For elastic contact, an exact analytical solution for the stresses and strains within two contacting bodies has been known since the 1880s. Despite this, there is no similar solution for elastic–plastic contact due to the integral nature of plastic deformations, and the few models that do exist develop approximate solutions for the elastic–perfectly plastic material model. In this work, the full transition from elastic–perfectly plastic to elastic materials in contact is studied using a bilinear material model in a finite element environment for a frictionless dry flattening contact. Even though the contact is considered flattening, elastic deformations are allowed to happen on the flat. The real contact radius is found to converge to the elastic contact limit at a tangent modulus of elasticity around 20%. For the contact force, the results show a different trend in which there is a continual variation in forces across the entire range of material models studied. A new formulation has been developed based on the finite element results to predict the deformations, real contact area, and contact force. A second approach has been introduced to calculate the contact force based on the approximation of the Hertzian solution for the elastic deformations on the flat. The proposed formulation is verified for five different materials sets.