Study of bifurcation in a pressurized hyperelastic membrane tube enclosed by a soft substrate

Accepted Manuscript Study of bifurcation in a pressurized hyperelastic membrane tube enclosed by a soft substrate N. Varatharajan, Anirvan DasGupta

PII: DOI: Reference:

S0020-7462(16)30297-9 http://dx.doi.org/10.1016/j.ijnonlinmec.2017.05.004 NLM 2844

To appear in:

International Journal of Non-Linear Mechanics

Received date : 4 November 2016 Revised date : 25 March 2017 Accepted date : 15 May 2017 Please cite this article as: N. Varatharajan, A. DasGupta, Study of bifurcation in a pressurized hyperelastic membrane tube enclosed by a soft substrate, International Journal of Non-Linear Mechanics (2017), http://dx.doi.org/10.1016/j.ijnonlinmec.2017.05.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Study of bifurcation in a pressurized hyperelastic membrane tube enclosed by a soft substrate N. Varatharajana,∗, Anirvan DasGuptaa,b a Centre b Department

for Theoretical Studies, Indian Institute of Technology Kharagpur, 721302, India of Mechanical Engineering, Indian Institute of Technology Kharagpur, 721302, India

Abstract We model a perivascular supported arterial tube as a uniform cylindrical membrane tube enclosed by a soft substrate, and derive the solution bifurcation criterion. We assume the surrounding soft substrate as an elastic foundation with distributed stiffness. We consider the tube to be a neo-Hookean material with isotropic and anisotropic (orthotropic) properties, and study solution bifurcation at a constant axial stretch. In the isotropic case, the surrounding soft substrate can substantially delay the onset of bifurcation through a subcritical jump in circular distension at bifurcation with increasing substrate stiffness. Introduction of anisotropy can significantly change the jump behaviour from subcritical to supercritical. Keywords: Bifurcation, Aneurysms, Hyperelastic membrane tube, neo-Hookean material.

1. Introduction Modeling and analysis of nonlinear bifurcation phenomena in biological soft tissues under stress is a challenging issue. It depend on various factors, including inhomogeneity, age, stresses, anisotropy, non-linearity, and incompressibility etc. Aneurysm is a localized bulging (bifurcation) in blood vessels which may lead to rupture, and even death [1, 2, 3]. Elastic and transitional arteries are susceptible to form aneurysms in the human circulatory system. Marfan’s syndrome is a disorder in the connective tissues, fibrillin, due to misfolding of fibers, which forms an abnormal structure [4, 5, 6]. It can be found in the skin, lungs, skeletal system, ocular system and cardiovascular system [7]. These nonlinear deformations are symptoms of dilation and weakening of biological tissues, and tear due to high blood pressure [7]. A healthy arterial wall is highly deformable and exhibits nonlinear stress-strain response with exponential stiffening at high pressures [8]. On the contrary, Marfan’s syndrome affected arterial tissues show an isotropic material behaviour and non-monotone uniaxial stress-strain relation with a double curvature, referred to as localized strain softening [9, 10, 11]. We can classify instability into two types, namely limit-point instability, and geometric instability. The limitpoint instability is a characteristic of a material. An elastomer material model exhibits limit-point instability when a local maximum exists in the pressure versus stretch curve [12, 13, 14, 15]. On the other hand, a geometric instability is characterized by a change in the nature of solution in the form of an unexpected localized bulging or change of global shape [9, 10, 16, 17, 18, 19]. This is commonly known as bifurcation. When a membrane tube is inflated, a solution with uniform radial distension is expected. However, this solution may eventually lose its geometric stability and exhibit bulging, buckling, twisting and more complex shapes [12, 20, 21, 22, 23]. It is important to note that a bifurcation may lead to localized solutions or blowup solutions as well. The criterion for the localized solutions and the range of existence of the localized solutions have been analyzed in [24]. It has been shown that a bifurcation can occur even if there is no limit point for an open ended tube with fixed axial stretch. An axisymmetric bulging occurs after the pressure maximum in the closed end tubes. An axisymmetric bifurcation is observed sooner in longer tube (compared to its diameter) than in a short tube for a given axial extension [25]. A bifurcation can occur before, at, or after the pressure maximum in a closed or open ended tube with fixed axial loading, depending on the length of the tube and axial stretch [25, 26]. Chater and Hutchinson [20] have described the propagation of a bulge along a long party balloon. The longitudinal growth of the bulge is due to a reduction in the internal pressure. The propagation pressure (required to maintain a quasi-static bulge) is lower than the pressure at the onset of bifurcation [26, 27, 28]. The importance of radial expansion and wall thickening (increase in thickness than in radius) in the stability of the aneurysms has also been studied [29]. The propagation of bifurcation has been observed as a competition between radial expansion and axial propagation [22, 24, 29]. ∗ Corresponding Author: N. Varatharajan, Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, 721302, India. E. Mail: [email protected], Tel:+91-9994725902

Preprint submitted to Elsevier

May 16, 2017

Haughton and Ogden [25] have obtained the bifurcation conditions corresponding to the bulging and prismatic modes. Later, Haughton [30] derived the integrated equations for the axisymmetric membrane tube from the equilibrium equations. Alternatively, the variational principle has been used to derive the equilibrium equations [22, 31]. Fu et al. [24] used the integrated equations to derive the bifurcation condition. Finite/infinite cylindrical membrane tubes with isotropy/anisotropy was studied in [32], and the bifurcation criterion was derived using incremental equation approach. The bifurcation condition for two cylindrical tubes with different radii joined by a single transition zone was obtained in [26]. Many authors have focused on the bifurcation of a membrane tube configuration subjected to axial loading and internal pressure [9, 10, 22, 24, 25, 28, 33]. Derivation of bifurcation condition [32], and numerical analysis of propagation of bifurcation [34] in neo-Hookean cylindrical membrane tube under constant axial stretch has been presented. The arterial prestretch are found as 1.33, 1.23, 1.08 and 1.05 for the age 20, 30, 60 and 70 years, respectively [35, 36]. The arterial strain and fiber extensibility are found to decrease with increase in age [37, 38]. The effect of rigid planar obstacle on saccular aneurysms has been studied [39]. Modeling and further study of perivascular constraints are worthy since many properties (the mechanical properties, interfacial conditions, boundary conditions and different tissues) are unclear [40, 41]. Pamplona and Mota [42] described an experimental and numerical study of a circular hyperelastic membrane over an elastic foundation. It is observed that the surrounding connective tissues with sufficient stiffening can arrest the growth of tears in an arterial model [43]. The interaction of an arterial wall surrounded by cerebral spinal fluid was modeled as a membrane tube on an elastic substrate [44]. Internally pressurized tubular structures are common in humans and other biological systems. They are usually surrounded by soft tissues that provide support and help in maintaining shape. In most of the previous studies, the effect of these surrounding tissues on the stability of internally pressurized tubular structures has not been considered. Here, we model an arterial tube with connective tissues as a membrane tube surrounded by an elastic substrate with the above mentioned biological age related axial prestretch to study the bifurcation of solution under circular distension. In this article, the kinematic conditions and the equilibrium equations are derived in Section 2. The bifurcation criterion is obtained using the equilibrium equations, in Section 3. In Section 4, the bifurcations in isotropic and anisotropic infinite cylindrical membrane tubes with surrounding soft substrate have been analyzed. The paper is concluded with Section 5. 2. Kinematics of deformation Consider a homogeneous, hyperelastic cylindrical membrane tube with the polar coordinates in the reference and deformed configurations given by (X 1 , X 2 , X 3 ) = (R, Θ, Z) and (x1 , x2 , x3 ) = (r, θ, z), respectively. The position vector of the cylindrical membrane tube in the reference (deformed) configuration is represented by eE ˆ R + Ze E ˆ Z (x = reˆ ˆ R, E ˆ Θ and E ˆ Z (ˆ X=R er + zeˆ ez ), 0 ≤ Θ, θ ≤ 2π, where E er , ˆ eθ and ˆ ez ) are the unit vectors in the radial, angular and axial directions of the undeformed (deformed) cylindrical membrane configuration. The latitude, meridian and normal directions to the deformed membrane tube are the principal directions. If the inflation is axisymmetric, then the deformation gradient for cylindrical tube is given by F= λr ˆ er ⊗ ˆ R + λθ ˆ ˆ Θ + λz ˆ ˆ Z , where λθ , λz and λr are the stretches in the circumferential, axial and radial E eθ ⊗ E ez ⊗ E directions, respectively. The strains are expressed in terms of the right Cauchy-Green deformation tensor 2ˆ ˆR ⊗ E ˆ R + λ2 E ˆ ˆ ˆ C = λ2r E θ Θ ⊗ EΘ + λz EZ ⊗ EZ . The principal stretches are obtained in the circumferential, axial √ e h 0 and thickness directions as λθ = Rree , λz = re02 + ze02 and λr = H e , respectively. Here, (·) denotes derivative with e respect to Z. 2.1. Material strain energy

We consider the cylindrical tube as an incompressible, thin and hyperelastic material. The corresponding e is defined over the volume in the reference configuration as strain energy U Z e e e c dZ, e U = 2π RH W (1) e L

c=W c (λθ , λz , λr ) is the strain energy per unit volume in the reference configuration. The incomprewhere W sibility property of the membrane material gives the constraint I3 = λθ λz λr = 1. Hence, the strain energy f (λθ , λz ) = W c (λθ , λz , 1 ). The first two invariants of the strain tensor density function may be expressed as W λθ λz

2

of the right Cauchy-Green deformation tensor are I1 =trC = λ2θ + λ2z + λ2r ,  1 1 1 1 I2 = (trC)2 − tr(C2 ) = 2 + 2 + 2 . 2 λθ λz λr

If the cylindrical tube is reinforced, then it is necessary to define further invariants, given by I4 = m · Cm, I5 = m · C2 m, I6 = n · Cn, I7 = n · C2 n, ˆ Θ + cos φE ˆ Z and n= − sin φE ˆ Θ + cos φE ˆ Z are two unit vectors in the reference configuration where m= sin φE of the cylindrical membrane tube. These vectors represent the directions of the two families of fibers, and φ is the angle between the fibers and the longitudinal axis direction. Here, I4 ≡ I6 = λ2θ sin2 φ + λ2z cos2 φ and I5 ≡ I7 = λ4θ sin2 φ + λ4z cos2 φ. 2.2. Substrate strain energy We assume that the arterial tube is surrounded by soft tissues which may be modeled as an elastic substrate. A linear distributed radial stiffness around the cylindrical membrane tube is considered. The pressure due to an elastic substrate Pef is given by Pef = e k w, e e e = k(e r − R),

(2)

e + w, where re = R e w e is the deflection in the radial direction, and e k (N/m3 ) is the linear stiffness of the substrate. The strain energy per unit volume is the area under the stress-strain curve and it can be written as  Z Z e f Pf d w e 2πe rde z, Wf = e L Z e 2 reze0 dZ, e = πe k (e r − R) (3) e L

e is the length of the cylinder in undeformed configuration. where L 2.3. Potential energy of the gas

If the gas pressure inside the cylindrical membrane tube is Peg and the total volume enclosed by the inflated tube is v, then the potential energyRofR the gas is given by Peg v. The volume in the deformed configuration for the uniform membrane tube is v = 2πe rde rde z . Thus, the potential energy of the inflating gas is given by Z fg = −Peg e W πe r2 ze0 dZ. (4) e L

The circumferential and axial components of the Cauchy stress tensors are denoted by σ eθθ and σ ezz respectively, and given as Ped re Ped re σ eθθ = and σ ezz = , (5) e h 2e h

where Ped (= Peg − Pef ) is the inflation pressure. That is, the inflation pressure Ped is the balanced pressure between the gas inside the cylindrical membrane tube and elastic substrate. It can be obtained from (5)1 as e W fλ H θ Ped = . e λ λ R θ z

The gauge pressure in the cylindrical membrane tube is given by e f e + H Wλθ . Peg = e k(e r − R) e λθ λz R

3

(6)

2.4. Derivation of equilibrium equations The equilibrium equations can be obtained from the total energy integral, which is the sum of the integrals (1), (3) and (4), as follows # Z " e Peg 2 0 k 2 0 e e f e e E =2π RH W− re ze + (e r − R) reze dZ, eH e eH e e 2R 2R L # Z " e fg P k 2 0 2 0 ˜= f− e reze dZ, e E W re ze + (e r − R) (7) eH e eH e e 2R 2R L

e= where E

E eH e. 2π R

Taking variation of the energy integral (7), we have

# e fg P k 2 0 2 0 f e e W− re ze + (e r − R) reze dZ eH e eH e e 2R 2R L ) !0 Z "( f e e fλ re0 fg W k k P Wλ θ z e rze0 + e 2 ze0 δe − reze0 + (e r − R)e (e r − R) r = − e eH e eH e eH e λz e R R R 2R L ( !0 ) # e fλ ze0 fg re2 W P k z e 2 re − − + (e r − R) δe z dZe eH e eH e λz 2R 2R "( ) ( ) # e fλ re0 fλ ze0 fg re2 W W P k z z 2 e + δe r+ − + (e r − R) re δe z . eH e eH e λz λz 2R 2R

e=δ δE

Z "

e L

e = 0 and gives the equilibrium equations as The above variation integral is extremized when δ E !0 e e fg fλ fλ re0 P k W W z θ e rze0 + k (e e 2 ze0 = 0, − − reze0 + (e r − R)e r − R) e e e e e e e λ z R RH RH 2R H e fλ ze0 fg re2 k W P z e 2 re = C1 , + (e r − R) − e e e e λz 2RH 2RH

where C1 is a constant. The boundary conditions are

fλ re0 W z = 0, λz

e fλ ze0 fg re2 W kf P z e 2 re = 0, − + (e r − R) e e eH e λz 2RH 2R

e = ±L. e For non-dimensionalization, we consider the following definitions in the interval Z

where

µ 2

e e e e → RH, e ze → Ru, re → Rw, Ze → RZ, H µ fg → µ p, p → Pg , e f → µ W, k→ P HK, W e 2 H 2 2R

is the material parameter (defined in Section 4). The equilibrium equations become Wλθ −



Wλz w0 λz

0

K (w − 1)2 u0 = 0, 2 Wλz u0 Pg w 2 K − + (w − 1)2 w = C1 , λz 2 2

− Pg wu0 + K(w − 1)wu0 +

(8)

along with the boundary conditions Wλz w0 = 0, λz Wλz u0 Pg w 2 K − + (w − 1)2 = 0 at Z = ±L. λz 2 2

4

(9)

The dimensionless gauge pressure is given by Pg = K(w − 1) +

Wλθ . λθ λz

3. Bifurcation condition We assume that the bifurcation initiates at an arbitrary point, which defines the origin Z=0. The bifurcation is assumed to be symmetric about the cylindrical tube at the origin. Hence, we can consider w Z=0 = w0 , u Z=0 = 0, (10) 0 0 0 w = 0, u = u0 ≥ 0, (11) Z=0

Z=0

at bifurcation. The trivial solution of the deformed cylindrical tube is w0 = w∞ and u00 = 1, where w∞ is the value of w at infinity for the uniform cylindrical tube. Equivalently, if the circular distension at bifurcation and at infinity are equal, then we have a usual uniformly deformed cylindrical tube configuration. The necessary condition for a non-trivial solution (bifurcation) to exist is when the equilibrium equations (8) have solution(s) other than w0 = w∞ and u00 = 1. Using the conditions in (10) - (11), the equilibrium equations in (8) may be rewritten as g1 ≡ Wλθ − Pg w0 u00 + K(w0 − 1)w0 u00 + g2 ≡ Wλz −

K (w0 − 1)2 u00 = 0, 2

Pg w02 K + (w0 − 1)2 w0 − C1 = 0. 2 2

(12)

Through (12), the values of w0 and u00 corresponding to a trivial equilibrium state are functionally related. Any perturbation to the equilibrium state should be trivial for stability. However, a non-trivial perturbed solution of (12) exists whenever the determinant of the Jacobian of (12), denoted by f , vanishes, i.e.,

We have

f ≡

∂g1 ∂w0 ∂g1 ∂u00

∂g2 ∂w0 ∂g2 ∂u00

= 0.

2  K f = Wλz λz [Wλθ λθ − Pg λz + 2K(λθ − 1)λz + Kλθ λz ] − Wλz λθ − Pg λθ + K(λθ − 1)λθ + (λθ − 1)2 , 2 where λθ = w0 , λz = u00 , and Pg = K(w0 − 1) + λ2z Wλz λz

Wλθ (value at equilibrium) at bifurcation. Then, w0 u00

2    K 2 2 λθ Wλθ λθ − Wλθ + K(λθ − 1)λθ λz + Kλθ λz − λθ λz Wλz λθ − Wλθ + (λθ − 1) λz = 0 2

(13)

at bifurcation. Equation (13) is the required bifurcation condition for an infinite membrane tube on an elastic substrate. For an infinite membrane tube with K = 0, we recover back the usual bifurcation condition [24, 25, 26, 32] 2

λ2z Wλz λz (λθ Wλθ λθ − Wλθ ) − λθ (λz Wλz λθ − Wλθ ) = 0.

(14)

4. Bifurcations in an infinite tube on elastic substrate The polymeric chain extensibility of a material differs in longitudinal and circumferential directions [45]. The limiting chain extensibility concept can be expanded to limiting extensibility in the fiber direction. The structure of the fiber reinforcement of an artery can be described in the two preferred directions. The fiber reinforced limiting extensibility models are suitable for explaining inflation and extension of an artery [46]. We consider the reinforced neo-Hookean strain energy density function, which can be derived using limiting fiber

5

extensibility, in the form   f = µ (I1 − 3) + ρ(Ii − 1)2 , W 2

i = 4, 5,

where µ > 0 is the shear modulus in the undeformed configuration and ρ > 0 is a constant that measures the strength of reinforcement in the fiber direction [13, 46, 47, 48, 49, 50, 51]. Firstly, we study the bifurcation of a perivascular supported arterial tube as an infinite isotropic membrane tube on an elastic substrate. To study the effect of the substrate, we fix λz and vary K in steps of ∆K = 0.001. For each increment in K, we numerically determine the circular distension λθ for bifurcation using the criterion (13). We consider the axial stretch values as λz = 1, 1.08, 1.23, 1.33 due to their physical significance [35, 36]. Next, we study the bifurcation of a neo-Hookean reinforced membrane tube enclosed by an elastic substrate. We fix λz and the fiber angle φ = 90◦ with the following cases (i) the material parameter is taken as ρ = 0.005, 0.05, and the substrate stiffness K is varied in steps of ∆K = 0.001 (ii) the substrate stiffness is fixed as K = 0.05 and ρ is varied in steps of ∆ρ = 0.001. We compare the circular distension and internal pressure at bifurcation for the fiber angles φ = 0◦ and 90◦ by fixing λz = 1, 1.33, and ρ = 0.05 (K = 0.05). We denote the B circular distension at bifurcation, and the corresponding gas pressure at bifurcation as λB θ and Pg , respectively. 4.1. Isotropic neo-Hookean model The bifurcation condition (14) for the neo-Hookean isotropic material model reads −λ8θ λ4z + 6λ4θ λ2z + 4λ2θ λ4z + 3 = 0. It can be easily checked that the bifurcation occurs at λB θ ≈ 1.67197 for the axial stretch λz = 1, and it approaches λB ≈ 1.260 when λ increases [32]. The bifurcation criterion (13) for the isotropic neo-Hookean z θ material with substrate stiffness K is given by  

1 2 3 3 2 4 4 3 4 2 (15) 2 λθ λz + 3 K (2λθ − 1) λθ λz + 8 − −2λθ λz + K (λθ − 1) λθ λz + 6 2 = 0. 2 The bifurcation function f is plotted with λB θ for certain values of the substrate stiffness K and λz in Figure 1.

200

K=0

200

100

f

f

100

K = 0.5

0

0

-100

-100 2

4

(a)

200

λθ

6

8

2

4

(b)

K=1

200

6

8

6

8

K=2

100

f

f

100

λθ

0

0

-100

-100 2

4

(c)

λθ

6

8

2

4

(d)

Figure 1: The bifurcation criterion f with λz = 1 ( ), 1.08 ( ), 1.23 ( (b) K = 0.5 (c) K = 1 (d) K = 2 for isotropic neo-Hookean material

), 1.33 (

λθ

) and the substrate stiffness (a) K = 0

B The roots of f represent the value of circular distension at bifurcation λB θ . The variation of λθ with K for B certain values of λz is presented in Figure 2(a). It is observed that λθ is triple valued up to certain values of K

6

10

10

7.5

7.5

PgB

λB θ

depending on λz . We considered the lowest branch for each λz for the value of circular distension at bifurcation. Beyond a certain value of K (as marked in Figure 2(a) by a broken vertical line for λz = 1), a subcritical jump in the value of λB θ can be observed. A further increment in the substrate stiffness K results in a gradual B reduction in λB . Initially, the value λB θ θ corresponding to λz = 1 is higher than the value of λθ corresponding to λz = 1.33 and the trend reverses twice, as observed in Figure 1. The corresponding pressure at bifurcation is shown in Figure 2(b), which also exhibits the subcritical jump phenomenon. From these figures, it is evident that, beyond a critical value of stiffness, the occurrence of geometric instability can be significantly delayed. As an interesting consequence of this phenomenon, we note that, if K reduces (due to tissue degeneration/aging), the tube can suddenly exhibit geometric instability in the form of local bulging.

5

2.5

5

2.5

0

0.5

1

1.5

2

0

(a)K

0.5

1

1.5

2

(b)K

B Figure 2: The plots (a) circular distension at bifurcation (λB θ ) with the substrate stiffness (K) (b) pressure at bifurcation (Pg ) with K, for the isotropic neo-Hookean material with λz = 1 ( ), 1.08 ( ), 1.23 ( ) and 1.33 ( )

4.2. Anisotropic neo-Hookean model The bifurcation condition (13) for an infinite incompressible neo-Hookean cylindrical tube with fiber anisotropy I4 (or I6 ) reads

2 2ρλ4θ λ4z sin2 2φ − 4λ4θ λ2z 2ρλ2θ sin4 φ − 2ρ sin2 φ + 1 + K (λθ − 1) 2 λ3θ λ3z + 12


− 8 8ρλ6θ λ2z sin4 φ + K (2λθ − 1) λ4θ λ3z + 8 6ρλ2θ λ6z cos4 φ + λ2θ λ4z 2ρλ2θ sin2 φ cos2 φ − 2ρ cos2 φ + 1 + 3 = 0.

In the following, we individually study the effects of variation of substrate stiffness and fiber properties on solution bifurcation. 4.2.1. Effect of varying substrate stiffness with fiber reinforced neo-Hookean model First we consider fiber anisotropy I4 (or I6 ). Figures 3(a) and 3(c) compare the effect of the material anisotropy parameter ρ for fiber angle φ = 90◦ (orthotropy with circumferential material symmetry axis) on B the variation of λB θ with K. For a lower value of ρ, the occurrence of a subcritical jump in λθ is observed in Figure 3(a). Interestingly, increase in ρ can completely change the jump to supercritical, as observed in Figure 3(c). Figures 3(b) and 3(d) show the corresponding pressure at bifurcation. It is to be noted that the pressure range shows a substantial increase with increase in ρ. ◦ ◦ Figure 4 shows the relative change in λB θ and pressure at bifurcation for two fiber angles φ = 0 and 90 ◦ with a fixed value of ρ = 0.05. The trends in the fiber angle φ = 0 case (orthotropy with longitudinal material symmetry axis) are very similar to that in the isotropic case with subcritical jump in λB θ . This, however, is in complete contrast to the trend with φ = 90◦ , which exhibits a supercritical jump in λB . θ Next, we consider an infinite neo-Hookean incompressible cylindrical membrane tube on substrate with fiber B anisotropy I5 (or I7 ). Figures 5(a) and 5(b) show the variations of λB θ and Pg with K for two values of ◦ anisotropy parameter ρ and fixed value of φ = 90 . The first observation is that the phenomenon of jump is completely absent in the considered range of K. Further, higher the anisotropy, lower is the range of circular distention at bifurcation. Increasing the value of ρ reduces the effect of λz on λB θ . This is, however, opposite in the case of PgB presented in Figures 5(b). Moreover, beyond a certain value of ρ, the pressure at bifurcation decreases with increase in axial stretch. It can be verified that λB θ approaches unity, and is independent of K for large values of ρ. B ◦ Figure 6 compares the effect of fiber angle φ on the variation of λB θ and Pg with K. Clearly, for φ = 0 , the ◦ variation exhibits subcritical jump. This behaviour is completely washed-out when φ = 90 in the considered 7

15

20

ρ = 0.005

ρ = 0.005 15

λB θ

PgB

10 10

5 5

0

0.5

1

(a)

1.5

2

0

0.5

K

1

15

1.5

2

6

8

K

(b) 100

ρ = 0.05

ρ = 0.05

PgB

λB θ

10 50

5

0

2

4

6

8

0

2

K (c)

4

(d)K

B B B Figure 3: The plots (a) λB θ with K and ρ = 0.005 (b) Pg with K and ρ = 0.005 (c) λθ with K for ρ = 0.05 (d) Pg with K and ρ = 0.05, for the fiber anisotropy I4 (or I6 ) neo-Hookean material with λz = 1 ( ), 1.08 ( ), 1.23 ( ) and 1.33 ( ), and φ = 90◦

60

10

ρ = 0.05 - φ = 0◦

ρ = 0.05

- φ = 90◦

- φ = 90◦

7.5

PgB

λB θ

40

5

20

R @ φ = 0◦

2.5

0

2

4

(a)

6

8

0

K

2

4

(b)

6

8

K

B Figure 4: The plots (a) λB θ with K and ρ = 0.05 (b) Pg with K and ρ = 0.05, for the fiber anisotropy I4 (or I6 ) neo-Hookean material with λz = 1 ( ), 1.33 ( ) and λz = 1 ( ), 1.33 ( ) for the fiber angles φ = 0◦ and 90◦ , respectively. The value of φ indicated by the arrow correspond to all curves intersected by the arrow.

range of values of K. It would be interesting to see the effect of intermediate fiber angles. However, this will require a more general model to account for torsional shear of the tube, which is presently absent in our model. 4.2.2. Effect of varying anisotropy parameter with fiber reinforced neo-Hookean model Figure 7(a) shows a non-monotonic variation in the circular distension at bifurcation for the fiber anisotropy I4 (or I6 ), and the fiber angle φ = 90◦ . The higher value of K exhibit the monotonically decreasing circular distension at bifurcation. Initially, the circular distension at bifurcation corresponding to λz = 1.33 is lower and the behaviour differs at higher values of ρ. An increment in the anisotropy parameter increases the pressure at bifurcation, as seen in Figure 7(b). 8

2.6 20

ρ = 0.005

2.4



16

2.2

λB θ

PgB

ρ = 0.05 

12

2

ρ = 0.05

1.8

@ R @ ρ = 0.005

8

 4

1.6 0

2

4

(a)

6

8

0

2

4

K

(b)

6

8

K

B Figure 5: The plots (a) λB θ with K and ρ = 0.005, 0.05 (b) Pg with K and ρ = 0.005, 0.05, for the fiber anisotropy I5 (or I7 ) neo-Hookean material with λz = 1 ( ), 1.08 ( ), 1.23 ( ) and 1.33 ( ), and the fiber angle φ = 90◦ . The value of ρ indicated by the arrow correspond to all curves intersected by the arrow.

10

ρ = 0.05

25

- φ = 0◦

20

7.5

@

15

PgB

λB θ

ρ = 0.05

5

R @ φ = 90◦

10 ◦

φ = 90

2.5

5

 0

2

4

(a)

6

8

- φ = 0◦

0

2

4

K

(b)

6

8

K

B Figure 6: The plots (a) λB θ with K and ρ = 0.05 (b) Pg with K and ρ = 0.05, for the fiber anisotropy I5 (or I7 ) neo-Hookean material with λz = 1 ( ), 1.33 ( ) and λz = 1 ( ), 1.33 ( ) for the fiber angles φ = 0◦ and 90◦ , respectively. The value of φ indicated by the arrow correspond to all curves intersected by the arrow.

K = 0.05

K = 0.05

5

1.75

PgB

λB θ

4

1.7

3

1.65 2

1.6 0

0.1

0.2

0.3

0.4

0.5

0

ρ (a)

0.1

0.2

0.3

0.4

0.5

ρ (b)

B Figure 7: The plots (a) λB θ with ρ and K = 0.05 (b) Pg with ρ and K = 0.05, for the fiber anisotropy I4 (or I6 ) neo-Hookean ), 1.08 ( ), 1.23 ( ) and 1.33 ( ), and φ = 90◦ material with λz = 1 (

◦ ◦ ◦ Figure 8(a) shows the variation of λB θ with the fiber angles φ = 0 and 90 . For the fiber angle φ = 0 , the ◦ value of λB increases monotonically which in contrast to the case with φ = 90 . However, the pressure increases θ with increase in ρ and the fiber angles, as seen in Figure 8(b). Figures 9(a) and 10(a) shows that the circular distension at bifurcation is non-monotonicity for the fiber anisotropy I5 (or I7 ) with φ = 90◦ , and monotonically increasing for φ = 0◦ in the considered range. Figures 9(b) and 10(b) shows the pressure at bifurcation found to increase with increase in anisotropy parameter for

9

5

K = 0.05

K = 0.05

1.85

φ = 90◦

φ = 0◦

4





PgB

1.75

λB θ

φ = 90◦ 

3

φ = 0◦

1.65



2

1.55 0

0.1

0.2

0.3

(a)

0.4

0.5

0

0.1

0.2

ρ

0.3

0.4

0.5

ρ

(b)

B Figure 8: The plots (a) λB θ with ρ and K = 0.05 (b) Pg with ρ and K = 0.05, for the fiber anisotropy I4 (or I6 ) neo-Hookean material with λz = 1 ( ), 1.33 ( ) and λz = 1 ( ), 1.33 ( ) for the fiber angles φ = 0◦ and 90◦ , respectively. The value of φ indicated by the arrow correspond to all curves intersected by the arrow.

12

2

K = 0.05

K = 0.05

1.9 9

λB θ

PgB

1.8 6

1.7

1.6

3

1.5 0

0.02

0.04

0.06

(a)

0.08

0.1

0

0.02

0.04

ρ

0.06

0.08

0.1

ρ

(b)

B Figure 9: The plots (a) λB θ with ρ and K = 0.05 (b) Pg with ρ and K = 0.05, for the fiber anisotropy I5 (or I7 ) neo-Hookean material with λz = 1 ( ), 1.08 ( ), 1.23 ( ) and 1.33 ( ), and φ = 90◦

12

2.4

K = 0.05

K = 0.05 φ = 0◦ @ I @

λB θ



9

@

1.8

PgB

φ = 90◦

2.1

φ = 90◦ 

6

@ @

φ = 0◦ 

3 1.5 0

0.02

0.04

0.06

(a)

0.08

0.1

0

ρ

0.02

0.04

0.06

(b)

0.08

0.1

ρ

B Figure 10: The plots (a) λB θ with ρ for K = 0.05 (b) Pg with ρ for K = 0.05, of the fiber anisotropy I5 (or I7 ) neo-Hookean material with λz = 1 ( ), 1.33 ( ) and λz = 1 ( ), 1.33 ( ) for the fiber angles φ = 0◦ and 90◦ , respectively. The value of φ indicated by the arrow correspond to all curves intersected by the arrow.

φ = 0◦ and 90◦ . Here, we have not observed the multiple circular distension values at bifurcation for the considered range of values of ρ.

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5. Conclusion The present work brings out the important role of perivascular tissues with isotropy and anisotropy in delaying geometric instability in internally pressurized membrane tubes. The presence of subcritical and supercritical jumps in solution bifurcation with variation of tissue stiffness and anisotropy was pointed out. We observed that the circular distension and the corresponding pressure at bifurcation increases up to the (sub or super) critical point for varying values of elastic substrate stiffness. Beyond the critical point, the circular distension at bifurcation decreases, and the corresponding pressure at bifurcation depends on membrane isotropic/anisotropy. The modeling and results presented in this work are relevant to the study of formation of aneurysm in blood vessels with aging. Acknowledgment: The authors would like to thank the editor and the anonymous reviewers for their valuable comments and suggestions for improving the manuscript. 6. References [1] T. B. Wilmink, C. R. Quick, C. S. Hubbard, N. E. Day, The influence of screening on the incidence of ruptured abdominal aortic aneurysms, Journal of Vascular Surgery 30 (2) (1999) 203–208. [2] B. P. Heather, K. R. Poskitt, J. J. Earnshaw, M. Whyman, E. Shaw, Population screening reduces mortality rate from aortic aneurysm in men, British Journal of Surgery 87 (6) (2000) 750–753. [3] J. A. Heller, A. Weinberg, R. Arons, K. Krishnasastry, R. T. Lyon, J. S. Deitch, A. H. Schulick, H. L. B. Jr, K. Kent, Two decades of abdominal aortic aneurysm repair: Have we made any progress?, Journal of Vascular Surgery 32 (6) (2000) 1091 – 1100. [4] R. Finkbohner, D. Johnston, E. S. Crawford, J. Coselli, D. M. Milewicz, Marfan syndrome: Long-term survival and complications after aortic aneurysm repair, Circulation 91 (3) (1995) 728–733. [5] M. Jonathan Yip, J.-A. Sawatzky, Cardiovascular management of marfan syndrome: Implications for nurse practitioners, The Journal for Nurse Practitioners 10(8) (2014) 594–602. [6] P. N. Robinson, M. Godfrey, The molecular genetics of marfan syndrome and related microfibrillopathies, Journal of Medical Genetics 37 (1) (2000) 9–25. [7] T. Mizuguchi, N. Matsumoto, Recent progress in genetics of marfan syndrome and marfan-associated disorders, Journal of Human Genetics 52 (2007) 1–12. [8] G. A. Holzapfel, T. C. Gasser, R. W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, Journal of elasticity and the physical science of solids 61 (1) (1996) 1–48. [9] D. Haughton, J. Merodio, The elasticity of arterial tissue affected by marfan’s syndrome, Mechanics Research Communications 36 (6) (2009) 659–668. [10] J. Merodio, D. Haughton, Bifurcation of thick-walled cylindrical shells and the mechanical response of arterial tissue affected by marfan’s syndrome, Mechanics Research Communications 37 (1) (2010) 1–6. [11] M. Destrade, J. Merodio, Compression instabilities of tissues with localized strain softening, International Journal of Applied Mechanics 3 (1) (2011) 69–83. [12] A. N. Gent, Elastic instabilities of inflated rubber shells, Rubber Chemistry and Technology 72 (2) (1999) 263–268. [13] L. Horn´ y, M. Netuˇsil, Z. Hor´ nk, Limit point instability in pressurization of anisotropic finitely extensible hyperelastic thin-walled tube, International Journal of Non-Linear Mechanics 77 (2015) 107–114. [14] A. Gent, Elastic instabilities in rubber, International Journal of Non-Linear Mechanics 40 (2-3) (2005) 165 – 175, special Issue in Honour of C.O. Horgan. [15] A. Goriely, M. Destrade, M. B. Amar, Instabilities in elastomers and in soft tissues, Q J Mechanics Appl Math 59(4) (2006) 615–630. [16] A. Needleman, Necking of pressurized spherical membranes, Journal of the Mechanics and Physics of Solids 24 (6) (1976) 339 – 359. 11

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Highlights • A cylindrical membrane tube enclosed by an elastic substrate is modeled. • The model has relevance to an arterial tube with perivascular support. • Bifurcation criterion for axisymmetric bulging mode is obtained. • Subcritical and supercritical jump phenomena are observed.

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