This paper presents a cubic model for the sphere–flat elastic–plastic contact without adhesion. In the cubic model, the applied load and the contact area are described by the cubic polynomial functions of the displacement to the power of 1/2 during loading and unloading, and the applied load is also expressed as the cubic polynomial function of the contact area to the power of 1/3 during loading. Utilizing these cubic polynomial functions, the elastic–plastic load (EPL) index, which is defined by the ratio between the dissipated energy due to plastic deformations and the work done to deform the sphere during loading, is calculated analytically. The calculated EPL index is just the ratio between the residue displacement after unloading and the maximum elastic–plastic displacement after loading. Using the cubic model, this paper extends the Johnson–Kendall–Roberts (JKR) model from the elastic regime to the elastic–plastic regime. Introducing the Derjaguin–Muller–Toporov (DMT) adhesion, the unified elastic–plastic adhesion model is obtained and compared with the simplified analytical model (SAM) and Kogut–Etsion (KE) model.