A JKR solution for a ball-in-socket contact geometry as a bi-stable adhesive system

In the present note, we start by observing that in the classical JKR theory of adhesion, using the usual Hertzian approximations, the pull-off load grows unbounded when the clearance goes to zero in a conformal "ball-in-socket" geometry. To consider the case of the conforming geometry, we use a recent rigorous general extension of the original JKR energetic derivation, which requires only adhesionless solutions, and an approximate adhesionless solution given in the literature. We find that depending on a single governing parameter of the problem, $\theta =\Delta R/\left(2\pi wR/E^{\ast }\right)$ where $E^{\ast }$ is the plane strain elastic modulus of the material couple, w the surface energy, $\Delta R$ the clearance and R the radius of the sphere, the system shows the classical bi-stable behaviour for a single sinusoid or a dimpled surface: pull-off is approximately that of the JKR theory for $\theta >0.82$ only if the system is not "pushed" strong enough, otherwise a "strong adhesion" regime is found. Below this value, $\theta <0.82$, a strong spontaneous adhesion regime is found similar to "full contact". From the strong regime, pull-off will require a separate investigation depending on the actual system at hand.