Non Local Mechanics in the Time and Space Domain--Fracture Propagation via a Peridynamics Formulation: A StochasticDeterministic Perspective

Novel numerical methods for treating fractional differential and integrodifferential equations arising in non local mechanics formulations are proposed. For fractional differential equations arising in modeling oscillatory systems incorporating viscoelastic elements governed by fractional derivatives, the devised scheme is based on the Grunwald-Letnikov fractional derivative representation, dual time meshing technique and Taylor expansion. The proposed algorithm transforms the governing fractional differential equation into a second order differential equation with appropriate effective coefficients. The enhanced efficiency of the scheme hinges upon circumventing the calculation of the non local fractional derivative operator. Several examples of application are provided.;Further, the concept of non locality, specifically viscoelasticity, governed by fractional derivatives is utilized to accurately model polyester materials. Specifically, the linear standard solid (Zener model) is extended to capture non linear viscoelastic behavior. Then, experimental data of polyester ropes are utilized using the Gauss Newton and Levenberg-Marquart minimization algorithm to determine the model parameters.;Next, for integrodifferential equations arising in peridynamics theory of mechanics, an approach is formulated based on the inverse multi-quadric (IMQ) radial basis function (RBF) expansion and the Kansa collocation method. The devised scheme utilizes interpolation functions and basis function expansion for the spatial discretization of the peridynamics equation. This significantly reduces the computational effort required to numerically treat the peridynamics equations. Further, the proposed method is extended to account for mechanical systems with random material properties operating under random excitation. For this, the stochastic peridynamics governing equation of motion is solved using the benchmark Monte Carlo analysis and tools of stochastic analysis. The stochastic analysis is done by numerical evaluation of the requisite Neumann expansion using pertinent Monte Carlo simulations.;Further, the usefulness of the radial basis function (RBF) collocation method in conjunction with a polynomial chaos expansion (PCE) is explored in stochastic mechanics problems. It is shown that the proposed approach renders further solution improvements in solving stochastic mechanics problems vis-a-vis the stochastic finite element method and the element free Galerkin method.