A collocation approach for spatial discretization of stochastic peridynamic modeling of fracture

Abstract

In this paper a collocation approach is presented for spatial discretization of the partial integrodifferential equation arising in a peridynamic formulation in stochastic fracture mechanics. In the formulation nodes are distributed inside the domain forming a grid, and the inverse multiquadric radial basis functions are used as interpolation functions inside the domain. Due to this discretization the peridynamic stiffness is generated in a manner similar to the finite element method. Further, any discontinuity in the domain is included in this discretized form and affects only the peridynamic stiffness of the adjacent nodes. Using this approach as a tool, the probability density function of the energy release rate can be determined at a given crack tip point for all possible crack paths. Thus, the crack propagation direction can be probabilistically identified. This is accomplished by numerical evaluation of the requisite Neumann expansion using pertinent Monte Carlo simulations. Specific examples of applications are included.

A fast collocation method for a static bond-based linear peridynamic model

“…In addition, the method reduces the computational work of evaluating and assembling the stiffness matrix, which often constitutes a major portion of the overall computational work, from O(N 2 ) to O(N ). In fact, the fast collocation method is nothing but the conventional one, and so naturally inherits the stability, convergence, and other numerical advantages of conventional collocation methods [12,22,30,33,36,42,52]. Furthermore, any numerical techniques developed for traditional collocation methods, such as the numerical quadra-tures for the accurate evaluation of singular integrals [2,18,26,27,29,36,51], can be applied to the fast collocation method.…”

“…where the indices i, i , j, and j and the indices m and n are related by (12). Thus to prove Theorem 1, it remains to prove the matrix blocks B j,j with |j − j | L + 1 are zeros and the matrix blocks…”

“…where the indices i, i , j, and j , and the indices m and n are related by (12). By the following translations…”

“…, and (w,w), where the second equality is obtained from the fact that (35) is equal to (37), the third equality is obtained by (38) or (29) and the indices i, i , j, and j as well as the indices m and n are related by (12).…”

Peridynamics review

“…Dynamic fracture has also been studied with PD for various types of structures such as polymers [185], fiber networks [186], concrete structures [187], anisotropic materials [188][189][190], functionally graded materials [191], polycrystalline materials [192] as well as geomaterials [62,[193][194][195][196]. It is worth mentioning that Evangelatos and Spanos [197] presented the application of PD to stochastic fracture modeling, with a novel spatial discretization of the PD equation. [198] presented a fatigue damage model to predict damage in laminates under cyclic loading.…”