This study presents a novel numerical method to simulate the time-dependent behavior of concrete materials. The deformation response of both traditional and fractional calculus (FC) viscoelastic models is analyzed by the proposed method, in which models’ one-dimensional constitutive relationships are extended to the three-dimensional form. By applying numerical and finite difference methods based on Caputo-type fractional integration, the stress-strain behavior of the FC viscoelastic model is discretized over the time scale. This method exhibits broad application prospects, and can be employed to represent various forms of viscoelastic models. For the concrete material, suitable FC viscoelastic models are developed. To facilitate practical engineering applications of the proposed model, a numerical solution algorithm is implemented in the finite element (FE) analysis through the User-defined Material Mechanical Behavior (UMAT) interface of the commercial FE software ABAQUS. Finally, FE results are compared with the experimental results from several concrete creep tests. The consistency between the FE results and experimental data confirms the effectiveness of the proposed model in describing concrete behavior. The proposed method and model provide theoretical and numerical support for a more profound understanding and simulation of the time-dependent deformation of RC specimens in practical engineering applications.