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Proc. R. Soc. A (2009) 465, 961–981
doi:10.1098/rspa.2008.0362
Published online 9 December 2008
The effect of preload on the pull-off force
in indentation tests of microfibre arrays
B Y R ONG L ONG
AND
C.-Y. H UI *
Department of Theoretical and Applied Mechanics, Cornell University,
Ithaca, NY 14853, USA
We determined how preload and work of adhesion control the force required to pull a
circular cylindrical indenter off a microfibre array. Five regimes, with different contact
behaviours, are identified for the unloading phase of indentation. These regimes are
governed by two dimensionless parameters. Above a critical preload, the pull-off force
and the pull-off stress reach a plateau value. The critical preload, as well as the plateau
pull-off force (stress), is found to depend on a single dimensionless parameter q, which
can be interpreted as a normalized work of adhesion.
Keywords: fibrillar adhesion; preload; pull-off force; indentation; work of adhesion
1. Introduction
Recent interest in bio-inspired adhesives has motivated many researchers to
fabricate microfibre arrays and to measure their adhesion (Sitti & Fearing 2003;
Peressadko & Gorb 2004; Kim & Sitti 2006; Northen & Turner 2006; Aksak et al.
2007; del Campo et al. 2007; Gorb et al. 2007; Greiner et al. 2007; Murphy et al.
2007; Noderer et al. 2007; Yao et al. 2007). Most of these bio-inspired adhesives
are made of soft elastomers to promote good contact. However, some arrays
using carbon nanotubes as fibres are found to exhibit excellent adhesion, as
demonstrated by the recent works of Tong et al. (2004), Jin et al. (2005), Zhao
et al. (2006) and Sethi et al. (2008).
Many of the theoretical works in this area have focused on contact mechanics
and adhesion of microfibre arrays (Jagota & Bennison 2002; Persson & Gorb
2003; Tang et al. 2005; Bhushan et al. 2006; Schargott et al. 2006; Tian et al.
2006; Yao & Gao 2006; Bhushan 2007; Glassmaker et al. 2007; Persson 2007;
Yao & Gao 2007). These works give important insights into design principles of
fibrillar interfaces. The present work concentrates on the interpretation of
experiments that characterize the adhesive properties of these arrays.
Indentation experiments based on the theory of Johnson Kendall and Roberts
( JKR; Johnson et al. 1971) have been used extensively to characterize the
adhesion of soft materials. An excellent review can be found in Shull et al. (1998).
* Author for correspondence ([email protected]).
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2008.0362 or via
http://journals.royalsociety.org.
Received 6 September 2008
Accepted 7 November 2008
961
This journal is q 2008 The Royal Society
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962
R. Long and C.-Y. Hui
Application of this technique to study the adhesion of microfibre arrays is
relatively recent. In these experiments, a compressive preload is applied to bring
a rigid smooth indenter into intimate contact with a microfibre array. The
microfibre array is usually part of a backing layer made of the same material.
Once contact is established, the indenter is retracted. As unloading begins, the
contact line is usually pinned (contact area fixed). Eventually, the contact line is
unpinned and the contact area starts to shrink stably until a critical tensile force
is achieved, which is known as the pull-off force.
To put our work into perspective, we briefly review works on measuring
adhesion of microfibre arrays. Sitti & Fearing (2003) characterized adhesion of
their array to silicon by measuring the force required to pull a silicon atomic
force microscope probe off it. Peressadko & Gorb (2004) and Gorb et al. (2007)
used the force needed to pull a flat glass off a microfibre array to characterize
adhesion. Northen & Turner (2006) measured the force required to pull a circular
‘flat punch’ (FP) off a microfibre array and defined the adhesive strength as the
pull-off force divided by the cross-sectional area of the punch. To avoid alignment
problems associated with a flat indenter, Kim & Sitti (2006) used a spherical
glass indenter to measure the pull-off force Fc. They also computed the energy
dissipated in a load cycle G (hysteresis) and suggested that Fc/Amax or G/Amax
can be used to characterize adhesion, where Amax is the maximum contact area.
They interpreted G/Amax as the work of adhesion, but did not study the
connection between Fc/Amax and G/Amax. The connection between hysteresis
and work of adhesion is investigated in more detail by Noderer et al. (2007) using
a film terminated microfibre array. Aksak et al. (2007) and Murphy et al. (2007)
used exactly the same methodology employed by Kim & Sitti (2006) to measure
the adhesion of their fibrillar arrays. In Yao et al. (2007), the pull-off force was
used to characterize the adhesion of microfibre arrays with different fibre
orientations. del Campo et al. (2007) and Greiner et al. (2007) also used spherical
indenters to measure the pull-off forces of their microfibre arrays.
The survey above indicates that the pull-off force is a widely accepted measure
of the adhesion. However, its usage becomes ambiguous if the pull-off force
depends on the preload. This dependence was demonstrated by Greiner et al.
(2007). They explained their observation using a theory developed by Schargott
et al. (2006). In this theory, the fibrils are modelled as a spring foundation, while
the indenter and the backing layer are assumed to be rigid. This assumption is
reasonable for the indenter, which is typically very stiff in comparison with the
highly deformable microfibres. However, the deformation of the backing layer
may not be small, since its compliance can be comparable to the fibre array.
Indeed, recent experiments of Kim et al. (2007) and a theory by Long et al.
(2008) have demonstrated that backing layer thickness can significantly affect
the pull-off force. It should be noted that the analysis of Long et al. (2008) cannot
be used to study the effect of preload on the pull-off force, since the indenter
was assumed to be flat. The aim of this work is to study how the pull-off force
depends on the preload, the geometry (e.g. the indenter radius) and material
properties (e.g. stiffness of backing layer). In the following, we will show that,
above a critical preload, the pull-off force is independent of preload. Also, for the
special case of a rigid backing layer, our results agree with those proposed by
Schargott et al. (2006).
Proc. R. Soc. A (2009)
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Effect of preload in indentation tests
(a)
963
(b)
load
elastic foundation
R
rigid cylindrical indenter
indenter
elastic half space
Figure 1. (a) Circular cylindrical indenter and (b) geometry and coordinate system. The fibrils are
modelled as an elastic foundation.
The plan of this paper is as follows. The governing equations are derived in §2.
Numerical and analytical results are presented in §3. Section 4 presents the
dependence of the pull-off force and the pull-off stress on preload and geometry.
The summary and discussion are in §5. A summary of notations is given in the
electronic supplementary material.
2. Formulation
We consider the problem of a circular cylindrical punch, despite the fact that
most tests are carried out with a spherical indenter. However, cylindrical
indenters have been used to measure adhesion (Chaudhury et al. 1996; She &
Chaudhury 2000) and in some situations it is more desirable. For example, Shen
et al. (submitted) have recently used a cylindrical indenter to study normal and
shear contact (friction); this specimen has the advantage that the history of a line
of fibres can be followed during sliding. In steady sliding, this line of fibres can be
taken to be representative of the entire contact region.
The geometry is shown in figure 1a. The indenter has radius R and is very long in
the out-of-plane direction. The deformation of the backing layer is in-plane strain.
Since the backing layer thickness is usually much thicker than the fibre height, we
model the backing layer as an elastic half space with Young’s modulus E and
Poisson’s ratio v. To account for finite thickness of the backing layer, we allow
the elastic half space to have a different elastic modulus to the microfibres. This
approximation can be justified by the recent work of Long et al. (2008). They showed
that the compliance of the backing layer can be modified by either decreasing
its thickness or by changing its modulus. For example, to study the response of
very thin backing layers, the modulus of the half space can be taken as infinite.
The geometry and the coordinate system are shown in figure 1b. The x -axis
coincides with the top surface of the elastic backing layer. The fibre array lies
between the indenter and the x -axis. The circular indenter has radius R and the
contact region is a long strip occupying j x j ! a; yZ 0. The vertical
displacement of the backing layer at yZ0 is denoted by v.
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964
R. Long and C.-Y. Hui
Since the spacing and the diameter of the fibres are very small in comparison
with the substrate thickness and punch radius, the fibre array is treated as an
elastic foundation that behaves as
ð2:1Þ
s ZKkd;
where s is the contact pressure on the foundation; k is its effective stiffness; and d
is the difference in normal displacement between the surface of the indenter that
is in contact with the fibre array and the surface of the backing layer. To justify
the usage of the foundation model, we have carried out simulations of normal
contact where the fibres are discrete. We found that the foundation model
produces essentially the same result as the discrete model, even for a very small
number of fibres (greater than four). Finally, it should be noted that our theory
does not take into account fibril buckling, which will occur at sufficiently large
preloads. When a fibre buckles, it loses contact and reduces adhesion (Hui et al.
2007). Fibre buckling can also dramatically increase the compliance of the fibre
array. Fortunately, such a significant change in compliance can usually be
observed from the loading data.
In this paper, we use the standard notation in contact mechanics, which is a
positive s is compressive. The effective stiffness k is related to the stiffness of a
fibre Kf by
k Z rKf ;
ð2:2Þ
where r is the number of fibres per unit area. For example, if the fibres are bars
with height L and cross-sectional area A, then the stiffness is
K f Z YA=L;
ð2:3Þ
where Y is Young’s modulus of the fibrils. The backing layer is assumed to be
linear elastic with Young’s modulus E and Poisson’s ratio v. Fibrils are assumed
to have identical pull-off strength, i.e. a fibril will be detached when the force
acting on it reaches Kf dc, where dc is the critical stretch a fibril can withstand.
Translating this to the foundation model, the interface will fail at a critical stress
kdc and the effective work of adhesion of the interface Weff is
Weff Z kd2c =2:
ð2:4Þ
This model for the behaviour of the microfibril array is similar to that proposed
by Schargott et al. (2006). In their calculation, they neglect the deformation of
the backing layer. If the deformation of the backing layer is taken into account,
then the contact condition is (following Johnson 1985)
K
x
s0
Z v0 C ;
R
k
j x j ! a;
ð2:5Þ
where s(x) denotes the contact pressure in jxj!a and a prime denotes
differentiation with respect to x. The displacement gradient v 0 along the x -axis
is related to the normal contact stress by (Johnson 1985)
v 0 ZK
Proc. R. Soc. A (2009)
2
pE ða
sðsÞds
;
Ka x Ks
ð2:6Þ
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Effect of preload in indentation tests
965
where E Z E=ð1Kv 2 Þ. Equation (2.6) is valid for all x. For j x j ! a, the integral
is interpreted as a principal integral. Combining equations (2.5) and (2.6) gives
ða
s;x
x
2
sðsÞds
K Z K
; j x j ! a:
ð2:7Þ
R
pE Ka x Ks
k
The contact stress is related to the applied normal indenter load F (FO0
compression) by
ða
sðsÞds Z F:
ð2:8Þ
Ka
It should be noted that all the forces in this work are actually force per unit outof-plane thickness. Additional constraints must be imposed on (2.7) to solve for
the contact width. This constraint depends on whether the contact area is
increasing (crack healing) or decreasing (crack growth).
(a ) Preload
During preload, the indenter is under compression and the contact line moves
outwards. A useful way to think about the mechanics of indentation is to view the
contact line as the front of an external crack that occupies the air gap between the
indenter and the substrate. In the preload phase, this external crack heals. Since
the goal of using microfibrils is to increase the adhesion, most arrays exhibit large
hysteresis, i.e. adhesion is usually small in the preload phase in comparison with
the retraction phase. Therefore, we assume no adhesion during preload to reduce
the number of parameters in our analysis. The absence of adhesion (Hertz contact)
implies that the fibres at the contact edge cannot bear tension. Continuity of
traction requires the normal stress to vanish at the edge, i.e.
sðx ZGaÞ Z 0:
ð2:9aÞ
Introduce the normalized variables
S Z as=Fp ;
X Z x=a
and S Z s=a;
ð2:9bÞ
where Fp is the applied preload and a is the contact width. Equations (2.7) and
(2.8) become
ð1
SðS ÞdS
ca3 X ZKS;X C a
;
ð2:10aÞ
K1 X KS
ð1
SðXÞdX Z 1;
ð2:10bÞ
K1
2ka
;
ð2:10cÞ
pE where a is the normalized contact width and 1/c is the normalized preload, i.e.
8Rk 2
1
ð2:10dÞ
F Z :
F p Z
3 p
c
ðpE Þ
Thus, c is known for a given preload. The normalized contact pressure S and the
normalized contact width a are determined by solving (2.10a) and (2.10b),
together with the normalized Hertz condition
ah
SðX ZG1Þ Z 0:
Proc. R. Soc. A (2009)
ð2:10eÞ
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966
R. Long and C.-Y. Hui
(b ) Unloading
Denote the contact width at the start of unloading by a 0. For a given preload
Fp, a 0 or its normalized form a0 is determined by solving (2.10a), (2.10b) and
(2.10e). During unloading, the contact width is non-increasing, a%a 0. In many
experiments, the contact line does not move until the indenter is under tension.
This contact or crack pinning is due to the fact that the work required to
advance the crack by a given amount is typically much greater than the energy
released when the crack closes by the same amount. In our case, contact is
adhesion-less in the preload phase, so the contact line will always be pinned at
the beginning of unloading, until a critical force Fu is achieved. We will call Fu
the unpinning force. Denote the force required to advance the contact line at a
given contact width a(a%a 0) by F. Physically, F and Fu should increase with dc
or, equivalently, with the effective work of adhesion Weff defined by equation (2.4).
The governing equations during unloading are still given by (2.7) and (2.8),
except (2.9a) is replaced by the condition of decreasing contact or crack
advance, i.e.
sðx ZGaÞ ZKkdc :
ð2:11Þ
In the unloading phase, we normalize the contact stress using
ð2:12Þ
f Z s=kdc :
With this new renormalization, (2.7), (2.8) and (2.11) become
ð1
fðS ÞdS
;
ð2:13aÞ
bX ZKf;X C a
K1 X KS
ð1
fðXÞdX Z 2F=Fmax ;
ð2:13bÞ
K1
where Fmax h 2akdc . The dimensionless parameter b in (2.13a) is defined by
a2
:
ð2:13cÞ
bZ
Rdc
The condition for crack advance (2.11) becomes
fð j X j Z 1Þ ZK1:
ð2:13dÞ
The retraction force F is normalized in the same way as the preload, i.e.
ð
8Rk 2
a3 1
a3 F
F Z 3 3 F Z
fðXÞdX Z
:
ð2:13eÞ
b K1
b kdc a
p ðE Þ
3. Results
Numerical results are presented first for the preload phase, followed by the
unloading phase.
(a ) Results for the preload phase
The solution is expected to lie between two limits. The first corresponds to the
case of stiff springs or Hertz contact, i.e. a[1. By setting cZ2/a2 and using
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Effect of preload in indentation tests
967
a[1, the deformation due to springs can be neglected in (2.10a). Equation
(2.10a) simplifies to
ð1
SðS ÞdS
XZ
;
ð3:1Þ
K1 X KS
which is the integral equation governing the indentation of a circular cylinder on
a flat half-space (no microfibre array; Johnson 1985). The solution of (3.1) is
2 pffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi
1KX ;
SZ
ð3:2aÞ
p
pE a2
:
ð3:2bÞ
4R
The other limit corresponds to a/0, i.e. when the spring is very soft in
comparison with the substrate. This is essentially the case considered by
Schargott et al. (2006). In this case, v z0 and the integral term in (2.13a) can be
neglected. Equation (2.13a) reduces to
Fp Z
ca3
ð3:3Þ
ð1KX 2 Þ; a/ 1;
2
where we have used Sð j X j Z 1ÞZ 0 to determine the constant of integration. To
find c, we use (2.10b), this gives
ca3 X ZKS;X 0 S Z
3
2ka 3
; a/ 1:
ð3:4Þ
or
F
Z
p
3R
2a3
In summary, the relations between preload F p ðFp Þ and contact width for soft and
hard springs are given by
c/
3
2a
F p Z
3
or Fp Z
2ka 3
;
3R
a/ 1
ð3:5aÞ
and
a2
pE * a2
F p Z
or Fp Z
; a[ 1;
ð3:5bÞ
2
4R
respectively. The intermediate cases are obtained by solving (2.10a) and (2.10b)
numerically. The normalized force versus normalized contact width is shown in
figure 2. This figure shows that the indenter force versus contact width can be
determined very accurately using the following expression:
10a3
pffiffiffi
;
ð3:6Þ
20a C 2 a C 15
where F p is the normalized indenting force during preload (2.10d ). Equation
(3.6) is the Hertz curve for the preloading of a microfibre array.
F p Z
(b ) Results for unloading a%a 0
Unloading takes place in two stages: in the first, the contact line is pinned
(aZa 0) and in the second, the contact line unpins and propagates inwards (crack
growth). The unpinning of the contact line will eventually lead to indenter pull-off.
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968
R. Long and C.-Y. Hui
normalized preload force
10 4
10 2
1
10 –2
10 – 4
10 – 6
10 –2
1
10
10 –1
normalized contact width
10 2
Figure 2. Normalized preload F p Z 8Rk 2 Fp =ðpE Þ3 versus the normalized contact width
a h 2ka=ðpE Þ. The solid line is (3.6) and the circles are numerical solutions of (2.10a) and (2.10b).
The asymptotic results (3.5a) and (3.5b) are plotted as dashed and dash-dotted lines, respectively.
In a load control test, the pull-off force Fc (Fc!0) is defined to be the minimum
of the retraction force F. The minimum can occur at aZa 0 or a!a 0. In the first
case, pull-off occurs at aZa 0, so the unpinning force is the same as the pull-off
force, i.e. Fu Z Fc . In the second case, Fu O Fc . We determine the load F to advance
the contact line using the following procedure. The contact width a 0 at the start
of unloading is known from the preload calculation. Note that b in (2.13a) can
be expressed in terms of a h 2ka=ðpE Þ, i.e.
pffiffiffi 2
pE
k
2
b Z qa ; q Z pffiffiffiffiffiffiffiffiffiffiffiffi
:
ð3:7Þ
R 2Weff 2k
Note that a and b are known once a is given. For a given a%a 0, we solve (2.13a) for
the normalized stress f, then evaluate (2.13e) to obtain the normalized retraction
The force required to unpin the crack, Fu, is obtained by setting aZa 0 in F.
force F.
Five regimes can be identified during unloading. Each regime corresponds
to different values of a h 2ka=ðpE Þ and b h a2 =ðRdc Þ. These regimes are
discussed below.
(i) Regime I a/1, b/1
This regime corresponds to very compliant fibres and a flat indenter. Equation
(2.13a) becomes approximately
Kf;X Z 0:
ð3:8Þ
The solution of (3.8), using the boundary condition fðX Z 1ÞZK1, is
fðXÞ Z K1;
K1% X % 1:
ð3:9Þ
Equation (3.9) states that the contact pressure is uniformly distributed. For this
reason, this regime is called ‘equal load sharing’ (ELS). The retraction force is
F ZK2kdc a
Proc. R. Soc. A (2009)
or F ZK2a3 =b ZK2a=q;
ð3:10Þ
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Effect of preload in indentation tests
969
where q is defined in (3.7) and is independent of the contact width. Since F
decreases with a, retraction in a force control test is unstable once the contact
line is unpinned. The pull-off force Fc is the same as the unpinning force Fu in this
regime. The relation between the pull-off force and the effective work of adhesion
Weff is obtained using (3.10) and (2.4), i.e.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fc Z Fu ZK2a 0 2kWeff :
ð3:11Þ
Since a/1, the pull-off force can be expressed in terms of the preload Fp using
(3.5a), i.e.
3RFp 1=3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fc ZK2
2kWeff :
ð3:12Þ
2k
A typical plot of normalized load versus normalized contact width in this regime
is shown in figure 3a. Similar plots are generated for all the other four regimes.
To summarize, in the ELS regime, the pull-off force increases with the preload,
and the pull-off occurs as the contact line unpins.
(ii) Regime II a/1, bzO(1)
This regime corresponds to soft springs, stiff backing layer or small
punch radius. Since we treated the fibrils as continuous spring foundation,
this regime is called the ‘spring dominated’ (SD) limit. As mentioned
in the introduction, Schargott et al. (2006) developed a model to explain how
the pull-off force depends on the preload. In their model, both the indenter
and the substrate were assumed to be rigid. In regime II, (2.13a) can be
approximated by
bX ZKf;X :
ð3:13Þ
The solution of equation (3.13) subject to the boundary condition (2.13d ) is
b
b
fðXÞ ZK X 2 K1 C :
2
2
Using (2.13e), the retraction force F is
3
a
1
1
3
F Z 2kdc
Ka
or F ZK2a
K :
3Rdc
qa2 3
ð3:14Þ
ð3:15Þ
The question of whether the unpinning force is the same as the pull-off
force can
pffiffiffiffiffiffiffiffi
be addressed by noting that F versus a has a unique minimum at ac Z Rdc . This
implies that retraction is stable for aO ac and unstable for a! ac in a force
control test. Therefore, Fu Z Fc if a 0 ! ac . The pull-off force is
3
pffiffiffiffiffiffiffiffi
a0
K a 0 ; a 0 % Rdc ;
Fc Z Fu Z 2kdc
ð3:16aÞ
3Rdc
pffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
4
Fc ZK kdc Rdc ; a 0 R Rdc :
ð3:16bÞ
3
Using (2.4) and (3.5a), the pull-off force can be expressed in terms of the work of
adhesion and preload Fp, i.e.
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970
R. Long and C.-Y. Hui
normalized load
0
– 0.05
– 0.10
– 0.15
– 0.20
0
(b) 0.030
0.025
preload
0.020
0.015
contact
0.010
line
0.005
pinning
0
retraction
–0.005
–0.010
–0.015
–0.020
0.02 0.04 0.06 0.08 0.10 0.12
0
normalized contact width
normalized load
(a) 0.05
contact
line
pinning
preload
stable
retraction
unstable
retraction
pull–off force
0.1
0.2
0.3
0.4
normalized contact width
Figure 3. Normalized load (F p in preload and F in retraction) versus normalized contact width
a h 2ka=ðpE Þ: (a) the ELS regime (qZ1). The preload curve is determined using (3.6) and the
retraction curve is obtained using (3.10) and (b) the SD regime (qZ20). The preload and retraction
curves are determined by (3.6) and (3.15), respectively.
3RFp
Fc Z Fu ZKFp C 2
2k
1=3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2kWeff ;
a0 !
pffiffiffiffiffiffiffiffi
Rdc ;
pffiffiffiffiffiffiffiffi
1=4 pffiffiffiffi
4
R; a 0 O Rdc :
8kW 3eff
3
The condition of stability in terms of the preload is
1=2
2
3=4
2k Rd3c
2k
2R Weff
:
Z
Fp O
3R
k
3
Fc ZK
ð3:17aÞ
ð3:17bÞ
ð3:18Þ
Equations (3.17a) and (3.17b) state that the pull-off force depends on the preload:
for small preloads, the magnitude of the pull-off force increases with preload
and is given by (3.17a); as the preload increases beyond 2kðRd3c Þ1=2 =3 (see
equation (3.18) above), it reaches a plateau value that is independent of preload,
given by (3.17b). The existence of a plateau pull-off force was observed by
Greiner et al. (2007).
A typical normalized load versus normalized contact width in this regime is
plotted in figure 3b. The magnitude of the pull-off force increases with the preload
and reaches a maximum given by (3.17b). For preloads satisfyingp(3.18)
ffiffiffiffiffiffiffiffi or
a 0 O a c , instability will occur when the contact width shrinks to ac Z Rdc . For
small preloads or a 0 ! ac , p
instability
will
occur
right after the crack unpins. Note
ffi
pffiffiffiffiffiffiffi
ffiffiffi
also that bZ1 and aZ 1= q at ac Z Rdc , consistent with our assumption that
b zOð1Þ. In this regime, the indenting displacement D is well defined since the
substrate is rigid. It can be shown that DZKdc =2 at instability.
(iii) Regime III a[1, b[1
This regime corresponds to an indenter with very large radius. Inside the
contact region, the curvature of the indenter can be neglected and the circular
indenter can be approximated by a FP. The condition a h 2ka=ðpE Þ/N
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Effect of preload in indentation tests
971
implies that the backing layer is much more compliant than the fibrils.
Therefore, this regime corresponds to a FP indenting on a elastic half-space, and
is called the FP limit, where the governing equation (2.13a) can be approximated by
ð1
fðS ÞdS
Z 0:
ð3:19Þ
K1 X KS
The solution of (3.19) satisfying (3.8) is given by the classical punch solution
(Johnson 1985)
fðXÞ Z
bF
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
pa3 1KX 2
ð3:20Þ
where F is the normalized retraction force. Note that the normalization condition
(2.13e) is satisfied. Equation (3.20) shows that the contact pressure goes to infinity as
the edge of the contact region is approached; therefore, the boundary condition
(2.13d ) cannot be satisfied by this solution. However, (3.20) is valid except close to
the contact edge, which can be treated as the tip of an external crack. The stress
intensity factor of this crack is
F
K ZK pffiffiffiffiffiffi :
pa
ð3:21Þ
The force required to move the contact line is determined by equating the energy
release rate, K 2 =ð2E Þ, to the effective work of adhesion, Weff Z kd2c =2. This
argument results in
pffiffiffiffiffiffi
pffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 2a
2a
F ZK pakE dc or F ZK
ZK
:
ð3:22Þ
b
q
Equation (3.22) implies that the unloading processp
isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
unstable once the contact line is
unpinned. Therefore, the pull-off force Fc Z Fu ZK pa 0 kE dc . The pull-off force can
be expressed in terms of the work of adhesion by
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fc ZK 2pa 0 E Weff :
ð3:23aÞ
Using (3.5b), since a[ 1 in this regime, the pull-off force is
1=4 pffiffiffiffiffiffiffiffiffi
Fc ZK2 pE RFp
Weff :
ð3:23bÞ
Equation (3.23b) shows how the pull-off force depends on preload. A schematic
normalized load versus normalized contact width curve in this regime is shown
in figure 4a.
(iv) Regime IV azO(1), b/1
In this regime, the circular indenter can be approximated as a FP, but
the stiffness of the fibrils is comparable to the backing layer. We call this
regime ‘FP with springs’ (FPS). The governing equation (2.13a) can be
approximated by
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972
R. Long and C.-Y. Hui
(a) 12
(b)
normalized load
10
8
6
4
contact
line
pinning
preload
2
contact
line
pinning
preload
0
retraction
retraction
–2
–4
0
1
2
3
4
5
normalized contact width
6
0
1
2
3
4
5
normalized contact width
6
Figure 4. Normalized load (F p in preload and F in retraction) versus normalized contact width
a h 2ka=ðpE Þ: (a) the FP regime with qZ1. The preload and retraction curves are obtained using
(3.6) and (3.22), respectively and (b) the FPS regime (qZ1). The preload and retraction curves are
obtained using (3.6), (3.26) and (3.29), respectively.
ð1
Kf;X C a
fðS ÞdS
Z 0;
K1 X KS
fðX Z 1Þ ZK1:
ð3:24Þ
ð3:25Þ
There is no simple analytic solution for (3.24) and (3.25). However, (3.24) and
(3.25) can be solved numerically. Equations (3.24), (3.25) and (2.13b) imply that
2F=Fmax depend only on a. The normalized retraction force F is related to
2F=Fmax by
a 2F
F Z
:
q Fmax
ð3:26Þ
Figure 5 shows how 2F=Fmax changes with a. Note 2F=Fmax /K2 in the ELS
limit, which is approached as a/0. It is also possible to generate a higher-order
asymptotic solution (see appendix A). Here, we state the result
2F=Fmax ZK2 C 2a C Oða2 Þ:
ð3:27Þ
p
ffiffiffiffiffiffi
As a/N, the FP limit is approached and (3.22) shows that F ZKa2 2a=b,
which implies
pffiffiffiffiffiffiffiffi
ð3:28Þ
2F=Fmax /K 2=a:
Figure 5 shows that
pffiffiffi
2 C 2a2
pffiffiffi
2F=Fmax ZK
ð3:29Þ
1 C a C a2 a
gives an excellent fit to our numerical results.
To determine the pull-off force Fc, we examine how F varies with a. Using
(3.25) and (3.29), we found
vF
! 0; c aR 0:
ð3:30Þ
va
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Effect of preload in indentation tests
973
normalized retraction force
– 0.2
– 0.4 equation (3.27)
– 0.6
– 0.8
–1.0
equation (3.28)
–1.2
–1.4
–1.6
–1.8
–2.0
0
2
4
6 8 10 12 14 16 18 20
normalized contact width
Figure 5. The dependence of 2F/Fmax on a. Asymptotic behaviours of 2F/Fmax, given by (3.27) and
(3.28), are indicated by the dashed lines. The circles (numerical results) are obtained by solving
(3.24), (3.25) and (3.26), respectively. The solid line is (3.29).
Equation (3.29) implies that F ! 0 for all a, so the retraction force is always
tensile. Furthermore, (3.30) shows that F is a decreasing function of a. This
indicates that unloading is unstable once the crack unpins, i.e. FcZFu. We can
also express the pull-off force in terms of effective work of adhesion Weff
pffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 C 2a20
Fc Z Fu ZK
ð3:31Þ
2kWeff a 0 ;
p
ffiffiffiffiffi
1 C a0 C a20 a0
where a0 Z 2ka 0 =ðpE Þ. We can use (3.31) and (3.6) to determine the
dependence of the pull-off force Fc on the preload Fp. A typical plot of the
normalized load versus normalized contact width is given in figure 4b.
(v) Regime V a[1, b[1
This regime corresponds to the classical JKR limit (Johnson et al. 1971),
where the fibrils are very stiff in comparison with the substrate. Equation (2.13a)
can be approximated by
ð1
fðS ÞdS
bX Z a
;
ð3:32Þ
K1 X KS
which is the governing equation for a circular cylindrical indenter in normal
contact with an elastic half space. The analytical solution of (3.32) is found to be
(Johnson 1985)
1 b ð2X 2 K1Þ
bF
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
fðXÞ ZK
ð3:33Þ
2p a
1KX 2
pa3 1KX 2
Similar to the FP limit, (3.33) cannot satisfy the boundary condition (2.13d ).
The force after the contact line unpins is determined using energy balance,
similar to the analysis of the FP limit. The stress intensity factor is found to be
rffiffiffiffi
E a2 p
F
KZ
K pffiffiffiffiffiffi :
ð3:34Þ
a
4R
pa
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974
R. Long and C.-Y. Hui
Equating the energy release rate, K 2 =2E , to the effective work of adhesion,
we obtain
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pE a 2
FZ
ð3:35Þ
K dc E kpa :
4R
Equation (3.35) implies that F achieves its minimum
1=3
3
Fmin ZK Rk 2 d4c pE ð3:36Þ
4
at
2 2 1=3
kdc R
ac Z
:
ð3:37Þ
pE p
ffiffiffiffiffiffiffi
Note that aZ 2b2 Z 1= 3 2q 2 at aZ ac , consistent with a[1 and b[1. As in the
SD limit, when a 0 % ac , retraction is unstable once the contact line unpins,
i.e.Fc Z Fu . On the other hand, if the preload is sufficiently large so that a 0 O ac ,
retraction is stable until the contact line shrinks to aZ ac , then pull-off occurs. In
this case, the pull-off force Fc reaches the plateau value Fmin and is independent
of preload. To summarize, the pull-off force is
1=3
3
; a 0 O ac ;
ð3:38aÞ
Fc Z 4pE RW 2eff
4
pE a 20 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fu ZK
ð3:38bÞ
C 2pE a 0 Weff ; a 0 % ac :
4R
Since a[ 1 in this regime, the pull-off force can be expressed as a function of
preload force Fp using (3.5b); this results in
1=3
3
; a 0 O ac ;
ð3:39aÞ
Fc Z 4pE RW 2eff
4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Fp R 1=4
Fc Z Fu ZKFp C 2pE Weff
a 0 % ac :
ð3:39bÞ
*
pE
The stability condition aO ac can be expressed in terms of the preload as
dc ðpE k 2 dc RÞ1=3
ð3:40Þ
4
A schematic load versus contact width plot in this regime is given in figure 6. We
summarize our results of the five regimes in table 1.
Since both the parameters depend on the contact width, different regimes can be
observed in a test. Specifically, both a and b decrease during unloading, so it is
possible that, at the start of unloading, the test is conducted in the JKR regime, but
pull-off can occur in the other regimes. Of course, it is impossible to start unloading
at a regime with a smaller a or b, and end up in a regime with a larger a or b. For
example, it is impossible to go from the SD regime to the FPS regime. An exception
is the ELS regime, this regime if it occurs, is the only one possible. This is because
a 0 O a implies a0 / 1; b0 / 10 a/ 1; b/ 1. ELS is possible if and only if the
condition a/ 1; b/ 1 is satisfied. In terms of the preload, these conditions are
sffiffiffiffiffiffiffiffiffiffiffiffi
3RFp 2=3
2k 3RFp 1=3
1
k
/ 1 and
/ 1:
ð3:41Þ
pE R 2Weff
2k
2k
Fp O
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975
Effect of preload in indentation tests
50
normalized load
40
contact
line
pinning
30
20
preload
10
0
unstable
retraction
–10
–20
–30
stable
retraction
0
2
pull–off
force
4
6
8
10
normalized contact width
12
Figure 6. Normalized load (F p in preload and F in retraction) versus normalized contact width
a h 2ka=ðpE Þ in the JKR regime (qZ0.1). The preload and retraction curves are obtained using
(3.6) and (3.35), respectively.
Table 1. Summary of the five contact regimes.
regimes
region of validity
physical interpretation
pull-off force (Fc)
ELS
a/1, b/1
equation (3.12)
SD
FP
a/1, bzO(1)
a[1, b/1
FPS
azO(1), b/1
JKR
a[1, b[1
all fibrils share the same load
(equal load shearing)
backing layer is rigid
indenter is nearly flat and
rigid backing layer
indenter is nearly flat, compliant
backing layer
JKR theory
equations (3.17a) and (3.17b)
equation (3.23b)
equation (3.31)
equations (3.39a) and (3.39b)
More than one regime may be valid for certain values of a and b. For example, the
FPS regime collapses to the ELS regime for small a, the SD regime also reduces to
the ELS regime for small b. Other examples are the JKR regime reduces to the FP
regime for small b and the FPS regime coincides with the FP regime for large a. It
should be noted that figures 3a,b, 4a,b and 6 were plotted assuming that one regime
dominates, ignoring the fact that this may not be always possible.
4. Effect of preload on the pull-off force and stress
(a ) Pull-off force
To illustrate how the pull-off force in a typical test depends on the preload, we
numerically evaluate the pull-off force for different preloads, assuming typical
values for the parameters R, k, Weff and E . Once these quantities are fixed, b is
completely determined by a through bZqa2, where q is defined by (3.7). Typical
values of q in the literature are between 0.01 and 10. The solution procedure
is as follows. For each a0, we solve (2.13a), (2.13b) and (2.13d ) to obtain the
pull-off force. The relation between preload and a0 is obtained using (3.6).
Figure 7 plots the pull-off force versus the preload for four different values of q.
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976
–1
(b) –10 –2
normalized pull-off force
normalized pull-off force
(a)
R. Long and C.-Y. Hui
–101
–102
–103
ELS / SD / FPS
FPS
1
JKR
10 5
ELS / SD / FPS
1010
(c) –10 – 4
(d ) –10 – 6
–10 –3
–10 –2
–10 –1
–1
10 –15
ELS / SD / FPS
10 –10
1
10 –5
normalized preload force
105
FPS
–101
10 –8 10 – 6 10 – 4 10 –2
normalized pull-off force
–10 –5
FP/FPS/JKR
–1
normalized pull-off force
–104
–10 –10
–10 –1
1
102
104
–10 –5
–10 – 4
–10 –3
ELS / SD / FPS
–10 –2
10 –15
SD
10 –5
10 –10
1
normalized preload force
105
Figure 7. Dependence of normalized pull-off force F c on normalized preload force F p for (a) qZ0.01;
(b) qZ1; (c) qZ10 and (d ) qZ500. Solid line, numerical result; crosses, ELS; circles, SD; pluses,
FP; triangles, FPS; squares, JKR.
Predictions of different regimes are also marked in these figures. The pull-off
forces Fc in the ELS, SD, FP, FPS and JKR regimes are given by (3.12),
(3.17a), (3.17b), (3.23b), (3.31), (3.39a) and (3.39b), respectively. As expected,
the magnitude of the pull-off force increases with the preload for small preloads.
For sufficiently large preloads, the pull-off force approaches a plateau value that
depends only on q. The existence of the plateau corresponds to the existence of a
smooth minimum in the force versus contact width curve. It should be noted
that the five regimes indicated above do not cover the entire parameter space
(a, b). Thus, the JKR and SD regimes are not the only regimes with a plateau
pull-off force.
Let us consider these figures in more detail. Figure 7a can be divided into four
regions. In each region, the numerical result can be well approximated by
analytical expressions of some regimes, which are indicated on the plot. Note
that F p monotonically increases with a (see (3.6) or figure 2). In the first region,
a0 and b0 Z 0:01a20 are both small; this means that pull-off is in the ELS, SD and
FPS regimes. In the second region, a0 is of order 1, but b0 Z 0:01a20 is still small,
so pull-off is in the FPS regime. Since qZ0.01, there is a small region where a0 is
large, but b0 remains small. In this narrow region, the FP, FPS and JKR regimes
are all valid. Finally, a0 becomes so large that only the JKR regime is valid. For
larger q, e.g. qZ1 and 10, the second region where FPS is valid disappears. In the
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977
normalized plateau pull-off force
Effect of preload in indentation tests
–10 – 6
SD regime
–10 – 4
–10 –2
–1
JKR regime
–10 2
–10 4
10 –3 10 –2 10 –1
1
10
10 2
dimensionless parameter q
10 3
pffiffiffiffiffiffiffiffiffiffiffiffi
Figure 8. Normalized plateau pull-off force FN versus qZ ðpE Þ2 = 4k 3=2 R 2Weff . The circles
(numerical results) are obtained by solving (2.13a) and (2.13c). The solid line is given by equation
(4.2). The dash-dotted (equation (4.1a)) and dashed (equation (4.1b)) lines are the predictions of
the JKR and SD regimes, respectively.
plateau region, neither the SD nor the JKR regime is approached. Of particular
interest is the case of large q (figure 7d ). For this case, the pull-off force is well
approximated by (3.17a) and (3.17b) (SD limit) for practically all preloads.
Denote the plateau pull-off force by FN and its normalization by
FNZ 8Rk 2 FN=ðpE Þ3 . The normalized plateau pull-off force depends only on
the dimensionless parameter q (see (3.7)), which is inversely proportional to the
square root of the effective work of adhesion. The dependence of j FN j on q is
shown in figure 8. Note that a large q implies small indenter radius, small
effective work of adhesion, soft fibrils or stiff backing layer. Of the five regimes,
only the SD and JKR regimes predict a plateau pull-off force. FN in these regimes
can be obtained using (3.17a), (3.17b) and (3.39a)
3 K4=3
ffiffiffi q
FN Z K p
zK0:94qK1:3
3
2 4
4
FN ZK q K3=2 zK1:3q K1:5
3
ðJKR regimeÞ;
ðSD regimeÞ:
ð4:1aÞ
ð4:1bÞ
Figure 8 shows that j FN j is a monotonically decreasing function of q. In
addition, for large and small q, FN is governed by (4.1a) and (4.1b), respectively.
We found that the numerical results for FN can be well approximated by
FN ZK
16 C 30q K4=3
:
12q 3=2 C 8:5q C 31:5
ð4:2Þ
Denote the minimum preload where the plateau pull-off force is first reached by
Fp . We shall call Fp the critical preload. Just as FN, the normalized critical
preload Fp depends only on q. Using (3.18) and (3.20), F p in the SD and JKR
regimes are
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978
normalized plateau pull-off stress
R. Long and C.-Y. Hui
–10 –2
–10 –1
JKR regime
–1
SD regime
–10
10 –3 10 –2 10 –1
1
10
10 2
dimensionless parameter q
10 3
pffiffiffiffiffiffiffiffiffiffiffiffi
Figure 9. Normalized plateau pull-off stress sc versus qZ ðpE Þ2 = 4k 3=2 R 2Weff . The circles
(numerical results) are obtained by solving (2.13a) and (2.13c). The solid line is equation (4.8). The
dash-dotted (equation (4.7a)) and dashed (equation (4.7b)) lines are the prediction of the JKR and
SD regimes, respectively.
1 K4=3
ffiffiffi q
F p Z p
z0:31q K1:3
3
2 4
2
F p Z q K3=2 z0:67q K1:5
3
ðJKRÞ;
ðSDÞ:
ð4:3aÞ
ð4:3bÞ
An expression that agrees well with these numerical results and is consistent with
(4.3a) and (4.3b) is given by
F p Z
2 C 2q K4=3
:
3q 3=2 C 2:75q 7=6 C 6:35
ð4:4Þ
(b ) Pull-off stress
Another measure of adhesion is the pull-off stress, which is the pull-off force
divided by the contact area at pull-off. The pull-off stress is harder to measure,
since it may not be possible to measure the contact area directly. These two
characterizations of adhesion are very different. For example, the pull-off force
can be very small in the ELS regime (figure 9), even though the theoretical upper
bound for the pull-off stress is achieved in the ELS regime. The existence of a
plateau pull-off force implies that the pull-off stress should also approach a
plateau. Denote the plateau pull-off stress by sc. Equations (2.13e) and (4.1a)
and (4.1b) imply that
sc
q F
2kac
Z N;
ac Z
:
ð4:5Þ
kdc
2ac
pE Since kdc is the maximum stress the interface can withstand, we normalize the
plateau pull-off stress by kdc, i.e.
s
sc Z c :
ð4:6Þ
kdc
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Effect of preload in indentation tests
979
Since ac depends only on q, equation (4.5) implies that sc depends only on q. As
shown in figure 8, large q favours the SD, whereas small q favours the JKR
regime. The behaviours of sc in these two limits are
3
ð4:7aÞ
sc ZK 7=3 q 1=3 ðJKRÞ; q/ 1;
2
2
ðSDÞ; q[ 1:
ð4:7bÞ
sc ZK
3
Figure 9 plots sc versus q. The asymptotic behaviour of sc predicted by (4.7a)
and (4.7b) is also plotted for comparison. The expression
6q C 12
sc ZK
ð4:8Þ
9q C 15:2q 1=3 C 20:16q K1=3
is consistent with (4.7a) and (4.7b) and provides an excellent fit to our
numerical results.
5. Summary
We studied the effect of preload on the force needed to pull a circular cylindrical
indenter off a microfibril array. The fibril array was modelled as an elastic
foundation and the backing layer was assumed to be infinitely thick. In the
preload phase, we obtained an approximate expression of normalized preload
as a function of a dimensionless parameter, a, which can be interpreted as a
normalized contact width. In the unloading phase, the force required to unpin
and to move the contact line is determined by two dimensionless parameters
a and b. We found five regimes that are characterized by different values of
a and b. These regimes are chosen because the dependence of the pull-off force on
preload can be determined by simple analytical expressions. We also show that
there exists a critical preload above which the pull-off force reaches a plateau
value. This critical preload, as well as the plateau pull-off force, depends on
a single dimensionless parameter q defined by (3.7). Simple analytical expressions are also obtained for the plateau pull-off force and the critical preload. In
the design of fibrillar adhesives, one is often interested in the pull-off stress
instead of the pull-off force. Equation (4.8) relates the plateau pull-off stress to
geometry and material properties through the dimensionless parameter q.
Our analysis suggests several ideas that can lead to more accurate
characterization of adhesion of microfibre arrays. First, it is important to
conduct tests in the plateau region, where the pull-off force is independent of the
preload. One can reach the plateau region by applying preload greater than the
critical preload Fp, which is given in (4.4) for all q. Second, the normalized
plateau pull-off force FN is also obtained as a function of q. Since q is inversely
proportional to the square root of the effective work of adhesion (3.7), we are able
to calculate the effective work of adhesion by simply measuring the plateau pulloff force. Third, it is useful to recognize that the pull-off force depends on both the
indenter radius (similar to the JKR theory) and the backing layer properties. For
example, the effect of a thin backing layer is to increase E, which increases q.
This will increase the pull-off load. Finally, while it is reasonable to characterize
adhesion using G/Amax (see §1, third paragraph), it is not useful to characterize
Proc. R. Soc. A (2009)
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980
R. Long and C.-Y. Hui
adhesion using Fc/Amax. Unlike the concept of the pull-off stress, Fc/Amax
approaches zero for large preload since Fc approaches a constant, whereas Amax
increases with preload. This characterization of adhesion may mislead one to believe
that adhesion is small, while in reality, it is not.
This work is supported by a grant from the Department of Energy (DE-FG02-07ER46463) and by
a grant from the National Science Foundation (CMS-0527785). C.Y.H. appreciates discussions
with Anand Jagota.
Appendix A. Derivation of (3.27)
Assume a perturbation solution of the form
fðXÞ ZK1 C af1 ðXÞ C/C ak fk ðXÞ C/:
ðA 1Þ
Substitute (A 1) into (2.13a) results in
ð1
fkK1 ðS ÞdS
df
Z k ; k Z 1; 2; .;
ðA 2Þ
X
KS
dX
K1
with f0ZK1. For kR1, fk satisfies fk(XZG1)Z0 and is obtained by integrating
(A 2). It can be shown that
fðXÞ ZK1 C a½2 ln 2Kð1 C XÞlnð1 C XÞKð1KXÞlnð1KXÞ C Oða2 Þ: ðA 3Þ
Equation (3.27) is obtained by integrating (A 3).
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