Abstract: Based on a linear combination of the solutions derived fromthe Sneddon method and the Lowengrub-Sneddon method, a general solution ofthe axisymmetric problem is obtained for the elastic half-space with mixedboundary conditions. And then, the frictionless and adhesive contact problembetween two general axisymmetric elastic objects is studied. For anarbitrary effective surface profile, i.e., the initial contact occurs at thecentral part, and an arbitrary surface adhesive interaction, a generalizedMaugis model is derived and it can be divided into two parts correspondingto the contributions of the surface profile and the surface adhesiveinteraction, respectively, and a coupling relation between the deformationand the adhesive interaction. Based on the Dugdale model for the surfaceadhesive interaction, a generalized M-D model is derived for an arbitraryeffective surface profile. Two extremes are found for this model. For ashort-range strong interaction or compliant material, its limiting form iscorresponding to the generalized JKR model. And for a long-range weakinteraction or stiff material, another limiting form is corresponding to ageneralized DMT model.