**High-Order Microplane Theory for Quasi-Brittle Materials with Multiple Characteristic Lengths**

__The heterogeneous internal structure of quasi-brittle materials governs several aspects of their behavior, especially in the nonlinear range. Size and spacing of weak spots (e.g., aggregate-matrix interfaces, flaws, and slip planes) where failure is likely to occur are two of the most important material characteristic lengths that can be used to characterize the micro- and meso-structure of these materials. Discrete (lattice and particle) models can be conveniently used to directly model these geometrical features, but they tend to be computationally expensive, and consequently, the derivation of macroscopic continuum-based approximations is often highly beneficial. The current study demonstrates that the continuum macroscale approximation of discrete fine-scale models leads naturally to a high-order microplane theory characterized by multiple characteristic lengths. The average weak spot spacing is shown to be associated with strain gradient effects; whereas the average weak spot size is shown to be associated with the Cosserat characteristics of the theory. The formulated microplane theory is compared with and contrasted to classical continuum theories available in the literature. Finally, for strain softening, known in the case of first-order local formulations to cause strain localization in an unrealistic vanishing size volume, a localization limiter capable of enforcing the minimum localization size to be of a finite value is formulated, exploiting the spectral wave propagation analysis approach.__

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