The document summarizes the Johnson-Kendall-Roberts (JKR) model of adhesion. The JKR model considers the effects of contact pressure and adhesion within the contact area between two elastic bodies. The model predicts that the contact area is larger than the Hertzian model when adhesion is present, and that there is a non-zero contact area even when the load is removed due to adhesion. The JKR model is commonly used to model adhesive contact and estimate surface energy from experimental data.

1. JKR (Johnson-Kendall-
Roberts)
model of adhesion
By:
Petra Jan, Ali Payami Golhin, Asif Quliyev, Ahmed Shumayal, Mohammadreza Shamshiri

2. Non-adhesive elastic contact
Hertzian theory : In non-adhesive contact Van der Waals interactions or
adhesive interactions are not allowed to occur within contact area, so
contacting bodies can be separated without pull-out forces.
Assumptions:
oThe strains are small and within the elastic limit
oThe surfaces are continuous and non-conforming
oEach body can be considered an elastic half-space
oThe surfaces are frictionless

3. Adhesion
Adhesion is a property of the system
adhesion depends on lots of things such as:
oTemperature
oTime
oSpeed
oDirection
oSurface dissipation
Steve Abbott

4. Rubber balls being pressed into
contact with a flat surface
No squashing when there is
no pressure ( F=0)
Hertzian Contact
In reality at zero pressure
rubber naturally deforms to
produce a finite contact width
because the surface energy
forces are sufficient to pull two
surfaces together
F
Steve Abbott

5. Adhesive elastic contact – JKR model
JKR : theory of adhesive contact using a balance between the stored elastic
energy and loss in surface energy, considers effect of contact pressure and
adhesion only inside area of contact.
o the area of contact is larger than that
predicted by Hertz theory,
o the area of contact has a non-zero value
even when the load is removed
o there is strong adhesion if the contacting
surfaces are clean and dry.
J.A. Greenwood, 1977

6. How does it compare?
oAll models are somehow evolved from Adhesion less Hertz
Model of Contact.
o DMT or JKR is dependent on specific type of interaction
between surfaces (energy).

7. Hertz
o For a sphere of radius R, the punch shape f(r)
is approximately
f(r) =
𝑅2
2𝑅
o The relation between penetration and contact
radius§ is δ (a)=
𝑎2
𝑅
The relation between the force and the contact radius is
4𝐸∗𝑎3
3𝑅

8. Bradley Model
oConsiders the full interaction potential V (z) between two surfaces,
including attractive and repulsive contributions to the potential.
oSummation of the interaction stresses over the curved bodies,
assumed rigid, provide the interaction force.
oSurface deformation is taken into account becomes “fully” self-
consistent method

9. Derjaguin Model
oModel for adhesion of spheres.
oConsiders attractive and repulsive parts of the potential separately.
oHe assumed that this repulsive contribution is so steep that the surfaces cannot
interpenetrate.
oboundary conditions specifies the surface normal displacement – instead of the surface
normal stress – inside
othe contact zone, exactly as in the Hertz model for the adhesion less elastic contact
oRestricted Self Consistent Method.
Assumes that the contact stresses and the gap shape are given by the Hertz predictions.

10. DMT Model –
Another Adhesive
Contact theory
Hertzian stress
distribution and
deformation fields still
apply
but that the adhesive
interaction stresses result
in an additional force,
which is now computed
in the manner of the
Derjaguin approximation.

11. Derjaguin Approximation
o While assuming that the adhesive interaction stresses derive from the interaction potential
δDMT = δH(a)
FDMT = FH(a) + Fext(a)
Real contact conditions can include strong adhesion as parameters.

12. JKR THEORY
Surface energy γ
o excess energy at solid material surface due to free
bonds, differs from material bulk energy
→ determines the ability to interact with other
counter surfaces/liquids and strength of interaction
o depends on surface roughness, properties and type
of bonds, can be modified by coatings

13. JKR model – adhesive contact (1/2)
P: external force on particle
𝐅𝐚𝐝: adhesion force
a: contact radius
d: ball diameter
a = ? Given by JKR model
1) 𝐏𝟏 = 𝟎
1) 𝐏𝟐 > 𝟎
2) 𝐏𝟑 = 𝟐 ∗ 𝐏𝟐
γ = const. in all cases
Ahmadi G. et al, n.d.
S. Abbott, 2018.

14. JKR model – adhesive contact (2/2)
4) 𝐏𝟒 < 𝟎
Force must be applied to lift ball
off the surface due to adhesion
(pull-off force)
However, pull-off force and
contact width of adhesive
contact also depend on
surface energy value
S. Abbott, 2018.

15. JKR curves
1)γ 1 = 0
Hertz case
2) γ 2 > 0
(same as in
previously shown
cases)
3) γ 2 ≫ 0
Pull-off force
increases a lot, due
to stronger adhesion
Higher surface
energy →
stronger
adhesion →
larger contact
width (a),
more pull-off
force needed
S. Abbott, 2018.

16. Total
Energy
ET
Stored Elastic
Energy
Mechanical
Potential Energy in
the Applied Load
Surface
Energy
Derivation of JKR Model 1/2
Equilibrium when
0
da
dET
2
3
)3(63 RRPRP
R
Ka
 

17. Derivation of JKR Model 2/2
a: contact radius
K: the elastic constant of the
sample
R: effective contact radius
P: indentation force
γ: indentation depth
E*: E(1-ν2) elastic moduli
δ: depth
Fad: maximum adhesive force
F: normal loading force
Δγ: work of adhesion
*
3
4
EK 
K
a
R
a
3
82




RP OffPull  5.1

18. Generalized JKR model 1/2
adhesive contact
non-adhesive contact
o by analogy to the derivation of JKR model,
Shull, (2002) developed the “generalized”
JKR model
o It describes the adhesive contact between
a rigid indenter (bead) and an elastic flat
sheet with finite thickness (h)
o the expressions for the energy release
rate ℊ is extended to the adhesion of
relatively thin layers by using the
approximation of compliance given by Eq:

19. Simple expressions are only available for incompressible materials with ν=0.5
as:
ℊ: energy release rate
P’: applied load
δ’: indentation depth (No adhesion)
H: finite thickness Geometric correction factors
Generalized JKR model 2/2

20. The JKR model – Some applications
oEstimation of Surface Free Energy
oJKR model allows direct estimation of the surface free energies of several
model low-energy surfaces.
oThis system provides a suitable experimental basis for systematic evaluation of
Young's equation.
oIn terms of analyzing surface energetics, it is complementary to measuring
contact angles.

21. The JKR model – Some applications
oThe JKR test
o Various experimental devices have been developed where the contact radius
can be monitored as a function of the applied load to infer the adhesive
properties of various types of surfaces.
oThe main application of such devices is to characterize surface modification
through adhesion.
o Using these devices we can record data in the stable adhesive contact region
(i.e. before contact rupture) and analyze them with a given contact model.

22. oThe dynamics of adhesive contact rupture
oThe JKR theory is very widely used for interatomic
oadhesion between isotropic elastic spheres.
oA typical curve for contact radius vs load PDMS lenses
o on a rigid substrate.
oThe macroscopic deformation of the PDMS lens very
o accurately conforms to JKR theory.
The JKR model – Some applications

23. oThe JKR can be used to measured surface energy instead of contact angles.
oUsing surface force apparatus and JKR formula, it is possible to find reliable
differences in surface energies depending on treatment level.
oIt provides relatively simple theoretical predictions of the effect of adhesive
forces in contact situations.
Conclusion

24. References
1. E. Barthel, Adhesive elastic contacts , JKR and more, Journal of Physics D: Applied Physics, IOP Publishing, 2008.
2. M. K. Chaudhury, G. M. Whitesides, Correlation Between Surface Free Energy and Surface Constitution, SCIENCE, VOL. 255.
3. V. L. Popov, Contact Mechanics and Friction, Springer, 2009
4. Ahmadi, G., Í Î, Í. & Ê, Ë. (n.d.) Particle Adhesion and Detachment Models. [Online]. Available from:
<https://webspace.clarkson.edu/projects/crcd/public_html/me437/downloads/6_JKR.pdf> [Accessed 7 May 2018].
5. S. Abbott (2018) JKR Curves | Practical Adhesion Science | Prof Steven Abbott [Online]. Available from: <https://www.stevenabbott.co.uk/practical-
adhesion/jkrcurves.php> [Accessed 7 May 2018].
6. S. Abbott (2018) JKR and Surface Adhesion Effects | Practical Adhesion Science | Prof Steven Abbott [Online]. Available from:
<https://www.stevenabbott.co.uk/practical-adhesion/jkr.php> [Accessed 7 May 2018].
7. J.Beetsma (2015) Surface Tension &amp; Surface Energy [Online]. Available from: <https://knowledge.ulprospector.com/3354/pc-surface-tension-surface-
energy/> [Accessed 7 May 2018].
8. Xinyao Zhu et al., Determination of work of adhesion of biological cell, under AFM bead indentation, Journal of the mechanical behavior of biomedical
materials, 56(2016), 77–86.
9. J.A. Greenwood, "Adhesion of Elastic Spheres",. Proc. R. Soc. Lond. A (1997) 453, 1277-1297
10. K. L. Johnson and K. Kendall and A. D. Roberts, Surface energy and the contact of elastic solids, Proc. R. Soc. Lond. A 324 (1971) 301-313