# A Viscoelastic-Plastic Creep Model for Initial Damaged Coal Sample Affected by Loading Rate
###### Abstract
Kachanov [23] proposed the concepts of continuous damage factor and effective stress, and studied creep failure in metals; Chan [24] developed a creep model to describe creep damage in shale, arguing that damage induces inelastic flow until the creep failure stage; Ayari [25] established a creep model under cyclic loading/unloading conditions based on continuous damage theory and cyclic loading creep tests; Tomanovic [26] developed a creep model for the time-dependent deformation of soft rock materials based on mudstone creep tests; Maranini [27] conducted creep tests on granite under confining pressures ranging from 0 to 40 MPa, established a non-associated constitutive equation and viscoplastic potential function considering instantaneous creep behavior, and validated the model's correctness; Mazzotti [28] proposed an isotropic model for creep damage in concrete under uniaxial compression, considering that nonlinear viscous strain changes are related to high stress levels, and studied the time-dependent damage and strength of concrete through numerical simulation; Hellmich [29] established an early-age creep constitutive model for concrete, considering the thermo-mechanical coupling mechanism in early pores, and identified the model parameters. Guo et al. [30] investigated the creep damage characteristics and constitutive modeling of pre-damaged coal. They proposed a novel creep damage variable incorporating both loading stress levels and temporal effects and further developed a fractional-order creep damage constitutive model to capture time-dependent nonlinear behaviors. Jiang et al. [31] investigated the failure mechanical properties and acoustic emission characteristics of soft rock-coal composites under dynamic disturbances, and systematically analyzed the damage evolution mechanisms of such composite systems under transient dynamic loading conditions. Yang et al. [32] investigated the creep rupture behavior and permeability evolution of sandstone containing pre-existing dual defects under high-temperature thermal treatment. Their findings provide critical theoretical support for optimizing underground coal gasification (UCG) and carbon capture and storage (CCS) technologies in deep geological formations. Ding et al. [33] analyzed the fracture response and damage evolution characteristics of creep-damaged coal under dynamic loading conditions, revealing the underlying mechanisms governing disaster initiation and propagation in deep mining environments. Bao et al. [34] investigated the time-dependent creep behavior of weakly layered rocks in slope strata under coupled damage interactions, and established a fractional-order nonlinear creep constitutive model that explicitly accounts for both initial damage and creep-induced progressive damage evolution. These studies on the creep characteristics of coal-rock masses under various conditions have provided critical theoretical and practical guidance for accurately predicting their long-term stability. Numerous creep models have been proposed to characterize these behaviors, with a predominant focus on how alterations in initial damage states influence creep time-dependent deformation stages. Furthermore, standardized specimen preparation methods for initially damaged coal-rock samples have been systematically developed to ensure experimental reproducibility and engineering relevance. However, research on the effects of varying loading rates on the creep characteristics of coal specimens remains notably insufficient. Current studies have largely neglected the critical influence of dynamic loading conditions on the long-term mechanical behavior of damaged coal pillars. However, research on the effects of varying loading rates on the creep characteristics of coal specimens remains notably insufficient. Current studies have largely neglected the critical influence of dynamic loading conditions on the long-term mechanical behavior of damaged coal pillars.