Fracture Dynamics Based on Finsler Differential Geometry

Crack paths are predicted using empirical mixed-mode criteria, while branching is attributed to a singular critical velocity. We show that fracture directionality and branching emerge from intrinsic thermodynamic topology. Mapping dissipation density to geometric distance defines a Finsler manifold from the interplay between the Eshelby tensor and directional fracture energy. Crack propagation becomes a Hamiltonian geodesic whose affine parameter is the physical crack advance. Classical mixed-mode criteria are linearized artifacts of this geodesic motion in isotropic (Riemannian) limits, enforced by macroscopic scale invariance. Analyzing Finsler Jacobi fields yields a geometric bifurcation condition: branching occurs when the curvature of the resistance field cancels the driving field. This framework predicts that materials with strong microstructural anisotropy undergo deterministic branching at quasi-static velocities, establishing velocity as a secondary parameter modifying the driving curvature rather than originating bifurcation.