Crack bridging using linear elastic spring distribution with power-law variation of stiffness

This article establishes a theoretical foundation and provides numerical evidence for a novel computational approach to simulate crack nucleation and propagation by finite crack advances within the framework of the coupled stress and energy criterion (CC) of finite fracture mechanics (FFM). The key innovation lies in bridging prospective crack paths, predicted by the stress criterion of CC, using a suitable Winkler-type continuous linear-elastic spring distribution. This allows a damage variable to be defined along these prospective finite crack paths, here called fictitious cracks. The effect of the spring stiffness variation on the elastic solution near a fictitious crack is analysed, focusing on how a power-law variation in spring stiffness affects the fictitious-crack tip stress singularity. Power-law variations of spring stiffness that eliminate this fictitious singularity are identified. Such a damageable spring distribution along the fictitious crack enables the seamless application of the energy criterion of CC to modelling virtual crack propagation through the progressive deactivation of springs within the bridged zone. Numerical tests show the effect of the power-law exponent of spring stiffness variation on both the stress state at the fictitious crack and on the energy release rate (ERR) calculated at the virtual-crack tip. Consequently, an optimal power law is proposed for modelling virtual crack propagation that successfully balances the elimination of the fictitious-crack tip singularity against the computational overhead associated with highly stiff spring distributions to achieve accurate ERR values. Finally, a new formulation of the path-independent J-integral is derived for a crack along a spring interface with varying stiffnesses.