Supporting Document For
Sliding-induced non-uniform pre-tension
governs robust and reversible adhesion–
a revisit of adhesion mechanisms of
gecko
Q.H. Cheng1, B. Chen1, H.J. Gao2 and Y.W. Zhang1,*
1
Department of Engineering Mechanics, Institute of High Performance Computing, Singapore
138632
2
School of Engineering, Brown University, Providence, RI, 02912 USA
Finite Element Model of Spatula Pad, Adhesion Interface and Substrate
We established a plane-strain finite element model as illustrated in Figure s1, with no deformation in
the y-direction which is the out-of-plane direction while x is the longitudinal direction of the spatula pad
and z is the vertical direction. This assumption allows us to use a single element in the y-direction for
spatula pad, adhesion Interface and substrate. The spatula pad is modelled by the 4-node quadrilateral
membrane element (M3D4) in ABAQUS (2009). This element is a surface element that transmits inplane forces only and has no bending stiffness, matching the material characteristics of the spatula pad.
The linear elastic material model is applied to the spatula with Young’s modulus E = 2 GPa, thickness H =
5 nm and length L in a range of 100 – 300 nm. The substrate is modelled as a flat rigid surface by the 8node linear brick element (C3D8) with rigid material property.
The cohesive interactions between the spatula pad and the substrate is considered as an adhesion
layer, which is modelled by the 8-node three-dimensional cohesive element (COH3D8) in ABAQUS. The
four nodes of the element’s top face are connected with the membrane element for the spatula, and
the four nodes of the element’s bottom face are connected with the brick element for the substrate.
The geometric thickness of the cohesive element is defined as hc = 0, although it is shown with a
dimension in Figure s1 for better vision. The zero thickness specification is a unique feature that is
impermissible for other structural elements, and enables direct measurement of the spatula’s distance
δ from the substrate (see Figure 4). In the simulations, the cohesive elements are included in the
computation only at locations where the spatula pad adheres to the substrate. The dashed boxes in
*
Corresponding author. Email: zhangyw@ihpc.a-star.edu.sg, Fax: 65 64674350
1
Figure s1 represent the cohesive elements that are to be gradually added to the model during the
adhesion process.
Membrane elements
Cohesive elements
x
x
hc=0
Elements to be added
Solid elements
Figure s1. Finite elements for spatula pad, adhesion layer and substrate.
Referring to Figure 4, initially when a concerned point in the spatula pad is beyond the cut-off distance
δc from the substrate, no cohesive element is taking effect for the point. When the point comes to
reach within the cut-off distance, a cohesive element for the point would commence to take action with
a vertical adhesion force. The force would increase up to the peak value fs while the distance decreases
to δ0. After that, the force would decrease as the distance continues to reduce. Once the distance
becomes zero, the point is deemed fully adhered while the adhesion force vanishes, and the horizontal
position of the point is registered as the initial reference position for calculating friction or shear force
of the cohesive element. When the point moves horizontally by a distance  from the reference
position, a friction force would be produced in the cohesive element. The friction force is calculated as
illustrated in Figure 3(b), in a similar manner as the calculation of adhesion force. The difference is that
the adhesion force employs an absolute distance δ from the substrate, but the friction force uses a
relative horizontal moving distance . The adhesion force always points downward, while the friction
force points to left or right.
Traction-Separation Law of the Cohesive Element
Mechanical behaviour of the cohesive element is governed by a traction-separation constitutive
relation. In our plane-strain model, only the normal traction f and shear traction τ along the interface
are considered, with no shearing in the y-direction. The normal traction f and shear traction τ follow the
following relations (refer to Figure 3)
 
 
0    0
fs ,
0    0
  f s ,

 0
0
f 
    f ,      ,     c    f ,     
 c
s
0
c
s
0
c

  0
 0
(s1)
where δ and  are the separation and slippage of the cohesive element, respectively, fs and τs are
normal and shear strengths of the adhesion, respectively, and δ0= δc/2 and δc is the cut-off distance for
both separation and slippage.
2
When the normal traction f and shear force τ increase, the mechanical behaviour of the cohesive
element would begin to degrade, implying initiation of damage to the material. Here a quadratic stress
criterion is applied for damage initiation of the cohesive element,
.
(s2)
Once the above initiation criterion is reached, the rate at which the material stiffness is degraded is
described by a damage evolution law. In this model, the damage evolution is defined based on the
energy dissipated as a result of the damage process, also called the fracture energy. The fracture energy
to cause failure in the individual mode of either normal or shear deformation is equal to the area under
the traction-separation curve, denoted as γ, which is assumed identical for normal and shear as shown
in Figure 4. Evolution of damage under a combination of normal and shear tractions is described by a
damage variable D. In the present study, the damage variable is defined by a mixed-mode condition in
the power law form:
,
(s3)
where a linear combination is applied by taking  = 1. When D = 1 for a specific cohesive element, the
element is deemed full failure and then removed from the model.
Force Equilibrium in the Spatula Pad
Figure 4 illustrates force equilibrium in the spatula pad at a specific time. The equilibrium conditions in
the y- and z-direction can be written as,
Px  P cos    ,
Pz  P sin   Fz , Fz  Fz  f s ,  c ,    fdx
C
A
(s4)
where Fz is the normal traction integrated over a range from point A to C in Figure 4. For given values of
fs and δc, Fz will increase with reducing angle .
In Type I simulations, when a desired pre-tension P0 at the right end of the spatula is reached, a shear
traction of τ0 =P0/L is present in the adhesion layer. Therefore, the horizontal force Px = τ0 increases
gradually with the adhesion length . In the case of a fixed vertical force Pz, the angle  will reduce
because tan =Pz/Px . However, a reduced angle  will let Pz increase as indicated in Eq. (s4).
Consequently, both forces Px and Pz have to increase, with the former increasing faster when the angle
 decreases.
Numerical Implementation
3
Numerical simulations are performed using ABAQUS/Standard version 9.2 (2009) with geometrically
nonlinear static analysis. Each simulation (naming a step in ABAQUS) is broken into increments so that
the nonlinear solution path can be followed. ABAQUS /Standard automatically chooses time steps of
the increments. In the increments, ABAQUS/Standard uses Newton's method to solve the nonlinear
equilibrium equations. At the end of each increment the system is in (approximate) equilibrium and
results are recorded. The equilibrium solution in an increment is pursued by a number of iterations. If
the model is not in equilibrium at the end of the iteration, ABAQUS/Standard tries another iteration.
With every iteration, the solution that ABAQUS/Standard obtains should be closer to equilibrium;
however, sometimes the iteration process may diverge—subsequent iterations may move away from
the equilibrium state. In such case, ABAQUS/Standard may terminate the iteration process and attempt
to find a solution with a smaller incremental time step. The linear system of algebraic equations that
arise at each iteration of the Newton procedure is solved by the Direct Sparse Solver.
Analytical Solution to Adhesion of a Spatula Pad with Linear Pre-tension
A theoretical explanation of the stability of adhesion with linear pre-tension in the spatula pad may be
derived based on the constant pre-tension model by Chen et al. (2009). Consider a linear distribution of
pretension in a thin pad adhered on a substrate as shown in Figure s2,
,
(s5)
where τ0 is a constant shear traction along the interface and L is the adhered length.
P

L
Figure s2.Peeling of a thin pad adhered to a substrate.
Following a derivation procedure similar to that in Chen et al. (2009), the local energy balance in the
vicinity of the adhesion front requires
,
(s6)
where EH is the elastic modulus of the thin film, γ is the adhesion energy of the interface, and P is the
critical peel-off force. For a constant shear τ0 and a specific adhesion length L, P can be determined as a
function of the peeling angle  as plotted in Figure s3. It can be seen that when the pretension P0 is
above certain value, there exists a critical angle cr (Chen et al. 2009). The attachment is stable within
the zone between the upper branch and the lower blanch, while detachment may occur beyond cr.
The critical angle cr is given as (Chen et al. 2009),
4
.
P0/EH = 0.12
P0/EH = 0.1
P0/EH = 0.07
P0/EH = 0.04
P0/EH = 0
0.15
P/EH
(s7)
0.1
0.05
0
0
10
20
 (degree)
30
Figure s3. Critical force for the stability of adhesion with a linear distribution of pre-tension in a thin
pad. (γ/EH = 0.001)
References
ABAQUS Analysis User's Manual, 2009 version 6.9, Dassault Systèmes Simulia Corp., Providence, RI,
USA.
Chen, B., Wu, P. D. & Gao, H. J. 2009 Pre-tension generates strongly reversible adhesion of a spatula pad
on substrate. J. R. Soc. Interface 6, 529-537. (DOI 10.1098/rsif.2008.0322)
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