Numerical Modeling of Brittle Fracture Using the Phase-Field Method

Abstract

A Master of Science in Mathematics by Ghaid Saeed entitled, “Numerical Modeling of Brittle Fracture Using the Phase-Field Method”, submitted in July 2025. Thesis advisor is Dr. Youssef Belhamadia. Soft copy is available (Thesis, Completion Certificate, Approval Signatures, and AUS Archives Consent Form).This thesis presents a numerical study of brittle fracture using the phase-field method. The models are based on Griffith’s energy principle, which formulates fracture as an energy minimization problem. In the phase-field method, cracks are described using a variable d, where d = 0 indicates intact material and d = 1 represents fully damaged zones. Two different mathematical models are considered. The first is a hybrid formulation that degrades the full strain energy isotropically but drives damage only from the tensile part. The second is an anisotropic model that separates the strain energy into deviatoric (shear) and volumetric (tensile and compressive) components, applying degradation only to the tensile and shear parts. All the obtained models consist of a system of coupled partial differential equations. Numerically, the governing equations are solved using the finite element method. Several fracture examples in two-dimensional structures are studied, including a square plate with two asymmetric notches, a notched plate with a circular hole, a dog-bone specimen under uniaxial tension, and a cruciform plate with an inclined notch. The results demonstrate how each formulation captures the fracture process and how the strain energy formulation influences crack initiation and propagation.College of Arts and SciencesDepartment of Mathematics and StatisticsMaster of Science in Mathematics (MSMTH