Technical CommunicationA novel damage-based creep model considering the complete creep process and multiple stress levels
Introduction
Modeling the time-dependent behaviors of natural rocks has become an important objective in rock engineering. As shown in Fig. 1(a), a complete creep strain curve can be divided into three stages after instantaneous elastic deformation, i.e., the primary (or initial decelerating creep) stage, the secondary (or stable creep) stage, and the tertiary (or accelerating creep) stage [10]. Many efforts have been devoted to establishing creep models for predicting the mechanical behavior of rock during the creep process. Generally, the most popular creep models can be divided into two groups: component models and empirical models [29], [1], [26]. Empirical models, such as the Rabotnov model and Hoff model, were developed to simulate rock creep behaviors by performing fitting analyses on existing experimental data [11], [14], [22], [3], [6]. Empirical models require very few parameters and are suitable for many applications. However, the parameters usually lack precise physical significance. Moreover, the creep time in experiments is usually much shorter than that in actual engineering practice.
Component models consider rocks to be a combination of several standard mechanical elements, such as Hooke springs, frictional elements, and Newton dashpots. The different time-dependent deformation behaviors of component models can be represented by a particular combination of these elements, which can generate elastic, plastic, and viscous deformation, respectively. Component models provide a feasible approach for establishing an analytical model based on the mathematical expression of standard elements. However, most of the previous creep models are concerned with the representation of the primary and secondary creep stages [25], and the previous models are unable to predict the accelerating creep stage and corresponding final failure of rock. Some researchers developed component models to capture the accelerating creep stage by adding mechanical elements [2], [8], [7], [5], [28], [27]. Nevertheless, the mathematical complexity and redundant parameters of these multicomponent systems introduce difficulties in the application of their models. The creep lifetime is an important characteristic of natural rock, which affects the creep process and is also dependent on the stress level. However, existing models have not considered the lifetime as a model parameter, which is the main reason why the accelerating creep stage is not well modeled in most of the existing models.
Another common problem of the existing creep models is that these models can only simulate the rock creep process at a specific stress level because all the model parameters are determined based on the creep test results at a specific stress level. This means that the model parameters are valid only under one stress level. Consequently, all the model parameters need to be redetermined if the stress level changes. The existing models do not have universal applicability in predicting creep behaviors at other stress levels.
Due to the complicated microscopic mechanism related to creep behavior, developing a unified creep model that can represent the complete creep process of rock under multiple stress levels is a challenge. In the proposed study, a new creep model has been established for this purpose, combining the time-hardening theory and damage theory. The proposed creep model is simple in mathematical expression and can fully capture all the typical creep behaviors for rock under multiple stress levels, including instantaneous elastic deformation, decelerating creep, stable creep, accelerating creep, and final failure.
Section snippets
Typical creep characteristics and basic assumptions
Investigating the typical creep characteristics of natural rocks and understanding the corresponding mechanism are the basis for modeling rock creep behaviors. Therefore, many efforts have been made to reveal the time-dependent deformation behaviors of rock specimens in laboratory works, such as uniaxial, triaxial and shear tests [24], [23], [18], [16], [20], [9], [13], [12]. According to previous experimental observations, the complete strain-creep time curve of rock under a particular stress
Novel damage-based creep model
Based on the typical creep deformation stages described in the last section, a novel creep model is established in this section based on the time-hardening theory and the damage theory. To deduce formulas for the proposed creep model, a cylindrical rock specimen under uniaxial compression stress is considered. Suppose the initial length and the initial section area of the specimen before loading are and . An instantaneous elastic strain of the specimen is generated under the axial
Determination of the model parameters under a specific stress level
The new creep model has already been established and formulated in Eq. (27) or Eq. (28), which contains three model parameters m, n, and . All three parameters can be determined according to the creep strain curve and creep strain rate curve under a specific stress level.
The minimum creep strain rate in the creep curve is reached at point G in Fig. 1(b) when the creep time is . Thus, Eq. (28) becomeswhere and are measured from the experimental
Determination of the model parameters under multiple stress levels
The three model parameters m, n, and under a specific stress level have been determined using the procedure described above. Nevertheless, it is necessary to determine how the parameters change if the stress level changes when considering the creep behavior of rock under multiple stress levels .
Model Validation and Discussion
Several groups of creep test results for different rock types, including one group of cement mortar [20], [30], [31], were chosen to validate the proposed creep model. All the model parameters determined according to the methods mentioned above are listed in Table 1. All the creep curves predicted by the proposed creep model compared with the experimental data are shown in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6.
Fig. 2, Fig. 3, Fig. 4, Fig. 5 show the experimental results with only one or two
Conclusions
In this study, the complete creep process of natural rock is divided into the deceleration region and the acceleration region based on the typical deformation characteristics and mechanism of rock creep behavior. The creep strengthening and weakening behaviors during the complete creep process under multiple stress levels can be effectively captured by the combination of the time-hardening theory and the damage theory.
- (1)
A unified creep model has been established with the advantages of a simple
CRediT authorship contribution statement
Wengui Cao: Conceptualization, Writing - review & editing. Ke Chen: Formal analysis. Xin Tan: Writing - original draft. He Chen: Data curation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (ID 51879104, 51378198) and Major Science and Technology Project of Hunan Province, China (ID 2019GK1011).
References (31)
- et al.
The steady state propagation of a damaged zone an elastic brittle solid
Fracture
(1984) A general constitutive equation for transient and stationary creep of rock salt
Int J Rock Mech Min Sci Geomech Abstracts
(1993)- et al.
A ten year creep experiment on small rock specimens
Int J Rock Mech Min Sci Geomech Abstracts
(1987) - et al.
Long-term creep experiment on some rocks observed over three years
Tectonophysics
(1980) - et al.
A statistical damage constitutive model for softening behavior of rocks
Eng Geol
(2012) - et al.
Time-dependent tests on intact rocks in uniaxial compression
Int J Rock Mech Min Sci
(2000) - et al.
Creep behaviour of a weak rock: experimental characterization
Int J Rock Mech Min Sci
(1999) - et al.
Complete creep curves under uniaxial compression
Int J Rock Mech Min Sci Geomech Abstracts
(1991) - et al.
Modelling of creep in rock materials in terms of material degradation
Comput Geotech
(2003) A study of creep of rocks
Int J Rock Mech Min Sci Geomech Abstracts
(1975)
A model for describing and predicting the creep strain of rocks from the primary to the tertiary stage
Int J Rock Mech Min Sci
A unified representation of the plasticity, creep and relaxation behavior of rocksalt
Int J Rock Mech Min Sci
A creep constitutive model for salt rock based on fractional derivatives
Int J Rock Mech Min Sci
Drying creep of concrete: constitutive model and new experiments separating its mechanisms
Mater Struct
A damage simulation technique of the full rock creep process considering acclerated creep
J Hunan Univ
Cited by (20)
A bounding surface viscoplastic damage model for time-dependent rock deformation
2023, Mechanics of MaterialsStudy on creep models and parameter inversion of columnar jointed basalt rock masses
2021, Engineering GeologyCitation Excerpt :Figs. 5 and 6 show that the instantaneous deformations of C36–1 and C36–2 are 248 μm and 274 μm, respectively; the creep deformations of C36–1 and C36–2 are 19 μm and 36 μm, respectively; the total deformations of C36–1 and C36–2 are 267 μm and 310 μm, respectively; the creep times of C36–1 and C36–2 are 570 h and 527 h, respectively; and the final residual deformations of C36–1 and C36–2 are 13 μm and 53 μm, respectively. As stated in the Introduction, the three empirical creep models (e.g., power function model, logarithmic model and exponential model) require very few parameters and are suitable for different engineering applications (Cao et al., 2020; Chen et al., 2020); therefore, the following paragraphs will analyze the applicability of the three empirical creep models for the creep characteristics of columnar jointed basalt rock masses in dam site areas. Based on the above equations, a total of 16 compression creep empirical models of the columnar jointed basalt rock masses can be obtained by using the least squares method (LSM), as listed in Tables 1-3.
Creep behaviour of saturated purple mudstone under triaxial compression
2021, Engineering GeologyCitation Excerpt :Liu et al. (2020) developed a nonlinear damage creep model for sandstone based on Norton's power law. Gao et al. (2020) studied a novel damage-based creep model that can simulate the overall creep process. As the most-used model, component models can flexibly combine basic components, namely, a Hooke body, a Newtonian body, and a Saint-Venant body, to reflect creep behaviours.