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Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials
Accepted Manuscript
Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials Yavar Anani , Gholam Hosein Rahimi PII: DOI: Reference:
S0020-7403(17)31480-7 10.1016/j.ijmecsci.2017.06.001 MS 3708
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
9 January 2017 25 May 2017 1 June 2017
Please cite this article as: Yavar Anani , Gholam Hosein Rahimi , Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.06.001
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Highlights
Field equations of axisymmetric FG incompressible hyperelastic shell are presented The solution is applied to find stress components in general form.
Both curvilinear and Cartesian coordinates are used to find general solution.
Proposed power law strain energy function is used to find stress components.
Effect of material inhomogeneity and structural parameter has been investigated.
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Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials Yavar Anania and Gholam Hosein Rahimia* Mechanical Engineering Department, Tarbiat Modares University, Tehran, Iran, * Corresponding Author [email protected] [email protected]
ABSTRACT
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a
In this paper, field equations and general solution for axisymmetric thick shell composed of
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functionally graded incompressible hyperelastic materials are presented. The solution is applied to find stress components in general form for these shells. Both curvilinear and Cartesian coordinates are used to find field equations and general solution. As a special case of the general axisymmetric problem, field equations and stress components of thick-walled
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hollow cylindrical shell composed of functionally graded material in a generalized plane strain condition are developed. For modeling hyperelastic behavior, power law strain energy
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function with variable material parameters is used. Material inhomogeneity is assumed to vary by a power law function in the radial direction and inhomogeneity parameter
) is power in
the mentioned power law function. Nonlinear regression method is used to find material
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constants of strain energy function from experimental data. As a result circumferential stretch, radial stress, circumferential stress and longitudinal stress through the radial direction is
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presented for different values of inhomogeneity parameter
. The achieved outcomes display that the material
) and structure parameter
: ratio of outer radius to inner radius)
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influence considerably on the mechanical behavior of thick-walled hollow cylindrical shell made of functionally graded materials. Accordingly with opting for a proper parameter
and structure
, particular FGM hollow cylinder that can meet some special requirements will
be designed by engineers. Keywords: Hyperelasticity; Field equations; Functionally graded material; General solution, Axisymmetric thick shell
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1.
INTRODUCTION
Generally, nonlinear behavior of rubber like materials modeled by hyperelasticity. Particular characteristics and financial benefits [1] of these materials are the main reason of their widely uses in various structures of diverse industries such as shells, spheres, tubes, rings and pads in, petrochemical, aerospace, biomedical and numerous other fields of human life. In order to
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mature rubber elasticity, hypothetical stress-strain relation that fits experimental data of hyperelastic materials is the subject of numerous works. For this purpose, two different phenomenological methods have been used to study rubber elasticity. Continuum mechanicsbased view point and statistical or kinetic theory method. The representative works can be found in the researches of Mooney [2], Blatz-Ko [3], Yeoh [4] and Ogden [5]. Review of
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mechanical behavior of rubber like materials can be found in the articles contributed by Beatty [6], Horgan and Polignone [7], Attard [8] and the monograph contributed by Fu and Ogden [9]. In the recent years several studies have been done on constitutive modeling of rubber like materials such as works by Anani and Alizadeh [10], Tomita et al. [11], Coelho et al. [12] and Santos et al. [13].
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The main concern of this paper is about field equations, general solution and stress components of general axisymmetric thick shell composed of functionally graded
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incompressible hyperelastic materials, therefore studies and investigations on different axisymmetric shells are carefully reviewed and their key notes are mentioned here.
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For the first time, Ikeda have made graded rubber like materials in the late 1980s [14], a while after these materials have appealed the care of scientists for modeling their behavior under
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geometrical and mechanical boundary conditions. Some significant and novel explores about stress analysis of inhomogeneous rubber like materials structures are stated here. For example, influences of material inhomogeneities on stress through-the-thickness of circular
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cylinders made of rubber like materials in mechanical and thermal load has been studied by Bilgili et al. [15]. In a different study, circular cylinder made of an inhomogeneous neoHookean material with circumferential displacements prescribed on the inner and the outer surfaces in the plane strain state has been investigated by Bilgili [16]. Torsion of a cylinder made of incompressible Hookean material with variable shear modulus along the axial direction has been studied by Batra [17] and twist of cross-section has been controlled by variation of the shear modulus.
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Research on thin-walled hyperelastic rubber-type structures has been done in the recent years. Some research of these structures are as follows. Non-linear radial oscillations of an incompressible hyperelastic spherical shell has been investigated by Roussos et al. [18]. nonlinear radial oscillations of a transversely isotropic hyperelastic incompressible tube is the subject of the research of Mason and Maluleke [19]. Mathematical modelling and stability
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analysis of an inflation of thin-walled hyperelastic tube with applications to abdominal aortic aneurysms has been done by Nikolova and Ivanov [20]. Fu et al. [21] have considered postbifurcation analysis of a thin-walled hyperelastic tube under inflation in their research.
Circular cylinder made of an inhomogeneous Mooney–Rivlin material by considering axisymmetric deformations in plane strain condition has been analyzed via Finite element
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method by Batra [22]. Batra and Bahrami [23] have analyzed cylindrical pressure vessel made of FG Mooney-Rivlin material where the material parameters are variable continuously through the radial direction either by a power law or an affine relation. Newly, Behavior of spherical shell and rotating cylindrical shell made of FG rubbers have been explored by Anani and Rahimi [24, 25]. In their studies, they have presumed power law function to model the
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variation of material properties in radial direction and distribution of stretch and stress components through the shell thickness have been presented.
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Review through the researches on different structures, it is observe that there is no literature and exploration about general solution of thick axisymmetric shell composed of isotropic FG
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rubber like materials. Therefore, presenting set of field equations, general solution and stress components of above general axisymmetric thick shells are the distinguished points of this
2.
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study.
Problem formulation
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The outcome of the rotation of a plane curve around an axis in its plane is a surface of revolution. The resulting curve is technically called a meridian. According to Fig. 1, coordination of each arbitrary point is indicated by , of rotation
− axis,
and
.
is the distance from the axis
represents the distance of its points from the
– axis and
between the axis of revolution and the normal to the meridian curve. two principal radii of curvature [26].
and
is the angle are also the
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Fig. 1 Meridian of the middle surface of a shell of revolution
In Fig. 2, the cross-section of an arbitrary variable thickness shell element is shown. In the presented curvilinear coordinates system
,
is the meridional coordinate,
normal distance from the midsurface to an arbitrary point
demonstrates the radial distance of the arbitrary point
and is presented as follows: ]
(1)
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[
is the circumferential
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angle. Moreover, according to Fig.2,
and
is the
Fig. 2 Cross-section of an arbitrary shell of revolution with variable thickness
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By using equation (1), the curvilinear coordinates
is transformed to Cartesian
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coordinates by:
(2)
Fig.3 shows the transformations between different coordinates. Cartesian coordinates in the reference and the current configurations are presented by
and
,
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respectively. Fig.3 Transformation between Cartesian and curvilinear coordinates
In view of that motion of a particle is
if Cartesian coordinates are chosen.
Curvilinear coordinates in the reference and the current configurations are represented by and a particle is
, respectively. Moreover, motion of
if curvilinear coordinates are chosen. Coordinate transformations
configurations are defined by
̂
and
By using equation (2), position vectors of point are described by equations (3.a) and (3.b): ( )
, respectively.
in the reference and current configurations
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( )
̂
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between the curvilinear and the Cartesian coordinates in the reference and current
( )
(3.a)
(3.b)
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Tangent ( ) and gradient vectors ( ) in the reference configurations are presented by [23]: (4)
Similarly, tangent ( ) and gradient ( ) vectors in the current configurations are obtainable
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by [23]:
(5)
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Additionally, covariant and contravariant metric coefficients for to the reference and current
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configurations are defined [23]: (6) (7)
By using equations (2), (3.a), (3.b), (4) and (5), tangent and gradient vectors in the reference and current configurations are presented by: {
(8) (9)
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{
(10) (11)
(12)
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{
{
(13)
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By using equations (2), (3.a), (3.b), (6) and (7), non-vanishing covariant and contravariant metric coefficients relative to reference and current configurations are defined by:
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(14)
The Christoffel symbols of the second kind, by symmetric condition
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[27]:
, are defined by
(16) are:
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Therefore, non-vanishing components of
(15)
,
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,
(17)
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,
Kirchhoff stress for incompressible hyperelastic material is [9,23]:
Where
(18) and
(
and Kirchhoff stress is defined by: therefore √
)
. In incompressible materials
. Physical components of stress tensor, √
. Relation between Cauchy stress and
, are presented [9]: (19)
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Mixed components of stress tensor are attained as follows: √
(20)
√
Using equations (14), (15), (19) and (20) resulted to:
(21)
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The equations of motion could be written compactly in tensor form [9]: ̈
Where the vertical bar (|) symbolizes covariant differentiation and Furthermore,
and ̈ are components of the body force vector
(22)
is the density.
per unit volume and
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covariant components of the acceleration of the volume in the deformed body. The covariant derivatives of a tensor of order two are also tensors and are represented as [28]:
where,
(23)
are the mixed components of a typical tensor of order two. By substituting
equations (17) and (21) into equation (22), the equations of motion in terms of the physical
)
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(
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components could be derived as follows:
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(
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(24) )
(25) (
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̈
̈
)
̈ (26)
In order to find Kirchhoff stress components in axisymmetric condition, non-vanishing components of (
)
of equation (18) are calculated: (
)
(
)
( )
(
)
( )
(27)
Therefore, non-vanishing stress components are presented as follows: (
)
*(
)
(
) +
(28)
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*(
)
( )
( ) + *(
(29)
)
( ) +
(30)
By implementing equations (28-30) to equations (24-26), set of field equations are derived as follows: (
)
(
) +
+
*((
̈ *(
(
(
( ) +
)
( ) +
)
*(
) ( )
+
)
(
(
)(
) +
̈ *(
)
( ) +
+
)
( ) )
*(
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*
*
)
)
(
( )
)
( ) +
*(
(31) )
) (32)
̈
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+
*(
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*
(33)
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In order to find stress components, equation (32) is used. By integrating from this equation it is found that: )
(
) +)
((
[
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)
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(
∫(
[
∫
( ) ]
(
*(
*
)
(
(
) +
*( )
) +))
̈ ] (34)
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By comparison of equation (29) and equation (34), hydrostatic pressure is calculated as follows:
(
) +)
*( (
[
)
( ) +
∫
( ) ]
*(
)
* (
(
* ) +)
(
) +
*( ) ̈ + (35)
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By substituting hydrostatic pressure in equations (28) and (30), other stress components are derived as follows:
(
) +
)
+
*( )
((
[
(
*(
)
( ) +
*(
)
(
) +)) (36)
) +
]
*( )
((
[
(
(
*
*(
)
) +)
( ) ]
*(
)
(
) +
̈ ]
∫
[
(
*
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(
[
) +)
( ) ]
[( )
∫
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*(
(
) +))
̈ ]
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(37)
As a result of above presented solution, field equations of general axisymmetric thick shell
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composed of functionally graded incompressible hyperelastic materials are presented by equations (31-33) and stress components are calculated by equations (34), (36) and (37).
3.
Case study: Axisymmetric circular cylinder
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In this section, thick-walled FGM hollow cylinder is considered as a special case of thick shell of revolution. Following simplifications are considered in Fig.2 and equation (1) for
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cylinder as a shell of revolution: (38)
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Thick-walled hollow cylindrical shell made of isotropic hyperelastic materials with an inner radius
and an outer radius
, in the plane strain condition, and internal pressure
is
considered. The cylinder is considered initially stress-free and assumed to be deformed statically. It is supposed that material is perfectly elastic and each point in the body goes from in Cartesian coordinate to coordinates
in the same coordinates. By considering, cylindrical
as a curvilinear coordinates in both reference and current configurations,
geometry of the cylinder in the reference and current configurations are described as follows:
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(39)
b,
(40)
Position vectors of each arbitrary point in the reference and current configurations are presented by: ,
(41)
,
shall be determined. By considering equations (4), (8), (11) and (41) tangent ( )
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Where
(42)
and gradient ( ) vectors in the reference configuration are presented by: [
]
[
]
(43)
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Also, by considering equation (5) and (42) tangent ( ) and gradient ( ) vectors in the current configuration are given by: [
]
[
]
(44)
Covariant and contravariant metric coefficients relative to reference and current
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configurations are calculated by using equations (6), (14) and (7), (15): ]
]
[
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[
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[
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Incompressibility condition
[
]
[
]
[
]
]
(45)
(46) implies that: (47)
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Resulting:
,
(48)
,
(49)
,
(50)
As a result by using equations (45), (46), (49) and (50) contravariant metric coefficients relative to reference and current configurations are described by:
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( [
)
( [
]
[
]
[
) ]
(51)
]
(52)
(
)
( )
( )
(
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By using equation (28-30) stress components are calculated as follows:
) ( )
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( )
(53) (54) (55) (56)
Equilibrium equation of the thick-walled cylinder in the radial direction and boundary conditions are expressed as:
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,
(57) (58)
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By substituting stress components from equation (53-56) in the equilibrium equation in the radial direction (equation (57)) and integrating, [(( )
(
) )]
[
( )
( ) ]
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∫
is expressed as follows: (59)
By comparison of equation (53) and equation (59), hydrostatic pressure is calculated: )
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(
( )
(60)
By implementing equation (60) into equations (54) and (55), other stress components are
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calculated:
( )
( )
( )
( )
( )
(61)
( )
(62)
is determined by using second boundary condition of equation (58): ∫
[(( )
(
) )]
3.1.1. Power law strain energy function
[
( )
( ) ]
(63)
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Power law strain energy function for incompressible materials is used as follows[29]: (64) As usual, this energy function vanishes in the undeformed configuration. strain energy is a convex function; increasing monotonically with attains a minimum in the reference configuration, so that
where
and
must contain no real root . In the
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which places restrictions on the values that the constants can take; i.e.: other word:
. The
, both W and resulted stress approach infinity for very large
deformations:
and
and there is no stress in the
undeformed configuration and strain energy is minimum in this state:
is material parameter which varies by power law
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. In equation (64),
,
( ) . Material constants " " and
function in radial direction, where
are
determined by using Levenberg–Marquardt nonlinear regression method for the rubber tested by experiment and
implementing them in equation (62) [ ( )
,
and
by
is calculated:
( )
(( )
ED
( )
∫
( )
energy function,
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mentioned strain
is material inhomogeneity parameter. By considering above
( ) )]
(65)
By integration of above equation, radial stress is achieved: )
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Where:
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(
)
Where
(66)
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(
(
) (
)
*(
( )) +
(67)
) is the Gauss-hypergeometric function [28]. Second boundary condition
of equation (58) is used to find outer radius of cylinder in the deformed configuration: (68) and
are calculated by using equations (64), (65) and (69), (70):
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4.
(( )
( )
)
(( )
(( )
( )
)
(
( ) )
(69)
( ) )
(70)
Result and Discussion
In this paper functionally graded rubber is considered and inhomogeneity occurs in the radial
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direction of the shell. It is presumed that, the inner surface of the shell is made of the rubber tested by Treloar [30]. Material properties of the FG rubber vary by a power law function in ( ) . For each special FG rubber,
the radial direction, i.e.:
for the outer
surface will be available (for example outer surface of the shell can be silicon rubber), therefore for each specific FG rubber, material inhomogeneity parameter ( ) will be found
Treloar [30],
MPa and
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( ) . For power law strain energy function and for the rubber tested by
by:
are achieved. For generality, material
inhomogeneity parameter is considered . It is obvious that Each specific FG rubber will be its specific
.
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Example: Two cylindrical shells with inner and outer radius are considered. The applied internal pressure is .
is calculated for different values of
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and
m,
m and
m,
. In this example, we considered by using equation (68).
Distribution of circumferential extension ratio, radial stress, circumferential stress and axial
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stress are presented in the below figures. Based on the presented results in Figs. 4-7, maximum value of extension ratio occurs in the inner radius then it decreases through the
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thickness and will be minimum in the outer radius. Moreover, extension ratio increases by increasing internal pressure and decreasing thickness of the vessel and material
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inhomogeneity parameter.
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and
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Fig.4. Distribution of extension ratio for different material inhomogeneity parameter ( ),
Fig.5. Distribution of extension ratio for different material inhomogeneity parameter ( ),
and
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and
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Fig.6. Distribution of extension ratio for different material inhomogeneity parameter ( ),
Fig.7. Distribution of extension ratio for different material inhomogeneity parameter ( ),
and
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According to Figs.8-11, magnitude of normalized radial stress increases by increasing internal pressure, structure parameter
and material inhomogeneity parameter ( ). Normalized
radial stress shows trivial differences for different material inhomogeneity parameter at points near the boundaries, while at points away from boundaries, the reverse holds true. Figs.12-15, show that normalized hoop stress in the inner surface of the cylinder increases by increasing internal pressure and decreasing structure parameter
and material inhomogeneity
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parameter ( ). In the other surface of the cylinder normalized hoop stress increases by increasing material inhomogeneity parameter ( ). In contrast with normalized radial stress which has the same distribution for different
, the most remarkable part of the outcomes is
distribution of normalized hoop stress which altering from monotonically increasing in as a function of gradient parameter
. Figs.16-19 demonstrate
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monotonically decreasing in
to
that longitudinal stress has the same distribution as hoop stress. Fig.8. Distribution of normalized radial stress for different material inhomogeneity parameter ( ),
and
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and
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Fig.9. Distribution of normalized radial stress for different material inhomogeneity parameter ( ),
Fig.10. Distribution of normalized radial stress for different material inhomogeneity parameter ( ),
and
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and
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Fig.11. Distribution of normalized radial stress for different material inhomogeneity parameter ( ),
Fig.12. Distribution of normalized hoop stress for different material inhomogeneity parameter ( ),
and
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Fig.13. Distribution of normalized hoop stress for different material inhomogeneity parameter ( ),
and
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Fig.14. Distribution of normalized hoop stress for different material inhomogeneity parameter ( ),
Fig.15. Distribution of normalized hoop stress for different material inhomogeneity parameter ( ),
and
and
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Fig.16. Distribution of normalized longitudinal stress for different material inhomogeneity parameter ( ),
and
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and
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Fig.17. Distribution of normalized longitudinal stress for different material inhomogeneity parameter ( ),
Fig.18. Distribution of normalized longitudinal stress for different material inhomogeneity parameter ( ),
and
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Fig.19. Distribution of normalized longitudinal stress for different material inhomogeneity parameter ( ),
and
For validation, finite element method is used to find the accuracy of the results. Inhomogeneity in radial direction is considered in Abaqus by using field variable. For this purpose, a vessel with inner and outer radius is considered.
and
m,
m, and internal pressure
are considered for modeling material inhomogeneity
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parameter. The results are presented in Figs.20-22. According to Figs.21-22, it is found that there is very good convergence between theoretical results and FEM results. Maximum difference between FEM results and theoretical results is happened in the internal diameter of the vessel. These differences are about 8.53% for
and 6.61% for
. Results show
that differences between theoretical results and FEM results decrease by increasing material
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inhomogeneity parameter. By considering these results it is concluded that there is good agreement between FEM and presented theoretical results, therefore theoretical solution can be applied for finding deformation fields and stress components of the axisymmetric thick
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vessel composed of FG hyperelastic material.
Fig.20. FEM results of FG hyperelastic cylinder (
7)
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2,
and
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Fig.21. Comparison between theoretical results and FEM results for
Fig.22. Comparison between theoretical results and FEM results for
2,
and
7
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5.
Conclusion
In the presented paper, general solution for axisymmetric revolution shell composed of functionally graded hyperelastic material is presented. Curvilinear coordinates and Cartesian coordinates are used to find set of field equations and stress components for above mentioned shell in axisymmetric condition. As a special case, the formulation is applied for thick hollow
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cylinder. Power law strain energy function is considered to determine stress components and displacement of the cylinder. The material properties vary by a power law function through the cylinder thickness. Stress components and stretches of the thick-walled cylinder in the plane strain state has been found by analytical method and related distribution has been presented.
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Based on presented results, It is found that, increasing material inhomogeneity parameter
)
causes decreasing in radial stress, extension ratio and deformed cylinder radius. Moreover by increasing internal pressure and structure parameter
, magnitude of normalized radial
stress increases. Additionally, by increasing internal pressure and decreasing structure parameter
normalized hoop stress increases. The distribution of normalized hoop stress
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and normalized longitudinal stress changing from monotonically increasing in monotonically decreasing in
as a function of gradient parameter
to
. This distribution is
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actual valuable for design of this kind of pressure vessel to postpone or evade failure. Finite element method is applied in order to validate theoretical results. By comparison between theoretical results and FEM results it is found that there is a good agreement between them.
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Above achieved results in investigating mechanical behavior of thick hollow cylindrical shell composed of functionally graded hyperelastic material with power law varying properties,
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demonstrate great effect of
and
in displacement field and stress components distribution.
Thus from a design approach, opting a proper
and
is a very useful method to control the
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stress and to optimal use of material.
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[4] Yeoh OH. Some forms of the strain energy function for rubber. Rubber Chem. Technology 1993; 66:754771. [5] Ogden RW. Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R.Soc. Lond. A., 1972. [6] Beatty MF. Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues—with
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[8] Attard MM. Finite strain-isotropic hyperelasticity. International Journal of Solids and Structures 2003; 40(17): 4353-4378.
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rubber and carbon-black-filled rubber under monotonic and cyclic straining. International Journal of Mechanical
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[12] Coelho M, Roehl D, Bletzinger K. Numerical and analytical solutions with finite strains for circular inflated membranes considering pressure-volume coupling. International Journal of Mechanical Sciences 2014; 82: 122-
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130.
[13] Santos T, Alves MK, Rossi R. A constitutive formulation and numerical procedure to model rate effects on
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porous materials at finite strains. International Journal of Mechanical Sciences 2015; 93:166-180. [14] Ikeda Y, Kasai Y, Murakami S, Kohjiya S. Preparation and mechanical properties of graded styrene-
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butadiene rubber vulcanizates. J. Jpn. Inst. Metals 1998; 62: 1013–1017. [15] Bilgili E. Controlling the stress–strain inhomogeneities in axially sheared and radially heated hollow rubber tubes via functional grading. Mech. Res. Commun.2003; 30:257–66. [16] Bilgili E. Modelling mechanical behaviour of continuously graded vulcanized rubbers. Plast Rubbers Compos 2004; 33(4):163-169. [17] Batra RC. Torsion of a functionally graded cylinder. AIAA J 2006; 44: 1363-1365.
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[18] Roussos N, Mason DP and Hill D. On non-linear radial oscillations of an incompressible hyperelastic spherical shell. Mathematics and Mechanics of Solids 2002;7:67-85. [19] Mason DP and Maluleke GH. Non-linear radial oscillations of a transversely isotropic hyperelastic incompressible tube. Journal of Mathematical Analysis and Applications 2007, 333(1): 365-380 [20] Fu YB, Pearce SP and Liu KK. Post-bifurcation analysis of a thin-walled hyperelastic tube under inflation,
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Int. J. of Non-Linear Mechanics 2008, 43(8): 697–706. [21] Nikolova EV and Ivanov T. Mathematical modelling and stability analysis of an inflation of thin-walled hyperelastic tube with applications to abdominal aortic aneurysms, Series on Biomechanics 2015, 29(4), 78-84 [22] Batra RC. Finite plane strain deformations of rubberlike materials. Int. J. Numer. Methods Eng. 1980; 15: 145-160.
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[23] Batra RC and Bahrami A. Inflation and eversion of functionally graded nonlinear elastic incompressible cylinders. Int. J. of Non-Linear Mechanics 2003; 44: 311-323.
[24] Anani Y and Rahimi GH. Stress analysis of thick pressure vessel composed of functionally graded incompressible hyperelastic materials, International Journal of Mechanical Sciences 2015; 104: 1-7
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[25] Anani Y and Rahimi GH. Stress analysis of rotating cylinder shell composed of functionally graded incompressible hyperelastic materials, International Journal of Mechanical Sciences 2016; 108-109: 122-128
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[26] Zamaninejad M, Rahimi GH, and Ghannad M. Set of field equations for thick shell of revolution made of functionally graded materials in curvilinear coordinate system. Mechanika 2009; 77: 18-26.
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[27] Chaudhry HR and Gupta US. Rotation of hyperelastic annular and solid disks of variable thickness. Int. J. of Non-Linear Mechanics 1992; 27: 341-346.
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[28] Arfken G. Mathematical Methods for Physicists. 3rd ed. Orlando: Academic Press; 1985. [29] Fatt MSH and Ouyang X. Integral-based constitutive equation for rubber at high strain rates. International
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Journal of Solids and Structures 2007; 44(20): 6491-6506. [30] Treloar LRG. Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc 1944; 40: 59-70.
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Graphical Abstract: Journal: International Journal of Mechanical Sciences (IJMS)
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Title: Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials Authors: Y.Anani, G.H.Rahimi Manuscript Number: SUBMIT2IJMS-D-17-00001R1
Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials Yavar Anani , Gholam Hosein Rahimi PII: DOI: Reference:
S0020-7403(17)31480-7 10.1016/j.ijmecsci.2017.06.001 MS 3708
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
9 January 2017 25 May 2017 1 June 2017
Please cite this article as: Yavar Anani , Gholam Hosein Rahimi , Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.06.001
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Highlights
Field equations of axisymmetric FG incompressible hyperelastic shell are presented The solution is applied to find stress components in general form.
Both curvilinear and Cartesian coordinates are used to find general solution.
Proposed power law strain energy function is used to find stress components.
Effect of material inhomogeneity and structural parameter has been investigated.
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Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials Yavar Anania and Gholam Hosein Rahimia* Mechanical Engineering Department, Tarbiat Modares University, Tehran, Iran, * Corresponding Author [email protected] [email protected]
ABSTRACT
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a
In this paper, field equations and general solution for axisymmetric thick shell composed of
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functionally graded incompressible hyperelastic materials are presented. The solution is applied to find stress components in general form for these shells. Both curvilinear and Cartesian coordinates are used to find field equations and general solution. As a special case of the general axisymmetric problem, field equations and stress components of thick-walled
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hollow cylindrical shell composed of functionally graded material in a generalized plane strain condition are developed. For modeling hyperelastic behavior, power law strain energy
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function with variable material parameters is used. Material inhomogeneity is assumed to vary by a power law function in the radial direction and inhomogeneity parameter
) is power in
the mentioned power law function. Nonlinear regression method is used to find material
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constants of strain energy function from experimental data. As a result circumferential stretch, radial stress, circumferential stress and longitudinal stress through the radial direction is
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presented for different values of inhomogeneity parameter
. The achieved outcomes display that the material
) and structure parameter
: ratio of outer radius to inner radius)
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influence considerably on the mechanical behavior of thick-walled hollow cylindrical shell made of functionally graded materials. Accordingly with opting for a proper parameter
and structure
, particular FGM hollow cylinder that can meet some special requirements will
be designed by engineers. Keywords: Hyperelasticity; Field equations; Functionally graded material; General solution, Axisymmetric thick shell
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1.
INTRODUCTION
Generally, nonlinear behavior of rubber like materials modeled by hyperelasticity. Particular characteristics and financial benefits [1] of these materials are the main reason of their widely uses in various structures of diverse industries such as shells, spheres, tubes, rings and pads in, petrochemical, aerospace, biomedical and numerous other fields of human life. In order to
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mature rubber elasticity, hypothetical stress-strain relation that fits experimental data of hyperelastic materials is the subject of numerous works. For this purpose, two different phenomenological methods have been used to study rubber elasticity. Continuum mechanicsbased view point and statistical or kinetic theory method. The representative works can be found in the researches of Mooney [2], Blatz-Ko [3], Yeoh [4] and Ogden [5]. Review of
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mechanical behavior of rubber like materials can be found in the articles contributed by Beatty [6], Horgan and Polignone [7], Attard [8] and the monograph contributed by Fu and Ogden [9]. In the recent years several studies have been done on constitutive modeling of rubber like materials such as works by Anani and Alizadeh [10], Tomita et al. [11], Coelho et al. [12] and Santos et al. [13].
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The main concern of this paper is about field equations, general solution and stress components of general axisymmetric thick shell composed of functionally graded
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incompressible hyperelastic materials, therefore studies and investigations on different axisymmetric shells are carefully reviewed and their key notes are mentioned here.
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For the first time, Ikeda have made graded rubber like materials in the late 1980s [14], a while after these materials have appealed the care of scientists for modeling their behavior under
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geometrical and mechanical boundary conditions. Some significant and novel explores about stress analysis of inhomogeneous rubber like materials structures are stated here. For example, influences of material inhomogeneities on stress through-the-thickness of circular
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cylinders made of rubber like materials in mechanical and thermal load has been studied by Bilgili et al. [15]. In a different study, circular cylinder made of an inhomogeneous neoHookean material with circumferential displacements prescribed on the inner and the outer surfaces in the plane strain state has been investigated by Bilgili [16]. Torsion of a cylinder made of incompressible Hookean material with variable shear modulus along the axial direction has been studied by Batra [17] and twist of cross-section has been controlled by variation of the shear modulus.
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Research on thin-walled hyperelastic rubber-type structures has been done in the recent years. Some research of these structures are as follows. Non-linear radial oscillations of an incompressible hyperelastic spherical shell has been investigated by Roussos et al. [18]. nonlinear radial oscillations of a transversely isotropic hyperelastic incompressible tube is the subject of the research of Mason and Maluleke [19]. Mathematical modelling and stability
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analysis of an inflation of thin-walled hyperelastic tube with applications to abdominal aortic aneurysms has been done by Nikolova and Ivanov [20]. Fu et al. [21] have considered postbifurcation analysis of a thin-walled hyperelastic tube under inflation in their research.
Circular cylinder made of an inhomogeneous Mooney–Rivlin material by considering axisymmetric deformations in plane strain condition has been analyzed via Finite element
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method by Batra [22]. Batra and Bahrami [23] have analyzed cylindrical pressure vessel made of FG Mooney-Rivlin material where the material parameters are variable continuously through the radial direction either by a power law or an affine relation. Newly, Behavior of spherical shell and rotating cylindrical shell made of FG rubbers have been explored by Anani and Rahimi [24, 25]. In their studies, they have presumed power law function to model the
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variation of material properties in radial direction and distribution of stretch and stress components through the shell thickness have been presented.
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Review through the researches on different structures, it is observe that there is no literature and exploration about general solution of thick axisymmetric shell composed of isotropic FG
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rubber like materials. Therefore, presenting set of field equations, general solution and stress components of above general axisymmetric thick shells are the distinguished points of this
2.
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study.
Problem formulation
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The outcome of the rotation of a plane curve around an axis in its plane is a surface of revolution. The resulting curve is technically called a meridian. According to Fig. 1, coordination of each arbitrary point is indicated by , of rotation
− axis,
and
.
is the distance from the axis
represents the distance of its points from the
– axis and
between the axis of revolution and the normal to the meridian curve. two principal radii of curvature [26].
and
is the angle are also the
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Fig. 1 Meridian of the middle surface of a shell of revolution
In Fig. 2, the cross-section of an arbitrary variable thickness shell element is shown. In the presented curvilinear coordinates system
,
is the meridional coordinate,
normal distance from the midsurface to an arbitrary point
demonstrates the radial distance of the arbitrary point
and is presented as follows: ]
(1)
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[
is the circumferential
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angle. Moreover, according to Fig.2,
and
is the
Fig. 2 Cross-section of an arbitrary shell of revolution with variable thickness
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By using equation (1), the curvilinear coordinates
is transformed to Cartesian
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coordinates by:
(2)
Fig.3 shows the transformations between different coordinates. Cartesian coordinates in the reference and the current configurations are presented by
and
,
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respectively. Fig.3 Transformation between Cartesian and curvilinear coordinates
In view of that motion of a particle is
if Cartesian coordinates are chosen.
Curvilinear coordinates in the reference and the current configurations are represented by and a particle is
, respectively. Moreover, motion of
if curvilinear coordinates are chosen. Coordinate transformations
configurations are defined by
̂
and
By using equation (2), position vectors of point are described by equations (3.a) and (3.b): ( )
, respectively.
in the reference and current configurations
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( )
̂
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between the curvilinear and the Cartesian coordinates in the reference and current
( )
(3.a)
(3.b)
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Tangent ( ) and gradient vectors ( ) in the reference configurations are presented by [23]: (4)
Similarly, tangent ( ) and gradient ( ) vectors in the current configurations are obtainable
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by [23]:
(5)
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Additionally, covariant and contravariant metric coefficients for to the reference and current
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configurations are defined [23]: (6) (7)
By using equations (2), (3.a), (3.b), (4) and (5), tangent and gradient vectors in the reference and current configurations are presented by: {
(8) (9)
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{
(10) (11)
(12)
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{
{
(13)
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By using equations (2), (3.a), (3.b), (6) and (7), non-vanishing covariant and contravariant metric coefficients relative to reference and current configurations are defined by:
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(14)
The Christoffel symbols of the second kind, by symmetric condition
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[27]:
, are defined by
(16) are:
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Therefore, non-vanishing components of
(15)
,
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,
(17)
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,
Kirchhoff stress for incompressible hyperelastic material is [9,23]:
Where
(18) and
(
and Kirchhoff stress is defined by: therefore √
)
. In incompressible materials
. Physical components of stress tensor, √
. Relation between Cauchy stress and
, are presented [9]: (19)
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Mixed components of stress tensor are attained as follows: √
(20)
√
Using equations (14), (15), (19) and (20) resulted to:
(21)
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The equations of motion could be written compactly in tensor form [9]: ̈
Where the vertical bar (|) symbolizes covariant differentiation and Furthermore,
and ̈ are components of the body force vector
(22)
is the density.
per unit volume and
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covariant components of the acceleration of the volume in the deformed body. The covariant derivatives of a tensor of order two are also tensors and are represented as [28]:
where,
(23)
are the mixed components of a typical tensor of order two. By substituting
equations (17) and (21) into equation (22), the equations of motion in terms of the physical
)
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(
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components could be derived as follows:
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(
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(24) )
(25) (
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̈
̈
)
̈ (26)
In order to find Kirchhoff stress components in axisymmetric condition, non-vanishing components of (
)
of equation (18) are calculated: (
)
(
)
( )
(
)
( )
(27)
Therefore, non-vanishing stress components are presented as follows: (
)
*(
)
(
) +
(28)
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*(
)
( )
( ) + *(
(29)
)
( ) +
(30)
By implementing equations (28-30) to equations (24-26), set of field equations are derived as follows: (
)
(
) +
+
*((
̈ *(
(
(
( ) +
)
( ) +
)
*(
) ( )
+
)
(
(
)(
) +
̈ *(
)
( ) +
+
)
( ) )
*(
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*
*
)
)
(
( )
)
( ) +
*(
(31) )
) (32)
̈
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+
*(
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*
(33)
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In order to find stress components, equation (32) is used. By integrating from this equation it is found that: )
(
) +)
((
[
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)
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(
∫(
[
∫
( ) ]
(
*(
*
)
(
(
) +
*( )
) +))
̈ ] (34)
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By comparison of equation (29) and equation (34), hydrostatic pressure is calculated as follows:
(
) +)
*( (
[
)
( ) +
∫
( ) ]
*(
)
* (
(
* ) +)
(
) +
*( ) ̈ + (35)
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By substituting hydrostatic pressure in equations (28) and (30), other stress components are derived as follows:
(
) +
)
+
*( )
((
[
(
*(
)
( ) +
*(
)
(
) +)) (36)
) +
]
*( )
((
[
(
(
*
*(
)
) +)
( ) ]
*(
)
(
) +
̈ ]
∫
[
(
*
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(
[
) +)
( ) ]
[( )
∫
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*(
(
) +))
̈ ]
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(37)
As a result of above presented solution, field equations of general axisymmetric thick shell
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composed of functionally graded incompressible hyperelastic materials are presented by equations (31-33) and stress components are calculated by equations (34), (36) and (37).
3.
Case study: Axisymmetric circular cylinder
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In this section, thick-walled FGM hollow cylinder is considered as a special case of thick shell of revolution. Following simplifications are considered in Fig.2 and equation (1) for
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cylinder as a shell of revolution: (38)
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Thick-walled hollow cylindrical shell made of isotropic hyperelastic materials with an inner radius
and an outer radius
, in the plane strain condition, and internal pressure
is
considered. The cylinder is considered initially stress-free and assumed to be deformed statically. It is supposed that material is perfectly elastic and each point in the body goes from in Cartesian coordinate to coordinates
in the same coordinates. By considering, cylindrical
as a curvilinear coordinates in both reference and current configurations,
geometry of the cylinder in the reference and current configurations are described as follows:
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(39)
b,
(40)
Position vectors of each arbitrary point in the reference and current configurations are presented by: ,
(41)
,
shall be determined. By considering equations (4), (8), (11) and (41) tangent ( )
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Where
(42)
and gradient ( ) vectors in the reference configuration are presented by: [
]
[
]
(43)
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Also, by considering equation (5) and (42) tangent ( ) and gradient ( ) vectors in the current configuration are given by: [
]
[
]
(44)
Covariant and contravariant metric coefficients relative to reference and current
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configurations are calculated by using equations (6), (14) and (7), (15): ]
]
[
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[
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[
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Incompressibility condition
[
]
[
]
[
]
]
(45)
(46) implies that: (47)
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Resulting:
,
(48)
,
(49)
,
(50)
As a result by using equations (45), (46), (49) and (50) contravariant metric coefficients relative to reference and current configurations are described by:
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( [
)
( [
]
[
]
[
) ]
(51)
]
(52)
(
)
( )
( )
(
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By using equation (28-30) stress components are calculated as follows:
) ( )
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( )
(53) (54) (55) (56)
Equilibrium equation of the thick-walled cylinder in the radial direction and boundary conditions are expressed as:
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,
(57) (58)
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By substituting stress components from equation (53-56) in the equilibrium equation in the radial direction (equation (57)) and integrating, [(( )
(
) )]
[
( )
( ) ]
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∫
is expressed as follows: (59)
By comparison of equation (53) and equation (59), hydrostatic pressure is calculated: )
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(
( )
(60)
By implementing equation (60) into equations (54) and (55), other stress components are
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calculated:
( )
( )
( )
( )
( )
(61)
( )
(62)
is determined by using second boundary condition of equation (58): ∫
[(( )
(
) )]
3.1.1. Power law strain energy function
[
( )
( ) ]
(63)
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Power law strain energy function for incompressible materials is used as follows[29]: (64) As usual, this energy function vanishes in the undeformed configuration. strain energy is a convex function; increasing monotonically with attains a minimum in the reference configuration, so that
where
and
must contain no real root . In the
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which places restrictions on the values that the constants can take; i.e.: other word:
. The
, both W and resulted stress approach infinity for very large
deformations:
and
and there is no stress in the
undeformed configuration and strain energy is minimum in this state:
is material parameter which varies by power law
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. In equation (64),
,
( ) . Material constants " " and
function in radial direction, where
are
determined by using Levenberg–Marquardt nonlinear regression method for the rubber tested by experiment and
implementing them in equation (62) [ ( )
,
and
by
is calculated:
( )
(( )
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( )
∫
( )
energy function,
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mentioned strain
is material inhomogeneity parameter. By considering above
( ) )]
(65)
By integration of above equation, radial stress is achieved: )
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Where:
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(
)
Where
(66)
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(
(
) (
)
*(
( )) +
(67)
) is the Gauss-hypergeometric function [28]. Second boundary condition
of equation (58) is used to find outer radius of cylinder in the deformed configuration: (68) and
are calculated by using equations (64), (65) and (69), (70):
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4.
(( )
( )
)
(( )
(( )
( )
)
(
( ) )
(69)
( ) )
(70)
Result and Discussion
In this paper functionally graded rubber is considered and inhomogeneity occurs in the radial
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direction of the shell. It is presumed that, the inner surface of the shell is made of the rubber tested by Treloar [30]. Material properties of the FG rubber vary by a power law function in ( ) . For each special FG rubber,
the radial direction, i.e.:
for the outer
surface will be available (for example outer surface of the shell can be silicon rubber), therefore for each specific FG rubber, material inhomogeneity parameter ( ) will be found
Treloar [30],
MPa and
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( ) . For power law strain energy function and for the rubber tested by
by:
are achieved. For generality, material
inhomogeneity parameter is considered . It is obvious that Each specific FG rubber will be its specific
.
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Example: Two cylindrical shells with inner and outer radius are considered. The applied internal pressure is .
is calculated for different values of
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and
m,
m and
m,
. In this example, we considered by using equation (68).
Distribution of circumferential extension ratio, radial stress, circumferential stress and axial
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stress are presented in the below figures. Based on the presented results in Figs. 4-7, maximum value of extension ratio occurs in the inner radius then it decreases through the
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thickness and will be minimum in the outer radius. Moreover, extension ratio increases by increasing internal pressure and decreasing thickness of the vessel and material
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inhomogeneity parameter.
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and
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Fig.4. Distribution of extension ratio for different material inhomogeneity parameter ( ),
Fig.5. Distribution of extension ratio for different material inhomogeneity parameter ( ),
and
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and
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Fig.6. Distribution of extension ratio for different material inhomogeneity parameter ( ),
Fig.7. Distribution of extension ratio for different material inhomogeneity parameter ( ),
and
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According to Figs.8-11, magnitude of normalized radial stress increases by increasing internal pressure, structure parameter
and material inhomogeneity parameter ( ). Normalized
radial stress shows trivial differences for different material inhomogeneity parameter at points near the boundaries, while at points away from boundaries, the reverse holds true. Figs.12-15, show that normalized hoop stress in the inner surface of the cylinder increases by increasing internal pressure and decreasing structure parameter
and material inhomogeneity
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parameter ( ). In the other surface of the cylinder normalized hoop stress increases by increasing material inhomogeneity parameter ( ). In contrast with normalized radial stress which has the same distribution for different
, the most remarkable part of the outcomes is
distribution of normalized hoop stress which altering from monotonically increasing in as a function of gradient parameter
. Figs.16-19 demonstrate
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monotonically decreasing in
to
that longitudinal stress has the same distribution as hoop stress. Fig.8. Distribution of normalized radial stress for different material inhomogeneity parameter ( ),
and
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and
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Fig.9. Distribution of normalized radial stress for different material inhomogeneity parameter ( ),
Fig.10. Distribution of normalized radial stress for different material inhomogeneity parameter ( ),
and
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and
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Fig.11. Distribution of normalized radial stress for different material inhomogeneity parameter ( ),
Fig.12. Distribution of normalized hoop stress for different material inhomogeneity parameter ( ),
and
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Fig.13. Distribution of normalized hoop stress for different material inhomogeneity parameter ( ),
and
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Fig.14. Distribution of normalized hoop stress for different material inhomogeneity parameter ( ),
Fig.15. Distribution of normalized hoop stress for different material inhomogeneity parameter ( ),
and
and
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Fig.16. Distribution of normalized longitudinal stress for different material inhomogeneity parameter ( ),
and
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and
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Fig.17. Distribution of normalized longitudinal stress for different material inhomogeneity parameter ( ),
Fig.18. Distribution of normalized longitudinal stress for different material inhomogeneity parameter ( ),
and
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Fig.19. Distribution of normalized longitudinal stress for different material inhomogeneity parameter ( ),
and
For validation, finite element method is used to find the accuracy of the results. Inhomogeneity in radial direction is considered in Abaqus by using field variable. For this purpose, a vessel with inner and outer radius is considered.
and
m,
m, and internal pressure
are considered for modeling material inhomogeneity
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parameter. The results are presented in Figs.20-22. According to Figs.21-22, it is found that there is very good convergence between theoretical results and FEM results. Maximum difference between FEM results and theoretical results is happened in the internal diameter of the vessel. These differences are about 8.53% for
and 6.61% for
. Results show
that differences between theoretical results and FEM results decrease by increasing material
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inhomogeneity parameter. By considering these results it is concluded that there is good agreement between FEM and presented theoretical results, therefore theoretical solution can be applied for finding deformation fields and stress components of the axisymmetric thick
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vessel composed of FG hyperelastic material.
Fig.20. FEM results of FG hyperelastic cylinder (
7)
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2,
and
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Fig.21. Comparison between theoretical results and FEM results for
Fig.22. Comparison between theoretical results and FEM results for
2,
and
7
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5.
Conclusion
In the presented paper, general solution for axisymmetric revolution shell composed of functionally graded hyperelastic material is presented. Curvilinear coordinates and Cartesian coordinates are used to find set of field equations and stress components for above mentioned shell in axisymmetric condition. As a special case, the formulation is applied for thick hollow
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cylinder. Power law strain energy function is considered to determine stress components and displacement of the cylinder. The material properties vary by a power law function through the cylinder thickness. Stress components and stretches of the thick-walled cylinder in the plane strain state has been found by analytical method and related distribution has been presented.
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Based on presented results, It is found that, increasing material inhomogeneity parameter
)
causes decreasing in radial stress, extension ratio and deformed cylinder radius. Moreover by increasing internal pressure and structure parameter
, magnitude of normalized radial
stress increases. Additionally, by increasing internal pressure and decreasing structure parameter
normalized hoop stress increases. The distribution of normalized hoop stress
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and normalized longitudinal stress changing from monotonically increasing in monotonically decreasing in
as a function of gradient parameter
to
. This distribution is
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actual valuable for design of this kind of pressure vessel to postpone or evade failure. Finite element method is applied in order to validate theoretical results. By comparison between theoretical results and FEM results it is found that there is a good agreement between them.
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Above achieved results in investigating mechanical behavior of thick hollow cylindrical shell composed of functionally graded hyperelastic material with power law varying properties,
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demonstrate great effect of
and
in displacement field and stress components distribution.
Thus from a design approach, opting a proper
and
is a very useful method to control the
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stress and to optimal use of material.
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Graphical Abstract: Journal: International Journal of Mechanical Sciences (IJMS)
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Title: Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials Authors: Y.Anani, G.H.Rahimi Manuscript Number: SUBMIT2IJMS-D-17-00001R1