  In the present note, we suggest a simple closed form approximate solution to
the adhesive contact problem under the so-called JKR regime. The derivation is
based on generalizing the original JKR energetic derivation assuming
calculation of the strain energy in adhesiveless contact, and unloading at
constant contact area. The underlying assumption is that the contact area
distributions are the same as under adhesiveless conditions (for an
appropriately increased normal load), so that in general the stress intensity
factors will not be exactly equal at all contact edges. The solution is simply
that the indentation is D=D1-Sqrt(2w A'/P") where w is surface energy, D1 is
the adhesiveless indentation, A' is the first derivative of contact area and P"
the second derivative of the load with respect to indentation. The solution
only requires macroscopic quantities, and not very elaborate local
distributions, and is exact in many configurations like axisymmetric contacts.
It permits also an estimate of the full solution for elastic rough solids with
Gaussian multiple scales of roughness, which so far was lacking, using known
adhesiveless simple results. The solution turns out to depend only on rms
amplitude and slopes of the surface, and in the fractal limit, slopes would
grow without limit, tends to the adhesiveless result - although in this limit
the JKR model is inappropriate. However, a more solid result is that the
solution would also go to adhesiveless result for large rms amplitude of
roughness h_{rms}, irrespective of the small scale details, and in agreement
with common sense and previous models by the author.
