Efficient stochastic fracture assessment of three-dimensional concrete structures: a machine learning-integrated phase field framework

Fracture is one of the dominant deterioration and failure mechanisms affecting the serviceability and safety of concrete infrastructure. In practice, unavoidable variations in material properties and geometric configurations introduce significant uncertainties into fracture-governed structural responses, leading to substantial variability in structural performance and failure risk. Reliable assessment of such responses under uncertainty is essential for performance-based design and risk-informed asset management. However, the inherent complexity of fracture behaviour and the high computational demand of uncertainty quantification make such assessments particularly challenging. To address this challenge, this study proposes a phase field-based framework for stochastic assessment of fracture-governed structural responses in three-dimensional (3D) concrete structures. The framework combines a phase field model with an extended support vector regression (X-SVR) algorithm to establish mappings between uncertain inputs and targeted structural responses. To further improve predictive capability, a novel automatic relevance determination-generalised Hermite kernel (ARD-GHK) is developed and embedded into the X-SVR formulation. The resulting surrogate models enable rapid uncertainty quantification and statistical characterisation of fracture-governed structural responses over large ensembles of random realisations. Three representative 3D examples are employed to validate the proposed framework. The results demonstrate that the framework achieves excellent agreement with benchmark Monte Carlo simulations while reducing computational cost by several orders of magnitude. The proposed ARD-GHK is also shown to outperform polynomial and Gaussian kernels in surrogate modelling. By integrating high-fidelity fracture simulation with data-driven surrogate modelling, the framework provides an efficient computational approach for uncertainty quantification and structural safety evaluation of concrete infrastructure.