Supplementary Material
1. Derivation of Energy Release Rate of plane strain membrane
for Full Friction (no-slip) and Frictionless contact
We are slightly modifying the calculations done by Rong et al. [1] to get the expressions for a
plane strain membrane. Most of the calculation is same and is reproduced here for
completeness.
Figure 1. Peeling a small segment of the membrane (shown by thick lines). (a) Contact length decreases
by an amount of c .Membrane configuration before (thick solid) and after (thick dashed) peeling is
shown. (b) Closer view of the contact edge during contact line receding. O2 denotes the right clamp. [1]
1.1 No-Slip Interface
Assume that a contact length of     , x  c has been achieved during inflation of the
membrane. The energy release rate can be calculated by peeling a small part of the membrane
and calculating the change in potential energy per unit new surface created.
A schematic of membrane peeling is shown in Figure 1a. During this process, a small
portion of membrane dc is detached. The detached membrane corresponds to (   d  ,   ) in
the undeformed configuration. As we mentioned in the main text, stretch ratio  is
discontinuous at the contact edge during deflation. Let   and   denote the stretch ratio just
inside and outside of the contact region at the contact edge, respectively. Using the same
nomenclature T  is the line tension just outside the contact edge and so on. By definition of
stretch we have
d   dc /   .
(1)
We call the membrane outside the original contact region, i.e., (  , a) , part 1 and the newly
detached membrane part 2 (see Figure 1b). The change in potential energy is then,
E  U  WP ,
(2)
where U is the increase of the elastic energy of the system and WP is the work done by the
pressure on the system.
As there is no slip in the contact region, the membrane is “locked” in the contact region and
hence, we can split the potential energy change as,
E  U  WP  U1  U2  WP1  WP 2
(3)
where U1 , U2 , WP1 and WP 2 are the increase in elastic energy and work done by the
pressure on part 1 and part 2, respectively. For part 1, energy balance requires that
WT  U1  WP1 ,
(4)
where WT is the work done by the tension T  on part 1 during contact line shrinking. A
simple calculation shows that the first order term of WT is


WT  U1  WP1  bT  • AA '  bT  ( cos ˆi  sin ˆj ) •((dc  dc* cos )iˆ  dc* sin ˆj )
 WT  bT  cos (dc  cos dc )  bT  sin dc* sin
(5)
 WT  bT  cos dc  bT dc ,
where dc * is the length of membrane part 2 after being detached from the substrate and b is
the length of membrane in the out of plane direction.
It can be shown that, to first order,

*
dc   dc.

Substituting (6) into (5), we obtain
(6)
 



(7)
WT  U1  WP1  bT cos dc  bT  dc  bT dc    cos .



The work done by pressure on membrane part 2 WP 2 is a second order quantity and we can
neglect it in equation (3). Due to discontinuity of stretch ratio  , the first order term of U2 is


U2  bh0d  W (  )  W (  )  ,
(8)
where W ( ) is the elastic energy density with respect to the undeformed configuration. Recall
that generally W  W ( ,  ) but  is the hoop stretch ratio and is equal to unity for plane strain
case.
Substitution of (7) and (8) in (3) gives us the change in potential energy:
E  bT  dc    /    cos   bh0d  W (  )  W (  ) .
(9)
Finally, the energy release rate for the process is (by definition):
 

E
d
G   lim
 T     cos   h0
W (  )  W (  ) 

dc  0 bdc
dc


(10)
 
 h0
 G  T    cos     W 

 
Note that for plane strain membrane under no slip contact,   and T  are uniform outside the
contact region and can be denoted as out and Tout . However because of friction the stretch
inside the contact region is not uniform.

1.2 Frictionless Interface
The calculations for this case are very similar to the no-slip case. The schematic of line receding
is same as in Figure 1. However because of frictionless contact, the part of membrane in
contact can now slide during detachment. A small portion of membrane dc is detached from
the substrate which corresponds to (   d  ,   ) in the undeformed configuration. Now the
change in the contact length consists of 2 parts:
dc  (dc)d  (dc)s ,
(11)
where (dc)d is the decrease in contact due to the detachment of part 2 and (dc)s is the decrease
due to slipping of the membrane still in contact ( [0,    d  ]) . By simple observation we can
see that (upto first order),
(dc)d  ind  ,
(dc)s  ( *  d  )din    *din ,
(12)
dc*  out d  .
Unlike the no-slip case, change in potential energy now consists of 3 parts:
E  U  WP  U1  U2  U3  WP1  WP 2 .
(13)
where U1 and U2 are the increase in elastic energy for part 1 and part 2 respectively. WP1
and WP 2 are the work done by the pressure on part 1 and part 2 respectively. The additional
term U3 is due to the change in elastic energy of the membrane still in contact
( [0,    d  ]) .
Similar to the no slip case, energy balance of part 1 (See (4)) requires,
WT  U1  WP1  bTout cos dc  bTout dc  .
(14)
The work done by pressure on membrane part 2 ( WP 2 ) is a second order quantity and can be
neglected. Due to discontinuity of stretch ratio  , the first order term of U2 is:
U2  bh0d  W(out )  W(in ).
(15)
Note that because there is no work done by pressure on the part that slips during contact line
shrinking, U3 is equal to the work done by Tin only.
U3  bTin ( *  d  )din  bTin (dc)s .
(16)
In absence of friction, balance of horizontal forces at the contact edge relates the inner and
outer tensions: Tin  Tout cos and we can write U3 as:
 U3  bTout cos (dc)s .
(17)
Substituting (14), (15) and (17) into (13), we get the change in potential energy:
E  bTout cos dc  bTout dc   bh0d  W (out )  W (in )  bTout cos (dc)s
 E  bTout cos (dc  (dc)s )  bTout dc*  bh0d W
 E  bTout cos (dc)d  bTout dc*  bh0d W .
Further simplification can be affected by using (12):
E  (Tout cos   Tout out  h0 W )bd 
Using the definition of energy release rate along with (12) we can write:
(T cosin  Tout out  h0 W )bd 
E
G   lim
  lim out
(dc )d 0 b(dc)
(dc )d 0
b(dc)d
d

 h
 G  Tout  out  cos   0 W
 in
 in
Comparison of (10) and (19) demonstrates that the expression for energy release rate is
identical for no-slip and frictionless case with   and   replaced by out and in .
(18)
(19)
2. Detailed Calculations for Pull-off Contact Length and
Adherence force
Complete description of the problem and definition of the terms and variables used is given in
the main text [2]. Here we show the detailed calculations for studying membrane pull-off.
Equations from part 1 [3] of the study will be referred as (Part 1-**) and similarly for part 2 [2]
as (Part 2-**).
The following non-dimensionalizations will be used in the calculations below,
w
c
d
R * 
W
T
c  ,d  ,R  ,   ,W  ,T 
, wad  ad
(20)
a
a
a
a

 h0
 h0
Pull-off on a No-slip Interface
Governing equations for the no-slip membrane in contact are reproduced here in normalized
form for completeness. case (, (Part 2-5), (Part 2-7a,b), (Part 1-1) and (Part 2-8)) are rewritten in
normalized form as:
The inner stretch just inside the contact edge is given by (Part 2-A1):
dc
(21)
  * ,
d
Stretch ratio in the outer portion of the membrane is uniform and can be expressed using
geometry (Part 2-5) as:
R ( m   0 )
 
,
(22)
1 
Membrane geometry gives us two more relations (Part 2-7a,b):
R (sinm  sin0 )  1  c ,
(23)
R (cos 0  cos m )  d ,
(24)
Force equilibrium of a small portion of the membrane in the outer region shows that tension is
uniform and related to radius of curvature and pressure (Part 1-1):
T   R ,
(25)
In addition to above, we also have the balance of energy at the contact edge (8):

1
(26)
T  (   cos0 )   W      W       wad .


Finally,
the neo-Hookean constitutive law ((Part 1-5) and (Part 1-4)) relates the tension and strain
energy density with stretch:
T  T ( )     3 ,
(27)
W  W ( )    2   2  2  / 2.
Assuming that the adhesion is high enough to cause the membrane to “pull-off” at a finite
contact, pressure and contact length at pull-off can be calculated as a function of d and wad .
Differentiating (22) - (26) with respect to c we get:
*
d   dR ( m   0 )
R  d m d 0  R ( m   0 ) d 




,

dc
dc 1  
1     dc
dc  1    2 dc
(28)
1

d 
d
dR

(sin m  sin 0 )  R  cos m m  cos 0 0   1,
(29)
dc
dc
dc 

d
d 
dR

(cos m  cos 0 )  R   sin 0 0  sin m m   0,
(30)
dc
dc
dc 

dT  d  dR
d
(31)

 R
,

d dc
dc
dc




d 0 
dT  d    
  d 
  1 d

cos


T


sin


0
0
   dc    2 dc
d   dc   
dc 



(32)







dW    d 
1 d
1 dW    d 
W      W        
  0.


2





d
c

d

dc
d

d
c



 

We can reduce these equations to a simpler form by enforcing the conditions for pull-off. At
pull-off, pressure is at its minimum:
d
 0.
(33)
dc
Also, in numerical simulations, outer stretch is observed to be at its minimum at pull-off:
d 
(34)
 0.
dc
Substituting (33) and (34) in (31), we get :
dR
(35)
 0.
dc
Numerical simulations show also that the pull-off point always corresponds to the clamp slope
going to zero (membrane locally flat at the clamped end):
m  0.
(36)
It should be noted that we observe that (36) holds true for frictionless pull-off as well.
Substituting (36) and (35) in (30) we get:
d 0
 0.
(37)
dc
Substitution of (34), (35) and (37) simplifies (28) to give:
*
0
0
R  d m  R ( m   0 ) d 
 d 

   m  
.
2
 



1    dc   1    dc
 dc  1   
(38)
1

Using (35) and (37) we can simplify (29) to get:
d
R m  1,
dc
(39)
Combining (38) and (39) we get an expression for inner stretch at the contact edge,
R0
(40)
 
.
1 


Using (22) and (36) we can evaluate the outer stretch at pull-off:
R  0  0 
 
1    
    
R0
1    
(41)
(42)
This analytical condition is also corroborated in our simulations where the pull-off point
coincides with the intersection of the curves for   and   . Note that (30), (31) and (32) are
automatically satisfied at the pull off point for (33)-(37).
Substituting (42) in the contact condition we get the simplified criterion for membrane pull-off:
T (1  cos0 )  wad ,
(43)
where T  T   T  . Equation (43) implies that the tension is continuous across the contact
edge at pull-off.
2.1 Transition from pinch-off to pull-off
Let us consider the case when we can fix the separation d between the unstressed membrane
and the substrate and vary the adhesion between them to obtain the critical adhesion wt
which identifies the point of transition from pinch-off to pull-off. This is the minimum value of
adhesion after which the membrane detaches unstably via pull-off.
Consider first the case of no-slip contact. At the critical value of adhesion, the
membrane has zero contact, hence c   *  0 . However, by definition of critical adhesion, this
detachment also corresponds to a pull-off with zero contact length, thereby requiring (33) - (42)
to be satisfied as well. Substituting these conditions and the constitutive law (27) in the
governing equations (22) - (26), we can simplify them as:
   R 0 ,
(44)
1
R 
(45)
sin 0
R (cos 0  1)  d
1
 
 R
 3

 
(46)
(47)


   1  (1  cos0 )  wad
(48)
 3








Note that    . Combining(44), (45) and (46), the outer stretch and contact edge slope at
detachment are given by,
2 tan  d 
 0t

,
sin 0t  sin 2tan1  d  
1
t 
where
 0t  2 tan1  d  .
(49)
(50)
The subscript ‘t’ denotes that quantities are calculated at the critical adhesion wt . Note that
(36) holds true irrespective of friction therefore, equations (49) and (50) are valid for both noslip and frictionless contact (with   replaced by out ) as they depend on geometry only.
Substituting (49) in (43) ,the critical adhesion value at which the membrane detachment mode
transitions from pinch-off to pull-off is found to be
3
 
  0t   2   0t 
0t
 sin   ,
wt  2 

(51)
 sin(0t )  sin(0t )  
2 



where 0t is given by (50).
For a frictionless interface, we do not have equality of stretches (42) at pull-off. Instead in
at pull-off is calculated using the expression for outer stretch (49) along with the force balance
( ) ( )
( )
condition at the contact edge (Part 2-6). We can then substitute out t , in t and 0 t in the
energy balance condition (Part 2-8) to obtain the critical adhesion wt .
3. Bibliography
1.
Long R., Shull K.R., Hui C.-Y. 2010 Large deformation adhesive contact mechanics of
circular membranes with a flat rigid substrate. Journal of the Mechanics and Physics of Solids
58, 1225-1242. (doi:10.1016/j.jmps.2010.06.007).
2.
Srivastava A., Hui C.-Y. 2013 Large Deformation Contact Mechanics of Long Rectangular
Membranes – part 2: Adhesive Contact.
3.
Srivastava A., Hui C.-Y. 2013 Large Deformation Contact Mechanics of Long Rectangular
Membranes – part 1: Adhesionless Contact.