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Derivation of Hertz contact law and associated pressures

N.M. Vriend

In contact problems without friction, the z-component of the displacement is the only componentof interest. The radius of the contact area can be approximated, based on a geometric derivation,as:

= l2

R, (1)

in which we assume

8/10/2019 Hertz.pdf

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and the depth of indentation is:

=

1

R

1/33Fn4E

2/3=

9F2n

16R(E)2

1/3(12)

Following the argument in Campbell, Powder Technology, 2005, the normal force exerted by twobodies of local curvature R in a contact:

Fn = 4

3R1/2E3/2. (13)

The normal stiffness is derived as the derivative of the normal force to the displacement :

k= dFn

d = 2R1/2E1/2 (14)

This allows writing the stiffness in terms of the normal force:

k= 61/3R1/3 (E)2/3

F1/3n (15)

The stiffness is linearly dependent on E, but also depends on R the local radius of the curvature.Bathurst and Rothenburg (J. Appl. Mech., 1988) derived that the bulk elastic modulusK =

E/(3 [1 2 ]) of a random granular material scales linearly with the contact stiffness k as:

K=f(n)k

R

R

1/2E (16)

with f(n) a function dependent on the contact number, the amount of neighbors that the particleis in contact with. Note that the bulk elastic modulus K scales not only linearly with the elasticmodulus Ethrough the stiffness, but also with the geometry of the contact R .

Now looking at the sound speed, or elastic wave speed, c, for pressure waves, we write this interms of the bulk modulus as:

c=

K1/2

(17)

with the bulk modulus Kand density. Consequently, the sound speed scales in Hertzian contactsas:

c K1/2 k1/2 1/4 F1/6n P1/6 (18)

From experiments, it has been observed that the scaling c P1/6 is only true in large confiningpressures. For lower pressures, the correct scaling should be c P1/4, which is inconsistent withthe above calculation.

Lets assume that the initial contact, at lower pressures, is closer to a conical situation of contact.This can be explained by interpreting the initial contact between roughness and asperities as a (non-Hertzian) conical contact, which is later transferred into a Hertzian sphere-on-sphere contact forhigher pressures. In the conical situation, a logarithmic singularity exists on the tip of the cone due

to the zero radius of curvature at the tip. The normal force can be expressed (not derived here) interms of the depth of indentation as:

Fn = 2

tan()E2 2. (19)

The normal stiffness is again derived as the derivative of the normal force to the displacement :

k=dFn

d =

4

tan()E . (20)

As a result, the bulk elastic modulus K scales linearly with the depth of indentation and thesound speed scales as:

c K1/2 k1/2 1/2 F1/4

n

P1/4, (21)

which agrees with the scaling for lower pressures as derived from experiments.

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