Research Article  Open Access
Hang Lin, Xing Zhang, Yixian Wang, Rui Yong, Xiang Fan, Shigui Du, Yanlin Zhao, "Improved Nonlinear Nishihara Shear Creep Model with Variable Parameters for RockLike Materials", Advances in Civil Engineering, vol. 2020, Article ID 7302141, 15 pages, 2020. https://doi.org/10.1155/2020/7302141
Improved Nonlinear Nishihara Shear Creep Model with Variable Parameters for RockLike Materials
Abstract
Creep property is an important mechanical property of rocks. Given the complexity of rock masses, mechanical parameters change with time in the creep process. In this work, a nonlinear function for describing the timedependent change of parameters was introduced and an improved variableparameter nonlinear Nishihara shear creep model of rocks was established. By creating rocklike materials, the mechanical properties of rocks under the shear creep test condition were studied, and the deformation characteristics and longterm shear strength of rocks during creep were analyzed. The material parameters of the model were identified using the creep test results. Comparison of the model’s calculated values and experimental data indicated that the model can describe the creep characteristics of rocks well, thus proving the correctness and rationality of the improved model. During shear creep, the mechanical properties of rocks have an aging effect and show hardening characteristics under low shear stress. Furthermore, according to the fact that G_{k} of the nonlinear model can characterize the creep deformation resistance, a method to determine the longterm shear strength is proposed.
1. Introduction
Creep property is an important mechanical property of rocks and the main factor affecting longterm stability of geotechnical engineering [1–7]. Obtaining the stress and deformation characteristics of rock masses from the perspective and method of their rheology is consistent with reality [8–10]. Thus, studying the creep characteristics of rocks is crucial. The creep behavior of rocks can be divided into three stages, namely, attenuated, stable, and accelerated creep. In the initial creep stage, the strain increases with time. In the steadystate creep stage, the strain increases uniformly with time, and the duration of this stage is relatively long. In the accelerated creep stage, the strain rate increases with time, and the rock is destroyed after reaching the critical fracture point. Numerous studies have shown that rocks exhibit typical nonlinear characteristics during creep, in which the accelerated creep stage is the most obvious [4, 11, 12]. To reveal the mechanical properties of rock rheological behavior, scholars have conducted targeted indepth research and obtained abundant research results [13–15]. By summarizing existing rheological models, Xia et al. [16] proposed a unified rheological mechanical model. Relevant studies have shown that model parameters vary during rock creep and are related to stress, time, temperature, and other factors [17, 18]. Referring to damage mechanics theory [19, 20], Jia et al. [21] proposed a nonlinear elasticviscoplastic damage model based on the modified Mohr–Coulomb criterion. Moghadamaab [22] used constitutive model to describe expansion and short and longterm damages in the creep process of rock salt and established a numerical model of the rock salt excavation underground chamber as it changes with time. Yang and Li [11] replaced the viscous element of the Nishihara model with a new element by using fractional calculus and established a nonlinear Nishihara model that describes rock rheology. Shao et al. [23] established a full stressstrain creep model from the perspective of microstructure evolution by considering various factors. These models effectively describe the nonlinear characteristics of model parameters during rock creep but focus on the creep mechanical properties of rocks under uniaxial and triaxial compression. Research on rock characteristics under direct shear creep conditions is relatively scarce. The primary failure mode of rocks under natural conditions is shear failure [24–30]. Therefore, the shear creep failure behavior of rocks should be studied. The classical Nishihara model can describe the mechanical properties of rocks in decay and stable creep stages [11, 31, 32], but its parameters are constant and cannot reflect the nonlinear characteristics of creep. To reveal the mechanism of rock mechanical behavior under shear creep conditions and establish a corresponding shear creep model, this study investigated the shear rheological mechanical characteristics of rocks with plasticelastic deformation by performing a shear test on a similar sandstone material. A timevarying factor was introduced to the Nishihara model to describe the functional relationship between the viscosity coefficient and time. On the basis of this factor, an improved nonlinear Nishihara shear creep model that can reveal the unsteady and nonlinear characteristics of parameters was established. The model’s rationality was verified through a comparison with experimental values.
2. Specimen Preparation and Test Methods
Natural rock masses contain many micro and macrocracks, which lead to different physical and mechanical characteristics [33–41]. Miller divided the stressstrain curves of rocks with different physical and mechanical properties into six categories [42, 43]. The third category is called the plasticelastic deformation feature, in which the stressstrain curve is concave in the initial stage and then becomes a straightline segment until destruction [44–46]. Rocks with many pores and microfractures, such as granite and sandstone, reflect this deformation characteristic [47, 48]. In this study, the mechanical behavior and deformation characteristics under shear creep conditions were studied by fabricating similar materials with the plasticelastic deformation feature of rocks [49].
Experiments were performed on a RYL600 computercontrolled rock shear rheometer, which is usually used for rheological testing of materials, such as rock and concrete, and composed of three components: the host, the test and control system, and the computer system. The rheometer can simultaneously apply verticalaxis compression test force and horizontalaxis shear test force to a specimen and control the loading by force or deformation. In the shear creep test, the rocklike specimens were under constant temperature and humidity conditions. The loading rate at each stage was 300 N/s. The specimens adopted the same type of material, mix ratio, curing time, and mold specification (10 cm × 10 cm × 10 cm). Through the uniaxial and triaxial compression tests, the mechanical parameters of rock materials with different cement, river sand, and water ratio were obtained. Finally, the sandstone materials with cement, river sand, and water ratio of 1 : 1 : 0.406 were selected as research objects. The mechanical parameters of the rocklike specimens are shown in Table 1, which is similar to the mechanical properties of sandstones with elastoplastic deformation characteristics. During pouring, the material was mixed according to the ratio, and the surface was smoothed after tamping in the steel mold to minimize adverse effects in the manufacturing process. The mold was taken apart after modeling for 24 hours and maintained for 28 days. All specimens are intact rocklike materials.

As the loading model shown in Figure 1, the specimens were subjected to a fast shear test to compare the strength characteristics of the rock’s fast shearing and shear creep conditions. The test results are shown in Figure 2. At the beginning of loading, the slope of the stressstrain curve increased with the loading stress, and this process lasted for a long time. After the microcracks were compacted completely, the slope of the curve entered the linear elastic stage with a constant size and then maintained this feature until failure. Significant plastic elastic deformation characteristics were exhibited. The fast shear strengths of specimens 1–4 were 4.1 MPa, 5.6 MPa, 6.45 MPa, and 7.33 MPa, respectively. In the grading loading creep test, normal stress was taken as 5%, 10%, 15%, and 20% of the uniaxial compressive strength of a specimen. The shear stress levels of specimen 1 were 30%, 40%, 50%, 60%, and 70% of the fast shear strength. The shear stress levels of specimen 2 were 30%, 40%, 50%, and 60% of the fast shear strength until destruction. The shear stress levels of specimen 3 were 30%, 40%, 60%, 70%, and 80% of the fast shear strength. The shear stress levels of specimen 4 were 30%, 50%, 60%, 70%, and 80% of the fast shear strength. The loading method is presented in Table 2, and the process is shown in Figure 3.
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3. Test Results and Analysis
3.1. Mechanical Properties with Time
The creep curves of graded shear loading under the same normal stress level were obtained through the creep shear test (Figure 4). In the creep test curve obtained, a straight line parallel to the longitudinal axis was made to intersect with the creep curve under the same normal stress and different shear forces, and the isochronal curve (Figure 5) was drawn by the stress and displacement values at each intersection point obtained. As can be seen from Figure 5, the slope of the isochronous shear stressshear displacement curve varies with stress. Stress is the factor that affects the mechanical properties of rocks with plasticelastic deformation characteristics, which are different from the characteristics of the microfracture compaction stage completed immediately after loading in most rocks. To study the time deterioration effect on the rock creep process, the deformation modulus under different creep time conditions was calculated through the slope of the curve section of the adjacent shear stress levels, and the relationship between deformation modulus and time was plotted and is shown in Figure 6.
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The formula for calculating the deformation modulus can be written aswhere τ_{1} and τ_{2} characterize the adjacent twostage shear stress and indicate the shear displacement at different moments under the respective shear stress conditions.
Figure 6 indicates that for the rocks with plasticelastic deformation characteristics, under the condition of low shear stress, the deformation modulus of rocks in creep behavior increased with time, which is contrary to the deterioration of the mechanical properties of most rocks with time [50, 51]. This result is due to the existence of many microcracks in plasticelastic rocks [52, 53]. Low shear stress cannot easily compact the microcracks inside the rock completely in a short time, and the microcracks that are not compacted continue to be compacted in the following creep behavior, thus showing a hardening effect. When the shear stress level was increased to a certain value, the deformation modulus weakened with time and showed aging deterioration. This phenomenon occurred because the shear stress at this time exceeded the critical shear stress when the microcrack was completely compacted, which not only stopped the hardening effect brought by microcrack compaction but also developed the internal crack of the specimen with time and produced significant aging degradation effects. When the shear stress continues to increase, the mechanical properties of the rock continued to deteriorate with time because the unfinished area of the rock when shear stress was applied to the upper stage continued to deteriorate under the new firstorder shear stress condition. Meanwhile, the increase in shear stress led to the deterioration of the area that is not easy to deteriorate under the condition of the shear stress of the upper level, resulting in continuous internal damage of the specimen.
3.2. Determination of LongTerm Strength under Different Normal Stress Conditions
According to the isochronous curve method which is commonly used in rheological mechanics [54], the stress corresponding to the second turning point of the isochronous curve after the fully compacted linear elastic phase was used as the longterm strength value of the rock. In Figures 5(a)–5(d), the shear stress value corresponding to the second turning point is 2.04 MPa, 2.8 MPa, 4.515 MPa, and 5.131 MPa, respectively. Thus, the longterm shear strength of the specimen under the normal stress condition of 0.78 MPa, 1.56 MPa, 2.34 MPa, and 3.12 MPa is 2.04 MPa, 2.8 MPa, 4.515 MPa, and 5.131 MPa, respectively. Correspondingly, the ratio of longterm shear strength to fast shear strength is 0.5, 0.5, 0.7, and 0.702, respectively. The phenomenon wherein the specimen had different ratios of longterm strength to fast shear strength under different normal stress conditions could be caused by the numerous microcracks in the rock with plasticelastic deformation characteristics. Under low normal stress, the internal damage of the rock accumulated continuously overtime and with the increasing shear stress. Damage occurred when the local damage reached the limit and lost the bearing capacity. Under high normal stress, the microcracks in the normal direction of the rock were compacted closely, thereby making the hardening effect caused by the compaction of microcracks more obvious than that under low stress. Meanwhile, under the condition of dense microfractures, the mechanical properties and internal continuity of the rock improved, thus strengthening the capability to transfer stress inside the rock, increasing the effective bearing area, and strengthening the capability to resist damage. Therefore, the ratio of longterm shear strength to fast shear strength of the rock under high normal stress is greater than that under low normal stress.
4. Improved Nonlinear Shear Creep Model Based on the Nishihara Model
The test results above illustrate that during the process of creep from the decay creep stage to the stable creep stage, the internal microcracks are continuously compacted, which causes irreversible plastic deformation and a hardening effect. Therefore, for rocks with an obvious hardening effect under low shear stress conditions, the traditional Nishihara model with linear parameters is no longer applicable. The model parameters must be described in a nonlinear manner. To reflect the mechanical properties of the rock in the creep process accurately, a timevarying factor was introduced to improve the traditional Nishihara model, and an improved variableparameter nonlinear Nishihara shear creep model of rock was established. The rationality of the model was verified by the creep test result of specimens.
4.1. Description of the Characteristics of the TimeDependent Parameter in the Nishihara Model
The Nishihara model is a rheological model composed of Hooke, Kelvin, and viscoplastic bodies in series, as shown in Figure 7. The model expression iswhere G_{M} is the Hooke body coefficient, which is the instantaneous elastic modulus; G_{K} is the Kelvin model elastic coefficient; η_{K} is the Kelvin model viscosity coefficient; η_{2} is the viscous element coefficient in the viscoplastic body; τ_{s} is the longterm shear strength of the rock; and t_{s} is the moment when the accelerated creep stage begins.
To describe the nonlinearity of parameters in the Nishihara model under shear loading, timevarying factor D was introduced to the Nishihara model by referring to the method of establishing fractionalderivative models [55], to describe the variation in element coefficients in the Nishihara model with time, and a nonlinear shear creep model with variable parameters was constructed (Figure 8).
A large number of studies have shown that the internal damage/hardening variable is closely related to time as an exponential function [56]. Therefore, on the basis of previous studies, the timevarying factor D is defined aswhere D ∈ [0, 1]; μ and α are material parameters representing the variation in element coefficients μ ∈ [−∞, 1]; and α ∈ [0, +∞].
Compared with the nonlinear function in the fractionalderivative model, which describes the aging characteristics of the model parameters with only one parameter [55, 57], in this paper, the function D used to describe the nonlinear characteristics of model parameters has two parameters, where μ represents the comparison of model parameters before and after creep occurs, and α reflects the trend of the model parameters over time. It is not as simple and easy to calculate as the function in fractionalderivate models, but it can describe the evolution characteristics of model parameters during the creep process more comprehensively.
In the Nishihara model, the Kelvin body represents the attenuation and stable creep stages, and the viscoplastic body represents the accelerated creep stage. In the actual creep process, all parameters change with time. Timevarying factor D was introduced to the Kelvin and viscoplastic bodies to describe the relationship between the viscosity coefficient and time.
The Kelvin model is composed of an elastic element and a viscous element in parallel, and its constitutive equation is
After introducing timevarying factor D, the Kelvin viscous element with variable parameters is expressed aswhere is the actual viscosity coefficient that changes over time, when t = ∞, = , is the viscosity coefficient of the final moment of rock stable creep; μ_{1} is the material parameter characterizing the damage/hardening degree of ; when μ_{1} is positive, hardens with time during rock creep; when μ_{1} ⟶ 1, hardens into a plastomer at t = ∞; when μ_{1} = 0, is constant, which is the original Kelvin model; when μ_{1} is negative, deteriorates with time during rock creep; when μ_{1} ⟶ ∞, softens into an ideal fluid at t = ∞; α_{1} is a parameter reflecting the trend of over time in the creep process; the larger α_{1} is, the longer it takes for the to stabilize.
Equation (7) indicates that when t = 0, the initial viscous element coefficient is
In accordance with (3)–(7), the constitutive equation of nonlinear Kelvin model with parameter changes is obtained as follows:
Combined with the initial conditions t = 0 and γ = 0, the Kelvin nonlinear shear creep model equation with parameter variation is obtained as follows:
Similarly, for the viscoplastic body introduced with timevarying factor D, the viscous element with variable parameters is expressed aswhere is the actual viscosity coefficient that changes over time, when t = ∞, = , is the viscosity coefficient of the final moment of rock accelerated creep; when μ_{2} > 0, η_{2} hardens with time, and the larger μ_{2} represents the higher degree of hardening; when μ_{2} < 0, η_{2} softens with time, and the smaller μ_{2} represents the higher degree of softness.
The constitutive equation of viscoplastic body of Nishihara model is
Combining (11) and (12), the constitutive equation of the viscous element with variable parameters is obtained as follows:
Combining (10) and (14), the constitutive model of improved nonlinear Nishihara shear creep model with variable parameters can be obtained as follows:
4.2. Model Verification
To test the correctness of the improved nonlinear shear creep model established in this study, the model was used to fit the test curve with the experimental data of specimens, which was compared with the fitting curve of the original Nishihara model to determine the advantages and disadvantages of the improved model. The parameters of the improved model are given in Table 3. The fitting results are shown in Figure 9, and it indicates that the improved Nishihara model was closer to the creep test curve than the original Nishihara model was, thereby reflecting the mechanical behavior of the specimen creep process and verifying the reasonableness of the model.

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4.3. Relationship between Creep Model Parameters and Stress
In rock rheology, constitutive model parameters have important physical and mechanical significance. Studying the relationship between model parameters and loading conditions (stress, loading rate, etc.) can effectively reveal the mechanical mechanism of rock rheology [58–60]. In order to better understand the mechanical properties characterized by D, on the basis of Table 3, the relationship between μ_{1}, α_{1}, and shear stress under the normal stress 1.56 MPa is plotted in Figure 10. It could be seen from Figure 10 that μ_{1} is greater than 0 under each shear stress condition, and the maximum value occurs when the shear stress is 1.68 MPa and then decreases with the increase of shear stress and finally stabilizes to 0.97, indicating that the nonlinear Kelvin viscosity coefficient has hardening under various shear stress conditions, and the hardening degree is the highest under low shear stress condition, and the hardening degree tends to be stable after shear stress increases. As shown in Figure 10, as the shear stress increases, α_{1} increases first, then decays, and then continues to increase. Combined with (7), it can be seen that the time required for the viscosity coefficient in the nonlinear Kelvin body to reach stability is relatively long under the condition of low shear stress and then fluctuates with the increase of shear stress, but all remain stable at a certain value. Finally, when the shear stress reaches the value beyond the longterm strength, it decreases rapidly, and this phenomenon is caused by the compaction of rocklike specimen under low shear stress conditions. The relationship between and time in accordance with (7) and Table 4 is plotted in Figure 11, which can better show the change process of represented by μ_{1} and α_{1}.

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The relationship between Kelvin model elastic coefficient and shear stress is plotted in Figure 12(b), and it shows that when the range of the shear stress was from 1.68 MPa to 2.24 MPa, Kelvin model elastic coefficient increased with stress, and this result indicates the obvious internal hardening effect of the specimen and the characteristics of the plasticelastic deformation of the rocklike specimen that not only hardens under lowstress fast shear but also under lowstress creep. When the shear stress was loaded to 2.8 MPa, the was attenuated, indicating that under this shear stress condition, the internal microcracks of the specimen began to develop and damage began to occur. After the shear stress reached 3.36 MPa, the specimen quickly entered the accelerated creep stage. The viscosity parameter of the viscous element in the viscoplastic body that describes the accelerated creep stage was negative, indicating that the internal damage of the specimen was continuously intensified and the creep rate was continuously accelerated until failure.
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4.4. LongTerm Strength Determination Method Based on Improved Nonlinear Model
It can be concluded from (10) that the rocklike specimen creep deformation is controlled by ; when is small, the mass of rock creep deformation is large; that is, the larger is, the smaller the creep deformation is. characterizes the ability to resist creep deformation. Therefore, in the improved nonlinear model, the author proposes a method to determine the longterm strength by using the curve of shear stress. This method uses the nonlinear Nishihara model proposed in this paper to fit shear displacementtime curve to obtain the curve of and then determines the shear stress of decay in the curve as the longterm strength. According to the parameters in Table 3, the curves of the specimen under different normal stress levels were drawn (Figure 12) and were compared with the longterm shear strength determined by the isochronous curve method, as shown in Table 5.

Table 5 shows that the longterm shear strength determined by these two methods is basically the same, which proves the effectiveness of the longterm strength determination method proposed in this paper. And compared with the isochronous curve method, the modulus inflection point method only needs to fit the experimental data to determine the under each stress condition. Then, the longterm strength of the rock can be obtained according to the attenuation of , which has a simpler and faster operation process.
5. Discussion and Conclusion
Research on the mechanical characteristics of plasticelastic rock deformation under the conditions of fast shear and shear creep indicates that under low shear stress, the mechanical properties of rocks with plasticelastic deformation are continuously strengthened during creep, thereby continuously enhancing the rock’s resistance to deformation. When the shear stress level increases, the microcracks inside the rock are completely compacted. At this time, the mechanical properties of the rock deteriorate with creep and aging. Meanwhile, the mechanical properties of rocks with plasticelastic deformation under different normal stress conditions differ. The rock bearing capacity and the damage resistance under high normal stress are better than those under low normal stress. Therefore, for rock mass engineering, the rock with plasticelastic deformation characteristics should be placed under low stress conditions, and the surrounding area should avoid highintensity stress field to enhance the stability of the engineering and the project.
In this work, a timevarying factor D was introduced to describe the viscous elements in a nonlinear manner, and an improved nonlinear shear creep model based on the Nishihara model was established. After fitting the experimental data, it is found that the model can reflect the shear creep behavior of rocklike specimens effectively, and the model has rationality. In addition, under the normal stress of 0.78 MPa and 1.56 MPa, the ratio of longterm shear strength to fast shear strength of rock samples with plasticelastic deformation characteristics is 0.5, indicating that the critical damage cumulative value required for rock reaches a failure state. Therefore, in a project, the longterm shear strength at which the cracks inside the rock begin to develop and expand must be determined to prevent rock destruction. At present, relevant scholars often monitor rock damage under creep conditions by calculating the creep rate, the deformation modulus, or the monitoring energy of the acoustic emission equipment. However, in practical engineering, calculating the rate at each moment or providing special equipment is difficult. By fitting the improved nonlinear Nishihara shear creep model to the surrounding rock displacementtime data monitored in the project, the damage can be discriminated according to the identified trend of the viscous modulus G_{k} of the surrounding rock so as to provide timely support and avoid the structural instability of the project, which is crucial in practical engineering. Furthermore, according to the fact that G_{k} of the nonlinear model can characterize the creep deformation resistance of rocklike specimen, a method to determine the longterm shear strength is proposed, and compared with the calculation results of isochronous curve method, the rationality and applicability of the method are verified.
Notations
σ_{c}:  Uniaxial compressive strength 
μ:  Poisson’s ratio 
φ:  Angle of internal friction 
c:  Cohesive force 
E:  The deformation modulus 
τ_{1}, τ_{2}:  Adjacent twostage shear stress 
:  The shear displacement at different moments 
G_{M}:  The Hooke body coefficient 
G_{K}:  The Kelvin model elastic coefficient 
η_{K}:  The Kelvin model viscosity coefficient 
η_{2}:  The viscous element coefficient in the viscoplastic body 
τ_{s}:  The longterm shear strength of the rock 
t_{s}:  The moment when the accelerated creep stage begins 
μ, α:  Material parameters representing the variation in element coefficients 
:  The element coefficient after entering the stable creep stage 
:  The realtime element coefficient 
μ_{1}, α_{1}:  Material parameters representing the variation in Kelvin model element coefficients 
μ_{2}, α_{2}:  Material parameters representing the variation in viscoplastic body element coefficients 
τ_{s}:  The longterm strength of the rock. 
Data Availability
All data generated or analyzed during this study are included in this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (51774322, 51774107, and 51774131), Hunan Provincial Natural Science Foundation of China (2018JJ2500), and Fundamental Research Funds for the Central Universities of Central South University (2019zzts669). The authors wish to acknowledge these supports.
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