Frictional Dynamics of Soft and Hard Solid Interfaces.pdf
1. Frictional dynamics of soft and hard solid
interfaces
Presented by:
Dr. Vinit Gupta
13-03-2021 1
2. Overview of present research works
Shear frictional interaction at soft-hard / hard-hard
solid interfaces (Rate and State dependent friction)
Molecular bonded contacts
(Tire-road contact, gel friction)
Asperity contacts
(Tectonic sliding)
Population balance frictional
modeling
Experimental validation using pure
gelatin hydrogels
Velocity strengthening friction regime
(Static friction, Steady dynamic friction,
Stress relaxation)
Unification of friction behaviour in certain set of frictional
parameters
Velocity weakening friction regime
(Stick-slip friction)
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3. Application of soft gel in sliding contact
Source: Google Images
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4. Source: 1. Google Images, 2-3. Singh et al. (GRC,2015), 4. Yamaguchi et al. (2009)
1. 2.
3. 4.
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5. Literature review
• Frictional theories for interfacial interaction.
1. Amontons’ Coulomb frictional laws:
• Force of friction seems to be independent of apparent area of contact. (Amontons’
first law)
• Force of friction at the sliding interface is directly proportional to applied load i.e.
where is the frictional force at the interface, is the coefficient of friction and is the
applied normal load. (Amontons’ second law)
• Kinetic friction is almost independent of sliding velocity and is less in magnitude
compared to static friction force. (Coulomb’s law)
2. Bowden and Tabor theory of asperity adhesion:
• A.C friction laws were first explained by asperity adhesion theory for rough surfaces.
• The theory proposes that asperities of rough interacting surfaces weld against each
other, resulting in shear plastic deformation of asperities and increase in real area of
contact, at the interface.
• Coefficient of friction is defined as a material property dependent on two other
material properties such as yield stress and shear stress and is independent of area of
contact.
• Bowden and Tabor theory was later modified and elastic deformation at the interfaces
was also included.
• Later, Tabor also accommodated the effect of aging time on interfacial friction in his
theory of asperity adhesion.
• This theory failed to explain the frictional behaviour at soft deformable surfaces and
liquid contacts.
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6. Contd.
3. Schallamach theory of bond rupture:
• Experiments by Schallamach on rubber solids, later by Milne on soft gel sheets and
other researchers established shear rate and aging time dependence of rubber and soft
solid friction.
• A.C friction laws were violated as kinetic friction was observed to be dependent on
shear sliding rate.
• Schallamach utilized the Eyring theory of rate reaction to model the friction at soft
solid interfaces considering the formation and rupture of molecular bonds between
the polymer chains and the opposing substrate.
4. Rate and State friction behaviour:
• Gradually, a rate and state dependent friction model for tectonic sliding was
established.
• Dietrich (1979) performed extensively experimentation on rock sliding and have
derived the empirical formula as:
• Ruina (1983) proposed that the friction in such surfaces is also dependent on time of
contact or hold time at the interface.
0
0
/ ln
v
A
v
0
0
0
/ ln ln
v
v
A B
v L
13-03-2021 6
2
0 / /(1 / )
F N MV t t
7. Contd.
5. Relationship between adhesion, friction and fracture
• Surface energy was used as an independent property to relate friction to adhesion.
• Gent and Petrich (1969) made measurements to find out the forces required to pull
off a thin layer of viscoelastic adhesive from a non-deformable substrate.
• An approximate relation between peel strength and stress-strain behaviour is
obtained on basis of interfacial bond strength.
• Johnson et al. (1971) have considered the effect of surface energy on the contact
interface if one of the body is highly deformable and subjected to low normal load
condition. They gave the expression for contact area and adhesive force based on
surface energy.
• JKR expression gives the external force required to separate surface on interfaces
with given surface energy and contact geometry.
• Roberts and Jackson (1975) reported experiments of sliding of rubber spheres on
glass. They analysed frictional energy based on JKR model. They reported that
sliding at rubber interface occurs as Schallamach waves and that the surface energy
is dependent on sliding rate.
• Spurr (1982) studied the frictional behaviour at polymeric interfaces. He also
correlated the friction and adhesion at the interface via. surface energy.
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8. Contd.
6. Origin of gel friction:
• Milne (1958) conducted sliding friction experiments on waxy grease. He observed that
the frictional stress depends on history of contact particularly time of loaded contact
prior to actual sliding.
• Rubio and Galeano (1994) sheared a transparent gelatin gel sheet at low shear rate in
between coaxial circular cylinder. They studied stress relaxation behaviour at the
interface and found it to be consistent with earthquake like dynamics.
• Gong et al. (1997) had investigated the friction behaviour of different hydrogels when
slid against a glass, steel or gel substrate. They through their experimental analysis
established that these hydrogel seems to violate Amonton-Coulomb’s friction laws and
shows negligible dependence on normal load in their velocity and load ranges.
• Dependence of dynamic friction on shear rate was an important finding.
• Spatio-temporal stick slip is observed normally in sliding of adhesive gel and elastomer
sheets on glass substrate.
• The stress drop events in spatio-temporal stick-slip behaviour are observed to follow a
power law and were later successfully fitted with Gutenberg-Richter’s law.
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9. Contd.
7. Rate and state dependent friction of gelatin hydrogels:
• Gelatin hydrogels when slid on smooth substrate shows shear rate dependence of static
and dynamic strength as was observed first by Baumberger et al. (2002,2003)
• It was reported that there exists logarithmic dependence of static frictional strength on
aging time.
• They also have shown rate dependence of static and dynamic frictional strength.
• Finally, it was established that there exist a critical transition velocity that separates both
stick-slip and steady sliding frictional regime.
• Singh (2010) have extensively studied rate and aging time dependence of pure gelatin
gels and gelatin gels mixed with glycerol and surfactants.
8. Population balance friction models:
• Singh and Juvekar (2010, 2011, 2014, 2016) have proposed the friction models for rate
dependent friction at soft-hard interface for dynamic friction, stress relaxation and then
for static friction using the concept of population balance of bonds formed and ruptured
at the contact interface.
• Population balance equation:
Assumptions: Large no. of bonds are formed and ruptured at the contact interface
amounting to interfacial frictional strength.
0
, ,
( ) ,
a a a
f a a b a
a
n t t n t t dt
r N N t t n t t r t
t t dt
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10. Contd.
• Singh and Juvekar (2011) have proposed the following non-dimensional
mathematical equation governing dynamic friction stress vs. sliding velocity
of soft solids as:
• During the sliding of a soft solid on a substrate, the dangling chains experience
another type of resistance owing to the surrounding chains which is known as
viscous retardation effect.
• The modified equation obtained after considering the effect of the viscous
retardation in the friction model is as follows:
0
̂ 0
ˆ
V
0 0
0 1 1
0 0 0
1
0 0 0
exp
ˆ ˆ
ˆ ln
ˆ ˆ ˆ
exp
ˆ ˆ ˆ
u u
V V u u u
G E
V V V
u u u
u E
V V V
0 0 0 0 0
0 0
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
1 1 0 1
0 0 0 0 0 0 0 0
0
ˆ ˆ
ˆ ˆ
1
0 0 0
1 1
ˆ
ˆ
exp ln exp
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ
1
1 exp
ˆ ˆ ˆ
V V V V V
V V
u u u u u u
e G e E e V e E e
V V V V V V V V
u u
e E e
V V V
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11. Contd.
• The FENE based dynamic friction model is proposed by Singh and Juvekar (2011) as:
• Later (2014) they have also proposed strong bond friction model for relaxation to
residual strength at the interface, which is slightly modified to consider the viscous
retardation in the current work, is as:
• Where,
2 2
0 0 0
0
2
0 0
ˆ
ˆ ˆ
ˆ
, ,
ˆ ˆ ˆ
ˆ
ˆ
ˆ
ˆ
1 ,
ˆ ˆ
m
m m
m
m
m
L u u
G L E L
V V V
L
L u
E L
V V
2 3
2 3
2 3
1 1 1 11 1 1 1 1 3 12 1 3
2 2
0 0
1 1 1 1 3
0
1 1 1 1 3
0
1
( ) ( ) log ( ) ( ) ( ) ( ) log ( ) ( )
ˆ ˆ
ˆ ( ) ( ) ( )
ˆ
1
1 ( ) ( ) ( )
ˆ
z z
z
e e
ws
z z
z
ws
z z
z
ws
e
e G z G z z E z E z G z z E z
V V
e
e E z E z E z
e
e E z E z E z
V
0 11 12 22 0
0 0 0 0
1 0 2 0 3 22 0
ˆ ˆ
ˆ ˆ
exp( ) , , , exp( )
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ
exp , * exp 1 , exp
w w w w
ws
ws ws
wsw wsw wsw ws
u u u u
z V z z z V
V V V u V u
z z V t z z V t z z V t
13-03-2021 11
12. Contd.
• Finally, Juvekar and Singh (2016) also evolved the expression of static friction at soft and
hard interface for creeping rupture of bonds that is expressed in dimensionless form as:
• Where,
ˆ
ˆ
1 exp
ˆ
ˆ 1
s
i
t
dX
r u X
dt X N
0
ˆ
ˆ
ˆ ˆ ˆ
1 exp
ˆ
1
ˆ
ˆ
ˆ 1 1
i
i
i s
t
V N X u
X N
d
dt N X r
0
0
ˆ , ,
i
s
i g
N N N M
N
N X r
N N K
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13. Outcome of literature review and Research
gap
1. Important outcomes of literature relevant to research area:
• When a soft solid such as gel or elastomer is sheared on a hard surface, it shows a
variety of interesting phenomenon, for example rate and aging time dependent
frictional strength, propagation of Schallamach waves, existence of stick-slip
instabilities.
• Many spatio-temporal events at soft/hard solid interfaces are analogous to the slip
events along the tectonic plates (Yamaguchi et al., 2009, 2011).
• As far as the modelling of friction of a soft and hard solid interface is concerned, the
expression for static frictional strength, dynamic and residual strengths has been
derived.
2. Research gap
• The friction models proposed by Singh and Juvekar (2014, 2016) have not validated
on hydrogels. However, they have validated the weak bond model for dynamic
friction on PDMS and glass interface.
• It is not known how the residual strength of a sliding interface affects the frictional
properties of hydrogels.
• There is no methodology or model available that can predict the critical velocity of
the sliding interface.
• There is no study in literature which intends to unify all shear frictional phenomena
on soft-hard contact interface in terms certain number of friction parameters.
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14. Aim and Objectives
1. Aim of the research work:
• The aim of the present research works is to unify all the frictional properties measured
in SHS experiments in terms of certain basic physical parameters related to the
population balance based friction models.
2. Objectives:
• To carry out dynamic friction experiments on hydrogels of varying gelatin
concentrations. Also to validate the WBF model using the steady friction data for
establishing the scaling laws of dynamic friction.
• To investigate stress relaxation behaviour of the hydrogel-glass interface in SHS tests.
Also to verify the SBF model using the stress relaxation data for proposing the scaling
laws.
• To study experimentally the static friction strength of soft-hard interfaces for
validating the static friction models.
• To perform SHS and SFS tests to compare the static, dynamic, residual strengths and
also critical velocity.
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15. Organization of research work
• Chapter 1 of the thesis gives overview of the work including important aim and
objectives.
• Chapter 2 gives the literature review, outcomes and research gap.
• Part of objective 1,2,3,4 are achieved in contributory research work that is organized
as:
Chapter 3
• SHS experiments are performed.
• Shear modulus, mesh size and properties of gel are worked out.
• Weak bond friction model is validated for steady dynamic friction.
• Scaling laws were established for density of adhering chains and viscous retardation
coefficient.
• Basic friction parameters were worked out paving way for unification of friction
phenomenon.
Chapter 4
• Strong bond friction model is validated for stress relaxation behaviour.
• Micromechanics of stress relaxation is discussed.
• Importance of residual stress is highlighted.
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16. Contd.
Chapter 5
• Static friction is studied using SHS experimental data.
• Limitation of Juvekar and Singh model is discussed in prediction of static friction.
• Experiments are validated considering weak bond theory.
Chapter 6
• SFS tests are performed.
• Effect of residual strength on frictional properties is studied.
• Critical sliding velocity of the interface is predicted using residual strength and is
experimentally validated.
Chapter 7 of the thesis work gives important conclusions, achievements and scope
of future work.
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18. Sample preparation and Experimentation
1. Sample preparation:
• Gelatin hydrogels are prepared using porcine skin gelatin of bloom strength 300, by
Sigma Aldrich® in double distilled water.
• Gelatin powder is used in concentrations, 6%, 8%, 10%, 12% and 15% wt./vol. in
water to prepare hydrogels.
• The mixture is then thoroughly stirred at around 80ºC for one hour and is then
allowed to cool at room temperature.
• It is then refrigerated in an aluminium mould at around 8°C for 24 hours.
• Resulting gel block is then cut into small rectangular pieces to have a contact area of
500 mm².
2. Experimentation:
• Sliding of soft adhesive hydrogel is actuated using servo-system on a smooth
relatively non-deformable glass surface at different pulling velocities in the range
of 0.3-5.0 mmps.
• Normal stress of around 0.5 kPa is applied on the top surface of gel block at the back
end of the plate to prevent lifting of gel block due to bending moment acting at the
trailing end of the gel block during sliding.
• SHS experiments were carried out at a fixed waiting time of 60s between two slide
steps.
• Also the experiments were carried out in an air conditioned atmosphere maintained at
humidity level of around 50-60% and at temperature of approximately 18ºC.
13-03-2021 18
20. Weak bond friction model and equation of
motion
• First, it is tried to fit the FENE chain based weak bond friction model, but the values of
were too high, for the model to correctly justify the sliding of hydrogels.
• Finally the weak bond Hookean chain model is fitted with experimental data which is
given by the following equation:
Here, non-dimensional terms are defined as:
where,
where,
Equation of motion:
However, as the inertia is negligible, spring force at the cantilever plate equals frictional
force at the interface.
ˆm
l
0 0 0 0 0
0 0
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
1 1 0 1
0 0 0 0 0 0 0 0
0
ˆ ˆ
ˆ ˆ
1
0 0 0
1 1
ˆ
ˆ
exp ln exp
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ
1
1 exp
ˆ ˆ ˆ
V V V V V
V V
u u u u u u
e G e E e V e E e
V V V V V V V V
u u
e E e
V V V
*
0 0
ˆ /
* * *
0 B
n k T KV t
*
0 0
ˆ /
V V V
*
B
V k T M
ˆ / M
0
( )
c c
K V V mV
13-03-2021 20
22. Weak bond model validation for steady
dynamic friction
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23. Properties of Gelatin hydrogels and frictional
parameters
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Table1. Properties of Gelatin Hydrogels
Table 2. Frictional parameters for model validation
25. Important findings in the study of steady dynamic
friction
• The plot between and mesh size is presented. It is observed that decreases with
increasing and the scaling law is estimated to be as
• This observation is expected as number dangling chains decreases with increase in
mesh size for a fixed area of contact.
• The estimated theoretical value of in Table 1 is expected to be larger than the
experimentally estimated value in Table 2 due to dynamic effect in the latter case.
• As mentioned earlier that hardly changes with gelatin concentrations. As a result is
fixed to be for scaling of experimental data.
• are found to be inversely related.
• Viscous retardation effect is important in the case of hydrogels with lower
concentrations.
• Weak bond friction model is validated well over varying concentrations of gelatin
hydrogel, basic friction parameters for unification of friction
behaviour are worked out.
0
n 0
n
3.86
0
n
0
n
*
V
-4 1
2.11 x 10 ms
and M
* * *
ˆ
, or , and
w
V V u
13-03-2021 25
26. Stress relaxation phenomenon
• Stress relaxation in soft gels occurs when the sliding gel block is brought to rest i.e
from a velocity above critical velocity, .
• Relaxation stress is fitted using the concept of strengthening of bonds during
relaxation process after a certain time interval.
• We have utilized the strong bond model proposed by Singh and Juvekar (2014) to
validate our experiments and undergo study on chain bonding behaviour during steady
relaxation.
• It is different from weak bond model as weak bond model does not considers stress
retention and relaxes to zero stress level.
• Weak bond model is thereby suitable during steady dynamic sliding where the
possibility of strong bond formation does not exist particularly in case of gelatin
hydrogels.
• Concept of both continuous strengthening and discontinuous strengthening can be
utilized however this work focuses only on concept of discontinuous strengthening.
0 0
V 0 c
V V
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27. Strong bond friction model
• Strong bond friction model is given by the following equation as:
Where,
2 3
2 3
2 3
1 1 1 11 1 1 1 1 3 12 1 3
2 2
0 0
1 1 1 1 3
0
1 1 1 1 3
0
1
( ) ( ) log ( ) ( ) ( ) ( ) log ( ) ( )
ˆ ˆ
ˆ ( ) ( ) ( )
ˆ
1
1 ( ) ( ) ( )
ˆ
z z
z
e e
ws
z z
z
ws
z z
z
ws
e
e G z G z z E z E z G z z E z
V V
e
e E z E z E z
e
e E z E z E z
V
0 11 12 22 0
0 0 0 0
1 0 2 0 3 22 0
ˆ ˆ
ˆ ˆ
exp( ) , , , exp( )
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ
exp , * exp 1 , exp
w w w w
ws
ws ws
wsw wsw wsw ws
u u u u
z V z z z V
V V V u V u
z z V t z z V t z z V t
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28. Model validation and table of parameters
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Table 3. Model validation parameters
30. Important findings in the study of stress
relaxation
• The present study identifies residual stress in SHS experiments on gelatin hydrogels
and also validates it with the strong bond model.
• It is concluded that a hydrogel with higher gelatin concentration results in less
transition time but high residual stress.
• Relaxation time constant corresponding to strong bond formation decreases with
increasing mesh size of the gelatin hydrogels.
• Moreover, activation length of the strong bonds increases with increase in mesh
size of the hydrogels.
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32. Juvekar and Singh Model
13-03-2021 32
0
0
0
exp
exp
c
g g c
f t kT
c
f t kT
g
c
g
g
g
df t
MV t
dt
F t N t f t
dF t d L t
K K V V t
dt dt
dF t u
N t f t e N t MV t
dt
u
K V N t f t e
V t
N t M K
dN t F t
u
N t
dt N t kT
F t
u
MV F t
N t N t kT
dF t
K N t
dt N t M K
It can be converted into non-dimensional form by replacing:
as:
0
0
0
ˆ , ,
ˆ
ˆ
1 exp
ˆ
ˆ 1
ˆ
ˆ
ˆ ˆ ˆ
1 exp
ˆ
1
ˆ
ˆ
ˆ 1 1
i
s
i g
s
i
i
i
i s
N N N M
N
N X r
N N K
t
dX
r u X
dt X N
t
V N X u
X N
d
dt N X r
34. Weak bond model to validate static friction
data
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35. Importance of residual strength
• With the study of stress relaxation phenomenon, it is established that residual
strength is a property that is independent of shear rate in rate dependent experiments.
• Also the residual strength corresponds to minima in force vs. velocity profile where
the transition from velocity weakening to velocity strengthening frictional regime
occurs.
• It, therefore, becomes important to study the effect of residual strength on frictional
properties.
• Also, an attempt is made to predict critical transition velocity using the residual
strength obtained in steady frictional regime.
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36. Slide-free-slide (SFS) experiments
• c=8%,10%,12% wt./vol. gelatin hydrogels.
• Both rate dependent and aging time dependent experiments.
• SFS and SHS experimental trends are compared to study the effect of residual
strength in SHS experiments.
1. Normal load 2. DAS 3. Load cell 4. Linear Slider 5. Bread Board 6. Glass plate 7. Servo motor 8. Gel
specimen.
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42. Important findings of study on effect of
residual strength
• Static friction and steady dynamic stress are higher for aging in presence of residual
strength.
• Static friction stress shows considerable dependence on aging time and varies as
log of aging time for experiments done in presence of residual strength.
• However, it is almost independent of aging time in absence of residual strength.
• Magnitude and amplitude of critical velocity is higher for SHS experiments as
compared to SFS experiments.
• Finally, the critical velocity is predicted for all three experimental concentrations
using residual strength of SHS experiments.
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43. Important Conclusions
• Weak bond formation and rupture resulting in steady dynamic friction
1. The Hookean model for dynamic friction is sufficient for analysing the frictional
properties of the gelatin hydrogel interface.
2. The rate dependent weak bond friction model fits well the experimental data for
varying concentrations of the hydrogels.
3. However, the viscous retardation effect becomes important in case of low
concentrations based gelatin hydrogels. It is also important during the low sliding
velocity.
4. The scaling law analysis show that stiffness of the dangling chains scales
inversely as relaxation time constant with mesh size.
5. The scaling laws are also established to correlate mesh size with the areal density
of dangling chains at the interface and viscous retardation coefficient.
• Bond strengthening during stress relaxation at the soft-hard interface
1. It is observed experimentally that irrespective of pulling velocity, the sliding block
relaxes to the same level of non-zero residual stress. However, the residual strength
increases with gelatin concentration in the hydrogels.
2. The stress relaxation process is steady upto the stage of residual strength. It is also
established experimentally that the weak bonds during steady relaxation gets
converted to strong bonds at the end of steady relaxation thereby resulting in the
residual strength.
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44. Contd.
3. Strong bond friction model is validated for different concentrations of hydrogels
and the residual stress behaviour is studied in light of strong bond friction
parameters. Weak bond parameters are also utilized in the friction model as are
obtained during analysis of steady sliding. It is believed that the present results
should be useful to understand the role of residual stress in stick-slip instability.
• Static frictional strength as a steady state process
1. Population balance friction model considering creeping rupture of bonds is
validated with experimental data. The stiffness ratio has to be rate dependent in
order to correctly validate the friction model.
2. Weak bond friction model is then utilized to validate the experimental data and
the model prediction shows close correlation with experimental findings.
3. Static friction is now seen as a steady state process involving formation and
rupture of bonds at equal rate.
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45. Contd.
• Dependence of static and dynamic friction on residual strength
1. Effect of residual strength on friction behaviour at the soft-hard contact interface is
studied by performing a comparative study using slide-hold-slide (SHS) and slide-
free-slide (SFS) experiments. It is observed that the static and dynamic frictional
strengths are larger in magnitude in the SHS experiments as compared to the SFS
experiments.
2. Amplitude and magnitude of critical transition velocity is also seen to be dependent
on presence or absence of residual stress.
3. Further, unlike the SHS experiments, static frictional strength is observed to be
independent of aging time in the SFS experiments. These observations are
attributed to the healing rate of the interfacial polymeric chains under hold and free
condition in the SHS and SFS tests respectively.
4. Finally, the significance of residual strength on the frictional properties of the
sliding surfaces is highlighted.
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47. Achievements of this research work
• This work validates and confirms the suitability of mathematical models based on
population balance concept for estimating the frictional strength at bonded contacts.
• Finally, this work achieves the aim by unifying all the frictional behaviours at the
contact interface in certain fixed number of frictional parameters.
13-03-2021 47
48. Scope of future work
• There is a need to develop an expression for the critical velocity without
considering the inertia of the soft solid.
• There is need of improvement in the existing model for static frictional strengths as
this model still over predicts the experimental data.
• More significantly, a comprehensive mathematical model is needed which could
explain the recent experimental observations (Viswanathan et al., 2016) on soft and
hard solid interfaces in the form of Schallamach wave, self-healing crack etc. in a
single set of experiments.
• Further, it would be interesting to correlate the spatio-temporal behaviour of soft
solid to tectonic sliding in a more realistic way.
• This study can be further extended on other soft-hard interfaces.
13-03-2021 48
49. Important References
1. A.K. Singh, Friction of Gels, Ph.D thesis, IIT Bombay (2010)
2. A.K. Singh, V.A. Juvekar, Steady dynamic friction at elastomer–hard solid interface: A model based on
population balance of bonds, Soft Matter 7, 10601 (2011)
3. A.K. Singh, V.A. Juvekar, A strong bond model for stress relaxation of soft solid interfaces.
arXiv:1412.0099v3 (2014)
4. V.A. Juvekar, A.K. Singh, Rate and aging time dependent static friction of a soft and hard solid interface.
arXiv: 1602.00973
5. F.P. Bowden, D. Tabor, Mechanism of metallic friction, Nature 150, 3798(1942)
6. Henry Eyring, Viscosity, plasticity and diffusion as examples of absolute reaction rates, J. Chem. Phys. 4,
283(1936)
7. A. Schallamach, The velocity and temperature dependence of rubber friction, Proc. Phys. Soc. B 66(5),
386(1953)
8. A.A Milne, Friction experiments with a soft solid, Wear 2, 28-39 (1958)
9. A. Schallamach, A theory of dynamic rubber friction, Wear 6, 375-382 (1963)
10. A.R. Savkoor, On the friction of rubber, Wear 8, 222-237 (1965)
11. A.N. Gent, R.P. Petrich, Adhesion of viscoelastic materials to rigid substrates, Proc. R. Soc. Lond. A. 310,
1502(1969)
12. K.L. Johnson, K. Kendall, A.D. Roberts, Surface energy and the contact of elastic solids, Proc. R. Soc. Lond.
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51. List of Publications and Conferences
• SCI Journal Publications
1. V. Gupta, A.K. Singh, Scaling laws of gelatin hydrogels for steady dynamic friction,
International Journal of Modern Physics-B 30, 1650198 (2016)
2. V. Gupta, A.K. Singh, Stress relaxation at a gelatin hydrogel-glass interface in direct
shear sliding, Modern Physics letters-B, 32, 1750345 (2018)
3. V. Gupta, A.K. Singh, Effect of residual strength on frictional properties of a soft and
hard solid interface, Proceedings of National Academy of Sciences-A, Manuscript
under review.
• Conferences
1. V. Gupta, A.K. Singh, A.A. Thakre, Complexities in frictional dynamics of gel-glass
interface: A comprehensive review, poster presented at IISc, Bangalore in APM-2015.
2. V. Gupta, A.K. Singh, N. Sinha, A friction model for rate dependent slip dynamics of
tire-road interaction, oral presentation at IPRoMM-2016 at VNIT, Nagpur.
3. V. Gupta, A.K. Singh, Shear rate dependence of static friction at a soft hydrogel-glass
interface, oral presentation at CHEMIX-2017 at VNIT, Nagpur.
4. V. Gupta, A.K. Singh, N. Sinha, Modeling and experimental validation of static
strength of a soft-hard solid interface, oral presentation at ICADVC-2018, NIT
Durgapur.
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