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1 Mechanics of Composie Maerials, Vol. 47, No. 2, May, 2 (Russian Original Vol. 47, No. 2, March-April, 2) VISCOELASTIC BEHAVIOR AND DURABILITY OF STEEL- WIRE - REINFORCED POLYETHYLENE PIPES UNDER A HIGH INTERNAL PRESSURE S. G. Ivanov,* A. N. Anoshkin 2, and V. Yu. Zuyko 2 Keywords: polyehylene pipe, seel wire skeleon, viscoelasiciy, elasoplasiciy, shor-erm srengh, durabiliy The srengh ess of seel-wire-reinforced polyehylene pipe specimens showed ha, under a consan inernal pressure exceeding 8% of heir shor-erm ulimae pressure, he fracure of he specimens occurred in less han 24 hours. A pressures slighly lower han his level, some specimens did no fail in a year and a half. The analyical model developed for describing he mechanical behavior of such pipes considers ha polyehylene is viscoelasic and seel is elasoplasic. This allows one o evaluae heir shor-erm srengh as well as heir durabiliy under a high inernal pressure. The experimenal resuls obained in srengh ess are explained by he redisribuion of sresses beween he wo maerials of he reinforced pipe. Calculaions were carried ou using he MahCAD sofware. Resuls of Pipe Tess The echnology of polyehylene pipes reinforced wih a seel wire skeleon has been developed and inroduced for he firs ime a he Mepos company (Ekaerinburg, Russia). A presen, such pipes are manufacured by several Russian companies, in paricular, Polimak JSC (Ekaerinburg) and Gazprom Transgaz Savropol LLC (Savropol). The wires in he skeleon are fasened by spo welding a heir inersecion poins (Fig. ). The resuls of a finie-elemen modeling of sresses in he wire skeleon and in he polyehylene body of pipe specimens, obained in ess for he shor-erm srengh, are discussed in []. The ess for he shor- and long-erm srengh of pipe specimens manufacured by Polimak JSC, wih an ouer diameer of 4 mm, including he nonreinforced hickened zone of a bu weld of polyehylene, were carried ou a he Perm Universiy of Twene, Neherlands 2 Perm Sae Technical Universiy, Russia *Corresponding auhor; ivanovsgen@gmail.com. Translaed from Mekhanika Kompozinykh Maerialov, Vol. 47, No. 2, pp , March-April, 2. Original aricle submied Sepember 23, // Springer Science+Business Media, Inc. 93

2 Fig.. Seel skeleon of a reinforced plasic pipe. Specimen Fig. 2. Specimen of a reinforced pipe afer shor-ime ess under inernal pressure. TABLE. Resuls of Tess on Pipe Specimens for he Long-Term Srengh Specimen p cons, МPа p cons /p lim % Time o failure and is characer h, a longiudinal crack wih a broken reinforcemen h, a longiudinal crack wih a broken reinforcemen Did no failed in 7 days (.37 4 h) Did no failed in 63 days (. 4 h) 8 6. (during 298 days (7. 3 h), hen increased o (during 298 days), hen increased o 7 Did no failed in 298 days a a pressure of 6. MPa; on increasing o 8.4 MPa, failed in hours a a polyehylene seem 94

3 Fig. 3. Failure of specimen 8 (according o Table ) along a welded unreinforced join. Sae Technical Universiy. Specimens, 2, and 3, loaded wih an inernal pressure a a rae of MPa/min, in shor-erm srengh ess, failed a, 3, and. MPa, respecively. The ulimae pressure describing he shor-erm srengh of he pipe, averaged over hree specimens, was p lim =.2 MPa. A ypical paern of failure of a pipe specimen is illusraed in Fig. 2, where a longiudinal crack wih broken circular coils of he reinforcemen is shown. In ess for he long-erm srengh (Table ), he inernal pressure grew o a cerain level p cons and was hen mainained a his level up o failure of specimens. We should noe ha, a a loading level higher han a cerain value (~8% of he shor-erm srengh), he failure occurred wihin 24 hours wih he break of reinforcemen, as in he case of ess for he shorerm srengh (see Fig. 2). However, a a pressure below his level, specimens 6 and 7 did no fail even in one year and a half. The purpose of he presen sudy is o explain his fac and o predic he long-erm srengh of such pipes a high loading levels. Taking ino accoun he fac ha polyehylene is viscoelasic, bu he greaer par of load is aken up by he seel wire, we may assume ha he long-erm srengh of he pipes a high load levels, in he course of ime, is caused by redisribuion of he load beween he polyehylene and he wire wound in he circumferenial direcion. The desrucion characer of specimen 8 indicaes ha, a load levels lower han ~8% of he shor-erm srengh, for he given sandard-size pipes, he creep failure of polyehylene in he axial direcion can occur in he region of a nonreinforced hickened join (Fig. 3). For predicing he failure according o his mechanism, compeing wih he break of he circular reinforcemen, daa on he nonlinear behavior of polyehylene in creep up o failure are necessary. Properies and Deformaion of he Wire in he Circular Direcion The wire skeleon of a pipe of ouer diameer 4 mm had he following parameers: wire diameer 2r = 3 mm, median diameer of he wire ring 2R = 34 mm, and he winding sep a = 8 mm. Figure 4 shows ension diagrams of he wire in he iniial sae and of a wire aken ou of he pipe, i.e., weakened by conac welding. For our calculaions, we used a simple piecewise linear wo-secion approximaion of he ension diagram (Fig. ). The elasic region is followed by a region of a consan sress σ = σ max a a srain εs ε εlim. Here, ε s = σ max E, where E is he elasic modulus of seel. The parameers σ max and ε lim are chosen so ha o describe he weakes wire, since is desrucion leads o he failure of neighboring wires and o he formaion of a longiudinal crack. I is seen ha he conac welding affecs he ension diagram significanly. Upon wising he wire ino a ring, some par of he maerial occurs in he region of plasic deformaion. The longiudinal deformaion of bending due o winding ε rr =.22 is disribued linearly over he cross secion of wire. Using 9

4 8, МPa a 7, МPa b Fig. 4. Deformaion diagrams of he wire in he iniial sae (a) and afer he conac welding (b). 6, МPa max 4 2. lim Fig.. Approximaion of he deformaion diagram of wire in acive loading., МPa y, mm 2 2 Fig. 6. Disribuion of he sress across he wire hickness a averaged srains equal o (), ε s 2 (2), ε s (3), 3ε s 2 (4), and 2ε s (). 96

5 F s,n / s Fig. 7. Tensile force F s as a funcion of he srain ε averaged over he cross secion of he wire. he ension diagram of he wire (see Fig. 4) and assuming ha he diagram σ ( ε) in compression is similar, we come o a relaion beween he longiudinal srain of he wire afer winding and he coordinae y. A he cener of cross secion of he wire, y =. Afer imposing on he wire he ensile deformaion from he inernal pressure in he pipe, an elasic unloading occurs in he compressed region of cross secion. Figure 6 shows he disribuion of sresses over he cross secion of he wire a differen values of he average srain (ensile srain from he inernal pressure). The disribuion of he sresses was described by he piecewise linear funcion σmax Eε f E y σmax f E y ε σmax, ε σ( y, ε) = + + R + R + E 2ε s, σmax, ε > 2ε s, () where ε is he srain averaged over he cross secion. f, x <, ( x)= x, x,, Inegraing Eq. () over he cross-secional area of he wire, we obain a relaion beween he force F s which sreches he wire and ε (Fig. 7). This relaion was approximaed by he piecewise linear funcion F s E E A A f sε, ε < 2ε s, ( ε)= s ( ε ( ε 2εs ))= 2 2 EAsεs = σmax, ε 2εs. (2) Viscoelasic Deformaion of Polyehylene In [2], he viscoelasic properies of he polyehylene used for pipes manufacured by winding of a polyehylene ape reinforced wih a seel wire were deermined from creep ess a six levels of consan sresses, from 4 o 4 MPa. The circumferenial srain in he reinforced pipes before originaion of a crack was raher small, of an order of magniude 2%. For such a level of srains, he nonlinear viscoelasic behavior of polyehylene was approximaed in [2] by he linear viscoelasic model 97

6 TABLE 2. Parameers of he Linear Viscoelasic Model [2] and he Calculaed Values of he Relaxaion Kernel in Uniaxial Tension Parameer i = i = i = 2 i = 3 i = 4 i = τ i, s 3 4 G i, MPa G /G i /β i A i /β i where s ij are componens of he sress deviaor, s from [2]): E The kernel K G σkk () δij εij ()= + ij G τ ij τ τ K G s ()+ K s ( ) d ( ), (3) 9 2 ij σ kk = σij δ ij. 9K () in Eq. (3) was aken as a sum of five exponens: K G G ()= exp. (4) i= Gi τi τi The values of τ i and G i in Eq. (4), found by he mehod of leas squares, are presened in Table 2. For he case of uniaxial ension, we have from relaion (3) ε ()= σ ()+ K τσ ( τ) dτ E ( ). () The insananeous modulus E can be expressed in erms of he consan G and he Poisson raio ν =.42 (aken = 2 ( + ν ) G. The parameers of he kernel K () in Eq. () can be expressed in erms of parameers of he kernel KG (): E K()= G K G () 3 E, KG ()= Bi exp( γ i), Bi = 3G τ, γ i = τ i. i= The funcional inverse o (), wih subscrips omied, is σ ()= E ε() Γ( τετ ) ( ) dτ, Γ()= Ai exp( β i). (6) i= The consans of he kernel Γ( ) in (6) were calculaed from hose of he kernel K () by using he known relaions [3] i i Ai = i= βi γ j, Bi = i= β j γi, j =,,, (7) and heir values are also presened in Table 2. Le us find wha esimae for he level of axial srain ε * reached in he polyehylene join by he insan of failure or he end of he ess is given by he linear model. For he diameer of bu join 64 mm, he calculaions by Eq. () give he following values of ε * for specimens 4-8 (see Table ): 2.4, 3.3, 3.8, 3., and 3.6%, respecively. 98

7 As seen, he srain level in specimen 8 is higher han in oher ones, excep for specimen 6, which did no fail. However, he disincion is insignifican and apparenly lies wihin he saisical sraggling of he ulimae srain for a welded polyehylene join. As already menioned, for predicing he failure in a nonreinforced hickened join, i is necessary o have daa on he nonlinear behavior of polyehylene in creep up o failure, as well as daa on he srengh of welded joins. Viscoelasoplasic Deformaion of a Reinforced Pipe Le us consider he deformaion of a pipe under an inernal pressure according o a simplified model, assuming ha he circular reinforcemen is preliminary deformed elasoplasically and he polyehylene is linearly viscoelasic (he presence of he longiudinal reinforcemen is negleced since he load from he inernal pressure is aken up mainly by he circular reinforcemen). Le us now examine he equilibrium of a nonuniform ring of recangular cross secion of widh a ( a is he disance beween wire coils in he pipe), hickness h, and inner radius R, loaded wih an inernal pressure p. The ring conains a seel wire ring wih a median radius R and cross-secional area As = π r 2 ; he radius of he wire is r. The cross-secional area of he ring occupied by polyehylene is A = ha A. The equilibrium equaion of such a ring has he form pe s pr a = Fs ( ε) + Fpe( ε ), (8) where F s ( ε ) is he force falling on he seel wire (2), and F pe ( ε ) is ha falling on he polyehylene par of he ring. For simpliciy, we assume ha he srain in he polyehylene ring is homogeneous and equal o he average crosssecional srain in he wire. Thereby we pass on o he consideraion of ension of a nonuniform rod wih a seel core, assuming ha boh he seel and he polyehylene in his core are in a uniaxial sress sae. Le us designae he longiudinal srain of he core by ε. F s ( ε ) is deermined by Eq. (2), whereas F pe ( ε ), wih accoun of Eqs. (6), is given by Fpe ( ε)= ApeE ε() Γ( τετ ) ( ) dτ. (9) Le us consider he case ε 2ε s. Equaion (8), wih accoun of Eqs. (2) and (9), can be pu ino he form where p()= ksε()+ kpe ε() Γ τετ ( ) dτ = k ε () Γ k s AsE = k 2 Ra pe ApeE = Ra ( ) ( τετ ( ) k = ks + kpe, Γ k ()= pe Γ(). k ) d τ, () A ε 2ε s, he deformaion can be presened as he funcional inverse o () ε ()= p()+ τ p( τ) dτ k K ( ), () where Γ () is he resolven of he kernel K (); he parameers of he kernels are conneced by relaions similar o Eqs. (7). The ime a which ε ( s )= 2ε s is designaed by s. Then, Eq. (8) can be wrien as p Fs ( ε ( )) Ra kpe ε Γ ( τ) ε( τ) dτ. (2) () = () 99

8 ( ). 2., s Fig. 8. Circumferenial srain ε as a funcion of ime a he consan loading rae of he pipe p ()= MPa/min. For he case ε > 2ε s, we have from Eq. () where p()= p() 2ε s ksṫ s ε()= p()+ ( τ) p( τ) k ε ( τ) dτ + ( τ) p( τ) k K ( s ) K dτ pe, (3) s Example of Esimaion of he Shor-Term Srengh and Durabiliy of a Reinforced Pipe a a High Pressure Le us consider he example of calculaion of a pipe of ouer diameer 4 mm wih he following geomerical parameers: R = 7 mm, R = 67 mm, h = 3 mm, a = 8 mm, and 2r = 3 mm. The lower esimae for he ulimae pressure upon is shor-erm increase can be found from Eq. (8) by neglecing he second erm in he righ-hand side, which is responsible for he conribuion of polyehylene o he load-carrying abiliy of he pipe: plim( ) = 2ε k s s s For he parameer σ max = 6 MPa of he approximaing diagram of he seel wire (see Fig. 4), his esimae gives p lim( s) = 9.3 MPa. A a lower pressure, no desrucion due o he redisribuion of sresses is possible. Now, we will ake ino accoun he conribuion of polyehylene o he load-carrying abiliy of he pipe. As a crierion of exhausion of is load-carrying abiliy, we choose ε ε, (4) () lim where ε lim =.8%. Le us consider he loading a a consan speed p = MPa/min = /6 MPa/s. Using Eqs. () and (3) and crierion (4), we find he ulimae pressure a a consan rae of is increase: p lim = 2.6 MPa. In his case, he conribuion of he seel wire o he magniude of breaking pressure in shor-erm loading up o failure makes 74%. The ime-dependen srain of he pipe a a consan rae of loading by inernal pressure is illusraed in Fig. 8. The kink in he srain diagram is explained by he kink in he calculaed diagram for he seel wire. The pressure a his poin is p ( s ) =.2 MPa. 2

9 . 2 lim. ( ) a... 2 lim. ( ) b, s.., s Fig. 9. Deformaion of he wire in long-erm srengh ess a p cons =.8 MPa (a) and p cons2 =.3 MPa (b). Now, we will consider he deformaion of he seel wire during long-erm ess, when he pressure is raised wih a consan rae p up o a given level p cons and hen is held consan a his level up o failure. Analyically, he pressure in such ess is described by he piecewise linear funcion of ime p p f ()= p cons. () p Insering Eq. () ino Eq. (), we obain a relaion beween he srain and ime for ε 2ε s. In calculaing he ime s a which he srain reaches he level 2ε s, we employ Eq. (3) o consruc he ime srain relaionship a ε > 2ε s. The relaionships obained are shown in Fig. 9a for p cons =.8 MPa and Fig. 9b for p cons2 =.3 MPa. Relaed o he esimaed ulimae pressure p lim = 2.6 MPa in shor-erm ess, hese pressure levels are pracically equal: p p cons lim % = 86%, p p cons lim 2 % = 82%. Using hen crierion (4), we come o a prediced durabiliy T = 47 min in he firs case and T 2 = 8 h in he second one. All he calculaions were carried ou in he MahCAD package. 2

10 Conclusions The analyical model suggesed for esimaing he shor-erm srengh and durabiliy of highly loaded pipes reinforced wih a seel skeleon yields quie good resuls in comparison wih experimenal daa. The calculaed shor-erm srengh is somewha overesimaed compared wih he experimenal one: 2.6 MPa agains.2 MPa, averaged over hree ess. For a more accurae esimae, more complee daa are needed on he deformaion of a wire weakened by conac welding. We should also poin o he wide scaer of experimenal daa on he shor-erm srengh. REFERENCES. S. G. Ivanov, L. L. Srikovskii, M. A. Gulyaeva, and V. Yu. Zuiko, Modeling he mechanical behavior of mealreinforced plasic pipes under inernal pressure, Mech. Compos. Maer., 4, No., (2). 2. M. P. Kruijer, L. L. Warne, and R. Akkerman, Modeling of he viscoelasic behavior of seel-reinforced hermoplasic pipes, Composies: Р A, 37, (26). 3. V. V. Moskviin, Srengh of Viscoelasic Maerials [in Russian], Nauka, Moscow (972). 22