Abstract: Geomaterials generally exhibit non-orthogonal deformation behaviour, i.e., the strain increment vector is not always normal to the yield surface during shearing. To address this issue, a fractional-order 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 element tests across a wide range of soil types, ensuring robustness and adaptability. Furthermore, a fractional-state hardening model is formulated based on a state-dependent hardening rule, establishing an effective link between the fractional order and the critical state parameters of soils according to the basic assumptions of critical state theory. 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 dilatancy-in overconsolidated clay, with validation provided through tests on Weald clay. While the modified Cam-clay model overestimates the peak strength, leading to some discrepancies, the fractional model consistently yields accurate predictions with the same set of parameters. The model also performs well in simulating, the constant-mean-stress (constant P) triaxial compression tests on Kaolinite P300 clay. 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 complex soil and rock materials.