and residual stresses are calculated. Numerical results are rresentedi in

graphical form.

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TABLE OF CONTENTSPage

INTRODUCTION I

ELASTIC SWAGING 1

SWAGING BEYOND THE ELASTIC LIMIT 3

UNLOADING ANALYSIS 4

NUMERICAL RESULTS AND DISCUSSIONS 5

REFERENCES 7 -

LIST OF ILLUSTRATIONS

1. Contact pressure and hoop stress at the interface as functions of 8interference for a section in zone I.

2. Contact pressure and hoop stress at the interface as functions of 9interference for a section in zone 2.

3. Contact pressure and hoop stress at the interface as functions of 10interference for a section in zone 3.

4. The hoop stress distributions at three sections. 11

5. The distributions of residual hoop stresses at three sections. 12

6. The distributions of residual displacements at three sections. 13

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'4 ' .. ." :

* . ....... a.. ~ *C~ >%~ ~ ~. ~ .~ V * ~S ~ V~ ~ .~., am. a ~ a..

INTRODUCTION

To increase the maximum pressure a cylinder can contain without plastic

deformation and to enhance its fatigue life, residual stresses are often pro-

duced in cylinders through autofrettage (ref 1). Many solutions have been

reported for the hydraulic autofrettage process (refs 2-6). The thick-walled

cylinders were subjected to uniform internal pressure of sufficient magnitude to

cause plastic deformation and then the pressure was removed.

A more economical way of producing residual stresses in thick-walled cylin-

ders is the swage autofrettage process. This process is carried out by a swage,

the diameter of which is greater than the inner diameter of the cylinder. This

swage is driven through the cylinder from one end to the other. A rigorous

analysis of this process is difficult. In this report a simple analysis of the

swage autofrettage process is presented. The swage mandrel and the cylinder are

made of tungsten carbide and steel, respectively. A two-dimensional plane-

strain analysis is used to determine the contact pressure at different locations

of the cylinder as a function of interference. The deformation and stress

distribution during autofrettage are obtained. At the end of the autofrettage

process, the permanent bore enlargement and residual stresses are calculated.

ELASTIC SWAGING

The swage mandrel is assumed to be a short cylindrical bar driven through a

long thick-walled cylinder from one end to the other. The diameter of the

mandrel (2c) is a constant, but the inner and outer diameters (2a and 2b) of the

tube are variables. When the difference between c and a is positive, we have

interference I. For small values of interference, the stress state in the

References are listed at the end of this report.

. o, ,'e , ,, '' , .', w 1 1 ', ' ' "'' " .',,, ' e,,''.'

swaging assembly is elastic. The stresses and displacement in the tube are

r a ... T.... a 2 a

oe - a2 /b2 b2 r2

u1 V2a a2

r 1 - a2/bi [(1+V) r + (1-v-2') ] (1)

and in the mandrel

r = o = -p

u/r = -(1-l-2v,2)p/El (2)

where E, v, and El, vI arc the material constants of the tube and mandrel,

respectively. At the interface, ua (tube) - ua (mandrel) = I by the com-

patibility requirement. The interference pressure p is a function of the inter-

ference I given by

El a2 a2 a2p= (1 - )/[(1+) + (1-v-202 ) 6 + (1-VI-2v1 2)(1 -')+)E/E 1 ] (3)

For sufficiently large values of the interference, the stresses in the tube

reach the yield limit. Assuming that Tresca's yield condition governs the

behavior of the material, the tube first becomes plastic at the interference

when the stresses satisfy a@ - ar = ao, where o is the initial tensile yield

stress. The solution for the critical interference pressure to cause incipient

plastic deformation is

P* : 0o (I - a2 /b2 ) (4)

and it follows from Eq. (3) that the interference for the onset of plastic flow

is .

0o a a- a21* [- ~ t(+) + (I-v-2 2 ) 22 (l-vl-2vl2)(1 - 6 )E/EI] (5)

which reduces to 1* = (1-0,) a 0o/E for the special case (E1 = E, = ).

2

N

SWAGING BEYOND THE ELASTIC LIMIT

For values of interference larger than that given by Eq. (5), a plastic

zone forms in the tube, so that for a r 4 p the tube is plastic, while for p

r 4 b the tube material is still in an elastic state. The elastic-plastic

interface raaius p is a function of the interference I.

We assume that the steel tube is elastically-ideally plastic, obeying

Tresca's yield criterion and the associated flow theory, but the tungsten car-

bide mandrel is elastic. This assumption is justified because the strength

ratio of tungsten carbide to steel is about three. For loading beyond theelastic limit, the closed-form solution has been found by Koiter (ref 2). The

expressions for the stresses and displacement in the tube arear/a o p2

() 1 + log r in (a r 4 p) (6)r/p 2 r

rlo

a(6; ) , in (p < r 4b) (7)ae/o

Er + 2E ~ - (1-2v)(1+v) - + (1-L,) (8)O rr 2

where the elastic-plastic interface p is related to the internal pressure p by

P/ao = %(1 - p2/b2 ) + log(p/a) (9)

For swaging beyond the elastic limit, the compatibility requires ua (tube) - ua

(mandrel) =I at the interface, i.e.,

E I = ( _ V 2) e - [ (1 -2) (l + ) - (1 - Vl -2 Vlz ) ] (1 0 )

CF a 1~2 a2 ao El-t(1i)1

Equations (9) and (10) give us a parametric representation of relating p to I

through the parameter p. The contact pressure at different locations can thus

be determined as a function of the interference I.

3

UNLOADING ANALYSIS

After swaging, the permanent bore enlargement and residual stresses can be

calculated by an unloading analysis. Let a double prime denote a component in -

the residual state, i.e., ao" = ae + 06'. Assuming elastic. unloading, the solu-

tion is given by

Oro

- . .. .. [ ± - - - ]' 1

06 b2 /a2 - 1 - r2

E u'/r - [(1-V) + (1+v)b 2/r2]p/(b2/aa-1) (12)

In a recent paper (ref 6), this author presented a more rigorous elastic-

plastic unloading analysis based on a new theoretical model considering the

Bauschinger and hardening effects during unloading. This mode is a very good S

representation for the material behavior of the high strength steel used in gun

tubes (ref 7). Taking into account the Bauschinger effect (f) and the strain- P

hardening during unloading (m'), we have obtained a closed-form solution. On

unloading, yielding will occur for a 4 r < p' with p' < p. The stresses in the

reverse yielding zone (a 4 r < p') are given by -. '

0r'/Oo = P/Co - V2'(1+f)(p'/a) 2 (j-a2/r2) - (1-A2')(l+f)log(r/a) (13)

0@'/ao = Or'/0o - (1+f)[1 + 2'(p'2 /r2 -1)] (14)

where(1-in')

'= (1-m')/m' + 2 (15) 2' = m'g1 '/(1-m') (15) .

The stresses in the elastic zone (p' 4 r 4 b) are

Or'/0o= %(l+f)[± (p'/r)2 - (p'/b) ] (16) •

The displacement for the entire tube (a < r < b) is

(Eao) u'/r = (1-2v)(l+v)(ar'/oo) - (1-v2)(1+f)(p'/r)2 (17)

4

Law I

The residual stresses and displacement are found by addition

ar" = ar + ar' , s" = Ce + ao' and u" = u + u' (18)

NUMERICAL RESULTS AND DISCUSSION "it

The material constants used in the calculations are E = 206.84 GPa, V =

0.3, oo = 1.29 GPa, m' = 0.3 for the high strength steel, and El = 610.19 GPa,

V1 = 0.258 for the tungsten carbide mandrel. The radius of the mandrel is a

constant c = 58.42 mm, but the thickness of the tube varies along the axial

direction with the inner radius (a) increasing slightly and the external radius 3

(b) tapering more rapidly. The values of a and b at four typical sections are

aj = 56.96, 57.82, 57.99, 58.63 mm and bj = 157.50, 106.75, 83.00, 83.00 mm, for

j = 1,2,3,4, respectively. The corresponding values of wall ratio are bj/aj =

2.765, 1.846, 1.431, 1.42 at four sections. The interference during swaging (I)

is the positive difference between c and a. The values of I at four sections

are Ij = 1.46, 0.60, 0.43, -0.21 mm for j = 1,2,3,4. The negative value of 14

means that there is no contact between the mandrel and the tube. For the posi-

tive values of interference, the contact pressure and the stress distribution

during swaging can be obtained using the methods presented previously in this

report. The information after swaging can be obtained by the unloading analysis

also presented previously.

The numerical results are presented in terms of the dimensionless quan-

tities defined by

r= r/a , p =p/ao , a@ = o/Co

I= (E/ao)I/a , u = (E/ao)u/a , etc. (19)

The contact pressure (p) and hoop stress (Or) at the interface are presented as

functions of the interference (I) in Figures 1, 2, and 3 for wall ratios b/a =

5p

2.765, 1.846, 1.431, respectively. The results for swaging within and beyond

the elastic limit are included. The pressure is a monotonous increasing func-

tion of the interference, but the maximum value of hoop stress occurs at the

onset of plastic flow as shown in the dotted curves. Initial yielding occurs at

I* = 0.774, 0.799, 0.830, and fully plastic flow occurs at I = 6.638, 2.909, 00

1.751 for three different wall ratios, respectively. The actual values of

interference (I) at the three chosen sections are II = 4.10, 12 = 1.66, 13 '

1.19. These values indicate that swaging is partially plastic at these sectionsS

in zones 1, 2, and 3. The corresponding locations of elastic-plastic boundary

are given by p/a = 2.2001, 1.4196, 1.19205, and the amounts of overstrain are

68, 49.6, and 44.6 percent, respectively. Also shown in Figures 1, 2, and 3 are

the values of contact pressure (p = 0.972, 0.555, 0.671) and the hoop stress at

the interface o0 = 1 - p. The distributions of hoop stresses during swaging are

shown in Figure 4 for typical sections in three zones. The maximum value of

hoop stress occurs at the elastic-plastic boundary. The information for the

displacement and stresses after swaging can be obtained by an unloading analy-

sis. The distributions of residual hoop stresses are shown in Figure 5 for the

chosen sections in three zones. Elastic unloading analysis is justified in zone

3, but reverse yieldings occur in zones 1 and 2 with p'/a = 1.305 and 1.014,

respectively. Finally, the distributions of residual displacements (u") at

typical sections in three zones are presented in Figure 6. Also shown in this

figure are the experimental data at the bore. The agreement between the calcu-

lated and experimental data is excellent in zone 1, but not so good in zones 2

and 3. This suggests that a more refined analysis is needed for sections with

smaller wall ratios. An investigation based on the finite element method is

being conducted and the results will be reported in the near future.

6

REFERENCES

1. Davidson, T.E. and Kendall, D.P., "The Design of Pressure Vessels for Veryl

High Pressure Operation," Mechanical Behavior of Materials Under Pressure,

(H.Ll.P. Pugh, ed.), Elsevier Co., 1970.

2. Hill, R., The Mathematical Theory of Plasticity, Oxford University Press,

London, 1950.

3. Bland, D.R., "Elastoplastic Thick-Walled Tubes of Work-Hardening Materials

Subject to Internal and External Pressures and Temperature Gradients,"

Journal of Mechanics and Physics of Solids, Vol. 4, 1956, pp. 209-229.

4. Franklin, G.J. and Morrison, J.L.M., "Autofrettage of Cylinders: Prediction

of Pressure/External Expansion Curves and Calculation of Residual Stresses,"

Proceedings of the Institute of Mechanical Engineers, Vol. 174, 1960,

pp. 947-974.

5. Chen, P.C.T., "The Finite Element Analysis of Elastic-Plastic Thick-Walled

Tubes," Proceedings of Army Symposium on Solid Mechanics, The Role of

Mechanics in Design-Ballistic Problems, 1972, pp. 243-253. ;

6. Chen, P.C.T., "The Bauschinger and Hardening Effect on Residual Stresses in

an Autofrettaged Thick-Walled Cylinder," Journal of Pressure Vessel

Technology, Vol. 108, February 1985, pp. 108-112.

7. Milligan, R.V., Koo, W.H., and Davidson, T.E., "TheBauschinger Effect in a

High Strength Steel," Journal of Basic Engineering, Vol. 88, June 1966, pp.

480-488.

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