Analysis of uniaxial tension and circumferential inflation on the mechanical property of arterial wall.
Sang, Jianbing ; Sun, Lifang ; Xing, Sufang 等
1. Introduction
Cardiovascular diseases are one of the major death factors in the
modern high civilized world. Therefore much effort is put to the
research aimed to explain the mechanisms that govern the cardiovascular
system in both healthy and pathological cases [1]. Nonlinear elasticity
is now extensively used to study the mechanical response of arterial
walls under certain conditions [2-3]. Understanding of arterial
wall's fundamental elastic properties and particularly their
nonlinear stress-strain characteristics becomes more and more important.
From uniaxial tests of arterial wall, we can see that arterial wall has
the phenomenon of non-linear stress strain relation and having higher
extensibility in the low stress and progressively lower with increasing
stretch, which is well known also in the framework of rubber-like
materials [4]. Therefore, based finite deformation theory, research on
arterial wall by adopting the method of rubber like materials is
rational [5, 6]. Many constitutive models have been constructed to
describe the physical characteristics of arterial wall based on
continuum mechanics. The prototype strain energy for isotropic materials
was proposed by Knowles [7] in 1977, which is the popular biomechanical
models. The Knowles' power-law strain-energy is given by:
W = [mu]/2b [[(1 + b/n([I.sub.1] - 3).sup.n] -1], (1)
where [mu] is the shear modulus, b and n are positive material
parameters and [I.sub.1] the first principal invariants of the
Cauchy-Green deformation tensor. When n > 1 and n [right arrow]
[infinity], we can get the Fung's strain energy function [8] as
shown in Eq. (2), which was a very popular model in biomechanics:
W = [mu]/2b {exp[b([I.sub.1] - 3}] -1], (2)
Gent simplified the Fung's strain energy function for
incompressible materials. One of the simplest strain-energies functions
[9] of Gent for incompressible materials is given by:
W = [mu]/2 [J.sub.m] ln(1 - [I.sub.1] - 3/[J.sub.m]), (3)
where [mu] is the shear modulus and [J.sub.m] is the constant
limiting value for [I.sub.1] - 3 .
From Eq. (3), Gent's strain energy function can be modified
as:
W = [mu][J.sub.m]/2 ln(1 - [I.sup.n.sub.1] - [3.sup.n]/[J.sub.m]),
(4)
where n is material parameter which can be determined from
experiments. In the Modified strain energy function Eq. (4), exponent n
has been introduced. When n = 1, the constitutive model Eq. (4) can be
simplified to the Gent Model. Simultaneously, the exponent n can reflect
the mechanical property of arterial wall for different age group.
Based on the elastic finite deformation theory, the Cauchy stress
tensor can be expressed as follows:
[sigma] = pI + n[mu][J.sub.m]/[J.sub.m] - ([I.sup.n.sub.1] -
[3.sup.n]) [I.sup.n-1.sub.1] B, (5)
where B - F [F.sup.T] is left Cauchy-Green deformation tensor,
[I.sub.1] is the first invariants of B and p is the undeterm-ined scalar
function that justifies the incompressible internal constraint
conditions.
The present work was carried out in order to analyze the uniaxial
tension and circumferential inflation on the mechanical property of
arterial wall based on the modified strain energy function from Gent. By
utilizing the nonlinear finite element software MSC. Marc, numerical
simulation on mechanical property of arterial wall was carried out,
which illustrate that modified constitutive model describes the finite
deformation property of arterial wall reasonably and the applied range
has been broadened.
2. Theoretical analysis
For uniaxial tension of cylinder arterial wall, the deformation
meets the following expressions:
r = f (R), [theta] = [THETA], z = [[lambda].sub.z]Z, (6)
where [[lambda].sub.z] is axial principal stretch; r , [theta], z
are cylindrical coordinate system at current configuration; R , [THETA],
Z are cylindrical coordinate system at initial configuration.
The deformation gradient and left Cauchy-Green deformation tensor
are as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
where [[lambda].sub.r] and [[lambda].sub.[theta]] are radial
principal stretch and circumferential principal stretch accordingly. For
incompressible condition
[[lambda].sub.r][[lambda].sub.[theta]][[lambda].sub.z] = 1.
The stress component can be achieved from the cons-titutive Eq. (5)
as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
For uniaxial tension, [[sigma].sub.rr] =
[[sigma].sub.[theta][theta]] = 0, we can get:
[[sigma].sub.zz] = n [mu][J.sub.m]([[lambda].sup.2.sub.z] -
[[lambda].sup.-1.sub.z])([[lambda].sup.2.sub.z] + 2
[[lambda].sup.-1.sub.z]/[J.sub.m]([([[lambda].sup.2.sub.z] +
2[[lambda].sup.-1.sub.z]).sup.n] - [3.sup.n]). (9)
From Eq. (9) axial force can be expressed as follow:
F = [[sigma].sub.zz] [[lambda].sup.2.sub.r] = [[sigma].sub.zz]
[[lambda].sup.-1.sub.z] =
= n [mu] [J.sub.m] ([[lambda].sub.z] - [[lambda].sup.-2.sub.z])
[([[lambda].sup.2.sub.z] + 2[[lambda].sup.- 1.sub.z]).sup.n-1]/[J.sub.m]
([([[lambda].sup.2.sub.z] + 2[[lambda].sup.-1.sub.z]).sup.n] -
[3.sup.n]). (10)
In order to discuss the effect of constitutive parameters [J.sub.m]
and n on the mechanical properties of material, non-dimensional stress
is introduced. From the Eq. (9), we can get:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)
[F.sup.*] = n [J.sub.m]([[lambda].sub.z] -
[[lambda].sup.-2.sub.z])[([[lambda].sup.2.sub.z] + 2[[lambda].sup.-
1.sub.r]).sup.n-1]/[J.sub.m] ([([[lambda].sup.2.sub.z] +
2[[lambda].sup.-1.sub.z]).sup.n] - [3.sup.n]). (11)
Figs. 1 and 2 have shown the computed result of Eq. (11), When the
parameter n is given (n = 1), as shown in Fig. 1. For Fig. 2, as the
constitutive parameter [J.sub.m] increases, the value of stress
decreases. For same axial stress, when the constitutive parameter
[J.sub.m] increases, the principal extension ratio becomes larger. On
the contrary, if the constitutive parameter [J.sub.m] decreases, the
principal extension ratio becomes smaller. Arterial wall of young people
has excellent elasticity, which can adapt dramatic changes of blood
pressure of human body. This reveals the new strain energy function from
Gent can be used to analyze the deformation of arterial wall under
external load. As is shown in Fig. 2. For the given [J.sub.m] = 2.289,
if the constitutive parameter n increases, the stress becomes greater
and it has the reinforcement feature apparently. Therefore, n is
considered as the material's reinforcement parameter.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
Considering the inflation of cylinder arterial wall, in cylindrical
coordinate system the initial geometry of the tube is given by:
A [less than or equal to] R [less than or equal to] B, 0 [less than
or equal to] [THETA] [less than or equal to] 2[pi], 0 [less than or
equal to] Z [less than or equal to] L. (12)
The deformation of the arterial wall can be expressed as:
r = f(R), [theta] = [THETA], z = [[lambda].sub.z] Z. (13)
The deformation gradient and left Cauchy-Green deformation tensor
are shown as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
From Eq. (14) we can get:
[[lambda].sub.1] = dr / dR, [[lambda].sub.2] = r / R,
[[lambda].sub.3] = [[lambda].sub.z]. (15)
Let [[lambda].sub.2] = r / R = [lambda] and the incompressible
condition [[lambda].sub.1] [[lambda].sub.2] [[lambda].sub.3] = 1 we can
get:
[[lambda].sub.1] = [([lambda][[lambda].sub.z]).sup.-1],
[[lambda].sub.2] = [lambda], [[lambda].sub.3] = [[lambda].sub.z], (16)
[I.sub.1] = trB = [([lambda][[lambda].sub.z]).sup.-2] +
[[lambda].sub.2] + [[lambda].sup.2.sub.z]. (17)
Substituting the Eqs. (14), (16) and (17) to Eq. (5), the following
expressions can be achieved:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
where [W.sub.1] = [partial derivative]W / [partial
derivative][I.sub.1].
For free inflation, [[sigma].sub.zz] = 0. Eq. (18) becomes as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
In the absence of body forces the equilibrium equation is expressed
as divT = 0. From the equilibrium equation, only one component is not
satisfied identically, namely, the radial component, which is:
d[[sigma].sub.zz]/dr + 1/r ([[sigma].sub.rr] -
[[sigma].sub.[theta][theta]] = 0 (20)
We consider the boundary conditions corresponding to a positive
pressure p on the inside of the arterial wall and no load on the
outside:
[[sigma].sub.rr] (a) = -P (P > 0), [[sigma].sub.rr] (b) = 0,
(21)
where P is internal pressure.
Substituting the Eq. (19) to Eqs. (20) and by utilizing the Eq.
(21), we can get:
P = n[mu][J.sub.m]/[J.sub.m] -([I.sup.n.sub.1] - [3.sup.n])
[I.sup.n-1.sub.1] [[[lambda].sup.2] - [([lambda]
[[lambda].sub.2]).sup.-2]] [epsilon]/[[lambda].sup.2][[lambda].sub.z],
(22)
where [epsilon] = (B - A)/A .
In order to discuss the effect of constitutive parameters [mu] and
[epsilon] on the mechanical properties of arterial wall, non-dimensional
pressure is introduced. From the Eq. (22), we can get:
[P.sup.*] = n[J.sub.m]/[J.sub.m] - ([I.sup.n.sub.1] -
[3.sup.n])[I.sup.n-1.sub.1][[[lambda].sup.2] - [([lambda]
[[lambda].sub.z]).sup.-2]] 1/[[lambda].sup.2][[lambda].sub.z] (23)
where [P.sup.*] = P / ([mu][epsilon]).
[FIGURE 3 OMITTED]
In order to discuss the influences of constitutive parameters
[J.sub.m], [[lambda].sub.z] and n on the mechanical properties of
arterial wall, the Eq. (23) is calculated. The result is shown in Figs.
3 and 4.
When the parameter n and [[lambda].sub.z] is given (n = 1,
[[lambda].sub.z] = 1), the relation between [P.sup.*] - [lambda] with
effect of [J.sub.m] as illustrated in Fig. 3. If the constant limiting
value [J.sub.m] increases, the rage of circumferential principal stretch
of arterial wall is enlarged obviously, which indicates that arterial
wall has strong inflation ability and has good toughness. This can
reflect the conditon of young people's arterial wall. In the
contrary, if the constant limiting value [J.sub.m] decreases, the rage
of circumferential principal stretch of arterial wall becomes nerrow,
which reflects the hardening of arterial wall. This can reflect the
conditon of old people's arterial wall.
[FIGURE 4 OMITTED]
As the parameter [J.sub.m] and [[lambda].sub.z] is given ([J.sub.m]
= 2.289, [[lambda].sub.z] = 1), the relation between [P.sup.*] -
[lambda] with effect of n as illustrated in Fig. 4. If the constitutive
parameter n increases, the circumferential principal stretch of arterial
wall becomes smaller under same internal pressure, which means the
inflation ratio of arterial wall decreases correspondingly. This can
reflect the conditon of young people's arterial wall. When the
value of n decreases, the rage of circumferential principal stretch of
arterial wall is enlarged, which means that arterial wall has strong
inflation ability. This can reflect the conditon of young people's
arterial wall. Comparing Fig. 2 with Fig. 1, it can be seen that
constitutive parameter n has more effect on the circumferential
principal stretch of arterial wall. Therefore, n is considered as the
material's reinforcement parameter.
[FIGURE 5 OMITTED]
Fig. 5 shows the relation between [P.sup.*] - [lambda] with effect
of [[lambda].sub.z]. When the parameter [J.sub.m] and n is given
([J.sub.m] = 2.289, [[lambda].sub.z] = 1). When the value of axial
principal stretch of arterial wall [[lambda].sub.z] increases, the the
circumferential principal stretch of arterial wall becomes smaller under
same internal pressure. In the same, if the the value of axial principal
stretch of arterial wall [[lambda].sub.z] decreases, the the
circumferential principal stretch of arterial wall becomes larger. It is
consisitent with he incompressible condition of material from arterial
wall.
3. Numerical simulation
Finite element analysis on nonlinear elastic deformation of
arterial wall have been proposed from [10-12]. The FEA software MSC.
Marc is employed in the numerical simulation of this research. The
strain energy function of arterial wall was assumed to be the strain
energy function of Eq. (4). In order to implement the modified strain
energy function from Gent into the finite element procedure, non-linear
finite element analysis of arterial wall was performed by a user
subroutine when defining the material properties, which allows the users
to define the derivatives of the strain energy functions with respect to
either the strain invariants or the principal stretches. In the
subroutine, w1, w2 and w3 are the first derivatives of the energy
function with respect to strain invariants and w11, w22, w33, ww12,
ww23, w31 are the second derivatives of the energy function with respect
to the strain invariants.
[FIGURE 6 OMITTED]
First, we think about the finite element analysis of uniaxile
tension of arterial wall. From [13], for adults inner diameter of
arterial wall is 10-30 mm, thickness is 2-3 mm. From that we take 24 mm
for outer diameter of arterial and 20 mm for inner diameter. So the
thichness of arterial wall is 2 mm. According to the symmetry of
structure, a quarter model of arterial wall is established. By fixing
constitutive parameter n = 1 and changing the value of constitutive
parameter [J.sub.m], the relation between axial tension stress and axial
pincipal stretch is shown as Fig. 6. From which, we can get for same
axial stress, when the constitutive parameter [J.sub.m] increases, the
axial principal extension ratio becomes larger. Similarly, when fixing
[J.sub.m] = 2.289 and changing the value of the constitutive parameter
n, with the constitutive para-meter n increases, the stress becomes
greater shown as Fig. 7. This agrees with the theoretical results shown
previously.
Arterial wall under pressure is analyzed by utilizing non-linear
finite element method. A quarter finite element models is also
considered as shown in Fig. 8.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
Figs. 9 and 10 show the stress contour of arterial wall under
internal pressure. From that, we can see the radial stress of arterial
wall is compress stress. The absolute value of radial stress increases
with the increase of pressure along the radius. At inside of arterial
wall, radial stress change obviously, but at outside of arterial wall,
radial stress entirely approachs to zero. Circumferential stress is
tensile stress, which also increases with the increase of pressure along
the radius. The maxmum circumferential stress appears at inside wall and
minimum circumferential stress appears outside wall, value of which is
all greater than radial stress at same internal pressure. The relation
between pressure and circumferential pincipal stretch from non-linear
finite element analysis is shown as Figs. 11 and 12. From which we can
get, when the parameter [[lambda].sub.z] and n is given, as the value of
[J.sub.m] increases, the circumferential principal stretch of arterial
wall becomes smaller under same internal pressure. Similarly as the
parameter [J.sub.m] and [[lambda].sub.z] is given. When the value of
constitutive parameter [[lambda].sub.z] increases, the the
circumferential principal stretch of arterial wall becomes smaller under
same internal pressure. This also agrees with the theoretical results
shown previously.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
4. Conclusion
This paper presents the analysis of uniaxial tension and
circumferential inflation on the mechanical property of arterial wall.
Based on the finite deformation theory. Non-linear elastic analysis of
uniaxial tension and inflation of arterial wall has been proposed. By
using the finite element software MSC. Marc, mechanical property of
arterial wall has been also analyzed. It is found that the new
constitutive model fulfills the requirement that the modified strain
energy function will transform into Gent model with n = 1. When n = 1
and [J.sub.m] [right arrow] [infinity], the modified strain energy
density function can be transformed into neo-hooken model. The
constitutive parameters [J.sub.m] can be considered as ultimate
elongation limit parameter of incompressible materials and n can be
considered as the material's reinforcement parameter. Both
constitutive parameters [J.sub.m] and n can reflect the conditon of
young people and old peopled arterial wall. The discussions illustrate
that modified constitutive model describes the finite deformation
property of arterial wall reasonably and the applied range has been
broadened by using the modified constitutive model.
http://dx.doi.Org/10.5755/j01.mech.21.2.8743
Acknowledgement
This paper is supported by Tianjin National Nature Science
Foundation (grant No. 12JCYBJC19600) and Scientific Research Key Project
of Hebei Province Education Department (grand No. ZD20131019).
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Jianbing Sang *, Lifang Sun **, Sufang Xing ***
* School of Mechanical Engineering, Hebei University of Technology,
Tianjin, 300130, China, E-mail: sangjianbing@126.com
** Hebei University of Technology, Tianjin, 300130, China, E-mail:
sunlifang@hebut.edu.cn
*** School of Mechanical Engineering, Hebei University of
Technology, Tianjin, 300130, China, E-mail: sangyajie@126.com
Received November 20, 2014
Accepted March 12, 2015