History-Dependent Fractional Plasticity for Cyclic Dilation in Granular Media

Abstract

The mechanical behavior of granular media under cyclic loading is characterized by complex phenomena such as progressive volumetric expansion, commonly referred to as cyclic dilation, and pronounced history dependence. Classical integer-order plasticity frameworks frequently struggle to accurately capture the fading memory effects and the accumulation of inelastic deformations over numerous loading cycles without resorting to excessively convoluted phenomenological rules. This paper introduces a comprehensive history-dependent fractional plasticity model specifically formulated to simulate cyclic dilation in granular materials. By replacing traditional integer-order derivatives with fractional-order derivatives in the evolution equations of internal state variables, the proposed framework intrinsically embeds the entire loading history into the constitutive response. The fractional continuum mechanics approach allows for a natural representation of non-local temporal effects, meaning that the current plastic flow direction and magnitude are determined not only by the instantaneous stress state but also by the weighted accumulation of past states. A rigorous thermodynamic basis is established, followed by the derivation of the fractional yield surface and the corresponding non-associated flow rule. The formulation is further complemented by a robust numerical integration scheme utilizing a modified fractional return mapping algorithm. Extensive parametric analyses and comparisons with representative cyclic triaxial test scenarios demonstrate that the fractional model offers superior capabilities in capturing the transition from initial contraction to ultimate cyclic dilation, providing a physically mathematically sound alternative to conventional bounding surface models.