A Monolithic Multiscale ESDA Formulation With Resistance‐Weighted Crack Tracking for Prestressed Reinforced Concrete Structures

ABSTRACT This work develops a variational and kinematic multiscale formulation for fracture in three‐dimensional prestressed reinforced concrete structural members ( 3D‐PRCSM ), complemented by a global crack tracking algorithm ( GCTA ) within an embedded strong discontinuity approach ( ESDA ). The multiscale framework couples macro‐ and micro‐scale fields through a monolithic variational structure and introduces a localization‐transfer tensor relating displacement discontinuities across scales. Metallic constituents, including longitudinal reinforcement, stirrups, and prestressing tendons are governed by a finite‐strain plasticity model, whereas the cement paste and aggregate phases follow an anisotropic cohesive‐damage formulation. Crack‐path determination is recast as a global shortest‐path problem over the topology of finite element integration points. Admissible integration points define graph nodes, while neighboring nodes are connected by non‐negative edge costs based on effective fracture‐propagation resistance and geometrical separation. A Dijkstra‐based procedure identifies the path of minimum accumulated resistance and reconstructs the crack trajectory through graph‐node predecessor relations. The formulation provides a mathematical and algorithmic basis for three‐dimensional crack tracking in heterogeneous prestressed concrete members without remeshing or modification of the underlying finite element topology. A controlled three‐dimensional toroidal material‐point benchmark assesses the tracking procedure independently of structural calibration. Finally, the GTA reconstructs an admissible minimum‐resistance trajectory through the prescribed heterogeneous resistance field and preserves the same global propagation branch over three material‐point discretizations, with corresponding geometrical path lengths differing by less than approximately 3.1%.