Peridynamics is a nonlocal mechanics theory using integro-differential equations without spatial derivatives. Unlike the classical continuum mechanics, peridynamics possesses certain advantages when solving problems involving cracks. In the beginning of the thesis, an analytical solution to the vibration problem via fixed horizon peridynamics is developed, including dynamic responses of a bar and of a beam. The analytical solution to the vibration of the bar is derived through a Taylor series expansion approximation. Numerical examples demonstrate the nonlocal dynamic behavior of the bar and its consistency, in the limit, with local behavior. Further, a new nonlocal beam theory is proposed. The proposed nonlocal beam equation is a generalization of the Euler-Bernoulli beam equation. An analytical solution for the beam deformation is derived. The numerical example of the nonlocal beam deformation shows that the fixed horizon peridynamics has boundary conditions related inaccuracy problems. Thus, a new numerical technique to reduce the discrepancy problem is introduced, which is called: Variable Horizon Peridynamics. This method is quite efficient and it does not require a pseudo-layer to be added outside the physical boundary.

Next, an efficient algorithm to model the bit-rock interaction process based on the variable horizon peridynamics is developed. This model iterates adaptively with the propagation of the crack and with the penetration of the drill bit. The crack propagation in the rock is captured in this model. The relationship between the penetration rate and other drilling parameters is investigated.

Finally, the Navier-Stokes equation is reformulated in a nonlocal sense via the variable horizon peridynamics. It is shown that the reformulated Navier-Stokes equation satisfies Newton’s second law. When the nonlocal parameter reduces to zero, the reformulated Navier-Stokes equation reduces to the classical Navier-Stokes equation. To elucidate the features of the approach, numerical examples of both local and nonlocal Navier-Stokes equations are used.