In the Maugis-Dugdale model of the adhesive contact of elastic spheres, the step cohesive stress sigma(0) is arbitrarily chosen to be the theoretical stress sigma(th) to match that of the Lennard-Jones potential. An alternative and more reasonable model is proposed in this paper. The Maugis model is first extended to that of arbitrary axisymmetric elastic objects with an arbitrary surface adhesive interaction and then applied to the case of a power-law shape function and a step cohesive stress. A continuous transition is found in the extended Maugis-Dugdale model for an arbitrary shape index n. A three-dimensional Johnson-Greenwood adhesion map is constructed. A relation of the identical pull-off force at the rigid limit is required for the approximate and exact models. With this requirement, the stress sigma(0) is found to be k(n)Delta gamma/z(0), where k(n) is a coefficient, Delta gamma the work of adhesion, and z(0) the equilibrium separation. Hence we have sigma(0) = 0.588 Delta gamma/z(0), especially for n = 2. The prediction of the pull-off forces using this new value shows surprisingly better agreement with the Muller-Yushchenko-Derjaguin transition than that using sigma(th) = 1.026 Delta gamma/z(0), and this is true for other values of shape index n. (c) 2007 Elsevier Inc. All rights reserved.