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We introduce a class of models based on near crack tip degradation of materials that can account for fracture growth under cyclic loads below the Griffith threshold. We incorporate the gradual degradation due to a cyclic load through a flow equation that decreases spatially varying parameters controlling the fracture toughness in the vicinity of the crack tip, with the phase and displacement fields relaxed to an energy minimum at each time step. Though our approach is phenomenological, it naturally reproduces the Paris law with high exponents that are characteristic of brittle fatigue crack growth. We show that the exponent decreases when the phase field dynamics is of the Ginzburg-Landau type with a relaxation time comparable to the cyclic loading period, or when degradation occurs on a scale larger than the process zone. In addition to reproducing the Paris law, our approach can be used to model the growth of multiple cracks in arbitrarily complex geometries under varied loading conditions as illustrated by a few numerical examples in two and three dimensions.

We investigate the fracture of Li-ion battery cathodic particles using a thermodynamically consistent phase-field approach that can describe arbitrarily complex crack paths and captures the full coupling between Li-ion diffusion, stress, and fracture. Building on earlier studies that introduced the concept of electrochemical shock, we use this approach to quantify the relationships between stable or unstable crack propagation, flaw size, and C-rate for 2D disks and 3D spherical particles. We find that over an intermediate range of flaw sizes, the critical flaw size for the onset of crack propagation depends on charging rate as an approximate power-law that we derive analytically. This scaling law is quantified in 2D by exhaustive simulations and is also supported by 3D simulations. In addition, our results reveal a significant difference between 2D and 3D geometries. In 2D, cracks propagate deep inside the particle in a rectilinear manner while in 3D they propagate peripherally on the surface and bifurcate into daughter cracks, thereby limiting inward penetration and giving rise to complex crack geometries.

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