Fate of a bulge in an inflated hyperleastic tube: theory and experiment
FFate of a bulge in an inflated hyperleastic tube: theory andexperiment
Masoud Hejazi ∗ York Hsiang † A. Srikantha Phani ‡ September 2020
Abstract
Mechanical instability in a pre-tensioned finite hyperelastic tube subjected to a slowly increasinginternal pressure produces a spatially localized bulge at a critical pressure. The subsequent fate of thebulge, under continued inflation, is critically governed by the end-conditions, and the initial tension inthe tube. In a tube with one end fixed and a dead weight attached to the other freely moving end, thebulge propagates axially at low initial tension, growing in length and the tube relaxes by extension.Rupture occurs when the tension is high. In contrast, the bulge formed in a tube, initially stretchedand held fixed at both its ends can buckle or rupture, depending on the amount of initial tension.Experiments on inflated latex rubber tubes are presented for different initial tensions and boundaryconditions. Failure maps in the stretch parameter space and in stretch-tension space are constructed,by extending the theories for bulge formation and buckling analyses to the experimentally relevantboundary conditions. The fate of the bulge according to the failure maps deduced from the theoryis verified; the underlying assumptions are critically assessed. It is concluded that buckling providesan alternate route to relieve the stress built up during inflation. Hence buckling, when it occurs, is aprotective fail safe mechanism against the rupture of a bulge in an inflated elastic tube. keyword
Localization, Stability, Bulge, Buckling, Aneurysms ∗ Department of Mechanical engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada † Department of Surgery, University of British Columbia, Vancouver, BC V6T 1Z4, Canada ‡ Corresponding Author: Department of Mechanical engineering, University of British Columbia, Vancouver, BC V6T1Z4, Canada. [email protected] . a r X i v : . [ phy s i c s . m e d - ph ] J a n Introduction
The objective of this work is to understand the fate of a spatially localized bulge, initiated at a criticalinternal pressure in a pre-stretched finite prismatic cylindrical tube made of a hyperelastic material.The bulge can propagate axially along the tube, grow radially in size and rupture, or buckle laterallyin a fail safe manner. A combination of these phenomena can also occur. There is considerable lit-erature on the initiation and propagation of bulges in pressure vessels [24, 2]. Despite their apparentsimplicity, inflated balloons and tubes have produced surprisingly relevant experimental observationsand illustrations for thermodynamic paradigms [32]. Interest in understanding the growth and rup-ture of aneurysms in human blood vessels motivated this study. Here, a local weakening of an arterialwall is presumed to initiate an axially localized bulge, called an aneurysm. In clinical practice, someaneurysms are observed in a laterally buckled state. Evidence suggests that such aneurysms do notrupture despite their size exceeding the minimum recommended size for surgical interventions [7] (seeFig. 1). The material properties of the arteries are complex and often difficult to measure and model in vivo , given the complexity of human vascular anatomy and anisotropic fiber reinforced structureof the blood vessel [20]. Pathogenesis of the aneurysm leads to collagen fiber degradation, reducesanisotropy and diminishes the exponential stiffening response under mechanical stress [37, 22]. Hence,An alternate path to qualitatively understand the phenomenon is to replicate buckled aneurysmsunder controlled settings in a laboratory. Such experiments on an inflated latex rubber tube ( rep-resenting the arterial aneurysm with hyperplastic properties) under an imposed tension are reportedhere, albeit within an isotropic elasticity setting of large deformations in the present study. The postbulge response in these tests is fracture or buckling, depending on the magnitude of pre-tension andboundary conditions. Outcomes in these controlled experiments are compared against the predictionsbased on mechanics models, incorporating diffuse interface modeling (DIM) [27, 28] and bifurcationanalysis [38, 17, 18, 40, 41]. Portraits spanned by axial and circumferential stretch axes and the axialtension and circumferential strain axes emerge from the present analyses, demarcating the dangerousrupture and fail safe buckling regimes.Pressurized cylindrical vessels are found in many technological applications: in oil and gas pipes [35,24], nuclear plants; and also in biology in blood vessels, for example. The stability of pressure vesselsunder combined loading is an active area of research despite its long history [2, 26, 40, 19], not all ofwhich can be surveyed here as the literature spans at least four decades. Pertinent results are recalledhere. That an internally pressurized India rubber tube exhibits a local bulge has been experimentallydemonstrated by Mallock in 1891, who also reported an initial analysis [31]. Bent submarine pipelinesalso show a localized kink buckling, which can transform and propagate when driven by an externalpressure [35, 21, 23]. The bulge problem in a hyperelastic tube has received wide attention since itsorigins in offshore pipelines. Inflating a cylindrical hyperelastic tube under prescribed volume startsfirst with a uniform expansion regime, followed by the formation of a long wavelength bulge along the2 a) (b)
Figure 1: (a) Buckled aneurysm, open source image from: epos.myesr.org/. (b) Unbuckled aneurrysm,open source image from: wikimedia.org.entire length of the tube [2, 23] which localizes at a critical (maximum) internal pressure. In the postlocalization regime, the bulge propagates radially and axially with an associated propagation pressure,below the bulge initiation pressure. This propagation pressure can be estimated using Maxwell’s equal-area rule [2]. Ideally, the propagating bulge invades the entire tube length. In reality, however, thetube can rupture or buckle depending on the boundary conditions at the ends and the initial tension.Experiments on bulge propagation in latex tubes have been conducted under a dead weight attachedto one end and the fluid (air) injected at the other end [25, 26]. They showed the effect of the axialtension on the shape of the bulge profile, maximum pressure, and the circumferential stretch at theonset of bifurcation and used a nonlinear-membrane model [38, 17, 18] to corroborate experiments.The analysis of inflation of a cylindrical tube under an axial load [17, 18] suggests that for a fixed-freecase, the length of the tube monotonically increases during inflation. For a fixed-fixed case, however,the pre-stretched length of the tube is held constant, and the axial force may increase, decrease,or remain constant depending on the pre-stretched length of the tube, or equivalently pre-imposedaxial tension. In the membrane approximation, the influence of bending stiffness on the bifurcationis negligible except for a very thick tube. Membrane assumption has been used in [27] to construct adiffuse interface model (DIM) to address large deformations, and it is asymptotically exact for tubeswith high length to diameter ratio. The application of this method to study bulge formation in tubeswith fixed-free end conditions is pursued in [27, 28]. In an extensive study by Fu and colleagues,3he bifurcation analysis has been revisited by developing a weakly non-linear model to study the post-bifurcation response of infinite tubes [10, 11, 36, 8]. Through these studies, imperfection sensitivity andloading (volume/pressure control) have been investigated. Following these works, a form of bifurcationcondition based on the Jacobian function of the internal pressure and axial force, which in turn areexpressed in terms of the axial and circumferential stretches, is obtained [9]. Using this method, theeffect of multilayering [29] and fiber-reinforcement [39] has been examined. In all of these studies,the footprint of boundary conditions is evident. For instance, it is shown by analysis [39] that thelocalized bulge formation with fixed-free boundary conditions is not possible for fiber reinforced tubes.Furthermore, the bifurcation conditions associated with the bulge have been verified by experiment forboth fixed-fixed and fixed-free boundary conditions [40]. A comparison between the weakly non-linearmodel and membrane theory can be found in [41]. Finally, in passing, one should mention the studieson the bulge formation in dielectric tubes, which use a variational method based on the thermodynamicfree energy of the system [30], see also [5] for an extensive discussion on experimental and analyticalmethods.In most the experiments reported thus far, the lateral buckling does not occur, since the end weightproducing tension is free to move to accommodate unloading in [25, 26], for example. It is possible,both in applications such as an aneurysm or supported pipes, to observe a different boundary conditionwhere both ends are held fixed. This can be accomplished, for example, by holding the tube betweenspatially fixed jaws of a material testing system to impose a predefined axial stretch. Although somestudies incorporated the fixed-fixed boundary conditions and observed post-bulge buckling in limitednumber cases [40, 13]. Very little seems to be known about the post-bulge buckle propagation andformation mechanism in this restrained inflation case, which is the subject of this article. We presentthe experiments first in section 2, followed by an analysis in section 3, a discussion comparing modeland measurements in section 4, and end with concluding remarks in section 5.
A series of quasi-static inflation tests on latex rubber tubes will now be presented to investigate theeffect of boundary conditions on the bulge profile and post-bulge response. First, quasi-static uniaxialtests with different tube lengths are performed to identify the constitutive constants ( α i ) in the Ogden’sstrain energy density function (SEF) W = (cid:88) i =1 s i α i ( λ α i + λ α i + λ α i − , (1)where W is the strain energy function, and λ j are the principal stretches. The measured stress-stretchcurves are fitted following the established procedures to calibrate the SEF parameters based on theuniaxial tensile test [34]. See Appendix A for details. The geometric and material properties of the4amples tested are as given in Table. 1. Note that the wall thickness to diameter ratio is small (Table 1: Geometric and material properties of the rubber tubes studied. The constitutive constants s i and λ i are Ogden’s SEF parameters arising in (1).Internal diameter, D mm Length, 2 L cm Tube wall thickness, H mm Ultimate uniaxial stretch, λ ut . ± . α = 1 . α = 1 . α = 4 . s = 704 . s = 373 . s = 2 . s i in kPa) H/D < .
23) so that the membrane approximation for bulge analysis is justified according to [9] .The bulge experiments fall into two groups with respect to axial loading and boundary conditions:(1) displacement control experiments with a fixed-fixed boundary condition in a material testing sys-tem; (2) force control experiments under a fixed-free boundary condition with a dead load attachedto the free end. The displacement control set up is shown in Fig. 2, where the upper and the lowerends represent the grips of a material testing system. The experiment proceeds in the following stagesas shown in Fig. 2: (a) a pre-stretch is applied to achieve the desired initial axial tension; (b) theinflation starts from the port at the top air inlet, while the pre-stretch is maintained by the machine;(c) the localized bulge forms and propagates; (d) the tube buckles (e) or ruptures (f), depending on theamount of initial tension. Similarly, the force control set up is shown in Fig. 3, and it has four stages:(a) a dead weight is attached to the tube to maintain the desired axial tension; (b) the inflation starts;(c) the localized bulge forms and propagates; (d) the tube ruptures. The main difference betweenthe two categories is the boundary conditions (BCs). In the displacement control (fixed-fixed BCs),the stretched length of the tube is held constant and the axial tension varies during the experiment.However, in the force control (fixed-free BCs) the axial tension is held constant by the dead weightand the length of the tube can change as one of the ends is attached to a slider and hence free to movevertically. A digital camera (Nikon DSLR D7200) camera records the deformation of the tube duringinjection and a pressure transducer records the internal pressure. An edge detection algorithm anddigital image correlation (DIC) in the MATLAB ® [1] image processing toolbox is used for calculatingthe circumferential and the axial stretches, the internal volume based on incompressibility assumption,and the bulge profile for further analysis. For the displacement control experiments, an Instron tensiletester (Model 5965) applies a prescribed pre-stretch by adjusting the relative positions of the grips,which are subsequently held fixed so that the final length is constant. A load cell measures the variableaxial tension during inflation. 5 L z 𝜃 lD GripLoad cell (a) (b) (c) (f)(e)Pressure transducer
Air inlet (inflation begins) l End seal Bulge(d) r 𝐹 e 𝐹 e 𝐹 e Figure 2: Bulging and rupture of a pre-stretched hyper-elastic rubber tube under gradual pneumaticinflation in a typical displacement control experiment. The experimental sequence is: (a) a pre-stretchis applied first and held constant throughout the experiment; (b) The stretched tube is graduallyinflated by increasing the internal air pressure ( P ); (c) Bulge forms at a critical pressure for a givenstretch; (d) Experimentally observed bulge which can either buckle or rupture, depending on the initialstretch; (e) Further inflation leads to buckling for low values of pre-stretch; (f) Further inflation leadsto rupture for higher values of pre-stretch. The axial force ( F e ) is measured in the tensile testingmachine (INSTRON, model 5969) and the pressure P is measured using a transducer near the moving(upper) grip of the tensile tester. Scale bars in (d), (e) and (f) are 1 cm. (Color online) This section considers the analysis of bulging, buckling and rupture. In the bulging analysis, the wallsof the inflated cylindrical vessel are idealized as membranes with negligible bending stiffness, given thelow wall thickness to diameter ratio. The buckling analysis takes bending stiffness, however small, intoaccount. Rupture analysis uses the strain energy density function based failure criterion. Two modelingframeworks are taken up to study bulging. The first, axisymmetric membrane theory relies on a set offour coupled ordinary differential equations (ODE). The second method, based on the diffuse interfacemodel (DIM), replaces the four ODEs by a single scalar master equation. The bifurcation point canbe evaluated efficiently using DIM, while the bulge profile of a finite tube given by the axisymmetricmembrane theory is more realistic. Both methods will be modified in the implementation of the6 L D Mg
Pressure transducerGrip (a) (b) ( d ) ( f )Air inlet End seal l M ( e )(c) ∆L Figure 3: Bulging and rupture of a pre-stretched hyper-elastic rubber tube under gradual pneumaticinflation in a typical force control experiment using a dead weight. The experimental sequence is: (a)Initial unstretched configuration with the upper end fixed; (b) Mass ( M ) is attached to maintain aconstant axial gravitational force; (c) Slow inflation of the tube leads to the formation of a localizedbulge for a given M ; (d) Observed bulge in a typical experiment; (e) Further inflation leads to rupturewithout buckling. Scale bars in (d) and (e) are 1 cm.boundary conditions. Adopting a cylindrical coordinate systems shown in Fig. 4, the triad R = ( R, Θ , Z ) locates a materialpoint in the undeformed configuration, and the triad r = ( r, θ = Θ , z ) the same point in the deformedconfiguration. Making use of symmetry it is sufficient to analyze the region 0 ≤ z ≤ L as sketchedin Fig. 4, where the origin of the co-ordinate system lies at the crown of the bulge.Explicitly accounting for the incompressibility, the thickness averaged principal stretches of theaxisymmetric membrane are ( λ , λ , λ ) = ( λ z , λ θ , ( λ z λ θ ) − ), where the associated thickness averagedstretches are λ z = ds/dS and λ θ = r/R . Note that s and S are respectively along the meridonial arclength of the deformed, and the undeformed tube as shown in Fig. 4. Hence forward, the subscripts1 and 2 are used interchangeably with z and θ , respectively. The constitutive strain energy densityfunction in the reference configuration and the Cauchy stresses follow as W ( λ z , λ θ , λ r ) = W ( λ z , λ θ , λ z λ θ ) ≡ ˆ W ( λ z , λ θ ) , σ i = λ i ∂ ˆ W∂λ i = λ i ˆ W λ i , i = 1 , , (2)where, σ = σ z , σ = σ θ , and the subscript denotes partial differentiation. Note that Lagrange’s7 r r R Z, zsS
UndeformedDeformed hr 𝛳 r o r i Figure 4: Deformed and undeformed configurations of a finite deformation in an axisymmetric mem-brane element sketched for a half of the tube. The total length of the tube is 2 L , and the region0 ≤ Z ≤ L is shown with the origin located at the crown of the bulge. The meridional coordinate ( S inundeformed and s in deformed) and radius R can be function of Z , providing the ability to introduceimperfection. The principal curvatures are κ = − dω/ds and κ = cos ω/r .multiplier is vanished by applying incompressibility condition ( λ r = 1 / ( λ z λ θ )Following Haughton and Ogden [18], one solves the equilibrium equation in r -direction with theboundary conditions at inner surface ( σ r | r = r i = − P ) and outer surface ( σ r | r = r o = 0). Define theinternal pressure as P = − (cid:90) λ o λ i ˆ W θ λ θ λ z − dλ θ , (3)where λ i = r i R i and λ o = r o R o . λ i and λ o are related by the following equation λ i λ z − (cid:18) R o R i (cid:19) ( λ o λ z − . (4)The resultant end force can be obtained by F e = πR i (cid:34) (1 − λ i λ z ) (cid:90) λ o λ i λ z ˆ W z − λ ˆ W θ (1 − λ θ λ z ) λ θ dλ θ (cid:35) . (5)The thin wall assumption ( R i >> H ) so that R i ≈ R o = R , and the incompressibility condition willlead to λ z rh = RH [25]. Consequently, (3) and (5) will reduce to P = 1 λ z λ θ HR ˆ W θ , (6) F e = 2 πRH ( ˆ W z − λ θ λ z ˆ W θ ) . (7)The above equations identify the bifurcation of the uniformly inflated tube to a localized bulge state.The bulge profile is to be obtained. A system of ordinary differential equations using thin wall assump-tion and the finite deformation of the axisymmetric membrane (discussed in [15]) have been derived8or the bulge profile in [26] for a prescribed volumetric inflation under quasi-static conditions: dλ z dZ = R ˆ W zz (cid:104) ( λ θ ˆ W zθ − ˆ W z ) dRdZ + sin( ω )( ˆ W θ − λ z ˆ W zθ ) dSdZ (cid:105) , dλ θ dZ = R (cid:2) λ z sin( ω ) dSdZ − λ θ dRdZ (cid:3) , dωdZ = R ˆ W z (cid:104) cos( ω )( ˆ W θ − λ z λ θ ˆ W z ) + F e πRH λ z λ θ (cid:105) dSdZ , dzdZ = − λ z cos( ω ) dSdZ , (8)where ω is the angle between the normal of the tube wall and r-axis (see Fig. 4), it is assumed that thetube is symmetrical with respect to the plane Z = 0, before and after the bulge formation. We canuse the above set of ordinary differential equations to determine the bulge profile using appropriateboundary conditions. Also, it is possible to include the effect of geometrical imperfection in the modelby introducing an initial nonuniform tube profile. It has been assumed that the strain energy densityfunction (SEF) is defined in terms of principal thickness averaged stretches ( ˆ W ( λ z , λ θ )). The fourboundary conditions ω (0) = 0 , z (0) = 0 , λ θ ( L ) = 1 , λ θ (0) = D max D , (9)define the boundary value problem to be solved for the bulge profile, where λ z depends on λ θ via thesecond differential equation in (8). Z = 0 at the crown of the bulge and Z = L at the end of thetube. Note that the tube is of an undeformed length 2 L . In the above, the tube is subjected to a fixedend force (i.e. dead weight). Accordingly, the axial force (end force F e ), is a constant parameter inderiving the bifurcation condition associated with uniform expansion to localized bulge.In the case of prescribed axial force where dead-weight acts at one end, the bulge can move axiallyand there is no buckling. There are only four unknowns ( λ z , λ θ , ω, z ) in (8), since F e is prescribed.Hence, the four boundary conditions in (9) together with four ODEs in (8) are sufficient. However,in our tests under prescribed volumetric inflation and under fixed grips (displacement control), weobserve that the crown of the bulge Z = 0 does not move axially, but instead the bulge grows insize and then buckles. The above equations in (8) remain valid; however, the boundary conditionsrequire careful implementation, since the end force F e is not constant in (8) but a function of λ θ and λ z at the stationary end z ( L ) = l . One now has to contend with five unknowns ( λ z , λ θ , ω, z, F e ) withan additional condition at z ( L ) = l . The varying F e is implemented incrementally in the numericalsolution using the shooting method. Starting from the beginning of inflation with an initially knowntension F e = F i , we proceed to the next step of incremental expansion by assigning an incrementalchange in the crown diameter D max in λ θ (0) = D max /D and solve for the remaining four unknowns.Then, using (7) we find F e for the next increment. As a side remark, we found that in the MATLAB ® ODE toolbox, the ODE113 solver was found to be more accurate compared with ODE45 and ODE23.After solving the ODE set (8) the associated values for internal pressure and end force can be obtainedfrom (6) and (7). 9 .2 Diffuse interface model
The precise identification of the bifurcation point is sensitive and computationally expensive by solvingthe four coupled differential equations in (8) together with the boundary conditions in (9). Particu-larly, for the fixed grip case, the end force increment size matters. Recent developments, within themembrane approximation, lead to the postulation of the so called diffuse interface model, in whichthe four ordinary equations in (8) are replaced by a single scalar master equation. Besides numeri-cal efficiency, the shooting method can give a more accurate bifurcation point, which motivates theregularized asymptotic problem in the slender tube limit.In the thin wall membrane approximation (
D >> H ), regularized models for slender structureswith non-convex ˆ W can be obtained through an asymptotic expansion by selecting the aspect ratio R/L as the parameter for expansion [27, 28]. The three-dimensional axisymmetric membrane modelis thus reduced to a one dimensional diffuse interface model by including the circumferential stretchgradient, through the relationship between Green-Lagrange strains and the principal stretches. TheGreen-Lagrange strain tensor is defined in terms of axial ( λ z ) and circumferential ( λ θ ) stretches by E = E θ E z = 12 λ θ − λ z + ( Rλ (cid:48) θ ) − , (10)where, by separating the gradient term ( λ (cid:48) θ = dλ θ dZ ) we have E = E ( λ z , λ θ ) + E ( λ (cid:48) θ ) = 12 λ θ − λ z − + 12 Rλ (cid:48) θ ) . (11)Note that here, the gradient of R are assumed to be small. Hence, ω in Fig. 4 is close to 0, s (cid:39) z and λ z = ds/dS (cid:39) dz/dZ . The strain energy density function can now be defined in terms of the first term( E ( λ z , λ θ )) if one neglects the circumferential stretch gradient ( λ (cid:48) θ ) asˆ W ( E ) = ˆ W ( λ z , λ θ ) = (cid:88) i =1 s i α i (cid:18) λ α i z + λ α i θ + ( 1 λ z λ θ ) α i − (cid:19) , (12)where λ z = √ E z + 1 and λ θ = √ E θ + 1.The total potential energy of the system is G = (cid:90) L ( ˆ W πRHdZ − π ( Rλ θ ) λ z P dZ − λ z F e dZ ) . (13)Accordingly, the scaled membrane energy per unit reference length is g ( P ∗ , λ z , λ θ ) = 1 S ini ˆ W − P ∗ e λ z λ θ − F ∗ λ z , (14)where the initial shear modulus in Ogden’s SEF is S ini = 12 (cid:88) i =1 s i α i , the scaled internal pressure P ∗ = P/S ini and the scaled axial force F ∗ = F e / (2 πRHS ini ) and e = R/H is the radius to the10hickness ratio. Note that F ∗ can depend on the boundary conditions and the axial equilibriumthrough (7). The equilibrium conditions follow from ∂g ∂λ z ( P ∗ , λ z , λ θ ) = 0 , (15) ∂g ∂λ θ ( P ∗ , λ z , λ θ ) = 0 . (16)One can solve equation (15) to find the equilibrium value for λ z which in turn can be expressed as afunction of P ∗ and λ θ : λ z = λ ( P ∗ , λ θ ) . (17)Inserting the above in (14) and (16), the membrane energy per unit length ( G ) and circumferentialequilibrium equation follow as: G ( P ∗ , λ θ ) = g ( P ∗ , λ , λ θ ) , (18)and n ( P ∗ , λ θ ) = − ∂g ∂λ θ ( P ∗ , λ ( P ∗ , λ θ ) , λ θ ) = 0 . (19)Solving the above reduced equilibrium condition ( n = 0) gives the homogeneous solution for theexpansion of the cylinder due to the internal pressure. To achieve the solution, we start with equation(15) by tabulating a set of roots in terms of P ∗ and λ θ , which gives us λ = λ ( P ∗ , λ θ ). Afterwards, byemploying the arclength method [27], the reduced equilibrium condition is solved using equation (17).In order to obtain the non-homogeneous solution, we can implement the contribution of the cir-cumferential stretch gradient by assuming E ( O ( (cid:15) )) a small correction to E ( O (1)). The order ofmagnitudes is inline with the finite elasticity theory and under the assumption that L >> R [27]. Thestrain energy density function can be expanded asˆ W ( E ) = ˆ W ( E + E ) = ˆ W ( E ) + ∂ ˆ W∂ E ( E ) . E + O ( (cid:15) ) . (20)By neglecting the higher order terms ( O ( (cid:15) )), an approximate the total potential energy is G ∗ = (cid:90) L G ( P ∗ , λ θ ( Z )) dZ + 12 (cid:90) L G ( P ∗ , λ θ ( Z )) λ (cid:48) θ ( Z ) dZ, (21)which includes the contribution of the membrane energy per unit length G and the contribution dueto stretch gradient G . The stretch gradient contribution evaluates to G ( P ∗ , λ θ ( Z )) = R (cid:34) λ S ini ∂ ˆ W ∂λ z ( λ z , λ θ ( Z )) (cid:35) λ z = λ ( P ∗ ,λ θ ( Z )) . (22)It should be noted that in deriving equation (21), we replace the axial stretch ( λ z ) with the axial stretchfunction (equation (17)) from the homogeneous solution. Accordingly λ z ( Z ) can be approximatedusing the scaled internal pressure ( P ∗ ) and the local circumferential stretch ( λ θ ( Z )). The non-linear11quilibrium condition is obtained from equation (21), with λ (cid:48) θ ( z = 0) = λ (cid:48) θ ( z = l ) = 0 as boundaryconditions, using the Euler-Lagrange method n ( P ∗ , λ θ ( Z )) − ∂G ∂λ θ ( P ∗ , λ θ ( Z )) λ (cid:48) θ ( Z ) + ddZ ( G ( P ∗ , λ θ ( Z )) λ (cid:48) θ ( Z )) = 0 . (23)The above equation can be solved using quadrature or direct numerical integration method to findthe relationship between λ θ and P ∗ . Assuming λ θ ( Z ) = λ θ + (cid:15) ( Z ) a perturbation from homogeneoussolution λ θ , the linear bifurcation analysis, with λ (cid:48) θ ( z = 0) = λ (cid:48) θ ( z = l ) = 0 as boundary conditions,yields the bifurcation condition ∂n ∂λ θ ( P ∗ , λ θ ) = G ( P ∗ , λ θ ) π L , (24)which has to be solved along with the circumferential equilibrium (16).For a fixed-fixed boundary condition, the end force does not contribute to the total potential energyduring inflation. Consequently, G in (13) reduces to G = (cid:90) L ( ˆ W πRHdZ − π ( Rλ θ ) λ z P dZ ) . (25)The effect of boundary conditions can be introduced by defining the constraint on the length ofthe tube as (cid:90) L ( λ z dZ ) − l = (cid:90) L ( λ z − lL ) dZ = 0 . (26)Not that this is the integral form of differential equation dz/dZ = λ z with z ( Z = o ) = 0 and z ( Z = L ) = l as boundary conditions. The Lagrangian of the above system of (26) and (25) with F † reads L = (cid:90) L ( ˆ W πRHdZ − π ( Rλ θ ) λ z P dZ − F † [( λ z − lL ) dZ ]) . (27)In the view of the Lagrangian, the new form of scaled membrane energy per unit length is in thesimilar form of (14) g † ( P ∗ , λ z , λ θ ) = 1 S ini ˆ W − P ∗ e λ z λ θ − F ∗ ( λ z − lL ) , (28)Where F ∗ = F † / (2 πRHS ini ) is the scaled Lagrange’s multiplier. Differentiation with respect to λ z and λ θ reads the same for of axial (15) and circumferential (16) equilibria. Differentiation with respectto F † , the Lagrange’s multiplier, reads λ z − l/L = 0, which leads to trivial solution λ z ( Z ) = l/L . Toavoid the trivial solution in numerical analysis, we use the alternative differential equation dz/dZ = λ Z , (29)where, z ( Z = o ) = 0 and z ( Z = L ) = l impose the boundary conditions at fixed ends. The axialequilibrium (15) should be solved with (29) to get the new form of (17) associated with fixed-fixed12oundary conditions. The diffuse interface model (21) and bifurcation condition (24) will have thesame form as before. However, they have to be solved along with the additional constraint condition(29). 𝐹 ! 𝑟𝑧𝑃 𝑃ℎ 𝜅 = 𝜅 " 𝜅 =0 𝑙 (a) (b) Figure 5: (a) Schematic for the buckling analysis of the uniform section of the tube of length l u . (b)the decoupled uniform section is considered as a tube subjected to axial compressive force F c andinternal pressure P .In this section, we study the lateral buckling of a tube subjected to an axial force and an internalpressure under fixed-fixed boundary conditions using an incremental hyperelastic formulation. In orderto simplify the analysis, we assume a uniform tube without the bulge, by decoupling buckling and bulgephenomena. This approximation is not valid in general but applicable in our case. In practice, bucklingfollows bulge formation. However, it is observed in our experiments that lateral buckling deformationoccurs in the uniform region of the tube, presumably due to the increased bending resistance of thebulged section. The conditions shown in Fig. 5 are assumed to hold. Note that the analysis tofollow does not account for twist-flexure coupling, which can be important if the interest is in postbuckling response. The present analysis is thus limited to the onset of buckling, and is not capableof predicting the post buckling shapes. Also, here we drop the membrane assumption in deriving thebuckling condition. In the numerical evaluation, the thickness averaged stretches with the thickness13veraged stretches ( λ z and λ θ ) can be used. The incompressibility condition is through a Lagrangemultiplier. To conduct the buckling analysis, we follow [14] and [3]. We assume W = W ( F ) as thestrain energy density function for a hyperelastic material, where F is the deformation gradient. Thestress-deformation relation is then given by S = ∂ W ∂ F − Γ F − . (30) σ = F ∂ W ∂ F − Γ I . (31)where Γ is the Lagrange multiplier associated with the incompressibility of hyperelastic materials, S is nominal stress tensor, and σ is the Cauchy stress tensor. Equilibrium equations for the incrementaldeformation field are [33] div ˙ S = 0 , (32)where ˙ S is the incremental nominal stress tensor. As the first-order approximation of the constitutiverelations, ˙ S is ˙ S = Bη + Γ η − ˙Γ I , (33)where ˙Γ is the increment in Lagrange multiplier, η = grad ( u ), u is the displacement field, and B isthe fourth-order tensor of instantaneous elastic moduli. I is the unit tensor. The components of B are given by: B piqj = F pα F qβ ∂ W iαjβ ∂F iα ∂F jβ . (34)For the deformation increment we have η = ∂u r ∂r r ( ∂u r ∂θ − u θ ) ∂u r ∂z∂u θ ∂r r ( ∂u θ ∂θ + u r ) ∂u θ ∂z∂u z ∂r r ∂u z ∂θ ∂u z ∂z . (35)Also, the incremental incompressibility condition is ∂u r ∂r + 1 r ( ∂u θ ∂θ + u r ) + ∂u z ∂z = 0 . (36)Based on the equilibrium at inner surface in the normal direction of the traction vector, the boundaryconditions at the inner surface are˙ S rr = P ∂u r ∂r , ˙ S rθ = P r ( ∂u r ∂θ − u θ ) , ˙ S rz = P ∂u r ∂z . (37)Defining Σ ij = r ˙ S ij , ∇ r = r∂/∂r , we can form a state space system [4] using equations (30), (32), (35),and (36): ∇ r U = M U , (38)where U = [ u r , u θ , u z , Σ rr , Σ rθ , Σ rz ] T is the state vector, with u r , u θ , u z , Σ rr , Σ rθ , and Σ rz being thestate variables, and M is the system matrix of order 6 ×
6, which contains partial derivatives with14espect to θ and z only. Note here that the origin Z = 0 is not at the crown of the bulge but is locatedat one of the ends L = l u as shown in Fig. 5). Assuming a smooth contact at both ends of the tube( z = 0 , L ), the boundary conditions on the incremental traction become T r ( z ) = 0 , T θ ( z ) = 0 , u z ( z ) = 0 , at z = 0 , z = L, (39)where T r ( z ) and T θ ( z ) are the incremental traction vector components in the polar co-ordinates, and u z ( z ) is the incremental axial displacement. To satisfy the above six boundary conditions in (39)and (37), we assume the following displacement vector in the cylindrical polar co-ordinates ( r , θ , z )shown in Fig. 5 U = au r ( ρ ) cos( mθ ) cos( nπζ ) au θ ( ρ ) sin( mθ ) cos( nπζ ) au z ( ρ ) cos( mθ ) sin( nπζ ) p a Σ rr ( ρ ) cos( mθ ) cos( nπζ ) p a Σ rθ ( ρ ) sin( mθ ) cos( nπζ ) p a Σ rz ( ρ ) cos( mθ ) sin( nπζ ) , (40)where a is the amplitude; m = 0 , , , ... denote the circumferential mode number; and n = 0 , , , ... isthe axial mode number; p is a pressure like quantity, having the unit of elastic modulus (i.e. N/m ); ρ = r/r i and ζ = z/L are the dimensionless radial and axial variables, respectively. The equilibriumequation (38) then turns into ∇ ρ U ( ρ ) = M ( ρ ) U ( ρ ) , (41)where ∇ ρ = ρ∂/∂ρ . To reduce the computation time and improve convergence, we can adopt ρ =exp( κ ) following [3]. Accordingly, (41) becomes ∂∂κ U ( κ ) = M ( κ ) U ( κ ) ( κ ∈ [0 , κ ]) , κ = ln ( r o r i ) . (42)The elements of M are given in Table 2 in Appendix B. By discretizing the tube wall uniformly into N number of layers, each of thickness t , we can write κ = N t . Hence, for the j th layer ( j ∈ [1 , N ]) κ j = jt . By employing a layer-wise method for laminated inhomogeneous tubes we can relate statevariables at the inner wall to the corresponding ones at the outer wall [6] as U ( κ ) = DU ( κ j = 0) , (43)where D = Π j = N exp ( M ( κ j ) t ) and Π is the product operator. Note that κ j = 0 corresponds to theinner surface r = r i , and κ = N t corresponds to the outer surface r = r o . In the absence of externalpressure, the boundary conditions at the inner and the outer surface readΣ rr = − Pp ( u θ + u r + αu z ) , Σ rθ = − Pp ( mu r + u θ ) , Σ rz = − Pp αu r ( κ j = 0 , inner surface) , Σ rr = 0 , Σ rθ = 0 , Σ rz = 0 ( κ j = κ = N t, outer surface) , (44)15here α = nπa/L . The above boundary conditions can also be expressed in terms of the displacementby using (44) and the resulting set of linear algebraic equations can be assembled into the form C [ u r ( κ j = 0) , u θ ( κ j = 0) , u z ( κ j = 0)] T = ⇒ det( C ) = 0 . (45)Elements of the matrix C are expressed in terms of m, n, P, λ z , λ θ (see table 3 in Appendix B). Here,we consider m = 0 and n = 1 , , ..., to obtain the bifurcation condition for the lateral mode observedin the experiment. Solving the bifurcation condition (equation (45)) for the applied internal pressuresone obtains a pressure contour plot in λ θ - λ z plane identifying the critical pressures that cause bucklingfor any chosen point in the λ θ - λ z plane.The analysis can be summarized as follows in two steps: (1) obtain the bulge profile of λ θ and λ z is obtained as a function of maximum bulge diameter ( D max /D ) and internal pressure ( P ) using (8)and (23); (2) use equation (45) to check the buckling condition for the internal pressure and the λ z and λ θ in the uniform region of the bulged tube. Note that l u is updated in each step to account forthe change in the length of the uniform section of the tube due to bulge propagation. The assumeduniform circumferential and axial stretches in (45) hold only in the uniform region, in Fig. 5. This section compares and contrasts the results from experiments and the analyses. Consider thefixed-free test shown in Fig. 3 where the tube is fixed at one end and a mass M is attached at theother end to impart initial tension. Quasi-static inflation of the tube shown in Fig. 6 begins with auniform expansion stage, followed by the initiation of the bulge. The bulge first expands mostly radiallygrowing in diameter and then axially growing in length. The engineering axial strain is plotted againstthe circumferential strain measured at the maximum diameter section in Fig. 6(a). The insets showdifferent stages of inflation, mentioned earlier. It should be noted that the increase in length withprogressive inflation is significant once the bulge is initiated around D max /D = 1 .
4. During thepropagation stage, there is a considerably high increase in length for almost constant circumferentialstrain. For the test shown, the end stage is axial yielding without rupture, since the applied initialtension is small. If one increases the tension, the qualitative picture remains the same except that athigh circumferential strains the rupture occurs. For higher values of initial weight (or pre-stretch) thebulge leads to rupture as shown in Fig. 6(b).The inflation test data is shown in Fig. 7 for a displacement control test (see Fig. 2). The onset ofbulging at a peak pressure is followed by rapid unloading first and then buckling in these tests, as iden-tified in the force-pressure plots. Increasing the initial axial pre-stretch λ i in the tube in Fig. 7(a) leadsto a significant decrease in the critical pressure for bulging and a somewhat reduced propagation pres-sure. Repeatability of experiments across three different specimens can be ascertained from Fig. 7(b).In these series of tests where the initial pre-stretch is low, the buckling protects against rupture as a16 𝐷𝐷 𝑚𝑚𝑚𝑚𝑚𝑚 / 𝐷𝐷 Bulge expansion Axial propagationUniform inflation 𝐷𝐷 𝑚𝑚𝑚𝑚𝑚𝑚 𝐷𝐷 ∆ 𝐿𝐿 / 𝐿𝐿 𝐷𝐷 𝑚𝑚𝑚𝑚𝑚𝑚 / 𝐷𝐷 ∆ 𝐿𝐿 / 𝐿𝐿 (a)(b) Uniform inflation Onset of rupture Ruptured
Figure 6: Deformation stages in a force control test (Fig.3): (i) Uniform inflation, (ii) Bulge expansion(radial propagation), (iii) Axial propagation or rupture. The added mass is M = 800 g in (a) andthe bulge propagates axially without rupture. In (b) the added mass is M = 3500 g and the ruptureoccurs during axial propagation. (Color online)fail safe mode. It is also possible to observe bulging and rupture at higher values of pre-stretch λ i asshown in Fig. 7(c).Fig. 8 shows the pressure variation during the inflation of the tube subjected to fixed-fixed boundaryconditions. As discussed above, the bulge forms when the pressure reaches its maximum value. Theentire deformation process is video recorded and the frames of this video are used for subsequent imageanalysis. For the three different stages indicated in (Fig. 8a), the axial and circumferential stretcheshave been calculated by performing digital image correlation (DIC) [1] in MATLAB (see Fig. 8b).At the onset of bulge formation (point A) in Fig. 8(b), we see the localization of the axial stretch atthe middle section of the pre-stretched tube, while the entire length of the tube is still under tension17 λ z > λ z < λ θ - λ z ) plane, measured at the uniform section A-A, are shown in Fig. 9. The analyticaltrajectory (dash-dotted line) for the bulge initiation has been constructed based on the axisymmetricmembrane analysis using (8) and the bulge initiation is based on DIM using (24). Repeating thisanalysis for multiple values of the pre-stretch will give the locus for the critical values of principalstretches associated with bulging. The resulting curve is shown as the dashed line which demarcates thebulging region. Similarly, the bifurcation condition (solid line) for lateral buckling is constructed basedon (45). The parameters listed in Table. 1 are used to construct these analytical curves. Experimentaldata for the principal stretches for three different specimens are also shown. DIC analysis is used todetermine λ z and λ θ , all with the same initial pre-stretch λ i = 1 .
05. From A to B, we have the uniforminflation and the experimental data points agree with the analytical prediction. Point B, is the onsetof bulging which occurs when the trajectory hits the bulging condition (dashed line). Note that theexperimental data points show a slightly larger λ θ at the bifurcation point which may be due to thefact that analytical prediction is stiffer as membrane approximation is used. After the bulge forms,the uniform section (A-A) relaxes and as we saw in Fig. 8, the compressive stress builds up and λ z decreases. The buckling occurs at a lower compressive axial stretch λ z , presumably due to imperfectboundary conditions.The uniform section (A-A) and bulged section (B-B) have dissimilar behavior in the post bulgeformation regime as shown in Fig. 10 for two different pre-tensions. The membrane axial tension( T = 2 πrhσ ) has been normalized using the initial shear modulus ( S ini ), tube length, and diameter.The differences can be observed in the deviation of membrane axial tension during the bulge expansionat these two sections as shown in Fig. 10(a). At section A-A outside the bulge, the membrane tensionis relieved by bulging and if the tensions F i is small, buckling sets in and relieves the axial tension.However, if the initial tension is large, for F i = 42 .
34 N, the initial decrease in the membrane tensiondue to bulging is not significant, and this lack of stress relief leads to rupture as there is no buckling.18 i = 1.3 λ i = 1.2 λ i = 1.1 λ i = 3F e (N) F e (N) Onset of rupture
Onset of bulging
Propagating bulge
𝑃 (𝑘𝑃𝑎)𝑃 (𝑘𝑃𝑎)λ i = 1.1 𝑃/𝑃 𝑚𝑎𝑥 𝐹 𝑒 /𝐹 𝑒𝑚𝑎𝑥 Onset of bulging Onset of buckling(a)(b)(c)
Figure 7: Bulging and buckling under displacement control: data from experiments ( P and F e aredefined in Fig.2): (a) Different values of pre-stretch ( λ i ), (b) Three different samples with an identicalpre-stretch, (c) Force-pressure curve associated with the rupture at high pre-stretch. (Color online)Note that at Section B-B in the bulged region the membrane tension increases, as it should due tothe inflation, and hence consistent with the growing radial size of the bulge. The initial tension, F i , isthus a critical parameter that governs the fate of the bulge. This observation leads to the constructionof the failure map shown in Fig. 10(b) which portrays the influence of F i . Four distinct regions exist19 e 𝜋 H 𝐷𝑆 𝑖𝑛𝑖 P 𝑆 𝑖𝑛𝑖 A: Onset of bulging B: Propagating bulgeC: Onset of b uckling λ i = 1.6 BB AA (b) 𝜆 𝜃 𝜆 𝑍 𝜆 𝜃 𝜆 𝜃 𝜆 𝑍 𝜆 𝑍 A B C (a)
Figure 8: (a) Axial tension-pressure data for displacement control test with bulging and buckling withthree stages identified as A, B and C for digital image correlation (DIC) analysis. (b) DIC contoursfor axial stretch ( λ z ) in the top row and the circumferential stretch ( λ θ ) in the bottom row. The labelsA, B,C correspond to the points identified in (a). Notice that compressive stress ( λ z <
1) emerges atC in the top row outside the bulged section. (Color online)for the post bulge response and they are as identified. At higher values of F i rupture is the fateof a bulge as the tube is inflated, or λ maxθ is increased. The rupture criterion is based on the SEF20 xisymmetric bifurcation conditionLateral bifurcation condition A-A
Bulge initiation λ θ λ z A BC D
Figure 9: Failure map for displacement control: Dotted line shows the behaviour of the intact section(A-A) during inflation and bulging on the plane of principal stretches. The experimental data pointsare shown along the analytical dotted line. The blue region (separated by dashed line) shows thebulging regime and the yellow region (separated by solid line) is the buckling regime. When dottedline reaches the bulging regime, it experiences compressive force induced by bulged section. Thecompressive force directs A-A to the buckling regime. (Color online)value at ultimate stretch ( λ ut ) in the uniaxial tensile test [16, 12]. These quantities are measuredexperimentally. According to [16], failure energy during the biaxial loading is half of the maximumenergy density of the uniaxial tensile test ˆ W max = 7 . J/mm = 0 . W ( λ ut ). Using this, we plot therupture envelope in F i - λ maxθ , shown as the dashed line in Fig. 10(b). The solid line shows the lateralbuckling envelope based on (45). The points shown for the experiments correspond to the end pointsof the test. The test is terminated when buckling or rupture occurs. Note that each experimentaldata point corresponds to different initial tensions F i , and there are 38 experiments with two rupturecases. It can be concluded that the calculated envelopes for buckling and rupture are in satisfactoryagreement with experiments. The post buckling can involve twisting of the buckled and bulged tube.This twisting is evident in Fig. 10 and is beyond the scope of the present analysis.We can show the stress relief due to buckling by comparing the von Mises stress on the bulgedsection at the onset of buckling and the early stage of buckled state (see Fig. 11). The principalstretches are calculated through the DIC method explained above. The stretches are calculated in theplane of buckling and the stretch distribution due to buckling has been approximated using the shapemodes in (40). Using the strain energy function, we obtain the principal stresses and von Mises stress.As shown in the Fig. 11, the peak stress at the crown of the bulge reduces once the buckling occurs.21 𝜋 H 𝐷𝑆 𝑖𝑛𝑖 A-A Exp. Ana.B-B Exp. Ana.
A-AB-B R up t u r e B u c k li ng 𝜆 𝜃 𝑚𝑎𝑥 Bulge Initiation F i = 42.43 N F i = 26.37 N (b) Experiment data points Buckling regionNo Rupture
No BucklingRupture region 𝐷 𝑚𝑎𝑥 𝐷 F i 𝜆 𝜃 𝑚𝑎𝑥 Buckling &Rupture region (a) RB 𝜋 H 𝐷𝑆 𝑖𝑛𝑖 Figure 10: (a) Axial membrane tension ( T = 2 πrhσ ) changes at sections A-A (open square markers)and B-B (open circular markers) until the bifurcation occurs due to bulge formation. Accordingly, themembrane axial tension increases in section B-B and decreases in A-A. In case of rupture (point R in(b)) the tension is preserved in A-A, however, in buckling case (point B in (b)) A-A loses the tensionand buckles due to compressive stresses. (b) Rupture-Buckling failure map. The experimental datapoints show λ maxθ = D max /D at the onset of rupture or buckling. Rupture and buckling zones areidentified by yellow and blue regions, respectively. The failure map in (b) shows that: (i) for a buckledbulge, the rupture is observed at a higher diameter ratio, (ii) by increasing the initial axial tension,rupture occurs without buckling. (Color online) The fate of a bulge initiated in an inflated cylindrical hyperleastic (latex rubber) tube has beenexamined through experiments and analyses. Within the membrane approximation limit, a justifiedassumptions for the experiments reported here, the bulge can either buckle or rupture, depending on(a) boundary conditions and (b) pre-existing axial stretch or tension. The diffuse interface model and22 a)(b) 𝜎 𝑣 /𝑆 𝑖𝑛𝑖 Figure 11: von Mises stress for (a) onset of buckling and (b) early stage of bucklingaxisymmetric membrane analyses, coupled with lateral buckling analysis produces a portrait spannedby the axial tension and circumferential strain axes demarcating the dangerous rupture and fail safebuckling regimes. Experimental verification is presented for the theoretical predictions. Bucklingprovides an alternate route to relieve the stress built up during inflation. Hence, buckling is a protectivefail safe mechanism and it delays rupture. Further analysis in the post buckling regime incorporatingtwist and experiments on rupture after buckling can be pursued in the future. While the present workhas been limited to isotropic elasticity, work is in progress on anisotropic reinforced layered tubes withbranches to make stronger connections with the biological problem of aneurysms.
Acknowledgement
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The parameters α i and s i in Ogden’s SEF defined in (1), are identified from a uniaxial tensile testconducted using Instron (Model 5965). The procedure followed is according to [34]. For the uniaxialtensile test we have λ = λ and from incompressibility we have λ = λ = λ − . The nominal stress isgiven by S = ∂ ˆ W ( λ, λ − ) ∂λ . (46)The experimental data is fitted to (1) as shown in Fig. 12 and the identified parameters are given in Axial stretch (λ) N o m i n a l s t r e ss ( S ) Ogden's SEF Model
Figure 12: Uniaxial tensile test data compared with calibrated Ogden’s SEF in (1).Table. 1. 28 ppendix B: Elements of the state space matrix for bucklinganalysis
The non zero elements of M ( ρ ) in (41) are given in Table. 2 following [3], where α = nπa/L , Γ is theLagrange’s multiplier associated with inncompressiblity, p is a pressure like quantity from (40), and B piqj can be obtained from (34). Note that m and n are the circumferential and axial mode numbersassociated with (40). To obtain elements of M ( κ ) in (42), we use ρ = exp( κ ) to affect the changevariable. Table 2: Non-zero elements of M ( ρ ).ine M = − M = − m M = − αρ ine M = ( B +Γ) mB M = B +Γ B M = P B ine M = ( B +Γ) mB αρ M = p B ine M = B + B − B +2Γ p + (cid:104) B p − ( B +Γ) B p (cid:105) m + (cid:104) B p − ( B +Γ) B p (cid:105) α ρ M = (cid:104) B + B + B − B +2Γ p − ( B + p ) B p (cid:105) mM = ( B + B − B − B +Γ) αρp M = 1 M = − ( B +Γ) mB M = − ( B +Γ) αρB ine M = (cid:104) ( B + B + B − B +2Γ) p − ( B +Γ) B p (cid:105) mM = B p − ( B + p ) B p + ( B + B − B +2 p ) m p + B p α ρ M = ( B + B + B − B − B +2Γ) p mαρ M = m M = − ( B +Γ) B ine M = − ( B + B − B − B +Γ) p αρ M = ( B + B + B − B − B +2Γ) p mαρM = B p m + ( B + B − B +2Γ) p α ρ M = αρ Non-zero elements of C in (45) are given in Table. 3, where P is the internal pressure, and D = Π j = N exp ( M ( κ j ) t ) in (34). Note that κ j = 0 corresponds to the inner surface r = r i , and κ = N t ( N is the number of wall layers of thickness t ) corresponds to the outer surface r = r o .Table 3: Non-zero elements of C .C i = D j − Pp ( D j + D j m + D j α ) C i = D j − Pp ( D j m + D j ) C i = D j − Pp D j α i = j − , j = 4 ,,
The non zero elements of M ( ρ ) in (41) are given in Table. 2 following [3], where α = nπa/L , Γ is theLagrange’s multiplier associated with inncompressiblity, p is a pressure like quantity from (40), and B piqj can be obtained from (34). Note that m and n are the circumferential and axial mode numbersassociated with (40). To obtain elements of M ( κ ) in (42), we use ρ = exp( κ ) to affect the changevariable. Table 2: Non-zero elements of M ( ρ ).ine M = − M = − m M = − αρ ine M = ( B +Γ) mB M = B +Γ B M = P B ine M = ( B +Γ) mB αρ M = p B ine M = B + B − B +2Γ p + (cid:104) B p − ( B +Γ) B p (cid:105) m + (cid:104) B p − ( B +Γ) B p (cid:105) α ρ M = (cid:104) B + B + B − B +2Γ p − ( B + p ) B p (cid:105) mM = ( B + B − B − B +Γ) αρp M = 1 M = − ( B +Γ) mB M = − ( B +Γ) αρB ine M = (cid:104) ( B + B + B − B +2Γ) p − ( B +Γ) B p (cid:105) mM = B p − ( B + p ) B p + ( B + B − B +2 p ) m p + B p α ρ M = ( B + B + B − B − B +2Γ) p mαρ M = m M = − ( B +Γ) B ine M = − ( B + B − B − B +Γ) p αρ M = ( B + B + B − B − B +2Γ) p mαρM = B p m + ( B + B − B +2Γ) p α ρ M = αρ Non-zero elements of C in (45) are given in Table. 3, where P is the internal pressure, and D = Π j = N exp ( M ( κ j ) t ) in (34). Note that κ j = 0 corresponds to the inner surface r = r i , and κ = N t ( N is the number of wall layers of thickness t ) corresponds to the outer surface r = r o .Table 3: Non-zero elements of C .C i = D j − Pp ( D j + D j m + D j α ) C i = D j − Pp ( D j m + D j ) C i = D j − Pp D j α i = j − , j = 4 ,, ,,