Nonlinear damage creep model based on variable-order fractional theory for rock materials

In this study, based on the variable-order fractional theory and continuous damage model with sum function, we proposed a nonlinear variable-order fractional damage creep model to describe the creep mechanical behavior of rock materials. For better exhibiting the evolution of consumption of viscosity in creep, the ‘two-area-continuous-variable-order-analytical-method’ is first presented. In different areas of creep (area 1 and 2), the consumption of viscosity in each area is expressed by the varying-order with an exponential function. The consumption of viscosity mainly happens in the area 1 within decaying and steady creep and most of viscosity is consumed to resist deformation of creep. And with the development of time, the proportion of consumption of viscosity in material gradually decreases, which is same as variation of order. During the area 2, comparing to the consumption of viscosity in the area 1, the viscosity play a relatively weak role in resisting deformation of creep, which is due to plasticity occupy the mail role. Finally, for verifying the applicability and rationality of proposed variable-order fractional damage creep model, except for experimental creep data of sandstone in this work, additional creep data of salt rock and mudstone are also achieved from other papers. The fitting curves of proposed model are well agreement with experimental data.