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Rigid-flexible contact analysis of an inflated membrane balloon with various contact conditions
Accepted Manuscript
Rigid-flexible contact analysis of an inflated membrane balloon with various contact conditions M.X. Liu , C.G. Wang , X.D. Li PII: DOI: Reference:
S0020-7683(18)30189-6 10.1016/j.ijsolstr.2018.05.004 SAS 9984
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
19 September 2017 16 April 2018 7 May 2018
Please cite this article as: M.X. Liu , C.G. Wang , X.D. Li , Rigid-flexible contact analysis of an inflated membrane balloon with various contact conditions, International Journal of Solids and Structures (2018), doi: 10.1016/j.ijsolstr.2018.05.004
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Highlights
Decoupled governing equations are obtained based on the force balance relationship. 2. Various contact conditions, especially the harmonic one, are discussed.
3. The numerical results from no-slip condition are verified by the designed
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test (DIC technology).
4. The maximum ordinate difference appears in the intersection of meridian
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stretch ratio.
5. The no-slip condition results in the maximum volume and minimum
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PT
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internal pressure.
1
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Rigid-flexible contact analysis of an inflated membrane balloon with various contact conditions M. X. Liu ab1 C. G. Wang *a2, and X. D. Li b3 a
Department of Engineering Mechanics, Tsinghua University, Beijing, China, 100084,
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b
Center for Composite Materials, Harbin Institute of Technology, Harbin, China, 150001
Abstract
Considering the Mooney–Rivlin hyperelastic model, a semi-analytical approach is
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introduced to analyze the rigid–flexible contact behaviors of an inflated membrane balloon between two plates with various interface conditions. This approach is based on the differential formulation and the coupling property of equilibrium equations are
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well-solved. In order to verify the reliability of the proposed theoretical model, an experimental test is designed, by which some important contact characteristics and
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patterns (no-slip condition) are obtained. Two special phenomena are observed for the
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meridian stretch ratio with different friction coefficients. One is that the intersection points of all curves fall in a small interval and the intersection of any two curves
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represents the same changing rate of the horizontal ordinate, resulting in the maximum
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difference. The other is the dividing point, where the stretch ratio decreases on the left of it and increases on the right due to the introduction of friction. Under the same contact angle, the larger displacement load should be applied to the balloon for the
1
Ph.D., Center for Composite Materials, Harbin Institute of Technology/[email protected].
2
Prof., Center for Composite Materials, Harbin Institute of Technology/[email protected]/*Corresponding
author. 3
2
Prof., Department of Engineering Mechanics, Tsinghua University/[email protected].
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small friction coefficient condition, resulting in the smaller contact area and internal pressure. In addition, the vulnerable position, direction and contact condition of the balloon are found during the contact process, which happen in the center along the circumferential direction under no-slip condition. These results provide solid guidance
membrane balloon.
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and support for our understanding of the rigid-flexible contact behaviors of an inflated
Keywords: Mooney–Rivlin hyperelastic membrane, rigid-flexible contact, force
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equivalent method, decoupling, stick-slip condition
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1. Introduction
As the typical membrane structure, inflated balloon has considerable importance
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in a number of scientific studies and technological applications. On the macro scale, it
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can be used in terrestrial and space structures due to the advantages of light-weight, quick and self-deployment, and compact storage properties (Jenkins, 2001). On the
CE
micro scale, it can be used as animal or plant cell, examples including endothelial cells
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lining on the vessel wall (Moretti et al., 2004), liposomes adhesion to glycolipid membranes (Bakowsky et al., 2008) and a soft bubble rolling on the stretched substrate (Chen and Gao, 2014), which supports the normal life activities. However, when contacting with the rigid substrate, variety of problems are involved for these applications. Especially, geometry and boundary condition nonlinearities are the major challenging issues in the contact process (Menga et al., 2016). In addition, the friction 3
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influence, stress distribution and deformation in the non-contact and contact regions are also important (Alberts et al., 1994; Wan and Liu, 2001; Tsang et al., 2006; Kumar and DasGupta, 2013; Patil et al., 2015; Chen and Chen, 2015). In this context, understanding the contact mechanics of an inflated membrane balloon and rigid plates
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is, thus, of considerable interest, which motivates the present study. The investigations on the contact behaviors of an inflated membrane can be summarized as two processes: geometry nonlinearity analysis and boundary condition
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nonlinearity analysis. The geometry nonlinearity is carefully considered by solving the membrane inflation problem. Based on the study of membrane large elastic deformation (Rivlin, 1948), various methods are proposed to deal with this problem,
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examples including the large elastic deformation theory (Adkinks and Rivlin, 1952), force balance relationship (Feng and Yang, 1970), nonlinearity (Tielking and Feng,
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1974), energy balance relationship (Patil and DasGupta, 2013) and critical energy
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release rate (Long et al., 2010). Moreover, a lot of works have been done to deal with the nonlinearity problem of the boundary condition. On the basis of membrane
CE
inflation, Feng and Yang (1973) analyzed the contact problem of inflated
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membrane-rigid plate system, while the interface friction is not taken into consideration. The Hertzian contact of circular membrane-rigid plates is addressed by Xu and Liechti (2010). In order to introduce the interface influence, Johnson, Kendall and Roberts (1971) developed the JKR technique to discuss the contact adhesion, and Johnson (1997) considered the relationship between adhesion and friction due to fracture mechanics. Adhesion of a fluid-filled membrane with a substrate is discussed 4
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by Sohail, et al (2013). The adhesive contact of a circular membrane-rigid substrate is studied by Xu and Liechti (2011) with analytical and experimental methods. After the interface friction is introduced, different contact conditions are discussed gradually. The no-slip contact condition is studied by using finite element method and finite strain
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theory (Charrier and Shrivastava, 1989; Long et al., 2010). The limited interface friction is considered by Kumar and DasGupta (2013) using energy balance relationship, which is called stick-slip condition.
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Based on the studies of geometrical and boundary condition nonlinearities above, it can be found the membrane inflation and contact problem are very complicated. Thus, it is difficult to obtain the analytic solution. Broadly, solution schemes proposed
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to this problem are divided into two categories: finite element analysis and semi-analytical approach. Based on the finite element method, the membrane large
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deformation problems, nonlinear static behaviors, inflation and contact characteristics
PT
are analyzed by Oden (1972), Leonard and Verma (1976) and Charrier and Shrivastava (1987). In addition, this method is applied to biomechanics, the behaviors of a cell
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adhering to a substrate were addressed by McGarry and Prendergast (2004). Moreover,
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a large number of works are devoted to the semi-analytical approach. On the basis of different contact models, which are inflated membrane-rigid plates (Yang and Feng, 1973), lipid membrane-solid elliptical cylinder substrates (Belay et al., 2016), elastic layer-rigid cylinder (Vasu and Bhandakkar, 2016), clamped edge membrane-rigid plates (Long, 2010) and elastic membrane-elastic cone (Patil and DasGupta, 2015) systems, the contact problems are simplified to a set of ordinary differential equations, 5
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which can be solved by numerical methods. In the existing literature, the membrane inflation problem including the geometry nonlinearity has been well resolved. For the complex boundary conditions, diverse methods based on the energy balance are proposed (Patil and DasGupta, 2013). In this
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reference, the method is based on the variational formulation, where the whole system is taken as the study object and the functional variational equations are built on account of the energy balance. However, the normal adhesive force and tangential friction force
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couple together because the energy is the centralized reflection of deformation. This will increase the difficulty of the solving process. Therefore, in this paper, a semi-analytical method is introduced to extend the modal of Feng and Yang (1973),
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which is originally devoted to describe the membrane free inflation and simple frictionless contact. This method is rooted in the differential formulation in elastic
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mechanics. A microelement is considered and the governing equations are then
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established on the basis of the equilibrium, deformation and constitutive. The equilibrium equations are derived from the non-moment theory of revolutionary thin
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shells, which consider the force balance in the meridian tangential and normal
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directions. The geometric equations are established based on the small deformation hypothesis and the Mooney–Rivlin hyperelastic model is introduced to construct the constitutive equations. Thus, the coupling problem is solved fundamentally, and a set of uncoupled ordinary differential equations can be easily obtained. These ordinary differential equations can be integrated numerically using the Runge-Kutta method with the given boundary values. In addition, more complex contact boundary 6
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conditions are studied, which is the extension of the wok of Feng and Yang (1973). Especially, for the stick-slip contact condition, the friction and slide are considered comprehensively. Moreover, the ideal gas thermodynamics state equation is assumed to be valid during the contact process in the studies of our research (Kumar and Patil, 2014;
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Lei and Chen et al, 2015; Kumar and Patil, 2015; Patil et al., 2015). Because of these concerns, this paper is organized as follows. The geometry and constitutive models, which are based on the Mooney–Rivlin hyperelastic membrane
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theory, are described in Section.2. The work of Feng and Yang (1973), where the frictionless contact condition of the membrane-rigid plate system, is firstly introduced in Section.3. Then, we make extensions of their work, including no-slip and stick-slip
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conditions when the interfacial friction force is taken into consideration. In Section.4, the inflatable and contact characteristics (no-slip condition) are predicted by the
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proposed model and verified by the experimental test. Finally, the influence of friction
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is discussed and results from three different contact conditions are compared. 2. Geometry and constitutive models
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A spherical balloon with uninflated radius r0 and uniform thickness h (state I
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the black line) is inflated to radius rs by pressure P0 (state II the red line). Then two parallel rigid plates are pressed by F into contact with the balloon (state III the green line). Half of the spherical balloon and one rigid plate are shown in Fig. 1. The inflated spherical balloon before contact is described by the spherical coordinates ( rs , , ). The cylindrical coordinates ( , , ) are used for the spherical balloon after contact. The pressure P0 before contact and the pressure P after contact are considered to be 7
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uniformly distributed in the balloon. Moreover, all thermodynamic processes are
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assumed to be reversible and isothermal during inflation and contact.
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Fig.1 the contact model of an inflated membrane and rigid plates (the initial state is I, state II indicates that the membrane comes into contact with rigid plates, state III shows that the membrane is compressed to the specified contact angle ), two main
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curvilinear directions (the circumferential ( ) and meridian ( ) directions) are shown
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on the right
The governing differential equations are built for the non-contact region and the
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contact one, separately.
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In the non-contact region: according to the geometric relation, the principal stretch ratios for the membrane can be written as , . Here, the subscripts and
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denote the meridian and circumferential directions.
'2 '2 dS , ds r0
. r0 sin
(1)
The prime in the foregoing and subsequent equations denotes the derivatives with respect to the angle . ds is the infinitesimal arc length of the uninflated spherical balloon. dS is the corresponding infinitesimal arc length of the compressed spherical 8
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balloon. The principal curvatures , can be obtained in terms of , and . ' '' 1 d ' '' , A dS ( '2 '2 ) 3 2
sin ' ， 1 B ( '2 '2 ) 2
(2)
where A and B are the Lame coefficients along and directions, respectively.
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Substituting Eq. (1) to Eq. (2), the principal curvatures , in terms of
, and are expressed as:
' ( sin )' ( sin )" [ ], 2 r0 2 ( sin )'2
1 r0 sin
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1
(3)
2 ( sin )'2 .
The structure is considered as the non-moment thin shell, which has no bending
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moment and torsion on the cross section, and the equilibrium differential equations can
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be expressed due to the constitutive relation.
(4)
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B A ( BT ) T ( AT ) ( AT ) ABq 0, A B ( AT ) T T ( BT ) ABq 0, T T qn，
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where T , T and T are the internal forces and q , q and qn are the external forces applied to the balloon along meridian, circumferential and normal directions, respectively. Because the inflated balloon can be considered as a revolution body, the equilibrium equations then will be simplified in meridian tangential and normal directions due to the axial symmetry. ( T = q =0, Q’M=B/sin , Q’O’=A (as shown in 9
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Fig.1))
dT T T P t , d cos T T Pn，
(5)
where Pn = qn and Pt = q are the external loads acting on the deformed surface in
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the normal and meridian tangential directions. In the present work, we consider the two-parameter Mooney-Rivlin hyperplastic constitutive model for the spherical balloon, which is incompressible, homogeneous
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and isotropic. This material model is based on the general theory of large elastic deformation and has been solved exactly by Mooney (1940). The strain energy density function for such a material is given by W C1 ( I1 3) C2 ( I 2 3) , where C1 +C2 =
G 2
1
2
2
and
I 2 2 2
1
2
1
2
are the equations of the
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I1 2 2
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and G is the shear modulus. The principal value of the deformable tensor
principal stretches , . Then the stress resultants of per unit length T and T in
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terms of the principal stretches , can be expressed as:
1 3 3 )(1 2 ),
(6)
1 T 2C1h( 3 3 )(1 2 )，
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T 2C1h(
where h is the thickness of the spherical balloon before inflation and
C2 . C1
In the non-contact region, it is assumed that Pn P and Pt 0 , where P is the internal pressure of the deformed spherical balloon. Then the equilibrium equations (Eq. (5)) can be rewritten in terms of , and .
10
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(7)
T T P. Substituting Eq. (1), Eq. (3) and Eq. (6) into Eq. (7), the equilibrium equations can also be expressed as 1
2 ( [
1
4
3
)(1 2 ) ' [(
3 3 4 )(1 2 ) 2
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' (
( sin ) ' 1 1 1 3 3 )] [ ( 3 )], sin 3
' ( sin )' ( sin )" 1 ][( 3 3 )(1 2 )] 2
(8)
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Pr0 2 ( sin ) '2 2 ( sin ) '2 1 [( 3 3 )(1 2 )] . sin 2hC1
Eq. (8) is a two-dimensional system of ordinary differential equations. In order to
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convert it to the integral form, we define four new variables.
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sin , ' , h
h P , p . r0 C1
(9)
Then Eq. (8) can be simplified to the iteration forms by substituting variables,
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which are three first-order ordinary differential equations in the non-contact region
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(Feng and Yang (1973)).
11
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( ' sin cos )(
'
3sin 2 ) 2 3 4
2 sin 3 ' cos sin ( 3 3 )( sin ) 1 sin sin 3 2 ' 3 3 1 2 sin sin ( ) sin
4 3 ( 4 2 sin 2 )sin
(10)
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' ,
,
1 sin 3 2 2 ( ) ' sin 3 3 ' . 2 4 2 2 p 2 ( sin )(sin ) (1 ) sin 2h 4 3
2
2
The boundary conditions for the non-contact region are:
nc
,
c
nc
*sin , (w
) nc
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c
, c
'
0,
(11)
2
where “nc” and “c” refer to the non-contact and contact regions, respectively. is
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the dividing point of the two regions.
The pressure and volume inside the spherical balloon before contact are related to
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s . They can be expressed as:
rs and V0 is the volume of the spherical balloon after inflation. The r0
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where s
(12)
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4 1 h V0 r03s 3 , P0 4C1 (1 6 )(1 s 2 ) , 3 s r0s
volume V of the balloon after contact can be obtained in terms of , and ,
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which is V 2 r03 2 2 2 '2 d . Because there is a small contact angle
increment from one contact state to the next one and the difference of the two adjacent contact states is very small, the contact process is assumed to be a quasi-steady state and the temperature change of the internal gas can be ignored. According to the ideal gas thermodynamics state equation, the pressure P inside the balloon after contact can be obtained: 12
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P
P0 V0 V
2 P0 s 3
3 2
2
. lim
(13)
x
d 2
' 2
3. Contact conditions In the contact region, three types of contact conditions are considered, which are
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named as frictionless contact, no-slip contact and stick-slip contact. 3.1 Frictionless contact condition
For the frictionless contact condition, the membrane in the contact region can
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slide freely over the rigid plates because of the tension induced by internal pressure. This has been considered and well-solved by Feng and Yang (1973). The stretch ratios along meridian and circumferential directions are equal at the pole (Eq. (15)) and
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successive between contact and non-contact regions (Eq. (11)). Because of the rigid-flexible contact, the ordinate of the membrane in the contact region keeps
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constant and geometry of the contact region requires that ' 0 . Then according to Eq. (1), the principal stretch ratios in the contact region can be reduced to
r0
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'
,
r0 sin
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. At the center of the balloon, the horizontal ordinate
reaches to the maximum, which results in the zero derivative of .Hence, the
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governing equations, which are two first order ordinary differential equations, can be expressed as:
13
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'
'
3 2 ( 2 3 4 ) 1 cos sin 1 1 , 2 3 3 1 3 2 ( )(1 ) 4 3 1 1 3 3 sin cos . sin
(14)
0
0
0 ,
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The boundary condition for the contact region is: (15)
where 0 is the stretch ratio at the pole ( 0 ). Eq. (15) is the initial condition for
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the governing equations (Eq. (14)) of the contact region. The initial condition for the non-contact region (Eq. (10)) can be obtained from the results of Eq. (14) and the boundary conditions given in Eq. (11).
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The solving process for this contact condition is shown in Fig.2, and the computation procedure is outlined as follow:
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1) Assigning the stretch ratio s after inflation and the contact angle , the pressure P0 and volume V0 before contact can be obtained from Eq. (12).
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2) Assume the stretch ratio 0 in the pole. Then Eq. (15) provides the initial
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integral values for Eq. (14), which are the governing equations in the contact region.
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Integrating the governing equations (Eq. (14)) with the initial condition (Eq. (15)) from
0 to , the stretch ratios ( , ) in the contact region can be obtained. 3) The results from Step 2 and Eq. (11) provide the initial conditions for Eq. (10),
which are the governing equations in the non-contact region. Guess a value for the pressure P after contact. Applying the Runge-Kutta method to Eq. (10) with the initial condition from to 14
2
, the stretch ratios ( , ) in non-contact
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region can be computed. 4) Adjust the guessed pressure P constantly by a small constant until the ' error1 caused by the boundary condition 0 is satisfied. Reassume the value 2
0 by a small constant constantly until the error2 from the thermodynamic
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PT
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equilibrium condition (Eq. (13)) is satisfied.
Fig.2 the flow chart of the solution procedure under frictionless contact condition
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3.2 No-slip contact condition In the case of the no-slip contact, the friction force between the plates and the
spherical balloon in the contact region is sufficiently large. Then materials in this region cannot slide over the plates. In this case, the contact depends on the history of the contact formation. The stretch ratios along the meridian and circumferential directions at the pole are equal to the initial stretch ratio due to the infinite friction 15
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force (
0
0
s ). The ordinate of the membrane in the contact region keeps
constant ( ' 0 ) and the horizontal ordinate reaches to the maximum at the center of the balloon ( ' 0 ), which are same as the conditions of frictionless and stick-slip. At the diving point of the contact and non-contact region, the circumferential stretch
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ratio and its derivative are successive. Because of the geometric relation and successive condition, the meridian stretch ratio after contact can be related to the parameter w before contact =
' r0
=(
)' = ' = w
. Then the boundary
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conditions between contact and non-contact region can be expressed as
w , , w w .
(16)
Here, the signs “-” and “+” indicate the values before and after contact. Therefore, the
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solution of this case can be determined incrementally as follow.
obtained in the pole: 0
PT
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The initial contact angle is set as 0 , so the initial integral condition can be
0
s ,
0
0w , 0 s
.
(17)
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Considering a small increment of the contact angle , the problem can be solved by integrating the governing equations (Eq. (10)) in the non-contact region. The
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results of this step can be stored in the contact region, which provide the initial integral conditions for the governing equations (Eq. (10)) in the non-contact region in the next
.
The solving procedure for this contact condition is shown in Fig.3 and the computation procedure is outlined as follow: 1) Assigning the stretch ratio s after inflation and contact angle , the 16
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pressure P0 before contact can be obtained from Eq. (12). 2) Initially, the contact angle and pressure are set as 0, P P0 . Then the configuration of the membrane can be determined by integrating the governing equations (Eq. (10)) in the non-contact region with the initial integral condition
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provided by Eq. (17). 3) Considering a small contact angle increment , the parameters for the diving point ( + ) between contact and non-contact regions can be expressed as
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, , w before contact. Integrate the governing equations (Eq. (10)) in the non-contact region with the guessed initial conditions (Eq. (16)). 4) Adjust the guessed pressure P constantly by a small constant until the ' error1 caused by the boundary condition 0 is satisfied. Reassume the guessed 2
initial values a small constant with ( w
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w
*(1 * c2 ),
w
*
Γ
, , w in the non-contact region from Step 4 are stored in the
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5) The results
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until the error2 from the thermodynamic equilibrium condition Eq. (13) is satisfied.
contact region. Then, repeat the steps from 1 to 5 until the contact angle reaches to the
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default value .
17
)
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Fig.3 the flow chart of the solution procedure under no-slip contact condition
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3.3 Stick-slip contact condition
In the contact condition of stick-slip, the sliding state of materials in the contact
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region is motivated by the membrane tension and resisted by the interfacial friction.
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Because the friction coefficient is limited, material will stick when the interfacial friction is greater than the membrane tension, while the others will slip.
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Considering the friction force, the equilibrium equation along the meridian
tangential direction (the first equation in Eq. (6)) of the spherical balloon in the contact region can be rewritten as
dT T T . Here is the friction force along d
the tangential direction, which can be also obtained by P . Hence, the equilibrium condition for the critical sliding state can be obtained: 18
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1 T T 1 ( ) (T T ) + f P =0, '
(18)
where f is the critical sliding friction coefficient. Therefore, the governing equations in this contact region are given by:
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3 2 2 3 4 1 cos sin 1 1 2 3 3 ' , 1 3 2 4 3 1 1 1 r0 1 f P sin 3 3 2C h 1 2 1 2 cos ' . sin
(19)
The governing equations in the non-contact region are the same as Eq. (10). The solving procedure for this contact condition is shown in Fig.4, and the computation
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PT
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procedure is outlined as follow:
19
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Fig.4 the flow chart of the solution procedure under stick-slip contact condition 1) The stretch ratio after inflation, contact angle and the critical friction
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by Eq. (12).
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coefficient are set to s , and f .The pressure before contact then can be obtained
2) Because of the interfacial friction, the problem is solved using the no-slip
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condition procedure until the contact angle becomes f , where the friction coefficient reaches to the critical value f . 3) Then sliding between plates and the inflated membrane occurs as the contact continues. Considering a small increment , integrate the governing equations in the contact region (Eq. (19)) from f to f i (i=1,2,3…) with initial conditions from the results of Step 2 ( ( 20
f
) s (
f
)ns , (
f
) s (
f
)ns ).
Here, “s”
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and “ns” denote the sliding and no-slip parts, respectively. Integrate the governing equations in the non-contact region (Eq. (10)) from f i to . The initial 2
conditions
1 f i
for
f i
nc
this
, c
integration
f i
*sin
i f
nc
f
c
are
i , w
f
i
given
f i
nc
.Here, c
by: “nc”
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and “c” indicate the non-contact and contact regions, respectively. 4) Adjust the pressure P by a small constant after contact constantly until ' the error1caused by the boundary condition 0 is satisfied. Reassume the initial 2
f
f
*(1 c2 ) with a small constant until the error2
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guessed value
from the thermodynamic equilibrium condition Eq. (13) is satisfied. 5) Because the adjusted value
' is related to f due to Eq. (1), the
, which means the meridian stretch ratio in the '
'
f
f
ns
s
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value must fulfill
f
no-slip part is larger than that in the sliding part. If this relationship is not satisfied, the
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material in the sliding part will slide to the no-slip part. This means the no-slip part will
PT
decreases ( f f ). If f 0 , the material in the contact region will all slip to the pole.
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6) If the material in the contact region becomes the sliding part, integral the
AC
governing equations in the contact region (Eq. (19)) with the initial condition
0
0
0 from 0 to . Integral the governing equations in the non-contact
region (Eq. (10)) from to of
contact
nc
, where the initial conditions are given by the results 2
region.
, c
nc
It
can
c
*sin , w
be
nc
expressed
as
. c
7) Adjust the pressure P after contact by a small constant constantly until 21
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guessed value 0 0 *(1 c2 ) by a small constant in Step 6 until the error2 from the thermodynamic equilibrium condition Eq. (13) is satisfied. 8) After the adjustment for 0 by a small constant in Step 7, this value needs
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to meet the condition 0 s , which means the material in the contact region is sliding. If this condition is not satisfied, material in the contact region will not slip to the pole. This means the no-slip part reappears in the contact region. Then return to the
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Step 1.
9) Repeat the steps from 1 to 8 until the contact angle reaches the default value . 4. Results and discussion
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In this section, the contact behaviors of the inflated membrane with different contact conditions are discussed. The contact characteristics will be obtained by
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solving the governing equations with the numerical methods discussed in details in
PT
Section 3. Results from the no-slip contact condition will be verified by the designed experiment (Section 4.1). The influence of friction coefficient on stretch ratios and the
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contour profiles of the balloon is discussed, and results from three contact conditions
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are compared in Section 4.2. 4.1 Experiment verification In order to verify the theoretical results, a testing scheme is proposed to measure
the patterns and some important contact characteristics of the inflated balloon. Because the friction coefficient is difficult to measure and the frictionless condition is hard to realize, the no-slip contact condition is carried out in this experiment. 22
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Tab. 1 Material and geometrical parameters Parameters
Magnitude
Initial radius (r0)
0.05 (m)
Young’s modulus of membrane(E) Poisson’s ratio ( )
0.3(mm)
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Thickness of the beam (h)
6 (MPa) 0.47 3
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Stretch ratio before contact ( s )
The material and structural parameters of the inflated balloon are shown in Table 1, which is a Polyvinylchloride thin membrane (PVC). The specimen in the experiment
CE
PT
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is shown in Fig. 5.
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Fig. 5 the specimen of the balloon in contact and the random speckles are drawn
on the surface (r0 is the initial radius, rs is the radius after inflated and l is the displacement load applied to plates)
Firstly, the principle of Digital Image Correlation Technology (DIC)measurement is briefly introduced. DIC is an optical measurement method and displacements and displacement gradients can be determined based on the video image (Bruck et al., 1989; 23
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Hild and Roux, 2006; Wang et al., 2012). The position of a characteristic point in the
subarea of the pre-deformation image is (x, y, z) and it is (x*, y*, z*) in the deformed image. Then the typical correlation function which measures how well the two images match is: F ( x, y, z ) * G( x* , y* , z * ) u u u v v v w w w , , , , , , , , ) 1 ， 1 x y z x y z x y z ( F ( x, y, z,)2 ) * (G( x* , y* , z * ) 2 ) 2
(20)
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S ( x, y, z, u, v, w,
where F ( x, y, z) and G( x* , y* , z* ) are the gray level values of images before and after deformation. Then the deformation information can be obtained by minimizing the
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correlation coefficient S.
Before the experiment, the speckle pattern on the balloon is reproduced artificially. To increase the contrast of the speckle, matte white paint and black paint are sprayed on
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the balloon surface evenly. Then the two rigid plates, which are two 5mm thick acrylic
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plates, are fixed on the loading shafts. In the experiment, the balloon with sprayed speckles is placed on the workbench and its position is adjusted so that it is on the
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compression axis. After that, the balloon is inflated by the pump and the diameter
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increases from 10cm to 30cm. The displacement load is applied to the balloon, which can be controlled precisely by the electronic universal testing machine. This load is
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noted down by a ruler on the machine and the internal pressure of the balloon is monitored by the barometer. The deformations of the balloon are tested using the digital image correlation (DIC) technology, which is a reliable mean to measure the displacement fields. There are mainly four steps in the DIC measurement process: (i) random speckles are made on the specimen surface; (ii) the calibration of CCD camera; (iii) capturing deformed images; (iv) images’ post-processing. The accuracy of the 24
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measurement precision is 1μm for our system.
Fig. 6 setup of the contact experiment
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The deformation contours of the balloon are obviously observed and tested in Fig. 7. It contains three stages, which are a) before contact, b) just in contact and c) in press.
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And the maximum displacement of the balloon (
) is compared in Tab. 2 2
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Tab.2. the experimental and theoretical results of the maximum displacement under
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maximum
displacement (
l ) r0
different displacement load Displacement load (cm)
0.63
2.13
4.25
theoretical results
2.05
2.18
2.34
experimental results
2.08
2.25
2.50
The balloon is firstly inflated by the pump before contact and the data is recorded when the internal pressure is 18.8KPa. Then, it is inflated to contact with the rigid plates, where the distance between the two plates is set to 30cm and the internal pressure monitored is 19.1KPa. After that, the balloon is compressed by the rigid plates 25
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and the internal pressure is 22.0 KPa when the displacement load is 2.10cm. When the inflated balloon is just in contact to plates, the maximum displacement appears in the middle (
). The tested deformation (10.4cm) agrees with the theoretical result 2
(10.0cm) with an error around 0.33%. When the displacement load is 4.25cm, the error
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reaches to the maximum, which is 6.8%.
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Fig. 7 three stages of contact (a) before contact b) just in contact c) in press)
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Tab. 3. the experimental and theoretical results of pressures under different displacement load
Displacement load (cm)
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Pr0 theoretical results Pressure ( ) C1h
experimental results
0.63
2.13
4.25
3.85
4.25
5.12
3.87
4.30
5.22
In addition, the pressures under different displacement load are tested and
compared with the theoretical predictions in Tab.3. Accompanied by the increasing displacement load, the dimensionless pressure
Pr0 increases. This means that the C1h
volume decreases with the rise of the displacement load, resulting in the increase of 26
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internal pressure. The tested pressure is close to the theoretical results within the maximum error of 3.41%. It also can be found that the error increases when the applied displacement keeps increasing. The difference between the experimental and theoretical results may because that the temperature increases in the balloon and the
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thermodynamic equilibrium is not satisfied. Moreover, some inevitable limitations in test techniques may also cause the difference between the results from experiments and theoretical model. When measuring the internal pressure of the balloon, the error occurs
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because the minimum scale of the barometer is 0.1KPa. Also, noise and truncation error will be introduced in the DIC calculation. In addition, the error may be caused by human factors when reading and recording data. The maximum error is 6.8% when comparing
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the theoretical and experimental results of the maximum displacement and it is 3.4% for measuring the internal pressure. In general, the differences will not exceed beyond 7%.
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In fact, the experiment results offer the relative trustworthy evidence to verify the
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proposed theoretical model and computation procedures. 4.2 Theoretical prediction
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4.2.1 The stick-slip contact results
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For the stick-slip contact condition, it becomes the frictionless contact condition
when the friction coefficient becomes zero. If the friction coefficient goes to infinity, it is converted into the no-slip contact condition. As a general case, the results from the stick-slip contact condition are mainly discussed in this section. For different friction coefficients, the changing trend of the meridian stretch ratio
with the angle is shown in Fig.8 a), b) and c), where the contact angles are 27
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30 , 45 and 60 .
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Fig.8 the changing trend of meridian stretch ratio with angle in different
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friction coefficient conditions: a), b) and c) represent that the contact angles are 30 , 45 and 60 , respectively.
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Taking the 60 contact angle as an example, two special phenomena can be seen in
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Fig.8 c). One is that the intersection points of all curves fall in a small interval, and the other is the dividing point ( angle 1.05 ). The intersection point of any two curves
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appears in the contact region, which represents that material have the same meridian stretch ratio under the conditions with corresponding friction coefficients at that point. Moreover, different from the condition of f 0 , a dividing point, which is the boundary of contact and non-contact regions, appears when friction is introduced. In the contact region, the meridian stretch ratio decreases when material is far away from the pole, while it increases in the non-contact region. In addition, with the increment of 28
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the friction coefficient, the meridian stretch ratio increases in the pole of the balloon, while it decreases in the middle ( 2 ). All these phenomena can be explained by the introduction of the friction, which prevents the material in the contact region from sliding to the pole and makes it hard for the material in the non-contact region slip to
and angle relationship with different friction r0
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Fig.9 a) horizontal ordinate
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the contact region.
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coefficients b) the horizontal ordinate difference
between introduced friction r0
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conditions (different coefficients) and frictionless one
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To better explain the intersection interval in Fig.8 c) (contact angle 60 ), the relationship between the horizontal ordinate and angle is counted in Fig.9 a) and the
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horizontal ordinate difference between the introduced friction conditions and the frictionless one is described in Fig.9 b). In the contact region, the relationship
' r0
is satisfied due to the expression
'2 '2 dS and ' =0 . Then the ds r0
intersection of any two curves represents the same changing rate of the horizontal ordinate 29
, resulting in the maximum difference at that point. Moreover, the maximum r0
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difference is calculated and shown in Table 4 when the contact angles are 30 , 60
45
and
. From this table, the maximum difference of horizontal ordinate all falls in a small
interval and the range is about 0.01rad in these three contact angle cases. The obtained results can prove that the intersection interval in Fig.8 is ubiquitous in the contact
Tab. 4. the maximum difference
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process. max of horizontal ordinate between the r0
introduced friction conditions and the frictionless condition with different contact angle
friction coefficient
0.01
contact angle 0.3740
45
0.5602
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60
0.05
0.09
0.1005
Range
0.3755
0.3756
0.3766
0.3861
0.0121
0.5513
0.5491
0.5494
0.5504
0.0111
0.7090
0.7052
0.7056
0.6929
0.0161
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30
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( 30 , 45 and 60 )
0.6967
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Moreover, a step point appears at angle 1.05 in Fig.8 c), which is
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corresponding to the dividing point between the contact and non-contact regions in Fig.10. This means the changing trend of the meridian stretch ratio is different in the
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contact and non-contact regions, though this stretch ratio changes continuously in these two regions. In addition, the first-order derivative of meridian stretch ratio is zero in frictionless and no-slip condition, while it is negative in the stick-slip condition where materials in the contact region all slide to the pole (Fig.10)). For the frictionless condition, the contact process is independent to the loading history. Then, the first
30
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derivative of meridian stretch can be expressed as '
'' r0
=
2 tan (1 tan ) due to r0
the geometrical relation = tan and the fact that is the constant. When friction is introduced, the balloon deformation is dependent on the history of contact. Especially for the no-slip condition, the meridian stretch ratio and its first variation in the pole are
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the same as those when the balloon is just inflated, where the first derivative is zero due to the constant stretch ratio. When friction coefficients are 0.01, 0.05 and 0.1005, materials in the contact region all slide to the pole. The first derivative of the meridian r0 1 f P 2C1h
in Eq. (19) (stick-slip condition)
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stretch is negative due to the one more item
compared to that in Eq. (14) (frictionless condition) and it decreases when friction
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coefficient increases.
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Fig.10 first-order derivative of meridian stretch ratio and angle relationship with
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different friction coefficients
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Fig.11 Dimensionless contact area and dimensionless pressure relationship with
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different friction coefficients under different contact angles
Fig.11 shows the internal pressure change with the contact area under different friction coefficients and contact angles. When displacement applied to plates keeps
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increasing, the contact area and the internal pressure increase. Under the same contact angle, the contact area and internal pressure are relatively small with the smaller
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friction coefficient. With the increase of the contact angle, the difference of contact
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area and internal pressure becomes more obvious with different friction coefficients. This phenomenon can be explained in Fig.15 that the contact area and volume decline
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because of the decreasing friction coefficient.
32
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Fig.12 Dimensionless contact area and dimensionless displacement load relationship with different friction coefficients under different contact angles It can be obviously seen in Fig.12 that the contact area will increase when the applied displacement load constantly increases under the same friction coefficient. In
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order to keep the same contact angle, the larger displacement load should be applied with the small friction coefficient, while the contact area is relatively small. In addition, with increasing the displacement load, the friction coefficient influence to contact area
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will be more obvious. This phenomenon can also be explained in Fig.15 below. For the small friction coefficient, the larger displacement load should be applied and small contact area can be obtained to keep the same contact angle.
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4.2.2 Result comparison of three contact conditions
In this section, results from the three contact conditions are compared. Here, the
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contact angle is set to 60 degrees. The friction coefficient is set to 0.1005, where the
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contact region is all slipping part.
Fig.13 the meridian and circumferential stretch ratio of three contact conditions (the contact angle is set to 60 degrees and the friction coefficient is set to 0.1005 for 33
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the stick-slip contact) Fig.13 compares the meridian and circumferential stretch ratio under three different conditions. For the frictionless contact condition, the stretch ratios vary continuously because material in the contact region can slide freely. For the no-slip and
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stick-slip contact conditions, the meridian stretch ratio has obvious dividing point between the contact and non-contact regions, which is explained in Section 4.2.1. For the meridian stretch ratio, it is smaller around the pole for the frictionless condition,
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while it is relatively larger close to the center position. This can be explained by the fact that the material will slide to the pole in the contact region and slide to the dividing point in the non-contact region. In addition, the circumferential stretch ratios are larger
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CE
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than the meridian ones on the whole.
Fig.14 the meridian and circumferential stress of three contact conditions
The comparison of stress under these three contact conditions are shown in Fig.14.
The changing trend of the stress is the same as that of the stretch ratio according to Eq. (6). This can also be interpreted that the stretch ratio is equal to strain to some extent. Due to the same material parameters and constitutive relation, the changing rule of 34
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stress can be obtained. Therefore, the vulnerable position of the balloon is found, which is in the center along the circumferential direction under the no-slip condition,
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with the largest stress.
Fig.15 contour profile of the inflated balloon with three contact conditions
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Lastly, the contour profiles of the balloon under three contact conditions are compared. Because of the rigid-flexible contact, in the contact region the ordinate
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keeps constant, while it is an arc in the non-contact region. In the pole and center of the
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balloon, the ordinate and horizontal ordinate are relatively larger for the no-slip condition. When the contact angle is the same, the volume of the balloon is the largest
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for the no-slip contact condition. As a result, the internal pressure of the balloon is the
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minimum according to the thermodynamic equilibrium. This can be explained by the fact that when the friction force between the balloon and plates reduces, the resistance to the deformation of the balloon is improved. Then to achieve the same contact angle, the volume of the balloon will decrease. Conclusions In this paper, a semi-analytical approach based on the force equivalent method is 35
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introduced to the Mooney–Rivlin hyperelastic membrane modal to characterize the rigid-flexible contact behaviors of an inflated membrane balloon. The non-moment theory of revolutionary thin shells is applied to the non-contact region, which is used to establish the constitutive equations. In the contact region, three different contact
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conditions including frictionless, no-slip and stick-slip are considered and the solving schemes are discussed in details. The inflatable and contact process can be tracked using the proposed model. The patterns and characteristics before contact, just in
conclusions are drawn as follows.
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contact and in press can be verified by the experimental tests. Several important
Considering the particularity of the stick-slip contact condition, the friction plays
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an important role. A small intersection interval appears in the meridian stretch ratio for different friction coefficients in the contact region and the horizontal ordinate changing
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ratio of any two conditions with different friction coefficients keeps the same, resulting
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in the maximum difference of this ordinate. Unlike the increasing meridian stretch ratio with the increment of the angle when friction coefficient is zero, a dividing point
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appears between the non-contact and contact regions when the friction is introduced. It
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declines in the contact region and increases in the non-contact region because of the interface friction, which prevents material of the balloon from sliding to the pole. Because the stretch ratios are equal to strain to some extent, the changing law of
the stress is similar to that of the stretch ratios due to the constitutive relation. The vulnerable position is then determined, which is in the center of the balloon along the circumferential direction, where the stress is the maximum. Because of the friction, 36
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materials in the non-contact region are difficult to slide to the dividing point, resulting in the largest stress under the no-slip condition. Because of the rigid-flexible contact, the balloon shape in the contact region keeps flat and it is an arc in the non-contact region. Under the same contact angle, the volume
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is relatively small for the frictionless condition, where the resistance to the deformation of the balloon is the weakest. As a result, the internal pressure is the maximum under this contact condition.
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Acknowledgements ： The authors gratefully acknowledge financial supports from National Natural Science Foundation of China, 11172079 and 11572099; Program for New Century Excellent Talents in University, NCET-11-0807; Natural Science
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Foundation of Heilongjiang Province of China, A2015002; the Fundamental Research Funds for the Central Universities, HIT.BRETIII.201209.
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Reference
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Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., Watson, J. D., 1994. Molecular Biology of the Cell, third ed. Garland, New York, pp. 949–1010
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Adkins, J. E., Rivlin, R. S., 1952. Large elastic deformation of isotropic materials IX.
AC
Philosophical transactions-Royal Society A. 244, 505–531. Belay, T., Kim, C. I., Schiavone, P., 2016. Interaction-induced morphological transitions of
lipid membranes in contact with an elliptical cross section of a rigid substrate. Journal of Applied Mechanics. 83, 011001. Bakowsky, H., Richter, T., Kneuer, C., Hoekstra, D., Rothe, U., Bendas, G., Ehrhardt, C., Bakowsky, U., 2008. Adhesion characteristics and stability assessment of lectinmodified liposomes 37
ACCEPTED MANUSCRIPT
for site-specific drug delivery. Biochimica Et Biophysica Acta-Biomembranes. 1778, 242–249. Bruck, H. A., McNeill, S. R., Sutton, M. A. et al., 1989. Digital image correlation using Newton-Raphson method of partial differential correction. Experimental mechanics. 29, 261-267. Chen, L., Chen, S. H., Gao, H. J., 2014. Biomimetic study of rolling transport through smooth
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muscle contraction. Colloids and Surfaces B. 123, 49–52. Chen, L., Chen, S. H., 2015. Adhesion of a gas-filled membrane on a stretched substrate. International Journal of Solids & Structures. 69, 189-194.
AN US
Charrier, J. M., Shrivastava, S., WU, R., 1987. Free and constrained inflation of elastic membranes in relation to thennoforming-axisymmetric problems. The Journal of Strain Analysis for Engineering Design. 22, 115-125.
M
Feng, W. W., Yang, W. H., 1973. On the contact problem of an inflated spherical nonlinear membrane. Journal of Applied Mechanics. 40, 209-214.
ED
Hild, F., Roux, S., 2006. Digital image correlation: from displacement measurement to
PT
identification of elastic properties–a review. Strain. 42, 69-80. Rivlin, R. S., 1948. Large elastic deformations of isotropic materials. I: fundamental concepts.
CE
Philosophical transactions-Royal Society A. 240, 459–490.
AC
Johnson, K. L., Kendall, K., Roberts, A. D., 1971. Surface energy and the contact of elastic
solids. Proceedings of the Royal Society of London A. 324, 301-313. Johnson, K. L., 1997. Adhesion and friction between a smooth elastic spherical asperity and a
plane surface. Proceedings of the Royal Society of London A. 453, 163–179. Jenkins, C., 2001. Gossamer spacecraft (membrane and inflatable structures technology for space applications). Progress in Astronautics and Aeronautics. 38
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Charrier, J. M., Shrivatava, S., 1989. Free and constrained inflation of elastic membranes in relation to thermoforming axisymmetric problem. The Journal of Strain Analysis for Engineering Design. 24, 55–74. Kumar, N., DasGupta, A., 2013. On the contact problem of an inflated spherical hyperelastic
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membrane. International Journal of Non-Linear Mechanics. 57, 130-139. Kumar, N., DasGupta, A., 2014. Contact mechanics and induced hysteresis at oscillatory contacts with adhesion. Langmuir. 30, 9107-9114.
AN US
Kumar, N., DasGupta, A., 2015. On the static and dynamic contact problem of an inflated spherical viscoelastic membrane. Journal of Applied Mechanics. 82, 121010. Lei, C., Chen, S., 2015. Adhesion of a gas-filled membrane on a stretched substrate.
M
International Journal of Solids & Structures. 69, 189-194.
Long, R., Hui, C. Y., 2012. Axisymmetric membrane in adhesive contact with rigid substrates:
ED
Analytical solutions under large deformation. International Journal of Solids & Structures. 49,
PT
672-683.
Long, R., Shull, K. R., Hui, C. Y., 2010. Large deformation adhesive contact mechanics of
CE
circular membranes with a flat rigid substrate. Journal of the Mechanics & Physics of Solids. 58,
AC
1225-1242.
Leonard, J. W., Verma, V. K., 1976. Double-curved element for Mooney-Rivlin membranes.
Journal of the Engineering Mechanics Division. 102, 625-641. Mooney, M., 1940. A theory of large elastic deformation. Journal of Applied Physics. 11, 582-592. Moretti, M., Prina-Mello, A., Reid, A. J., Barron, V., Prendergast, P. J., 2004. Endothelial cell 39
ACCEPTED MANUSCRIPT
alignment on cyclically-stretched silicone surfaces. Journal of Materials Science: Materials in Medicine. 15, 1159–1164. Menga, N., Afferrante, L., Carbone, G., 2016. Effect of thickness and boundary conditions on the behavior of viscoelastic layers in sliding contact with wavy profiles. Journal of the Mechanics
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& Physics of Solids. 95, 517-529. McGarry, G. J., Prendergast, P. J., 2004. A three-dimensional finite element model of an adherent eukaryotic cell. European cells & materials. 7, 27–34.
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Oden, J. T., 1972. Finite elements of nonlinear continua. New York.
Patil, A., DasGupta, A., Eriksson, A., 2015. Contact mechanics of a circular membrane inflated against a deformable substrate. International Journal of Solids & Structures. 67, 250-262.
M
Patil, A., DasGupta, A., 2013. Finite inflation of an initially stretched hyperelastic circular membrane. European Journal of Mechanics-A/Solids. 41, 28-36.
ED
Patil A, DasGupta A., 2015. Constrained inflation of a stretched hyperelastic membrane inside
PT
an elastic cone. Meccanica. 50, 1495-1508. Sohail, T., Tang, T., Nadler, B., 2013. Adhesive Contact of a Fluid-filled Membrane Driven by
CE
Electrostatic Forces. International Journal of Solids & Structures. 50, 2678-2690.
AC
Tsang, P. H., Li, G., Brun, Y. V., Freund, L. B., Tang, J. X., 2006. Adhesion of single bacterial
cells in the micro-Newton range. Proceedings of the National Academy of Sciences. 103, 5764– 5768.
Tielking, J. T., Feng, W. W., 1974. The application of the minimum potential energy principle to nonlinear axisymmetric membrane problems. Journal of Applied Mechanics. 41, 491–496. Vasu, T. S., Bhandakkar, T. K., 2016. A Study of the Contact of an Elastic Layer–Substrate 40
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System Indented by a Long Rigid Cylinder Incorporating Surface Effects. Journal of Applied Mechanics. 83, 06. Wan, K. T., Liu, K. K., 2001. Contact mechanics of a thin-walled capsule adhered onto a rigid planar substrate. Medical and Biological Engineering and Computing. 39, 605–608.
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Wang Y., Lava P., Coppieters S., et al., 2012. Investigation of the uncertainty of DIC under heterogeneous strain states with numerical tests. Strain. 48, 453-462.
Xu, D., Liechti, K. M., 2010. Analytical and experimental study of a circular membrane in
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Hertzian contact with a rigid substrate. International Journal of Solids & Structures. 47, 969-977. Xu, D., Liechti, K. M., 2011. Analytical and experimental study of a circular membrane in adhesive contact with a rigid substrate. International Journal of Solids & Structures. 48, 2965-2976.
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Yang, W. H., Feng, W. W., 1970. On axisymmetrical deformations of nonlinear membranes.
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CE
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Journal of Applied Mechanics. 37, 1002–1011.
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Graphical abstract
42
Rigid-flexible contact analysis of an inflated membrane balloon with various contact conditions M.X. Liu , C.G. Wang , X.D. Li PII: DOI: Reference:
S0020-7683(18)30189-6 10.1016/j.ijsolstr.2018.05.004 SAS 9984
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
19 September 2017 16 April 2018 7 May 2018
Please cite this article as: M.X. Liu , C.G. Wang , X.D. Li , Rigid-flexible contact analysis of an inflated membrane balloon with various contact conditions, International Journal of Solids and Structures (2018), doi: 10.1016/j.ijsolstr.2018.05.004
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Highlights
Decoupled governing equations are obtained based on the force balance relationship. 2. Various contact conditions, especially the harmonic one, are discussed.
3. The numerical results from no-slip condition are verified by the designed
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test (DIC technology).
4. The maximum ordinate difference appears in the intersection of meridian
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stretch ratio.
5. The no-slip condition results in the maximum volume and minimum
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internal pressure.
1
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Rigid-flexible contact analysis of an inflated membrane balloon with various contact conditions M. X. Liu ab1 C. G. Wang *a2, and X. D. Li b3 a
Department of Engineering Mechanics, Tsinghua University, Beijing, China, 100084,
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b
Center for Composite Materials, Harbin Institute of Technology, Harbin, China, 150001
Abstract
Considering the Mooney–Rivlin hyperelastic model, a semi-analytical approach is
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introduced to analyze the rigid–flexible contact behaviors of an inflated membrane balloon between two plates with various interface conditions. This approach is based on the differential formulation and the coupling property of equilibrium equations are
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well-solved. In order to verify the reliability of the proposed theoretical model, an experimental test is designed, by which some important contact characteristics and
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patterns (no-slip condition) are obtained. Two special phenomena are observed for the
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meridian stretch ratio with different friction coefficients. One is that the intersection points of all curves fall in a small interval and the intersection of any two curves
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represents the same changing rate of the horizontal ordinate, resulting in the maximum
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difference. The other is the dividing point, where the stretch ratio decreases on the left of it and increases on the right due to the introduction of friction. Under the same contact angle, the larger displacement load should be applied to the balloon for the
1
Ph.D., Center for Composite Materials, Harbin Institute of Technology/[email protected].
2
Prof., Center for Composite Materials, Harbin Institute of Technology/[email protected]/*Corresponding
author. 3
2
Prof., Department of Engineering Mechanics, Tsinghua University/[email protected].
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small friction coefficient condition, resulting in the smaller contact area and internal pressure. In addition, the vulnerable position, direction and contact condition of the balloon are found during the contact process, which happen in the center along the circumferential direction under no-slip condition. These results provide solid guidance
membrane balloon.
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and support for our understanding of the rigid-flexible contact behaviors of an inflated
Keywords: Mooney–Rivlin hyperelastic membrane, rigid-flexible contact, force
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equivalent method, decoupling, stick-slip condition
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1. Introduction
As the typical membrane structure, inflated balloon has considerable importance
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in a number of scientific studies and technological applications. On the macro scale, it
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can be used in terrestrial and space structures due to the advantages of light-weight, quick and self-deployment, and compact storage properties (Jenkins, 2001). On the
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micro scale, it can be used as animal or plant cell, examples including endothelial cells
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lining on the vessel wall (Moretti et al., 2004), liposomes adhesion to glycolipid membranes (Bakowsky et al., 2008) and a soft bubble rolling on the stretched substrate (Chen and Gao, 2014), which supports the normal life activities. However, when contacting with the rigid substrate, variety of problems are involved for these applications. Especially, geometry and boundary condition nonlinearities are the major challenging issues in the contact process (Menga et al., 2016). In addition, the friction 3
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influence, stress distribution and deformation in the non-contact and contact regions are also important (Alberts et al., 1994; Wan and Liu, 2001; Tsang et al., 2006; Kumar and DasGupta, 2013; Patil et al., 2015; Chen and Chen, 2015). In this context, understanding the contact mechanics of an inflated membrane balloon and rigid plates
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is, thus, of considerable interest, which motivates the present study. The investigations on the contact behaviors of an inflated membrane can be summarized as two processes: geometry nonlinearity analysis and boundary condition
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nonlinearity analysis. The geometry nonlinearity is carefully considered by solving the membrane inflation problem. Based on the study of membrane large elastic deformation (Rivlin, 1948), various methods are proposed to deal with this problem,
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examples including the large elastic deformation theory (Adkinks and Rivlin, 1952), force balance relationship (Feng and Yang, 1970), nonlinearity (Tielking and Feng,
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1974), energy balance relationship (Patil and DasGupta, 2013) and critical energy
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release rate (Long et al., 2010). Moreover, a lot of works have been done to deal with the nonlinearity problem of the boundary condition. On the basis of membrane
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inflation, Feng and Yang (1973) analyzed the contact problem of inflated
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membrane-rigid plate system, while the interface friction is not taken into consideration. The Hertzian contact of circular membrane-rigid plates is addressed by Xu and Liechti (2010). In order to introduce the interface influence, Johnson, Kendall and Roberts (1971) developed the JKR technique to discuss the contact adhesion, and Johnson (1997) considered the relationship between adhesion and friction due to fracture mechanics. Adhesion of a fluid-filled membrane with a substrate is discussed 4
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by Sohail, et al (2013). The adhesive contact of a circular membrane-rigid substrate is studied by Xu and Liechti (2011) with analytical and experimental methods. After the interface friction is introduced, different contact conditions are discussed gradually. The no-slip contact condition is studied by using finite element method and finite strain
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theory (Charrier and Shrivastava, 1989; Long et al., 2010). The limited interface friction is considered by Kumar and DasGupta (2013) using energy balance relationship, which is called stick-slip condition.
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Based on the studies of geometrical and boundary condition nonlinearities above, it can be found the membrane inflation and contact problem are very complicated. Thus, it is difficult to obtain the analytic solution. Broadly, solution schemes proposed
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to this problem are divided into two categories: finite element analysis and semi-analytical approach. Based on the finite element method, the membrane large
ED
deformation problems, nonlinear static behaviors, inflation and contact characteristics
PT
are analyzed by Oden (1972), Leonard and Verma (1976) and Charrier and Shrivastava (1987). In addition, this method is applied to biomechanics, the behaviors of a cell
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adhering to a substrate were addressed by McGarry and Prendergast (2004). Moreover,
AC
a large number of works are devoted to the semi-analytical approach. On the basis of different contact models, which are inflated membrane-rigid plates (Yang and Feng, 1973), lipid membrane-solid elliptical cylinder substrates (Belay et al., 2016), elastic layer-rigid cylinder (Vasu and Bhandakkar, 2016), clamped edge membrane-rigid plates (Long, 2010) and elastic membrane-elastic cone (Patil and DasGupta, 2015) systems, the contact problems are simplified to a set of ordinary differential equations, 5
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which can be solved by numerical methods. In the existing literature, the membrane inflation problem including the geometry nonlinearity has been well resolved. For the complex boundary conditions, diverse methods based on the energy balance are proposed (Patil and DasGupta, 2013). In this
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reference, the method is based on the variational formulation, where the whole system is taken as the study object and the functional variational equations are built on account of the energy balance. However, the normal adhesive force and tangential friction force
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couple together because the energy is the centralized reflection of deformation. This will increase the difficulty of the solving process. Therefore, in this paper, a semi-analytical method is introduced to extend the modal of Feng and Yang (1973),
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which is originally devoted to describe the membrane free inflation and simple frictionless contact. This method is rooted in the differential formulation in elastic
ED
mechanics. A microelement is considered and the governing equations are then
PT
established on the basis of the equilibrium, deformation and constitutive. The equilibrium equations are derived from the non-moment theory of revolutionary thin
CE
shells, which consider the force balance in the meridian tangential and normal
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directions. The geometric equations are established based on the small deformation hypothesis and the Mooney–Rivlin hyperelastic model is introduced to construct the constitutive equations. Thus, the coupling problem is solved fundamentally, and a set of uncoupled ordinary differential equations can be easily obtained. These ordinary differential equations can be integrated numerically using the Runge-Kutta method with the given boundary values. In addition, more complex contact boundary 6
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conditions are studied, which is the extension of the wok of Feng and Yang (1973). Especially, for the stick-slip contact condition, the friction and slide are considered comprehensively. Moreover, the ideal gas thermodynamics state equation is assumed to be valid during the contact process in the studies of our research (Kumar and Patil, 2014;
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Lei and Chen et al, 2015; Kumar and Patil, 2015; Patil et al., 2015). Because of these concerns, this paper is organized as follows. The geometry and constitutive models, which are based on the Mooney–Rivlin hyperelastic membrane
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theory, are described in Section.2. The work of Feng and Yang (1973), where the frictionless contact condition of the membrane-rigid plate system, is firstly introduced in Section.3. Then, we make extensions of their work, including no-slip and stick-slip
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conditions when the interfacial friction force is taken into consideration. In Section.4, the inflatable and contact characteristics (no-slip condition) are predicted by the
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proposed model and verified by the experimental test. Finally, the influence of friction
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is discussed and results from three different contact conditions are compared. 2. Geometry and constitutive models
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A spherical balloon with uninflated radius r0 and uniform thickness h (state I
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the black line) is inflated to radius rs by pressure P0 (state II the red line). Then two parallel rigid plates are pressed by F into contact with the balloon (state III the green line). Half of the spherical balloon and one rigid plate are shown in Fig. 1. The inflated spherical balloon before contact is described by the spherical coordinates ( rs , , ). The cylindrical coordinates ( , , ) are used for the spherical balloon after contact. The pressure P0 before contact and the pressure P after contact are considered to be 7
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uniformly distributed in the balloon. Moreover, all thermodynamic processes are
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assumed to be reversible and isothermal during inflation and contact.
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Fig.1 the contact model of an inflated membrane and rigid plates (the initial state is I, state II indicates that the membrane comes into contact with rigid plates, state III shows that the membrane is compressed to the specified contact angle ), two main
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curvilinear directions (the circumferential ( ) and meridian ( ) directions) are shown
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on the right
The governing differential equations are built for the non-contact region and the
PT
contact one, separately.
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In the non-contact region: according to the geometric relation, the principal stretch ratios for the membrane can be written as , . Here, the subscripts and
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denote the meridian and circumferential directions.
'2 '2 dS , ds r0
. r0 sin
(1)
The prime in the foregoing and subsequent equations denotes the derivatives with respect to the angle . ds is the infinitesimal arc length of the uninflated spherical balloon. dS is the corresponding infinitesimal arc length of the compressed spherical 8
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balloon. The principal curvatures , can be obtained in terms of , and . ' '' 1 d ' '' , A dS ( '2 '2 ) 3 2
sin ' ， 1 B ( '2 '2 ) 2
(2)
where A and B are the Lame coefficients along and directions, respectively.
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Substituting Eq. (1) to Eq. (2), the principal curvatures , in terms of
, and are expressed as:
' ( sin )' ( sin )" [ ], 2 r0 2 ( sin )'2
1 r0 sin
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1
(3)
2 ( sin )'2 .
The structure is considered as the non-moment thin shell, which has no bending
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moment and torsion on the cross section, and the equilibrium differential equations can
ED
be expressed due to the constitutive relation.
(4)
CE
PT
B A ( BT ) T ( AT ) ( AT ) ABq 0, A B ( AT ) T T ( BT ) ABq 0, T T qn，
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where T , T and T are the internal forces and q , q and qn are the external forces applied to the balloon along meridian, circumferential and normal directions, respectively. Because the inflated balloon can be considered as a revolution body, the equilibrium equations then will be simplified in meridian tangential and normal directions due to the axial symmetry. ( T = q =0, Q’M=B/sin , Q’O’=A (as shown in 9
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Fig.1))
dT T T P t , d cos T T Pn，
(5)
where Pn = qn and Pt = q are the external loads acting on the deformed surface in
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the normal and meridian tangential directions. In the present work, we consider the two-parameter Mooney-Rivlin hyperplastic constitutive model for the spherical balloon, which is incompressible, homogeneous
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and isotropic. This material model is based on the general theory of large elastic deformation and has been solved exactly by Mooney (1940). The strain energy density function for such a material is given by W C1 ( I1 3) C2 ( I 2 3) , where C1 +C2 =
G 2
1
2
2
and
I 2 2 2
1
2
1
2
are the equations of the
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I1 2 2
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and G is the shear modulus. The principal value of the deformable tensor
principal stretches , . Then the stress resultants of per unit length T and T in
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terms of the principal stretches , can be expressed as:
1 3 3 )(1 2 ),
(6)
1 T 2C1h( 3 3 )(1 2 )，
AC
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T 2C1h(
where h is the thickness of the spherical balloon before inflation and
C2 . C1
In the non-contact region, it is assumed that Pn P and Pt 0 , where P is the internal pressure of the deformed spherical balloon. Then the equilibrium equations (Eq. (5)) can be rewritten in terms of , and .
10
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(7)
T T P. Substituting Eq. (1), Eq. (3) and Eq. (6) into Eq. (7), the equilibrium equations can also be expressed as 1
2 ( [
1
4
3
)(1 2 ) ' [(
3 3 4 )(1 2 ) 2
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' (
( sin ) ' 1 1 1 3 3 )] [ ( 3 )], sin 3
' ( sin )' ( sin )" 1 ][( 3 3 )(1 2 )] 2
(8)
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Pr0 2 ( sin ) '2 2 ( sin ) '2 1 [( 3 3 )(1 2 )] . sin 2hC1
Eq. (8) is a two-dimensional system of ordinary differential equations. In order to
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convert it to the integral form, we define four new variables.
ED
sin , ' , h
h P , p . r0 C1
(9)
Then Eq. (8) can be simplified to the iteration forms by substituting variables,
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which are three first-order ordinary differential equations in the non-contact region
AC
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(Feng and Yang (1973)).
11
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( ' sin cos )(
'
3sin 2 ) 2 3 4
2 sin 3 ' cos sin ( 3 3 )( sin ) 1 sin sin 3 2 ' 3 3 1 2 sin sin ( ) sin
4 3 ( 4 2 sin 2 )sin
(10)
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' ,
,
1 sin 3 2 2 ( ) ' sin 3 3 ' . 2 4 2 2 p 2 ( sin )(sin ) (1 ) sin 2h 4 3
2
2
The boundary conditions for the non-contact region are:
nc
,
c
nc
*sin , (w
) nc
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c
, c
'
0,
(11)
2
where “nc” and “c” refer to the non-contact and contact regions, respectively. is
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the dividing point of the two regions.
The pressure and volume inside the spherical balloon before contact are related to
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s . They can be expressed as:
rs and V0 is the volume of the spherical balloon after inflation. The r0
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where s
(12)
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4 1 h V0 r03s 3 , P0 4C1 (1 6 )(1 s 2 ) , 3 s r0s
volume V of the balloon after contact can be obtained in terms of , and ,
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which is V 2 r03 2 2 2 '2 d . Because there is a small contact angle
increment from one contact state to the next one and the difference of the two adjacent contact states is very small, the contact process is assumed to be a quasi-steady state and the temperature change of the internal gas can be ignored. According to the ideal gas thermodynamics state equation, the pressure P inside the balloon after contact can be obtained: 12
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P
P0 V0 V
2 P0 s 3
3 2
2
. lim
(13)
x
d 2
' 2
3. Contact conditions In the contact region, three types of contact conditions are considered, which are
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named as frictionless contact, no-slip contact and stick-slip contact. 3.1 Frictionless contact condition
For the frictionless contact condition, the membrane in the contact region can
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slide freely over the rigid plates because of the tension induced by internal pressure. This has been considered and well-solved by Feng and Yang (1973). The stretch ratios along meridian and circumferential directions are equal at the pole (Eq. (15)) and
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successive between contact and non-contact regions (Eq. (11)). Because of the rigid-flexible contact, the ordinate of the membrane in the contact region keeps
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constant and geometry of the contact region requires that ' 0 . Then according to Eq. (1), the principal stretch ratios in the contact region can be reduced to
r0
PT
'
,
r0 sin
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. At the center of the balloon, the horizontal ordinate
reaches to the maximum, which results in the zero derivative of .Hence, the
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governing equations, which are two first order ordinary differential equations, can be expressed as:
13
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'
'
3 2 ( 2 3 4 ) 1 cos sin 1 1 , 2 3 3 1 3 2 ( )(1 ) 4 3 1 1 3 3 sin cos . sin
(14)
0
0
0 ,
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The boundary condition for the contact region is: (15)
where 0 is the stretch ratio at the pole ( 0 ). Eq. (15) is the initial condition for
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the governing equations (Eq. (14)) of the contact region. The initial condition for the non-contact region (Eq. (10)) can be obtained from the results of Eq. (14) and the boundary conditions given in Eq. (11).
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The solving process for this contact condition is shown in Fig.2, and the computation procedure is outlined as follow:
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1) Assigning the stretch ratio s after inflation and the contact angle , the pressure P0 and volume V0 before contact can be obtained from Eq. (12).
PT
2) Assume the stretch ratio 0 in the pole. Then Eq. (15) provides the initial
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integral values for Eq. (14), which are the governing equations in the contact region.
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Integrating the governing equations (Eq. (14)) with the initial condition (Eq. (15)) from
0 to , the stretch ratios ( , ) in the contact region can be obtained. 3) The results from Step 2 and Eq. (11) provide the initial conditions for Eq. (10),
which are the governing equations in the non-contact region. Guess a value for the pressure P after contact. Applying the Runge-Kutta method to Eq. (10) with the initial condition from to 14
2
, the stretch ratios ( , ) in non-contact
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region can be computed. 4) Adjust the guessed pressure P constantly by a small constant until the ' error1 caused by the boundary condition 0 is satisfied. Reassume the value 2
0 by a small constant constantly until the error2 from the thermodynamic
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PT
ED
M
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equilibrium condition (Eq. (13)) is satisfied.
Fig.2 the flow chart of the solution procedure under frictionless contact condition
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3.2 No-slip contact condition In the case of the no-slip contact, the friction force between the plates and the
spherical balloon in the contact region is sufficiently large. Then materials in this region cannot slide over the plates. In this case, the contact depends on the history of the contact formation. The stretch ratios along the meridian and circumferential directions at the pole are equal to the initial stretch ratio due to the infinite friction 15
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force (
0
0
s ). The ordinate of the membrane in the contact region keeps
constant ( ' 0 ) and the horizontal ordinate reaches to the maximum at the center of the balloon ( ' 0 ), which are same as the conditions of frictionless and stick-slip. At the diving point of the contact and non-contact region, the circumferential stretch
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ratio and its derivative are successive. Because of the geometric relation and successive condition, the meridian stretch ratio after contact can be related to the parameter w before contact =
' r0
=(
)' = ' = w
. Then the boundary
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conditions between contact and non-contact region can be expressed as
w , , w w .
(16)
Here, the signs “-” and “+” indicate the values before and after contact. Therefore, the
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solution of this case can be determined incrementally as follow.
obtained in the pole: 0
PT
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The initial contact angle is set as 0 , so the initial integral condition can be
0
s ,
0
0w , 0 s
.
(17)
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Considering a small increment of the contact angle , the problem can be solved by integrating the governing equations (Eq. (10)) in the non-contact region. The
AC
results of this step can be stored in the contact region, which provide the initial integral conditions for the governing equations (Eq. (10)) in the non-contact region in the next
.
The solving procedure for this contact condition is shown in Fig.3 and the computation procedure is outlined as follow: 1) Assigning the stretch ratio s after inflation and contact angle , the 16
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pressure P0 before contact can be obtained from Eq. (12). 2) Initially, the contact angle and pressure are set as 0, P P0 . Then the configuration of the membrane can be determined by integrating the governing equations (Eq. (10)) in the non-contact region with the initial integral condition
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provided by Eq. (17). 3) Considering a small contact angle increment , the parameters for the diving point ( + ) between contact and non-contact regions can be expressed as
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, , w before contact. Integrate the governing equations (Eq. (10)) in the non-contact region with the guessed initial conditions (Eq. (16)). 4) Adjust the guessed pressure P constantly by a small constant until the ' error1 caused by the boundary condition 0 is satisfied. Reassume the guessed 2
initial values a small constant with ( w
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w
*(1 * c2 ),
w
*
Γ
, , w in the non-contact region from Step 4 are stored in the
PT
5) The results
ED
until the error2 from the thermodynamic equilibrium condition Eq. (13) is satisfied.
contact region. Then, repeat the steps from 1 to 5 until the contact angle reaches to the
AC
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default value .
17
)
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Fig.3 the flow chart of the solution procedure under no-slip contact condition
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3.3 Stick-slip contact condition
In the contact condition of stick-slip, the sliding state of materials in the contact
PT
region is motivated by the membrane tension and resisted by the interfacial friction.
CE
Because the friction coefficient is limited, material will stick when the interfacial friction is greater than the membrane tension, while the others will slip.
AC
Considering the friction force, the equilibrium equation along the meridian
tangential direction (the first equation in Eq. (6)) of the spherical balloon in the contact region can be rewritten as
dT T T . Here is the friction force along d
the tangential direction, which can be also obtained by P . Hence, the equilibrium condition for the critical sliding state can be obtained: 18
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1 T T 1 ( ) (T T ) + f P =0, '
(18)
where f is the critical sliding friction coefficient. Therefore, the governing equations in this contact region are given by:
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3 2 2 3 4 1 cos sin 1 1 2 3 3 ' , 1 3 2 4 3 1 1 1 r0 1 f P sin 3 3 2C h 1 2 1 2 cos ' . sin
(19)
The governing equations in the non-contact region are the same as Eq. (10). The solving procedure for this contact condition is shown in Fig.4, and the computation
AC
CE
PT
ED
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procedure is outlined as follow:
19
M
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Fig.4 the flow chart of the solution procedure under stick-slip contact condition 1) The stretch ratio after inflation, contact angle and the critical friction
CE
by Eq. (12).
PT
coefficient are set to s , and f .The pressure before contact then can be obtained
2) Because of the interfacial friction, the problem is solved using the no-slip
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condition procedure until the contact angle becomes f , where the friction coefficient reaches to the critical value f . 3) Then sliding between plates and the inflated membrane occurs as the contact continues. Considering a small increment , integrate the governing equations in the contact region (Eq. (19)) from f to f i (i=1,2,3…) with initial conditions from the results of Step 2 ( ( 20
f
) s (
f
)ns , (
f
) s (
f
)ns ).
Here, “s”
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and “ns” denote the sliding and no-slip parts, respectively. Integrate the governing equations in the non-contact region (Eq. (10)) from f i to . The initial 2
conditions
1 f i
for
f i
nc
this
, c
integration
f i
*sin
i f
nc
f
c
are
i , w
f
i
given
f i
nc
.Here, c
by: “nc”
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and “c” indicate the non-contact and contact regions, respectively. 4) Adjust the pressure P by a small constant after contact constantly until ' the error1caused by the boundary condition 0 is satisfied. Reassume the initial 2
f
f
*(1 c2 ) with a small constant until the error2
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guessed value
from the thermodynamic equilibrium condition Eq. (13) is satisfied. 5) Because the adjusted value
' is related to f due to Eq. (1), the
, which means the meridian stretch ratio in the '
'
f
f
ns
s
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value must fulfill
f
no-slip part is larger than that in the sliding part. If this relationship is not satisfied, the
ED
material in the sliding part will slide to the no-slip part. This means the no-slip part will
PT
decreases ( f f ). If f 0 , the material in the contact region will all slip to the pole.
CE
6) If the material in the contact region becomes the sliding part, integral the
AC
governing equations in the contact region (Eq. (19)) with the initial condition
0
0
0 from 0 to . Integral the governing equations in the non-contact
region (Eq. (10)) from to of
contact
nc
, where the initial conditions are given by the results 2
region.
, c
nc
It
can
c
*sin , w
be
nc
expressed
as
. c
7) Adjust the pressure P after contact by a small constant constantly until 21
ACCEPTED MANUSCRIPT ' the error1caused by the boundary condition 0 is satisfied. Reassume the initial 2
guessed value 0 0 *(1 c2 ) by a small constant in Step 6 until the error2 from the thermodynamic equilibrium condition Eq. (13) is satisfied. 8) After the adjustment for 0 by a small constant in Step 7, this value needs
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to meet the condition 0 s , which means the material in the contact region is sliding. If this condition is not satisfied, material in the contact region will not slip to the pole. This means the no-slip part reappears in the contact region. Then return to the
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Step 1.
9) Repeat the steps from 1 to 8 until the contact angle reaches the default value . 4. Results and discussion
M
In this section, the contact behaviors of the inflated membrane with different contact conditions are discussed. The contact characteristics will be obtained by
ED
solving the governing equations with the numerical methods discussed in details in
PT
Section 3. Results from the no-slip contact condition will be verified by the designed experiment (Section 4.1). The influence of friction coefficient on stretch ratios and the
CE
contour profiles of the balloon is discussed, and results from three contact conditions
AC
are compared in Section 4.2. 4.1 Experiment verification In order to verify the theoretical results, a testing scheme is proposed to measure
the patterns and some important contact characteristics of the inflated balloon. Because the friction coefficient is difficult to measure and the frictionless condition is hard to realize, the no-slip contact condition is carried out in this experiment. 22
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Tab. 1 Material and geometrical parameters Parameters
Magnitude
Initial radius (r0)
0.05 (m)
Young’s modulus of membrane(E) Poisson’s ratio ( )
0.3(mm)
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Thickness of the beam (h)
6 (MPa) 0.47 3
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Stretch ratio before contact ( s )
The material and structural parameters of the inflated balloon are shown in Table 1, which is a Polyvinylchloride thin membrane (PVC). The specimen in the experiment
CE
PT
ED
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is shown in Fig. 5.
AC
Fig. 5 the specimen of the balloon in contact and the random speckles are drawn
on the surface (r0 is the initial radius, rs is the radius after inflated and l is the displacement load applied to plates)
Firstly, the principle of Digital Image Correlation Technology (DIC)measurement is briefly introduced. DIC is an optical measurement method and displacements and displacement gradients can be determined based on the video image (Bruck et al., 1989; 23
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Hild and Roux, 2006; Wang et al., 2012). The position of a characteristic point in the
subarea of the pre-deformation image is (x, y, z) and it is (x*, y*, z*) in the deformed image. Then the typical correlation function which measures how well the two images match is: F ( x, y, z ) * G( x* , y* , z * ) u u u v v v w w w , , , , , , , , ) 1 ， 1 x y z x y z x y z ( F ( x, y, z,)2 ) * (G( x* , y* , z * ) 2 ) 2
(20)
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S ( x, y, z, u, v, w,
where F ( x, y, z) and G( x* , y* , z* ) are the gray level values of images before and after deformation. Then the deformation information can be obtained by minimizing the
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correlation coefficient S.
Before the experiment, the speckle pattern on the balloon is reproduced artificially. To increase the contrast of the speckle, matte white paint and black paint are sprayed on
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the balloon surface evenly. Then the two rigid plates, which are two 5mm thick acrylic
ED
plates, are fixed on the loading shafts. In the experiment, the balloon with sprayed speckles is placed on the workbench and its position is adjusted so that it is on the
PT
compression axis. After that, the balloon is inflated by the pump and the diameter
CE
increases from 10cm to 30cm. The displacement load is applied to the balloon, which can be controlled precisely by the electronic universal testing machine. This load is
AC
noted down by a ruler on the machine and the internal pressure of the balloon is monitored by the barometer. The deformations of the balloon are tested using the digital image correlation (DIC) technology, which is a reliable mean to measure the displacement fields. There are mainly four steps in the DIC measurement process: (i) random speckles are made on the specimen surface; (ii) the calibration of CCD camera; (iii) capturing deformed images; (iv) images’ post-processing. The accuracy of the 24
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measurement precision is 1μm for our system.
Fig. 6 setup of the contact experiment
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The deformation contours of the balloon are obviously observed and tested in Fig. 7. It contains three stages, which are a) before contact, b) just in contact and c) in press.
ED
And the maximum displacement of the balloon (
) is compared in Tab. 2 2
CE
PT
Tab.2. the experimental and theoretical results of the maximum displacement under
AC
maximum
displacement (
l ) r0
different displacement load Displacement load (cm)
0.63
2.13
4.25
theoretical results
2.05
2.18
2.34
experimental results
2.08
2.25
2.50
The balloon is firstly inflated by the pump before contact and the data is recorded when the internal pressure is 18.8KPa. Then, it is inflated to contact with the rigid plates, where the distance between the two plates is set to 30cm and the internal pressure monitored is 19.1KPa. After that, the balloon is compressed by the rigid plates 25
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and the internal pressure is 22.0 KPa when the displacement load is 2.10cm. When the inflated balloon is just in contact to plates, the maximum displacement appears in the middle (
). The tested deformation (10.4cm) agrees with the theoretical result 2
(10.0cm) with an error around 0.33%. When the displacement load is 4.25cm, the error
M
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reaches to the maximum, which is 6.8%.
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Fig. 7 three stages of contact (a) before contact b) just in contact c) in press)
CE
PT
Tab. 3. the experimental and theoretical results of pressures under different displacement load
Displacement load (cm)
AC
Pr0 theoretical results Pressure ( ) C1h
experimental results
0.63
2.13
4.25
3.85
4.25
5.12
3.87
4.30
5.22
In addition, the pressures under different displacement load are tested and
compared with the theoretical predictions in Tab.3. Accompanied by the increasing displacement load, the dimensionless pressure
Pr0 increases. This means that the C1h
volume decreases with the rise of the displacement load, resulting in the increase of 26
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internal pressure. The tested pressure is close to the theoretical results within the maximum error of 3.41%. It also can be found that the error increases when the applied displacement keeps increasing. The difference between the experimental and theoretical results may because that the temperature increases in the balloon and the
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thermodynamic equilibrium is not satisfied. Moreover, some inevitable limitations in test techniques may also cause the difference between the results from experiments and theoretical model. When measuring the internal pressure of the balloon, the error occurs
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because the minimum scale of the barometer is 0.1KPa. Also, noise and truncation error will be introduced in the DIC calculation. In addition, the error may be caused by human factors when reading and recording data. The maximum error is 6.8% when comparing
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the theoretical and experimental results of the maximum displacement and it is 3.4% for measuring the internal pressure. In general, the differences will not exceed beyond 7%.
ED
In fact, the experiment results offer the relative trustworthy evidence to verify the
PT
proposed theoretical model and computation procedures. 4.2 Theoretical prediction
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4.2.1 The stick-slip contact results
AC
For the stick-slip contact condition, it becomes the frictionless contact condition
when the friction coefficient becomes zero. If the friction coefficient goes to infinity, it is converted into the no-slip contact condition. As a general case, the results from the stick-slip contact condition are mainly discussed in this section. For different friction coefficients, the changing trend of the meridian stretch ratio
with the angle is shown in Fig.8 a), b) and c), where the contact angles are 27
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30 , 45 and 60 .
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Fig.8 the changing trend of meridian stretch ratio with angle in different
ED
friction coefficient conditions: a), b) and c) represent that the contact angles are 30 , 45 and 60 , respectively.
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Taking the 60 contact angle as an example, two special phenomena can be seen in
CE
Fig.8 c). One is that the intersection points of all curves fall in a small interval, and the other is the dividing point ( angle 1.05 ). The intersection point of any two curves
AC
appears in the contact region, which represents that material have the same meridian stretch ratio under the conditions with corresponding friction coefficients at that point. Moreover, different from the condition of f 0 , a dividing point, which is the boundary of contact and non-contact regions, appears when friction is introduced. In the contact region, the meridian stretch ratio decreases when material is far away from the pole, while it increases in the non-contact region. In addition, with the increment of 28
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the friction coefficient, the meridian stretch ratio increases in the pole of the balloon, while it decreases in the middle ( 2 ). All these phenomena can be explained by the introduction of the friction, which prevents the material in the contact region from sliding to the pole and makes it hard for the material in the non-contact region slip to
and angle relationship with different friction r0
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Fig.9 a) horizontal ordinate
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the contact region.
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coefficients b) the horizontal ordinate difference
between introduced friction r0
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conditions (different coefficients) and frictionless one
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To better explain the intersection interval in Fig.8 c) (contact angle 60 ), the relationship between the horizontal ordinate and angle is counted in Fig.9 a) and the
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horizontal ordinate difference between the introduced friction conditions and the frictionless one is described in Fig.9 b). In the contact region, the relationship
' r0
is satisfied due to the expression
'2 '2 dS and ' =0 . Then the ds r0
intersection of any two curves represents the same changing rate of the horizontal ordinate 29
, resulting in the maximum difference at that point. Moreover, the maximum r0
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difference is calculated and shown in Table 4 when the contact angles are 30 , 60
45
and
. From this table, the maximum difference of horizontal ordinate all falls in a small
interval and the range is about 0.01rad in these three contact angle cases. The obtained results can prove that the intersection interval in Fig.8 is ubiquitous in the contact
Tab. 4. the maximum difference
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process. max of horizontal ordinate between the r0
introduced friction conditions and the frictionless condition with different contact angle
friction coefficient
0.01
contact angle 0.3740
45
0.5602
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60
0.05
0.09
0.1005
Range
0.3755
0.3756
0.3766
0.3861
0.0121
0.5513
0.5491
0.5494
0.5504
0.0111
0.7090
0.7052
0.7056
0.6929
0.0161
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30
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( 30 , 45 and 60 )
0.6967
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Moreover, a step point appears at angle 1.05 in Fig.8 c), which is
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corresponding to the dividing point between the contact and non-contact regions in Fig.10. This means the changing trend of the meridian stretch ratio is different in the
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contact and non-contact regions, though this stretch ratio changes continuously in these two regions. In addition, the first-order derivative of meridian stretch ratio is zero in frictionless and no-slip condition, while it is negative in the stick-slip condition where materials in the contact region all slide to the pole (Fig.10)). For the frictionless condition, the contact process is independent to the loading history. Then, the first
30
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derivative of meridian stretch can be expressed as '
'' r0
=
2 tan (1 tan ) due to r0
the geometrical relation = tan and the fact that is the constant. When friction is introduced, the balloon deformation is dependent on the history of contact. Especially for the no-slip condition, the meridian stretch ratio and its first variation in the pole are
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the same as those when the balloon is just inflated, where the first derivative is zero due to the constant stretch ratio. When friction coefficients are 0.01, 0.05 and 0.1005, materials in the contact region all slide to the pole. The first derivative of the meridian r0 1 f P 2C1h
in Eq. (19) (stick-slip condition)
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stretch is negative due to the one more item
compared to that in Eq. (14) (frictionless condition) and it decreases when friction
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coefficient increases.
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Fig.10 first-order derivative of meridian stretch ratio and angle relationship with
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different friction coefficients
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Fig.11 Dimensionless contact area and dimensionless pressure relationship with
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different friction coefficients under different contact angles
Fig.11 shows the internal pressure change with the contact area under different friction coefficients and contact angles. When displacement applied to plates keeps
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increasing, the contact area and the internal pressure increase. Under the same contact angle, the contact area and internal pressure are relatively small with the smaller
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friction coefficient. With the increase of the contact angle, the difference of contact
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area and internal pressure becomes more obvious with different friction coefficients. This phenomenon can be explained in Fig.15 that the contact area and volume decline
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because of the decreasing friction coefficient.
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Fig.12 Dimensionless contact area and dimensionless displacement load relationship with different friction coefficients under different contact angles It can be obviously seen in Fig.12 that the contact area will increase when the applied displacement load constantly increases under the same friction coefficient. In
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order to keep the same contact angle, the larger displacement load should be applied with the small friction coefficient, while the contact area is relatively small. In addition, with increasing the displacement load, the friction coefficient influence to contact area
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will be more obvious. This phenomenon can also be explained in Fig.15 below. For the small friction coefficient, the larger displacement load should be applied and small contact area can be obtained to keep the same contact angle.
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4.2.2 Result comparison of three contact conditions
In this section, results from the three contact conditions are compared. Here, the
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contact angle is set to 60 degrees. The friction coefficient is set to 0.1005, where the
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contact region is all slipping part.
Fig.13 the meridian and circumferential stretch ratio of three contact conditions (the contact angle is set to 60 degrees and the friction coefficient is set to 0.1005 for 33
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the stick-slip contact) Fig.13 compares the meridian and circumferential stretch ratio under three different conditions. For the frictionless contact condition, the stretch ratios vary continuously because material in the contact region can slide freely. For the no-slip and
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stick-slip contact conditions, the meridian stretch ratio has obvious dividing point between the contact and non-contact regions, which is explained in Section 4.2.1. For the meridian stretch ratio, it is smaller around the pole for the frictionless condition,
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while it is relatively larger close to the center position. This can be explained by the fact that the material will slide to the pole in the contact region and slide to the dividing point in the non-contact region. In addition, the circumferential stretch ratios are larger
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than the meridian ones on the whole.
Fig.14 the meridian and circumferential stress of three contact conditions
The comparison of stress under these three contact conditions are shown in Fig.14.
The changing trend of the stress is the same as that of the stretch ratio according to Eq. (6). This can also be interpreted that the stretch ratio is equal to strain to some extent. Due to the same material parameters and constitutive relation, the changing rule of 34
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stress can be obtained. Therefore, the vulnerable position of the balloon is found, which is in the center along the circumferential direction under the no-slip condition,
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with the largest stress.
Fig.15 contour profile of the inflated balloon with three contact conditions
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Lastly, the contour profiles of the balloon under three contact conditions are compared. Because of the rigid-flexible contact, in the contact region the ordinate
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keeps constant, while it is an arc in the non-contact region. In the pole and center of the
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balloon, the ordinate and horizontal ordinate are relatively larger for the no-slip condition. When the contact angle is the same, the volume of the balloon is the largest
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for the no-slip contact condition. As a result, the internal pressure of the balloon is the
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minimum according to the thermodynamic equilibrium. This can be explained by the fact that when the friction force between the balloon and plates reduces, the resistance to the deformation of the balloon is improved. Then to achieve the same contact angle, the volume of the balloon will decrease. Conclusions In this paper, a semi-analytical approach based on the force equivalent method is 35
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introduced to the Mooney–Rivlin hyperelastic membrane modal to characterize the rigid-flexible contact behaviors of an inflated membrane balloon. The non-moment theory of revolutionary thin shells is applied to the non-contact region, which is used to establish the constitutive equations. In the contact region, three different contact
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conditions including frictionless, no-slip and stick-slip are considered and the solving schemes are discussed in details. The inflatable and contact process can be tracked using the proposed model. The patterns and characteristics before contact, just in
conclusions are drawn as follows.
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contact and in press can be verified by the experimental tests. Several important
Considering the particularity of the stick-slip contact condition, the friction plays
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an important role. A small intersection interval appears in the meridian stretch ratio for different friction coefficients in the contact region and the horizontal ordinate changing
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ratio of any two conditions with different friction coefficients keeps the same, resulting
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in the maximum difference of this ordinate. Unlike the increasing meridian stretch ratio with the increment of the angle when friction coefficient is zero, a dividing point
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appears between the non-contact and contact regions when the friction is introduced. It
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declines in the contact region and increases in the non-contact region because of the interface friction, which prevents material of the balloon from sliding to the pole. Because the stretch ratios are equal to strain to some extent, the changing law of
the stress is similar to that of the stretch ratios due to the constitutive relation. The vulnerable position is then determined, which is in the center of the balloon along the circumferential direction, where the stress is the maximum. Because of the friction, 36
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materials in the non-contact region are difficult to slide to the dividing point, resulting in the largest stress under the no-slip condition. Because of the rigid-flexible contact, the balloon shape in the contact region keeps flat and it is an arc in the non-contact region. Under the same contact angle, the volume
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is relatively small for the frictionless condition, where the resistance to the deformation of the balloon is the weakest. As a result, the internal pressure is the maximum under this contact condition.
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Acknowledgements ： The authors gratefully acknowledge financial supports from National Natural Science Foundation of China, 11172079 and 11572099; Program for New Century Excellent Talents in University, NCET-11-0807; Natural Science
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Foundation of Heilongjiang Province of China, A2015002; the Fundamental Research Funds for the Central Universities, HIT.BRETIII.201209.
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Reference
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Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., Watson, J. D., 1994. Molecular Biology of the Cell, third ed. Garland, New York, pp. 949–1010
CE
Adkins, J. E., Rivlin, R. S., 1952. Large elastic deformation of isotropic materials IX.
AC
Philosophical transactions-Royal Society A. 244, 505–531. Belay, T., Kim, C. I., Schiavone, P., 2016. Interaction-induced morphological transitions of
lipid membranes in contact with an elliptical cross section of a rigid substrate. Journal of Applied Mechanics. 83, 011001. Bakowsky, H., Richter, T., Kneuer, C., Hoekstra, D., Rothe, U., Bendas, G., Ehrhardt, C., Bakowsky, U., 2008. Adhesion characteristics and stability assessment of lectinmodified liposomes 37
ACCEPTED MANUSCRIPT
for site-specific drug delivery. Biochimica Et Biophysica Acta-Biomembranes. 1778, 242–249. Bruck, H. A., McNeill, S. R., Sutton, M. A. et al., 1989. Digital image correlation using Newton-Raphson method of partial differential correction. Experimental mechanics. 29, 261-267. Chen, L., Chen, S. H., Gao, H. J., 2014. Biomimetic study of rolling transport through smooth
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muscle contraction. Colloids and Surfaces B. 123, 49–52. Chen, L., Chen, S. H., 2015. Adhesion of a gas-filled membrane on a stretched substrate. International Journal of Solids & Structures. 69, 189-194.
AN US
Charrier, J. M., Shrivastava, S., WU, R., 1987. Free and constrained inflation of elastic membranes in relation to thennoforming-axisymmetric problems. The Journal of Strain Analysis for Engineering Design. 22, 115-125.
M
Feng, W. W., Yang, W. H., 1973. On the contact problem of an inflated spherical nonlinear membrane. Journal of Applied Mechanics. 40, 209-214.
ED
Hild, F., Roux, S., 2006. Digital image correlation: from displacement measurement to
PT
identification of elastic properties–a review. Strain. 42, 69-80. Rivlin, R. S., 1948. Large elastic deformations of isotropic materials. I: fundamental concepts.
CE
Philosophical transactions-Royal Society A. 240, 459–490.
AC
Johnson, K. L., Kendall, K., Roberts, A. D., 1971. Surface energy and the contact of elastic
solids. Proceedings of the Royal Society of London A. 324, 301-313. Johnson, K. L., 1997. Adhesion and friction between a smooth elastic spherical asperity and a
plane surface. Proceedings of the Royal Society of London A. 453, 163–179. Jenkins, C., 2001. Gossamer spacecraft (membrane and inflatable structures technology for space applications). Progress in Astronautics and Aeronautics. 38
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Charrier, J. M., Shrivatava, S., 1989. Free and constrained inflation of elastic membranes in relation to thermoforming axisymmetric problem. The Journal of Strain Analysis for Engineering Design. 24, 55–74. Kumar, N., DasGupta, A., 2013. On the contact problem of an inflated spherical hyperelastic
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membrane. International Journal of Non-Linear Mechanics. 57, 130-139. Kumar, N., DasGupta, A., 2014. Contact mechanics and induced hysteresis at oscillatory contacts with adhesion. Langmuir. 30, 9107-9114.
AN US
Kumar, N., DasGupta, A., 2015. On the static and dynamic contact problem of an inflated spherical viscoelastic membrane. Journal of Applied Mechanics. 82, 121010. Lei, C., Chen, S., 2015. Adhesion of a gas-filled membrane on a stretched substrate.
M
International Journal of Solids & Structures. 69, 189-194.
Long, R., Hui, C. Y., 2012. Axisymmetric membrane in adhesive contact with rigid substrates:
ED
Analytical solutions under large deformation. International Journal of Solids & Structures. 49,
PT
672-683.
Long, R., Shull, K. R., Hui, C. Y., 2010. Large deformation adhesive contact mechanics of
CE
circular membranes with a flat rigid substrate. Journal of the Mechanics & Physics of Solids. 58,
AC
1225-1242.
Leonard, J. W., Verma, V. K., 1976. Double-curved element for Mooney-Rivlin membranes.
Journal of the Engineering Mechanics Division. 102, 625-641. Mooney, M., 1940. A theory of large elastic deformation. Journal of Applied Physics. 11, 582-592. Moretti, M., Prina-Mello, A., Reid, A. J., Barron, V., Prendergast, P. J., 2004. Endothelial cell 39
ACCEPTED MANUSCRIPT
alignment on cyclically-stretched silicone surfaces. Journal of Materials Science: Materials in Medicine. 15, 1159–1164. Menga, N., Afferrante, L., Carbone, G., 2016. Effect of thickness and boundary conditions on the behavior of viscoelastic layers in sliding contact with wavy profiles. Journal of the Mechanics
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& Physics of Solids. 95, 517-529. McGarry, G. J., Prendergast, P. J., 2004. A three-dimensional finite element model of an adherent eukaryotic cell. European cells & materials. 7, 27–34.
AN US
Oden, J. T., 1972. Finite elements of nonlinear continua. New York.
Patil, A., DasGupta, A., Eriksson, A., 2015. Contact mechanics of a circular membrane inflated against a deformable substrate. International Journal of Solids & Structures. 67, 250-262.
M
Patil, A., DasGupta, A., 2013. Finite inflation of an initially stretched hyperelastic circular membrane. European Journal of Mechanics-A/Solids. 41, 28-36.
ED
Patil A, DasGupta A., 2015. Constrained inflation of a stretched hyperelastic membrane inside
PT
an elastic cone. Meccanica. 50, 1495-1508. Sohail, T., Tang, T., Nadler, B., 2013. Adhesive Contact of a Fluid-filled Membrane Driven by
CE
Electrostatic Forces. International Journal of Solids & Structures. 50, 2678-2690.
AC
Tsang, P. H., Li, G., Brun, Y. V., Freund, L. B., Tang, J. X., 2006. Adhesion of single bacterial
cells in the micro-Newton range. Proceedings of the National Academy of Sciences. 103, 5764– 5768.
Tielking, J. T., Feng, W. W., 1974. The application of the minimum potential energy principle to nonlinear axisymmetric membrane problems. Journal of Applied Mechanics. 41, 491–496. Vasu, T. S., Bhandakkar, T. K., 2016. A Study of the Contact of an Elastic Layer–Substrate 40
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System Indented by a Long Rigid Cylinder Incorporating Surface Effects. Journal of Applied Mechanics. 83, 06. Wan, K. T., Liu, K. K., 2001. Contact mechanics of a thin-walled capsule adhered onto a rigid planar substrate. Medical and Biological Engineering and Computing. 39, 605–608.
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Wang Y., Lava P., Coppieters S., et al., 2012. Investigation of the uncertainty of DIC under heterogeneous strain states with numerical tests. Strain. 48, 453-462.
Xu, D., Liechti, K. M., 2010. Analytical and experimental study of a circular membrane in
AN US
Hertzian contact with a rigid substrate. International Journal of Solids & Structures. 47, 969-977. Xu, D., Liechti, K. M., 2011. Analytical and experimental study of a circular membrane in adhesive contact with a rigid substrate. International Journal of Solids & Structures. 48, 2965-2976.
M
Yang, W. H., Feng, W. W., 1970. On axisymmetrical deformations of nonlinear membranes.
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CE
PT
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Journal of Applied Mechanics. 37, 1002–1011.
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Graphical abstract
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