The Unified Calculation Formula of Non-orthogonal Elastoplastic Models for Geomaterials

The deformation behavior of soil and rock materials often exhibits non-orthogonal characteristics. During shear deformation, the strain increment vector may deviate from being normal to the yield surface, necessitating advanced constitutive modeling. To address this issue, a fractional differential approach is adopted, leading to the development of a non-orthogonal elastoplastic model built upon conventional elastoplastic theory. This model introduces a novel concept of plastic flow and loading direction within a normalized formulation framework and incorporates an error-controlled stress integration algorithm to enhance computational efficiency. The proposed E-S method enables modular testing across a wide range of soil types, ensuring robustness and adaptability. Furthermore, a fractional state hardening model is formulated based on the state-dependent hardening rule, establishing an effective link between the fractional order and the critical state parameters of soils. Notably, the modified Cambridge model emerges as a special case within this generalized framework. Simulation results along various stress paths demonstrate that the fractional model successfully captures essential behaviors—such as strain softening and shear dilation—in overconsolidated clay, with validation provided through tests on Weald clay. While the modified Cambridge model tends to overestimate peak strength, resulting in some discrepancies, the fractional model consistently yields accurate predictions using the same set of parameters. By accounting for non-orthogonal deformation, the proposed model significantly enhances predictive accuracy. These findings underscore the effectiveness of the normalized fractional framework and support its potential application in future finite element analyses of geomaterials.