A Meshfree Peridynamic Model for Brittle Fracture in Randomly Heterogeneous Materials

TL;DR

This paper develops a stochastic meshfree peridynamic model to simulate brittle fracture in heterogeneous materials, incorporating randomness in microstructure parameters and validated through numerical experiments and real-world glass-ceramic fracture analysis.

Contribution

It introduces a unified stochastic peridynamic framework with rigorous convergence analysis and applies it to complex heterogeneous materials, including glass-ceramics.

Findings

The scheme converges with algebraic or sub-exponential rates in random space.

The model accurately predicts crack initiation and growth in heterogeneous materials.

Numerical fracture toughness estimates align well with experimental data.

Abstract

In this work we aim to develop a unified mathematical framework and a reliable computational approach to model the brittle fracture in heterogeneous materials with variability in material microstructures, and to provide statistic metrics for quantities of interest, such as the fracture toughness. To depict the material responses and naturally describe the nucleation and growth of fractures, we consider the peridynamics model. In particular, a stochastic state-based peridynamic model is developed, where the micromechanical parameters are modeled by a finite-dimensional random vector, or a combination of random variables truncating the Karhunen-Lo\`{e}ve decomposition or the principle component analysis (PCA). To solve this stochastic peridynamic problem, probabilistic collocation method (PCM) is employed to sample the random field representing the micromechanical parameters. For each sample, the deterministic peridynamic problem is discretized with an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and demonstrate its convergence for a number of benchmark problems, showing that it sustains the asymptotic compatibility spatially and achieves an algebraic or sub-exponential convergence rate in the random space as the number of collocation points grows. Finally, to validate the applicability of this approach on real-world fracture problems, we consider the problem of crystallization toughening in glass-ceramic materials, in which the material at the microstructural scale contains both amorphous glass and crystalline phases. The proposed stochastic peridynamic solver is employed to capture the crack initiation and growth for glass-ceramics with different crystal volume fractions, and the averaged fracture toughness are calculated. The numerical estimates of fracture toughness show good consistency with experimental measurements.