  We have employed a numerical procedure to analyze the adhesive contact
between a soft elastic layer and a rough rigid substrate. The solution of the
problem is obtained by calculating the Green's function which links the
pressure distribution to the normal displacements at the interface. The problem
is then formulated in the form of a Fredholm integral equation of the first
kind with a logarithmic kernel, and the boundaries of the contact area are
calculated by requiring that the energy of the system is stationary. The
methodology has been employed to study the adhesive contact between an elastic
semi-infinite solid and a randomly rough rigid profile with a self-affine
fractal geometry. We show that, even in presence of adhesion, the true contact
area still linearly depends on the applied load. The numerical results are then
critically compared with the prediction of an extended version of the Persson's
contact mechanics theory, able to handle anisotropic surfaces, as 1D
interfaces. It is shown that, for any given load, Persson's theory
underestimates the contact area of about 50% in comparison with our numerical
calculations. We find that this discrepancy is larger than what is found for 2D
rough surfaces in case of adhesionless contact. We argue that this increased
difference might be explained, at least partially, by considering that
Persson's theory is a mean field theory in spirit, so it should work better for
2D rough surfaces rather than for 1D rough surfaces. We also observe, that the
predicted value of separation is in very good agreement with our numerical
results as well as the exponent of the power spectral density of the contact
pressure distribution and of the elastic displacement of the solid. Therefore,
we conclude that Persson's theory captures almost exactly the main qualitative
behavior of the rough contact phenomena.
