In the present thesis a three-dimensional finite element formulation for concrete failure under high-energy impact loadings is presented. The formulation is based on the theoretical framework of continuum mechanics and irreversible thermodynamics. In order to obtain a non-unilateral contact/impact description, the spatial regions occupied by the bodies under collision are discretized with linear tetrahedral elements. The temporal discretization is traditionally conducted by finite difference method. During the time of contact, the contact constrains are satisfied with the Lagrange multiplier method adapted for the explicit time integration. The frictional behaviour on the contact interface is assumed to be of kinematic type described with the Coulomb friction model. At the numerical implementation level, the dissipative nature of friction forces is introduced via the relaxation of tangential displacement on the contact interface. The quantitative description of expected finite strains is conducted according to the Update Lagrange formulation of motion. Following the standard notations used in contact mechanics, the concrete body is denoted as target body and the body that transmits its kinetic energy, as the contractor body. The hyperelastic material model is used to define the constitutive behaviour of the contractor body. On the other hand, in order to simulate the mechanical behavior of the concrete body, the stress-strain relationship is carried out via the rate sensitive microplane material model with relaxed kinematic constraint (co-rotational formulation). It is important to note that the concrete strain rate effect can be particularly evidenced in impact loadings. Thus, to numerically replicate the concrete response in a realistically way, the strain rate effect is introduced by means of the energy activation theory of bond ruptures. For this purpose, the macroscopic strain measure is performed with the Green-Lagrange strain tensor. Damage and cracking phenomena are modeled within the concept of smeared cracking. Furthermore, the crack band method is used with the aim of assuring the objectivity of the analysis with respect to the size of the finite elements. Finally, in order to validate the developed formulation, the free fall drop weight experiment is numerically replicated. The retrieved numerical results are evaluated, discussed and compared with the experimental results. A parametric study aimed at numerical investigation of the influence of loading rate on the failure mode of beams under impact loadings, is carried out. It is shown that the plain beam resistance and failure modes strongly depend on the loading rate. Indeed, the numerical formulation predicts correctly the beam failure patterns investigated with the experimental methodology. Namely, for relative low impact velocities the numerical results coincide with the experimental results providing the expected bending mode of failure (mode-I). On the other hand, by increasing the loading rate, there is a transition of the failure mechanism from bending to shear mode (mixed mode). Based on the retrieved numerical predictions, one may conclude that the developed numerical formulation is adequate for investigating the response of concrete under high-energy impact loadings. Indeed, due to the fact that the failure process in concrete occurs in a very short period of time, the numerical study is useful and necessary for better understanding of damage phenomena occurring under impact loadings.