Anomalous rheology is a material behavior that presents the fingerprint of power-laws, arising from anomalous diffusion in microstructures, and observed in a range of complex materials. Such microstructures often display a fractal nature with sub-diffusive dynamics, e.g., of entangled polymer chains, and defect interactions such as dislocation avalanches, cracks, and voids. The corresponding macroscopic non-exponential behavior makes integer-order models to lack a compact representation of the small-scale physics. Furthermore, classical linear viscoelastic models require arbitrary arrangements of Hookean/Newtonian elements, introducing a limited number of exponential relaxation modes that, at most, represent a truncated power-law approximation. While this may be satisfactory for short times at engineering accuracy, such models often yield high-dimensional parameter spaces and lack predictability for multiple time/length-scales. In this scenario, Fractional Calculus (FC) becomes an attractive modeling alternative since it naturally accounts for power-law kernels in its integro-differential operators. This allows accurate and predictive modeling of soft materials for multiple timescales, in which most standard models fail or become impractical.In this work, a data-driven framework for efficient, multi-scale fractional modeling and failure of anomalous materials is proposed. The overarching goal is to identify/construct efficient fractional rheological models, especially for soft materials, undergoing nonlinear response and failure. To this purpose, a fractional linear and nonlinear viscoelastic existence study is developed and employed for the first time to urinary bladder tissues undergoing large strains. The framework is extended to account for power-law viscoplastic behavior, and aiming for applications to larger systems, the resulting models are solved through a new approach called fractional return-mapping algorithm, that generalizes existing predictor-corrector schemes of classical elastoplasticity. Regarding the effects of fractional constitutive laws on structural dynamics, a few developed models are incorporated to beam and truss structures, where the effects of evolving constitutive laws on the anomalous dynamics of systems are analyzed. Although FC became an effective modeling tool in the last few decades, it requires careful considerations to satisfy basic thermodynamic conservation/dissipation laws. To this end, the thermodynamic consistency of the developed visco-elasto-plastic models with the addition of damage effects is proved. Furthermore, the associated energy release rate due to crack/void formation is consistent with the employed fractional rheological elements, which naturally introduces memory effects on damage evolution.Fractional differential equations (FDEs) inherently carry a functional nonlocal dependency and near-singular behaviors at bounded domains, which increases the computational complexity and degenerates the global accuracy of many existing numerical schemes. Therefore, two numerical contributions are proposed in the last part of the framework. The first one is a data-driven singularity-capturing approach that automatically addresses the low solution regularity and yields high accuracy for long time-integration. In the second contribution, fast implicit-explicit (IMEX) schemes are developed for stiff/nonlinear FDEs, which are shown to have larger stability regions than existing approaches.
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