A nonlinear rheological shear constitutive model of bolted joints considering initial damage and damage evolution

Abstract

Understanding the shear mechanics mechanism of bolted joints is of great significance for predicting and preventing geological disasters. Most current studies seldom consider the rheological effects of bolted joints. In this paper, a comprehensive rheological constitutive model is proposed, accounting for initial damage and damage evolution across different rheological stages and bolt characteristics. The model incorporates an elastoplastic Hooke body for instantaneous deformation, parametric nonlinear Kelvin and viscous models for attenuation and steady creep stages, and a visco-plastic model based on time-dependent shear strength for accelerated creep stage. Additionally, a bolt-rock cooperative deformation model is introduced, considering the evolution of the bolt's elastic modulus. The resulting elasto-viscoplastic constitutive model effectively describes the shear rheological behavior of bolted joints, with its validity and superiority demonstrated through comparisons with shear creep tests and the Maxwell model. This research aims to provide valuable theoretical guidance for the construction and reinforcement of rock mass engineering projects.

Keywords

Bolted joint shear creep constitutive model instantaneous deformation initial damage

Introduction

Rock mass widely exists in engineering construction, due to the presence of joints, rock mass as engineering material exhibits high anisotropy and heterogeneity (Sarfarazi and Haeri, 2016; Yuan et al., 2022; Zhu et al., 2022). In addition, fractures initiated in the rock would interact with the sliding of existing joints, which significantly affects the deformation mechanism and strength characteristics, eventually failing (Wu et al., 2021; Zhao et al., 2019; Zhu et al., 2023). Therefore, the stability assessment of rock mass engineering is usually based on the analysis of joints. To reinforce the joint pre-existing in rock mass engineering, bolt technology is widely used (Liao et al., 2021; Wang et al., 2022), as shown in Figure 1. Over the past decades, numerous studies of bolted joints have been carried out. Based on the elastic theory of semi-infinite beam and the approximate differential equation of the deflection curve, Chen et al., (2020) proposed an analytical method that can predict the mechanical behavior of a fully grouted bolt subjected to pull-and-shear load. Similarly, Singh and Spearing (2021) established an analysis model to predict the shear characteristics of the bolt under large deformation, and proposed a method to consider the effect of post-elastic strain hardening on the bending stiffness of bolted rock. Recently, considering the relationship between the dilation effect and the bolt deformation and the influence of joint surface abrasion, Zheng et al. (2021) proposed an analytical model for predicting the shear behaviors of bolted rough joints. To estimate the effectiveness of the rock reinforcement system, on the basis of analytically-derived interface behavior between a rock bolt and the rock material for grouted rock bolts, Nie et al. (2014) proposed a rock bolt element by coupling a rock bolt element into the DDA(2D).

The above research has well revealed the shear mechanics mechanism of bolted joints, but most of the researchers have rarely considered the rheological effect of the rock mass. Whereas, research showed that the characteristics of joints are closely related to time (Li et al., 2022; Zhao et al., 2017, 2017), then the instability of rock mass engineering is induced (Mansouri and Ajalloeian, 2018; Singh et al., 2018). The rheological dislocation displacement of the joints leads to the obvious stress concentration of anchor bolts, and may cause shear cracks in the protective layer of the concrete (Liu et al., 2017; Yu et al., 2021; Zhang et al., 2019). Therefore, it is necessary to study the shear creep behavior of bolted rock joints. With the constitutive model, the shear rheological mechanical mechanism of the rock mass can be revealed effectively, which has been proved by many studies (Liao et al., 2018; Peng et al., 2021; Zhao et al., 2017). Hence, this study based on the division of three stages of rock mass rheology and the principle of bolt-rock cooperative deformation, proposes a rheological shear constitutive model to obtain the stress and deformation characteristics of bolted joints during the rheological process. The validity and superiority of the model are verified through a series of laboratory shear rheological tests on joints and comparisons with the Maxwell model. The results can help contribute to a better understanding of the rheological deformation mechanism of bolted joints, and provide theoretical benchmarks for mitigating disasters induced by the shear rheological failure of bolted joints.

Constitutive modeling

Under the creep conditions, as the space-time evolution of microstructure, the stress and deformation fields in the rock mass are constantly adjusted and reorganized, which is one of the causes of rock mass damage. According to the degree of crack propagation especially the characteristics of deformation evolution, the rock creep process can be divided into three stages: attenuation creep stage, steady creep stage, and accelerated creep stage (Zhao et al., 2019). The typical shear creep curve of rock masses is shown in Figure 2. The section of OA is initial instantaneous deformation, which is caused by instantaneous loading. Then with the evolution of time, enter the attenuation creep stage AB, where the $\frac{d γ}{d t} > 0$ , $\frac{d^{2} γ}{d t^{2}} < 0$ . It is worth noting that if the stress does not exceed the threshold, there is a possibility that only elastic strain will occur – once unloaded, the strain will recover completely. The BC segment is referring to the steady creep stage, where the $\frac{d γ}{d t} \geq 0$ , $\frac{d^{2} γ}{d t^{2}} = 0$ , strain occurring in this stage is increases linearly. The accelerated stage is the CD segment, and certain plastic deformation has been caused in this stage, rock damage will evolve vigorously, resulting in the accelerated growth of plastic deformation until failure.

In most rheological cases, the deformation of bolted rock mass exhibits strong elasticity, viscosity and plasticity characteristics (Xia and Sun, 2002; Zhao et al., 2021) or a combination of them: elasticity, viscoelasticity, viscoplastic and viscoelastic-plastic characteristics. These different deformation properties can be characterized by the combination of three elements: elastic element, viscous element, and plastic element (Chen et al., 2023; Tang et al., 2018). To reflected rheological mechanical properties effectively, early scholars proposed four representative models (Liu et al., 2021), which laid the foundation for the investigation of the rheological constitutive model, as shown in Figure 3. This paper will use this research idea for reference, improve the existing model or propose new models, to establish the constitutive model describing the rheological properties of bolted joints.

Elastic-plastic body considering hardening and damage effects of instantaneous deformation

There are many micro-cracks in the rock mass. With the occurrence of shearing, these micro-cracks will be compacted or further propagated, resulting in plastic deformation and the evolution of mechanical properties (Guan et al., 2018), as shown in Figure 4. By identifying three special stress thresholds (crack closure stress τ_cc, yield stress τ_y, peak stress τ_p), the process of compaction and propagation of cracks can be defined as follows: (1) hardening effect stage – the stress is less than τ_cc and the mechanical properties are enhanced due to the compaction of original micro-cracks; (2) elastic stage – the stress is in the range from τ_cc to τ_y and the mechanical properties remain stable; (3) damage softening stage – the stress is in the range from τ_y to τ_p and damage occurs due to the propagation of micro-cracks; (4) post-peak stage – the stress is greater than σ_p and the mechanical properties deteriorate rapidly.

Generally, the rheological model describes the instantaneous deformation by the traditional elastic Hooke body, which has the limitation of the reflection of the instantaneous hardening and damage characteristics. Due to the complexity of rock masses, the instantaneous hardening/damage stress thresholds are different, characteristics of instantaneous deformation can be plastic or completely elastic. Considering the possible plastic deformation caused by the micro-crack effect is rarely studied in rheological research, this paper proposes an elastoplastic improved Hoke body to fully reflect the instantaneous deformation. The schematic diagram of the improved Hoke body is shown in Figure 5.

The constitutive equation is:

γ_{I} = \frac{τ}{G_{M} (1 - D_{0})}

(1)

where G_M is the instantaneous shear modulus when the first-level shear stress load. D₀ is the micro-crack effect factor. When D₀ is less than 0, it indicates that the rock micro-cracks are compacted under this level of shear stress, and the rock mass is hardened; when D₀ is greater than 0, it indicates that under this level of shear stress, the rock is damaged, and the shear modulus is softened compared to the first level of shear stress. It should be noted that the G_M in the elastoplastic improved Hoke body is a constant, which is obtained by the first-level shear stress and the corresponding instantaneous shear displacement.

Nonlinear viscoelastic model based on Kelvin model

To describe the behavior of the attention creep stage, the classical Kelvin model is usually used. This traditional model obtained the corresponding creep deformation by combining the elastic and viscous components in parallel (Xia and Sun, 2002).

The constitutive equation of the Kelvin model is:

γ_{K el} = γ_{1} = γ_{2}

(2)

τ = τ_{1} + τ_{2} = G_{K} γ + η_{K} \dot{γ}

(3)

γ_{K el} = γ_{1} = γ_{2} = \frac{τ_{1}}{G_{K}} = \frac{τ_{2}}{η_{K}} t

(4)

γ_{K el} = \frac{τ}{G_{K}} (1 - e^{- \frac{G_{K}}{η_{K}} t})

(5)

where γ_Kel is the total shear displacement of Kelvin body;

\dot{γ}

is the first order derivative of γ, which represents the shear displacement rate; γ₁ and γ₂ are the shear displacement of elastic and viscous element respectively.

G_{K}

is the elastic modulus.

η_{K}

is the viscosity coefficient.

It can be seen from the above equations that the parameters of the Kelvin model are constant. However, many researches show that the parameters nonlinear evolve during rheological process (He et al., 2018; Li et al., 2015; Zhao et al., 2019). To reflect the nonlinear rheological characteristics of rock masses, we improve the Kelvin model by introducing the time damage factor D_K, to describe the nonlinear characteristics of $η_{K}$ , as shown in Figure 6.

A series of studies (Chen et al., 2021; Lin et al., 2021; O'Neill et al., 2018; Zhang et al., 2021) show that rock damage evolves exponentially during rheological process. Therefore, the expression of D_K can be defined as (Lin et al., 2020a):

D_{K} = χ e^{- β t}

(6)

where

χ

and

β

are the material parameters,

χ \in [- \infty, 1]

β \in [- \infty, + \infty]

By introducing the time damage factor D_K, the viscous element $η_{K}$ in Kelvin model can be expressed as:

η_{K}^{t} = η_{K}^{S} (1 - D_{K}) = η_{K}^{S} (1 - χ e^{- β t})

(7)

where

η_{K}^{S}

is the maximum viscous coefficient during the attention creep stage;

η_{K}^{t}

is the time-dependent viscous coefficient.

After the description of the nonlinear characteristics of parameters. The constitutive model of improved Kelvin model is obtained:

τ = τ_{1} + τ_{2} = G_{K} + η_{t} \dot{γ}

(8)

η_{t} \dot{γ} = η_{\infty} (1 - χ e^{- β t}) \dot{γ} = τ - G_{K} γ

(9)

Combined with the initial conditions t = 0, γ = 0, the shear displacement of model can be written as:

γ_{K el} = \frac{τ}{G_{K}} (1 - e^{- \frac{G_{K}}{η_{K}^{S}} (\frac{\ln (e^{β t} - χ) - \ln (1 - χ)}{β})})

(10)

Nonlinear viscous model considering shear stress

By combining the improved Hoke body and nonlinear Kelvin body, a constitutive model describing the instantaneous deformation and attenuation rheological stage can be obtained, as shown in Figure 7.

The equation of constitutive model is:

γ = γ_{I} + γ_{K el}

(11)

γ = \frac{τ}{G_{M} (1 - D_{0})} + \frac{τ}{G_{K}} (1 - e^{- \frac{G_{K}}{η_{K}^{S}} (\frac{\ln (e^{β t} - χ) - \ln (1 - χ)}{β})})

(12)

This equation shows that when t is large enough, that is, the rock mass enters steady creep stage, the shear displacement is a certain value, indicating that the displacement rate is 0. However, the displacement rate of rock mass increases slowly with a value. The constitutive model shown in Figure 7 cannot reflect the characteristics of steady creep stage. Recently, a series of shear rheological tests of rock joints found that there is an exponential function relationship between the displacement rate and the shear stress level during steady creep stage (Jia et al., 2018). Based on this, a nonlinear viscous model is proposed in this chapter to describe the characteristics of steady creep stage, as shown in Figure 8.

The constitutive equation is:

τ = η_{2} e^{λ τ} {\dot{γ}}_{s}

(13)

where γ_s is the displacement at the steady creep stage, η₂ is the viscosity coefficient and λ is the material parameter.

The shear displacement expression is obtained by transformation:

γ_{s} = \frac{τ}{η_{2} e^{λ τ}} t

(14)

Visco-plastic rheological model considering initial damage

According to the theory of damage mechanics, microscopic damage can be effectively reflected by the evolution of shear strength (Asadizadeh et al., 2018; Zhao, 1997). The shear strength is equal to the shear stress at the beginning of accelerated creep, and then decreases with time until failure (Lin et al., 2020b). We based on the time-dependent characteristic of shear strength, establish a viscoplastic model which can accurately reflect the mechanical behavior of the accelerated creep stage, as shown in Figure 9. This model is composed of a plastic element representing shear strength and a viscous element representing rheological rate in parallel.

Its constitutive equation is:

τ - τ_{d} = η_{V P} {\dot{γ}}_{V P}

(15)

The time function of the shear strength represented by the plastic element can be written as:

τ_{d} = τ (1 - D_{C})

(16)

where τ_d is the shear strength of rock masses, τ is the loading shear stress. D_C is the damage variable in the accelerated creep stage, D_C ∈ [0,1].

In actuality, the instability failure of rock is generally induced by initial damage, such as joints, holes, etc (Hu et al., 2020; Huang et al., 2022; Wang et al., 2021a, Wang et al., 2021b). Therefore, from the perspective of the engineering application of the model, it is necessary to consider both the initial damage and the microscopic time-dependent damage caused by rheology. Under strain equivalence theory, the coupled damage of the rock mass can be expressed as (Liu et al., 2015):

D_{C} = D_{m a} + D_{m i} - D_{m a} D_{m i}

(17)

where D_ma is the macroscopic initial damage of the rock mass; and D_mi is the microscopic damage of the rock mass. It can be seen from the expression that when the rock mass only has microscopic damage, that is, when the macroscopic damage D_ma = 0, the coupling damage D_C is equal to the microscopic damage D_mi. When the microscopic damage D_mi = 0, that is, when the rock mass is only macroscopically damaged, the coupled damage D_C = D_ma. This shows that the coupling damage expression can be well adapted to the study of mechanical characteristics under coupling conditions.

The macroscopic damage D_ma can be calculated by the following equation (Chen et al., 2020):

D_{m a} = 1 - \frac{E_{C}}{E_{o}}

(18)

where E_C is the elastic modulus of a rock mass with natural macroscopic defects and E_O is the elastic modulus of the complete rock mass.

Regarding the rheological state, Kachanov proposed Kachanov's law (Zhang et al., 2021), which can better describe the microscopic damage of rock masses, the corresponding expression is:

D_{m i} = 1- {(1 - \frac{t}{t_{C}})}^{\frac{1}{δ + 1}}

(19)

where

δ

is the external stress,

t_{C}

is the time corresponding to the failure.

Since this model describes the mechanical behavior of the rock mass during the accelerated creep stage, when the damage starts to occur, $t = t - t_{s}$ in equation (19). Correspondingly, the true failure time is ${\tilde{t}}_{C} = t_{R} - t_{s}$ . Thus, the expression of D_mi can be written as:

D_{m i} =1- {(1 - \frac{t - t_{s}}{\tilde{t_{C}}})}^{\frac{1}{δ + 1}} = 1- {(\frac{t_{R} - t}{t_{R} - t_{s}})}^{\frac{1}{δ + 1}}

(20)

where t_s is the time of rock mass as it enters the accelerated creep stage; and t_R is the moment when the rock mass is failure. When t = t_s, D_mi = 0; and when t = t_R, D_mi = 1.

Substituting equations (18) and (20) into equation (17), the rock mass damage considering the initial damage can be calculated as:

D_{C} = 1 - \frac{E_{C}}{E_{o}} {(\frac{t_{R} - t}{t_{R} - t_{S}})}^{\frac{1}{δ + 1}}

(21)

Substituting equation (21) into equation (16), the constitutive equation of this model is:

τ_{d} = τ (1 - D_{C}) = τ \frac{E_{C}}{E_{o}} {[ϕ (t)]}^{\frac{1}{δ + 1}}

(22)

where,

ϕ (t) = {\begin{array}{l} 1 & t < t_{s} \\ \frac{t_{R} - t}{t_{R} - t_{s}} & t > t_{s} \end{array}

(23)

Combined with the initial conditions: t = t_s, γ = 0, the shear displacement equation of the model can be obtained as follows:

γ_{V P} = \frac{τ}{η_{V P}} [(t - t_{s}) + (t_{R} - t_{s}) \frac{δ + 1}{δ + 2} [\frac{E_{C}}{E_{o}} {[ϕ (t)]}^{\frac{δ + 2}{δ + 1}} - 1]]

(24)

where γ_vp is the shear displacement of the nonlinear viscoplastic element.

These series of models describing different creep stages were combined to establish the nonlinear rheological constitutive model of rock joints (Figure 10), which can reflect the mechanical properties of the three rheological stages.

Combining equations (12), (14) and (24), the elastic-visco-plastic nonlinear rheological constitutive equation of rock joint surface can be obtained:

γ = γ_{I} + γ_{K el} + γ_{s} + γ_{V P}

(25)

γ = {\begin{array}{l} \frac{τ}{G_{M} (1 - D_{0})} + \frac{τ}{G_{K}} (1 - e^{- \frac{G_{K}}{η_{K}^{s}} (\frac{\ln (e^{β t} - χ) - \ln (1 - χ)}{β})}) + \frac{τ}{η_{2} e^{λ τ}} t & τ < τ_{s} \\ \frac{τ}{G_{M} (1 - D_{0})} + \frac{τ}{G_{K}} (1 - e^{- \frac{G_{K}}{η_{K}^{s}} (\frac{\ln (e^{β t} - χ) - \ln (1 - χ)}{β})}) + \frac{τ}{η_{2} e^{λ τ}} t + \frac{τ}{η_{v p}} [(t - t_{s})+(t_{R} - t_{s}) \frac{δ + 1}{δ + 2} [\frac{E_{C}}{E_{o}} {[ϕ (t)]}^{\frac{δ + 2}{δ + 1}} - 1]] & τ > τ_{s} \end{array}

(26)

Shear rheological model of rock mass joint surface considering the bolt

Rock bolts are widely used in the field of engineering for the high stiffness of both axial and shear loads, and its inherent improvement of rock integrity. The influence of bolts on the stability of rock mass is generally manifested in two aspects: firstly, by sharing the confining pressure with the rock joint, the bolt can reduce the stress of surrounding rock and improve safety. On the other hand, the bolt can also improve the stability of rock mass properties, such as the “pile bolt effect” (Fang et al., 2020). Due to the shear movement of rock along the joint, bolts installed in rock mass undergo bolt-rock cooperative deformation (He et al., 2018), which is the component of the whole rheological deformation. During the rheological process, the pre-load of the bolt is gradually reduced, and the time-dependent characteristic can be expressed as (Yang et al., 2020):

ℕ = {\begin{array}{l} ℕ_{0} - K t & t \leq t_{s} \\ A_{1} e^{B_{1} (}^{t - t_{h})} + C_{1} & t > t_{s} \end{array}

(27)

where

ℕ_{0}

is the initial pre-load of the bolt,

K, A_{1}, B_{1}, C_{1}

are the constant.

Then the shear stress and shear modulus of bolt can be calculated as:

τ = ℕ (t) / A_{r}

(28)

G_{b} = ℕ (t) / γ A_{r}

(29)

where A_r is the area corresponding to the effective anchorage range of bolted joints.

Thus, the shear modulus of the bolt $G_{b}$ can be obtained:

G_{b} = {\begin{array}{l} G_{s} - K t & t \leq t_{s} \\ A_{1} e^{B_{1} (}^{t - t_{h})} + C_{1} & t > t_{s} \end{array}

(30)

Considering the coupling effect of bolt and joint. This paper based on the characteristic of the shear modulus of the bolt in equation (30), establishes a rheological model of bolt-rock cooperative deformation, as shown in Figure 11.

Accordingly, the constitutive equation of this model is:

τ = G_{b} γ_{b} = {\begin{array}{l} G_{b} (F_{1}) γ_{b} = (G_{s} - K t) γ_{b} & t < t_{h} \\ G_{b} (F_{2}) γ_{b} = (A_{1} e^{B_{1} (}^{t - t_{h})} + C_{1}) γ_{b} & t > t_{h} \end{array}

(31)

where G_b is the shear modulus of the bolt;

γ_{b}

is the shear displacement of bolt-rock cooperative deformation.

The shear displacement expression can be obtained by solving equation (31):

γ_{b} = \frac{τ}{G_{b}} = {\begin{array}{l} \frac{τ}{(G_{s} - K t)} & t \leq t_{h} \\ \frac{τ}{(A_{1} e^{B_{1} (}^{t - t_{h})} + C_{1})} & t > t_{h} \end{array}

(32)

By the combination of the model shown in equation (26) and this model, a shear rheological constitutive model of bolted joints can be obtained, as shown in Figure 12.

The corresponding shear displacement expression is:

γ = γ_{I} + γ_{K el} + γ_{s} + γ_{V P} + γ_{b}

(33)

γ = {\begin{array}{l} \frac{τ}{G_{M} (1 - D_{0})} + \frac{τ}{G_{K}} (1 - e^{- \frac{G_{K}}{η_{K}^{s}} (\frac{\ln (e^{β t} - χ) - \ln (1 - χ)}{β})}) + \frac{τ}{η_{2} e^{λ τ}} t + \frac{τ}{(G_{s} - K t)} & τ < τ_{s} \\ \frac{τ}{G_{M} (1 - D_{0})} + \frac{τ}{G_{K}} (1 - e^{- \frac{G_{K}}{η_{K}^{s}} (\frac{\ln (e^{β t} - χ) - \ln (1 - χ)}{β})}) + \frac{τ}{η_{2} e^{λ τ}} t + \frac{τ}{η_{v p}} [(t - t_{s})+(t_{R} - t_{s}) \frac{δ + 1}{δ + 2} [\frac{E_{C}}{E_{o}} {[ϕ (t)]}^{\frac{δ + 2}{δ + 1}} - 1]] + \frac{τ}{(A_{1} e^{B_{1} (}^{t - t_{h})} + C_{1})} & τ > τ_{s} \end{array}

(34)

Model verification

In this section, to verify the correctness of the constitutive model equations (26) and (34) proposed in this study, a series of creep tests were selected for comparative analysis, and the Maxwell model is compared with the proposed model to illustrate the superiority of our model. Besides, the root-mean-square-error (RMSE) is used to evaluate the consistency between the model and the test data. The lower RMSE value (→ 0) is indications of models with better performance. The statistical indices can be obtained by:

RMSE = \sqrt{\frac{\sum_{i = 1}^{n} {(γ_{_{test}}^{i} - γ_{_{\mod}}^{i})}^{2}}{n}}

(35)

where n is the total number of test data; γ_test and γ_mod are the test data and predicted value of shear displacement, respectively.

Shear creep test with anchor bolt of Liu (Liu, 2020)

Liu (Liu, 2020) conducted the shear creep tests of bolts with the joint angles of 15°, 30°, 45° and 60°, respectively, as shown in Figure 13. The artificial prefabricated through-through joints were adopted in this experiment.

Figure 14 presents the comparison between the proposed theoretical curve using equation (34) and the experimental results. Table 1 lists the parameter values for the constitutive model. Table 2 shows the comparison of RMSE values between the different models. There is a good consistency between the model results and the experimental results, and the proposed model is better than the Maxwell model in describing the shear behavior of bolted joints. This indicates that the model can better approximate the shear creep deformation behavior of the bolt under the different joint angles.

Table 1.

Model parameters of joint with bolt.

Joint angle/°	Shear stress/MPa	G_M/MPa·mm⁻¹	D_O	G_K/MPa·mm⁻¹	η_K/MPa·mm⁻¹·h	β	χ	η2/MPa·mm⁻¹·h	λ	G_b/MPa·mm	K/MPa·mm·h⁻¹
0	0.5 MPa	1.357	0	13.61	13.79	0.005	−3.38	89.63	59.94	138557	23931
	1.0 MPa		−0.046	15.88	67.57	0.121	0.97	89.63	29.18	138557	1763
	1.5 MPa		−0.017	16.81	190.4	0.524	0.83	89.63	19.89	138557	5592.6
	2.0 MPa		−0.067	11.78	161.48	0.58	0.69	89.63	14.74	138557	5288.8
30	0.5 MPa	1.533	0	8.12	600074	−0.217	0.001	91.56	53.55	124968	4937
	1.0 MPa		0.03	20.87	58162	−0.26	−0.001	91.56	8.51	124968	5892
	1.5 MPa		−0.04	24.98	1702948	0.128	5 × 10⁻⁵	91.56	469.7	124968	7446
	2.0 MPa		−0.011	14.56	1676.7	5.373	0.8	91.56	12.63	124968	951
45	0.5 MPa	1.662	0	12.55	0.0028	6 × 10⁻⁵	0.005	90.33	92.10	89153	13138
	1.0 MPa		0.032	22.12	1663.3	5.23	0.996	90.33	25.93	89153	3494
	1.5 MPa		0.015	29.15	2223.8	5.43	0.826	90.33	24.81	89153	6518
	2.0 MPa		0.018	13.96	860279	−0.196	9 × 10⁻⁵	90.33	14.61	89153	3272
	2.5 MPa		−0.07	10.69	459	1.81 × 10⁻⁵	0.999	90.33	1.48	89153	1752
60	0.5 MPa	1.572	0	6.521	254186	−0.253	8 × 10⁻⁵	102.85	15.73	75519	8782
	1.0 MPa		0.046	19.03	160.14	0.398	0.736	102.85	29.98	75519	2914
	1.5 MPa		0.021	1.057	210.14	−0.095	−0.212	102.85	6.982	75519	2711
	2.0 MPa		0.0076	13.12	121.85	0.384	0.859	102.85	16.68	75519	1460

Table 2.

Comparison of RMSE values (shear creep test with anchor bolt).

Joint angle/°	Shear stress/MPa	RMSE
Joint angle/°	Shear stress/MPa	Proposed model	Maxwell model
0	0.5 MPa	2.45 × 10⁻⁴	7 × 10⁻³
	1.0 MPa	9.33 × 10⁻⁴	9.9 × 10⁻³
	1.5 MPa	2.7 × 10⁻³	1.6 × 10⁻²
	2.0 MPa	2.5 × 10⁻³	3.4 × 10⁻²
30	0.5 MPa	5.85 × 10⁻⁴	1 × 10⁻²
	1.0 MPa	1.8 × 10⁻⁴	8.5 × 10⁻³
	1.5 MPa	5.4 × 10⁻⁴	8.7 × 10⁻³
	2.0 MPa	1.5 × 10⁻³	2.5 × 10⁻²
45	0.5 MPa	3.5 × 10⁻⁵	8 × 10⁻³
	1.0 MPa	1 × 10⁻³	9.6 × 10⁻³
	1.5 MPa	4.26 × 10⁻⁴	1 × 10⁻²
	2.0 MPa	1.5 × 10⁻³	2.11 × 10⁻²
	2.5 MPa	5.3 × 10⁻³	3.51 × 10⁻²
60	0.5 MPa	1.3 × 10⁻³	1.6 × 10⁻²
	1.0 MPa	5.63 × 10⁻⁴	9.9 × 10⁻³
	1.5 MPa	2.07 × 10⁻⁴	1 × 10⁻²
	2.0 MPa	2.8 × 10⁻³	2.7 × 10⁻²

2. Shear creep test without bolts of Zhu and Jia (Jia et al., 2018; Zhu et al., 2019)

Zhu and Jia (Jia et al., 2018; Zhu et al., 2019) conducted a series of shear creep tests to investigate the rheological properties of natural rough joints. all joints are taken from natural discontinuity, as shown in Figure 15.

These test results are also selected to conduct the validation of the proposed model under no bolt conditions. Table 3 lists the parameter values for the constitutive model. Table 4 shows the comparison of RMSE values between the different models. In contrast to the Maxwell model, the theoretical curves obtained from equation (26) match well with the test results in Figure 16. This indicates the proposed model can also effectively predict the shear creep behavior without anchor bolts, the validity and superiority of the model are clearly demonstrated.

Table 3.

Model parameters of joint without bolt.

τ/MPa	G_M/MPa·mm⁻¹	G_K/MPa·mm⁻¹	η_K/MPa·mm⁻¹·h	β	χ	η_vp/MPa·mm⁻¹·h	δ
3.385	10.85421	1.12	1.75	0.99	201	0.16	1.78
1.330	0.99295	3.33	2.99	0.95	1.6	0.007	12.8
0.725	0.33572	1.1	0.18	0.99	591	0.039	−0.23
1.095	0.4147	0.66	2.09	0.82	8.68	0.011	35.2

Table 4.

Comparison of RMSE values (shear creep test without bolts).

τ/MPa	RMSE
τ/MPa	Proposed model	Maxwell model
3.385	2.43 × 10⁻⁴	1 × 10⁻²
1.330	4.5 × 10⁻³	8.9 × 10⁻²
0.725	4.2 × 10⁻³	3.2 × 10⁻²
1.095	4.3 × 10⁻³	3 × 10⁻²

Conclusions

In this study, based on the mechanical properties of three rheological stages and the deformation characteristics of bolts, an elasto-viscoplastic constitutive model of bolted joints considering initial damage and damage evolution is proposed, which is composed of several rheological elements: (I) elastoplastic improved Hooke body of transient deformation considering hardening and damage effects. (II) parametric nonlinear Kelvin model of attenuation stage. (III) nonlinear viscous model considering shear stress of steady creep stages. (IV) visco-plastic model based on time-dependent shear strength considering initial damage of accelerated stage. (V) rheological model of bolt-rock cooperative deformation. Subsequently, with the combination of these series of models, an elasto-viscoplastic constitutive model describing the shear rheological behavior of bolted joints is established.

The validity and superiority of this proposed model are verified by comparison analysis among the predictive results of the proposed model, shear creep tests and the Maxwell model. The model inversion results show that the main rheological characteristics (Hardening and damage effects of instantaneous deformation, nonlinear characteristic of parameters, time-dependent characteristic of shear strength caused by damage evolution, cooperative deformation characteristics of bolted rocks) can be well reflected in this model. This research is expected to provide theoretical guidance for the construction and reinforcement of rock mass engineering.

Footnotes

Data availability statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper gets its funding from Projects (42277175) supported by National Natural Science Foundation of China; Science and Technology Progress and Innovation Plan of Hunan Provincial Department of Transportation(202120); Hunan Civil Air Defense Research Project (HNRFKJ-2021-07); Hunan provincial key research and development Program(2022SK2082), Postgraduate Scientific Research Innovation Project of Hunan Province (CX20220195), and Fundamental Research Funds for the Central Universities of Central South University (2022ZZTS0107). The authors wish to acknowledge these supports.

ORCID iD

Hang Lin

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