An analytical self-consistent model for the adhesion of Gibson solid

A full self-consistent model (FSCM) for the adhesive contact between an axisymmetric rigid punch and a Gibson solid (an incompressible, linear graded elastic material) is established, which gives a self-consistent relation between the surface gap and interaction. Power-law shaped indenters and the Lennard-Jones interaction law are studied as representative cases, and the self-consistent equation is expressed in a dimensionless form with two independent parameters, namely the shape index and the Tabor number. The self-consistent equation for Gibson solid is a higher order polynomial equation with respect to the surface gap, which differs from the nonlinear integral equation for power-law graded elastic material. By taking the surface gap as the independent variable instead of the radius, the self-consistent equation is solved explicitly, which gives the first explicit form of the solution to an FSCM. When the Tabor number exceeds a critical value, jump-in instability and adhesion hysteresis occur, and the folding phenomenon that the Gibson solid surface is flipped and folded along the radial direction is observed. The critical Tabor number is determined in an explicit form and it is found to be independent of the surface index. The extended Maugis-Dugdale (M-D) model for Gibson solid is invalid when the Tabor number is large enough and the extended Johnson-Kendall-Roberts (JKR) model does not present the soft limit, due to their assumption of simple contact. An asymptotic solution is derived for the soft limit of the FSCM, which gives a power-law asymptotic relation between the dimensionless pull-off force and the Tabor number. This study provides a self-consistent toy model for the adhesive contact of Gibson solid and may deepen the understanding on the adhesion of graded materials.