Auxetic materials are increasingly used in the field of sports, defense, bio-medicine, and acoustic-filtering due to their unconventional behavior of exhibiting a negative Poisson's effect. The prospective use of auxetics to mitigate shock and impact forces have proven the relevance of research on auxetics. It becomes of paramount importance to understand their behavior and mechanics of deformation, especially regarding their response to dynamic loading conditions. Although auxetics have caught attention from several researchers, a detailed overview of computational modeling of auxetics is still missing and the deformation behavior of auxetics subjected to different loading conditions is an unfolded topic in structural mechanics. In this thesis, computational tools are developed and validated to investigate the effect of auxeticity in a linear elastic continuum and a re-entrant honeycomb structure under different static and dynamic loading conditions. The possibility to assess the global behavior of an assembly of re-entrant honeycomb unit cells by the analysis of an equivalent continuum model with the same global dimensions and homogenized mechanical properties has been examined. The dynamic analysis of a continuum model of an isotropic auxetic material presented the influence of auxeticity in increasing the ratio of shear wave velocity to dilatation wave velocity. The finite element analysis of a simply supported auxetic beam subjected to impact at mid-span affirmed the hypothesis that in auxetics, the material flows towards the point of impact, thus increasing the indentation resistance. Computational challenges in analyzing a beam impact problem are discussed through the comparison of results from the standard finite element analysis, modal analysis, and the B-Bar method. It is recommended to further develop the proposed models to have a more robust approach on analyzing the continuum beam impact problem. The assembly of a re-entrant honeycomb structure used in the finite element simulations were modelled with different unit cells by the variation of the two dominant geometrical parameters of the unit cell members. The results highlighted the strong correlation between the effective mechanical properties and the geometrical parameters, therefore indicating the possibility of achieving the desired values of the Poisson's ratio by simply modifying the necessary geometrical parameters. The results also showed that it is possible to obtain a Poisson's ratio more negative than -1 which can be exploited to design a displacement amplifier especially suitable for defense applications. The close comparison of the analytical and the computational results provided a groundwork to extend the numerical study to geometrical nonlinear analysis and other loading conditions like bi-axial loading and bending at the structural level. The comparison of results obtained from the analysis of an assembled cellular structure and an equivalent continuum model subjected to bi-axial loading and bending demonstrated the possibility of assessing the global behavior of cellular structure using the continuum model. Dynamic analyses of a re-entrant honeycomb structure revealed that the wave propagation phenomenon in a re-entrant honeycomb structure is a result of several reflections, transmissions, mode conversions and superposition of waves occurring at the connection of cell members. The proportion of reflection, transmission and mode conversion of incident wave when it passes the connection can be controlled by varying the re-entrant angle between the cell members. For the lower frequency regime, the speed of wave propagation in re-entrant structures can be estimated using the continuum approach. Furthermore, in a secondary study, the influence of the geometrical parameters of a re-entrant honeycomb structure to exhibit the stop-pass band was examined. Recommendations are provided for the further development of computational tools to analyze the three-dimensional and disordered re-entrant structures. In addition, it is also suggested to investigate the beam impact problem in the cellular model, the effect of rigidity of connection of cell members and research the possibility of having an isotropic re-entrant honeycomb structure as the present model is orthotropic in nature.