Structure of kink in buckle propagation

on elastic pipeline

N Sugimoto

Depar咄ent of Mechanical Engineering, Fac141tyof Engineering Science，

University of osa㎞，0saka 560, 翔)an

　　Reprintedfrom ASME Book NO AMR137

1UTAM Symposium 爽NonlinearWaves in SOlids

ｉ

Structure of kink in buckle propagation on elasticpipeline

N　Sugimoto

Z)9a?4mesi Q/4fecんa7z･icalj7り171ee7｀1?1･,9，.Facs&yθ/'f7z夕i7z･eer17り･S'cie刄cej

ひy1‘i‘ue7｀s≒ofosa･kG｡ 0sa,kG，560，J叩a'n･

This paPerconsiders麟eady buckle propagationon anelastic pipeline sl!bjected to

a combined action of bending， axial tensionand external hydrostatic side pressure.

A structu:re of a kink i芦examined by applying the idea of the ma,tched-asymptotic

expansion method to thenonnnear wave equations for the beam4exural mode coupled

with the ring-flexural mode derived previously. Their “outer solutions" are nothing but

theonesto the equations simpnfied 晦the approximation that the thecro8sgsectional

ovanzation bal゛"ces with the cirmumferential bending. Although they suggest

propagation of a kink，the appr(xximation becomes invalid at the very kink so that the

iuter solutions cannot be employed there. For the kink，“lnner solutions" aresought by

taking account of the lo4itudinal stretching as a shellin addition to the circumferential

bendhlg. lt then becomes possible to give the lowest innerstructure of the kink a･nd also

to provide u㎡formly valid solutions through6ut the pipenne. But a problem whether or

not the apPropriate outer solutions can be singled out stm remains unresolved.

INTRODUCTION

ltis known as the buckle propagation that when a subma･

rine pipeline is subjected toacombined action of bend-

ing， axial tension and hydrostatic side pressure exter-

nally,a buckle (i.e. a local khlk)can be propagated along

the pipenne， entailiilgits catastrophic conapse (Palmer

and Martin (1975)，Mesloh ei al･ (1976)).lmportant

point is that itoccurs under the side Pressure even well

below the bucknng pressure of the pipe due to the side

pressure. lts phenolnenological aspect has beencla,rified

to some extent by the experiments due to Kyriakides and

Babcock(1981)and Kyriakides and Chang (1992)，but

its physical mecha･nism for initiation and propagation of

the buckle has been left open.

　This motivated the author to consider theoretically

such a very interesting phenomenon (Sugimoto(1987，

1989a，b)).Admittedly,both elasticand plasticbeha,vior

of the pipeline are engaged in highly nonlhlear manner.

But it is conjectured that the proPagation mechanism

associated globany with the pipenne wm be maintained

elasticany，whereas the plastic collapse wmensueafter

propagation of buckle so that the plasticitywm not take

part in the propagation mechanism.　Focusing only on

the aspect of propagation， the formulation was given in

the framework of the thjree-dimensional theory of elas-

tic but finitedeformations to derive the nonlinear wa,ve

equations .forthe beam-flexural modecoupled with the

ring-flexural mode. While steady propagation of a kink

is suggested by the equationssimplified，the purposeof

this paper is toexamine an inner structure of the kink

by solving the fu,llequations.

　ln the beginningμt is instructive to l?capitulate the

results wi仏in the elastic theory so far obtained. The

mechanism for steady propagation is found to be given

by the tension efrect due not only to the axial tension

applied externany but also to the side pressure. 'When

this tension eflectazld the bending balancefor the beam-

flexural mode， the ring･.flexuralmodecanbe simplified

into that the cross-sectionalovalization bal＆nces with

the circumferential bendhlg. 'Wlth this simpnfication，in

passing，the Brazier efect can be described by the equa･

tions in a static case without the axial tension. As the

Brazier eflectis explained by the energy method (Calla-

dine(1983))，the present equations ca‘nprovide a new

approach based on the dynamical equations. The steady-

wave solutions to the simplmed equations suggest prop-

agation of a kink in the de皿ection. Further it is shown

that the propagation velocity and the“propagation pres-

sure" necessary for initiation of propaliation of the kink

agree quantatively with the experimental results in spite

of neglect of the plasticity･

　At the kink，however，it is found that the above sim-

plmcation becolnesinvand so that those solutionsshould

not be employed there. T徊khlk appears to be so in an

axial scale much longer tha!l a typical length of a buckle.

‘W'hen this situation is viewed from a standpoint of the

matched-asymptotic expansion method (see，e･g.Nayfeh

(1973))，the solutions to the simplified equations corrre-

spond to“outer solutions" to the full equatigns， while

“inner solutions" provide a structure of the khlk now

to be solved by taking full account of the ring-flexural

mode. ln this connection，it is asked whether or not the

piecewise valid outer solutions separated by the kink can

be singled out by examing the inner solutions.　But this

problem stm remains unresolved. lt is obvious that only

the lowest-order inner solutions in this paper are insu伍-

cient for this purpose.

parl ot Nonlinear waves in solids, edited by JL Wegner and FR Norwood

ASME Book NO AMR137　　　　　　　　　　　　　　　　　　　　　　　　　　346 ＠1995 American Society ofMechanical Engineers

GovERNING EqUATloNs

At the outset， we present the governing equations for

elastic buckle propagation derived previously by Sugl-

moto(1987， 1989a･).when the pipenne is subjected to

finite bending as a beam， it gives rise to the hlextensional

ovaJization of the cross-section due to the geometrical

nonlinearity. ‘W'hne this，in turn， reduces the moment of

1【lertiaof the cross-section， the cross-sectional deforma･-

tion is supported by the circumferential bending and the

longitudinal stretching ai a shell in addition to the a.xial

tension appned.

　Denoting by λand j， respectively，the de旦ection of

the pipeline as a beam and thecoe伍cient of thesecond-

harmonic component of the ovalized contour decomposed

injtothe R)urier series with respect to the circumferential

angle of the cross-section，the goveming equations are

given in the Lagrangian formulation as fonows:

　∂り1　.ga,2

ρ゜'･Iir十‾i‾

ρ0

∂2／

㎜

∂t2＋

２

㎜∂z2

ぐｒ－－‐Ｌ

１＋

　　一

ぐ

四

--

３

㎜２

う　ー

ｊ　ｐ一Ｒ

∂2λ

㎜∂z2

ぎ－71，

3－

㎜　５

(倣)2

１－－－．″

-

㎜

7･;z

∂2j.ｊｌぐ

(2)

NONLINEAR WAVES IN SOLIDS　　347

assmed to be of comparable order of o(Z″y3)，while a

magnitude of r is comparable with pa･/ll･，i.e. 0(.＆y2).

　For steady-wa･ve solutions pi'opagating with a constant

velocity lj(＞O)，we look for ,j4.and ,Zr which depend on

z a"d t only though a combination z － ･ut.Normalizing

the variables by setting ,A/G ＝λ″－･d(z － ･ut)/1＝ぐ，it

then fonows from Eqs (1)axld(2)that

9a

ぐＬ 　ｎり

　一一

「ｑＪ

う　ー

３ＩＮ

　＋

1 d4j″　　d2j″　3

召‘司ｒ‘りqF゛一５

ぐ 訃･====･’ 　八‥｀。

　一一

ｊ

３　　　　　４

with c＝d２,Ａ' ７dぐ2，9，＝(1･2 －‘t･1)/ε2‘t･g，96＝(1･2－

痙)/占いnd4＝(e4/72)(1－gｱ2)(1 － p7‘P。)－1，where lJO

is the speed of the longitudinal waves given by ，/(j;/po)，

whne lla and ‘u&are the speeds of waves associated with

the eflective tension and given, respectively,by･,ZCr,7po)

and ･ぴ(TI,7p，o).Equations(3)and(4)provide the basis

for the foUowing analysis.

PROPAGATION OF KINK

We begin the analysis with noting that ga and 9みare

very small by the assumption that 7/Z ajndpa7 Ek a;x:e

ofO(72).ln considering propagation of a kink， it wa,s

emphasized by Sugimoto (1989b)that gaC in Eq (3)must

not be neglected because it is the very eflect that balances

with the bending on the second term and drives the kink

forward.This balance occurs not over the a,xial scale ぐ

associated with the typical length l of the buckle， but

overa longer scale Z defined by l 9a l1/2 ぐ.Rewriting Eq

(3)in terms of Z， hl fact，two terms balajnce ea.ch other，

while the first two terms in Eq (4)are found to be of

O(qJ1,qaqb)a‘nd to be very s:mall compared with the last

term.Thus jy is givenapproximately as

　　　　　　　　　　　　ぎ＝－S r!2.　　　　　　　　　　(5)

　Substituting thisinto Eq (3)，we have the single equa･

tion for C. Setting 9δC2/2＝j，it is simpnfied to

(1一心-

AI -

㎜ 士

　〃λ

１ ぐ ／㎜

dZ

ｊ２　十4(sgn9,)j＝o

゜ぞ(1－1)‘μ･

(6)

∂ｚ２１

＋

Ｚａ２∂４３″

一一２０　∂ｚ４

∂2Zμ

㎜　∂z2

１

3Zll,*2

㎜　　5a4

with .ZEμ＝j/a where z and t denote, resp ectively, the a.x-

ial coordinate along｀the pipenne and the time， whne a,瓦

aTld h，*denote，reSpeCtiVely,the pipe゛S radlUS at the mid-

dlesurface，the pipe゛s thickness a7爽ldits“eflective thick-

ness" denned by h,7,ﾉ(1－(y2)，cr the Poisson゛s ratio. With

the densitypo in the undeformed state and the‘Young゛s

modulus Z，p aJld pe represent， respectively，the hydro-

static side pressure and the bucknng pressure given by

pc＝Z(ll･/a)3/[4(1－･72)].The side pressure also increa,ses

the a.xial tension in addition to the one 7 appUed exter-

nally through the “efrective tension" 7･lz(＝7十pa/2瓦)

and 7あ(＝7'十･pa75k)｡

　For the sake of simplicity， the higher-order eflect in

the “ovanzation" j∃μhas been neglected in Eqs (1)and

(2)because l Ｂ″lmust always be less tha.n unity by

the assumption that the pipe°s periphery doei not touch.

specmcany，zl″2/16"｀d 53″/6 have already been dis-

carded in Eqs(1)"ld(2)，respectively(see Sugimoto

(1989a)).Although the discard of the latter might seem

to be inco】!lsistent，itprovides the higher-order modmca･･

tion to the lowest nonlinear term on the right-hand side

of Eq (2)(i.e. of cubic order as a whole)so its neglect is

consistent with that oＩＢ″２７１6in Eq (1)｡

　Here the fonowing points should bere:rna,rked.Deriva,･

tion of Eqs (1)aTld(2)rests on the scaling assump-

tion that the two small parameters ぞ(≡G,7 l≪1)and

7(≡瓦/a≪1)，lbeing a typical axial length of the

buckle，satisfy the order relation 7 ～62. 1n addition，

a ma･gnitude of p is smaner than pc， of course， but p is

十j

㎜2/

･Wlth j。･‘1″and .Zμare derived as follows:

with j4q｡j､Z6’

　　　　　　　　　　　３’＝一弘　　　　　（7）　　　　　　　　　　　　　　9

Here the sign“士”inλ″indicates two possibnty of posl-

tiveornegative deflection，but only positive one iscon-

cerned in the following. The solutions to Eq（6）have al-

ready been obtained analyticaUy and discussed fully by

Sugimoto（1987）.lt is found that two types of the solu-

tionsexist depending on the sign of ga，the“flexural wave

… 一 一

348　　NONLINEAR WAVES IN SOLIDS

mode” for ga ＞O and the“buckle propagation mode” for

ga＜0. 0f course，the latter is concerned here.lnpartic-

ular，ifthe boundary conditions for the undeformed state

are applied at the i�inity of Z，j is given by the inverse

of the fonowing functions:

､/(2－f)一石tanh‾1･y(1－//2)－､zo＝士z(8)

Here an integration constajnt Zo is taken to be Zo ＝ 1－

(1/y2)tanh‾1(1/､/2)so as to set j＝1 at the origin of

Z. The sign“士"indicates a palr of solutions symmetric

with respect to Z｡

　As the point / ＝1 is a shlgular Point of Eq (6)、dj/dぐ

diverges there.　Since / represents physically thecross-

０．５

0.25

０

一２百

　　　　丿

　　　　　　　　一石’

ｌ　　　　　　ｌ　　　　　　ｌ-1.0 -0.5 Ｏ

　Ｚ

０．５ 1.0

FIG 1. ProjUes of the outer solutions for,the ddlection

λslnd the cross-sectional ovalization －jEI″as the func-

tion of the axial cooridnate Z (＝19， 11/2 ぐ)where only

the subcritical bza,nches are ta,ken and the smaJl blank

circles represent the ki:akin the outer solutions.

0.5

0.25

　　　　　　　　　　　　－B3

２

㎜　　　　　　　　　　　　　ｌ　　　　　　　　　　　　　　　　　　　竃　　　　　　　　　　　，９

　　　　　　　　　一一-Ｆ　　　　　　　　　　　　　　　　　　　９

　　　　　　　　－　　　　　　　ｊ

　一ZO　　　　　ZO

「

｜

－1.0 －0.5 Ｏ　Ｚ

０．５

FIG 2. Profiles of the outel: solutions for the ddlection j，

s171dthe cross-sectional ovalization －j″＆s the fllnction

of theaxial cooridna,teZ(＝19， 11/2ぐ)where both the

sub- and. suPerczitical bra;xlchesareta,ken and the small

blank circles represent the kink in the outer solutions.

sectionalovalization jEμ，thedivergence of its derivatives

invalidates the neglect of the first two terms in Eq

(4)and therefore the aPproximation (5)，nomatter how

small9a and 9ゐmay be.

　Except for / ＝1，0f course，Eq(6)and therefore the

solutions(8)remaill to be vand approximately.　Thus

(8)are to be interpreted as piecewise vand solutions for

O＜j＜1 and 1 ＜/≦2，respectively. lt is reme:mbered

that the point / ＝1 corresponds to the nmit point of

the Brazier eflect，above which the static bending be-

comes unstable (see Sugimoto (1987)).ln view of this，

we call the solutions below j'＝1 “SUbcritiCal braTlch"，

while the ones above it the“supercritical branch"/Wlth

these two br゛-nches，we can buUd up any solution by com-

bining them appropriately so as to satisflythe bounda,ry

conditions at the infinity，for example， as shown in Figs

l and 2 (Sugimoto(1989b)).Among them， the solutions

consisting only of the subcriticalbranches in Fig l a･ppear

to be realized in reality.But it is emphasized that there

a.reno means to single out the appropriate solutions only

within the approximation based on Eq (6).ln any case，

a kink appears hl the deflection at j ＝1.

STRUCTURE OF KINK

Let us now consider inner solutions for a structure of

the kink by discarding the simplification. ″lothis end， it

must be the best if Eqs (3)and(4)could be solved fu.ny

without any approximation. But it is very di伍cult to do

so analyticiny and even numericany for very small values

of ga and 9&.By regarding ga and 9みas sman asymptotic

parameters，then， the idea of the matched-asymptotic

expansion method (called simply MAE hereafter)issuc-

cessfuny applicable.

　lnnersolutions to Eqs (3)a･nd(4)are sought so as

to be matched asymptoticany with the outer solutions

(8).The matching conditions are given by expanding the

solutions(8)in terms of Z around Z ＝o. For example，

suppose that the subcritical outer solutiot1 / increases

from the minus infinity of Z to approach j 士1asZ→

o一(by taking the plus sign in (8)).Then the asymptotic

solutions as Z→O－ are given by

　　　　　　　　/＝1－21Z11/2十〇(ｚ)，

･･1'＝町ざF7(1-11剔十〇(z2))，

RI＝－

０

Ｃ -

㎜

l(1－21剔1/2十〇(z))，

(1- l z ll/2 十〇(z))・ (9)

一 一 一 一

､/2

㎜3φ

Here note that these expressions are also yand for the

subcriticalbranch with the minus sign in (8)asz→0十.

　Accordhlg to the idea of MAE， the inhity asぐ→－(x)

corresponds to Z →O一 by assuming a suitable matchhlg

０

一 一

regionand the inner solutions ＆re to besought so as to

approach asymptoticaUy the expressions (9)asぐ→－cx).

with z＝19.11/2ぐsubstituted in (9)，they contain the

small parameter l 9a l1/4. Thus it is 4ppropriate to seek

the innersolutions for j″ and C in the form of power

SerieS expaTlsion With reSpeCt tO 1 9a l1/4｡

　Then the lowest problem becomes

１一９″

㎜　12

２ ぐＥ

d4召″

㎜　dぐ4

　ｎり

　一一

Tｊ　a:1

eQIN

　＋

十ｊ″十δC2＝0，

where the･ term with ･gihi.s

(1o)

３１１一一

ぐ

been discarded because 9&

is of comparable order with ga. liot this problem， the

matching conditions are simply given as follows:

3･ －，一旦１

Ｃ→谷　asぐ→－(ｘ)．

　Equation(10)can be immediately integrated，on

posing the matching conditions(12)，togive

ト　＝

　Ｃ

う　ｐ″

３一２

2φ

㎜9φ

(12)

lm一一

(13)

Using this relation to enminate C in Eq (11)，we have

　δ

㎜12

４３″

㎜

dぐ4＋

(３'十2/9)2(3'十8/9)

　　(3'十6/9)2

This equation can be integrated once into

δ

㎜12

Ｌ

dBl a3B’

一一－dぐdぐ3

ular solution:

ぐ１一２

差dぐ4

　　＋

ｒ　　　．

　ｊ

＝０

(ｊ'十2/9)3

2(j'十6/9)

＝０

(14)

－0，(15)

(16)

(17)

(18)

－β’

1.0

６一９

0.5

０

NONLINEAR WAVES IN SOLIDS 349

一

！

－

1.

£ ● ㎜ ●

｜

ｆ

　 　 　 　 　 　 　 　 　 　 　 　 　 　 ｔ

｜　　　　｜ ｜-３ ０ －1.0

ｏ　ａＪ

1.0　　2.0

２一９

-2.0

FIG 3. ProlUe of the ixlner solution for the cross-

sectional ovalization －3″ as the function ofぐwhere

δ｀＝2/9貪-7･d the blank circles represent the point

一召″＝6/9.

－２．０

-0.4

LO

-1.0

-1.0

０

20.0

０

-0.8 -0 -

-20.0

－10.0

10.0

０

３．０

一

り一 ｊ

＿！　９

＿２　９

ｌ .|１．０

-O｡8 -0.6 -0.4 ０．２

盛

dζ

　　　　　　　　　　　　　　　y

FIG 4. Trajectory of the inner solution l)r j″ pro-

jected in the plane (djEI″/dぐ,召″)where the blank

circlesrepzesent the point ,Z?″＝－6/9.

d2.召″

皿　dぐ2

Where a･Tlintegration COnSta･nt haS been determined by

assuming that ぎ"｀d its derivatiyes vanish asぐ→－(x).

Solutions around Ｂ″＝－2/9

″lk)see゛71asymptotic behavior of jμasぐ→－(x〉，we

setjEI″＝－2/9十＆and substitute it into (14).Since

＆vanishes as ぐ→－(x)，we lhlearize the equation with

respect to &.　But the resulting equation d4&/dぐ4＝0

gives no inforn51ationthan ＆＝0.Thus the linearization

is abandoned aTld the quadratiC term in ＆isretained as

4&㎜

dぐ4＋a&2＝0，

whereα＝81/2δ.This equation has a non-trivial partic-

＆＝λ1ぐ‾4，

withλ1＝－840/a＝－560δ/27. ″lofind a general solu-

tion around it，we set ＆＝λ1ぐ‾４十i;(ぐ)and substitute it

into Eq (16)to have

十験s十as2

包

dζ2

　　　　　　　　　　　　　　β'

FIG 5. Trajectory of the innex' solution for j″ Pro-

jected in the plue (d2j″/dぐ2,jEI')where d2 jEI″/dぐ2

diverges as j″→－6/9.

350 NONLINEAR WAVES IN SOLIDS

Here we linearize Eq (18)a711d take a,n elementary solu-

tion in the form ofぐ″｀，sbeing a const"｀t. Then the

proper values of g are given by s ＝8,－5, 1.5 士iへ/39.75.

Among them， only the solution with g ＝－5 satisfiesthe

mat(ihing condition as ぐ→－(x).

　We note that solutions to Eq (15)(axld therefore Eq

(16))anow azl arbitrary choic芦of the origin of ぐ.Thus

(17)may be taken as λ1(ぐーぐo)‾4，ぐo be1ﾛ,g an arbi-

　　　　　　　・　　　　　　　　　　　　　　　　　　　　j　　　　　　　　　　・　　　　　　　　　・trary constazlt. Maldng use of thii arbitraziness， ＆just

obtained can be absorbed into the solution (17)so that

＆may be taken to be zero. From this，it is found that

(17)is the only asymptotic solution erna,nating from the

minus i�inity ofぐ．

　Owing to this solution，we can now integrate the fuII

equation(15)numericany. ln doing so，the higher-oder

terms than (17)are included in the expansion as fonows:

＝λ1ぐ‾4＋λ2ぐ‾８＋…， (19)

withλ2＝－548800δ2/3159. Figure 3 shows －.Zr as the

function ofぐ.Here the fuU solution to Eq (15)is drawn

by taking δ＝2/9 for definiteness (so that / ＝c2)･

Figures 4 and 5 show the projection of the trajectory of

the solution in the phase space (が,dy/dぐ,d２ Ｂ'７dぐ2).

The arrows indicate the dlrection of the solution startklg

fi'om BI＝－2/9 at the minus i�inity ofぐtowardぐ＝0.

Solutions around BI＝－6/9

lt is found from these figures that the solution which has

started fi:om β″＝－2/9 crosses j″＝－6/9.As j?″ap-

proaches it，Figs 4"｀d 5 show that dj″/dぐremains to

be finite but d2,ZI″/dぐ2tends to diverge. This is under-

stood from that the point .Z?″＝－6/9 is a singular point

of Eq (14).ln view of the numerical results，we consider

゛71asymptotic solutionaroundj″＝－6/9.　Supposing

that lμcrosses-6/9 at a certain value ofぐ，sayぐ1jt is

suggested to be of the fbnowing form of the power series

combined with the logarithmic singularity:

RI十こ＝μoξ十μ1ξ21og1 4 1十μ2ξ2十μ3ξ3(log lξ1)2十…･　　9　　　　　　　　　　　　　　　　　　　

(2o)

with 4 ＝ぐーぐ1 where μo is ゛71 arbitray constantl

whneμ1＝64/243δμg，μ2＝352/729δμg and μ3＝

4096/17714yδ2μg. The principle of advancing the expan-4096/1

●　　　　●sion is to determinecoefncients of r(1og lξ1)゛－1(m,n:

positive integers)so as to satisflyEq (15)successively

from the lower oider terms toward the higher-oderones.

Asμo is arbitrary，this expansion may contain other ar-

bitrary const゛nts than μo. These constajntscannot be

specmed by such a local ajnalysisaloneand should be de-

termined globally hl co7｀711ectionWith the matching con-

ditions.

　Apart from this sohltion，wehave a,ngther asymptotic

solution around jr ＝－6/9.　supposing that ，ZI″takes

－6/9 at a certain point ぐ＝ぐ2，itis given in the form of

the power series of ?74/3as follows:

jy＋;1＝z/1 ?74/3＋μ2 ?78/3＋…lＩＩｎ

Ｚ・

ぐ

with 77 ＝ぐーぐ2 where z/1＝－(16/15δ)1/3 and z/2 ＝

(43740/343δ2)1/3.Beca,use the second-order deivative

diverges as ?7→0，wec"｀not foUow (21)to solve Eq (15)

numericany. By trialand error in view of (21)，however，

we can have a solution wh:ich tends to (21)as ?7→0，

while to the other asymptotic solution (20)as 77tends to

a finitevalue of ?7.1f Eq (15)is solved in η＜0，then it

is possible to find such a sohltion that μo becomes con-

sistent with the one (μoQゴー1.310)already obtaixled for

the solution starting from 3″＝－2/9. 1t shows that －RI

increases from 6/9 toward the negative direction of ?7to

attain the maximum －B″ Q;S0.735 and then returns to

6/9 from above at ?7 s －0.260.　VVith this solution，we

ca71connect the solution startingfrom召″＝－2/9 at the

minus innnity ofぐby adjusting the origin ofぐ.Further-

more since the solution (21)is symmetric with respect

to ?7＝0，it can be continued i11to the positive region

of ?7 andeven connected beyond BI＝－6/9 nkewise.

W'ith such continuation，the fun inner solution －,1ris

drawn with the a,xisof symmetry atぐ＝O. Asぐtends to

the plus infinity,thissolution approaches asymptotically

j″＝－2/9｡

　The curvature C of the deflection ｡･47is easny calcu-

lated by (13).According to －jr ＜6/9 or －Ｂ″＞6/9，

C is positive or negative， respectively，so that the de-

flectionis found to be convex downward or upward. 0f

course，the deflection can be obtained by integrating C

twice withrespect toぐ by using the matching conditions

(9).The curva,ture starts from y2/(3φ)at the minus

infinity ofぐ(note that this value is derived from the

qua･dratic term in z for the deflection in (9)，though not

expncitly given there).From the matching conditions，

the deflection takes a very large value へ/2/(919al,,ﾉ6)

SlnCe 9a is sman. But the deflectionderived from (13)gives no more than the higher-oder modification. Thus

as far as the deflection is concerned， the outer solution

prevailsinto the kink to the lowest approximation.

DISCUSSIONS

The hmer solution in the kink has been obtained. But

it should berelnarked that as jr approaches －6/9， C

diverges as (13)suggests. Then the neglect of gaC in

Eq(3)is inadmissible and the solution just obtained be-

comes invalid i】lthe vicinity of the connection point. ln

this case, the fun equations (3)and(4)must be solved by

considering a further“inner region" around j″＝－6/9in

the hlner solutions. ln the verycoreof this region，how-

ever，d23″/dぐ2 and y in (4)may be negl9cted because

the former exhibits the weak logarithmic singularity and

the latter is only just －6/9 as foUows:

ｉ

－19,1C十

　　　１

１ ２

一一2dξ2

d43″一一2o d£4

ぐＥ 　ｎり

　一一

Ｊ－。

う　ー

３一２

　＋

　０

　一一

　Ｃ

３一５

　＋

Their exact solutions can be found as follows:

ｊ″＋４一３

　一一 19， 1ξ2,1oglξ|十z)14十z)242，

Ｃ 一

一

､/(219.1)

３ξ　　’

(22)

(23)

(24)

where Z)l and Z)2＆re arbitrary constants to be deter-

mined in connection with the asymptotic solution (20)･

The solutions (24)are considered to give its hlner solu-

tionsat the very point ，ZEI″＝－6/9.

solutions around(24)，it win be possi

solutions th:rough j″＝－6/9.

By gxpanding the

ble to connect the

CONCLUSION

R)r the stea･dy buckle propagationon anelastic pipeline。

the uniformly valid solutions have been obtained by ap-

plying the idea of the matched-asymptotic expansion

method to the governi】lgequations derived previous￥

The outer solutions suggest propagation of a kink in an

axial scale much longer than a typicallength of a buckle，

whne the inner solutions give the structure of the kink.

But it is stm open within the present lowest innersolu-

tions how the approPriate outer solutions canbe singled

out.

　lnside of the kink， the cross-sectional ovalization is sup-

ported not only by the circumferential bending as in the

outer solutions but also the longitudinal stretching as a

shell.lt is found that the a,bsolute value of the ova,liza-

tion B″ txcetds beyond 2/9 as the critical vaJue given by

the Brazier eflect and attains the maximum value O｡735

approximately. ln contrast with the outer solutions， the

eflective tensions a･nd the inertia do not play a primary

role except for the vicinity oＩＢ’＝－6/9. Around this，a

further inner structure appears where the curvature dl-

verges. These results are the very merit of the dynamical

equations(1)and(2)that cannot be described by the

NONLINEAR WAVES IN SOLIDS 351

energy method used in the Brazier eflect｡

　Fhlally itis concluded that propagation of a kink along

the pipenne can be explained even within the theory of

nonlinear elasticity.But the pipenne resumes its original

shape after the kink has been propaga･ted. Therefore， of

courselno coUapse can be covered by the present theory，

which is the inherent demerit. ″lbestabnsh a physical

mecha”is再nnking between propagation of a kink j゛-71d

ensuing plastic conapse seems to beavery di伍cult prob-

lem but a chanenging one.

ACKNO'VV'LEDGEIMIENT

The author wishes to thank ProfessorT. Kakutani for

his comments on the manuscript.

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