Fractional-order critical state model for soils in characteristic stress space

The fractional derivative possesses the properties of describing the zero-order differential to the first-order one consecutively, and the fractional gradient direction of the curve is no longer perpendicular to its tangent. The non-orthogonal gradient law of fractional differential can be used to describe the non-associated flow rules between the plastic flow direction and the yield surface of soils. Based on the concept of characteristic stress and the critical state theory, the fractional-order flow law is used to describe the direction of plastic strain increment of soils in the characteristic stress space, and the fractional-order critical state model is then established. The fractional-order differential and the characteristic stress are unified by using the established constitutive model in the framework of the critical state theory. On the one hand, the deformation and strength properties of soils under triaxial compression and extension can be directly described by the established constitutive model. On the other hand, the method for determining the fractional order by triaxial tests is also given. There are only 5 material parameters with clear physical meanings in the established constitutive model, and they can be easily determined by the conventional triaxial tests. Moreover, the proposed constitutive model can also be simplified into the modified Cam-clay model. Finally, the new model is validated by means of the test results from the conventional triaxial tests.