The methodology for identification of mechanical characteristics of the nonlinear material model for concrete, taking into account the elastic-plastic deformation and the damage accumulation under monotonous and cyclic loading, is proposed. The using such improved models of concrete deformation is actual for carrying out finite-element computations of the most important elements of unique and responsible buildings and structures. The proposed methodology is verified for three different types of concrete (B45, B25, B5), including also their preliminary heat treatment at 200 °C, 300 °C, 400 °C and 600 °C. The experiments were carried out on standard specimens of cubic and prismatic form under compression, as well as on dog-bone-shaped specimens under tension. Elasticity and plasticity moduli, ultimate strengths in compression and tension, damage evolutions during deformation process were obtained in tests. Particular attention has been paid to the search for reliable and effective methods for determining damage based on cyclic deformation curves in the pre-peak and after-peak loading regimes. Comparison of simulation results with experimental data under monotonic and cyclic compression demonstrates a good agreement for regular and for overheated concrete.