Stochastic meshless methods for deformation and fracture in materials and structures

Crack propagation is an important failure mechanism in structural and mechanical systems requiring accurate numerical models to implement essential simulation supporting failure prediction. The existence of system uncertainty and risk in loads, material properties, and crack size requires a reliability-based fracture-mechanics analysis to be taken into account. The objective of this study is to develop new stochastic meshless methods for predicting probabilistic structural response and reliability, with particular emphasis on structures containing crack-like defects. The effort is based on: (1) deterministic linear-elastic fracture of homogeneous and functionally graded materials; (2) stochastic meshless methods in linear elasticity; (3) stochastic linear-elastic fracture of homogeneous materials; and (4) stochastic nonlinear fracture of homogeneous materials.; To accomplish the objective of this study the following contributions are made: First, an efficient meshless method was developed to analyze linear-elastic cracked structures subject to single- or mixed-mode loading conditions. The method involves an element-free Galerkin formulation in conjunction with an exact implementation of essential boundary conditions, a new weight function, and a new basis function for meshless fracture of orthotropic materials. A coupled meshless-finite element method was also developed for mode-I and mixed-mode loading conditions. In addition, meshless method involving two new interaction integrals was developed for calculating the stress-intensity factors for a stationary crack in two-dimensional functionally graded materials of arbitrary geometry. Second, stochastic element-free Galerkin method was developed for predicting probabilistic second-moment response and reliability of linear-elastic structures subject to spatially varying random material properties. Third, meshfree method was developed to predict first-order derivatives of stress-intensity factors with respect to crack size. Based on these derivatives, probabilistic fracture-mechanics analysis can be conducted for linear-elastic cracked structures. Fourth, a new enriched meshless method is presented for nonlinear-elastic fracture analysis. Subsequently, a stochastic meshless method was developed for sensitivity and probabilistic fracture-mechanics analysis of nonlinear cracked structures. A total of twenty-seven numerical examples are presented to illustrate the methods developed in this study.