The Greenwood–Williamson (GW) model has been one of the commonly used contact models to study rough surface contact problems during the past decades. While this has been a successful model, it still has a number of restrictions: (i) surface asperities are spheres; (ii) the overall deformation must be assumed to be small enough, such that there is no interaction between asperities, i.e., they are independent of each other; and (iii) asperity deformation remains elastic. This renders the GW model unrealistic in many situations. In the present work, we resolve above restrictions in a discrete version of the GW model: instead of spherical asperities, we assumed that the surface consists of three-dimensional sinusoidal asperities which appear more similar to asperities on a rough surface. For single asperity mechanical response, we propose a Hertz-like analytical solution for purely elastic deformation and a semi-analytical solution based on finite element method (FEM) for elastic–plastic deformation. The asperity interaction is accounted for by discretely utilizing a modified Boussinesq solution without consideration of asperity merger. It is seen that the asperity interaction effect is more than just the delay of contact as shown in the statistical model, it also contributes to the loss of linearity between the contact force and the contact area. Our model also shows that: for elastic contact, using spherical asperities results in a larger average contact pressure than using sinusoids; when plasticity is taken into account, using a sphere to represent asperities results in a softer response as compared with using sinusoids. It is also confirmed that sinusoidal asperities are a much better description than spheres, by comparison with fully resolved FEM simulation results for computer-generated rough surfaces.