Fractional derivative viscoelastic models based on L2-1σ formula: modelling and numerical application

This study develops two three-dimensional fractional derivative viscoelastic models and introduces a novel L_2-1σ difference formula for numerical implementation. In deriving the fractional derivative models, a spring-pot element, defined by Caputo fractional calculus, replaces the dashpot or spring in the classical Kelvin-Voigt model. Using the Laplace transform approach, the creep compliance and stress relaxation modulus of the fractional derivative model are derived. Additionally, the models are extended into a generalized three-dimensional form, and a novel L_2-1σ difference algorithm is employed to discretize the constitutive equations. This method has been shown to provide superior computational efficiency and accuracy compared to the GL algorithm, with a convergence order of 3-α. Subsequently, one of the models was compiled and implemented through the UMAT interface of ABAQUS, facilitating both element-level and large-scale project simulations. The numerical predictions from the model were compared with the results of triaxial creep tests conducted on Hangzhou clay and Hong Kong marine clay. The results confirm a reasonable degree of agreement between the predicted and experimental results, validating the accuracy and feasibility of both the model and the L_2-1σ difference algorithm. Furthermore, when applied to a practical case involving the long-term deformation of an artificial island construction project, the predictions from this model exhibit strong consistency with the measured data. This work provides valuable insights into understanding and predicting the time-dependent behavior of viscoelastic materials.