Supported contact mechanics models
The relation between indentation and force depends on the shape of the tip and the thickness
of the sample. AtomicJ supports several tip shapes – Sphere, Cone, Pyramid, Power-shaped,
Hyperboloid, Blunt cone, Blunt pyramid, Truncated (flat-ended) cone and Truncated pyramid.
For sphere and cone it also implements corrections for finite thickness of the sample. All
equations assume that the material behaves as linear elastic (i.e. that the indentations are
relatively small).
a
b
Fig. 1. Tip shapes: a. Paraboloid that approximates sphere in the Hertz’s equation. R – radius of
curvature at the apex. b. Hyperboloid. R – radius of curvature at the apex, θ – half angle
between the asymptotes.
c
d
e
Fig. 1. (cont.). Tip shapes. c. Cone. θ –
half angle. d. Blunt cone (cone capped
by a sphere). R- radius of curvature at
the apex, b – tip radius at the level of
transition between the capping sphere
and the cone, θ – half angle. If the
transition from sphere to cone is
smooth, then b = RCos[θ]. e. Truncated
cone. b – truncation radius, θ – half
angle
In the equations, we will use the following symbols:
E – Young’s modulus;
ν – Poisson’s ratio
δ – depth of indentation;
a – contact radius
h – sample thickness
Tip geometry parameters are explained in Fig. 1.
1. Sphere (Hertz). An approximation of the sphere by a paraboloid. It is accurate when the
contact radius a is much smaller than the sphere radius R, which means that
4E
P
3 1  2


R
R
R.
3
2
2. Sphere (Sneddon). Unlike Hertz’s, Sneddon’s solution (Sneddon 1965) does not require that
R
R.
P

E  R2  a2
 Ra
Log 
  aR 
2 
1   2
 Ra 



δ
a
 Ra
Log 

2
 Ra 
3. Sphere, thin sample (Dimitriadis et al. 2002)
3
 2
4E
4 2 2 8  3 4π 2  3 16  3 3π 2  4 
2
P
R  1 
  2   3  
    4  
  
π
π
π 
15 
π 
5  
3 1  2 

where χ 
R
h
Coefficients α and β depends on whether the sample is adherent to the substrate or not.
For adherent sample:
3  2
1 
5  2
  0.056
1 
  0.347
For non-adherent sample:
1.2876  1.4678  1.3442 2
 
1 

0.6387  1.0277  1.5164 2
1 
4. Hyperboloid (Akhremitchev and Walker 1999)
P
R Cot  
 1   
Ea3  2 
2 π


1



ArcTan

where










a
 2 2   
1  2  R  2
2

 1  
a2  π
   ArcTan    
2R  2
 2 2  
5. Cone (Harding and Sneddon 1945)
P
2 E Tan  

π 1 
2

2
6. Cone, thin sample (Gavara and Chadwick 2012)
P
8E Tan  
3π

2Tan  

π2h
 2 1  
 16 2Tan  
2
2 

h2 
For adherent sample, α = 1.7795, for non-adherent α = 0.388.
7. Power-shaped (Galin 1946). The tip is modeled as a solid obtained by revolving a power
function along the y axis. It is thus axisymmetric and its profile is given by the power function
f(r) = A rλ. The load P and indentation depth δ can be expressed in terms of contact radius a as:

 
2
E A 
2
P
2   a  1
2
1  2   1    
2

 
 2  a
2
A 2

 2
  
where Γ is Euler’s gamma function. Hertz’ equation for sphere and Sneddon’s equation for cone
are special cases of the equation for the power-shaped tip, with λ = 2, A = 1/(2R) and λ = 1, A =
1/Tan[θ], respectively. In the AtomicJ interface, A is termed the factor and λ is termed the
exponent.
8. Blunt cone (Briscoe et al. 1994). The Hertz solution for the sphere is applied for small
indentations, for which the contact radius a is smaller than the transition radius b. This is the
case as long as  
P
2E
1  2


b2
. For larger indentations:
R

a2
 a 
2 Tan  


3
π
b  a
2
2

ArcSin


 a   3R  a  b
2



a π
b  a  2 2
  ArcSin      a  b
Tan    2
a R



1
2
1/2

b
a 2  b 2  



3R  
 2 Tan  
 a

9. Truncated cone (Briscoe et al. 1994). Truncated cone can be treated as a special case of
blunted cone, for which the radius of curvature at the apex is infinite. Substituting R = ∞ into
the equations for blunt cone, we get:
P
2E
1  2



a2
 a 
2 Tan  

 
π
b 
2
2
  ArcSin     a  b
2
a
 



1
2


b

 
 2 Tan    
a π
b 
  ArcSin   
Tan θ   2
a
10. Pyramid, regular, four sided (Bilodeau 1992)
P
1.4906 ETan[ ] 2

2 1  2


11. Blunt pyramid, regular, four-sided (Rico et al. 2005). The Hertz solution for the sphere is
applied for small indentations, for which the contact radius a is smaller than the transition
b2
radius b. This is the case as long as  
. For larger indentations
R
3
2E 
21/2 a 2  π
b  a
P
 a 2  b2
 a 
  ArcSin    
2
π Tan    2
1  
 a   3R

23/2 a  π
b  a  2 2
  ArcSin      a  b
π Tan    2
a R




 

1
2
 21/2 b
a 2  b 2  



3R  
 π Tan  
1/2
 a

12.Truncated pyramid (Rico et al. 2005). Truncated pyramid can be treated as a special case of
blunted pyramid, for which the radius of curvature at the apex is infinite. Substituting R = ∞ into
the equations for blunt pyramid, we get:
P
2E
1  2



21/2 a 2  π
b 
2
2
a



  ArcSin     a  b
π Tan    2
a


 
23/2 a  π
b 
  ArcSin   
π Tan    2
a

1
2
 21/2 b  

 
 π Tan    
HYPERELASTIC MODELS
13. Sphere, Fung’s hyperelastic model (Fung 1979) .The equation for force – indentation depth
relation was derived by Lin et al (2009), who found that the equation for sphere’s contact radius
a
from Hertz’z model (used here) holds for hyperelastic materials as long as  0.4 .
R
P
  a 3  15Ra 2  
20 E0  a5  15Ra 4  75R 2 a 3 
exp
b 

2
2
3 
2
3 
9 1  ν 2   5Ra  50 R a  125R 
  25R a  125R  
a  R
14. Sphere, Ogden’s hyperelastic model (Ogden 1972).The equation for force – indentation
depth for single-term Ogden’s model relation was derived by Lin et al (2009):
 /2 1
 1

a
a 

P
 1  0.2  
1  0.2 
R
R  
9 1  ν 2  

40 E0 a 2


a  R
ADHESIVE CONTACT
15. Derjaguin-Muller-Toporov (DMT) (Derjaguin et al. 1975). DMT model was derived for
indentation with a sphere (approximated by a paraboloid) in the presence of adhesion forces.
We implemented the most commonly referenced form of the DMT model, in which the
additional load due to adhesive forces equals 2πγR and is independent of indentation. This
formulation is due to Maugis (1992). DMT model gives good approximation of the load –
indentation relationship only for stiff samples, small values of tip radius R and small surface
energy γ.
P
4E R
 3/2  2 R
2
3 1 


16. Johnson-Kendall-Roberts (JKR) (Johnson et al. 1971). JKR model was derived for indentation
with a sphere (approximated by a paraboloid) in the presence of adhesion forces. JKR model is
a good approximation of the load – indentation relationship for soft samples, large value of
radius R and large energy of adhesion (Maugis 1992).
P=
4Ea 3
E a 3
2
2π
3R 1- 2
1- 2



a2

R


2 a
E
(1  2 )
17. Sphere, Maugis solution. Maugis (1995) derived the load – indentation relationship in the
presence of adhesion, which takes into account the true shape of a spherical tip (as opposed to
the JKR model, which uses parabolic approximation).
2𝐸𝑎 𝑅 2 + 𝑎2
𝑅+𝑎
𝑅
2𝜋𝛾𝑎
𝑃=
Log [
]− −√
2
𝐸
1−𝜈
4𝑎
𝑅−𝑎
2
1 − 𝜈2
(
)
𝛿=
𝑎
𝑅+𝑎
2π𝛾𝑎
Log [
]−√
𝐸
2
𝑅−𝑎
1 − 𝜈2
18. Hyperboloid, Sun-Akhremitchev-Walker (SAW). Sun et al (2004) derived the load – contact
radius and indentation – contact radius relations for indentation with a hyperboloidal tip in the
presence of adhesion.
𝑎 2
(𝐴 ) − 1
2𝐸
𝐴
𝑎2 − 𝐴2 𝜋
2𝑎𝛾𝜋(1 − 𝜈 2 )
√
𝑃=
(
𝑎𝐴 +
( + ArcSin [
]) − 𝑎
)
𝑎 2
1 − 𝜈 2 2𝑅
2
2
𝐸
(𝐴 ) + 1
(
)
𝑎 2
(𝐴) − 1
𝑎𝐴 𝜋
2𝜋𝑎𝛾(1 − 𝜈 2 )
√
𝛿=
( + ArcSin [
]) −
𝑎 2
2𝑅 2
𝐸
(𝐴) + 1
where 𝐴 = 𝑅Cot[𝜃]
References:
1. Akhremitchev BB, Walker GC, Finite Sample Thickness Effects on Elasticity Determination Using
Atomic Force Microscopy. Langmuir 15: 5630 – 5634 (1999).
2. Bilodeau G Regular Pyramid Punch Problem. ASME J Appl Mech. 59: 519 – 523 (1992).
3. Briscoe DJ, Sebastian KS, Adams MJ, The effect of indenter geometry on the elastic response to
indentation. J Phys D: Appl Phys 27: 1156 – 1162 (1994).
4. Derjaquin BV, Muller VM, Toporov YP, Effect of contact deformations on the adhesion of
particles. J Colloid Interface Sci 53: 314-326 (1975).
5. Dimitriadis EK, Horkay F, Maresca J, Kachar B, Chadwick RS, Determination of elastic moduli of
thin layers of soft material using the atomic force microscope. Biophys J 82:2798–2810 (2002).
6. Fung YC, Fronek K, Patitucci P, Pseudoelasticity of arteries and the choice of its mathematical
expression. Am J Physiol-Heart C 237:H620–H631 (1979).
7. Galin LA, Spatial contact problems of the theory of elasticity for punches of circular shape in
planar projection. J. Appl. Math. Mech. (PMM) 10: 425–448 (1946).
8. Gavara N and Chadwick RS, Determination of the elastic moduli of thin samples and adherent
cells using conical AFM tips. Nat Nanotechnol. 7: 733–736 (2012).
9. Harding JW, Sneddon IN, The elastic stresses produced by the indentation of the plane surface of
a semi-infinite elastic solid by a rigid punch. Proc Camb Philol Soc. 41:16 (1945).
10. Johnson KL, Kendall K, Roberts AD, Surface energy and the contact of elastic solids. Proc. R. Soc.
Lond. A 324: 301-312 (1971).
11. Lin DC, Schreiber DI, Dimitriadis EK, Horkay F, Spherical indentation of soft matter beyond the
Hertzian regime: numerical and experimental validation of hyperelastic models. Biomech Model
Mechanobiol 8:345–358 (2009).
12. Maugis D Extension of the Johnson-Kendall-Roberts theory of the elastic contact of spheres to
large contact radii. Langmuir 11: 679 – 682 (1995).
13. Ogden RW, Large deformation isotropic elasticity—on the correlation of theory and experiment
for incompressible rubberlike solids. Proc R Soc Lond A Math Phys Sci 326: 565–584 (1972).
14. Rico F, Roca-Cusachs P, Gavara N, Farre R, Rotger M and Navajas D, Probing Mechanical
Properties of Living Cells by Atomic Force Microscopy With Blunted Pyramidal Cantilever Tips.
Phys Rev E 72, 021914 (2005).
15. Sneddon IN, The relation between load and penetration in the axisymmetric Boussinesq problem
for a punch of arbitrary profile. Int J Eng Sci 3: 47 – 57 (1965).
16. Sun Y, Akhremitchev B, Walker G, Using the adhesive interaction between atomic force
microscopy tips ad polymer surfaces to measure the elastic modulus of compliant samples.
Langmuir 20: 5837 – 5845 (2004).