# On the long and short-range adhesive interactions in viscoelastic contacts

G. Violano<EMAIL_ADDRESS>Department of Mechanics, Mathematics and
Management, Polytechnic University of Bari, Via E. Orabona, 4, 70125, Bari,
Italy  L. Afferrante  Department of Mechanics, Mathematics and Management,
Polytechnic University of Bari, Via E. Orabona, 4, 70125, Bari, Italy

###### Abstract

Recently, tribologists have shown increasing interest in rate-dependent
phenomena occurring in viscoelastic fractures. However, in some cases,
conflicting results are obtained despite the use of similar theoretical
models. For this reason, we try to shed light on the effects that long and
short-range adhesion has on the pull-off force in the contact of viscoelastic
media by exploiting a recently developed numerical model.

We find that, in the limit of long-range adhesion, the unloading velocity has
little effect on the pull-off force, which is close to the value predicted by
Bradley for rigid bodies. In such case, the detachment process is
characterized by a uniform bond-breaking of the contact area, and viscous
dissipation involves the bulk material.

For medium(short)-range adhesion, the pull-off force is instead a monotonic
increasing function of the pulling velocity and, at high speeds, reaches a
plateau that is a function of the adiabatic surface energy. In this case, the
detachment process is similar to the opening of a circular crack, and viscous
dissipation is localized at the contact edge.

##  I Introduction

In the 1970s, two pioneering contact mechanics theories were formulated to
study the adhesion between elastic spheres. First, Johnson, Kendall & Roberts
(JKR) JKR1971  exploited a thermodynamic approach to include adhesion in the
Hertz theory. JKR solution assumes infinitely short-range adhesive
interactions inside the contact area, that cause a deformation of the Hertzian
contact profile. Later, Derjaguin, Muller & Toporov (DMT) DMT1975  proposed a
different solution where weak long-range adhesive interactions are localized
outside the contact area, without deforming the Hertzian gap.

Both JKR and DMT theories predict a pull-off force, i.e., the maximum tensile
force reached during the detachment, that is independent of the elastic
properties of the half-space and equal to $1.5\pi\Delta\gamma R$ and
$2\pi\Delta\gamma R$, respectively, being $R$ the radius of the sphere and
$\Delta\gamma$ the adiabatic surface energy.

Tabor Tabor1977  first clarified that JKR theory is accurate in the limit of
high values of a dimensionless parameter $\mu$ (known as Tabor parameter),
namely for soft materials with high surface energy and radii of curvature. On
the contrary, DMT theory is accurate in the limit of $\mu\ll 1$. Notice DMT
pull-off force is the same calculated by Bradley Bradley1932  in the limit of
rigid bodies.

Maugis Maugis1992 , exploiting the Dugdale cohesive law, formulated a theory,
known as Maugis-Dugdale (MD) theory, which allows to capture the JKR-DMT
transition in terms of $\mu$. Therefore, he showed that, moving from JKR to
DMT limit, the pull-off force ranges in between $1.5\pi\Delta\gamma R$ and
$2\pi\Delta\gamma R$, as also confirmed by a successive numerical study
Greenwood1997 .

In modern adhesive systems, soft materials are widely used to enhance adhesive
features. Such materials exhibit rate-dependent adhesion as a consequence of
their intrinsic viscoelasticity, while JKR, DMT, and MD theories cannot
capture rate effects and viscous dissipation. In fact, experiments have shown
that the viscoelastic pull-off force may be order of magnitudes larger than
the value predicted by elastic adhesion theories (see, for example, Ref.
Violano2021a ).

Recently, Das & Chasiotis (DC) DC2021  proposed to extend MD model to the case
of a rigid sphere indenting a viscoelastic half-space. In their works,
unloading starts from a relaxed state of the viscoelastic material and is
performed at fixed rate of the applied force.

In the limit of short-range adhesive interactions, DC model predicts the pull-
off force to be a monotonic increasing function of the unloading rate. If
$F_{\mathrm{PO,0}}$ denotes the elastic pull-off force, the maximum pull-off
force that can be reached at high unloading rate is
$F_{\mathrm{PO,\max}}=(F_{\mathrm{PO,0}}E_{\infty})/E_{0}$, being $E_{\infty}$
and $E_{0}$ the instantaneous and relaxed moduli of the viscoelastic material,
respectively. However, such value cannot be reached for any value of the Tabor
parameter (see, for example, Refs. ViolanoSIZE ; Muser2021 ; Ciavarella2021 ).

Moreover, when long-range adhesive interactions are predominant, namely when
$\mu\rightarrow 0$, DC model continues to predict
$F_{\mathrm{PO,\max}}=(F_{\mathrm{PO,0}}E_{\infty})/E_{0}$ at high unloading
rate, while in Ref. Ciavarella2021  it is observed that
$F_{\mathrm{PO,\max}}\rightarrow F_{\mathrm{PO,0}}$ independently of the value
of the unloading rate.

In this work, we try to clarify such qualitative and quantitative discordance
by performing contact simulations with the fully deterministic model developed
in Ref. Afferrante2022 , where adhesive interactions are modeled with Lennard-
Jones based potential and viscoelasticity with the standard linear solid.

##  II Statement of the problem

Consider a rigid sphere, with radius of curvature $R$, pulled off from a
viscoelastic substrate. Detachment starts from static equilibrium conditions
as the sphere initially approaches the substrate at low speed up to a maximum
penetration $\delta_{\max}$. Retraction is then performed at a constant speed
$V=-d\delta/dt$.

The contact problem is solved using the finite element (FE) model presented in
Ref. Afferrante2022 , at which the reader is referred for further details. We
here recall that: i) interface interactions are modelled by a traction-gap law
based on Lennard-Jones potential, according to the proximity approximation
(i.e., the force acting between two bodies is expressed in terms of the force
between two semi-infinite planes); ii) the viscoelastic behaviour of the
substrate is modelled with the standard linear solid according to the Maxwell
representation.

We are interested in studying how the pull-off force is affected by the
retraction speed $V$, in the cases of short, medium, and long range adhesive
interactions. In the elastic case, the Tabor parameter is defined as

$\mu=\frac{R^{1/3}}{\epsilon}\left[\frac{\Delta\gamma(1-\nu^{2})}{E_{0}}\right]^{2/3}\text{,}$
(1)

where $E_{0}$ is Young’s modulus of the substrate, $\epsilon$ is the range of
action of attractive forces, and $\nu$ is Poisson’s ratio.

For a viscoelastic material, Young’s modulus $E(\omega)$ is a function of the
frequency of excitation and hence varies during the detachment process.
Viscoelastic modulus $E(\omega)\rightarrow E_{0}$ only when detachment occurs
in quasi-static conditions, namely when $V\rightarrow 0$. As a result, while
in the elastic case we are sure that $\mu$ uniquely describes the trend of the
pull-off force, in the viscoelastic case this it is not necessarily true.
However, we can reasonably assume that long-range adhesion is expected for low
values of the adiabatic surface energy $\Delta\gamma$, resulting in $\mu\ll
1$, while medium and long-range adhesion occurs at higher $\Delta\gamma$. In
the following, we shall discuss our results in terms of $\Delta\gamma$ rather
than $\mu$.

##  III Results and Discussion

All plots are given for $E_{\infty}/E_{0}=10$ and $\nu\approx 0.5$, and in
terms of the following dimensionless quantities:
$\hat{F}_{\mathrm{PO}}=F_{\mathrm{PO}}/(\pi\Delta\gamma R)$;
$\Delta\hat{\gamma}=\Delta\gamma/(E_{0}R)$; $\hat{V}=V\tau/\epsilon$, being
$\tau$ the relaxation time of the viscoelastic material. Furthermore, the
unloading phase started after reaching the penetration
$\hat{\delta}_{\max}=\delta/\epsilon=17.7$.

Figure 1: The normalized pull-off force $\hat{F}_{\mathrm{PO}}$ in terms of
the dimensionaless pulling velocity $\hat{V}$. Finite element calculations
(markers) are shown for $\Delta\hat{\gamma}=1.35\times 10^{-7}$, $3.73\times
10^{-6}$, $6.75\times 10^{-4}$, $1.35\times 10^{-3}$. DMT and JKR elastic
limits are also shown as a reference.

Figure 1 shows the normalized pull-off force $\hat{F}_{\mathrm{PO}}$ in terms
of the pulling velocity $\hat{V}$, which is constant during the retraction of
the sphere. Results are obtained for different values of the adiabatic surface
energy $\Delta\hat{\gamma}$ and for $\hat{V}$ ranging from $0.0033$ to $100$.
First, let us consider what happens when $V\sim 0$; for the lower values of
$\Delta\hat{\gamma}$, the DMT limit is returned, being
$\hat{F}_{\mathrm{PO}}\sim 2$. As a result, adhesion is basically described by
long-range interactions. On the contrary, for the higher values of
$\Delta\hat{\gamma}$, $\hat{F}_{\mathrm{PO}}\sim 1.5$ and the JKR limit is
recovered being dominant the short-range adhesion. In all cases,
$\hat{F}_{\mathrm{PO}}$ is a monotonic increasing function of $\hat{V}$ and
reaches a maximum asymptotic value $\hat{F}_{\mathrm{PO,\max}}$ at high
pulling velocities. However, notice $\hat{F}_{\mathrm{PO,\max}}$ is strongly
affected by the value of $\Delta\hat{\gamma}$.

In this regard, fracture mechanics and cohesive zone models (see, for example,
Refs. greenwood2004 ; PB2005 ) suggest that
$F_{\mathrm{PO,\max}}/F_{\mathrm{PO,0}}=E_{\infty}/E_{0}$, being
$F_{\mathrm{PO,0}}$ the pull-off force in the elastic limit. Our calculations,
instead, returns $F_{\mathrm{PO,\max}}/F_{\mathrm{PO,0}}\sim
0.8E_{\infty}/E_{0}$ for $\Delta\hat{\gamma}=$ $1.35\times 10^{-3}$ and
$F_{\mathrm{PO,\max}}/F_{\mathrm{PO,0}}\sim 0.14E_{\infty}/E_{0}$ for
$\Delta\hat{\gamma}=$ $1.35\times 10^{-7}$. These results agree with the
theoretical findings given in Ref. Ciavarella2021 , where it is suggested that
there is an upper bound of the pull-off force that depends on the Tabor
parameter. However, this is not the only reason for which
$F_{\mathrm{PO,\max}}/F_{\mathrm{PO,0}}$ is less than $E_{\infty}/E_{0}$;
indeed, in a recent study ViolanoSIZE , we have shown that geometric and
finite-size effects influence $F_{\mathrm{PO,\max}}$, which decreases when
retraction starts from smaller initial contact radii $a_{\max}$.

Figure 2: The maximum pull-off force $\hat{F}_{\mathrm{PO,\max}}$ in terms of
the dimensionaless adiabatic surface energy $\Delta\hat{\gamma}$.
$\hat{F}_{\mathrm{PO,\max}}$ is calculated at high pulling speed
($\hat{V}=100$). Green markers refer to finite element calculations, red and
blue dashed lines are DMT and JKR elastic limits, respectively, and black
dashed line is the theoretical limit predicted at high speeds by fracture
mechanics models.

To highlight the effect of the adiabatic surface energy on the maximum pull-
off force, Fig. 2 shows $\hat{F}_{\mathrm{PO,\max}}$ in terms of
$\Delta\hat{\gamma}$ and for $\hat{V}=100$. Results disagree with DC findings
DC2021 , where $\hat{F}_{\mathrm{PO,\max}}$ is predicted to increase when we
move from short-range to long-range adhesive interactions for all rates of
unloading (see Figure 6C in Ref. DC2021 ). Therefore, in DC calculations,
$\hat{F}_{\mathrm{PO,\max}}$ is surprisingly found to be a decreasing function
of the Tabor parameter.

However, we have to observe that our outcomes agree with results given in Ref.
Ciavarella2021 , where a similar model to DC one, based on Maugis-Dugdale
approach, is adopted. Our results are also in agreement with recent numerical
findings of Muser & Persson (MP) Muser2021 , who performed calculations on the
detachment of a rigid cylinder from a viscoelastic substrate. They found a
decrease in $\hat{F}_{\mathrm{PO,\max}}$ when reducing the Tabor parameter
$\mu$. However, they changed the size of the radius $R$ of the indenter
maintaining the same value of $\Delta\hat{\gamma}$. In such conditions, size
effects due to $a_{\max}$ may arise, leading to additional variations in the
pull-off force ViolanoSIZE ; moreover, at small scales, a transition of the
detachment mode from crack propagation to uniform bond breaking can also occur
ViolanoSIZE ; Muser2021 ; Gao2004 .

In addition, Van Dokkum et al. VanDokkum2021  observed, in a similar problem,
the normalized effective surface energy
$\Delta\gamma_{\mathrm{eff}}/\Delta\gamma$ (or equivalently
$\hat{F}_{\mathrm{PO}}$) decreases with the Tabor parameter. Furthermore,
Jiang et al. Jiang2021 , performing FE calculations on the pull-off force of a
spherical indenter from a soft viscoelastic stamp, observed a reduction of
$\hat{F}_{\mathrm{PO}}$ when decreasing $\Delta\hat{\gamma}$. Similarly to our
approach, they changed the value of $\Delta\hat{\gamma}$ maintaining the same
maximum penetration $\delta_{\max}$.

Our results find confirmation also in the experimental measurements of Ahn &
Shull (AS) Ahn1998 , who performed axisymmetric adhesion tests between a
hemispherical elastomeric cap and a variety of flat substrates with different
values of interface energy. Moving from the JKR formalism, they used a
standard fracture mechanics approach to estimate the effective surface energy
$\Delta\gamma_{\mathrm{eff}}$ according to the phenomenological equation
$\Delta\gamma_{\mathrm{eff}}=\Delta\gamma[1+f(V_{\mathrm{c}})]$ MB1980 , where
the quantity $f(V_{\mathrm{c}})$ takes into account the increase in surface
energy due to viscous dissipation and depends only on the contact line
velocity $V_{\mathrm{c}}=-da/dt$ (provided the process is assumed occurring at
constant temperature).

Figure 3: The increase in the dissipated energy density at pull-off. Results
are given for a pulling speed $\hat{V}=1$, with $\Delta\hat{\gamma}=$
$1.35\times 10^{-3}$ (A) and $\Delta\hat{\gamma}=1.35\times 10^{-7}$ (B).

Figures 3A-B show the increase in the dissipated energy density
${\Delta}U_{\mathrm{D}}$ at pull-off, for $\hat{V}=1$ and $\Delta\hat{\gamma}$
equal to $1.35\times 10^{-3}$ and $1.35\times 10^{-7}$, respectively. For high
values of the adiabatic surface energy (Fig. 3A), viscous dissipation is
localized at the edge of contact, suggesting that the detachment process is
similar to the mechanism of opening of a circular crack Afferrante2022 . For
low $\Delta\hat{\gamma}$ (Fig. 3B), a much lower viscous dissipation is
observed. It mainly involves the bulk material, suggesting the occurrence of a
different debonding mechanism, as clarified in Figs. 4A-B, where the normal
displacement (net of the rigid one) is shown at three different times close to
the pull-off instant (namely the instant at which the tensile force is
maximum).

Figure 4: Displacement fields in the vicinity of the pull-off instant, for a
pulling speed $\hat{V}=1$, with $\Delta\hat{\gamma}=$ $1.35\times 10^{-3}$ (A)
and $\Delta\hat{\gamma}=1.35\times 10^{-7}$ (B). The dimensionless time
$\hat{t}=t/\tau$ increases in the direction of the arrow. The displacements
are given net of the rigid one.

Results are given for the two cases corresponding to Figs. 3A-B. For
$\Delta\hat{\gamma}=$ $1.35\times 10^{-3}$, the debonding mechanism is clearly
related to crack propagation; for $\Delta\hat{\gamma}=$ $1.35\times 10^{-7}$,
the displacement field instead moves homogeneously leading to a quasi-uniform
bond breaking.

##  IV Conclusions

In this paper, we investigate the effect of the adhesive interactions on the
pull-off force of viscoelastic bodies when the detachment process starts from
a relaxed equilibrium state of the material.

In the limit of long-range adhesion ($\Delta\gamma\rightarrow 0$), quasi-
uniform bond-breaking describes the mechanism of detachment, and viscous
dissipation occurs in the bulk material. As a result, the pull-off force is
not affected by the pulling speed and tends to the rigid limit of Bradley
Bradley1932 .

When medium(short)-range interactions occur, the detachment process is similar
to that of a propagating circular crack. In this case, the pull-off force is a
monotonic increasing function of the pulling speed. At high speeds, the pull-
off load reaches a plateau, which depends on $\Delta\gamma$.

Therefore, we can state that the pull-off force is limited by the strength of
adhesion ($\Delta\gamma$), rate of unloading and finite-size effects
($a_{\max}$). This is the reason why the ratio between the actual pull-off
force and its elastic value does not approach the theoretical limit
$E_{\infty}/E_{0}$ at high pulling speeds.

Such outcomes also have important implications in problems of viscoelastic
fracture mechanics.

## ACKNOWLEDGEMENTS

G.V. and L.A. acknowledge support from the Italian Ministry of Education,
University and Research (MIUR) under the programme ”Departments of Excellence”
(L.232/2016).

## DECLARATIONS

### Conflict of interest

The authors declare that there are no conflicts of interests within the
outlined work.

### Fundings

The authors declare that no funds, grants, or other support were received
during the preparation of this manuscript.

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