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Research
Cite this article: Liu Y, Gao Y. 2015
Non-uniform breaking of molecular bonds,
peripheral morphology and releasable
adhesion by elastic anisotropy in bio-adhesive
contacts. J. R. Soc. Interface 12: 20141042.
http://dx.doi.org/10.1098/rsif.2014.1042
Received: 17 September 2014
Accepted: 20 October 2014
Subject Areas:
biomechanics, mathematical physics
Keywords:
interface fracture mechanics, molecular bonds,
releasable adhesion
Authors for correspondence:
Yan Liu
e-mail: [email protected]
Yanfei Gao
e-mail: [email protected]
Non-uniform breaking of molecular
bonds, peripheral morphology and
releasable adhesion by elastic anisotropy
in bio-adhesive contacts
Yan Liu1,2 and Yanfei Gao2
1
2
Tianjin First Central Hospital, Tianjin Medical University, Tianjin 300192, People’s Republic of China
Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA
Biological adhesive contacts are usually of hierarchical structures, such as the
clustering of hundreds of sub-micrometre spatulae on keratinous hairs of
gecko feet, or the clustering of molecular bonds into focal contacts in cell
adhesion. When separating these interfaces, releasable adhesion can be accomplished by asymmetric alignment of the lowest scale discrete bonds (such as
the inclined spatula that leads to different peeling force when loading in different directions) or by elastic anisotropy. However, only two-dimensional
contact has been analysed for the latter method (Chen & Gao 2007 J. Mech.
Phys. Solids 55, 1001–1015 (doi:10.1016/j.jmps.2006.10.008)). Important questions such as the three-dimensional contact morphology, the maximum to
minimum pull-off force ratio and the tunability of releasable adhesion
cannot be answered. In this work, we developed a three-dimensional cohesive
interface model with fictitious viscosity that is capable of simulating the
de-adhesion instability and the peripheral morphology before and after the
onset of instability. The two-dimensional prediction is found to significantly
overestimate the maximum to minimum pull-off force ratio. Based on an
interface fracture mechanics analysis, we conclude that (i) the maximum
and minimum pull-off forces correspond to the largest and smallest contact
stiffness, i.e. ‘stiff-adhere and compliant-release’, (ii) the fracture toughness
is sensitive to the crack morphology and the initial contact shape can be
designed to attain a significantly higher maximum-to-minimum pull-off
force ratio than a circular contact, and (iii) since the adhesion is accomplished
by clustering of discrete bonds or called bridged crack in terms of fracture
mechanics terminology, the above conclusions can only be achieved when
the bridging zone is significantly smaller than the contact size. This
adhesion-fracture analogy study leads to mechanistic predictions that can be
readily used to design biomimetics and releasable adhesives.
1. Introduction
Two clean and atomically smooth surfaces, when being brought together, will
have strong adhesive force because of intermolecular interactions such as van
der Waals force. Macroscopic surfaces do not retain this property because surface
roughness results in a tiny fraction of surface area in true contact, and thus the
overall adhesion force is negligibly small. Biological systems have evolved into
intriguing surface structures to maximize the adhesion/de-adhesion forces by
hierarchical structures. For instance, some animals and insects have keratinous
hairs, called setae, on their feet, which then have hundreds of small hairs,
called spatulae, of sub-micrometre diameters [1,2]. The small size in each hair
enables a nearly uniform stress distribution during de-adhesion, thus avoiding
crack-like behaviour that usually corresponds to low pull-off force. Also the
long hairs can easily bend to conform to the surrounding rough surface and
thus to increase the fraction of true contact area. For another example, cell
adhesion is accomplished by micrometre-sized focal contacts, each of which
& 2014 The Author(s) Published by the Royal Society. All rights reserved.
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2. Problem definition and interface model
As shown by the boundary value problem in figure 1, we aim
to calculate the pull-off force that separates a finite contact
(circular, elliptical or arbitrary shape) that is connected to an
elastically anisotropic substrate by discrete bonds. Rather
than treating these bonds individually, we assume there is a
uniform distribution of these bonds and the interface
constitutive law can be described by an interface traction –
separation relationship in figure 2. The fracture mechanics
analysis dictates that the de-adhesion force depends on the
fracture energy and the fracture strength of the interface, but
not on the detailed shape of the traction –separation law
[12,17]. The use of bilinear model in figure 2 applies well for
long range forces such as van der Waals interaction, or
nonlinear springs such as spatulae or micromachined pillar
arrays. As pointed out by Freund & Lin [18], the bio-adhesive
contacts in cells are realized by mobile receptor–ligand bonds
on the interface, but not by any kind of chemical signalling
process. Therefore, cell adhesion is primarily governed by
thermodynamic and mechanical driving forces. The receptor –ligand bonding energy is about 1–10 times kBT with
kB being the Boltzmann constant and T being absolute
2
J. R. Soc. Interface 12: 20141042
which corresponds to the Griffith crack limit [16]. Since many
bio-adhesive contacts will be close to the uniform stress limit,
it is unclear whether the degree of orientation dependence of
pull-off forces in the above calculations still remains large.
Second, both elastic anisotropy and actual bio-adhesive contacts are of three-dimensional nature. The pull-off force is
closely connected to the evolution of crack morphology
which in turn depends on elastic anisotropy and initial contact zone shape. For example, the long-chain molecular
bonds may break non-uniformly in space so that an initial circular contact may evolve into an elliptical-shaped crack. The
synergy of all these factors can certainly lead to a releasable
adhesion that is very different from predictions in [14].
In this work, we develop a cohesive interface model that
describes the interface fracture process of the long-chain molecules in bio-adhesive contacts. It should be noted that the
Griffith limit corresponds to a sudden release of applied
energy and a snap-back instability will occur, while no
instability occurs for the uniform stress limit. To examine
the onset of instability and simulate the post-stability evolution, we have adopted a fictitious viscosity approach in
the numerical implementation of the cohesive interface
model into a nonlinear finite-element framework [12], as
will be discussed in §2. A parametric study has been conducted to study (i) the transition from Griffith to uniform
stress limits as governed by the ratio of crack bridging zone
to contact radius, (ii) the degree of elastic anisotropy on the
ratio of maximum-to-minimum pull-off forces, and (iii) the
evolution of the crack front or peripheral morphology
with respect to the crack bridging characteristics and the
degree of elastic anisotropy. Representative examples include
the twofold symmetric surface of Cu (110) in §3 and the
transverse isotropic material in §4. Limitations of the twodimensional results in [14] have been identified. Asymptotic
limits of interface fracture and contact stiffness analysis discussed in §5, so as to develop design rules that can be used
to tune the orientation dependence of pull-off forces.
Conclusions are given in §6.
rsif.royalsocietypublishing.org
contains the clustering of a larger number of molecular
receptor–ligand bonds. The adhesion and de-adhesion of
these focal contacts lead to elastic deformation of the surrounding material, which is termed mechanosensitivity [3–6]. There
are also many other examples in which the adhesive contacts
comprise discrete ‘bonds’ in biological systems such as placental
villi in embryo implantation, and in artificial systems such
as micromachined pillar arrays or microfludics with subsurface
microstructures [7], and in geological systems such as
multi-asperity contact [8].
A central problem in adhesion and de-adhesion of biological surface structures is how these long-chain discrete
bonds (being spatulae or receptor –ligand bonds) are
broken. Would these bonds be broken simultaneously, or
the debonding starts somewhere and propagates into a certain spatial pattern? How do these discrete bonds feel the
neighbouring bonds and the faraway applied load? We
note that the de-adhesion process from a finite contact area
is equivalent to an interface fracture problem, where the
crack is external and can be of various shapes. In the limit
of Griffith crack, the stress near the contact edge diverges to
infinity, and the fracture occurs when the applied energy
release rate (that is, the energy drop with a unit length
increase of the crack front) reaches the interface fracture
energy [9]. This scenario applies to breaking brittle materials
such as glass and ceramics. When the interface is made of
long-chain molecules, the low interface stiffness may significantly reduce the stress singularity at the contact edge or
crack front, with the limit case being that the entire interface
has a uniform stress that equals the interface theoretical
strength [10–12]. This scenario applies to tearing a Velcro
patch or pulling off gecko toes that comprise a huge number
of hairs. The transition from the Griffith limit to the uniform
stress limit is governed by the ratio of the crack bridging
zone size to the contact radius [10–12], which is also called
the stress concentration index [13]. Therefore, how the longchain bonds break spatially is closely connected to the above
transition, which has important consequences in designing
surface structures to achieve optimized adhesion force.
Another central problem in bio-adhesive contact is the
orientation-dependent adhesion strength, so that releasable
adhesion can be easily achieved when loading in various
directions. In the hierarchical structure, this is accomplished
by the alignment of discrete bonds (such as seta –spatula
structure in gecko toes) in inclined directions, so that the
pull-off force varies significantly with respect to the pulling
direction [2]. This is like peeling a Velcro or Scotch tape
at different attack angles from the substrate. Besides this
structural point of view, the orientation dependence of deadhesion force can be obtained by the elastic anisotropy of
the substrate [14,15]. In the context of fracture energy, the
energy release rate is inversely proportional to the moduli
in the plane spanned by the crack growth direction and the
crack plane normal. This line of thought has been used in
natural systems such as the attachment pad of cicada and
grasshopper which have anisotropic microstructure and
thus elastic anisotropy. A plane-strain elastic analysis of circular adhesive contact with a substrate of transverse
isotropy found that the maximum pull-off force can be
achieved if pulling in the direction normal to the plane of
transverse isotropy [2,14]. However, there are two severe
limitations of these works. First, their solutions are based
on the so-called Johnson –Kendall– Roberts adhesive contact,
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(a)
cell focal contact
F
b
Ft
molecular bonds
substrate
applied force
x3
x2
F
and
bF
aF
x1
Dta
smax
,
þ zD_ ta
D
dmax
where the total separation is
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
< D2n þ D2t1 þ D2t2 ,
D ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
: D2 þ D2 ,
t1
t2
substrate
(c)
Tta ¼ T(D)
applied displacement
x3
x2
U
bU
aU
substrate
Figure 1. (a) Schematic of the cell focal contact by clustering of receptor –
ligand bonds. (b,c) An initially circular contact may evolve into an elongated
morphology when subjected an inclined force F or displacement U. The major
axis of the adhesive region generally does not align with the applied F or U
because of elastic anisotropy. (Online version in colour.)
temperature. Therefore, one immediate consequence of thermodynamic fluctuation is the stochastic behaviour of the
receptor–ligand molecular bonds [19,20]. The lifetime of an
individual bond is of the order of 100 s, while the de-adhesion
process is much faster. Consequently, a mean-field stochastic
elastic model of dissociation and association of receptorligand bond [13] will be adopted. Details of this model will
not be pursued here, since essential features can also be well
captured in the traction –separation law in figure 2. It should
be noted that the same constitutive parameters are used
across the entire interface, which implies that we do not consider a spatial variation of the bond density. In reality, the
bond density can be higher at the contact boundary because
of entropic effects. Such an observation can be easily incorporated in our model by assigning different constitutive
if Dn 0
(2:2)
(2:3)
if Dn , 0:
In the above model, kn is taken as orders of magnitude higher
than kc. When implementing the above relationship into a
nonlinear finite-element framework, we need to use the
material tangent, given by Jkl ¼ @ Tk/@ Dl, where subscripts k
and l run as (n, t1, t2). A fictitious viscosity, z, is introduced
in order to successfully simulate the snap-back instability
that is intrinsic to Griffith-type cracks [12,24] and adhesive
contacts at the Griffith limit [16].
In the geometric set-up in figure 1, we differentiate the
applied force or displacement conditions, which are not
necessarily in the same direction because of the elastic anisotropy. Note that our finite-element method is displacement
based. We consider a circular or elliptical surface contact on
a half space, and the boundary extends 20 times the contact
radius. The normalized fictitious viscosity is optimized to be
_ dmax 103 ; a smaller value than this will lead to smaller
zU=
time steps and longer calculation time, and a larger value will
lead to excessive stress due to the viscous force term in
equations (2.1) and (2.2). Particularly, this choice of normalized
fictitious viscosity only introduces negligible effect on the
calculated pull-off or adhesion force [12]. The shape of the
interface traction–separation law is fixed, e.g. dmax/dc ¼ 0.5,
and the interface fracture energy, G ¼ 12smax dc , is kept constant
in our simulations so that the variations of smax and dmax will
be inversely proportional to each other. A total number of
about 50 000 three-dimensional eight-node elements are
used. The cohesive interface model is implemented into a commercial finite-element software, ABAQUS, by the user defined
element subroutine.
Our numerical simulations will be specifically interested
in the following questions. First, the pull-off force depends
on the loading direction because of the elastic anisotropy.
We will explore representative anisotropic materials including cubic crystal and transversely isotropic material. One
question that naturally arises is in which loading directions
we can find the pull-off force extrema. Second, during deadhesion, the contact periphery will evolve into a shape dictated by the elastic anisotropy, which can be monitored by
J. R. Soc. Interface 12: 20141042
(b)
3
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Fn
parameters across the interface, which however is not pursued
in this work for the sake of clarity [21,22]. We also note that the
adhesive interaction can also be modelled in a body force
formulation [23], which will not be pursued in this work.
The bilinear model in figure 2 has three governing
parameters: the interface strength smax, the characteristic
length scale dmax and the decay length dc. The total separation
can be projected into the normal and two tangential directions in local coordinates. When Dn , 0, the two contacting
surfaces are not penetrable, so only tangential separations
contribute to the total separation. Following [24], we write
the relationship as
8
Dn
smax
<
, if Dn 0
T(D) þ zD_ n
Tn ¼
(2:1)
D
dmax
:
if Dn , 0
kn Dn ,
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(a)
4
T
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smax
kc
dc
dmax
tn, Dn
tt, Dt
local n-direction
surface S+
local t2-direction
GP
local t1-direction
surface S–
Figure 2. (a) A general traction– separation model that accounts for a variety of ‘molecular’ adhesions in biological systems. (b) Finite-element simulation requires
the projection of traction and separation on the Sþ/S2 plane. GP denotes Gaussian integration point in the finite-element method.
finding the contours of D ¼ dmax. It is particularly interesting
to find the relationship between the evolution of peripheral
morphology and the pull-off forces.
3. Anisotropic elastic substrate with cubic
symmetry
s33 (r) ¼
Representative load–displacement curves are plotted in figure 3
for Cu (110) surface with respect to various loading directions
and interface properties. The elastic constants are given table 1
as Material (A), where the Voigt contracted form, cij, is used.
The twofold surface of this material exhibits a significant elastic
anisotropy. Note that the elastic constants of typical biological
materials are of the order of kPa–GPa, as opposed to approximately 100 GPa in metals and ceramics. However, the purpose
of our study is merely to illustrate the role of the degree of elastic
anisotropy, and the absolute value of elastic constants will be
embedded in the normalization factor.
As the first observation, after extensive numerical tests, we
found that an applied tangential displacement along the [110]
direction gives the minimum pull-off force, and an applied
normal displacement in the [110] direction corresponds to
the maximum pull-off force. To explain this observation,
we need to refer to the analytical solutions of the contact problem [16]. Regardless of the indenter shape, a circular contact
under shear loading is equivalent to an external circular crack
under shear force, with the interface shear stress distribution
given by
F1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
(3:1)
2pa a2 r2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where r is the radial coordinate, r ¼ x21 þ x22 . At the contact
edge, the stress singularity is the same as that in a crack. The
mode II (in-plane shear) and mode III (anti-plane shear)
stress intensity factors are
s13 (r) ¼
KII ¼
F1
F1
pffiffiffiffiffiffi cos u and KIII ¼ pffiffiffiffiffiffi sin u,
2a pa
2a pa
where u ¼ tan1 (x2 =x1 ). The non-adhesive normal contact
depends on the indenter shape, but a normal pull-off force
applied on an adhesive contact is of the same form as the
flat-ended punch contact problem. We have the normal
stress distribution and the mode I (in-plane tension) stress
intensity factor
(3:2)
F3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pa a2 r2
and
KI ¼
F
p3ffiffiffiffiffiffi :
2a pa
(3:3)
The above solutions apply to any arbitrarily anisotropic solid
as long as the contact shape is circular. The de-adhesion process is thus equivalent to a fracture problem with mixed
modes as in equations (3.1)–(3.3).
In the linear fracture mechanics (i.e. the Griffith limit of the
crack characteristics), fracture occurs when the applied energy
release rate, G, reaches the interface fracture energy, G. Although
the stress intensity factors are purely governed by the applied
load, crack size and crack shape, the energy release rate depends
additionally on the elastic constants. For instance, the isotropic
elasticity gives
1 n2 2
1
2
2
G¼
(3:4)
K ,
KI þ KII þ
1 n III
E
where E and n are Young’s modulus and Poisson’s ratio, respectively. Substituting equations (3.2) and (3.3) into (3.4), one can
easily prove that for an isotropic elastic solid subjected to a constant force with varying loading directions, the tangential load
will give the highest G and the normal load gives the lowest
G. Therefore, de-adhesion is the most difficult in normal direction, and the easiest in tangential direction. Our simulations
for isotropic elasticity (not shown here for brevity) have confirmed this conclusion, in which the pull-off force depends on
b and n. For an anisotropic elastic solid, the relationship between
G and K is quite complicated, which is out of the scope of this
work. Considering the limit cases of normal and tangential loading conditions, an elementary tensor transformation shows that
the tensile modulus in normal direction (which enters in the
relationship of G and KI) is higher than other directions, and
the shear modulus in (x2, x3) plane is higher than that in (x1,
J. R. Soc. Interface 12: 20141042
(b)
Δ
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[110] x3
5
x2 [001]
copper
0.07
0.06
Cu (110) surface
load in aU = 60o, bU =30o
F/(c11a2)
0.05
smax/c11 = 0.04
dmax/a = 0.005
normal load
0.04
smax/c11 = 0.02
dmax/a = 0.01
0.03
smax/c11 = 0.01
dmax/a = 0.02
0.02
0.01
shear in [110]
direction
0
0
0.01
0.02
0.03
0.04
0.05
0.06
U/a
Figure 3. De-adhesion of a circular contact from the Cu (110) surface. At a given set of interface parameters, depending on the direction of the applied U, the pulloff force and the load – displacement stiffness vary. Calculations that discontinue on or slightly after the peak force are those cases which intentionally do not use
the fictitious viscosity method, so that snap-back instability leads to numerical divergence. (Online version in colour.)
x3) plane (which enter in the relationship of G and KII, KIII). It is
thus anticipated that tangential loading in x1-direction gives the
largest G that corresponds to the lowest pull-off force Fmin, and
the normal loading gives the lowest G that corresponds to the
largest pull-off force Fmax. However, the contact deformation
fields are not simply directional or shear, and a more detailed
analysis based on contact stiffness will be given in §5.
The second important observation is the role played by the
crack bridging characteristics. The stress singularity in the
linear elastic fracture mechanics will be limited by the interface
strength. The relative significance of the surrounding elastic
stiffness and the interface stiffness defines a length scale,
c11dmax/smax, which determines the size of the crack bridging
zone. If c11 dmax =smax a, i.e. the small-scale bridging (SSB)
behavior, the crack approaches the Griffith crack. The largescale bridging (LSB) behaviour, when c11dmax/smax approaches
or is larger than the crack size, approaches the uniform stress
limit. The releasable adhesion, as characterized by the ratio of
Fmax/Fmin, varies significantly with respect to the ratio of crack
bridging zone size to contact radius, c11dmax/smaxa. As shown
in figure 4a, the SSB limit at c11 dmax =smax a 1 corresponds to
the Griffith crack. From equations (3.1)–(3.4), one can derive
the following dimensionless relationship:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F c12 c44
c11 dmax
¼g
,
, a, b
,
(3:5)
psmax a2 SSB
c11 c11
smax a
where the dimensionless pre-factor, g, depends on the elastic
anisotropy and loading direction. In the LSB limit, the
pull-off force simply approaches the product of uniform
stress and contact area
F ¼ 1,
(3:6)
psmax a2 LSB
which is independent of elastic anisotropy and loading direction.
Obviously, the degree of releasable adhesion, Fmax/Fmin, only
deviates from unity when the controlling parameter c11dmax/
smaxa is far away from the LSB limit, as shown in figure 4b.
The transition from SSB to LSB correlates closely to the
crack shape, as the third important observations from our
numerical simulations. As shown in figure 5, the peripheral
morphology, or the crack front shape, is illustrated by the contours of crack opening displacement, D/dmax. The left column
is close to the LSB asymptote, so that the crack bridging zone
is large and the crack shape exhibits a strong dependence on
loading direction and elastic anisotropy. That is, the tangential
loading in figure 5a leads to an elliptical shape with major axis
in the compliant direction in shear, the inclined loading in
figure 5b leads to a rotated elliptical shape (but the major axis
does not necessarily align with the loading direction), and
the normal loading in figure 5c gives rise to an elliptical
shape with major axis in the compliant direction in tension.
The right column is close to the SSB asymptote, so that the
J. R. Soc. Interface 12: 20141042
smax/c11 = 0.08
dmax/a = 0.0025
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x1 [110]
npt ¼ 1.67 1026
Fmax: normal load
F/(psmaxa2)
mt ¼ 4 kPa
c66 ¼ 7.27 kPa
mt ¼ 4 kPa
npt ¼ 0.0167
c44 ¼ 4.0 kPa
c66 ¼ 7.27 kPa
6
1
Fmin: shear load in ·110Ò direction
J. R. Soc. Interface 12: 20141042
c44 ¼ 4.0 kPa
(a)
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Cu (110) surface
10–1
ntp ¼ 0.375
1.5
1.4
1.2
1.1
1.0
c12 ¼ 121.4 GPa
c33 ¼ 459.2 kPa
Et ¼ 450 kPa
c33 ¼ 4.5 GPa
Et ¼ 4.5 GPa
c11 ¼ 168.4 GPa
c11 ¼ 23.6 kPa
Ep ¼ 20 kPa
c11 ¼ 23.27 kPa
Ep ¼ 20 kPa
(A) copper
(B) transversely isotropic material (L22/L11 ¼ 4.41)
(C) transversely isotropic material (fictitious) (L22/L11 ¼ 440)
0.9
10–2
elastic constants
Cu (110) surface
Fmax: normal load
Fmin: shear load in ·110Ò direction
1.3
Fmax/Fmin
np ¼ 0.375
ntp ¼ 0.375
c13 ¼ 12.0 kPa
np ¼ 0.375
c12 ¼ 8.727 kPa
c44 ¼ 75.4 GPa
c12 ¼ 9.055 kPa
c13 ¼ 12.25 kPa
(b)
materials
Table 1. Three types of elastically anisotropic materials are investigated in this work. The fictitious transversely isotropic material has an extraordinarily large Et, in order to test the effects of ratio of Et/Ep or the parameter in equation
(4.1) on the pull-off force.
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10–1
1
c11dmax/(smaxa)
10
Figure 4. The maximum and minimum pull-off forces are calculated for Cu
(110) surface with respect to the ratio of the crack bridging zone size to the
contact radius, c11dmax/smaxa. (a) Double logarithmic plot of pull-off force
with respect to c11dmax/smaxa approaches 1/2 slope at the Griffith asymptote, and approaches unity at the LSB limit. (b) The releasable adhesion
by elastic anisotropy, as described by Fmax/Fmin, can only be achieved
when the Griffith asymptote is approached, i.e. c11 dmax =smax a 1.
crack bridging zone is narrow but the degree of eccentricity
and rotation of the elliptical crack morphology is the same as
that in the left column. In all these contour plots, the limits
are set as D/Ddmax [ [0.9, 1.1] for better visualization.
4. Anisotropic elastic substrate with transverse
isotropy
Many biological materials are transversely isotropic [5,6].
Following the notations in Bower [25] as shown in figure 6a,
the elastic constants include the directional modulus in p and t
directions, Ep and Et, in-plane and out-of-plane Poisson’s ratios,
np and ntp, and shear modulus mt. The degree of elastic anisotropy
can be characterized primarily by the following ratio [14]:
qffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffi
1 n2p
L22
Et
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
(4:1)
L11
E
p
1 n2 E =E
tp p
t
A representative set of parameters is given in table 1 as
Material (B) [6]. Keeping all other elastic constants the same
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(a)
7
Y
X
J. R. Soc. Interface 12: 20141042
Y
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field-1
8.384
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
0
field-1
2.138
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
0
X
Z
Z
(b)
field-1
1.887
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
0
field-1
7.000
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
0
Y
Y
X
Z
X
Z
(c)
field-1
1.788
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
0
field-1
6.350
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
0
Y
Y
X
Z
X
Z
left column:
x2 [001]
right column:
smax/c11 = 0.01
dmax/a = 0.02
smax/c11 = 0.04
x1 [110]
dmax/a = 0.005
Figure 5. Representative peripheral morphologies of the contact when pulling off the contact in various directions: (a) shear displacement in [110] direction,
(b) inclined displacement with aU ¼ 608 and bU ¼ 308, and (c) normal displacement. The contour plots are normalized crack opening displacement,
D/dmax, at the peak load, which indicates the location of the crack front. Since the left (or right) column is for a weak (or strong) interface, the cohesive bridging
zone size is large (or small) and D/dmax in the range of [0.9,1.1] can be found in a broad (or narrow) band.
but varying Et, we get a fictitious Material (C) with a very
large degree of elastic anisotropy, L22/L11 ¼ 440.
To ensure a two-dimensional contact problem between a
cylinder and the transversely isotropic material, the cylinder
axis lies along the x1-direction in figure 6a, but the plane of transverse isotropy can rotate about x1. Chen & Gao [14] solved the
corresponding adhesive contact problem at the Griffith crack
limit and determined the ratio of Fmax/Fmin when loading in various directions in the (x2, x3) plane. It is found that Fmax/Fmin is
about 4 when L22/L11 ¼ 100 and about 12 when L22/L11 ¼ 1000,
and the scaling relation can be approximated by Fmax/Fmin ln(L22/L11). However, in three-dimensional contacts, because
Downloaded from http://rsif.royalsocietypublishing.org/ on July 31, 2017
(a)
8
t out-of-plane direction
x [00 1] t-direction
2
X2 = [010]
p in-plane direction
x1 [100] p-direction
X1 = [100]
(b)
0.35
transversely isotropic substrate loaded on · p, tÒ surface
elongated in p-direction
shear in t-direction
L /L = 440
22
0.30
11
smax/c11 = 0.08
elongated in t-direction
dmax/a = 0.0025
0.25
F/(c11a2)
shear in p-direction
0.20
elongated in p-direction
0.15
0.10
0.05
0
0.005
0.010
0.015
0.020
U/a
Figure 6. (a) Schematic of a transversely isotropic material with p being the in-plane direction and t being the out-of-plane direction. The contact is applied on
k p, tl plane. (b) Representative load – displacement curves with the change of the loading direction and the contract shape (circle and ellipses with major axis lying
along p- or t-directions). (Online version in colour.)
of the finite size in the x1-direction and the possibility of loading
in arbitrary direction not in the (x2, x3) plane, their conclusions
need to be revisited.
For a circular contact loaded on the k p, tl plane, our extensive
simulations find that the maximum pull-off force corresponds to
tangential loading in the t-direction and the minimum one to
tangential loading in the p-direction in figure 6a. As shown in
figure 7, the dependence of Fmax/Fmin on c11 dmax =smax a that
describes the cracking bridging behaviour is exactly the same
as that for Cu (110) surface. For the Material (C), the transition
from LSB to SSB occurs at a very low ratio of c11 dmax =smax a,
suggesting that the degree of elastic anisotropy actually plays
a minor role in a wide range of cohesive zone size.
The ratio of Fmax/Fmin from a three-dimensional contact
analysis is much smaller than the prediction from [14]. We
now examine if the two-dimensional limit solution can be
approached if the initial contact shape is elongated as
opposed to circular. This line of thought is motivated by
the crack morphology study in figure 8. In calculations of
figures 7 and 8, L22/L11 ¼ 440 so that Et Ep . Consequently, when loaded in t-direction (here a tangential load in
x2-direction), the crack will initiate at u ¼ 08 and 1808, and
fracture will be extremely difficult at u ¼ +908 because the
extremely large Et leads to a very small energy release rate.
For tangential loading along x1-direction, the shear modulus
anisotropy leads to a lens-shaped crack morphology in
figure 8a. In order to maximize (or minimize) the pull-off
force, we can ensure a large fraction of the contact edge lying
in the stiff (or compliant) direction. For instance, the right
column plot in figure 8a can be changed to an elliptical contact
J. R. Soc. Interface 12: 20141042
plane of transverse isotropy
rsif.royalsocietypublishing.org
[010] x3
X3 = [001]
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(a)
9
t-direction
x2
rsif.royalsocietypublishing.org
–
x2 [001]
x2
x1 [100]
p-direction
x1
x1
(b)
J. R. Soc. Interface 12: 20141042
3.0
transversely isotropic substrate loaded on ·p, tÒ surface
2.5
Fmax /Fmin
ellipse (amajor /aminor = 4), L22 /L11 = 440
2.0
ellipse (amajor/aminor = 4), L22/L11 = 4.41
circle, L22 /L11 = 4.41
1.5
circle, L22 /L11 = 440
1.0
circle, Cu (110) surface
10–3
10–2
10–1
c*11dmax/(smaxa)
1
10
Figure 7. The releasable adhesion by elastic anisotropy, as described by Fmax/Fmin, can only be achieved for a small ratio of c11 dmax/smaxa. Note that c11 is taken as
c11 of Material (A) for calculations of both Materials (A) and (B). The two-dimensional results by Chen & Gao [14] for separating a cylinder (with the axis lying down
along the x1-axis) and a half space can only be approached for amax =amin 1 and c11 dmax =smax a 1. (Online version in colour.)
(with the same contact area and the major axis lying in the compliant direction), so that the uncracked fraction will be
maximized at the peak pull-off force. This leads to a much
larger pull-off force in the right column plot in figure 8b than
that in the right column plot in figure 8a. Similarly, the left
column plot in figure 8b increases the fraction of the contact
edge with low shear modulus, which leads to much lower
pull-off force than the left column plot in figure 8a. It should
be noted that the contour plots in figure 8 have different
bounds of D/dmax for better illustration of the crack fronts.
Our results find that the elongated shape with amajor/
aminor ¼ 4 increases the ratio of Fmax/Fmin by 60%, as shown
in figure 7b. However, the limit values at c11 dmax =smax a 1
are still far less than the two-dimensional results in [14].
between G and K is complicated for anisotropic solids and
varies significantly with respect to the crack orientations
[9]. The contact problem by itself also leads to complicated
stress fields that combine both tensile/compressive and
shear components. All the above simulation results and the
contact-fracture analogy suggest the validity of the ‘stiffadhere and compliant-release’ concept [2], in which the
orientation-dependent adhesion maximizes in the stiffest
direction—for strong bonding, and minimizes in the most
compliant direction—for easy de-adhesion. The contact
stiffness analysis has been given in [26] by
and
5. Contact stiffness governs the directions
of Fmax and Fmin
In the above sections, the effects of elastic anisotropy are discussed in terms of the directional modulus and shear
modulus, which will enter the energy release rate representation (e.g. equation (3.4)). The detailed relationship
rffiffiffiffi
A
F1 ¼ 8Geff,x1
U1 ,
p
rffiffiffiffi
A
F3 ¼ 2Eeff
U3 ,
p
F2 ¼ 8Geff,x2
rffiffiffiffi
A
U2
p
(5:1)
where A ¼ pa2 ¼ pamajor aminor is the contact area, Geff,x1 and
Geff,x2 are effective shear moduli, and Eeff is the effective
normal stiffness. These effective moduli depend on the elastic
constants and the orientations of the three axes and can be
calculated from a line integral along the contact periphery
[26]. Referring to figure 1 where the direction of the applied
Downloaded from http://rsif.royalsocietypublishing.org/ on July 31, 2017
(a)
10
Y
Y
X
J. R. Soc. Interface 12: 20141042
field-1
1.100
1.010
1.008
1.005
1.003
1.000
0.998
0.995
0.993
0.990
0
rsif.royalsocietypublishing.org
field-1
3.406
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
0
X
Z
Z
(b)
field-1
6.260
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
0
field-1
1.0200
1.0010
1.0007
1.0005
1.0002
1.0000
0.9997
0.9995
0.9992
0.9990
0
Y
Y
X
X
Z
Z
(c)
field-1
2.020
1.100
1.075
1.050
1.025
1.000
0.975
0.950
0.925
0.900
0
field-1
1.3800
1.0100
1.0075
1.0050
1.0025
1.0000
0.9975
0.9950
0.9925
0.9900
0
X
Y
Y
Z
X
Z
–
x2 [001]
left column:
shear in [100]
direction
right coloumn:
–
shear in [001]
direction
x1 [100]
Figure 8. Contour plots of the normalized crack opening displacement, D/dmax, at the peak load, which correspond to the six cases in figure 6b: (a) circular contact,
(b) elliptical contact with major axis lying in p-direction, and (c) elliptical contact with major axis lying in t-direction. The left column is for shear displacement in
p-direction, and the right one in t-direction. Note that the right column has different upper and lower limits for D/dmax contours. These peripheral morphologies
correlate well to the search of Fmax/Fmin.
displacement is described by aU and bU, we have
rffiffiffiffi
A
U
F¼2
p
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(4Geff,x1 cos bU cos aU )2 þ(4Geff,x2 cos bU sin aU )2 þ(Eeff sin bU )2 :
(5:2)
Therefore, finding the pull-off force extrema is equivalent to
finding the extrema of stretching stiffness, dF/dU, as shown
in figure 9.
For isotropic material, Eeff ¼ E=(1 n2 ) and Geff ¼
E=[2(1 þ n)(2 n)], and the stretching stiffness is given in
the polar plot in figure 9a. The normal contact gives the
Downloaded from http://rsif.royalsocietypublishing.org/ on July 31, 2017
(a)
stretching stiffness, dF/dU, for isotropic elasticity, n = 0.3
(b)
11
90
1.0
0.774
3
72
0.6
0.
82
5
0.622
3
0.67
23
0.774
150
30
30
0.4
0.4
0.825
0
[1-10]
0.937
240
1. 0
37
210
330
0. 97
radial coordinate: cos(b)
angular coordinate: a
0.774
240
300
270
300
270
radial coordinate: cos(b)
angular coordinate: a
(c)
330
0.825
0. 7
1.002
(d)
stretching stiffness, dF/dU, for transverse isotropy (L22/L11 = 4.41)
[00-1] 90 1.0
60
120
150
stretching stiffness, dF/dU, for transverse isotropy (L22/L11 = 440)
[00-1] 90 1.0
1.352
1.212
1.073
0.934
0.6
1.212
49.68
1.073
30
150
0.4
0.934
5
180
[010]
0.795
0.934
210
1.212
240
radial coordinate: cos(b)
angular coordinate: a
0.6
37.42
0.4
25.17
cos(b) = 0.2
0
[100]
30
25.17
180
0
[010]
12.91
12.91
25.17
25.17
37.42
210
330
37.42
49.68
[100]
330
49.68
61.94
240
300
270
49.68
37.42
12.91
12.91
5
0.79
0. 79
5
0.934
1.073
1.212
1.352
1.073
0.8
61.94
cos(b) = 0.2
0.79
60
120
0.8
radial coordinate: cos(b)
angular coordinate: a
300
270
Figure 9. The normalized load – displacement stiffness, (dF/dU )/(2aE) or (dF/dU)/(2ac11), plotted against the loading direction aU and bU for representative
materials. Maximum pull-off force, Fmax, corresponds to the highest stiffness, and Fmin to the lowest stiffness. (Online version in colour.)
highest contact stiffness so that the pull-off force maximizes,
while the minimum pull-off force corresponds to the tangential loading. For Cu (110) surface in figure 9b, the stiffest
direction is normal contact and the most compliant one is
shear in [110] direction. For the two transversely isotropic
materials, the stiffest direction is always shear in t-direction
and the most compliant direction is shear in p-direction.
Note that these contact stiffness calculations in figure 9 are
all for circular contacts. One can optimize the shape of the
contact so as to obtain a pull-off range [Fmin, Fmax] that is
much wider than that of the circular contact.
—
—
—
6. Conclusion
Interface fracture mechanics and cohesive interface simulations have been conducted to analyse the non-uniform
breaking of molecular bonds in bio-adhesive contacts. Motivated by a recent two-dimensional analytical solution of
pulling off circular contacts from a transverse isotropic
material, we systematically investigate the roles played by
three-dimensional contact shape, the degree of elastic
anisotropy and the crack bridging characteristics.
— Since the adhesion is accomplished by clustering of discrete bonds or so-called bridged crack, the de-adhesion
process is controlled by the transition from SSB behaviour
—
(i.e. the Griffith crack limit) to LSB behaviour (i.e. the
uniform stress limit), as the dimensionless parameter
c11dmax/smaxa increases.
The maximum and minimum pull-off forces correspond to
the largest and smallest contact stiffness, i.e. ‘stiff-adhere
and compliant-release’, which confirms the previous
two-dimensional investigations in Yao & Gao [2].
The pull-off extrema are closely related to the crack morphology. That is, the compliant direction will fracture
more easily, and the resulting crack morphology corresponds to, but inversely, the contact stiffness contours.
The two-dimensional analysis in Chen & Gao [14] significantly overpredicts the pull-off force extrema, due to the
restricted contact shape into an infinitely long strip and
due to the mere consideration of the Griffith limit.
These predictions shed interesting light on the design of
biomimetics and releasable adhesives. The maximum (or
minimum) pull-off force can be optimized by searching the
largest (or lowest) contact stiffness, and by arranging
the initial contact shape so that a large fraction of the contact
periphery lies in the stiffest (or most compliant) directions.
Funding statement. Y.L. acknowledges the support from Tianjin Bureau
of Public Health ( project 2012KZ025), as well as a visiting scholar
position provided by the University of Tennessee during the course
of this study. Y.G. is grateful for the financial support from NSF
CMMI 0900027 and CMMI 1300223.
J. R. Soc. Interface 12: 20141042
37
0.9
7
23
0.7 73
0.6
7
1.06 034
1.
34
0.9
1. 0
0. 9
210
[110]
0.774
23
0.67
3
0.622
0.774
0.622
180
0
cos(b) = 0.2
02
1. 0
1.002
180
cos(b) = 0.2
0.937
0.97
1. 0 0 2
4
1.03
67
1.0
67
0.9
7
0.8
0. 7
37
0.9
150
60
120
60
0.93
0.970.8 7
1.002 0.6
1.034
0.
120
rsif.royalsocietypublishing.org
stretching stiffness, dF/dU, for Cu (110) surface
[001] 90 1.0
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1.
12