Fractional Calculus based Elasticity obtained from Homogenization

Abstract

Anomalous phenomena, such as diffusion in biological systems, scale-dependent plasticity in thin

films, and electromagnetic wave propagation in complex media, often defy explanation by conventional

theories rooted in integer-order calculus. Models based on fractional calculus have proven

effective in capturing such behaviors, yet the fractional exponents they employ are typically introduced

as empirical fitting parameters without grounding in the underlying physics. In this work, we

propose that fractional differential equations can emerge naturally through homogenization of materials

with complex microstructures, which are themselves the source of the anomalous behavior that

necessitates a fractional description. Focusing on linear elasticity, we identify precise microstructural

conditions that give rise to emergent fractional behavior. In the static regime, homogenization reveals

that the order of the fractional derivative is directly linked to the power-law exponent of the microstructure’s

autocovariance function. By connecting microscale structural variation to macroscale

material response, our study opens new avenues for designing architectured materials with tailored

anomalous properties. The implications of thiswork extend broadly across physics, biology, materials

science, and beyond.