UCLA

The objective of this research is to develop the multi-scale mathematical formulation and the associated numerical techniques for development of a micro-crack informed stochastic damage model for brittle materials with application to fragment-impact modeling of concrete materials.

A Generalized Stochastic Chaos (GSC) method has been developed in this work. In this approach, the radial basis (RB) and reproducing kernel (RK) approximations have been introduced to represent the stochastic process which provide flexibility in adjusting smoothness and locality in the finite dimensional stochastic spaces. In conjunction with of RK and RB approximations, a collocation method has been employed for solving the stochastic partial differential equation. In this manner, the stochastic system is discretized into a set of deterministic systems that can be solved separately, and the approximation can be tailored according to the characteristics of the stochastic systems under consideration.

The GSC method has been applied to the development of stochastic damage law based on homogenization of microstructures with random voids. In this development, an extrinsic enrichment method under the framework of mesh-free methods has been introduced for modeling micro crack propagation in the RVE. The near-tip field is discretized by utilizing the visibility criterion in conjunction with crack-tip enrichment while preserving the Partition of Unity (POU) property, which guarantees the conservation of mass. Furthermore, an integration method which meets the so called integration constraint has been proposed to enhance the solution accuracy near the crack tip without using the inefficient higher order Gauss quadrature rules.

In the homogenization processes, the characteristics of the stochastic representative volume element (SRVE) have been investigated. The existence of SRVE has been confirmed through the satisfaction of the Hill condition, ergodicity and the principle of the minimum potential energy. The size effect and mesh dependency of the damage model are also demonstrated by using the principle of the minimum potential energy. The mesh dependency issue has been resolved by introducing a length-scale into the homogenized damage evolution equation. Finally, a two-parameter multi-scale damage model has been developed under the framework of the SRVE. The proposed model is then validated through the comparison between numerical simulations and experimental observations of an ultra-high strength concrete subjected to trial axial compression with various levels of confinement.