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Study on stress distribution and extrusion load threshold of compressed filled rock joints

  • Pengpeng Wang , Shigui Du , Gan Li and Zhanyou Luo EMAIL logo
Published/Copyright: December 31, 2023
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Applied Rheology
From the journal Applied Rheology Volume 33 Issue 1

Abstract

The distribution of stress and the normal extrusion load threshold in weak interlayer are crucial for direct shear test of filled rock joints, but there is a lack of theoretical research in this area. First, an analytical solution for stress distribution was derived using a semi-inverse method. Then, it is compared by the numerical simulation method. Finally, the influence of the width and thickness of weak interlayer on the extreme values of stress components was analyzed, and the distribution pattern of the normal extrusion load was discussed. The results show that under the same conditions, the analytical solution and the numerical simulation results are in good agreement. The maximum horizontal stress in the weak interlayer decreases with increasing width and increases with increasing thickness, while the change of the minimum is opposite. The normal extrusion load increases first and then decreases along the width direction of the weak interlayer. By comparing the normal extrusion load with the empirical value, the mechanism of extrusion failure in the weak interlayer is revealed.

Keywords: rock joints; weak interlayer; stress distribution; analytical solution; numerical simulation

1 Introduction

Joints commonly govern the overall mechanical behavior of rock masses [1 2 3]. Due to the presence of weaker material filling within the joints, filled rock joints exhibit lower shear strength compared to unfilled [4,5,6,7], which has a dominant effect on the shear behavior [8]. And the normal stress is a significant factor affecting the shear strength of filled rock joints [9,10,11,12].

In terms of theoretical models, Papaliangas et al. [13] established a predictive model for peak and residual friction coefficients of filled rock joints. Indraratna et al. [14] formulated a normalized mathematical semi-empirical model for predicting the shear strength of filled joint. And Zhao et al. [15] proposed an improved shear strength model for filled joint based on the negative exponential relation between friction angle and infill thickness to asperity height ratio. All these models indicate a positive correlation between shear strength of filled joints and normal stress. In the realm of experimentation, Abolfzali and Fahimifar [16] conducted direct shear tests on 96 samples of various sizes and rock materials, revealing that with an increase in normal stress, both the shear and normal stiffness of joints increase. Zhu et al. [17] conducted laboratory direct shear tests on two parallel coplanar intermittently jointed rock samples and observed that the influence of normal stress on residual stress was significant during the residual phase. Feng et al. [18] found that the shear strength of filled rock joints increased linearly with the increase of normal stress through shear experiments under four groups of different normal stresses. However, Sun and Zhao [19] proposed that during the loading of normal stress in indoor tests, weak interlayer is often extruded or crushed, leading to errors in the shear strength of weak interlayer rock joints. Therefore, it is significant to understand and investigate the normal extrusion load threshold of weak interlayer within filled rock joints by revealing the stress distribution and extrusion failure mechanisms.

In conclusion, while several scholars have studied the impact of normal stress on shear strength, there has been limited research on the stress distribution and the normal extrusion load threshold for weak interlayer within filled rock joints. This study uses the semi-inverse method to derive the stress distribution function of weak interlayer within rock joints and validates it through numerical simulation experiments. The influence of the width and thickness of the weak section on the extreme value of the stress component is analyzed, and the distribution of the normal extrusion load within the interlayer is discussed.

2 Analytical solution of stress distribution function in compressed weak interlayer

2.1 Fundamental assumption

Due to the different sedimentary environment and tectonic movement of the rock, its occurrence is complicated. In order to obtain the analytical solution of the weak interlayer stress distribution function, the following assumptions are made:

  1. The rock joints are flat.

  2. The weak interlayer is homogeneous, isotropic, and behaves as a perfectly elastic material.

  3. Small deformation compressive strain occurs in the weak interlayer before extrusion.

  4. Frictional forces exist at the upper and lower interfaces of the weak interlayer, while the left and right interfaces have no lateral confinement.

  5. The normal stress on a unit within the interlayer depends only on its vertical position.

The simplified weak interlayer model is shown in Figure 1.

Figure 1 Mechanical model for the extrusion of weak interlayer.
Figure 1

Mechanical model for the extrusion of weak interlayer.

2.2 Development of an analytical solution

In the given boundary conditions or internal action of the object, the semi-inverse solution is often used to solve. Semi-inverse solutions can be obtained in all unknowns by presupposing that some are known, while others can be obtained from the fundamental equations and boundary conditions. The equilibrium differential equation for the extrusion of a weak interlayer is given by:

(1) ∂ σ x ∂ x + ∂ τ x y ∂ y = 0 ∂ σ y ∂ y + ∂ τ x y ∂ x = 0 .

There exists a stress function Φ that all stress components satisfy the equilibrium differential equation (1).

(2) σ y = ∂ 2 Φ ∂ x 2 σ x = ∂ 2 Φ ∂ y 2 τ x y = − ∂ 2 Φ ∂ x ∂ y .

Assuming that the normal stress within the interlayer unit is only dependent on the vertical position, we can derive the following equation:

(3) ∂ 2 Φ ∂ x 2 = f ( y ) .

Integrating the aforementioned equation with respect to x twice yields the following equation:

(4) Φ = x 2 2 f ( y ) + x f 1 ( y ) + f 2 ( y ) ,

where f ( y ) , f 1 ( y ) , and f 2 ( y ) are all undetermined functions.

The stress function indicated by equation (4) needs to satisfy the compatibility equation, which leads to the following equation:

(5) ∂ 4 Φ ∂ x 4 = 0 , ∂ 4 z ∂ x 2 ∂ y 2 = d 2 f ( y ) d y 2 ∂ 4 Φ ∂ y 4 = x 2 d 4 f ( y ) 2 d y 4 + x d 4 f 1 ( y ) d y 4 + d 4 f 2 ( y ) d y 4 1 2 d 4 f ( y ) d y 4 x 2 + d 4 f 1 ( y ) d y 4 x + d 4 f 2 ( y ) d y 4 + 2 d 2 f ( y ) d y 2 = 0 .

The compatibility condition necessitates that equation (5) has an infinite number of roots. Consequently, both the coefficients and the constant term of this quadratic equation must be equal to zero. This leads to the following equation:

(6) d 4 f ( y ) d y 4 = 0 , d 4 f 1 ( y ) d y 4 = 0 , d 4 f 2 ( y ) d y 4 + 2 d 2 f ( y ) d y 2 = 0 .

From equation (6), the stress function can be obtained as follows:

(7) Φ = x 2 2 ( A y 3 + B y 2 + C y + D ) + x ( E y 3 + F y 2 + G y ) .

Substituting equation (7) into equation (2), we obtain the stress components:

(8) σ x = x 2 ( 6 A y + 2 B ) / 2 + x ( 6 E y + 2 F ) − 2 A y 3 − 2 B y 2 + 6 H y + 2 K σ y = A y 3 + B y 2 + C y + D τ x y = − x ( 3 A y 2 + 2 B y + C ) − ( 3 E y 2 + 2 F y + G ) .

Due to the symmetry of the interlayer, equation (8) can be simplified to:

(9) σ x = B x 2 − 2 B y 2 + 6 H y + 2 K σ y = B y 2 + D τ x y = − 2 B x y .

The boundary conditions for the upper and lower interfaces of the weak interlayer are as follows:

(10) ( σ y ) y = ± h = − Q .

As the results are consistent with symmetrical distribution in each interval, the analysis can be focused on the interval where x ≥ 0 and y = − h :

(11) ∫ 0 L 2 B h x d x = f ⋅ Q ⋅ L ,

where f represents the friction coefficient at the interface between the weak interlayer and the rock, Q is the normal load, L is the cross-sectional length of the weak interlayer, and h is the thickness of the weak interlayer.

Due to the non-existence of a solution for the original equation when σ x = ± L = 0 , it becomes imperative to contemplate the synthesis of an equilibrium force system at the boundaries on both the left and right sides:

(12) ∫ − h h σ x d y = 0 ,

(13) ∫ − h h σ x ⋅ y d y = 0 .

By combining equations (10)–(13), we can obtain the following：

(14) B h 2 + D = − Q ∫ 0 L 2 B h x d x = f ⋅ Q ⋅ L ∫ − h h σ x d y = 0 ∫ − h h σ x ⋅ y d y = 0 .

The coefficients are determined as follows:

B = f Q 2 L h ; D = − Q ( L + f h ) L ; K = f Q h 3 L − f Q L 2 h

Substituting the solved coefficients into equation (10), the stress components can be expressed as:

(15) σ x = f Q 2 L h ( x 2 − 2 y 2 ) + f Q ( 2 h 2 − 3 L 2 ) 3 L h σ y = f Q 2 L h y 2 − Q ( L + f h ) L τ x y = − f Q L h x y .

3 Finite element model and results of compressed filled rock joints

3.1 Model establishment

A numerical model was developed by FLAC 3D software. The model consists of three components: the upper rock mass, the lower rock mass, and the weak interlayer. To avoid any lateral displacement, we applied displacement constraints to the lateral sides of the upper and lower rock models. Because the weak interlayer is extruded under low normal stress conditions, the upper and lower rocks are set to be completely elastic with great rigidity. And the Mohr-Coulomb elastic–plastic model was adopted in the weak interlayer. The model sets two contact surfaces with the same friction angle between the upper and lower walls of the rock and the weak interlayer. The numerical model established is shown in Figure 2.

Figure 2 Numerical experimental model of filled rock joints: (a) numerical experimental model and (b) internal contact surface of the model.
Figure 2

Numerical experimental model of filled rock joints: (a) numerical experimental model and (b) internal contact surface of the model.

3.2 Verification of stress distribution within the weak interlayer

The upper and lower rock measure 100 mm × 50 mm × 10 mm, as does the weak interlayer. Table 1 lists the physical and mechanical parameters of the rock masses, as well as the weak interlayer. According to equation (15), it can be inferred that the normal stress variation in weak interlayers is insignificant. However, the horizontal stress and shear stress in the interlayer change greatly, so it is necessary to verify their laws.

Table 1

Physical and mechanical parameters of filled rock joints

Parameter Elastic modulus (GPa) Poisson’s ratio Friction angle (°) Cohesion (MPa)
Rock 25.7 0.286 41.6 18.65
Weak interlayer 0.027 0.35 25.7 0.1

After applying a normal load of 0.25 MPa, the numerical simulation results are shown in Figure 3(a) and (c). By putting the stress component function into MATLAB software for calculation, the resulting stress nephogram is depicted in Figure 3(b) and (d).

Figure 3 Comparison between numerical simulation and analytical solution results: (a) numerical simulation stress nephogram of horizontal stress, (b) analytical solution stress nephogram of horizontal stress, (c) numerical simulation stress nephogram of shear stress, and (d) analytical solution stress nephogram of shear stress.
Figure 3

Comparison between numerical simulation and analytical solution results: (a) numerical simulation stress nephogram of horizontal stress, (b) analytical solution stress nephogram of horizontal stress, (c) numerical simulation stress nephogram of shear stress, and (d) analytical solution stress nephogram of shear stress.

Figure 3(a) and (b), respectively, displays the horizontal stress nephogram within the weak interlayer obtained using numerical simulation and analytical solution. A comparison between the two plots reveals the following observations:

  1. The distribution pattern of horizontal stress obtained from numerical simulation and analytical solution is similar. Horizontal stress within the interlayer symmetrically distributes along the width of the weak interlayer, transitioning from compression in the middle toward tension at the edges.

  2. The deviation between numerical simulation and analytical solution results is relatively small. The maximum value in the analytical solution stress nephogram is 10 kPa, which is 2 kPa less than the numerical simulation result. The minimum value is −211 kPa, 10 kPa larger than the numerical simulation result.

Figure 3(c) and (d) depicts the shear stress nephogram within the weak interlayer obtained through numerical simulation and analytical solution. Observations from these plots include the following:

  1. The distribution pattern of shear stress obtained from numerical simulation and analytical solution is similar. The shear stress reaches the extreme value at the four corners of the weak interlayer and is 0 on the symmetry axis.

  2. The deviation between numerical simulation and analytical solution results for shear stress is small. The analytical solution indicates that the maximum shear stress is 0.125 MPa, which is higher than the numerical simulation result by 0.025 MPa.

4 Effect of the weak interlayer section size on the stress component

From the analysis of equation (15), it can be seen that the normal load and the friction coefficient of the interface are significantly proportional to the stress component of the weak interlayer, and the extreme value of the shear stress is also only related to them. However, the impact of the weak interlayer cross-sectional dimensions on the distribution of horizontal and normal stresses within the interlayer remains unclear. To investigate this relationship, baseline parameters for the weak interlayer are set as shown in Table 2.

Table 2

Baseline parameters for weak interlayer

Parameter Interlayer thickness (mm) Interlayer width (mm) Friction coefficient Normal load (MPa)
Interlayer thickness 10 50 0.5 0.25

4.1 Effect of the weak interlayer width on the stress component

Figure 4 depicts the relationship between weak interlayer width and horizontal as well as normal stresses. The following observations can be made from the graph:

  1. The maximum horizontal (tensile stress) stress within the weak interlayer decreases as interlayer width increases, and the rate of change gradually diminishes. This is because as the width of the interlayer increases, the interface friction also increases, resulting in a reduction in the potential energy for interlayer extrusion, thereby decreasing the horizontal stress.

  2. The minimum horizontal stress (compressive stress) within the weak interlayer increases as interlayer width increases, and the rate of change remains relatively constant. Increasing the interlayer width may lead to a rise in interfacial friction, which can result in higher compressive stress at the core of the interlayer.

  3. The maximum normal compressive stress within the weak interlayer decreases with an increase in interlayer width, and the rate of change gradually decreases. The overall variation is also relatively small.

Figure 4 Relationship between weak interlayer width and horizontal and normal stresses: (a) relationship between maximum horizontal stress and width, (b) relationship between minimum horizontal stress and width, and (c) relationship between maximum normal compressive stress and width.
Figure 4

Relationship between weak interlayer width and horizontal and normal stresses: (a) relationship between maximum horizontal stress and width, (b) relationship between minimum horizontal stress and width, and (c) relationship between maximum normal compressive stress and width.

4.2 Effect of the weak interlayer thickness on the stress component

Figure 5 illustrates the relationship between weak interlayer thickness and horizontal as well as normal stresses. The graph provides the following insights:

  1. The maximum horizontal stress (tensile stress) within the weak interlayer increases with an increase in interlayer thickness, and the rate of change remains constant. This is because as the interlayer thickness increases, so does the free surface area, making the interlayer more prone to extrusion and causing the maximum horizontal stress at the boundary to keep increasing.

  2. The minimum horizontal stress (compressive stress) within the weak interlayer decreases as interlayer thickness increases, and the rate of change gradually reduces. This could be attributed to the dispersion of the normal load caused by the increase in soil mass.

  3. The maximum normal compressive stress within the weak interlayer increases with an increase in interlayer thickness, with a relatively small overall variation.

Figure 5 Relationship between weak interlayer thickness and horizontal and normal stresses: (a) relationship between maximum horizontal stress and thickness, (b) relationship between minimum horizontal stress and thickness, and (c) relationship between maximum normal compressive stress and thickness.
Figure 5

Relationship between weak interlayer thickness and horizontal and normal stresses: (a) relationship between maximum horizontal stress and thickness, (b) relationship between minimum horizontal stress and thickness, and (c) relationship between maximum normal compressive stress and thickness.

5 Distribution patterns of normal extrusion load in weak interlayer

According to equation (15), it is evident that stress components at all locations within the interlayer vary, resulting in the normal extrusion load being unique for each point within the interlayer. Due to the complexity of the expression, it is difficult to solve the whole display solution, so the most extrudable surface (y = 0) in the weak interlayer is taken as the representative, and the normal extrusion stress distribution of the weak interlayer is studied with x as the variable. Based on the Mohr-Coulomb failure criterion, the function for the normal extrusion load can be expressed as equation (16). Bringing parameters from Tables 1 and 2 into the function yields Figure 6.

(16) Q ( x ) = 2 c ⋅ cos φ ( 1 − sin φ ) f x 2 L h − f h L − f 2 h 2 − 3 L 2 3 L h − 1 .

Figure 6 Distribution of normal extrusion load.
Figure 6

Distribution of normal extrusion load.

Figure 6 depicts the variation curve of the normal extrusion load along the width direction of the interlayer. From the graph, the following observations can be made:

The normal extrusion load exhibits symmetric distribution along the width direction of the weak interlayer, increasing initially and subsequently decreasing.

The rate of change in the normal extrusion load experiences an initial increase followed by a decrease.

The maximum value of the normal extrusion load is reached at x = 0.

Combining with Figure 5, it is known that the interlayer experiences tensile stress at the boundaries. On the one hand, due to the Poisson effect, the internal interlayer expands toward the free surface, resulting in tensile stress on interlayer’s surface. On the other hand, as there is no lateral stress constraint, the compression of the rock on the interlayer leads to tensile stress on the interlayer surface. The combination of these two factors results in the extrusion failure on the interlayer surface. Engineering experience indicates that the typical stress threshold for clay extrusion ranges from 0.15 to 0.2 MPa [16]. Figure 6 demonstrates that slight extrusion on the weak interlayer surface does not significantly impact the overall state, and entirety extrusion only takes place once the core area of the weak interlayer is damaged.

6 Conclusions

This study simplifies the physical model of a flat-filled rock joints and derives analytical solutions for stress component distribution within the weak interlayer. These solutions are validated through numerical simulation experiments, and the influence of weak interlayer width and thickness on stress component extremities is analyzed. The distribution patterns of the normal extrusion load are also discussed. The main conclusions are as follows:

  1. The analytical solutions for stress distribution within filled account for factors such as weak interlayer width, thickness, and contact surface friction coefficient. A comparison between the analytical solutions and numerical simulation results demonstrates good agreement.

  2. When the thickness of the weak interlayer remains constant, the maximum horizontal stress and maximum normal compressive stress in the weak interlayer decrease with an increase in weak interlayer width. Conversely, the minimum horizontal stress in the weak interlayer increases with an increase in width.

  3. When the width of the weak interlayer remains constant, the minimum horizontal stress and maximum normal compressive stress in the weak interlayer increase with an increase in weak interlayer thickness. Conversely, the minimum horizontal stress in the weak interlayer decreases with an increase in thickness.

  4. The normal extrusion load varies at different points within the weak interlayer. Along the most extrudable surface, the normal extrusion load is symmetrically distributed and increases first and then decreases along the width of the weak interlayer. The interlayer becomes unstable and the filling is squeezed out when the core area is damaged. Such systems can find some useful applications in the design of direct shear test of filling rock joint.

Acknowledgement

The authors acknowledge the financial support of the National Natural Science Foundation of China.

  1. Funding information: This research was funded by the National Science Foundation of China (Grant Number No. 42277147) and Special Support Plan for High Level Talents of Zhejiang Province (Grant Number No. 2020R52028).

  2. Author contributions: Methodology, software, validation, writing – original draft preparation, P.W.; resources, supervision, project administration, S.D.; investigation, G.L.; conceptualization, writing – review and editing, Z.L. All authors have read and agreed to the published version of the manuscript.

  3. Conflict of interest: The authors declare no conflict of interest.

  4. Informed consent: Informed consent was obtained from all individuals included in this study.

  5. Ethical approval: The conducted research is not related to either human or animal use.

  6. Data availability statement: The authors confirm that the data supporting the findings of this study are available within the article.

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Received: 2023-09-09
Revised: 2023-10-27
Accepted: 2023-11-12
Published Online: 2023-12-31

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/arh-2023-0113
Keywords for this article
rock joints; weak interlayer; stress distribution; analytical solution; numerical simulation
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  34. Study on the risk of seepage field of Qiantang River underground space excavated in water-rich rheological rock area
  35. Numerical analysis of the impact of excavation for undercrossing Yellow River tunnel on adjacent bridge foundations
  36. Deformation rules of deep foundation pit of a subway station in Lanzhou collapsible loess stratum
  37. Development of fiber compound foaming agent and experimental study on application performance of foamed lightweight soil
  38. Monitoring and numerical simulation analysis of a pit-in-pit excavation of the first branch line of Lanzhou Metro
  39. CT measurement of damage characteristics of meso-structure of freeze-thawed granite in cold regions and preliminary exploration of its mechanical behavior during a single freeze-thaw process
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