The numerical analysis of spontaneously formed discontinuities such as cracks is a long-standing challenge in the engineering field. Approaches based on the mathematical framework of classical continuum mechanics fail to be directly applicable to describe discontinuities since the theory is formulated in partial differential equations, and a unique spatial derivative, however, does not exist on the singularities. Peridynamics is a reformulated theory of continuum mechanics. The partial differential equations that appear in the classical continuum mechanics are replaced with integral equations. A spatial range, which is called the horizon, is associated with material points, and the interaction between two material points within a horizon is formed in terms of the bond force. Since material points separated by a finite distance in the reference configuration can interact with each other, the peridynamic theory is categorized as a nonlocal method.The primary focus in this research is the development of the discretized bond-based peridynamics for solid mechanics. A connection between the classical elasticity and the discretized peridynamics is established in terms of peridynamic stress. Numerical micromoduli for one- and three-dimensional models are derived. The elastic responses of one- and three-dimensional peridynamic models are examined, and the boundary effect associated with the size of the horizon is discussed. A pairwise compensation scheme is introduced in this research for simulations of an elastic body of Poisson ratio not equal to 1/4. In order to enhance the computational efficiency, the research-purpose peridynamics code is implemented in an NVIDIA graphics processing unit for the highly parallel computation using a high-level implicit programming model. Numerical studies are conducted to investigate the responses of brittle and ductile material models. Stress-strain behaviors with different grid sizes and horizons are studied for a brittle material model. A comparison of stresses and strains between finite element analyses and peridynamic solutions is performed for a ductile material. To bridge material models at different scales, a multiscale procedure is proposed.An approach to couple the discretized peridynamics and the finite element method is developed to take advantage of the generality of peridynamics and the computational efficiency of the finite element method. The coupling of peridynamic and finite element subregions is achieved by means of interface elements. Two types of coupling schemes, the VL-coupling scheme and the CT-coupling scheme respectively, are introduced. Numerical examples are presented to validate the proposed coupling approaches including one- and three-dimensional elastic problems and the mixed mode fracture in a double-edge-notched concrete specimen.A numerical scheme for the contact-impact procedure ensuring compatibility between a peridynamic domain and a non-peridynamic domain is developed. A penalty method is used to enforce displacement constraints for transient analyses by the explicit time integration. In the numerical examples, the impact between two rigid bodies is examined to validate the contact algorithm. The ballistic perforation through a steel plate is investigated, and the residual velocities of the projectile are compared with the results by an analytical model.Peridynamics is applied to study porous brittle materials. An algorithm is developed to generate randomly distributed cubic voids and spherical voids for a given porosity. The material behaviors at the macroscopic level including the resultant Young's modulus and the strength are studied with varying amounts of porosities. The degradations of Young's modulus and strength are compared with empirical and analytical solutions.
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