AD-A24 9 491

TECHNICAL REPORT ARCCB-TR-92010

ANALYTICAL SOLUTION OFELASTIC-PLASTIC THICK-WALLED

CYLINDERS WITH GENERAL HARDENING

PETER C. T. CHEN

APR2 9 ' I9

MARCH 1992

US ARMY ARMAMENT RESEARCH.DEVELOPMENT AND ENGINEERING CENTER

CLOSE COMBAT ARMAMENTS CENTERBEN9T LABORATORIES

WATERVLIET, N.Y. 12189-4050

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4. TITLE AND SUBTITLE S. FUNDING NUMBERSANALYTICAL SOLUTION OF ELASTIC-PLASTICTHICK-WALLED CYLINDERS WITH GENERAL HARDENING AMCMS: 6111.02.H610.0

PRON: IA11ZICANMBJL AUTHOR(S)

Peter C. T. Chen

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U.S. Army ARDEC ARCCB-TR-92010Benet Laboratories, SMCAR-CCB-TLWatervliet, NY 12189-4050

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U.S. Army ARDECClose Combat Armaments CenterPicatinny Arsenal, NJ 07806-5000

11. SUPPLEMENTARY NOTESPresented at the Ninth Army Conference on Applied Mathematics and Computing,University of Minnesota, Minneapolis, MN, 18-21 June 1991.Published in Proceedings of the Conference.

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13. ABSTRACT (Maximum 200 words)This report presents an analytical solution for the elastic-plastic behavior ofthick-walled cylinders under internal pressure. The general hardening lawemployed in this investigation is a piecewise linear representation of arbitrarystress-strain curves in unixial form. Closed-form analytical solutions aredeveloped for the stresses, the elastic and plastic strains, and the displacementsby using Tresca's yield criterion and its associated flow rule. Experimental dataobtained from cylinders made of either SAE 1045 steel, OFHC copper, or aluminumalloy 1100 are used to determine the material constants. Numerical results forpartially-plastic and fully-plastic cylinders are presented for the radialdistributions of plastic hoop strain, radial, and tangential stresses.

14. SUBJECT TERMS 15. NUMBER OF PAGESThick-Walled Cylinders, Elastic-Plastic, General Hardening, 14Fully-Plastic, Internal Pressure 16. PRICE CODE

17. SECURITY CLASSFICATION 13. SECURITY 43ASSIFICATION 19. SECU 'RITY CLASSIFICATION 20. UMITATION OF ABSTRACTOF REPORT I OF THIS PAGE I OF ABSTRACT

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TABLE OF CONTENTS

Pace

ABSTRACT .. . ................................................ 1

INTRODUCTION.............................................. 1

BASIC EQUATIONS ............................................. 1

GENERAL HARDENING ........................................... 4

MATERIAL PROPERTIES .......................................... 6

NUMERICAL RESULTS ........................................... 7

REFERENCES .................................................. 7

TABLES

1. ELASTIC CONSTANTS FOR THREE METALS ........................ 6

2. PLASTIC CONSTANTS FOR THREE METALS ........................ 6

LIST OF ILLUSTRATIONS

la. Effective stress-strain diagram ................................... 8

1 b. Effective stress-plastic strain diagram .............................. 8

2. Variations of stresses and plastic hoopstrains in a steel tube .......................................... 9

3. Variations of stresses and plastic hoopstrain in a copper tube ......................................... 10

4. Variations of stresses and plastic hoopstrain in an aluminum alloy tube .................................. 11

5. Comparisons of stresses and plastic hoopstrain in a partially-plastic tube ................................... 12

6. Comparisons of stresses and plastic hoopstrain in a fully-plastic tube ...................................... 13

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ANALYTICAL SOLUTION OF ELASTIC-PLASTIC THICK-WALLEDCYLINDERS WITH GENERAL HARDENING

Peter C.T. ChenU.S. Army Armament Research, Development, and Engineering Center

Close Combat Armaments CenterBenet Laboratories

Watervliet, NY 12189-4050

ABSTRACT. This paper presents an analytical solution for the elastic-plastic behavior of thick-walled cylinders under internal pressure. The generalhardening law employed in this investigation is a piecewise linear represen-tation of arbitrary stress-strain curves in uniaxial form. Closed-form analyti-cal solutions are developed for the stresses, the elastic and plastic strains,and the displacements by using Tresca's yield criterion and its associated flowrule. Experimental data obtained from cylinders made of either SAE 1045 steel,OFHC copper, or aluminum alloy 1100 are used to determine the materialconstants. Numerical results for partially-plastic and fully-plastic cylindersare presented for the radial distributions of plastic hoop strain, radia 1 , andtangential stresses.

1. INTRODUCTION. Of all the available elastic-plastic solutions, theoroblem of pressurized thick-walled cylinders has received the greatest atten-tion. This is because of the symmetric nature of the problem and its practicalimportance to pressure vessels and the autofrettage process of gun tubes. Manysolutions for this problem have been reported over the last four decades [1-3].Analytical solutions can be obtained only when simplifying assumptions are maderegarding material properties. Bland [2] developed analytical solutions formaterials with linear hardening prooerties. Recently, Megahed (3] considered anonlinear hardening law a = Y + A.6 on in uniaxial form and developed an approxi-mate solution for any value of the strain-hardening exponent n. Closed-formanalytical solutions for the plastic hoop strain can be obtained only for fouroarticular values (n = 1, 1/2, 1/3, and 1/4), and the integral has to be eval-uated numerically for n = 1/3 and 1/4.

The general hardening law employed in this investigation is a oiecewiselinear representation of actual stress-strain curves in uniaxial form. A finitenumber of straight lines can represent arbitrary curves with greater accuracythan other representations [4]. The problem is formulated in a manner similarto (2,3] by using Tresca's yield criterion and the associated flow rule.Closed-form analytical solutions are developed for the stresses, the elastic andolastic strains, and the displacements.

2. BASIC EQUATIONS. Consider a long thick-walled cylinder, internalradius a and external radius b, that is subjected to internal pressure ocausing partial plastification. Assuming small strain and no body forces in theaxisymmetric state of generalized plane-strain, the radial and tangentialstress, ar and aq, must satisfy the equilibrium eauation

r(dar/dr) = ag - ar (1)

and the corresponding strains, Cr and £e , are given in terms of the radialdisplacement, u, by

Cr = du/dr , C6 = u/r (2)

1

Total strains are decomposed into elastic and plastic components and the strain-

stress relations are

Cr. = lar - v(ae+az)]/E + CrP (3a)

eg = Lag - v(ar+taz)]/E + cp(3b)

cz = [z- L(ar+ag)]/E + * (3c)

where E and v are elastic constants. Subject to ag ) az r Tresca's cri-

terion states that yielding occurs when

a- 0'r = a(c) (4)

where the yield stress a is a function of plastic strain cP. The associated

flow rule states that

dee = dcrp >, and dczP=O (5)

Hence, from Eq. (3c)

a= U(ar+ae) + Ecz (6a)

and the total axial force on any section is

F = 27wva 2p + 7IE(b2-a 2)cz (6b)

There are three cases of importance: first, plane-strain, cz = 0; second, a tubewith open ends, F = 0; and third, a tube with closed ends, F = irazp. In thelatter two cases, substitution into Eq. (6b) determines cz. Since cz is nowknown, Eas. (3a) and (3b) are inverted in order to express stresses in terms oftotal strains and plastic hoop strain as follows:

a= tVO+ (lv))cr + (1-2v.)c 6p + 1)Cz] (7a)

a= El)C~. + (l-V)C6 (1-2v)cg + 1Vez] (7b)

wnere E= E/E(1+v~)(1-2v)]. Substitution of Eqs. (7a) and (7b) into Eqs. (1) and(2) yields the following differential equation:

d2u 1 du u 1-2v 2coP dc9P(8

Integrating with respect to r leads to

du/dr +. u/r =-(1:3-)(2J+coO) + 2C (9a)

where

J r COP r-ldr (9b)a

2

Integrating again yields the analytical solution

1-2

U = -(-:-)rJ + Cr + D/r I0)

where C and D are integration constants to be determined from boundary con-ditions; ar = -P at r = a and ar = 0 at r = b.

Upon substitution of the resulting values of C and D into the expressionsof displacement, radial and hoop stresses, the following distributions areobtained:

ar = -P - EJ + (P+EJo)(1-a2/r2 )/(1-a 2/b2 ) (Ila)

a@ = -P - E(J+C6p ) + (P+EJo)(+a 2/r2 )/(1-a 2/b2 ) (11b)

a 2

EU/r = (l+v)(1- 2v)ar + 2(lV2)(aZ/r2)(p+EJo)/(1 - a7) - vEc z (11c)

where E = E/(1-0 2 ), and Jo is the value of the integral J at the plastic front,r =p, i.e.,

p p/a

Jo f Eqp r-, dr = f Cop P-Idt (lid)a 1

Note that ce = 0 and J = Jo throughout the outer elastic zone defined byp < r < b. At the plastic front, the Tresca effective stress a = a6 - ar = Y.where Y is the initial yield stress and also c£p = 0. Therefore, using Ea.(11b) to provide a, one readily obtains

o p2 a2

S+ E = 2 (1 - 6-) (12)

Using Eq. (7), the distributions of ar, a@, and u can be written in simplerforms as follows:

ar = -D - EJ + ! (pz/a 2 - p2/r2 ) (13a)

ag = ar + Y(p 2 /r2 r 0 P) (13b)a0

Eu/r = (1+v)(1-2v)ar + Y(l-u 2 )p2 /r2 - vEez (13c)

It is obvious from Eq. (13b) that Tresca effective stress a is simply given by

a = Y(P 2/r2 cop ) (14)

Therefore, if the radial variation of plastic hoop strain is known, the integralJ and all field quantities can be determined.

3

3. GENERAL HARDENING. The general hardening law employed in this investi-gation is a piecewise linear representation. Arbitrary stress-strain curves inuniaxial form can be approximated by a finite number of straight lines [4]. Thestraight line through the origin is given by the relation

a = EZ (15)

where E is Young's modulus. All of the other straight lines are given by therelation

a = (1-mi)aoi + miEe (16)

where aoi is the stress at the intersection of the two straight lines given byEqs. (15) and (16), and miE is the slope of the straight lines given by Eq. (16).Let a i, ei be the stress and strain at the intersection of two straight lineswith slope mi_iE and miE as shown in Figure la. Then

ai = (1-mi-I)aoi I + mi-lEe i = (1-mi)aoi + miEc i

which leads to Li and ai in terms of aoi and mi

Eci = [(1-mi)aoi - (1-mij)aoij]/(mij-mi) (17a)

and

ai = [mi-1(1-mi)aoi - mi(1-mij)aoi..]/(mi-l-mi) (17b)

Eq. (16) can be written also as a function of effective plastic strain CP asshown in figure lb.

a = aoi + hiEc Ei < Fi+1

where hi = mi/(1-mi).

Since COP = -Cr0 and ezp = 0, the effective plastic strain P is determinedas 2cgp/V5, and hence, Eq. (14) is rewritten in terms of the plastic strain inthe tube as

V3 E ) 9

a = Y(pz/rz - 2 P) (19)

A comparison between the expressions for effective stresses provided by Eas.(18) and (19) yields the following explicit equation for eP:

E -p-P = (p2/r2-Ci)/b i (20)

which is valid in EiO C P i+1p and ri ) r ) ri+ 1 and

bi= (V312)I(1-v2') + hi , Ci = aoila0 (21)

4

The values of ri and ri+i can be determined by

ri = p(bi E-p + ci]

E -i~p ]_ri+ 1 = p(bi p + Ci (22)

If C1p = 0 and ao, = Y, then r, = p. This is true for most materials. Sinceri ) a, the calculation of ri for i = 1,2.. .m should stop when the above rela-tion is violated, i.e., rm+l < a. Let us define V i for i = 1,2 .... m by thefollowing integral:

rf E -p qr f C) drr~ e r bi r~

ri~+1 r~ r br

1 0 2 (p_2= 6 (

(A -- ) - ) -7) 2C i in(ri/ri+l)] (23)bi ri+ 1 ri

Then

r E- dr rm E dr ,r E -o drV a Pr f a E - Vm-1 +...+ vi 1 + c r

1 2 0 2 rme 2 (( (pI)n2 Cm a + Vm_1 .+. Vi 1= bm a(5 - 2 m Jn.

+ 1 2 -- ) - )2 - 2Ci In (r/ril)] (24)bi ri 1 r

and the maximum value of V is

m-11E- dr 1 p 2

2 C nVo= f Cp 2 Cm In aT] + Vi (25)

i=1

The integrals V i (i=1,2....m-l), V, and Vo given analytically by Eqs. (23),(24), and (25) can be easily evaluated. The integral J is related to theintegral V by

IiV yJ = - - V 26

2 E

All field quantities u, Er, Ce, ar , ag, az., and ceg can now be calculated.

5

4. MATERIAL PROPERTIES. Test members were made from three differentmetals as follows: SAE 1045 steel, OFHC copper, and aluminum alloy 1100 [4].The values of the elastic constants (E and v) for the three metals are shown inTable 1. The values of the constants (aoi. m, oi , ci) approximating the plasticportion of the stress-strain diagram for three metals are shown in Table 2.

TABLE 1. ELASTIC CONSTANTS FOR THREE METALS

Material E, Ksi V

SAE 1045 steel 30,000 0.29OFHC copper 16,000 0.35aluminum alloy 1100 10,250 0.33

TABLE 2. PLASTIC CONSTANTS FOR THREE METALS

(1 1 1 1Straight Line aoi, Ksi I mi ai, Ksi I %i,

SAE 1045 Steel

1 43.4 0.05083 43.4 0.1452 54.0 0.02858 66.924 1.6873 80.0 0.00847 90.638 4.4534 95.0 0.00309 103.542 9.5325 111.0 0.00128 122.280 29.745

OFHC CooDper

1 2.50 0.17125 2.50 0.016

2 3.25 0.07063 3.686 0.059

3 4.00 0.03125 4.E53 0.136 14 5.37 0.01991 7.'00 0.7655 8.40 0.01313 14.151 2.7906 21.0 0.00450 27.484 9.1377 39.0 0.00078 42.757 30.350

Aluminum Alloy 1100

1 8.0 0.67024 8.0 0.0782 11.0 0.32683 11.942 0.1353 13.0 0.09561 13.557 0.1844 14.7 0.01590 15.007 0.3325 16.1 0.00210 16.310 1.131

Each of the stress-strain diagrams can be approximated by a finite number ofstraight lines with extreme accuracy. The error introduced by the approximationis less than 1 percent for all cases.

6

5. NUMERICAL AESULTS. Typical results for the analytical solution arepresented first by means of prescribing a plastic front and determining thecorresponding plastic hoop strain and radial and hoop stresses in the tube. Atube with b/a - 2 is employed, and the plastic front is prescribed at p/a = I.0,1.2, 1.4, 1.6, 1.8. and 2.0. Figure 2 illustrates the stresses and plastic hoopstrans obtained using the material constants for SAE 1045 steel. Figures 3 and4 present similar results for OFHC copper and aluminum alloy 1100, respectively.Figure 5 shows a comparison of stresses and olastic hoop strains for threepartially-plastic tubes at p/a = 1.6. Figure 6 presents a similar comparisonfor three fully-plastic tubes at p/a = 2.0. Future work related to the resultsobtained here will look into the elastic-plastic behavior of the tube duringpressure release. The influence of phenomena such as the Bauschinger effect onresidual stresses should be modelled (5,6].

REFERENCES

1. Hill, R., The Mathematical Theory of Plasticity, Oxford University Press,London, 1950.

2. Bland, D.R., "Elastoplastic Thick-Walled Tubes of Work-Hardening MaterialsSubject to Internal and External Pressures and Temperature Gradients,"Journal of Mechanics and Physics of Solids, Vol. 4, 1956, pp. 209-229.

3. Megahed, M.M., "Elastic-Plastic Behavior of a Thick-Walled Tube With GeneralNonlinear Hardening Properties," International Journal of MechanicalSciences, Vol. 32, 1990, pp. 551-563.

4. Sidebottom, O.M. and Chu, S.C., "Bursting Pressure of Thick-Walled CylindersAxial Load, and Torsion," Experimental Mechanics, Vol. 15, 1975, pp.209-218.

5. Chen, P.C.T., "The Bauschinger and Hardening Effect on Residual Stresses inan Autofrettaged Thick-Walled Cylinder," Journal of Pressure VesselTechnoloay, Vol. 108, pp. 108-112.

6. Chen, P.C.T., "Stress and Deformation Analysis of Autofrettaged HighPressure Vessels," ASME PVP, Vol. 110, 1986, pp. 61-67.

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