on the inflation of poro-hyperelastic annuli · journal of the mechanics and physics of solids 107...

Journal of the Mechanics and Physics of Solids 107 (2017) 229–252

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Journal of the Mechanics and Physics of Solids

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On the inflation of poro-hyperelastic annuli

A.P.S. Selvadurai 1 , ∗, A.P. Suvorov

2

Department of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke Street West, Montréal, QC H3A 0C3, Canada

a r t i c l e i n f o

Article history:

Received 17 April 2017

Revised 8 June 2017

Accepted 15 June 2017

Available online 17 June 2017

Keywords:

Poro-hyperelasticity

Fluid-saturated media

Canonical analytical solutions

Large deformations

Time-dependent phenomena

Calibration of computational results

a b s t r a c t

The paper presents the radially and spherically symmetric problems associated with the

inflation of poro-hyperelastic regions. The theory of poro-hyperelasticity is a convenient

framework for modelling the mechanical behaviour of highly deformable materials in

which the pore space is saturated with fluids. Including the coupled mechanical responses

of both the hyperelastic porous skeleton and the fluid is regarded as an important con-

sideration for the application of the results, particularly to soft tissues encountered in

biomechanical applications. The analytical solutions for radially and spherically symmet-

ric problems involving annular domains are used to benchmark the accuracy of a standard

computational approach. The paper also generates results applicable to the hyperelastic

solutions when coupling is eliminated through the presence of a highly permeable pore

structure.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

A mathematically consistent theory of poroelasticity that describes the mechanics of a fluid-saturated porous medium

undergoing infinitesimal elastic deformations was first proposed by Biot (1941) . The theory has been extensively applied

to describe the coupled hydro-mechanical behaviour of geomaterials such as competent rock, bone and other engineering

materials and is now a well-established theory that has contributed significantly to the engineering geosciences ( Rice and

Cleary, 1976; Selvadurai, 1996, 2007; Coussy, 1995; Cowin, 2001; Verruijt, 2015; Cheng, 2015; Selvadurai and Suvorov, 2016a ).

The assumptions of the classical linear theory of poroelasticity restricts the applications to fluid-saturated materials that can

undergo only small elastic strains where the elastic behaviour of the porous skeleton is defined by Hooke’s law and fluid

migration within the accessible pore space is defined by Darcy’s law. In addition, the partition of the total stresses to the

stresses sustained by the porous skeleton and the pore fluid takes into consideration the compatibility of volume changes

between the two phases. The applications of the classical theory of poroelasticity to several areas of engineering, including

materials engineering and environmental geosciences, are well-documented in the volumes cited previously and in the pro-

ceedings of the Biot Conferences held periodically. The limitations of the classical theory of poroelasticity become evident

when the theory is used to describe the mechanics of fluid-saturated materials that can undergo large strains. This aspect

was recognized by Biot (1972) , leading to the development of a generalized theory to account for the large strains in the

porous skeleton, which can be described by a theory of hyperelasticity. Analogous theories to describe diffusive phenomena

in hyperelastic materials were developed by researchers including Adkins (1964) and Green and Adkins (1964) and the ad-

∗ Corresponding author.

E-mail address: [email protected] (A.P.S. Selvadurai). 1 William Scott Professor and James McGill Professor. 2 Research Associate in Applied Mechanics.

http://dx.doi.org/10.1016/j.jmps.2017.06.007

0022-5096/© 2017 Elsevier Ltd. All rights reserved.


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230 A.P.S. Selvadurai, A.P. Suvorov / Journal of the Mechanics and Physics of Solids 107 (2017) 229–252

vances that specifically relate to fluid-saturated hyperelastic skeletal materials were developed by a number of investigators

and references to these studies are given by Selvadurai and Suvorov (2016b) and Suvorov and Selvadurai (2016) .

There are several definitions of what constitutes a pore fluid occupying the pore space of a hyperelastic material, partic-

ularly if the porous medium has several scales of porosity both directly accessible (inter-cellular) and directly inaccessible

(intra-cellular), with mass transfer by either flow and/or diffusion that could also be controlled by chemical and electrical

potentials in the fluids. In soft biological tissues, the fluid can exist at several scales and the theory of poro-hyperelasticity

examined here is limited to the interactive fluid movement within the accessible pore space, and the flow processes are

described in relation to a reduced Bernoulli potential-based form of Darcy’s law. Incorporating the influence of the fluid

saturating the hyperelastic porous skeleton is desirable from the point of view of providing a justifiable basis for hyperelas-

ticity observed in biological tissues. The complete removal of the fluid can, on occasions, render biological tissues void of

hyperelasticity.

Since the initial developments of a theory of poro-hyperelasticity, advances have been made in incorporating poro-

hyperelasticity within the framework of computational schemes such as the finite element method. The work of Simon

and Gaballa (1988), Simon (1992), Simon et al., (1996) and a recent study by Ayyalasomayajula et al., (2010) apply finite el-

ement approaches to the study of poro-hyperelasticity, which can be applied to complex geometries and loading conditions,

particularly those associated with biological tissues. Recent studies by Selvadurai and Suvorov (2016b) and Suvorov and

Selvadurai (2016) have focused on developing canonical analytical solutions with numerical approaches to time integration

in order to examine poro-hyperelasticity problems dealing with one-dimensional compression, spherically symmetric com-

pression and poro-hyperelastic shear. In particular, extensive references are given to the application of the multi-physical

theories of fluid-saturated porous media in the context of both hyperelasticity and poromechanics. Most importantly, these

results provide the much needed analytical results that can serve as benchmarks for the validation of computational ap-

proaches for modelling fluid-saturated hyperelastic media.

The constraints of the infinitesimal theory restrict the application of the classical theory of poroelasticity to situations

where the medium can experience large strains. The classical theory of poroelasticity developed by Biot (1941) was extended

( Biot, 1972 ) to include finite strains but the application of the approach has not progressed beyond this initial development.

Informative expositions of the topic are given by Baek and Srinivasa (2004), Baek and Pence (2011) and Pence (2012) . Re-

cently there has been an increased focus on modelling soft tissues encountered in the biological and medical sciences and

in the highly deformable porous materials used in tactile sensor applications (here the fluid is air). The concerted efforts in

modelling such materials are largely by appeal to classical theories of hyperelasticity ( Humphrey, 2002 ; Taber, 2008; Huyghe,

2015 ) generally without considering the influence of the pore fluid pressure and its diffusive effects encountered in the cou-

pled poro-hyperelasticity problem. The present paper is a continuation of the basic approaches adopted by Selvadurai and

Suvorov (2016b) and Suvorov and Selvadurai (2016) for the study of the classical Lamé-type problems for a poro-hyperelastic

material with a specific form of the skeletal hyperelastic constitutive equation. The boundaries of the annular regions under

inflation are hydraulically constrained to initiate one-dimensional fluid flow in the deformed region. The mathematical so-

lutions are complemented by computational results obtained via a finite element technique. This paper will deal with the

poroelastic deformation problem and an investigation of instabilities in the annular regions will be presented in a subse-

quent article ( Suvorov and Selvadurai, 2017 ).

2. Constitutive modelling

We consider the total Cauchy stress σi j in the fluid-saturated poro-hyperelastic material, which is assumed to be com-

posed of the Cauchy stress relevant to the porous hyperelastic fabric ( σ ′ i j

) and the isotropic stress in the interstitial fluids,

denoted by p. We assume that the effective stress relationship is of the form

σi j = σ ′ i j − p δi j , (2.1)

where δi j is Kronecker’s delta. This form of the effective stress relationship for the fluid-saturated hyperelastic material is

an adaptation of the relationship proposed in soil mechanics literature and is a limiting case of the result developed by

Biot (1941) , which takes into consideration the compressibilities of both the pore fluid and the porous skeleton. In instances

where the compressibility of the porous skeleton is small in comparison to the compressibility of the pore fluid, Biot’s

result reduces to ( 2.1 ). The effective or skeletal stress σ ′ i j

can be represented in terms of its deviatoric component s ′ i j

and an

isotropic stress (effective pressure) p ′ such that

σ ′ i j = s ′ i j − p ′ δi j , p ′ = −σ ′

ii / 3 (2.2)

Following conventional formulations of problems in finite elasticity ( Green and Zerna, 1968; Spencer, 1970, 2004; Og-

den, 1984; Batra, 2005 ), we consider the initial coordinates of a point in the deformable porous skeleton with Carte-

sian coordinates X i (i = 1 , 2 , 3) (with X 1 = X ; X 2 = Y ; X 3 = Z), which moves to a new position with coordinates defined

by x i (i = 1 , 2 , 3) (with x 1 = x ; x 2 = y ; x 3 = z). The deformation gradient tensor F is given by

F = f i j =

∂ x i ∂ X j

. (2.3)

A.P.S. Selvadurai, A.P. Suvorov / Journal of the Mechanics and Physics of Solids 107 (2017) 229–252 231

The left Cauchy–Green strain tensor B is defined by

B = F F T , (2.4)

where

F = J −1 / 3 F ; J = det F . (2.5)

The strain energy function U for the isotropic hyperelastic material is assumed to be of the form

U = U( I 1 , I 2 , J) , (2.6)

where I 1 and I 2 are, respectively, the first and second invariants of the strain tensor B . The deviatoric component of the

effective stress s ′ is defined as

s ′ =

2

J de v

{(∂U

∂ I 1 + I 1

∂U

∂ I 2

)B − ∂U

∂ I 2 B · B

}. (2.7)

The isotropic component of the effective stress is defined as

p ′ = −∂U

∂ J . (2.8)

Extensive discussions of the various forms of strain energy functions that have been proposed in the literature to de-

scribe hyperelastic materials are given by Mooney (1940), Rivlin and Saunders (1951), Gent and Rivlin (1952a, 1952b ), Doyle

and Ericksen (1956), Rivlin (1960) (see also Barenblatt and Joseph, 1997 ), Adkins (1961), Varga (1966), Hart-Smith and Crisp

(1967), Alexander (1968), Spencer (1970), Green and Adkins (1970), Treloar (1975, 1976 ), Ogden (1972), Carlson and Shield

(1980), Beatty (1987, 20 01a, 20 01b, 20 08 ), Carroll and Hayes (1996), Gent (1996), Libai and Simmonds (1998), Hill (2001),

Gent and Hua (2004), Antman (2005) and Horgan (2015) . The reader is also referred to Lur’e (1990), Drozdov (1996), Dorf-

mann and Muhr (1999), Fu and Ogden (2001), Hayes and Saccomandi (2001), Saccomandi and Ogden (2004), Selvadurai

(2006), Selvadurai and Yu (200 6a,200 6b ), Yu and Selvadurai (2007), Wineman and Rajagopal (2011), Selvadurai and Shi

(2012) and particularly Selvadurai and Suvorov (2016b) who provide extensive reviews of past and recent literature on the

topic of hyperelasticity and poromechanical applications of hyperelasticity. In this study, we consider the finite deformation

problem that is formulated with reference to a second-order reduced polynomial of the form

U = C 10 ( I 1 − 3) + C 20 ( I 1 − 3) 2 +

1

D 1

(J − 1) 2 +

1

D 2

(J − 1) 4 , (2.9)

where C 10 , C 20 , D 1 and D 2 are material parameters. We note that in the strain energy function ( 2.9 ) the terms containing

strain invariants I 1 , I 2 are uncoupled from the terms containing the strain invariant J, which implies that there is no coupling

between shear and volumetric deformations. The constants C 10 and D 1 can also be defined in terms of the linear elastic shear

modulus (G ) and the bulk modulus (K) of the porous skeleton as

2 C 10 = G, D 1 =

2

K

. (2.10)

For this particular material, the deviatoric component of the effective stress is given by

s ′ =

2

J [ C 10 + 2 C 20 ( I 1 − 3)] dev ( F F T ) , (2.11)

and the isotropic component of the effective stress (effective pressure) can be expressed in the form

p ′ = − 2

D 1

(J − 1) − 4

D 2

(J − 1) 3 . (2.12)

If C 20 = 0 , D 2 → ∞ , the strain energy function for the neo-Hookean material can be recovered from ( 2.9 ).

To complete the constitutive modelling, we need to consider fluid flow through a poro-hyperelastic material. Attention is

restricted to the case where the entire pore space of the hyperelastic skeleton is saturated with a fluid and we assume that

fluid flow takes place as a result of a gradient in the Bernoulli potential. In the case of slow flows through the pore space, the

velocity potential can be neglected in comparison to the other contributions and the datum potential can also be neglected

provided that the potential is measured with reference to a fixed datum ( Selvadurai, 20 0 0; Ichikawa and Selvadurai, 2012 ).

Considering the principle of mass conservation, we have

−∇ · ϕ( v f − v s ) = ∇ · ∂u

∂t (2.13)

where ϕ is the porosity, v f is the velocity of the fluid in the pore space and v s is the velocity of the solid skeleton of the

porous material, ∇ is the gradient operator referred to the coordinates of a particle of fluid in the deformed configuration,

and u is the displacement vector defined as u = x − X . The derivation of Eq. (2.13) is presented in Selvadurai and Suvorov

(2016b) and will not be repeated here.


Fig. 1. Initial and deformed configurations of the porous fluid-saturated cylindrical shell. The pressure is applied at the inner boundary of the shell.

We assume that flow of the fluid through the isotropic hyperelastic skeleton can be described by an isotropic form of

Darcy’s law as

ϕ( v f − v s ) = − k

η∇p, (2.14)

where k is the permeability, which is assumed to be a constant and η is the dynamic fluid viscosity. We note that when

porous hyperelastic media are subjected to large strains, the porosity will change with the strain and the permeability will

evolve with the alteration of porosity. In this study, however, we do not attempt to introduce the strain dependency in the

porosity and consequently the permeability of the porous hyperelastic material is assumed to remain constant.

By combining ( 2.13 ) and ( 2.14 ) we can obtain the governing equation for the fluid pressure

k

η∇

2 p = ∇ · ∂u

∂t (2.15)

In ( 2.15 ) ∇

2 is Laplace’s operator referred to the coordinates in the deformed configuration, i.e.

∇

2 =

∂ 2

∂ x 2 +

∂ 2

∂ y 2 +

∂ 2

∂ z 2 . (2.16)

Using the familiar chain rule for differentiation, the second derivatives of Eq. (2.16) can be expressed in terms of deriva-

tives with respect to coordinates in the initial configuration. For example:

∂ 2

∂ x 2 =

∂ 2

∂ X

2

(∂X

∂x

)2

+

∂ 2

∂ Y 2

(∂Y

∂x

)2

+

∂ 2

∂ Z 2

(∂Z

∂x

)2

+ 2

∂ 2

∂ X ∂ Y

∂X

∂x

∂Y

∂x

+2

∂ 2

∂ X ∂ Z

∂X

∂x

∂Z

∂x + 2

∂ 2

∂ Y ∂ Z

∂Y

∂x

∂Z

∂x +

∂

∂X

∂ 2 X

∂ x 2 +

∂

∂Y

∂ 2 Y

∂ x 2 +

∂

∂Z

∂ 2 Z

∂ x 2 (2.17)

…..etc.

3. The cylindrical annular region

3.1. Problem description

We consider the response of a poro-hyperelastic material consisting of a porous fabric where the pores contain an in-

compressible fluid. The material of the solid phase and the fluid itself are assumed incompressible. However, due to the

existence of pores, the whole material can display the traits of a compressible hyperelastic material once the fluid is al-

lowed to flow in and out of the porous region.

We examine a poro-hyperelastic annular region, which, in the undeformed configuration , occupies a cylindrical annular

region ( Fig. 1 )

A ≤ R ≤ B (3.1)

referred to the cylindrical polar coordinate system (R, �, Z) . Attention is restricted to the radially-symmetric plane strain

deformation of this cylindrical shell region described by

r = r(R, t) ; φ = � ; z = Z. (3.2)


As a result of this deformation, the inner and outer boundaries of the shell move to new positions r = a and r = b

respectively, and the deformed shell will occupy the volume ( Fig. 1 )

a ≤ r ≤ b. (3.3)

The principal stretches associated with the radially-symmetric deformation are

λ1 =

∂r

∂R

; λ2 =

r

R

, λ3 = 1 , (3.4)

where the indices 1, 2 and 3 refer to the radial, circumferential and axial directions respectively, and plane strain deforma-

tions of the annular region are implied by ( 3.4 ).

If the radial displacement function is introduced

u = r − R, (3.5)

then the principal stretches in the radial and circumferential directions can be evaluated as

λ1 = 1 +

∂u

∂R

λ2 = 1 +

u

R

. (3.6)

We assume that a total radial compressive stress of magnitude P A > 0 is applied at the inner boundary of the shell and

is caused, for example, by pressurization of the interior region of the annulus due to the injection of fluid with the same

properties as the fluid saturating the annular domain ( Fig. 1 ). The inner boundary is assumed to allow fluid flow into and

out of the shell. Thus, the boundary condition at the inner boundary of the shell can be written as

σrr (R, t) = −P A H(t ) , p(R, t ) = p A H(t) , R = A, ∀ t > 0 (3.7)

where p is the pore fluid pressure inside the shell, p A is the pore fluid pressure at R = A , and H(t) is the Heaviside step

function of time. Due to the free drainage condition for the inner boundary surface R = A , the fluid pressure p A must be

equal to P A .

The outer boundary is assumed to be free of externally applied stress and impervious or undrained. The boundary con-

ditions at the outer boundary can be written as

σrr (R, t) = 0 , ∂ p

∂R

= 0 , R = B, ∀ t > 0 (3.8)

The second boundary condition of ( 3.8 ) is obtained from Darcy’s law ( 2.14 ) since the relative fluid velocity is zero at the

impervious boundary.

The solid skeleton of the cylindrical shell is assumed to be a hyperelastic material of the neo-Hookean type. Assuming

that the Jacobian is J = λ1 λ2 λ3 , the strain energy for this material is given by

U = C 10 ((λ2 1 + λ2

2 + λ2 3 ) J

−2 / 3 − 3) +

1

D 1

(J − 1) 2

= C 10 (λ4 / 3 1

λ−2 / 3 2

λ−2 / 3 3

+ λ−2 / 3 1

λ4 / 3 2

λ−2 / 3 3

+ λ−2 / 3 1

λ−2 / 3 2

λ4 / 3 3

− 3) +

1

D 1

( λ1 λ2 λ3 − 1) 2 . (3.9)

Here C 10 and D 1 are the elastic constants related to the initial shear and bulk moduli of the material as

G = 2 C 10 , K = 2 / D 1 . (3.10)

Thus, D 1 is a measure of compressibility of the material. Poisson’s ratio is defined as

ν =

3 K − 2 G

6 K + 2 G

. (3.11)

3.2. Equations governing the radially symmetric problem

From the strain energy function, the total Cauchy stress components can be obtained as

σrr = σ ′ rr − p =

λ1

J

∂U

∂ λ1

− p

σθθ = σ ′ θθ − p =

λ2

J

∂U

∂ λ2

− p

σzz = σ ′ zz − p =

λ3

J

∂U

∂ λ3

− p (3.12)

where p is a pore fluid pressure. This gives us

σrr = C 10 ( 4

3

λ1 / 3 1

λ−5 / 3 2

− 2

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

) +

2

D 1

( λ1 λ2 − 1) − p


σθθ = C 10 (−2

3

λ1 / 3 1

λ−5 / 3 2

+

4

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

) +

2

D 1

( λ1 λ2 − 1) − p

σzz = C 10 (−2

3

λ1 / 3 1

λ−5 / 3 2

− 2

3

λ−5 / 3 1

λ1 / 3 2

+

4

3

λ−5 / 3 1

λ−5 / 3 2

) +

2

D 1

( λ1 λ2 − 1) − p (3.13)

In ( 3.13 ) we take into consideration the constraint λ3 = 1 imposed by the plane strain assumption indicated in ( 3.2 ). For

radially-symmetric deformations, the stress equilibrium equation in the radial direction can be written as

∂ σ ′ rr

∂r +

σ ′ rr − σ ′

θθ

r − ∂ p

∂r = 0 . (3.14)

Consequently, we evaluate

σ ′ rr − σ ′

θθ = 2 C 10 (λ1 / 3 1

λ−5 / 3 2

− λ−5 / 3 1

λ1 / 3 2

) .

Also,

∂ σ ′ rr

∂r =

∂ σ ′ rr

∂ λ1

∂ λ1

∂r +

∂ σ ′ rr

∂ λ2

∂ λ2

∂r =

∂ σ ′ rr

∂ λ1

∂ 2 r

∂ R

2

1

λ1

+

∂ σ ′ rr

∂ λ2

(1 − λ−1 1

λ2 )

R

since

∂ λ1

∂r =

∂ 2 r

∂ R

2

1

λ1

, ∂ λ2

∂r =

1 − λ−1 1

λ2

R

.

Therefore,

∂ σ ′ rr

∂r =

∂ 2 r

∂ R

2

(C 10

{

4

9

λ−5 / 3 1

λ−5 / 3 2

+

10

9

λ−11 / 3 1

λ1 / 3 2

+

10

9

λ−11 / 3 1

λ−5 / 3 2

}

+

2

D 1

λ−1 1 λ2

)+

(1 − λ−1 1

λ2 )

R

(C 10

{

−20

9

λ1 / 3 1

λ−8 / 3 2

− 2

9

λ−5 / 3 1

λ−2 / 3 2

+

10

9

λ−5 / 3 1

λ−8 / 3 2

}

+

2

D 1

λ1

).

Thus, the single equation of equilibrium ( 3.14 ) can be written as

∂ 2 r

∂ R

2 R

(C 10

{

4

9

λ−5 / 3 1

λ−2 / 3 2

+

10

9

λ−11 / 3 1

λ4 / 3 2

+

10

9

λ−11 / 3 1

λ−2 / 3 2

}

+

2

D 1

λ−1 1 λ2

2

)+(1 − λ−1

1 λ2 ) (

C 10

{

−20

9

λ1 / 3 1

λ−5 / 3 2

− 2

9

λ−5 / 3 1

λ1 / 3 2

+

10

9

λ−5 / 3 1

λ−5 / 3 2

}

+

2

D 1

λ1 λ2

)+2 C 10 (λ

1 / 3 1

λ−5 / 3 2

− λ−5 / 3 1

λ1 / 3 2

) − λ2 R

∂ p

∂r = 0 . (3.15)

Using the simplification

∂ f

∂r =

∂ f

∂R

∂R

∂r = λ−1

1

∂ f

∂R

,

we can rewrite the equilibrium Eq. (3.15) as

2

D 1

(R

∂ 2 r

∂ R

2 λ2

2 + λ1 λ2 ( λ1 − λ2 )

)+ C 10 R

∂ 2 r

∂ R

2

(4

9

λ−2 / 3 1

λ−2 / 3 2

+

10

9

λ−8 / 3 1

λ4 / 3 2

+

10

9

λ−8 / 3 1

λ−2 / 3 2

)+ C 10

(−20

9

λ1 / 3 1

λ−5 / 3 2

− 2

9

λ−5 / 3 1

λ1 / 3 2

+

10

9

λ−5 / 3 1

λ−5 / 3 2

)( λ1 − λ2 )

+2 C 10 (λ4 / 3 1

λ−5 / 3 2

− λ−2 / 3 1

λ1 / 3 2

) − λ2 R

∂ p

∂R

= 0 . (3.16)

The traction boundary conditions are given by

C 10

(4

3

λ1 / 3 1

λ−5 / 3 2

− 2

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

)+

2

D 1

( λ1 λ2 − 1) − p = −P A ; at R = A

C 10

(4

3

λ1 / 3 1

λ−5 / 3 2

− 2

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

)+

2

D 1

( λ1 λ2 − 1) − p = 0 ; at R = B (3.17)

The mass conservation principle ( 2.13 ) applied to the present problem takes the form

−(

∂

∂r +

1

r

)ϕ( v f − v s ) =

(∂

∂r +

1

r

)∂u

∂t , (3.18)

where u is the radial displacement, v f is the velocity of the fluid in the pore space, v s is the velocity of the solid skeleton.

Here ∂ ∂r

+

1 r is the divergence operator written in the cylindrical coordinate system.


Now, using Darcy’s law ( 2.14 ) and the mass conservation principle ( 3.18 ), we can eliminate the relative velocity term

ϕ( v f − v s ) to obtain the governing equation for the fluid pressure as

k

η∇

2 p =

(∂

∂r +

1

r

)∂u

∂t , (3.19)

where k is the permeability and η is the dynamic fluid viscosity. In ( 3.19 ) ∇

2 is Laplace’s operator referred to the coordinates

in the deformed configuration, i.e.,

∇

2 =

∂ 2

∂ r 2 +

1

r

∂

∂r =

1

r

∂

∂r (r

∂

∂r ) . (3.20)

All derivatives in ( 3.19 ) can be found in terms of R instead of r using the relationships

∂

∂r = λ−1

1

∂

∂R

; ∂ 2

∂ r 2 = λ−2

1

∂ 2

∂ R

2 − λ−3

1

∂ 2 r

∂ R

2

∂

∂R

; 1

r

∂

∂r =

1

λ1 λ2

1

R

∂

∂R

. (3.21)

Therefore, the Eq. (3.19) governing the fluid flow can be rewritten as

k

η

(λ−2

1

(∂ 2 p

∂ R

2 − λ−1

1

∂ 2 r

∂ R

2

∂ p

∂R

)+ λ−1

1

1

r

∂ p

∂R

)=

(λ−1

1

∂ r

∂R

+

˙ r

r

), (3.22)

where ˙ r =

∂r ∂t

=

∂u ∂t

.

Since the inner boundary of the cylindrical shell allows fluid flow and the outer boundary is assumed to be impervious,

the fluid flow boundary conditions are given by

p(A, t) = P A ;(

∂ p(R, t)

∂R

)R = B

= 0 (3.23)

We note that the boundary condition p(R = A ) = P A cannot be satisfied initially, at t = 0 , if the pressure P A is applied

suddenly as a step function. In this case, the response of the fluid-saturated cylindrical shell will be undrained.

3.3. Initial response of the radially symmetric inflation

The initial response of the annular region is undrained, which implies that no loss or gain of fluid occurs at time t = 0 .

In turn, this implies that since the constituents are incompressible the volume of any region will be preserved immediately

after application of the internal pressure P A ; i.e.

r 2 − a 2

R

2 − A

2 = 1 . (3.24)

Thus, the Jacobian is equal to unity at all points of the cylindrical shell, i.e.,

J = λ1 λ2 = 1 . (3.25)

As a consequence of ( 3.25 ), we can derive the relationship between the stretches λ2 at the boundaries of the shell

denoted by λ2 a and λ2 ab ,

λ2 2 a = 1 +

B

2

A

2 (λ2

2 b − 1) , λ2 2 b = 1 +

A

2

B

2 (λ2

2 a − 1) . (3.26)

The effective stresses ( 3.13 ) and the difference σ ′ rr − σ ′

θθcan now be found as

σ ′ rr = C 10

(4

3

λ−2 2 − 2

3

λ2 2 −

2

3

), σ ′

θθ = C 10

(−2

3

λ−2 2 +

4

3

λ2 2 −

2

3

)σ ′

rr − σ ′ θθ = 2 C 10 (λ

−2 2 − λ2

2 ) , (3.27)

which can then be substituted into the stress equilibrium Eq. (3.14) . Before doing that, it is convenient to change the inde-

pendent variable from r to λ2 and use the following identity

d

dr =

1

R

(1 − λ2 2 )

d

d λ2

, (3.28)

which holds if λ1 λ2 = 1 . By substitution of σ ′ rr − σ ′

θθinto the equilibrium Eq. (3.14) , it can be represented as

d σ ′ rr

d λ2

+ 2 C 10

λ−2 2

− λ2 2

λ2 (1 − λ2 2 )

=

dp

d λ2

. (3.29)

After some simplifications, ( 3.29 ) reduces to

d σ ′ rr

d λ2

+ 2 C 10

1 + λ2 2

λ3 =

dp

d λ2

. (3.30)
2


Fig. 2. Dependence of the applied pressure on the stretch of the outer surface of the porous hyperelastic cylindrical shell. The response of the shell is

undrained. [The solid line corresponds to the analytical result and the open circles correspond to the computational results obtained using ABAQUS TM ].

After integration of ( 3.30 ), the fluid pressure can be obtained as

p = σ ′ rr + C 10 ( ln λ2

2 − λ−2 2 + E) , (3.31)

where E is a constant of integration. This constant can be found from the traction boundary condition at the surface R = B .

The total radial stress σrr = σ ′ rr − p is equal to zero on the surface R = B , and thus, the fluid pressure is

p = σ ′ rr + C 10

(ln

λ2 2

λ2 2 b

− λ−2 2 + λ−2

2 b

)

= C 10

(4

3

λ−2 2 − 2

3

λ2 2 −

2

3

)+ C 10

(ln

λ2 2

λ2 2 b

− λ−2 2 + λ−2

2 b

), (3.32)

where we have used the expression for the effective radial stress ( 3.27 ). The total radial stress σrr = σ ′ rr − p can be obtained

from ( 3.32 ) as

σrr = −C 10

(ln

λ2 2

λ2 2 b

− λ−2 2 + λ−2

2 b

). (3.33)

From the traction boundary condition at the surface R = A , σrr ( λ2 a ) = −P A , we obtain the applied pressure as

P A = C 10

(ln

λ2 2 a

λ2 2 b

− λ−2 2 a + λ−2

2 b

). (3.34)

By using the relationship ( 3.26 ) between the stretches λ2 a , λ2 b , we can express the applied pressure as a function of the

stretch λ2 b or λ2 a only. For example, by using λ2 b , we obtain

P A = C 10

(

ln

1 +

B 2

A 2 (λ2

2 b − 1)

λ2 2 b

−(

1 +

B

2

A

2 (λ2

2 b − 1)

)−1

+ λ−2 2 b

)

(3.35)

By taking the limit of ( 3.35 ) for λ2 b → ∞ , we can find an asymptotic value of the applied pressure as

P cr = C 10 ln

B

2

A

2 = G ln

B

A

(3.36)

This value is the maximum value that the applied pressure can attain and thus constitutes a critical value of the applied

pressure, P cr .

Fig. 2 shows the applied pressure P A (normalized with respect to the shear modulus G ) as a function of the stretch

λ2 = b/B for ratios of inner to outer radii equal to A/B = 0 . 8333 and A/B = 0 . 6 6 67 . The response of the shell is assumed to

be undrained. The analytical result obtained from ( 3.35 ) is shown with a solid line and open circles correspond to the finite


Fig. 3. The variation of fluid pressures at the boundaries of the cylindrical shell as a function of the circumferential stretch at the outer boundary of the

shell. The response of the shell is undrained. [The solid line corresponds to the analytical result and the open circles correspond to the computational

results obtained using ABAQUS TM ].

C

element solution obtained by using the finite element program ABAQUS TM . Comprehensive validations of the ABAQUS TM

software for the study of problems in poromechanics are given in Selvadurai and Suvorov (2012, 2014 ) and validations with

applications to poro-hyperelasticity are given by Selvadurai and Suvorov (2016b) and Suvorov and Selvadurai (2016) .

Naturally, for a fixed value of the applied pressure, a larger stretch will be observed in thinner shells. An asymptote

λ2 → ∞ on the curve corresponds to the critical value of the applied pressure P cr (dashed line), which is indicative of

the loss of stability. The results suggest that the initial or undrained response of the shell to internal pressurization does

not depend on compressibility of the skeleton of the shell, i.e., does not depend on the elastic constant D 1 . To use the

ABAQUS TM program, it is necessary to specify both elastic constants C 10 and D 1 ; D 1 was chosen to correspond to various

values of Poisson’s ratio (e.g., 0.2, 0.4) but the response obtained was practically independent of this value.

Fig. 3 shows the pore fluid pressure p in the shell as a function of the circumferential stretch at the outer boundary

of the shell (the fluid pressure is normalized with respect to the shear modulus). The ratio A/B is set equal to 0 . 8333 or

0 . 6 6 67 . The response of the shell is assumed to be undrained. The analytical result obtained from ( 3.32 ) is shown with

a solid line and the open circles correspond to the finite element solution obtained using ABAQUS TM . Note that the pore

fluid pressure in the shell is negative and its magnitude is generally higher than the applied pressure, especially for thinner

shells. Also, the fluid pressure is non-uniform across the thickness of the shell and | p(A ) | > | p(B ) | . Again, the fluid pressure

for the undrained response is independent of the compressibility of the skeleton.

Example 1. As a simple illustrative example, consider a hyperelastic material model for aortic tissue . Raghavan and Vorp

(20 0 0) used a reduced polynomial form of the strain energy function with C 10 ≈ 200 kPa, C 20 ≈ 20 0 0 kPa for an AAA (ab-

dominal aortic aneurysm) wall and assumed that the material was incompressible (see also Wang et al., 2001 ). Although

the reduced polynomial material differs from the present neo-Hookean material model, we can still use the same value for

10 for purposes of illustration. The radius of a healthy aorta can be taken as 10 mm and its wall thickness is approximately

1.5 mm to 2 mm ( Toungara and Geindreau, 2013 ). Thus, for this case, radii B = 10 mm, A = B − 2 = 8 mm and A/B = 0 . 8 .

The tissue of the aorta has very low permeability and thus the response of the aorta to internal pressure application can

be considered as undrained. Using the graph for A/B = 0 . 8333 shown in Figures 2 and 3 , we can see that the critical ap-

plied pressure is approximately equal to 0 . 18 G which is about 72 kPa. This value is, however, well above the typical adult

maximum blood pressure 120/80 mm of Hg or 16 kPa. Since the reduced polynomial material is even more stable than the

present neo-Hookean material, any loss of stability is not expected for this case.

In an unhealthy dilated state, the diameter of the AAA increases substantially and can reach 50 mm to 80 mm ( Raghavan

and Vorp, 20 0 0 ). If we take radius B = 40 mm and A = B − 2 = 38 mm, then A/B = 0 . 95 and we find that the critical applied

pressure is about 20 kPa given that C 10 is still 200 kPa. Thus, given that the applied internal pressure is in the same range

(16 kPa) and assuming weakening of the material of the aorta itself, it is likely that in an unhealthy artery the loss of

stability will lead to rupture of the AAA.


3.4. Long term response

In the long term, since the outer boundary of the shell is impervious, the fluid pressure everywhere is equal to the

applied pressure at the inner boundary, i.e.,

p = P A . (3.37)

Therefore, the stresses ( 3.13 ) can be expressed as

σrr = C 10

(4

3

λ1 / 3 1

λ−5 / 3 2

− 2

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

)+

2

D 1

( λ1 λ2 − 1) − P A

σθθ = C 10

(−2

3

λ1 / 3 1

λ−5 / 3 2

+

4

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

)+

2

D 1

( λ1 λ2 − 1) − P A (3.38)

In view of ( 3.37 ), the equilibrium Eq. (3.16) can be written as

2

D 1

(R

d 2 r

d R

2 λ2

2 + λ1 λ2 ( λ1 − λ2 )

)+ C 10 R

d 2 r

d R

2

(4

9

λ−2 / 3 1

λ−2 / 3 2

+

10

9

λ−8 / 3 1

λ4 / 3 2

+

10

9

λ−8 / 3 1

λ−2 / 3 2

)+ C 10

(−20

9

λ1 / 3 1

λ−5 / 3 2

− 2

9

λ−5 / 3 1

λ1 / 3 2

+

10

9

λ−5 / 3 1

λ−5 / 3 2

)( λ1 − λ2 ) + 2 C 10 (λ

4 / 3 1

λ−5 / 3 2

− λ−2 / 3 1

λ1 / 3 2

) = 0 (3.39)

The traction boundary conditions ( 3.17 ) for the long-term response take the form

C 10

(4

3

λ1 / 3 1

λ−5 / 3 2

− 2

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

)+

2

D 1

( λ1 λ2 − 1) = 0 ; at R = A

C 10

(4

3

λ1 / 3 1

λ−5 / 3 2

− 2

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

)+

2

D 1

( λ1 λ2 − 1) = P A ; at R = B (3.40)

The problem is thus reduced to solving a nonlinear second-order differential Eq. (3.39) that must satisfy two boundary

conditions ( 3.40 ). After obtaining the solution of the boundary value problem for r(R ) , we can find the stretches λ1 and λ2

and the pore fluid pressure from either of the two traction boundary conditions ( 3.40 ).

The solution of the Eq. (3.39) can most conveniently be obtained with the help of the built-in function bvp4c in

MATLAB

TM . This function gives a solution to the boundary value problem for a system of first-order ordinary differential

equations. In order to use this algorithm, the Eq. (3.39) is represented as a system of two first-order differential equations:

i.e. ⎧ ⎪ ⎪ ⎨

⎪ ⎪ ⎩

dr

dR

= λ1

d λ1

dR

= −d −1 0

(

2

D 1

λ1 λ2 ( λ1 − λ2 ) +

2

9

C 10 (−10 λ1 / 3 1

λ−5 / 3 2

− λ−5 / 3 1

λ1 / 3 2

+ 5 λ−5 / 3 1

λ−5 / 3 2

)( λ1 − λ2 )+

+2 C 10 (λ4 / 3 1

λ−5 / 3 2

− λ−2 / 3 1

λ1 / 3 2

)

)

⎫ ⎪ ⎪ ⎬

⎪ ⎪ ⎭

(3.41)

where

d 0 =

2

D 1

Rλ2 2 +

2

9

C 10 R (2 λ−2 / 3 1

λ−2 / 3 2

+ 5 λ−8 / 3 1

λ4 / 3 2

+ 5 λ−8 / 3 1

λ−2 / 3 2

)

Here, the two unknown functions are r(R ) and λ1 (R ) .

As before, in order to find the critical applied pressure corresponding to the long-term response of the shell, it is useful

to have a plot of the applied pressure versus the stretch of the outer surface of the shell as was evaluated for the undrained

response. An asymptote on that curve will correspond to the critical pressure P cr . In practice, the codes used for computa-

tions (e.g., MATLAB

TM or ABAQUS), are typically terminated by default or slowed down considerably once that asymptote is

reached because of poor convergence; this termination can be interpreted as an indication of loss of stability. The critical

applied pressure can be determined in this way for several shell thicknesses and/or the compressibility parameter for the

shell indicated by G/K.

Fig. 4 shows the dependence of the critical applied pressure on the compressibility parameter G/K for A/B = 0 . 8333 . The

results are presented for the long-term response, assuming that the applied pressure is sustained permanently at the inner

surface of the shell, and for the undrained (instantaneous) response. For the long-term response, the critical applied pressure

is shown for the poro-hyperelastic shell, where the fluid pressure is equal to the applied pressure, i.e., p = P A , (solid line),

and for the purely hyperelastic shell with no fluid pressure effects (dashed line). As was shown earlier, the critical applied

pressure for the undrained response does not depend on compressibility of the shell (horizontal line). In contrast, over the

long-term the critical pressure shows a significant dependence on the compressibility of the shell and is always lower than

the critical pressure for the undrained (initial) response, suggesting that the duration of the load plays an important role in

destabilization of the compressible poro-hyperelastic shell. Moreover, the critical applied pressure for the long-term response

is reduced as the shell becomes more compressible. The existence of positive fluid pressure in the poro-hyperelastic shell

leads to a slight reduction in the critical applied pressure compared to the case of a purely hyperelastic shell. However, in

the case of a shell with an incompressible skeleton, ν = 0 . 5 , the critical applied pressure is the same regardless of the fluid

pressure value.


Fig. 4. Dependence of the critical applied pressure on the compressibility of the porous fluid-saturated hyperelastic cylindrical shell. The ratio A/B = 0 . 8333 .

The initial and long-term responses are indicated. [The solid and dotted lines are the analytical results and the open and solid circles correspond to the

computational results obtained using ABAQUS TM ].

Fig. 5. Dependence of the critical applied pressure on the compressibility of the porous fluid-saturated hyperelastic cylindrical shell. The ratio A/B = 0 . 6667 .

The initial and long-term responses are indicated. [The solid and dotted lines are the analytical results and the open and solid circles correspond to the


Fig. 5 shows similar results for thicker shells with A/B = 0 . 6 6 67 . Here, the value of the critical applied pressure is nat-

urally higher both for the undrained and the long-term responses. Again, we observe that the critical applied pressure in

the long-term is lower than that observed initially and the differences between the critical pressures is more significant

for shells with high values of the compressibility parameter G/K. Since the shell is now thicker, in a relative sense, the

reduction of the critical applied pressure in the poro-hyperelastic shell, compared to the purely hyperelastic shell, is now

stronger than that observed in Fig. 4 . Again, for the incompressible shell, the critical applied pressure is independent of the

fluid pressure.

Example 2. As a further example, consider a reduced polynomial material model for ILT (Intra-Luminal Thrombus), sug-

gested by Toungara and Geindreau (2013) . Here C ≈ 9 kPa, C ≈ 8 kPa and Poisson’s ratio is 0.4. The permeability of the
10 20


ILT is approximately 1 × 10 −15 m

2 ( Ayyalasomayajula et al., 2010 ) but theoretically can be as high as 1 × 10 −12 m

2 ( Toungara

and Geindreau, 2013 ). Since an ILT is a relatively thick shell, we can take A/B = 2 / 3 . Again, using the neo-Hookean material

model instead of the reduced polynomial model, we consider only C 10 ≈ 9 kPa and thus G ≈ 18 kPa. From the graph shown

on Figure 5 for A/B = 2 / 3 we can deduce that in the long term, stability loss will occur when P A ≥ 0 . 32 G = 5 . 76 kPa while in

the short term it will occur when P A ≥ 0 . 4 G = 7 . 2 kPa, which is well below the typical maximum applied pressure of 16 kPa.

In reality, however, ILT is expected to show a more stable behavior since it is usually constrained or supported by the outer

AAA wall, which has a much higher stiffness. In addition, the use of a reduced polynomial material model instead of the

neo-Hookean material will enhance the stability of the shell.

3.5. Transient response

To obtain the transient response of the porous fluid-saturated cylindrical shell, the equilibrium equation to be solved

takes the form

2

D 1

(R

∂ 2 r

∂ R

2 λ2

2 + λ1 λ2 ( λ1 − λ2 )

)+ C 10 R

∂ 2 r

∂ R

2

(4

9

λ−2 / 3 1

λ−2 / 3 2

+

10

9

λ−8 / 3 1

λ4 / 3 2

+

10

9

λ−8 / 3 1

λ−2 / 3 2

)+ C 10

(−20

9

λ1 / 3 1

λ−5 / 3 2

− 2

9

λ−5 / 3 1

λ1 / 3 2

+

10

9

λ−5 / 3 1

λ−5 / 3 2

)( λ1 − λ2 )

+2 C 10 (λ4 / 3 1

λ−5 / 3 2

− λ−2 / 3 1

λ1 / 3 2

) − λ2 R

∂ p

∂R

= 0 [3 . 16]

The second governing equation is the fluid flow equation

k

η

(λ−2

1

(∂ 2 p

∂ R

2 − λ−1

1

∂ 2 r

∂ R

2

∂ p

∂R

)+ λ−1

1

1

r

∂ p

∂R

)=

(λ−1

1

∂ r

∂R

+

˙ r

r

), [3 . 22]

with

˙ r =

∂r

∂t =

∂u

∂t .


C 10

(4

3

λ1 / 3 1

λ−5 / 3 2

− 2

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

)+

2

D 1

( λ1 λ2 − 1) − p = −P A ; at R = A

C 10

(4

3

λ1 / 3 1

λ−5 / 3 2

− 2

3

λ−5 / 3 1

λ1 / 3 2

− 2

3

λ−5 / 3 1

λ−5 / 3 2

)+

2

D 1

( λ1 λ2 − 1) − p = 0 ; at R = B [3 . 17]

The fluid flow boundary conditions are given by

p(A, t) = P A ;(

∂ p

∂R

)R = B

= 0 . [3 . 23]

In the following we consider the case where the internal pressure is applied only for a short time duration �T and

reduced to zero after this period, i.e.,

P pulse (t) = H(t) P A − H(t − �T ) P A , (3.42)

where H(t) is the Heaviside step-function. It is our objective to show how the duration of the pulse �T affects the value of

the critical pressure. It is obvious, however, that if �T → ∞ we will recover the results for the long-term response studied

earlier. Similarly, when �T → 0 , the undrained response will be obtained. We showed that the difference between the

critical pressures for these two extreme scenarios depends on the compressibility of the shell. Thus, for intermediate values

of �T , we examine a compressible shell with a fixed value for Poisson’s ratio, say ν = 0 . 2 or G/K = 0 . 75 .

The critical pressure for the pulse loading can be found using the following rule: If the loss of stability does not occur at

any time during application of the pulse load P pulse (t) , the applied pressure P A is assumed smaller than the critical pressure,

i.e., P A < P cr . If the behavior was unstable at some time during application of the pulse load P pulse (t) , the applied pressure P A is considered equal to or higher than the critical pressure, i.e., P A ≥ P cr .

By introducing dimensionless variables

R = R/A, t = t/ �T (3.43)

it can be shown that for a fixed value of the radial coordinate and time, the solution for the problem of pulse loading is a

function of the parameter k �T ηA 2

, and consequently, the critical pressure is a function of this parameter.

Fig. 6 shows how the critical pressure, applied as a pulse loading at the inner boundary of the shell, depends on the

permeability of the shell k and on the pulse duration �T . The compressibility of the shell is G/K = 0 . 75 and A/B = 0 . 8333 ;

the fluid viscosity is set to η = 0 . 001 Pa sec. We can immediately see that for a fixed duration of the pulse loading, the

critical pressure becomes higher as the permeability decreases, and thus, a reduction in the permeability has a stabilizing


Fig. 6. Dependence of the critical applied pressure P cr on permeability of the shell k , radius of the shell A and the time duration of the pulse loading �T .

Parameters G/K = 0 . 75 and A/B = 0 . 8333 .

effect on the compressible porous hyperelastic shell. Naturally, the critical pressure decreases as the load duration �T is

increased and, in the limit �T → ∞ , loss of stability will occur in the long term for the lowest value of the critical applied

pressure (dashed horizontal line). Also, for the pulse loading, the critical pressure depends on the value of the radius A (or

B ) and for a smaller radius, the applied critical pressure decreases.

As mentioned previously, the dependencies of the critical applied pressure P cr on k/η, �T and A can be combined into a

single parameter k �T /ηA

2 and then

P cr = P cr

(k �T

ηA

2

). (3.44)

Therefore, it is sufficient to plot only a single curve showing how P cr /G depends on k �T /ηA

2 .

Fig. 7 shows evolution of the stretch λ2 b = b/B of the outer surface of the porous hyperelastic cylindrical shell subjected

to a pulse loading of duration �T . The ratio A/B = 0 . 6 6 67 and the shell is assumed to be compressible with G/K = 0 . 75 , i.e.,

ν = 0 . 2 . The magnitude of the applied pressure is P A = 0 . 272 G . The stretch is plotted for different values of the parameter

k �T / A

2 [sec] and it can be seen that for sufficiently high values of permeability k and/or for long pulse durations, the stretch

becomes unbounded during application of the pulse loading, which signifies a loss of stability. However, for sufficiently low

permeability values and for short pulse durations, the stretch is bounded and the response is stable. In addition, Fig. 7 shows

the response of the pure hyperelastic shell subjected to the same pulse loading (dashed line). Since the pore fluid pressure

is equal to zero for this material, the response is rate independent and does not depend on the duration of the loading �T .

Although the response of the pure hyperelastic shell is stable for this level of loading, the behavior of the poro-hyperelastic

shell, in which fluid pressure boundary conditions ( 3.23 ) are imposed, can be either stable or unstable, depending on the

parameter k �T / A

2 .

4. The spherical annular region

4.1. Problem description

We consider a porous solid body that initially occupies a spherical annular region ( Fig. 8 )

A ≤ S ≤ B, (4.1)

referred to the spherical coordinate system (S, �, ) . This spherical shell region then undergoes a spherically-symmetric

deformation described by

s = s (S) ; φ = �; θ = . (4.2)


Fig. 7. Evolution of the stretch λ2 b = b/B of the outer surface of the fluid-saturated porous hyperelastic cylindrical shell subjected to a pulse loading of

magnitude P A = 0 . 272 G and duration �T . The ratio A/B = 0 . 6667 and G/K = 0 . 75 .

Fig. 8. Initial and deformed configurations of the porous fluid-saturated spherical shell; pressure is applied at the inner boundary of the shell.

As a result of this deformation, the inner and outer boundaries of the shell will move to new coordinates a and b

respectively, and the deformed shell will occupy the region

a ≤ s ≤ b. (4.3)

The principal stretches associated with this deformation are

λ1 =

∂s

∂S ; λ2 = λ3 =

s

S (4.4)

If the radial displacement function is introduced

u = s − S, (4.5)

then the principal stretches can be evaluated as

λ1 = 1 +

∂u

∂S ; λ2 = λ3 = 1 +

u

S . (4.6)

Assume that a pressure P A > 0 is applied at the inner boundary of the shell caused, for example, by injection of fluid

( Fig. 8 ). It is also assumed that the inner boundary allows fluid flow into and out of the shell. Thus, the boundary condition


at the inner boundary of the shell can be written as

σss (S, t) = −P A H(t ) , p(S, t ) = p A H(t) , S = A, ∀ t > 0 (4.7)

where p is the pore fluid pressure inside the shell and, as before, p A = P A . The outer boundary is assumed to be free of

stress and impervious or undrained. The boundary conditions at the outer boundary can be written as

σss (S, t) = 0 , ∂ p

∂S = 0 , S = B, ∀ t > 0 (4.8)

Since the relative fluid velocity is zero at the impervious boundary, the second boundary condition of ( 4.8 ) can be ob-

tained from Darcy’s law ( 2.14 ).

The solid skeleton of the spherical shell is assumed to be a hyperelastic material of neo-Hookean type. The strain energy

for this material is given by

U = C 10 ((λ2 1 + 2 λ2

2 ) J −2 / 3 − 3) +

1

D 1

(J − 1) 2

= C 10 (λ4 / 3 1

λ−4 / 3 2

+ 2 λ−2 / 3 1

λ2 / 3 2

− 3) +

1

D 1

( λ1 λ2 2 − 1) 2 (4.9)

assuming that λ2 = λ3 and the Jacobian is J = λ1 λ2 2 . Here C 10 and D 1 are the elastic constants defined by ( 3.10 ).

4.2. Governing equations

From the strain energy function, the total Cauchy stress components can be found as

σss = σ ′ ss − p =

λ1

J

∂U

∂ λ1

− p; σθθ = σ ′ θθ − p =

1

2

λ2

J

∂U

∂ λ2

− p (4.10)

where p is a pore fluid pressure. This gives us

σss =

4

3

C 10 (λ1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) − p

σθθ =

2

3

C 10 (−λ1 / 3 1

λ−10 / 3 2

+ λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) − p (4.11)

For spherically-symmetric deformations, the stress equilibrium equations in the radial direction can be written as

∂ σ ′ ss

∂s +

2

s ( σ ′

ss − σ ′ θθ ) − ∂ p

∂s = 0 . (4.12)

Consequently, we can evaluate

σ ′ ss − σ ′

θθ = 2 C 10 (λ1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

)

∂ σ ′ ss

∂s =

∂ σ ′ ss

∂ λ1

∂ λ1

∂s +

∂ σ ′ ss

∂ λ2

∂ λ2

∂s =

∂ σ ′ ss

∂ λ1

∂ 2 s

∂ S 2 1

λ1

+

∂ σ ′ ss

∂ λ2

(1 − λ−1 1

λ2 )

S

∂ σ ′ ss

∂s =

∂ 2 s

∂ S 2 1

λ1

(4

9

C 10 (λ−2 / 3 1

λ−10 / 3 2

+ 5 λ−8 / 3 1

λ−4 / 3 2

) +

2

D 1

λ2 2

)+

(1 − λ−1 1

λ2 )

S

(8

9

C 10 (−5 λ1 / 3 1

λ−13 / 3 2

+ 2 λ−5 / 3 1

λ−7 / 3 2

) +

4

D 1

λ1 λ2

)Therefore, the equilibrium equation can be written as

d 2 s

d S 2 S

(4

9

C 10 (λ−5 / 3 1

λ−7 / 3 2

+ 5 λ−11 / 3 1

λ−1 / 3 2

) +

2

D 1

λ−1 1 λ3

2

)+(1 − λ−1

1 λ2 ) (

8

9

C 10 (−5 λ1 / 3 1

λ−10 / 3 2

+ 2 λ−5 / 3 1

λ−4 / 3 2

) +

4

D 1

λ1 λ2 2

)+4 C 10 (λ

1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

) − λ2 S ∂ p

∂s = 0 (4.13)

After simplifications and using

∂ f

∂s =

∂ f

∂S

∂S

∂s = λ−1

1

∂ f

∂S ,


we can rewrite the equilibrium Eq. (4.13) as

2

D 1

(S d 2 s

d S 2 λ3

2 + 2 λ1 λ2 2 ( λ1 − λ2 )) +

4

9

C 10 S d 2 s

d S 2 (λ−2 / 3

1 λ−7 / 3

2 + 5 λ−8 / 3

1 λ−1 / 3

2 )

+

8

9

C 10 (−5 λ1 / 3 1

λ−10 / 3 2

+ 2 λ−5 / 3 1

λ−4 / 3 2

)( λ1 − λ2 ) + 4 C 10 (λ4 / 3 1

λ−10 / 3 2

− λ−2 / 3 1

λ−4 / 3 2

) − λ2 S ∂ p

∂S = 0 (4.14)


4

3

C 10 (λ1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) − p = −P A ; at S = A

4

3

C 10 (λ1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) − p = 0 ; at S = B (4.15)

Now consider the mass conservation principle

−(

∂

∂s +

2

s

)φ( v f − v s ) =

(∂

∂s +

2

s

)∂u

∂t , (4.16)

where u is the radial displacement, v f is the velocity of the fluid in the pore space, v s is the velocity of the solid skeleton.

Here ∂ ∂s

+

2 s is the divergence operator written in the spherical coordinate system.

Using Darcy’s law ( 2.14 ) and the mass conservation principle ( 4.16 ), we can eliminate the relative velocity term φ( v f − v s )and obtain the governing equation for the fluid pressure

k

η∇

2 p =

(∂

∂s +

2

s

)∂u

∂t , (4.17)

where k is the permeability and η is the fluid viscosity. In ( 4.17 ) ∇

2 is Laplace’s operator referred to the coordinates in the

deformed configuration, i.e.,

∇

2 =

∂ 2

∂ s 2 +

2

s

∂

∂s =

1

s 2 ∂

∂s ( s 2

∂

∂s ) . (4.18)

All derivatives in ( 4.17 ) can be found in terms of S instead of s using

∂

∂s = λ−1

1

∂

∂S ; ∂ 2

∂ s 2 = λ−2

1

∂ 2

∂ S 2 − λ−3

1

∂ 2 s

∂ S 2 ∂

∂S ; 2

s

∂

∂s =

1

λ1 λ2

2

S

∂

∂S . (4.19)

Therefore, the fluid flow Eq. (4.17) can be rewritten as

k

η

(λ−2

1

(∂ 2 p

∂ S 2 − λ−1

1

∂ 2 s

∂ S 2 ∂ p

∂S

)+ λ−1

1

2

s

∂ p

∂S

)=

(λ−1

1

∂ ˙ s

∂S +

2

s

s

), (4.20)

where ˙ s =

∂s ∂t

.

The fluid flow boundary conditions are given by

p(A, t) = P A ;(

∂ p(S, t)

∂S

)S= B

= 0 (4.21)

since the inner boundary of the shell is open for fluid flow but the outer boundary is assumed to be impervious.

We note that the boundary condition p(S = A ) = P A cannot be satisfied initially, at t = 0 , if the pressure P A is applied

suddenly as a step function. In this case, the initial response of the fluid-saturated spherical shell will be undrained.

4.3. Initial response

The initial response of the annular region is undrained, which implies that there is no loss or gain of the fluid at time

t = 0 . In turn, this implies that the volume of any region will be preserved immediately after application of the internal

pressure P A , since the constituents are incompressible ( Fig. 8 ), i.e.,

s 3 − a 3

S 3 − A

3 = 1 . (4.22)

Eq. (4.22) implies that the Jacobian is equal to unity at all points of the shell, i.e.,

J = λ1 λ2 2 = 1 . (4.23)

As a consequence of ( 4.22 ) or ( 4.23 ), we can obtain the relationship between the stretches λ2 at the surfaces of the

spherical shell denoted by λ2 a and λ2 b ,

λ3 a 2 = 1 +

B

3

A

3 (λ3

b2 − 1) , λ3 b2 = 1 +

A

3

B

3 (λ3

a 2 − 1) . (4.24)


Using ( 4.23 ), we can obtain from ( 4.11 ) the effective stresses as

σ ′ ss =

4

3

C 10 (λ−4 2 − λ2

2 ) , σ ′ θθ =

2

3

C 10 (−λ−4 2 + λ2

2 ) ,

σ ′ ss − σ ′

θθ = 2 C 10 (λ−4 2 − λ2

2 ) . (4.25)

Before substitution of σ ′ ss − σ ′

θθinto the stress equilibrium Eq. (4.12) , we note that the following relation must hold

between the derivatives with respect to s and λ2 , due to ( 4.23 ),

d

ds =

1

S (1 − λ3

2 ) d

d λ2

. (4.26)

After substitution of σ ′ ss − σ ′

θθinto ( 4.12 ), changing the independent variable from s to λ2 , we obtain the equilibrium

equation in the following form,

d σ ′ ss

d λ2

+ 4 C 10

λ−4 2

− λ2 2

λ2 (1 − λ3 2 )

− dp

d λ2

= 0 . (4.27)

After some simplifications,

d σ ′ ss

d λ2

+ 4 C 10

1 + λ3 2

λ5 2

− dp

d λ2

= 0 . (4.28)

After integration, we obtain the fluid pressure as

p = σ ′ ss + 4 C 10 (− 1

4 λ4 2

− 1

λ2

+ E) , (4.29)

where E is a constant of integration. The total radial stress is given by σss = σ ′ ss − p and it must be equal to zero at the

surface R = B on which λ2 = λ2 b . The constant of integration E can now be found from that condition, and this leads to the

following expression for the fluid pressure

p = σ ′ ss + 4 C 10 (− 1

4 λ4 2

− 1

λ2

+

1

4 λ4 2 b

+

1

λ2 b

)

=

4

3

C 10 (λ−4 2 − λ2

2 ) + 4 C 10 (− 1

4 λ4 2

− 1

λ2

+

1

4 λ4 2 b

+

1

λ2 b

) . (4.30)

Once we use the expression for the effective stress σ ′ ss given by ( 4.25 ), we can completely determine the fluid pressure

from ( 4.30 ). Consequently, the total radial stress σss = σ ′ ss − p can be found as

σss = −4 C 10 (− 1

4 λ4 2

− 1

λ2

+

1

4 λ4 2 b

+

1

λ2 b

) . (4.31)

On the surface R = A the radial stress σss is equal to −P A , and thus, from ( 4.31 ) we can obtain the applied pressure as a

function of the stretches on the boundaries of the spherical shell, i.e.,

P A = 4 C 10 (− 1

4 λ4 2 a

− 1

λ2 a

+

1

4 λ4 2 b

+

1

λ2 b

) . (4.32)

Using the relationship that exists between the stretches at the boundaries ( 4.24 ), the applied pressure can be expressed

as a function of either the stretch λ2 a or λ2 b . For example, using the stretch λ2 b , gives us

P A = 4 C 10

(

−1

4

(1 +

B

3

A

3 (λ3

b2 − 1)

)−4 / 3

−(

1 +

B

3

A

3 (λ3

b2 − 1)

)−1 / 3

+

1

4 λ4 2 b

+

1

λ2 b

)

. (4.33)

Note that as λ2 b → ∞ , the applied pressure P A goes to zero, and for λ2 b = 1 the applied pressure is equal to zero as

well. Thus, between λ2 b = 1 and λ2 b = ∞ there must be a value of the stretch for which the applied pressure reaches

the maximum value. This value of the applied pressure corresponds to the critical value P cr . Due to the complexity of the

expression ( 4.33 ), the critical value of the applied pressure can perhaps be obtained only numerically.

Fig. 9 shows the applied pressure P A (normalized by the shear modulus G ) as a function of the stretch λ2 b = b/B of the

outer surface for ratios A/B = 0 . 8333 and A/B = 0 . 6 6 67 . The shell is assumed to be undrained. Naturally, for a fixed value of

the applied pressure, higher stretches will be observed in thinner shells. The maximum on the P A ( λ2 ) curve corresponds to

the critical value of the applied pressure P cr , and thus, to the loss of stability. The analytical result ( 4.33 ) is shown with a

solid line and the circles correspond to the finite element solution obtained by using the finite element program ABAQUS TM .

It is observed that the initial or undrained response does not depend on the compressibility of the shell, i.e., does not

depend on the elastic constant D 1 . In the ABAQUS TM program, the elastic constant D 1 must be specified anyway and is

chosen to correspond to various values of Poisson’s ratio (e.g., 0.2, 0.4) but the results were practically independent of this

value.


Fig. 9. Dependence of the applied pressure on the stretch of the outer surface of the poro-hyperelastic spherical shell. The response of the shell is

undrained. [The solid line corresponds to the analytical result and the open circles correspond to the computational results obtained using ABAQUS TM ].

Fig. 10. The variation of fluid pressures at the boundaries of the spherical shell as a function of the circumferential stretch at the outer boundary of the

shell. The response of the shell is assumed to be undrained. [The solid line corresponds to the analytical result and the open circles correspond to the


Fig. 10 shows the pore fluid pressure p in the shell as a function of the circumferential stretch at the outer boundary

of the shell (the fluid pressure is normalized by the shear modulus). The ratios A/B are 0 . 8333 and 0 . 6 6 67 . The response

of the shell is assumed to be undrained. The analytical result obtained from ( 4.30 ) is shown with a solid line and the

circles correspond to the finite element solution obtained using ABAQUS TM . Note that the pore fluid pressure in the shell

is negative and its magnitude is in general higher than the applied pressure, especially for thinner shells. Also, the fluid

pressure is non-uniform across the thickness of the shell and | p(A ) | > | p(B ) | . The fluid pressure for the undrained response

is independent of compressibility of the skeleton.


4.4. Long term response

In the long term the fluid pressure everywhere is equal to the applied pressure at the inner boundary, i.e.,

p = P A . (4.34)

Therefore, the stresses can be found as

σss =

4

3

C 10 (λ1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) − P A

σθθ =

2

3

C 10 (−λ1 / 3 1

λ−10 / 3 2

+ λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) − P A . (4.35)

Since the fluid pressure is uniform, the equilibrium Eq. (4.14) is given by

2

D 1

(S d 2 s

d S 2 λ3

2 + 2 λ1 λ2 2 ( λ1 − λ2 )) +

4

9

C 10 S d 2 s

d S 2 (λ−2 / 3

1 λ−7 / 3

2 + 5 λ−8 / 3

1 λ−1 / 3

2 )

+

8

9

C 10 (−5 λ1 / 3 1

λ−10 / 3 2

+ 2 λ−5 / 3 1

λ−4 / 3 2

)( λ1 − λ2 ) + 4 C 10 (λ4 / 3 1

λ−10 / 3 2

− λ−2 / 3 1

λ−4 / 3 2

) = 0 (4.36)

The traction boundary conditions for the long-term response take the form

4

3

C 10 (λ1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) = 0 ; at S = A

4

3

C 10 (λ1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) − P A = 0 ; at S = B (4.37)

The problem is thus reduced to solving a nonlinear second-order differential Eq. (4.36) that must satisfy two boundary

conditions ( 4.37 ). After finding the solution s (S) of the boundary value problem, we can find the stretches λ1 and λ2 and

the pore fluid pressure from either of the two traction boundary conditions ( 4.37 ).

This boundary-value problem can conveniently be solved with the help of the MATLAB

TM function bvp4c . This function

finds a solution to boundary value problems for a system of first-order ordinary differential equations. In order to use this

function, the Eq. (4.36) is represented as a system of two first-order differential equations ⎧ ⎪ ⎪ ⎨

⎪ ⎪ ⎩

ds

dS = λ1

d λ1

dS = −d −1

(

4

D 1

λ1 λ2 2 ( λ1 − λ2 ) +

8

9

C 10 (−5 λ1 / 3 1

λ−10 / 3 2

+ 2 λ−5 / 3 1

λ−4 / 3 2

)( λ1 − λ2 )+

+4 C 10 (λ4 / 3 1

λ−10 / 3 2

− λ−2 / 3 1

λ−4 / 3 2

)

)

(4.38)

where

d =

2

D 1

Sλ3 2 +

4

9

C 10 S(λ−2 / 3 1

λ−7 / 3 2

+ 5 λ−8 / 3 1

λ−1 / 3 2

) .

Here, the two unknown functions are s (S) and λ1 (S) .

To find a critical applied pressure corresponding to the long-term response, it is useful to have a plot of the applied

pressure versus the stretch of the outer surface of the shell, as was done for the undrained response. A turning point on

that curve will correspond to the critical pressure P cr . As indicated previously, the codes used can auto-terminate once the

instability point is reached because of poor convergence; this termination can be interpreted as an indication of the loss of

stability. In this way, the critical values of the applied pressures can be found for several values of the thickness of the shell

and/or its compressibility parameter G/K.

Fig. 11 shows dependence of the critical applied pressure on the compressibility parameter G/K for A/B = 0 . 8333 . The

results are presented for the long-term response, assuming that the applied pressure is sustained permanently at the inner

surface of the shell, and for the undrained (instantaneous) response. For the long-term response, the critical applied pressure

is shown for the poro-hyperelastic shell, with a positive fluid pressure that is equal to P A , (solid line) and for the purely

hyperelastic shell with zero fluid pressure (dashed line). As was shown earlier, the critical applied pressure for the undrained

response does not depend on the compressibility of the spherical shell (horizontal line). In contrast, in the long-term, the

critical pressure shows a significant dependence on the compressibility of the shell and is always lower than the critical

pressure for the undrained (initial) response; this suggests that load duration plays an important role in destabilization of

the compressible poro-hyperelastic shell. Moreover, the critical applied pressure for the long-term response is reduced as

the shell becomes more compressible. The positive pore fluid pressure that exists in the poro-hyperelastic shell reduces the

critical applied pressure to a certain extent compared to the purely hyperelastic shell. However, for an incompressible shell,

no reduction occurs and the critical applied pressure becomes independent of the pore fluid pressure.

Fig. 12 shows similar results for a thicker shell with A/B = 0 . 6 6 67 . Here, the value of the critical applied pressure is

naturally higher both for the undrained and the long-term responses. Again, we observe that the critical applied pressure

in the long-term is lower than that observed initially and the difference between the critical pressures is more significant

for shells with high compressibility parameters. The decrease in the critical applied pressure for the poro-hyperelastic shell,


Fig. 11. Dependence of the critical applied pressure on the compressibility of the porous fluid-saturated hyperelastic spherical shell. The ratio A/B = 0 . 8333 .

The initial and long-term responses are indicated. [The solid and dotted lines are the analytical results and the open circles correspond to the computational


Fig. 12. Dependence of the critical applied pressure on the compressibility of the porous fluid-saturated hyperelastic spherical shell. The ratio A/B = 0 . 6667 .

The initial and long-term responses are indicated. [The solid and dotted lines are the analytical results and the open circles correspond to the computational


as compared to the purely hyperelastic shell, becomes more significant here due to the increased thickness of the shell.

However, in the case of the incompressible shell, the critical applied pressure is the same for both materials.

4.5. Transient response

To find the transient response of the porous fluid-saturated spherical shell, the equilibrium equation to be solved takes

the form

2

D

(S d 2 s

d S 2 λ3

2 + 2 λ1 λ2 2 ( λ1 − λ2 )) +

4

9

C 10 S d 2 s

d S 2 (λ−2 / 3

1 λ−7 / 3

2 + 5 λ−8 / 3

1 λ−1 / 3

2 )

1


+

8

9

C 10 (−5 λ1 / 3 1

λ−10 / 3 2

+ 2 λ−5 / 3 1

λ−4 / 3 2

)( λ1 − λ2 ) + 4 C 10 (λ4 / 3 1

λ−10 / 3 2

− λ−2 / 3 1

λ−4 / 3 2

) − λ2 S ∂ p

∂S = 0 [4 . 14]

The second governing equation is the fluid flow equation

k

η

(λ−2

1

(∂ 2 p

∂ S 2 − λ−1

1

∂ 2 s

∂ S 2 ∂ p

∂S

)+ λ−1

1

2

s

∂ p

∂S

)=

(λ−1

1

∂ ˙ s

∂S +

2

s

s

), [4 . 20]

with ˙ s =

∂s ∂t

.


4

3

C 10 (λ1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) − p = −P A ; at S = A

4

3

C 10 (λ1 / 3 1

λ−10 / 3 2

− λ−5 / 3 1

λ−4 / 3 2

) +

2

D 1

( λ1 λ2 2 − 1) − p = 0 ; at S = B [4 . 15]

The fluid flow boundary conditions are

p(A, t) = P A ;(

∂ p(S, t)

∂S

)S= B

= 0 . [4 . 21]

In the following we consider a pulse load in which the pressure is removed after a certain time period �T , i.e.,

P pulse (t) = H (t) P A − H (t − �T ) P A , (4.39)

where H(t) is a Heaviside step-function. The objective is to show how the duration of the pulse �T affects the value of

the critical pressure. It is obvious, however, that if �T → ∞ we will recover the results for the long-term response studied

earlier. On the other hand, when �T → 0 , the undrained response will be obtained. We showed that the difference between

the critical pressures for these two extreme scenarios depends on the shell’s compressibility. Thus, for intermediate values

of �T , it is reasonable to examine a compressible shell with a fixed value of Poisson’s ratio, say ν = 0 . 2 or G/K = 0 . 75 .

The critical pressure for the pulse loading can be found using the following rule: If the loss of stability did not occur at

any time during application of the pulse load P pulse (t) , the applied pressure P A is assumed smaller than the critical pressure.

If the behavior was unstable at some time during application of the pulse load P pulse (t) , the applied pressure P A is equal to

or higher than the critical pressure.

By introducing the dimensionless variables

S = S/A, t = t/ �T , (4.40)

we can show that, for a fixed value of radial coordinate and time, the solution for the problem of pulse loading becomes a

function of the parameter k �T ηA 2

, and consequently, the critical pressure is a function of this parameter.

Fig. 13 shows how the critical pressure, applied as a pulse load at the inner boundary of the shell, depends on the

permeability of the shell k and on the pulse duration �T . The compressibility of the shell is G/K = 0 . 75 and A/B = 0 . 8333 .

The fluid viscosity is set to η = 0 . 001 Pa ·s. It is immediately evident that for a fixed duration of the pulse loading, the

critical pressure becomes higher for lower values of permeability; thus, lowering the permeability has a stabilizing effect on

the compressible porous hyperelastic shell. Naturally, the critical pressure decreases as the load duration �T is increased

and in the limit �T → ∞ loss of stability will occur in the long term for the lowest value of the critical applied pressure

(dashed horizontal line). Also, for the pulse loading, the critical pressure depends on the value of the radius A (or B ) and,

for a smaller radius, the applied critical pressure is decreased. As was mentioned previously, the dependency of the critical

applied pressure P cr on k , �T and A can be combined into a single parameter k �T /ηA

2 , and then

P cr = P cr

(k �T

ηA

2

). (4.41)

Therefore, it is sufficient to plot only one curve showing how P cr /G depends on k �T /ηA

2 .

Fig. 14 shows the evolution of the stretch λ2 b = b/B of the outer surface of the poro-hyperelastic spherical shell subjected

to a pulse loading of duration �T . The ratio A/B = 0 . 6 6 67 and the shell is compressible with G/K = 0 . 75 , i.e., ν = 0 . 2 . The

applied pressure is P A = 0 . 33 G . The stretch is plotted for different values of the parameter k �T / A

2 [sec] and it can be seen

that for sufficiently high values of permeability k and/or for long pulse durations, the stretch becomes unbounded during

application of the pulse loading, signifying the loss of stability. However, for sufficiently low permeability values and for

short pulse durations, the stretch is bounded and the response is stable. Moreover, Fig. 14 shows the response of the purely

hyperelastic shell subjected to the same pulse loading (dashed line). Since the pore fluid pressure is equal to zero for this

material, the response is rate independent and does not depend on the duration of the loading �T . Although the response

of the pure hyperelastic shell is stable for this magnitude of the applied pressure, the behavior of the poro-hyperelastic

shell, in which fluid pressure boundary conditions ( 4.21 ) are imposed, can be either stable or unstable, depending on the

parameter k �T / A

2 .


Fig. 13. Dependence of the critical applied pressure P cr on permeability of the shell k , radius of the shell A and the time duration of the pulse loading �T .

Parameters G/K = 0 . 75 and A/B = 0 . 8333 .

Fig. 14. Evolution of the stretch λ2 b = b/B of the outer surface of the poro-hyperelastic spherical shell subjected to the pulse loading of magnitude P A =

0 . 33 G and duration �T . The ratio A/B = 0 . 6667 and G/K = 0 . 75 .

5. Concluding remarks

The conventional approach to modelling soft tissues that can undergo large elastic deformations is based on the direct

application of the theories of hyperelasticity developed in connection with the modelling of rubberlike elastic materials.

While this allows easy application of a large body of theoretical approaches in non-linear continuum mechanics to the

study of soft tissue mechanics, the inability to consider the mechanics of all the phases composing a highly deformable

material places a restriction on the wider and realistic application of theories of non-linear continuum approaches to soft

tissue mechanics. Consideration of the coupled processes of the mechanics of a highly deformable porous skeleton and

the fluid saturating such a medium is a useful way to examine the mechanics of soft tissues. The present paper clearly

demonstrates that the analytical methods of a theory of hyperelasticity of the porous skeleton can be combined with fluid


flow in the pore space to develop a large class of solutions to canonical problems that are amenable to analytical treatment.

In this study attention is devoted to the problem of the inflation of annuli, both cylindrical and spherical. While much

of the analytical developments can move forward, there are obvious limitations because the governing non-linear partial

differential equations have to be solved by appeal to a numerical procedure based on the use of solution schemes available

in MATLAB

TM . Although the MATLAB solution can be obtained, important results concerning the role of the porous skeleton-

pore fluid couplings on the development of instabilities due to the pressurization of the annuli can only be inferred from

the numerical results. In the context of the applications of the general approaches presented through poro-hyperelasticity

to practical situations, recourse needs to be made to robust computational methods such as the finite element method. The

accuracy and reliability of such computational approaches needs to be verified if they are to be unconditionally accepted.

In this study, the results for the annuli inflation problems obtained analytically are used to benchmark the results obtained

using the computational schemes available in the general-purpose finite element code ABAQUS TM . The results show excellent

agreement between the two solutions. The results presented in the paper are for a poro-hyperelastic material with a special

form of a strain energy function. Nonetheless, the results are encouraging enough to state that the computational scheme

provides reliable and accurate results for the study of fluid-saturated porous materials undergoing more general hyperelastic

behaviour.

Acknowledgements

The work described in this paper was supported through an NSERC Discovery Grant and the James McGill Research

Chairs research support awarded to the first author. The authors are grateful to the reviewers for their valuable comments.

In particular, the authors would like to express their appreciation to Professor Thomas J. Pence, Department of Mechanical

Engineering, Michigan State University, East Lansing, MI, for his careful observations and highly constructive comments that

led to significant improvements in the modelling of the poro-hyperelasticity problems reported in this paper.

References

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