A finite strain axially-symmetric solution for elastic tubes

hr. J. Soli& Srrucnucr Vol. 24. No. 7. pp. 675-682. 1988 Prmcd I” Great Bnum.

ootlF768188 13.00+ .m C 1988 Pqnmoa F-res pk

A FINITE STRAIN AXIALLY-SYMMETRIC SOLUTION FOR ELASTIC TUBES DAVID Department

of Aeronautical

Engineering.

( Rccriced 2 I

.\IuJ

DURBAN

Technion-Israel Israel

I987

: in

Institute of Technology.

revised form 5 November

Haifa

3X00.

1987)

Abstract-An exact solution is given for the finite straining of elastic tubes under the action of internal and external pressure at the boundaries. The material model is described by Cauchy-elastic constitutive relations for isotropic Hookean-like materials. The governing equations are reduced to quadratures which provide a simple solution for the problem. Numerical results reveal the existence of a bursting pressure. Some further results are given for incompressible solids. Also discussed is the problem of a pressurized cylindrical cavity embedded in an infinite medium.

INTRODUCTION

Large deformation ticity[l].

analysis of prcssurizcd elastic tubes is a standard topic in finite elas-

Exact solutions

though arc still rare, depending on the details of the constitutive

relations. Hcrc a simple solution is prescntcd for that problem using a special model ofa Cauchyelastic solid. The true stresses are rcluted to the strains by the classical Hookcan relations, but the strain components arc taken as the finite logarithmic

measures. The same model

has been cmploycd in Ref. [2] for the analogous spherical shell model, and in Refs [3.4] in the context of finite plasticity analysis. A judicious

choice of a new variable enables a direct solution,

the governing equations. Sample numerical computations (nondimensionalized

by quadratures.

of

show that the internal pressure

with rcspcct to the shear modulus) reaches a maximum point which-

for tubes of modcrate thickness -is almost independent of Poisson’s ratio. That observation is supported by an approximate solution for thin-walled tubes. Some further simplifications are made for incompressible materials. Finally

the pressurized cavity, embedded in an infinite

medium, is considered. It is

shown that the applied pressure reaches an asymptotic value which is of the order of the elastic modulus.

GOVERNING

EQUATIONS

The particular material model employed here is that of a Cauchy-elastic isotropic solid the constitutive

relations of which are 6, = 2GIno,+i_InJ,

i=

1.23

(1)

whcrc 6, arc the principal Cauchy stress components, a, the principal stretches, and J the volume ratio dcfincd by

J =

(2)

u,u$?,.

The elastic constants G. i. arc expressed by the usual Hookean connections

G=

E

.

VE

Z(lfv)‘A= (l+v)(I-2v)

where E is the elastic modulus and v Poisson’s ratio.

(3)

D. DLXBAS

676

Consider now a thick-walled tube. deforming in a plane-strain axially-symmetric pattern. under the action of internal or external uniform normal stresses. Let r stand for the Lagrangean radial coordinate with (a.b) the undeformed internal and external radii. respectively. Introducing the non-dimensional coordinate

p’r

(4)

cl

the principal

logarithmic

strains can be written

as

c,=In(l+u’),~~=In

I+” (

P>

where II is the radial displacement, nondimensionalized with respect to u, and the prime denotes differentiation with respect to 1’. The constitutive relations (I) can now bc put in the form a,=2GIn(I+~‘)+i.lnJ

I+ I’ +i.lnJ /’ 1

r7,,= ZGln (

whcrc (fl,. fl,,, f7:) arc the usual cylindrical

(6)

(7)

polar stress components.

J=(l+fr’) (

and

I+ U I’ 1

Equations (6.(9) are compatible with the plane-strain constraint, but there should be no diliiculty in cxtcnding the subscqucnt analysis to combinations of radial boundary stresses and a uniformly imposed axial strain. Turning to theequilibrium rcquircments one hasjust the single radial equation, written in the undcformcd rcfcrencc configuration

pa;+

-‘Iv!-(6,-g,,)

I+ This equation

is supplcmcntod

u/p

by the boundary

= 0.

data

fl,(P = 1) = -p, a,(p = If) = where TVis the initial

I

(11)

radii ratio h I]

=

-- . II

(I’)

Since the volume ratio J is expressed by eqn (9) one can regard the three equations. eqns (6). (7) and (IO). with the three unknowns (a,.a,,,~). as the governing system. It will be shoun that this system admits a simple solution by quadratures.

A finite strain axially-symmetric solution for elastic tubes

677

SOLuTION The solution centres on the new variable X defined by

-1‘+U = pIn’(p+u). +u/p

x=

(13)

Now. subtracting eqn (7) from eqn (6) gives 6,--b@ = 2GInX and the equilibrium

(‘4

equation. eqn (IO), becomes pi+

2GXln X = 0.

(15)

Combining eqns (6) and (9) one obtains

u,=

(2G+i.)In(I+u’)+A_In

(

I,%

>

(‘6)

or. with the aid of eqn (13)

a,=2(G+L)ln

I+:

(

>

+(2G+%)InX.

(17)

The radial Jcrivativc of cqn ( 17) is

a; = 2(G+i.)ln’ Observing now the itlcntity,

(

I + ;- +(2G+A) )

z.

(‘8)

from cqn (13)

I+;

In’ (

=:(X-l)

(19)

)

cqn (18) can be written as

0; = 2(G+i_)

X-l I_ P

, +(2G+A)

Inserting oqn (20) in cqn (IS) results in the diffcrcntial

2.

(20)

relation

where

g(S)

and

= -

I x(X2 -X)+/?X’InX

(22)

678

Combining

eqns (21) and (15) gives another differential

relation. namely

da, = .?Gf‘(.Y, d.4

(14)

where

f(X) = -y(,Y),Yln

.Y.

(25)

The solution of the problem is provided by the integrals ofeqns (21) and (23). Denoting by ‘y, and X;, the boundary values of variable X, one has t.6

Inq =

J

g(X) d.Y

‘,

‘I

r+p = 2G

I*.f~x)d.\

(77)

i,.

Thcsc cqu;ltions can hc solved (using the method cmployod in Ref. [2]) without difficulty. for any givcrt ri~~ltcri~~~ constants, radii ratio and oxtcrnal loads. Once the vatucs of ,u,,. ,U,, have been dctcrmincd

The circltmferenti;Il usual plain-strain

it is possible to got the mdial stress. from cqn (24). as

stress follows from cqn (14). while the axial stress is obtained from the relation

IT1 =

v(r7,+cfs)*

(29)

To find the radiaf displ~ccRicnt u one can rewrite eqn (I?) in the form

In

I+:: (

=(I-2,*)$

- ( I - v) In X

1

and use the expression for o, from eqn (38). The tr~Insformi!tion Lagrangcan

coordinate

1’ is given by tho intcgral ofcqn

(II),

from variable X to the

nnmcly

The fundamental functions j1.U) and ,y(.U) arc shown in Fig. I for S> 0. FuncLion j’(X) has an asymptote at .t’= 0 and its bchaviour for small values of X is given csscntially by

.f(.Y) - The asymptote ,I’ = 0 is upproachcd

’ In .Y.

Y

by function g(.U) as well with the leading term

(31)

A finite strain axially-symmetric

solution

for elastic tubes

679

I

4

I I

I

0

’

SIX)

10 -

,

0.J

1.0

*

I I

I I -s-

I

I I

I I I

-10 -

I

I I I

I Fig. I. The l’undamcntd

functions/(X)

and g(X)

(33) Function y(X) however has a second ;~symptote at X = 1, and it can easily be verified via eqn (22) that, in the ncighbourhood of A’= I I

YW) - g--q.

(34)

These expansions will be used in the next section for investigating certain aspects of the solution. Note also (Fig. I) that the influence of Poisson’s ratio v is appreciable only for low values of X. DlSCUSSION

Equations (26) and (27) have been solved for the case of internal pressure only (t = 0) using a standard numerical proccdurc. The calculations covered a wide range of radii ratios and different values of Y.The boundary values of variable X were found always in the range 0 -CC A’, c X, < 1. Tracing the load history of the tube revealed the expected existence of a maximum pressure. PmaXrknown also as the bursting pressure. The precise location of the maximum pressure point along the deformation path is obtained from the rate form of eqns (26) and (27). namely

680

D.

DLXBAN

02-

01-

Fig. 2. Variation

of the maximum

pressure with radii ratio

where the superposed dot denotes differentiation the point of instability

with respect to a time-like

parameter. At

one has 0 = 0 and eqns (35) and (36) may be conjoined, with the

aid of eqn (25). to give

The solution ofcqns (37) und (26) yields the boundary values of variable Xthat correspond Rcprcscntativc results for the maximum prcssurc arc shown in Fig. 2. It is seen that to 1’111.1<. Poisson’s ratio has ;I small clli.ct. on the non-dimensional bursting pressure, cvcn for rclativcly thick tubes. A further simplitication

of cqns (26) and (27) is possible for incomprcssiblc

mntcrials

with L*= 1/2. 1:rom cqns (23) it can bc found that z = 2. /I = 0, 2nd function i/(,1’) from ccln (22) bccomcs

Y(X) = Equation

I z(x--n.

(36) has now the exact integral

(39) and the internal pressure p? from cqn (27), is likewise

(40)

where G = E/3. Of course, the calculation of the maximum pressure is now much simpler since eqns (39) and (37) can be combined to a single equation with one unknown. When the tube is thin-walled eqns (26) and (27) may be replaced by the first-order approximations

In rl

=

.c7(Xo) Wh- x,1

(41)

(42) where 3.U,, = X,+X,,.

Thus.

with the aid of cqn (25)

A finite strain axially-symmetric

Fig. 3. Variation

681

solution for elastic tubes

of cavity pressure with X0. The asymptotic

pressure is reached at Xd = 0.

(43) This expression has a milxi~~un~ at ,I’(, = e-’ given by

Note that within this approximation.

the non-dimensional

bursting pressure is indcpcndcnt

of I’. Results

obtained from eqn (44) are shown in Fig. 2 conlirll~in~ tube approximation up to about 11= 3.5.

the validity of thin

At the other extrcmc one has the problem of a prcssurizcd cavity cmbcddcd in an infinite with

medium. Here tt -+ Y;

ilfld.

any 0 < X,, c I provided

in view of expression (34). condition

that

k;, = I. The

internal

prcssurc,

(26) is satisfied

eqn (27).

is now

exprcsscd as

with X,, dccrcasing from 1, at zero load, to 0. Figure 3 shows the dependence of the nondimensional cavity pressure. cyn (45), on the value of X,, for different values of Y. At the limit of X, + 0 the pressure reaches an asymptotic value which is equal to the area under the curvef‘(X) that singularity

in Fig. I over the range 0 < X < I. It is worth

mentioning

in this context

(32) has a bounded intcgral in the vicinity of X = 0. Note. however. that

the displacement at the cavity, eqn (30), bccomcs unbounded as the asymptotic prcssurc is approached. For incompressible materials one has the exact result, for the asymptotic value ofp

Lymp1 ._._ _ -_ ‘G

’

InX I-__-_ 3(X-l)

dX = ;;

I

z 0.8225

(46)

or, put diffcrcntly

P.trympt = ;

E 2 0.5483E.

(37)

That result may be compared to the necking stress in uniaxial tension of material (I). with v = l/Z, given by b,,k = E, and to the asymptotic pressure in a spherical cavity, embedded in an infinite medium, given by

682

D.

DCRFIAS

= + Pal~l7lpl

0.7311E.

,2laterial compressibility increases the magnitude of the asymptotic pressure sustained the medium. With Y= l/3 one obtains pJsymptr 0.702E. while for Y = 1,~. P JIympr z 0.788E which is considerably above eqn (47). by

A~klino~Ir@?wn~--I am indebted to Mr Michael Kubi who performed was supported by the Technion-V.P.R. Fund-L. Kraus Research Fund.

the numerical computations.

The work

REFERENCES I. A. E. Green and W. Zema. Thoorericol E/~.sricirr. Oxford (197%. 2. D. Durban and M. Baruch. Behaviour of an incrementally elastic thick-\ralled sphere under internal and external pressure. Inr. J. Non-Lineor Mwh. 9. IO-C-1 IV (IY7-t). 3. D. Durban, Large strain solution for pressurized elastoplastic tubes. J. .4pp/. .Jfwh. 4. X9-230 (1978). 4. D. Durban. Finite straining ofpressurizedcompressibleelasto-plastic tubes. /III. J. Enwy Ser. (to be published).