A new model of the stationary electrostatic plasma sheath in front of divertor plates is developed, which takes into account strong inelastic processes. Using particle-in-cell simulations and analytic estimates it is demonstrated, that the properties of the tokamak divertor plasma sheath can significantly deviate from the properties of the classical sheath model. The most significant deviations are the increased energy flux to the plates and non-monotonic potential and ion velocity profiles in the presheath. Two main reasons for these deviations are identified: strong inelastic collisionality of the plasma presheath and the presence of super-thermal plasma particles originating from the upstream scrape-off layer.