Modeling Fatigue-Induced Anisotropic Quasi-Brittle Damage Based on the Endurance Surface Concept

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This work proposes a novel continuum damage framework for fatigue based on the endurance-surface concept and uses the energy-release rate as the driving force. Damage evolution is governed by the distance of the thermodynamic driving force from the endurance surface. In contrast to classic failure surfaces, this allows damage to accumulate over many cycles even under constant-amplitude loading. The endurance surface therefore directly relates to the physical endurance limit of the material. To obtain mesh-objective results, the formulation is regularized by a micromorphic gradient enhancement. The incorporation of anisotropic damage evolution and the microcrack-closure-reopening effect extends the framework to multiaxial fatigue and loading-path-dependent degradation. The chosen prototype damage evolution fulfills three requirements: reasonable physics, computational robustness, and calibration flexibility. The model is successfully calibrated to both the monotonic response of plain concrete and to the high-cycle fatigue behavior of low-alloy steel. The numerical examples cover monotonic failure of an L-shaped concrete specimen, stress-life behavior under cyclic loading, and combined axial-torsional fatigue. These cases demonstrate how the proposed formulation applies to practical scenarios ranging from standard quasi-brittle fracture benchmarks to classical fatigue characterization and complex multiaxial damage evolution. The examples demonstrate that the formulation captures progressive degradation over many cycles and reproduces characteristic stress-life behavior. The influence of anisotropic degradation becomes especially relevant under multiaxial loading conditions during the near-failure phase. Overall, the approach provides a thermodynamically consistent, gradient-enhanced, and computationally robust framework for simulating fatigue-driven damage in the high-cycle regime.

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