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On the contact problem of an inflated spherical hyperelastic membrane
International Journal of Non-Linear Mechanics 57 (2013) 130–139
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On the contact problem of an inflated spherical hyperelastic membrane Nirmal Kumar n, Anirvan DasGupta Department of Mechanical Engineering and Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
art ic l e i nf o
a b s t r a c t
Article history: Received 17 January 2013 Received in revised form 12 June 2013 Accepted 28 June 2013 Available online 5 July 2013
In this paper, the mechanics of contact of an inflated spherical non-linear hyperelastic membrane pressed between two rigid plates has been studied. We have considered the membrane material to be a homogeneous and isotropic Mooney–Rivlin hyperelastic solid. All three cases, namely frictionless, no-slip and stick–slip conditions have been considered separately in the plate-membrane contact region. The stretch of the membrane, and the surface traction (for no-slip contact) has been determined. For the stick–slip case, the sliding front is observed to be initiated at the contact periphery which moves towards the pole. The state at which the impending wrinkling condition occurs has been determined analytically. It is observed that the impending wrinkling state occurs at the periphery of the contact. Based on this, the minimum initial stretch (inflation) required to prevent wrinkling at any point in the membrane has been determined. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Hyperelastic membrane Large deformation Contact problem Friction Stick–slip Wrinkling
1. Introduction Closed inflated membranes are routinely used in airbags and suspensions for cushioning and absorbing shocks. Such applications fundamentally involve a variable contact between an inflated membrane and a rigid or flexible surface. The membrane has the characteristics of a non-linear spring. The contact problem is interesting and challenging due to large deformations which involve geometric non-linearity, material non-linearity and complex contact conditions due to friction and adhesion. This work is motivated by some of these fundamental issues in the mechanics of deformation of inflated membranes with contact. Some large deformation problems without contact can be found in [1–7]. Models for contact problem with or without friction and adhesion can be found in [8–22]. Modeling of membranes with wrinkling has been discussed in detail in [23–28]. An approach based on the conversion of a boundary value problem into an initial value problem for axisymmetric membrane problems has been developed by Yang and Feng [29]. This method has been extended by Patil and DasGupta [30] for the free inflation problem. Feng and Yang [31] investigated the deformation of pressurized spherical balloon kept between two large rigid plates considering Mooney–Rivlin model. However, they have not considered adhesion and friction between the membrane and the plate during contact. Another frictionless contact problem has
n
Corresponding author. Tel.: +91 933 384 6992; fax: +91 322 225 5303. E-mail addresses: [email protected], [email protected] (N. Kumar), [email protected] (A. DasGupta). 0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.015
been addressed by Nadler [32] in which the condition and domain of wrinkling has been analyzed assuming that the wrinkling can occur only in contacting region. Charrier and Shrivastava [33,34] have related thermoforming with inflation of axisymmetric and non-axisymmetric membranes against a rigid contact (cylindrical, conical and flat) with no-slip, and no-friction conditions. Recently, Long et al. [35] have considered the adhesive contact problem of circular membranes with a flat rigid plate. However, for no-slip condition they have used an approximated value of the infinitesimal increment in contact radius without considering the thermodynamic properties of the gas inside the membrane. In our problem, we have considered closed system instead of open system considered in [35]. In no-slip case, we found increment in contact radius by considering thermodynamic properties of the gas inside the membrane. The contact problem of an uninflated membrane with frictional sliding condition has been solved using the finite element method in [36,37]. In the existing literature, while free inflation problems have been well studied, the contact problem of inflated membranes has not been explored completely. In particular, contact problems with stick–slip contact has not been solved yet. Contact problems of an inflated membrane considered as a closed thermodynamic system remains to be addressed. Further, the occurrence of impending wrinkling condition in such problems is also of importance which has not been discussed in detail. In the present work, we are interested in understanding the mechanics of deformation of inflated membranes with contact. To keep the geometry simple, we have considered a completely spherical inflated membrane pressed symmetrically between two rigid plates. We have formulated the problem as a variational
N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
A1 A0
Γ
ψ Q0
131
Q1 Γ
ψ
rs
Γc
Ac
η
ρ
Q2
η
r0
Fig. 1. Geometry of spherical membrane before and after inflation and contact with two large rigid plates.
problem using the Mooney–Rivlin strain energy function for the membrane and the potential energy of the inflating gas. The deformation process is assumed to be isothermal. We have studied the problem with frictionless, no-slip and stick–slip contact conditions. The stick–slip case is actually a non-conservative problem, and does not have a variational formulation as such. However, as proposed in this paper, one can formulate a sequence of variational problems while checking and allowing for intermediate slip using the force equilibrium condition of the membrane at the contact. The stretches and contact traction (for no-slip contact) have been determined. The occurrence of the impending wrinkling state has also been determined analytically to find the limits of the solutions obtained with the present formulation.
The internal energy of the inflating gas can be written as Z 2π Z π=2 p 2 W p ¼ 2 η′ρ dψ dθ: 2 0 0
ð5Þ
The total potential energy function is given by Π ¼ UW p . 2.3. Governing equation In the following, we derive the governing equations for the non-contacting region ψϵ½Γ; π=2, and the contacting region ψϵ½0;p Γffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (discussed later) separately. Using a substitution ffi λ1 ¼ ðλ′2 sin ψ þ λ2 cos ψ 2 þ η′2 one can express the potential energy as a function of λ2 , λ2 ′ and η′. The governing equations for the non-contacting region can be obtained as
2. Problem formulation
d ∂Π ∂Π ¼ 0; dψ ∂λ′2 ∂λ2
Consider a spherical balloon of uninflated radius r0 and uniform thickness h0 (state I), which is inflated to a radius rs (state II) by a pressure p0. Only the upper half of the balloon is shown in Fig. 1. Consider two rigid plates coming in contact and pressing the inflated balloon symmetrically, as shown in Fig. 1 (state III). The material points of the membrane are parameterized through the angles θ (azimuthal angle) and ψ. The extent of contact is measured by the contact angle Γ c (see point Ac in Fig. 1 state III) in deformed membrane, which corresponds to the material points with ψ ¼ Γ (see points A0 and A1 in Fig. 1 states I and II, respectively). We assume that the problem is axisymmetric and the membrane material is homogeneous and isotropic. Further, all thermodynamic processes are assumed to be reversible and isothermal during inflation and contact.
Defining w ¼ η′, and v ¼ λ′2 r 0 sin ψ in Eq. (6), the governing equations for the non-contacting region read
2.1. Kinematics of deformation The principal stretch ratios for the membrane is written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 sin ψ ρ ρ′2 þ η′2 λ1 ¼ ð1Þ ; λ3 ¼ p0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; λ2 ¼ r 0 sin ψ r0 ρ ρ′2 þ η′2 where we have used the incompressibility condition λ1 λ2 λ3 ¼ 1. The prime denotes derivative with respect to ψ. The strain invariants in terms of the stretch ratios are given by þ
λ22
þ
λ23
and
1 1 1 I2 ¼ 2 þ 2 þ 2 λ1 λ2 λ3
ð2Þ
ð6Þ
v′ ¼ f 1 ðλ2 ; v; w; ψÞ;
ð7Þ
w′ ¼ f 2 ðλ2 ; v; w; p; ψÞ;
ð8Þ
v : r 0 sin ψ
ð9Þ
λ′2 ¼
The boundary conditions for the non-contacting region are vjψ ¼ Γ ¼ v0 ;
vjψ ¼ π=2 ¼ 0;
wjψ ¼ Γ ¼ 0;
λ2 jψ ¼ Γ ¼ λc ;
ð10Þ
where v0 and λc are as yet unknown. They will be decided from the junction condition discussed later. The pressure and volume inside the contact free spherical membrane are related to λs as ! h0 4 3 3 1 ; ð11Þ V 0 ¼ πr 0 λs ; p0 ¼ 4C 1 1 6 1 þ αλ2s 3 r 0 λs λs where λs ¼ r s =r 0 and V0 is volume of the spherical membrane after inflation and before contact. The volume inside the membrane after contact is given by (using Eq. (1)) Z η Z π=2 V ¼ 2π ρ2 dη ¼ 2π ðr 0 λ2 sin ψÞ2 η′dψ ð12Þ 0
I 1 ¼ λ21
d ∂Π ¼0 dψ ∂η′
Γ
where η is vertical distance of the rigid plate from the equator of the balloon after contact. Assuming isothermal compression of the gas. The pressure after contact may be written as p ¼ p0 V 0 =V.
2.2. Potential energy functional 2.4. Contact conditions For a Mooney–Rivlin material, the strain energy density function may be written as U^ ¼ C 1 ðI 1 3Þ þ C 2 ðI 2 3Þ ¼ C 1 ½ðI 1 3Þ þ αðI 2 3Þ;
ð3Þ
where C1 and C2 are material constants and α ¼ C 2 =C 1 . The strain energy of the membrane is then obtained as Z Z 2π Z π=2 U^ sin ψ dψ dθ: ð4Þ U ¼ U^ dV ¼ 2r 20 h0 V
0
0
We have considered three types of contact conditions between the plate and the membrane, namely frictionless contact, no-slip contact and stick–slip contact. In the frictionless contact, the material points of the membrane in the contacting region can slide freely over the plate. The governing equations are considered separately in the non-contacting and contacting regions. In the contacting region, the pressure work Wp vanishes because w¼0 throughout. Hence, the governing equation in this region is given
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by d ∂U ∂U ¼ 0: dψ ∂λ′2 ∂λ2
ð13Þ
This is a second order ordinary differential equation. We convert this equation into two first order ordinary differential equations using a substitution v ¼ λ′2 r 0 sin ψ to obtain v′ ¼ f c1 ðλ2 ; v; ψ Þ;
λ′2 ¼
v : r 0 sin ψ
ð14Þ
The initial conditions for this problem are given by vjψ ¼ 0 ¼ 0;
λ2 j ψ ¼ 0 ¼ λ0 :
ð15Þ
The governing equation for the non-contacting region is given in Eqs. (7), (8) and (9). The initial conditions for this region are obtained from the solution of Eq. (14). In the case of no-slip contact, the surface of the plate provides zero displacement kinematic constraint for all contacting material points of the membrane. This requires sufficient friction to be present between the membrane and the plate. One can determine the required coefficient of friction as follows. Let, T1 and T2 be the principal stress resultants in the meridional and circumferential directions, respectively. These stress resultants are related to the principal stretches as ! λ1 1 T 1 ¼ 2h0 C 1 ð16Þ 1 þ αλ22 ; λ2 λ31 λ32 T 2 ¼ 2h0 C 1
! λ2 1 3 3 1 þ αλ21 ; λ1 λ1 λ2
"
! λ1 3 þ 3 4 1 þ αλ22 þ λ2 λ1 λ2
! # λ1 1 3 3 2αλ2 λ2 λ1 λ2
ð19Þ
In the contacting region of the membrane, the traction τ along the meridional direction is given by dT 1 1 dT 1 dT 1 1 1 þ ðT 1 T 2 Þ; þ ðT 1 T 2 Þ ¼ λ′1 þ λ′2 ð20Þ τ¼ ρ ρ′ ρ dρ dλ1 dλ2 where (since w¼0 in contacting region) λ1 ¼ λ′2 sin ψ þ λ2 cos ψ:
ð21Þ
If μ is the minimum required coefficient of friction for no-slip condition, then μ¼
jτj : p
slip case is written as v′ ¼ f s λ2 ; v; μf p; ψ ; λ′2 ¼
ð22Þ
If the coefficient of friction between the plate and membrane is less than μ then the traction τ exceeds the maximum frictional resistance. Consequently, the stick–slip condition arises which is the most practical contact condition. The governing equation of the membrane in the equilibrium sliding (or impending sliding) state is derived using the free body diagram shown in Fig. 2 as dT 1 dT 1 1 1 þ ðT 1 T 2 Þ þ μf p ¼ 0; λ′1 þ λ′2 ð23Þ ρ′ ρ dλ1 dλ2 where μf is the coefficient of friction between the plate and membrane. It may be noted that the direction of frictional force is taken away from the pole, since it is found (see also [31]) that the material points in the contacting region move towards the pole (in frictionless condition) as the contact proceeds. Using Eqs. (16)–(19), (21) and v ¼ λ′2 sin ψ in Eq. (23), the governing equations of the membrane in the contacting region for the stick–
v : r 0 sin ψ
ð24Þ
The governing equations in non-contacting region remain the same as in the frictionless and no-slip cases.
3. Solution procedure 3.1. Frictionless contact The initial conditions for the non-contacting region are obtained from the junction conditions at ψ ¼ Γ as ðvÞnc ¼ ðvÞc ;
ð17Þ
where h0 is the thickness of the undeformed membrane. Hence, ! dT 1 1 3 ¼ 2h0 C 1 þ ð18Þ 1 þ αλ22 ; λ2 λ41 λ32 dλ1 dT 1 ¼ 2h0 C 1 dλ2
Fig. 2. Free body diagram of an elementary part of the membrane in contacting region.
ðλ2 Þnc ¼ ðλ2 Þc ;
ð25Þ
where the subscripts nc and c refer to, respectively, the noncontacting and contacting regions. We assigned value of initial stretch λs and contact angle Γ C . Initially, we take some guesses, p ¼ p0 . We begin the solution procedure by assigning a value of Γ c and λs , and assuming Γ ¼ Γ c , p ¼ p0 and λ0 ¼ λs . With the initial conditions Eq. (15) we integrate Eq. (14) (contacting region) from ψ ¼ 0 to ψ ¼ Γ. Using the junction condition Eq. (25), we integrate Eqs. (7), (8) and (9) (non-contacting region) from ψ ¼ Γ to ψ ¼ π=2. In general the boundary conditions vðπ=2Þ ¼ 0 and pV ¼ p0 V 0 are not satisfied, and hence, we iteratively adjust p and λ0 to satisfy the corresponding boundary condition. The height of the material Rψ points in non-contacting region is obtained as η ¼ π=2 w dψ. Therefore, the height of the plate from equator is η ¼ ηjψ ¼ Γ . Finally, we adjust Γ so that tan 1 ðρc =ηÞ ¼ Γ c , where ρc is the contact radius, which is calculated as ρc ¼ ðr 0 λ2 sin ψÞjψ ¼ Γ . 3.2. No-slip contact In the no-slip contact condition, the contact area depends on the history of contact formation. Therefore, we determine the solution incrementally as follows. The governing equations for non-contacting region are given by Eqs. (7)–(9). Initially, we set the contact angle Γ ¼ 0 and the pressure p ¼ p0 . Therefore, at the pole, vjψ ¼ 0 ¼ 0;
wjψ ¼ 0 ¼ 0;
λ2 jψ ¼ 0 ¼ λs :
ð26Þ
Considering a small increment ΔΓ due to pressing, we integrate Eqs. (7)–(9) from ψ ¼ Γ þ ΔΓ to ψ ¼ π=2 with the initial conditions defined from the guess value for the variables λ2 and v as λþ 2 ðΓ þ þ ΔΓÞ ¼ λ 2 ðΓ þ ΔΓÞ and v ðΓ þ ΔΓÞ ¼ v ðΓ þ ΔΓÞ. Due to contact geometry wþ ðΓ þ ΔΓÞ ¼ 0. Here and + signs indicate the variable values before and after contact, respectively. We adjust the pressure value p to satisfy the boundary condition vðπ=2Þ ¼ 0. The isothermal condition pV ¼ p0 V 0 is satisfied by adjusting þ þ the value of λþ 2 ðΓ þ ΔΓÞ as λ2 ðΓ þ ΔΓÞ ¼ λ2 ðΓ þ ΔΓÞð1γc2 Þ and þ þ þ v ðΓ þ ΔΓÞ is calculated as v ðΓ þ ΔΓÞ ¼ ðλþ 2 ðΓ þ ΔΓÞλ2 ðΓÞÞr 0 sin ðΓ þ ΔΓÞ=ΔΓ. Where c2 ¼ pV=p0 V 0 1. We keep increasing Γ until tan 1 ðρc =ηÞ ¼ Γ c .
N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
3.3. Stick–slip contact Let the coefficient of friction be μf . We first solve the deformation problem using the procedure used for no-slip condition up to the contact angle Γ ¼ Γ f where the required value of μ (calculated from Eq. (22)) is μf . For further deformation, the solution procedure is given as a flow chart in Fig. 3. In this flow chart, ψ ¼ Γ s is the last material point to be in contact and ψ ¼ Γ st is the last material point which slides. The subscripts ns and ss refer to, respectively, the no-slip condition and stick–slip condition. The flow chart contains three modules, namely Module-1, Module-2 and Module-3 which are elaborated in Figs. 4–6, respectively. In Module-1 integration has been carried out from ψ ¼ Γ f to ψ ¼ π=2 for stick–slip contact condition. In the flow chart in Fig. 3, the lines and boxes in magenta color form the first loop in which the gas pressure p inside the membrane is modified to satisfy the boundary condition vðπ=2Þ ¼ 0. The lines and boxes of blue color form the second loop in which the isothermal condition ðpV ¼ p0 V 0 Þ is satisfied by adjusting vjψ ¼ Γf . During this adjustment, the condition ðρ′jψ ¼ Γf Þns ≥ðρ′jψ ¼ Γf Þss is to be satisfied since the material points tend to slide towards the pole leading to a reduction in the value of ρ′ at ψ ¼ Γ f . If the adjustment of vjψ ¼ Γf at a point does not satisfy the impending sliding/no-slip condition, it implies that the slip front has progressed further towards the center of the membrane. Consequently, we progress to an interior contact point (by setting Γ f ¼ Γ f ΔΓ) to adjust vjψ ¼ Γf . In case the slip front moves up to the pole (i.e., the conditional statement in the red box in flow chart shown in Fig. 3 is not satisfied), we switch to Module-2 which handles the condition when all the
133
material points in contacting region slide right up to the pole. In Module-2, the condition λ0 ≥λs implies that the no-slip condition arises right after the material point at ψ ¼ Γ st . Module-3 captures the recurrence of the no-slip condition. In Module-3, the failure
Fig. 4. Flow chart of Module-1.
Fig. 3. Flow chart for stick–slip condition. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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Fig. 5. Flow chart of Module-2.
of condition μ≤μf (red colored decision box shown in Fig. 6) signifies the recurrence of the stick–slip condition.
3.4. Wrinkling
ð29Þ
which has the solution
A membrane reaches an impending wrinkling state when either T1 or T2 (Eqs. (16) and (17)) vanishes [23]. In the no-slip case, the material points in the contacting region are not allowed to move. Therefore, T1 and T2 can vanish independently, contrary to that in the frictionless case (as shown in [32]). We consider these two cases separately. If we assume that the membrane reaches the impending wrinkling state in the meridional direction, i.e., T 1 ¼ 0, Eq. (16) simplifies (using Eq. (21)) to 1 1 pffiffiffiffiffi λ2 cos ψ : λ′2 ¼ ð27Þ sin ψ λ2 The solution of this first order differential equation is 0 12=3 3 2 1 π @ 5 4 ψ j2 þ 1:7972103521033889A ; λ2 ¼ cosec ψ 3E 2 2 ð28Þ where E½ϕjm is the elliptic integral of second kind with amplitude 2 ϕ and modulus m ¼ k and is defined as Z ϕ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1k sin t dt: E½ϕjm ¼ 0
On the other hand if we assume T 2 ¼ 0, Eq. (17) reads ! 1 1 λ cos ψ : λ′2 ¼ 2 sin ψ λ22
λ2 ¼ cosec ψð1:5ψ0:75 sin 2ψÞ1=3 :
ð30Þ
The analytical solutions (28) and (30) (shown graphically in Fig. 7) provide us the values of the circumferential stretch λ2 at any angle ψ which if reached will lead to wrinkling in the material at that angular location. As observed in the figure, since the limiting value of λ2 corresponding to the condition T 1 ¼ 0 is lower for all ψ, it is more critical. Thus, we expect to see circumferential (axisymmetric) wrinkles appearing first. 4. Results and discussion We consider membrane materials with α ¼ 0, 0.1 and 0.2. In the following, we use the non-dimensional pressure P r ¼ pr 0 =C 1 h0 . The relation between the spherical membrane stretch λs and the initial inflating pressure p0 is given by Eq. (11), and shown in Fig. 8. We study the deformation process for two values of the initial stretch λs ¼ 1:3 and λs ¼ 3 which are, respectively, in the precritical and post-critical stretch regions of the curves shown in Fig. 8. We have taken β ¼ 102 , γ ¼ 102 and ϵ1 ¼ ϵ2 ¼ 104 , while ΔΓ is chosen as 103 radian (see Section 3). Integration of the governing equations has been performed using the Runge–Kutta method in all frictionless, no-slip and stick–slip cases.
N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
135
Fig. 6. Flow chart Module-3. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
1.5
5
1.4
4
1.3
3
1.2
2
1.1 1
1 0
12
6
4
3
5 12
0
2
1
2
3
4
5
) (for
Fig. 8. Pressure ratio versus λs for different values of α ¼ 0(−−−−−), 0.1 (− − −) and 0.2(————).
Fig. 9 shows the stretch ratios ðλ1 ; λ2 Þ for the frictionless and noslip cases for different contact angles Γ c . In the frictionless case, as the contact proceeds, the stretch ratios ðλ1 ; λ2 Þ decrease near the pole. This reveals that the material points of the membrane in the contacting region move towards the pole. The points encircled in red in Fig. 9 are the points where the contact ends. At these points, the profile of the meridional stretch ðλ1 Þ changes its behavior. This is because, the
curvature in the meridional direction goes from zero in the contacting region to a positive value in the non-contacting region. The motion of the material points in the contacting region in the frictionless contact case has interesting implications that is observed in Fig. 10, which compares the contact profile for the frictionless and no-slip cases. For each values of Γ c , the pressure ratio (Pr) in the no-slip case has a slightly lower value than that in
Fig. 7. Variation of λ2 with ψ for the solution given by Eq. (28) ( T 1 ¼ 0) and Eq. (30) (−−−−−) (for T 2 ¼ 0).
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3.4
3.4
3.2
3.2
3.0
3.0
2.8
2.8
2.6
2.6
2.4
0
12
6
4
3
5 12
2.4 2
0
12
6
4
5 12
3
2
Fig. 9. Principal stretch ratios λ1 ( ) and λ2 (−−−−−) of an inflated spherical balloon after contact (λs ¼ 3 and α ¼ 0:2). (a) Frictionless case and (b) No-slip case. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
2
2.5
3
0
3.5
0
0.5
1
1.5
2
2.5
3
3.5
Fig. 10. Profiles of spherical membrane before and after contact for different contact angles Γ c ¼ 01 (————), 151 (− − −), 301(−−−−−), 451 ( (α ¼ 0:2 and λs ¼ 3). Stick–slip condition profile in which Pr ¼ 5.326, for μf ¼ 0:122 and Γ c ¼ 601 (- - - - -). (a) Frictionless case. (b) No-slip case.
) and 601 (
)
0 0.6 0.5 0.4
1
0.2
1.5
2
0
12
6
4
3
0
0
12
6
4
3
Fig. 11. Surface traction and minimum required coefficient of friction of spherical membrane after contact [fλs ¼ 3; α ¼ 0gð Þ, fλs ¼ 3; α ¼ 0:2gð−−−−−Þ, fλs ¼ 1:3; α ¼ 0g; ð− − −Þ, fλs ¼ 1:3; α ¼ 0:2gðÞ]. (a) Traction force along the meridional direction. (b) Minimum required coefficient of friction.
N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
3.0
137
3.6 3.4
2.8
3.2 3.0
2.6
2.8 2.6
2.4
2.4 2.2
0
12
6
4
3
5 12
2.2 2
0
Fig. 12. Principal stretch ratios in stick–slip condition for Γ s ¼ π=3 and μf ¼ 0ðÞ, 0.0267 ( slip (————) (α ¼ 0:2 and λs ¼ 3).
12
6
4
5 12
3
2
), 0.0788 (− − −), 0.1025(−−−−−) and no-
), 0.0475(
3.6
Table 1 Critical coefficient of friction μfc . α
λs
μfc
0 0 0.2 0.2
1.3 3 1.3 3
0.313 0.262 0.256 0.103
3.4 3.2 3.0 2.8
frictionless case. In the frictionless contact case, since the material points in the contacting region move towards the pole, the effective volume is reduced leading to slightly higher pressure. In stick–slip condition, pressure value lies between the pressure value found in no-slip and frictionless condition (shown in Fig. 10). The traction force τ defined in Eq. (20) is plotted in Fig. 11(a). It is observed that the maximum magnitude of the traction force is higher for the higher value of α. It is interesting to note that, for higher initial stretch λs , the magnitude of the traction force is lower. The minimum required coefficient of friction μ defined in Eq. (22) is shown in Fig. 11(b). It may be noted in this figure that, even though the surface traction magnitude is low for the case of λs ¼ 3; α ¼ 0, the required coefficient of friction for no-slip is high. This is because, this point is in the post-critical regime where the inflation pressure in the membrane is low (see Fig. 8). We have incorporated frictional sliding for different values of λs and α. The material points in the contacting region do not slide in the no-slip condition, and slide freely in the frictionless condition. Therefore, the results for stretches found in the stick–slip condition lie between the no-slip and the frictionless conditions, as shown in Fig. 12. For the results corresponding to the stick–slip case in this figure, the values of μf are taken small enough for the frictional sliding to occur throughout the contacting region (see Section 3.3). In other words, the sliding front reaches the pole of the membrane. In the stick–slip condition, as we increase the value of μf the stretch ratio at the vicinity of pole ðλ0 Þ goes near to λs . The value of μf for which the condition λ0 ¼ λs is achieved is said to be the critical coefficient of friction μfc . If μf 4μfc then sliding does not occur throughout the contacting region for the value of Γ s ≤π=3. In this case, the sliding front is arrested, and cannot reach the pole. We have calculated the values of μfc for different values of λs and α, some of which are given in Table 1. In Fig. 13, the stretch ratios of the membrane after contact for stick–slip condition is shown and compared with the stretch ratios for the no-slip condition. In this result μf ¼ 0:122 4 μfc ðΓ f ¼ 0:08 radÞ,
2.6 2.4 2.2
0
12
6
4
3
5 12
2
Fig. 13. Principal stretch ratios in stick–slip condition for μf ¼ 0:122 [λ1 ð−−−−−Þ, λ2 ð Þ] and for no-slip condition [λ1 ðÞ; λ2 ðÞ], in which Γ s ¼ π=3.
and the no-slip condition recurs (see Section 3.3) at ψ ¼ 0:965 rad when the sliding front is arrested. We determine the state of impending wrinkling by solving the contact problem for no-slip condition for λs ¼ 1:1; 1:2; 1:3 and α ¼ 0, 0.2. In these solutions, the contact angle proceeds from Γ ¼ 0 up to the value where the wrinkling just sets in. This corresponds to the contact angle where ðλ2 Þc touches the λ2 solution (for T 1 ¼ 0) curve (given by Eq. (28)), as shown in Fig. 14. It is observed that, in all cases, the critical value of λ2 is reached at the edge of the contact. Hence, the wrinkling will always occur first at the periphery of the membrane-plate contact. The conditions for impending wrinkling are tabulated in Table 2. From Fig. 14, it is interesting to observe that the value of λ2 remains almost constant in the contacting region. Therefore, for λ2 jψ ¼ 0 ¼ λs ≥1:5, we will not have any wrinkling in the membrane. The corresponding initial inflation pressure can be determined from Eq. (11). 5. Outlook The present work deals with the contact problem of an inflated spherical membrane pressed symmetrically between two rigid
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Table 2 Conditions for wrinkling. α
λs
Γ ðradÞ
Γ c ðradÞ
ðλ2 Þc
Pr
0 0 0 0.2 0.2 0.2
1.1 1.2 1.3 1.1 1.2 1.3
0.807 1.103 1.31 0.805 1.083 1.311
0.963 1.255 1.394 0.940 1.234 1.4
1.10006 1.19993 1.30034 1.09953 1.19171 1.3009
1.933 4.115 7.91044 2.406 5.1334 10.883
1.5
1.5
1.4
1.4
1.3
1.3
1.2
1.2
1.1
1.1
1
0
12
6
4
3
5 12
1 2
0
12
6
4
3
5 12
2
Fig. 14. Variation of λ2 in contacting region (− − −) and in non-contacting region (−−−−−) with ψ up to the impending wrinkling state. Variation of λ2 with ψ for the solution given by Eq. (28) ( ). (a) For α ¼ 0. (b) For α ¼ 0:2.
plates. Three possible contact conditions, namely frictionless, noslip and stick–slip conditions have been considered. For the former two idealized contact conditions, the problem is conservative. In the case of the stick–slip contact, the problem can be formulated as a sequence of variational problems with intermediate slip, which is governed by the force equilibrium equation of the membrane in the contacting region. The main observations are summarized below. 1. In the frictionless contact problem, the principal stretches at the vicinity of the pole decrease as the contact proceeds. This phenomenon indicates that, the material points in the contacting region move towards the pole. 2. The deformed profiles for the frictionless and no-slip cases do not differ significantly. However, in the no-slip case, the pressure is less than that in the frictionless case for the same contact angle condition. In the frictionless case, the motion of the material points towards the pole tends to reduce the volume thereby increasing the pressure. The results found in stick–slip condition lie between the results of no-slip and frictionless conditions. 3. The minimum required coefficient of friction has been estimated for no-slip. In the case of the stick–slip contact, the minimum coefficient of friction that arrests the progress of sliding front for a given level of pressing has also been estimated. 4. The condition for occurrence of wrinkling has been determined analytically. It is observed that for λs 4 1:5, wrinkling will not occur anywhere in the contacting region. Based on the formulation presented in this work, one can obtain some effective properties of the spherical membrane under
contact. The dynamic response of a membrane with contact would be a challenging problem for the future.
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[15] W.W. Feng, J.T. Tielking, P. Huang, The inflation and contact constraint of a rectangular Mooney membrane, Journal of Applied Mechanics 41 (1974) 979–984. [16] P.V.M. Rao, S.G. Dhande, Deformation analysis of thin elastic membranes in multiple contact, Journal of Advance in Engineering Software 30 (1999) 177–183. [17] M. Shillor, M. Sofonea, J.J. Telega, Analysis of viscoelastic contact with normal compliance, friction and wear diffusion, Journal of Comptes rendus Mecanique 331 (2003) 395–400. [18] W. Szyszkowski, P.G. Glockner, Finite deformation and stability behaviour of spherical inflatables under axisymmetric concentrated loads, International Journal of Non-Linear Mechanics 19 (1984) 489–496. [19] W. Szyszkowski, P.G. Glockner, Spherical membranes subjected to concentrated loads, Journal of Engineering Structures 9 (1987) 45–52. [20] W.H. Yang, K.H. Hsu, Indentation of a circular membrane, Journal of Applied Mechanics 38 (1987) 227–230. [21] A. Konyukhov, K. Schweizerhof, Contact formulation via a velocity description allowing efficiency improvement in frictionless contact condition, Computational Mechanics 33 (2004) 165–173. [22] A. Konyukhov, K. Schweizerhof, Covariant description for frictionless contact problem, Computational Mechanics 35 (2005) 190–213. [23] A.C. Pipkin, The relaxed energy density for isotropic elastic membranes, IMA Journal of Applied Mathematics 36 (1) (1986) 85–99. [24] D.J. Steigmann, A.C. Pipkin, Axisymmetric tension field, Journal of Applied Mathematics 42 (1989) 427–440. [25] D.J. Steigmann, A.C. Pipkin, Finite deformation of wrinkled membranes, Journal of Applied Mechanics and Physics (ZAMP) 40 (1989) 526–542. [26] D.J. Steigmann, Tension-field theory, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 429 (1876) (1990) 141–173.
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[27] E.M. Haseganu, D.J. Steigmann, Analysis of partly wrinkled membranes by the method of dynamic relaxation, Computational Mechanics 14 (6) (1994) 596–614. [28] Y.W. Wong, S. Pellegrino, Wrinkled Membranes, Textiles Composites and Inflatable Structures, CIMNE, Barcelona, 2003. [29] W.H. Yang, W.W. Feng, On axisymmetrical deformations of nonlinear membranes, Journal of Applied Mechanics 37 (4) (1970) 1002–1011. [30] A. Patil, A. DasGupta, Finite inflation of an initially stretched hyperelastic circular membrane, European Journal of Mechanics/A Solids 41 (2013) 28–36. [31] W.W. Feng, W.H. Yang, On the contact problem of an inflated spherical nonlinear membrane, Journal of Applied Mechanics 41 (1973) 209–214. [32] B. Nadler, On the contact of a spherical membrane enclosing a fluid with rigid parallel planes, International Journal of Non-Linear Mechanics 45 (3) (2010) 294–300. [33] J.M. Charrier, S. Shrivatava, Free and constrained inflation of elastic membranes in relation to thermoforming non-axisymmetric problem, The Journal of Strain Analysis for Engineering Design 22 (1987) 115–125. [34] J.M. Charrier, S. Shrivatava, Free and constrained inflation of elastic membranes in relation to thermoforming axisymmetric problem, The Journal of Strain Analysis for Engineering Design 24 (1989) 55–74. [35] R. Long, K.R. Shull, C.Y. Hui, Large deformation adhesive contact mechanics of circular membrane with flat rigid substrate, Journal of the Mechanics and Physics of Solid 58 (2010) 1242–12215. [36] K.J. Bathe, A. Chaudhary, A solution method for planar and axisymmetric contact problems, International Journal for Numerical Method in Engineering 21 (1985) 65–88. [37] A.P.S. Selvadurai, Q. Yu, On the indentation of a polymeric membrane, Proceeding of Royal Society A 462 (2006) 189–209.
Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
On the contact problem of an inflated spherical hyperelastic membrane Nirmal Kumar n, Anirvan DasGupta Department of Mechanical Engineering and Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
art ic l e i nf o
a b s t r a c t
Article history: Received 17 January 2013 Received in revised form 12 June 2013 Accepted 28 June 2013 Available online 5 July 2013
In this paper, the mechanics of contact of an inflated spherical non-linear hyperelastic membrane pressed between two rigid plates has been studied. We have considered the membrane material to be a homogeneous and isotropic Mooney–Rivlin hyperelastic solid. All three cases, namely frictionless, no-slip and stick–slip conditions have been considered separately in the plate-membrane contact region. The stretch of the membrane, and the surface traction (for no-slip contact) has been determined. For the stick–slip case, the sliding front is observed to be initiated at the contact periphery which moves towards the pole. The state at which the impending wrinkling condition occurs has been determined analytically. It is observed that the impending wrinkling state occurs at the periphery of the contact. Based on this, the minimum initial stretch (inflation) required to prevent wrinkling at any point in the membrane has been determined. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Hyperelastic membrane Large deformation Contact problem Friction Stick–slip Wrinkling
1. Introduction Closed inflated membranes are routinely used in airbags and suspensions for cushioning and absorbing shocks. Such applications fundamentally involve a variable contact between an inflated membrane and a rigid or flexible surface. The membrane has the characteristics of a non-linear spring. The contact problem is interesting and challenging due to large deformations which involve geometric non-linearity, material non-linearity and complex contact conditions due to friction and adhesion. This work is motivated by some of these fundamental issues in the mechanics of deformation of inflated membranes with contact. Some large deformation problems without contact can be found in [1–7]. Models for contact problem with or without friction and adhesion can be found in [8–22]. Modeling of membranes with wrinkling has been discussed in detail in [23–28]. An approach based on the conversion of a boundary value problem into an initial value problem for axisymmetric membrane problems has been developed by Yang and Feng [29]. This method has been extended by Patil and DasGupta [30] for the free inflation problem. Feng and Yang [31] investigated the deformation of pressurized spherical balloon kept between two large rigid plates considering Mooney–Rivlin model. However, they have not considered adhesion and friction between the membrane and the plate during contact. Another frictionless contact problem has
n
Corresponding author. Tel.: +91 933 384 6992; fax: +91 322 225 5303. E-mail addresses: [email protected], [email protected] (N. Kumar), [email protected] (A. DasGupta). 0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.015
been addressed by Nadler [32] in which the condition and domain of wrinkling has been analyzed assuming that the wrinkling can occur only in contacting region. Charrier and Shrivastava [33,34] have related thermoforming with inflation of axisymmetric and non-axisymmetric membranes against a rigid contact (cylindrical, conical and flat) with no-slip, and no-friction conditions. Recently, Long et al. [35] have considered the adhesive contact problem of circular membranes with a flat rigid plate. However, for no-slip condition they have used an approximated value of the infinitesimal increment in contact radius without considering the thermodynamic properties of the gas inside the membrane. In our problem, we have considered closed system instead of open system considered in [35]. In no-slip case, we found increment in contact radius by considering thermodynamic properties of the gas inside the membrane. The contact problem of an uninflated membrane with frictional sliding condition has been solved using the finite element method in [36,37]. In the existing literature, while free inflation problems have been well studied, the contact problem of inflated membranes has not been explored completely. In particular, contact problems with stick–slip contact has not been solved yet. Contact problems of an inflated membrane considered as a closed thermodynamic system remains to be addressed. Further, the occurrence of impending wrinkling condition in such problems is also of importance which has not been discussed in detail. In the present work, we are interested in understanding the mechanics of deformation of inflated membranes with contact. To keep the geometry simple, we have considered a completely spherical inflated membrane pressed symmetrically between two rigid plates. We have formulated the problem as a variational
N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
A1 A0
Γ
ψ Q0
131
Q1 Γ
ψ
rs
Γc
Ac
η
ρ
Q2
η
r0
Fig. 1. Geometry of spherical membrane before and after inflation and contact with two large rigid plates.
problem using the Mooney–Rivlin strain energy function for the membrane and the potential energy of the inflating gas. The deformation process is assumed to be isothermal. We have studied the problem with frictionless, no-slip and stick–slip contact conditions. The stick–slip case is actually a non-conservative problem, and does not have a variational formulation as such. However, as proposed in this paper, one can formulate a sequence of variational problems while checking and allowing for intermediate slip using the force equilibrium condition of the membrane at the contact. The stretches and contact traction (for no-slip contact) have been determined. The occurrence of the impending wrinkling state has also been determined analytically to find the limits of the solutions obtained with the present formulation.
The internal energy of the inflating gas can be written as Z 2π Z π=2 p 2 W p ¼ 2 η′ρ dψ dθ: 2 0 0
ð5Þ
The total potential energy function is given by Π ¼ UW p . 2.3. Governing equation In the following, we derive the governing equations for the non-contacting region ψϵ½Γ; π=2, and the contacting region ψϵ½0;p Γffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (discussed later) separately. Using a substitution ffi λ1 ¼ ðλ′2 sin ψ þ λ2 cos ψ 2 þ η′2 one can express the potential energy as a function of λ2 , λ2 ′ and η′. The governing equations for the non-contacting region can be obtained as
2. Problem formulation
d ∂Π ∂Π ¼ 0; dψ ∂λ′2 ∂λ2
Consider a spherical balloon of uninflated radius r0 and uniform thickness h0 (state I), which is inflated to a radius rs (state II) by a pressure p0. Only the upper half of the balloon is shown in Fig. 1. Consider two rigid plates coming in contact and pressing the inflated balloon symmetrically, as shown in Fig. 1 (state III). The material points of the membrane are parameterized through the angles θ (azimuthal angle) and ψ. The extent of contact is measured by the contact angle Γ c (see point Ac in Fig. 1 state III) in deformed membrane, which corresponds to the material points with ψ ¼ Γ (see points A0 and A1 in Fig. 1 states I and II, respectively). We assume that the problem is axisymmetric and the membrane material is homogeneous and isotropic. Further, all thermodynamic processes are assumed to be reversible and isothermal during inflation and contact.
Defining w ¼ η′, and v ¼ λ′2 r 0 sin ψ in Eq. (6), the governing equations for the non-contacting region read
2.1. Kinematics of deformation The principal stretch ratios for the membrane is written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 sin ψ ρ ρ′2 þ η′2 λ1 ¼ ð1Þ ; λ3 ¼ p0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; λ2 ¼ r 0 sin ψ r0 ρ ρ′2 þ η′2 where we have used the incompressibility condition λ1 λ2 λ3 ¼ 1. The prime denotes derivative with respect to ψ. The strain invariants in terms of the stretch ratios are given by þ
λ22
þ
λ23
and
1 1 1 I2 ¼ 2 þ 2 þ 2 λ1 λ2 λ3
ð2Þ
ð6Þ
v′ ¼ f 1 ðλ2 ; v; w; ψÞ;
ð7Þ
w′ ¼ f 2 ðλ2 ; v; w; p; ψÞ;
ð8Þ
v : r 0 sin ψ
ð9Þ
λ′2 ¼
The boundary conditions for the non-contacting region are vjψ ¼ Γ ¼ v0 ;
vjψ ¼ π=2 ¼ 0;
wjψ ¼ Γ ¼ 0;
λ2 jψ ¼ Γ ¼ λc ;
ð10Þ
where v0 and λc are as yet unknown. They will be decided from the junction condition discussed later. The pressure and volume inside the contact free spherical membrane are related to λs as ! h0 4 3 3 1 ; ð11Þ V 0 ¼ πr 0 λs ; p0 ¼ 4C 1 1 6 1 þ αλ2s 3 r 0 λs λs where λs ¼ r s =r 0 and V0 is volume of the spherical membrane after inflation and before contact. The volume inside the membrane after contact is given by (using Eq. (1)) Z η Z π=2 V ¼ 2π ρ2 dη ¼ 2π ðr 0 λ2 sin ψÞ2 η′dψ ð12Þ 0
I 1 ¼ λ21
d ∂Π ¼0 dψ ∂η′
Γ
where η is vertical distance of the rigid plate from the equator of the balloon after contact. Assuming isothermal compression of the gas. The pressure after contact may be written as p ¼ p0 V 0 =V.
2.2. Potential energy functional 2.4. Contact conditions For a Mooney–Rivlin material, the strain energy density function may be written as U^ ¼ C 1 ðI 1 3Þ þ C 2 ðI 2 3Þ ¼ C 1 ½ðI 1 3Þ þ αðI 2 3Þ;
ð3Þ
where C1 and C2 are material constants and α ¼ C 2 =C 1 . The strain energy of the membrane is then obtained as Z Z 2π Z π=2 U^ sin ψ dψ dθ: ð4Þ U ¼ U^ dV ¼ 2r 20 h0 V
0
0
We have considered three types of contact conditions between the plate and the membrane, namely frictionless contact, no-slip contact and stick–slip contact. In the frictionless contact, the material points of the membrane in the contacting region can slide freely over the plate. The governing equations are considered separately in the non-contacting and contacting regions. In the contacting region, the pressure work Wp vanishes because w¼0 throughout. Hence, the governing equation in this region is given
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by d ∂U ∂U ¼ 0: dψ ∂λ′2 ∂λ2
ð13Þ
This is a second order ordinary differential equation. We convert this equation into two first order ordinary differential equations using a substitution v ¼ λ′2 r 0 sin ψ to obtain v′ ¼ f c1 ðλ2 ; v; ψ Þ;
λ′2 ¼
v : r 0 sin ψ
ð14Þ
The initial conditions for this problem are given by vjψ ¼ 0 ¼ 0;
λ2 j ψ ¼ 0 ¼ λ0 :
ð15Þ
The governing equation for the non-contacting region is given in Eqs. (7), (8) and (9). The initial conditions for this region are obtained from the solution of Eq. (14). In the case of no-slip contact, the surface of the plate provides zero displacement kinematic constraint for all contacting material points of the membrane. This requires sufficient friction to be present between the membrane and the plate. One can determine the required coefficient of friction as follows. Let, T1 and T2 be the principal stress resultants in the meridional and circumferential directions, respectively. These stress resultants are related to the principal stretches as ! λ1 1 T 1 ¼ 2h0 C 1 ð16Þ 1 þ αλ22 ; λ2 λ31 λ32 T 2 ¼ 2h0 C 1
! λ2 1 3 3 1 þ αλ21 ; λ1 λ1 λ2
"
! λ1 3 þ 3 4 1 þ αλ22 þ λ2 λ1 λ2
! # λ1 1 3 3 2αλ2 λ2 λ1 λ2
ð19Þ
In the contacting region of the membrane, the traction τ along the meridional direction is given by dT 1 1 dT 1 dT 1 1 1 þ ðT 1 T 2 Þ; þ ðT 1 T 2 Þ ¼ λ′1 þ λ′2 ð20Þ τ¼ ρ ρ′ ρ dρ dλ1 dλ2 where (since w¼0 in contacting region) λ1 ¼ λ′2 sin ψ þ λ2 cos ψ:
ð21Þ
If μ is the minimum required coefficient of friction for no-slip condition, then μ¼
jτj : p
slip case is written as v′ ¼ f s λ2 ; v; μf p; ψ ; λ′2 ¼
ð22Þ
If the coefficient of friction between the plate and membrane is less than μ then the traction τ exceeds the maximum frictional resistance. Consequently, the stick–slip condition arises which is the most practical contact condition. The governing equation of the membrane in the equilibrium sliding (or impending sliding) state is derived using the free body diagram shown in Fig. 2 as dT 1 dT 1 1 1 þ ðT 1 T 2 Þ þ μf p ¼ 0; λ′1 þ λ′2 ð23Þ ρ′ ρ dλ1 dλ2 where μf is the coefficient of friction between the plate and membrane. It may be noted that the direction of frictional force is taken away from the pole, since it is found (see also [31]) that the material points in the contacting region move towards the pole (in frictionless condition) as the contact proceeds. Using Eqs. (16)–(19), (21) and v ¼ λ′2 sin ψ in Eq. (23), the governing equations of the membrane in the contacting region for the stick–
v : r 0 sin ψ
ð24Þ
The governing equations in non-contacting region remain the same as in the frictionless and no-slip cases.
3. Solution procedure 3.1. Frictionless contact The initial conditions for the non-contacting region are obtained from the junction conditions at ψ ¼ Γ as ðvÞnc ¼ ðvÞc ;
ð17Þ
where h0 is the thickness of the undeformed membrane. Hence, ! dT 1 1 3 ¼ 2h0 C 1 þ ð18Þ 1 þ αλ22 ; λ2 λ41 λ32 dλ1 dT 1 ¼ 2h0 C 1 dλ2
Fig. 2. Free body diagram of an elementary part of the membrane in contacting region.
ðλ2 Þnc ¼ ðλ2 Þc ;
ð25Þ
where the subscripts nc and c refer to, respectively, the noncontacting and contacting regions. We assigned value of initial stretch λs and contact angle Γ C . Initially, we take some guesses, p ¼ p0 . We begin the solution procedure by assigning a value of Γ c and λs , and assuming Γ ¼ Γ c , p ¼ p0 and λ0 ¼ λs . With the initial conditions Eq. (15) we integrate Eq. (14) (contacting region) from ψ ¼ 0 to ψ ¼ Γ. Using the junction condition Eq. (25), we integrate Eqs. (7), (8) and (9) (non-contacting region) from ψ ¼ Γ to ψ ¼ π=2. In general the boundary conditions vðπ=2Þ ¼ 0 and pV ¼ p0 V 0 are not satisfied, and hence, we iteratively adjust p and λ0 to satisfy the corresponding boundary condition. The height of the material Rψ points in non-contacting region is obtained as η ¼ π=2 w dψ. Therefore, the height of the plate from equator is η ¼ ηjψ ¼ Γ . Finally, we adjust Γ so that tan 1 ðρc =ηÞ ¼ Γ c , where ρc is the contact radius, which is calculated as ρc ¼ ðr 0 λ2 sin ψÞjψ ¼ Γ . 3.2. No-slip contact In the no-slip contact condition, the contact area depends on the history of contact formation. Therefore, we determine the solution incrementally as follows. The governing equations for non-contacting region are given by Eqs. (7)–(9). Initially, we set the contact angle Γ ¼ 0 and the pressure p ¼ p0 . Therefore, at the pole, vjψ ¼ 0 ¼ 0;
wjψ ¼ 0 ¼ 0;
λ2 jψ ¼ 0 ¼ λs :
ð26Þ
Considering a small increment ΔΓ due to pressing, we integrate Eqs. (7)–(9) from ψ ¼ Γ þ ΔΓ to ψ ¼ π=2 with the initial conditions defined from the guess value for the variables λ2 and v as λþ 2 ðΓ þ þ ΔΓÞ ¼ λ 2 ðΓ þ ΔΓÞ and v ðΓ þ ΔΓÞ ¼ v ðΓ þ ΔΓÞ. Due to contact geometry wþ ðΓ þ ΔΓÞ ¼ 0. Here and + signs indicate the variable values before and after contact, respectively. We adjust the pressure value p to satisfy the boundary condition vðπ=2Þ ¼ 0. The isothermal condition pV ¼ p0 V 0 is satisfied by adjusting þ þ the value of λþ 2 ðΓ þ ΔΓÞ as λ2 ðΓ þ ΔΓÞ ¼ λ2 ðΓ þ ΔΓÞð1γc2 Þ and þ þ þ v ðΓ þ ΔΓÞ is calculated as v ðΓ þ ΔΓÞ ¼ ðλþ 2 ðΓ þ ΔΓÞλ2 ðΓÞÞr 0 sin ðΓ þ ΔΓÞ=ΔΓ. Where c2 ¼ pV=p0 V 0 1. We keep increasing Γ until tan 1 ðρc =ηÞ ¼ Γ c .
N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
3.3. Stick–slip contact Let the coefficient of friction be μf . We first solve the deformation problem using the procedure used for no-slip condition up to the contact angle Γ ¼ Γ f where the required value of μ (calculated from Eq. (22)) is μf . For further deformation, the solution procedure is given as a flow chart in Fig. 3. In this flow chart, ψ ¼ Γ s is the last material point to be in contact and ψ ¼ Γ st is the last material point which slides. The subscripts ns and ss refer to, respectively, the no-slip condition and stick–slip condition. The flow chart contains three modules, namely Module-1, Module-2 and Module-3 which are elaborated in Figs. 4–6, respectively. In Module-1 integration has been carried out from ψ ¼ Γ f to ψ ¼ π=2 for stick–slip contact condition. In the flow chart in Fig. 3, the lines and boxes in magenta color form the first loop in which the gas pressure p inside the membrane is modified to satisfy the boundary condition vðπ=2Þ ¼ 0. The lines and boxes of blue color form the second loop in which the isothermal condition ðpV ¼ p0 V 0 Þ is satisfied by adjusting vjψ ¼ Γf . During this adjustment, the condition ðρ′jψ ¼ Γf Þns ≥ðρ′jψ ¼ Γf Þss is to be satisfied since the material points tend to slide towards the pole leading to a reduction in the value of ρ′ at ψ ¼ Γ f . If the adjustment of vjψ ¼ Γf at a point does not satisfy the impending sliding/no-slip condition, it implies that the slip front has progressed further towards the center of the membrane. Consequently, we progress to an interior contact point (by setting Γ f ¼ Γ f ΔΓ) to adjust vjψ ¼ Γf . In case the slip front moves up to the pole (i.e., the conditional statement in the red box in flow chart shown in Fig. 3 is not satisfied), we switch to Module-2 which handles the condition when all the
133
material points in contacting region slide right up to the pole. In Module-2, the condition λ0 ≥λs implies that the no-slip condition arises right after the material point at ψ ¼ Γ st . Module-3 captures the recurrence of the no-slip condition. In Module-3, the failure
Fig. 4. Flow chart of Module-1.
Fig. 3. Flow chart for stick–slip condition. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
Fig. 5. Flow chart of Module-2.
of condition μ≤μf (red colored decision box shown in Fig. 6) signifies the recurrence of the stick–slip condition.
3.4. Wrinkling
ð29Þ
which has the solution
A membrane reaches an impending wrinkling state when either T1 or T2 (Eqs. (16) and (17)) vanishes [23]. In the no-slip case, the material points in the contacting region are not allowed to move. Therefore, T1 and T2 can vanish independently, contrary to that in the frictionless case (as shown in [32]). We consider these two cases separately. If we assume that the membrane reaches the impending wrinkling state in the meridional direction, i.e., T 1 ¼ 0, Eq. (16) simplifies (using Eq. (21)) to 1 1 pffiffiffiffiffi λ2 cos ψ : λ′2 ¼ ð27Þ sin ψ λ2 The solution of this first order differential equation is 0 12=3 3 2 1 π @ 5 4 ψ j2 þ 1:7972103521033889A ; λ2 ¼ cosec ψ 3E 2 2 ð28Þ where E½ϕjm is the elliptic integral of second kind with amplitude 2 ϕ and modulus m ¼ k and is defined as Z ϕ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1k sin t dt: E½ϕjm ¼ 0
On the other hand if we assume T 2 ¼ 0, Eq. (17) reads ! 1 1 λ cos ψ : λ′2 ¼ 2 sin ψ λ22
λ2 ¼ cosec ψð1:5ψ0:75 sin 2ψÞ1=3 :
ð30Þ
The analytical solutions (28) and (30) (shown graphically in Fig. 7) provide us the values of the circumferential stretch λ2 at any angle ψ which if reached will lead to wrinkling in the material at that angular location. As observed in the figure, since the limiting value of λ2 corresponding to the condition T 1 ¼ 0 is lower for all ψ, it is more critical. Thus, we expect to see circumferential (axisymmetric) wrinkles appearing first. 4. Results and discussion We consider membrane materials with α ¼ 0, 0.1 and 0.2. In the following, we use the non-dimensional pressure P r ¼ pr 0 =C 1 h0 . The relation between the spherical membrane stretch λs and the initial inflating pressure p0 is given by Eq. (11), and shown in Fig. 8. We study the deformation process for two values of the initial stretch λs ¼ 1:3 and λs ¼ 3 which are, respectively, in the precritical and post-critical stretch regions of the curves shown in Fig. 8. We have taken β ¼ 102 , γ ¼ 102 and ϵ1 ¼ ϵ2 ¼ 104 , while ΔΓ is chosen as 103 radian (see Section 3). Integration of the governing equations has been performed using the Runge–Kutta method in all frictionless, no-slip and stick–slip cases.
N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
135
Fig. 6. Flow chart Module-3. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
1.5
5
1.4
4
1.3
3
1.2
2
1.1 1
1 0
12
6
4
3
5 12
0
2
1
2
3
4
5
) (for
Fig. 8. Pressure ratio versus λs for different values of α ¼ 0(−−−−−), 0.1 (− − −) and 0.2(————).
Fig. 9 shows the stretch ratios ðλ1 ; λ2 Þ for the frictionless and noslip cases for different contact angles Γ c . In the frictionless case, as the contact proceeds, the stretch ratios ðλ1 ; λ2 Þ decrease near the pole. This reveals that the material points of the membrane in the contacting region move towards the pole. The points encircled in red in Fig. 9 are the points where the contact ends. At these points, the profile of the meridional stretch ðλ1 Þ changes its behavior. This is because, the
curvature in the meridional direction goes from zero in the contacting region to a positive value in the non-contacting region. The motion of the material points in the contacting region in the frictionless contact case has interesting implications that is observed in Fig. 10, which compares the contact profile for the frictionless and no-slip cases. For each values of Γ c , the pressure ratio (Pr) in the no-slip case has a slightly lower value than that in
Fig. 7. Variation of λ2 with ψ for the solution given by Eq. (28) ( T 1 ¼ 0) and Eq. (30) (−−−−−) (for T 2 ¼ 0).
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N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
3.4
3.4
3.2
3.2
3.0
3.0
2.8
2.8
2.6
2.6
2.4
0
12
6
4
3
5 12
2.4 2
0
12
6
4
5 12
3
2
Fig. 9. Principal stretch ratios λ1 ( ) and λ2 (−−−−−) of an inflated spherical balloon after contact (λs ¼ 3 and α ¼ 0:2). (a) Frictionless case and (b) No-slip case. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
2
2.5
3
0
3.5
0
0.5
1
1.5
2
2.5
3
3.5
Fig. 10. Profiles of spherical membrane before and after contact for different contact angles Γ c ¼ 01 (————), 151 (− − −), 301(−−−−−), 451 ( (α ¼ 0:2 and λs ¼ 3). Stick–slip condition profile in which Pr ¼ 5.326, for μf ¼ 0:122 and Γ c ¼ 601 (- - - - -). (a) Frictionless case. (b) No-slip case.
) and 601 (
)
0 0.6 0.5 0.4
1
0.2
1.5
2
0
12
6
4
3
0
0
12
6
4
3
Fig. 11. Surface traction and minimum required coefficient of friction of spherical membrane after contact [fλs ¼ 3; α ¼ 0gð Þ, fλs ¼ 3; α ¼ 0:2gð−−−−−Þ, fλs ¼ 1:3; α ¼ 0g; ð− − −Þ, fλs ¼ 1:3; α ¼ 0:2gðÞ]. (a) Traction force along the meridional direction. (b) Minimum required coefficient of friction.
N. Kumar, A. DasGupta / International Journal of Non-Linear Mechanics 57 (2013) 130–139
3.0
137
3.6 3.4
2.8
3.2 3.0
2.6
2.8 2.6
2.4
2.4 2.2
0
12
6
4
3
5 12
2.2 2
0
Fig. 12. Principal stretch ratios in stick–slip condition for Γ s ¼ π=3 and μf ¼ 0ðÞ, 0.0267 ( slip (————) (α ¼ 0:2 and λs ¼ 3).
12
6
4
5 12
3
2
), 0.0788 (− − −), 0.1025(−−−−−) and no-
), 0.0475(
3.6
Table 1 Critical coefficient of friction μfc . α
λs
μfc
0 0 0.2 0.2
1.3 3 1.3 3
0.313 0.262 0.256 0.103
3.4 3.2 3.0 2.8
frictionless case. In the frictionless contact case, since the material points in the contacting region move towards the pole, the effective volume is reduced leading to slightly higher pressure. In stick–slip condition, pressure value lies between the pressure value found in no-slip and frictionless condition (shown in Fig. 10). The traction force τ defined in Eq. (20) is plotted in Fig. 11(a). It is observed that the maximum magnitude of the traction force is higher for the higher value of α. It is interesting to note that, for higher initial stretch λs , the magnitude of the traction force is lower. The minimum required coefficient of friction μ defined in Eq. (22) is shown in Fig. 11(b). It may be noted in this figure that, even though the surface traction magnitude is low for the case of λs ¼ 3; α ¼ 0, the required coefficient of friction for no-slip is high. This is because, this point is in the post-critical regime where the inflation pressure in the membrane is low (see Fig. 8). We have incorporated frictional sliding for different values of λs and α. The material points in the contacting region do not slide in the no-slip condition, and slide freely in the frictionless condition. Therefore, the results for stretches found in the stick–slip condition lie between the no-slip and the frictionless conditions, as shown in Fig. 12. For the results corresponding to the stick–slip case in this figure, the values of μf are taken small enough for the frictional sliding to occur throughout the contacting region (see Section 3.3). In other words, the sliding front reaches the pole of the membrane. In the stick–slip condition, as we increase the value of μf the stretch ratio at the vicinity of pole ðλ0 Þ goes near to λs . The value of μf for which the condition λ0 ¼ λs is achieved is said to be the critical coefficient of friction μfc . If μf 4μfc then sliding does not occur throughout the contacting region for the value of Γ s ≤π=3. In this case, the sliding front is arrested, and cannot reach the pole. We have calculated the values of μfc for different values of λs and α, some of which are given in Table 1. In Fig. 13, the stretch ratios of the membrane after contact for stick–slip condition is shown and compared with the stretch ratios for the no-slip condition. In this result μf ¼ 0:122 4 μfc ðΓ f ¼ 0:08 radÞ,
2.6 2.4 2.2
0
12
6
4
3
5 12
2
Fig. 13. Principal stretch ratios in stick–slip condition for μf ¼ 0:122 [λ1 ð−−−−−Þ, λ2 ð Þ] and for no-slip condition [λ1 ðÞ; λ2 ðÞ], in which Γ s ¼ π=3.
and the no-slip condition recurs (see Section 3.3) at ψ ¼ 0:965 rad when the sliding front is arrested. We determine the state of impending wrinkling by solving the contact problem for no-slip condition for λs ¼ 1:1; 1:2; 1:3 and α ¼ 0, 0.2. In these solutions, the contact angle proceeds from Γ ¼ 0 up to the value where the wrinkling just sets in. This corresponds to the contact angle where ðλ2 Þc touches the λ2 solution (for T 1 ¼ 0) curve (given by Eq. (28)), as shown in Fig. 14. It is observed that, in all cases, the critical value of λ2 is reached at the edge of the contact. Hence, the wrinkling will always occur first at the periphery of the membrane-plate contact. The conditions for impending wrinkling are tabulated in Table 2. From Fig. 14, it is interesting to observe that the value of λ2 remains almost constant in the contacting region. Therefore, for λ2 jψ ¼ 0 ¼ λs ≥1:5, we will not have any wrinkling in the membrane. The corresponding initial inflation pressure can be determined from Eq. (11). 5. Outlook The present work deals with the contact problem of an inflated spherical membrane pressed symmetrically between two rigid
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Table 2 Conditions for wrinkling. α
λs
Γ ðradÞ
Γ c ðradÞ
ðλ2 Þc
Pr
0 0 0 0.2 0.2 0.2
1.1 1.2 1.3 1.1 1.2 1.3
0.807 1.103 1.31 0.805 1.083 1.311
0.963 1.255 1.394 0.940 1.234 1.4
1.10006 1.19993 1.30034 1.09953 1.19171 1.3009
1.933 4.115 7.91044 2.406 5.1334 10.883
1.5
1.5
1.4
1.4
1.3
1.3
1.2
1.2
1.1
1.1
1
0
12
6
4
3
5 12
1 2
0
12
6
4
3
5 12
2
Fig. 14. Variation of λ2 in contacting region (− − −) and in non-contacting region (−−−−−) with ψ up to the impending wrinkling state. Variation of λ2 with ψ for the solution given by Eq. (28) ( ). (a) For α ¼ 0. (b) For α ¼ 0:2.
plates. Three possible contact conditions, namely frictionless, noslip and stick–slip conditions have been considered. For the former two idealized contact conditions, the problem is conservative. In the case of the stick–slip contact, the problem can be formulated as a sequence of variational problems with intermediate slip, which is governed by the force equilibrium equation of the membrane in the contacting region. The main observations are summarized below. 1. In the frictionless contact problem, the principal stretches at the vicinity of the pole decrease as the contact proceeds. This phenomenon indicates that, the material points in the contacting region move towards the pole. 2. The deformed profiles for the frictionless and no-slip cases do not differ significantly. However, in the no-slip case, the pressure is less than that in the frictionless case for the same contact angle condition. In the frictionless case, the motion of the material points towards the pole tends to reduce the volume thereby increasing the pressure. The results found in stick–slip condition lie between the results of no-slip and frictionless conditions. 3. The minimum required coefficient of friction has been estimated for no-slip. In the case of the stick–slip contact, the minimum coefficient of friction that arrests the progress of sliding front for a given level of pressing has also been estimated. 4. The condition for occurrence of wrinkling has been determined analytically. It is observed that for λs 4 1:5, wrinkling will not occur anywhere in the contacting region. Based on the formulation presented in this work, one can obtain some effective properties of the spherical membrane under
contact. The dynamic response of a membrane with contact would be a challenging problem for the future.
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