E* = 115400 MPa (steel-on-steel equivalent modulus). Decreasing d means stronger pressing and a larger real contact area ratio.
Top: asperities with reference-plane separation d (red dots = asperities in contact). Bottom: F_0(d/σ), A_r/A_0 and P/A_0 vs d/σ.
In the GW model the asperity height z follows a Gaussian distribution φ(z), and each asperity is compressed as a Hertzian sphere. Only asperities with z > d are in contact, with indentation δ = z − d.
Asperity height distribution (Gaussian with std. dev. σ):
$$\varphi(z) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\!\left(-\frac{z^2}{2\sigma^2}\right)$$Real contact count per unit area; η is the asperity density:
$$n(d) = \eta \int_d^\infty \varphi(z)\,dz$$Real contact area ratio (A_c = π β δ per contact):
$$\frac{A_r}{A_0} = \pi\beta\eta \int_d^\infty (z-d)\,\varphi(z)\,dz$$Nominal contact pressure (P_c = (4/3) E* β^{1/2} δ^{3/2} per contact):
$$\frac{P}{A_0} = \frac{4}{3}E^*\sqrt{\beta}\,\eta \int_d^\infty (z-d)^{3/2}\varphi(z)\,dz$$A key observation is that as d decreases (more pressing), n(d), A_r/A_0 and P/A_0 grow at almost the same rate. This is the microscopic basis of Amontons-Coulomb friction.