Part - I: Coupling of peridynamics and the finite element method and modeling of creepOne of the important modes of unpredicted failure for large complex structures such as power stations, turbines, processing plants, refineries is a failure due to creep. The ability to predict the evolution of damage due to creep is important. Classical continuum based damage models are used widely for this purpose along with some numerical techniques such as the Finite Element Method (FEM). Although this gives a better prediction of creep strain, when the material is near failure, the finite element method is not easily able to predict crack initiation and propagation. Special care has to be taken for crack propagation such as, re-meshing or use of extended FEM which either lose accuracy or increase the computational cost. These disadvantages can be overcome by using Peridynamics (PD) instead of the finite element method due to its integral formulation and the ability to model the crack as the material response. Therefore one of the aims of this dissertation is to develop a peridynamic formulation equivalent to classical stress-based damage models. We have chosen the Liu-Morakami damage model for our study. In order to check the robustness of the new method, numerous examples are simulated and the results are compared with finite element simulations and show good agreement.Although peridynamics is a very powerful method in predicting crack propagation, it still has some disadvantages such as very high computational cost and an error in solution near the boundary of the domain. Also, the application of loads and boundary conditions in peridynamics is a tedious process. Therefore, in order to remove these disadvantages, peridynamics can be coupled with the finite element discretization. The aim is to use PD near crack and the FEM everywhere else. This allows advantage to be taken of both methods giving accurate results with optimum resources. The main challenge in the coupling of peridynamics and the finite element method is a generation of spurious reflections of waves at the interface of peridynamics and the finite element domain. To understand this problem in detail, we use an analytical approach to study the propagation of a plane wave and its spurious reflection in a peridynamic bar using the two different methods. In the first method, a coupled peridynamic--finite element approach is used in which the peridynamic formulation is used in one part of the domain and the finite element is used in the other part. In the second method, the peridynamic formulation is used in the entire domain, but the bar is discretized by two grids of different sizes. In both cases, the size of the grid of each zone does not change and the two grids share one node with each other. The incident wave travels from the finer grid toward the coarser grid. For the case when peridynamics is used on the entire domain, the size of the peridynamics horizon changes based on the size of the gird. For both cases, we investigate the impact of the relative size of the girds on the amplitude and energy of the transmitted and reflected waves. Our analytical and numerical results show that more spurious reflections occur when the size mismatch between the two grids is large. In both cases, the issue of spurious wave reflection becomes more severe when the peridynamic horizon size increases. For the case of a coupled peridynamic--finite element grid, even when the size of the two grids is the same, spurious wave reflection occurs which is due to the change in the formulation from a nonlocal to a local continuum. The spurious reflection reduces when the wavelength of the incident wave is large compared with the coarse grid. Part - II: Study of the damping property of polymer compositesVisco-elastic materials are used in many applications such as building dampers, bunkers or protective casing for an external hard drive. This is because visco-elastic materials have very good damping capacity and can absorb a huge amount of energy without failure due to their polymeric nature. Polyurea is an example of one such material which is commercially available. Two main problems with the use of such materials are, i) their behavior is very complex and hence it is hard to develop a constitutive model which is simple to use and also accurate, and ii) these materials lack strength and hence need an addition of some fillers in order to provide strength. In this dissertation, a nonlinear hyper--viscoelastic constitutive model obtained by the superposition of a hyperelastic and a viscoelastic model is proposed to model the behavior of polyurea under both tensile and compressive loading conditions at various strain rates. The incompressible Ogden model is used to model the strain-dependent response of polyurea while a three parameter standard linear solid (SLS) model and nonlinear K-BKZ modes are used to model the strain rate dependent behavior of polyurea. The material parameters of the model are found by curve fitting of the proposed model to the experimental data. Comparison of the proposed model and the experimental data shows that the proposed model can closely reproduce the stress-strain behavior of polyurea under a wide range of strain rates (-6500 to 294 /s).We also use finite element modeling to investigate the damping property of polymer composites consisting of a viscoelastic matrix and randomly dispersed elastic particles. The dynamic correspondence principle of viscoelasticity is used to solve boundary value problems. The impact of vibration frequency and inclusion size and volume fraction on the damping capability of polymer composites are studied. Results obtained by finite element simulations are compared with results obtained from popular micromechanics methods such as the rules of mixtures, Halpin-Tsai, Hashin-Shtrikman, and Mori-Tanaka. It is shown that micromechanics methods give accurate predictions only when inclusion volume fraction is small. Our results show that the loading frequency and inclusion volume fraction significantly impact the damping capability of polymer composites. In contrast, the impact of the size of inclusions on the damping properties of polymer composites is negligible. The effect of domain boundary conditions on the simulation results is studied by conducting finite element modeling on representative volume elements (RVEs) which are subjected to periodic or mixed boundary conditions. The modeling results indicate that both boundary conditions lead to similar predictions for damping. Sensitivity analyses are conducted to assess how material properties influence the damping property. The sensitivity analyses show that an increase in the stiffness of the composite matrix leads to a reduction in the damping capability of polymer composites. In contrast, an increase in the stiffness of inclusions results in an increase in the damping capability of polymer composites. Moreover, the damping properties are improved if the relaxation time of the viscoelastic matrix is increased.