Effect of axial prestretch on inflation instability in finitely extensible thin-walled tube

Transcription

1 BULLETIN OF APPLIED MECHANICS 0(36), 5-0 (04) 5 Effect of axial pestetch on inflation instability in ftely extensible thin-walled tube L. Honý Abstact Mechanical esponse of elastic ateials undegoing lage defoations exhibit seveal kinds of a loss of stability. Such a situation ay occu, fo instance, when a thin-walled cylinde is inflated by an intenal pessue. The loss of stability is anifested by a non-onotonic elationship between the inflating pessue and the cicufeential stetch of the tube. The esults, known fo the liteatue, show that hypeelastic ateials with so called liiting chain extensibility popety always exhibit a stable esponse fo sufficiently sall values of the liiting extensibility paaete. In othe wods, apid lage stain stiffening pevents elastoeic ateials fo an onset of instability. The pesent study deonstates how axial pestetch of the thin-walled tube affects the stability of the defoation. It is shown that fo axial pestetches = (non-pestetched),.05,.,.,.5,, 3, and 5 the stable inflation is obtained fo ateials with = 9., 0.3,.55, 4.06, 3., 47.43, 87.3, and 07.8, espectively. Fo which can be concluded that the highe the pestetch is, the oe copliant ateial will show stable inflation. It was also found that the highe the pestetch is, the lowe is the pessue at which the instability occus. Index Tes Axial pestetch, Hypeelasticity, Inflation instability, Liiting chain extensibility, Pessuiation. M I. INTRODUCTION ATERIALS undegoing lage elastic defoations exhibit any non-self-evident phenoena in thei echanical behavio. Soe of the ae linked to a loss of stability duing defoation. Fo instance, non-onotonic inflation of thin-walled cylindes and sphees, kink foation on a twisted od, o winkling of the suface of the bent block ae good exaples [-4]. Taditionally, they have been discussed with efeence to ubbe-like ateials; howeve, possible applications to the echanics of soft tissues (ateies, veins, skin), which inheently undegoing lage defoations, has also been ecently ecognied [4-0]. A typical exaple is the instability of a pessuied thinwalled cylindical tube, which can anifest in two diffeent ways. The fist is the onset of buckling instability anifested by a tansvese deflection (in the bioechanics of blood vessels this is usually efeed to as totuosity) siila to the buckling seen in a long slende colun [6-0]. H.C. Han and cowokes investigated this poble in detail and developed atheatical expessions elating citical pessue to axial load, tosion couple, and geoety [6, 8-0]. Anothe echanis by which pessuied tubes becoe unstable is though non-onotonic inflation [,, -5]. It is a well-known phenoenon to all who have inflated a patyballoon. In this situation the adius of the balloon onotonically inceases in contast to applied pessue. The balloon appeas to be stiff at tial defoations, howeve, the pessue-stetch elationship eaches a local axiu at a cetain defoation. Afte the local axiu is passed, the pessue-stetch elationship fos a deceasing function and ay each a local u, which can be followed again by an inceasing function, depending on the constitutive odel used. In expeients, non-onotonic inflation ay be accopanied with localied defoation (bulging a loss of constant adius along the axis of the tube) which esebles aneuys foation in ateies [-5]. Non-onotonic inflation is one of the ost studied instabilities in the context of ubbe-like ateials and will be of ou concen. We will focus on the behavio of thin-walled cylindical tubes with closed ends ade fo a ateial undegoing apid lage stain stiffening. Kanne and Hogan [] ecently investigated this poble in isotopic hypeelastic tubes exhibiting liiting chain extensibility. Liiting chain extensibility involves statistical polye echanics that explain lage stain stiffening of elastoe ateials based on the fte extensibility of acoolecula chains. In [], a phenoenological analogue of olecula odels based on a logaithic fo of the stain enegy density function, which was oiginally poposed by A.N. Gent [6] (see also [7] fo olecula statistical intepetation, and [8, 9] fo details of the application in ateial bioechanics), was used. It has been shown that, depending on the specific value of the liiting extensibility paaete, non-onotonic inflation of thin-walled cylindical tubes ay o ay not occu []. To be oe specific, cylindical tubes ade fo the ateial with < 8. ae always stable duing a pessuiation and tubes with > 8. exhibit local axiu followed by local u. Kanne and Hogan [] investigated tubes opeating in thei natual state that is without any axial foce except the foce induced by the pessue acting on the closed ends of the tube. Thee ae, howeve, situations when tubes ae axially Manuscipt eceived August 0, 04 and accepted fo publication Septebe, 04. This wok has been suppoted by the Cech Msty of Health in the poject no. NT 330. Lukáš Honý is with the Faculty of Mechanical Engineeing, Cech Technical Univesity in Pague, Technická 4, 66 07, Pague, Cech Republic. Tel.: , eail: lukas.hony@fs.cvut.c

2 BULLETIN OF APPLIED MECHANICS 0(36), 5-0 (04) pestetched in a ounting. One nice exaple ae blood vessels. In ou bodies, ateies ae longitudinally pestained. This fact, although known since 880 due to Roy [0], did not eceive lage scientific attention in copaison with the cicufeential stiffness and distensibility of ateies, but at pesent its significant physiological ole coes to light [9, -7]. In the pesent study, the effect of axial pestetch on the inflation instability will be investigated fo two points of view. It will be shown that axial pestetch enables ateials with > 8. to be stable in the inflation. Fo one specific ( = 5), it will be also shown that inceased pestetch leads to the deceased intenal pessue in the onset of the instability. A. Constitutive odel II. METHODS The ateial of the tube will be consideed to be incopessible and hypeelastic chaacteied by the stain enegy function W defined pe unit efeence volue. In such a case the constitutive equation can be witten in the fo () [8]. W T p I F () F Hee denotes the Cauchy stess tenso. F is the defoation gadient defined as F = x/ X, whee x and X, espectively, denote the position vecto of a ateial paticle in the defoed and the efeence configuation. p plays the ole of a Lagangean ultiplie, which epesents the hydostatic contibution to, not captued by W, due to the incopessibility constaint. Ove the last decades, seveal odels fo W have been developed to descibe the echanical behavio of elastoes and soft tissues unde lage stains. We will focus ouselves on the so called liiting chain extensibility odel poposed by Gent in 996 [6]. It will be used in the fo () which has appeaed in the studies of ateial echanics [8,9]. 3 ln I W Hee is the shea odulus at inftesial stains. I denotes the fist invaiant of the ight Cauchy-Geen stain tenso C (I = tace(c)), whee C = F T F (altenatively one can also use left Cauchy-Geen stain tenso b, b = FF T, because I (C) = I (b)). is efeed to as the liiting extensibility (diensionless) paaete because it esticts adissible defoations of the ateial to the doain whee I 3 < applies. In othe wods, I 3 iplies W. Thus fte extensibility of a acoolecula chain is, in the phenoenological appoach, captued by a suitable atheatical fo of the stain enegy (logaithic function). () Regading inflation instability, Kanne and Hogan [] found that thee is a citical value of, = 8., which disciinates behavios of pessuied thin-walled cylindical tubes. The inflation is stable (onotonically inceasing pessue fo inceasing cicufeential stetch) fo ateials with < 8.. Wheeas fo ateials with > 8., thee is a local axiu followed by local u that is subsequently followed by a steeply inceasing section of the pessue-stetch cuve (see Figue 5 in []). Substituting fo () into () one obtains constitutive equations in the fo (3). (3) 3 b p I I B. Inflation-extension of the thin-walled tube Conside a long thin-walled cylindical tube with closed ends that, in the efeence configuation, has a iddle adius R, thickness H, and length L. Assue that duing pessuiation, the otion of the ateial paticle located oiginally at (R,, Z), which is sufficiently distant fo ends, is descibed by the equations suaied in (4). R h H Z (4) Hee and h espectively denote defoed iddle adius and thickens. The equations (4) expess the fact that the tube is unifoly inflated and extended (o shotened) and that it is not twisted. The stetches k (k =,, ) ae the coponents of the defoation gadient F, F = diag[,, ], and fo the ight Cauchy-Geen stain tenso C = diag[,, ] applies. In this kineatics, the invaiant I has the fo (5). I (5) The ateial of the tube is consideed to be incopessible, thus the volue atio, = det(f), gives equation (6) expessing =. (6) Equilibiu equations of a thin-walled tube with closed ends loaded by an intenal pessue P and axial (pestetching) foce F ed can be witten in the fo (7). Hee,, and espectively, denote the adial, cicufeential and axial coponent of the Cauchy stess tenso. P P Fed P (7) h h h The cobination of constitutive equation (3), I (5), and equilibiu equations (7) gives the syste of equations govening the inflation-extension esponse (8 9). 6

3 BULLETIN OF APPLIED MECHANICS 0(36), 5-0 (04) 7 P p 3 P p 3 h Fed P p 3 h h (8) (9) (0) 3 P P 3 (4) Fo the fist equation, (9), it can be concluded that the Lagangean ultiplie p has the fo (). p P 3 () Now we will substitute () into (9) and (0). Pio doing so, the application of the equations (4a) and (4b) enable us to wok with known efeence geoety. Beaing in ind the incopessibility condition (6), we also eliinate explicit dependence on, = //. Afte soe siple algeba, we aive at equations govening the inflation-extension esponse: () fo cicufeential diection, and (3) fo axial diection. 3 P PR H 3 3 P Fed RH 3 PR H () (3) As the final step, the equations () and (3) will be divided by to obtain diensionless pessue. In (4) and (5), the notation = H/R is also intoduced. 3 P 3 Fed P R C. Siulation of the inflation-extension esponse (5) The equations (4) and (5) will be used to siulate echanical esponse of intenally pessuied and axially pestetched thin-walled tube. The siulations will show how axial pestetch affects cicufeential stetch and intenal pessue at the onset of instability (SIM ), and that iposing of the pestetch elevates the value of which disciinates between stable and unstable echanical esponse of the tube (SIM ). All the coputations wee pefoed nueically in the copute algeba syste MAPLE 8 using the coand fsolve and inceenting by The siulations wee conducted in two steps. In the fist step, tial cicufeential stetch and pestetching axial foce F ed wee solved with pescibed axial pestetch and intenal pessue P = 0. In the second step, was inceented wheeas F ed hold constant, and unknown P and wee obtained by the fsolve pocedue. SIM was pefoed fo =,.0,.05,.05,.075,.,.5,.,.3 and.3 with = 5. In SIM, =,.,.,.5,, 3, and 5 wee used and was inceented fo 8. to the value at which local axiu appeaed fo the fist tie. In SIM, fou significant digits wee consideed in the pocedue.

4 BULLETIN OF APPLIED MECHANICS 0(36), 5-0 (04) 8 III. RESULTS AND DISCUSSION Pedicted inflation-extension esponses fo SIM ae in the Figue (P/ in the panel A, and P/ in B). The Figue depicts the esults obtained in SIM (P/ in the panel A, and P/ in B). Figue The inflation-extension esponse in SIM. In SIM, = 5 was pescibed. Note that inceased axial pestetch led to deceased P/ and at the onset of instability which is indicated by solid cicles. Fo >.3, the instability does not occu. The last cuve, bown colo, was copute fo =.3. Figue The inflation-extension esponse in SIM. When tubes opeate axially pestetched, they will exhibit stable esponse fo highe than in non-pestetched case. Pesented values of ae uppe bounds showing stationay point. They wee obtained in the coputations with inceented by 0.00 and with fou significant digits consideed. The positions of local axia with espect to cicufeential and axial stetch, noalied pestetching foce and diensionless intenal pessue ae detailed in the Figue 3 fo SIM, and fo SIM in the Figue 4.

5 BULLETIN OF APPLIED MECHANICS 0(36), 5-0 (04) 9 Figue 3 SIM esults. A Diensionless pessue and cicufeential stetch at the local axiu. B Cicufeential stetch and axial stetch at the local axiu. C Noalied pestetching foce and diensionless pessue at the local axiu. D Axial and cicufeential stetch duing the pessuiation. The esults of the siulations deonstate that the highe the axial pestetch is, the lowe is the intenal pessue at which the instability occus. The fact that local axiu pessue deceases in SIM could be expected because the cuves ae obtained fo diffeent values of. Odinaily, inceases with inceasing pestetetch which iplies that the doain of adissible defoations extends and the ateial, fo the cetain point of view, could be egaded as oe copliant. Howeve, the pessue at local axiu deceases also in SIM. The siulations clealy deonstate that by the iposing of the sufficient axial pestetch, ftely extensible tube is pevented fo the onset of instability (SIM, Figue ). Although the intenal pessue at the local axiu deceases with axial pestetch, the cicufeential stetch, and thus distensibility of the tube, inceases (SIM, Figue and 3B). In case of SIM, thee is, howeve, local u fo =. and =.55 (Figue 4B). The dependence of the uppe bound of fo stable esponse was fitted by quadatic polynoial = 6.464( ) with coefficient of deteination equal (Figue 4A). Fo the sake of copleteness, it should be noted that the pestetching foce inceases with the inceent in axial pestetch. Figue 4 SIM esults. A Dependence of axiu with the stable inflation on axial pestetch. Obtained data is fitted by = 6.464( ) B The position of the local axia in the cicufeential and axial stetch diaga. C Dependence of the position of local axia on pestetching foce and diensionless intenal pessue. D Axial and cicufeential stetch duing the pessuiation. In contast to Kanne and Hogan [], u ensuing stable esponse of the non-pestetched tube was found to be 9. (in [] 8. is pesented). The diffeence in the esults coes fo two souces. The fist is the fte thickness of the tube enteing ou coputation via = 0.. Anothe eason fo utual isatch is a diffeent handling of the adial stess. In the pesent study, it was assued that = ½P, howeve in [] plane stess condition = 0 is used. Finally it should be einded that the onset of inflation instability ay in expeients be accopanied with a localied defoation (bulging of the tube). Thus pesented shapes of esulting cuves in post-instability doain should be stictly consideed as only estiates of the tue behavio. Siilaly to post-buckling behavio analyses, the deteination of the tue defoed geoety should leave the assuption of unifo cylindicality [-5].

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