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).

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.

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 .

2. Sphere (Sneddon).Unlike Hertz’s, Sneddon’s solution (Sneddon 1965) does not require that .

3. Sphere, thin sample (Dimitriadis et al. 2002)

Coefficients α and β depends on whether the sample is adherent to the substrate or not.
For adherent sample:

For non-adherent sample:

4. Hyperboloid (Akhremitchev and Walker 1999)

5. Cone (Harding and Sneddon 1945)

6. Cone, thin sample (Gavara and Chadwick 2012)

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) = Arλ. The load Pand indentation depth δcan be expressed in terms of contact radius a as:

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-shapedtip, withλ = 2, A = 1/(2R) and λ = 1, A = 1/Tan[θ], respectively.In the AtomicJ interface, A is termed the factorand 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 . For larger indentations:

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:

10. Pyramid, regular, four sided (Bilodeau 1992)

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 radius b. This is the case as long as . For larger indentations

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:

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 from Hertz’z model (used here) holds for hyperelastic materials as long as .

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):

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.

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).

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).

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.

where

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