  Motion of an object near a soft wall with intervening fluid is a canonical
problem in elastohydrodynamics, finding presence in subjects spanning biology
to tribology. Particularly, motion of a sphere towards a soft substrate with
intervening fluid is often encountered in the context of scanning probe
microscopy. While there have been fundamental theoretical studies on this
setup, they have focussed on specific applications and thus enforced suitable
simplifications. Here we present a versatile semi-analytical framework for
studying the elastohydrodynamics of axisymmetric loading of a rigid sphere near
a soft elastic substrate coated on a rigid platform mediated by an aqueous
electrolytic solution. Three loading modes are considered - approach, recession
and oscillatory. The framework incorporates - large oscillation frequency and
amplitude, two-way coupling between pressure and substrate deformation, and
presence of DLVO forces. From computations using the framework, we gain
insights on the effects of DLVO forces, substrate thickness, substrate material
compressibility (quantified by Poisson's ratio) and high oscillation frequency
for different physical setups encountered in SPM and the likes. We list some
key observations. A substrate that is thicker and more compressible allows for
larger deformation, i.e. is effectively softer. Presence of DLVO forces lead to
magnification in force of upto two orders of magnitude and in substrate
deformation of upto an order of magnitude for oscillatory loading at low
frequencies and approach/recession loading at low speed. For oscillatory
loading at high frequencies, DLVO forces do not appreciably affect the force
and deflection behaviour of the system. Having demonstrated the versatility and
utility of our framework, we expect it to evolve into a diverse and useful tool
for solving problems of soft-lubrication.
