Plane Strain 2D Crack Propagation In a Heterogeneous Matrix: A Phase-Field Study of Inclusion Stiffness and Proximity

Abstract

This thesis presents a computational phase-field study of crack propagation in brittle composite materials containing a single embedded inclusion. The central question is how inclusion stiffness and proximity to the crack path govern crack-inclusion interaction in plane-strain Mode I fracture. A parametric sweep is conducted across stiffness ratios Eratio = Einclusion/Ematrix ∈ {0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 1.0, 5.0} and vertical offsets yoffset ∈ {4ℓ, 8ℓ, 16ℓ} for both square and circular inclusion geometries, yielding a master phase diagram that classifies each configuration as arrested, deflected, or passed-through. The results demonstrate that compliant inclusions (Eratio < 1) positioned close to the crack path attract and arrest the crack through a fracture energy landscape mechanism: reduced local stiffness lowers the strain energy density within the inclusion, creating an energetic minimum that draws the crack tip toward the compliant region. Stiff inclusions (Eratio = 5.0) produce the opposite effect, repelling the crack slightly upward without deflecting it in any sustained sense. For the square inclusion geometry, the arrest–deflection phase boundary lies between Eratio = 0.3 and Eratio = 0.4 at yoffset = 4ℓ. For the circular geometry, this boundary shifts to between Eratio = 0.2 and Eratio = 0.3, reflecting the smaller effective cross-section presented by a curved boundary to the approaching crack front. At large offsets (yoffset = 16ℓ), only extreme compliance contrasts produce crack arrest, placing a practical bound on the range of crack-inclusion interaction. A preliminary stochastic extension models the matrix Young’s modulus as a spatially correlated lognormal random field and evaluates crack outcomes across ten independent seeds near the deterministic phase boundary. Configurations well inside the arrested or passed-through regimes show consistent outcomes across seeds, confirming that the deterministic phase boundaries are robust under moderate microstructural heterogeneity, while configurations near the boundary exhibit sensitivity to the random field realization. These results provide a foundation for future probabilistic characterization of crack-inclusion interaction in heterogeneous brittle composites.