  The total elastic stiffness of two contacting bodies with a microscopically
rough interface has an interfacial contribution K that is entirely attributable
to surface roughness. A quantitative understanding of K is important because it
can dominate the total mechanical response and because it is proportional to
the interfacial contributions to electrical and thermal conductivity in
continuum theory. Numerical simulations of the dependence of K on the applied
squeezing pressure p are presented for nominally flat elastic solids with a
range of surface roughnesses. Over a wide range of p, K rises linearly with p.
Sublinear power-law scaling is observed at small p, but the simulations reveal
that this is a finite-size effect. We derive accurate, analytical expressions
for the exponents and prefactors of this low-pressure scaling of K by extending
the contact mechanics theory of Persson to systems of finite size. In agreement
with our simulations, these expressions show that the onset of the low-pressure
scaling regime moves to lower pressure as the system size increases.
