Open AccessArticle

Regularities of Plastic Deformation Zone Formation Around Unsupported Shafts in Tectonically Disturbed Massive Rock

Petr A. Demenkov

and

Ekaterina L. Romanova

Department of Construction of Mining Enterprises and Underground Structures, Empress Catherine II St. Petersburg Mining University, 21st Line 2, 199106 St. Petersburg, Russia

Author to whom correspondence should be addressed.

Geosciences 2025, 15(1), 23; https://doi.org/10.3390/geosciences15010023

Submission received: 24 October 2024 / Revised: 9 December 2024 / Accepted: 6 January 2025 / Published: 10 January 2025

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Abstract

In the presented paper, an approach to assessing the size of the plastic deformation zone around a circular cavity intersecting a crushed disintegrated layer in tectonically stressed massive rock is suggested. Fractured rock zones of different quality and spatial configurations are investigated in order to predict the size of dangerous plastic deformation zones or zones of potential rock collapses. The analysis is performed by means of numerical modeling after preliminary verification of the model by in situ and monitoring data. The paper explores the impact of such parameters of the fractured rock zone as the GSI index, the true thickness of the zone, and its inclination angle relative to the plane perpendicular to the axis of the excavation. It was found that with the increase in the thickness and angle of inclination of the fractured rock zone, the size of the hazardous zone in the vicinity of the excavation increases, while with the increase in its strength characteristics, the size of the potential failure zone decreases. According to the results of the study, qualitative dependencies are established, which localize and predict the size of the danger zone, or the potential failure zone in the vicinity of an unfixed excavation.

Keywords:

fractured rock zone; tectonically disturbed massif; vertical excavation; stress–strain state; rock pressure; rock massif

1. Introduction

Mining activity nowadays is conducted to depths of up to 4 km, although not without some difficulties. Excavations are hosted in heterogeneous, disturbed, tectonically stressed rock masses. These factors complicate excavation significantly. Falling or sliding of blocks of rock from the boundary contour is quite common there. For this reason, mining in tectonically stressed regions requires both the knowledge of the mechanical properties of rock and the analytical capacity to predict rock mass performance under load.

Pre-mining state of stress is formed under gravity, historical tectonism, and neotectonics activity, which directly affects displacements and geological processes in the earth’s crust [1,2,3]. Due to the complexity of the crustal structure, the stress field distribution in it is heterogeneous [4,5,6]. Stresses in the earth’s crust are caused by displacement and deformation of rocks and subsequent loss of stability of underground mining structures [7,8,9]. Understanding the stress state of the host rock mass is crucial for predicting [10,11] and preventing [12,13] possible expression of tectonic forces, as far as the structure’s stability under loads. In situ stress measurements are quite vital for these aims, although it is often complicated in some regions of rocks [14]. Despite this, system monitoring is one of the most promising directions of research in this area of subsurface construction development [15,16].

Falls of blocks of rock and irregular contours of vertical shafts’ cavities are a common problem in disturbed rocks. It is becoming more complicated when excavating in tectonically stressed masses, in particular, when the shaft crosses into fractured discontinuous rock zones. Research in this area is mainly represented by articles based on in situ measurements [17,18,19]. In research [20], the relationship between the stress field and the structure of certain regions of the earth’s crust was investigated and a model of stresses in it was created. In [21], the relationship between vertical and horizontal stress components for different rock types depending on depth, lithology, and fault condition was suggested. In [22], a description of the current state of tectonic stresses in the Darling Basin (Australia) was proposed. In [23], the distribution features of the present-day stress field in the mine area were determined, and the relationship between the stress field and tectonic stresses was systematically investigated. In [24], three basic types of faulting (normal, strike-slip, and reverse) were explained in terms of the shape of the causative stress tensor and its orientation relative to the earth’s surface. The issues of subsurface construction in tectonically complicated massifs have also been investigated [25,26,27]. In [28], an analysis of the experience in predicting the stress state of a blocky, tectonically stressed rock mass is given and recommendations for solving similar problems are provided. The study [29] presents the results of analyzing the forms of rock failures during mining operations in high-stress conditions. The paper [30] presents the justification of rational parameters of mine supports in complicated mining and geological conditions. The main danger for underground excavations crossing fractured crushed layers is the formation of plastic deformation zones in disintegrated rocks and their subsequent falls.

The Hoek–Brown criterion [31] has found wide practical application in defining the stress conditions under which a rock mass will deform inelastically and might collapse. This approach was later implemented into a “RocLab” program (www.rocscience.com). This program estimates the uniaxial compressive strength of the intact rock elements, the material constants, and the Geological Strength Index (GSI), which are further needed to assess the plastic zone size around the opening. Based on [31], failure criterion expressions were given for the extent of plastic behavior and the related stress fields [32]. The dimensionless parameter for assessing a plastic region of radial extent that develops around the opening was suggested. In [33], some typical plastic zone shapes for different average horizontal-to-vertical stress ratios were demonstrated as far as an approach to estimating plastic zone size in soft rocks was given. The research was based on the RMR method of determining rock mass class.

Despite the fact that scientists all over the world have paid a lot of attention to the study of stress distribution patterns in massifs with local tectonic settings, an approach to assessing the influence of crushed zone parameters on the size of the plastic deformation zone has not yet been proposed. The study investigates the regularities and features of stress distribution around a vertical circular excavation while crossing the fractured crushed rock zones of different thicknesses and spatial configurations, in order to predict the boundaries and geometry of the potential failure zone around the cavity. An approach to estimate the plastic zone radius is suggested by taking into account the thickness, angle of inclination of the crushed zone, and its GSI parameter based on the Hoek–Brown failure criterion.

2. Materials and Methods

To solve the problem of determining the stress–strain state of the massif around the excavation in the fractured disintegrated rock zone (crushed zone) and the size of the potential collapse zone, the presented study uses numerical modeling, which is now broadly applicable in geomechanics [34,35] to forecast the development of dangerous processes in the massif [36,37]. Its undoubted advantage is the possibility of representing the model geometry in an explicit way and capturing the step-by-step development of stresses during excavation [38,39]. It affords to consider such factors as structural and geological features of the massif, physical, and mechanical properties of rocks, the initial stress field of the massif, properties of the support material, and the features of its plastic deformation [40,41]. In this study, more than a hundred solid numerical models with variable characteristics of crushed zone parameters were created in the Abaqus CAE software package. Numerical models take into account the geology and tectonics of the studied rock massif.

In all models built, the physical and mechanical characteristics of the rock mass are unchangeable, while the characteristics of the fractured rock zones (thickness and angle of inclination to the intersected excavation) are varied. The strength and deformability of discontinuous jointed rock mass was varied based on the Hoek–Brown strength criterion [42] by changing the GSI parameter of the zone. The ranges of investigated parameter variation are given in Table 1 (the range of values was chosen in accordance with [7] as the most typical for disturbed zones in the studied massif).

The rock strength criterion is determined from the following formula [42]:

σ_{1}^{'} = σ_{3}^{'} + σ_{c i} {(m_{b} \frac{σ_{3}^{'}}{σ_{c i}} + s)}^{a},

(1)

where

σ_{1}^{'}

and

σ_{3}^{'}

are the major and minor effective principal stresses at failure;

σ_{c i}

is the uniaxial compressive strength of the intact rock material;

m_{b}

is a reduced value of the material constant

m_{i}

and is given by [42]:

m_{b} = m_{i} e x p (\frac{G S I - 100}{28 - 14 D});

(2)

m_{i}

is a curve-fitting parameter derived from triaxial testing of intact rock;

s and a are constants for the rock mass given by the following relationships for

G S I

> 25 [38]:

s = e x p (\frac{G S I - 100}{9 - 3 D}),

(3)

a = \frac{1}{2} + \frac{1}{6} (e^{- G S I / 15} - e^{- 20 / 3}),

(4)

and for

G S I

< 25,

s = 0,

(5)

a = 0.65 - G S I / 200

(6)

D is a factor that depends upon the degree of disturbance to which the rock mass has been subjected by blast damage and stress relaxation and should be estimated from Table 1 of [38]. It varies from 0 for undisturbed in situ rock masses to 1 for very disturbed rock masses.

Based on the range of fractured rock zone strength data obtained, the other parameters required for modeling were determined [42]:

-: friction angle

φ^{'} = a r c s i n [\frac{6 a m_{b} {(s + m_{b} σ_{3 n}^{'})}^{a - 1}}{{2 (1 + a) (2 + a) + 6 a m_{b} (s + m_{b} σ_{3 n}^{'})}^{a - 1}}];

(7)

-: cohesion

c^{'} = \frac{σ_{c i} {[(1 + 2 a) s + (1 - a) m_{b} σ_{3 n}^{'}] (s + m_{b} σ_{3 n}^{'})}^{a - 1}}{(1 + a) (2 + a) \sqrt{1 + (6 a m_{b} {(s + m_{b} σ_{3 n}^{'})}^{a - 1})} / ((1 + a) (2 + a))},

(8)

where

σ_{3 n}^{'} = σ_{3 m a x}^{'} / σ_{c i}

σ_{3 m a x}^{'} —

upper limit of confining stress that is found in the equation [42]:

\frac{σ_{3 m a x}^{'}}{σ_{c m}} = 0.47 {(\frac{σ_{c m}}{σ_{1}})}^{- 0.94},

(9)

where

σ_{1}

is the horizontal stress value (as far as the horizontal stress value is higher than the vertical stress in the surveyed massif [42]);

σ_{c m}

is the rock mass strength, defined by the equation:

σ_{c m} = σ_{c i} \frac{(m_{b} + 4 s - a (m_{b} - 8 s)) {(m_{b} / 4 + s)}^{a - 1}}{2 (1 + a) (2 + a)}

(10)

Rock mass deformation modulus was determined from the equation:

E_{m} = E_{i} (0.02 + \frac{1 - D / 2}{1 + e^{(60 + 15 D - G S I) / 11}}),

(11)

where

E_{i}

is deformation modulus of the sample.

In the study, the authors propose a formula for determining the plastic deformation zone around the excavation crossing the fractured zone.

Following the calculations, a regression analysis method was used to identify the relationship between the fractured rock zone parameters and the size of the hazard zone in the vicinity of the disturbance. Regression analysis is a statistical data processing technique used to establish a relationship between a dependent variable (in this case, the size of the plastic deformation zone, or collapse zone) and several independent variables (disintegrated crushed zone thickness and its inclination angle, as far as its

G S I

To assess the influence of variables on the plastic deformation zone size, this study uses the regression analysis method. Polynomial regression of the second order is applied to take into account the quadratic dependence between the variables. It extends the traditional linear regression by including several independent variables to assess the influence of each of them and their interactions on the outcome at the same time.

The expression yields the relations:

l_{p c z} = (β_{0} + β_{1} X_{1} + β_{2} X_{2} + β_{3} X_{3} + β_{4} X_{1}^{2} + β_{5} X_{2}^{2} + β_{6} X_{3}^{2} + β_{7} X_{1} \cdot X_{2} + β_{8} X_{1} \cdot X_{3} + β_{9} X_{2} \cdot X_{3}) + R

(12)

where

l_{p c z}

is predicted value (size of plastic deformation zone);

β_{0}

is constant (free term);

β_{1}, β_{2}, β_{3}

are regression coefficients determining the influence of each variable

X_{1}, X_{2}, X_{3}

;

R

is the radius of the cylindrical cavity.

The dimensionless value of the plastic zone proportion size

ξ

should be defined as follows (Figure 1):

ξ = \frac{l_{p c z}}{R},

(13)

The suggested Equation (13) results were compared with one of the approaches to the mathematical solutions of assessing the plastic zone size around the opening based on the Hoek–Brown criterion [32]. This approach suggests dimensionless parameter

ξ,

which expresses the size of the failed region as a proportion of the radius of the unsupported cavity and is given by:

ξ = \exp [\frac{1}{k + 1} \sqrt{1 + 4 \frac{σ_{1}}{m_{b} σ_{c i}} {(\frac{k + 1}{k})}^{2}} - \frac{1}{k + 1}]

(14)

where k is a parameter that defines the cavity’s shape, and k = 1 for cylindrical opening.

Before proceeding to the study of the influence of the variability of disintegrated crushed zone parameters on the stress–strain state of a circular excavation, the model of the rock mass was verified. The verification was based on the results of diagnostics [7] of the vertical shaft support of the mine located in the Talnakh ore deposit according to its geological and structural data [43]. A three-dimensional model of the shaft and the host rock mass massif was created. The stress state of the rock mass model was selected in such a way that the magnitude of the load on the support at a certain depth corresponded to the monitoring data [7].

During verification, changes in support loads reflected by changes in the ratio of components of the rock mass stress field were determined. In massive rock, subject tectonic stresses prevail [7,43], and one sub-horizontal stress component,

σ_{x}

, is significantly greater than both the overburden stress

σ_{z}

and the other horizontal component

σ_{y}

. This way of stress distribution in the massif is caused by the faults that are induced by a tectonic regime of a region. According to Anderson’s faulting theory [24], this stress state is supposed to be produced by strike-slip faulting (SS), as the vertical stress is greater than the minimum horizontal stress and less than the maximum horizontal stress.

The ratio of the maximum horizontal and vertical components of rock pressure is also named lateral stress coefficient

λ

[7,44,45]

λ = \frac{ν}{1 - ν} = \frac{σ_{x m a x}}{σ_{z}}

(15)

where

σ_{x m a x}

is maximum horizontal stress, MPa, and

σ_{y}

is the vertical stress, MPa, and

ν

is Poisson’s ratio for the rock mass.

The ratio of the greatest horizontal stress

σ_{x}

to the vertical stress

σ_{z}

varied from 1 to 1.3, and the ratio of horizontal components

σ_{y}

σ_{x}

varied from 0.4 to 0.7 during verification. As a result of the selection of the parameters of the stress state of the rock massif, the following dependencies were determined at the depth of 900 m: the ratio of the maximum horizontal and vertical components of rock pressure

σ_{x} / σ_{z} =

1.3 (lateral stress coefficient

λ

); the ratio of horizontal stress components

σ_{y} / σ_{x} =

0.5. Absolute values of stress components were determined:

σ_{x} = 17.16 MPa, σ_{z} = 13.20 MPa σ_{y} = 8.58 MPa .

(16)

It is also important to note that there is another approach to describing the stress components ratio in the massive rock that is quite widely used worldwide [45,46,47]. The stress components ratio is described by coefficient k—the mean horizontal-to-vertical stress ratio, or ‘at-rest’ coefficient of earth pressure:

k = \frac{σ_{x} + σ_{y}}{{2 σ}_{z}},

(17)

where

σ_{x}

and

σ_{y}

are horizontal stress components, and

σ_{z}

is the vertical stress.

Introducing the values of stress components

σ_{x}

σ_{z}

σ_{y}

in Equation (10) gives the expression

k = 0.975

(18)

Variations in the k ratio with depths can be observed from results of the measurement of the pre-mining state of the stress results comparison, accumulated by Brown and Hoek [48] In this paper, two curves were presented as bounds for lateral stress coefficient k (Figure 1)

\frac{100}{z} + 0.3 \leq k \leq \frac{1500}{z} + 0.5,

(19)

where

z

is the depth in meters.

Other researchers later collected more data on stress measurements from different regions and modified the lower bound to K = 1/3 [47,49]

0.3 \leq k \leq \frac{1500}{z} + 0.5

(20)

Based on [47,48,49] and Equations (19) and (20), the k ratio of the Talnakh ore deposit at the depth of 900 m is supposed to vary in bounds:

0.3 \leq k \leq 2.1

(21)

It is evident that introducing the k ratio value from Equation (17) in Equation (21) yields a fair inequality. Expression (16) demonstrates that the determined stress components in a rock mass correlate well with the theory of stress contribution in rock massifs and field data (Figure 2) [45,46,47].

Figure 2a [50] shows variations in vertical stresses upon in situ data [47] and Figure 1b [50] shows the values of the coefficient k corresponding to the vertical stresses that are bounded by the two lines. Authors compiled values of the rock mass subject stress field components with the diagrams [50] and plotted the matched values in Figure 2a,b (red circles). As is shown in Figure 2a, values of vertical stresses in the massif are lower than the average values based on the results of field data [47], Figure 2a. This is explained by the nonlinear dependence of vertical stress variation in the massif with depth (due to the local tectonic setting) [43]. However, the coefficient k value for the rock mass subject correlates well with Anderson’s theory [24], which proposes that for strike-slip faulting (SS), the k ratio converges to unity.

To achieve a more complete picture of the convergence of the calculated values of stresses in the concrete support with monitoring data [7], it was necessary to investigate another parameter of the model—the rock to the support stiffness ratio. At this stage of verification, the deformation modulus of the massif was varied in the range of 54,000–84,000 MPa. The characteristics of the concrete support remained unchanged and were determined according to [7]. The shaft’s support was modeled as a double-layer cylindrical shell where the outer layer is a 0.3 m thick concrete shell and the inner one is cast iron tubing support with a back side thickness of 60 mm according to its design [7]. The behavior of the vertical shaft concrete support was described by means of the Simplified Concrete Damage Plasticity (SCDP) model [34], based on the theories of plastic flow and damage mechanics, and the behavior of the cast iron tube shell was described as linear elastic. The SCDP model represents a set of equations to describe the elastic and plastic behavior of concrete and represents the evolution from tensile to compressive failure, simulating the cracking and failure processes of concrete. With the ratio of the deformation modulus of concrete to the rock mass

E_{c} / E_{r m} =

0.47, the deformation modulus of the historical rock mass was assumed to be 74,000 MPa. The result of model verification is presented in Figure 3.

The results of verification showed high convergence: it is obvious that the load on the support obtained during modeling almost completely matches the trend line of the values of field data, indicated in Table 5.5 of [7], which speaks to the fairness of the conducted studies and the correct selection of the relationship between the components of the main stresses in the massif.

According to the results of the verification of the model with in situ data of the studied rock massif, the transition was made to study the influence of various characteristics of fractured rock zones on the unfixed circular vertical excavation in order to predict dangerous zones, or zones of potential collapse in the places where the excavation crosses fractured zones. The criterion for determining the boundary of the potential collapse zone is the boundary of the plastic deformation zone (Figure 1).

The model geometry represents a 50 × 50 × 100 m massive rock with a vertical circular cavity in the center. The horizontal dimensions of the model exclude the influence of boundary conditions (at least 5 diameters of the cavity per each side). Boundary conditions of the model include the prohibition of orthogonal displacements on the outer edges of it. The stress state of the rock mass is formed by introducing a principal stress field by selected components (expression (16)). Geological disturbances (crushed zones with disintegrated rock) cross the vertical shaft in the middle. The thickness of the fractured rock zone varies from 2 to 8 m; the angle of inclination to the plane perpendicular to the axis of the excavation at the point of its intersection with the fractured rock zone varies from 0 to 60 degrees. A schematic representation of the model is presented in Figure 1. According to Equation (14), the plastic deformation zone size was determined without taking into account the inclination angle and crushed zone thickness for the investigated massif for comparison of the obtained data. In total, 147 numerical models were built.

3. Material Characteristics

The difference in the physical–mechanical characteristics of the fractured rock zone was accounted for by changing the geological strength index GSI in the interval from 15 to 30, which is equal to very poor and poor conditions of the surface according to the classification [31].

The physical and mechanical parameters of the rock massif and crushed disintegrated zone are presented in Table 2. The elastic–plastic Coulomb–Mohr model is used to describe the behavior of the rock massif.

4. Results

The results of the study are displayed in Figure 4, Figure 5, Figure 6 and Figure 7.

Figure 4, Figure 5, Figure 6 and Figure 7 illustrate the changes in the size of the plastic deformation zone associated with changes in the strength characteristics, thickness, and inclination angle of the fractured rock zone. The first group of graphs, (a), illustrates the relationship between the relative dimensionless size of the plastic deformation zone

ξ

and the GSI value. The second group of graphs, (b), illustrates the relationship between the plastic zone proportion size

ξ

and the inclination angle of the fractured rock zone.

All the diagrams demonstrate that higher GSI values correspond to smaller plastic deformation zone sizes. For example, with a GSI increase from 15 to 30, the proportion size of the fractured rock zone varies from 3.5 to 2.5 for the crushed zone 8 m thick with an inclination of 15° (Figure 7a). This pattern holds for all crush zone thicknesses from 2 to 8 m. The spike in the size of the plastic deformation zone is observed when the GSI of the fractured rock zone goes from 15 to 20.

As the inclination angle of the fractured rock zone rises to the plane perpendicular to the shaft axis, the size of the plastic deformation zone increases. The spike in the size of the plastic deformation zone is observed when the inclination angle of the fractured rock zone goes from 30 to 45 degrees or more. This trend is more evident at low GSI values. For example, for GSI 15, the size of the crushed zone increases rapidly from 3 (0°) to 4 (60°) for the crushed zone 8 m thick (Figure 7b).

When the thickness of the fractured rock zone changes from 2 to 6 m, the size of the plastic deformation zone increases in a regular manner. However, when it reaches the value of 8 m, plastic deformations explode, which is also explained by the increase in the impact of the fractured disintegrated rock mass’s own weight. The fact of expansion of the plastic deformation zone with the increasing inclination angle of the fractured rock zone is also noticeable, which allows us to conclude that the strain-weakening of the rock intensifies when the layer of disintegrated rocks has an inclined occurrence. At higher GSI values, this extension is less evident, emphasizing the damping effect of stronger rock massifs on stress propagation and, consequently, its deformations.

The biggest size of the plastic deformation zone in the vicinity of the excavation (

ξ

value is up to four) is observed when the following factors coincide: fractured rock zone thickness 8 m, inclination angle relative to the plane perpendicular to the axis of the excavation 60 degrees, and geological strength index GSI 15.

Nearly all obtained values of the relative size of the plastic deformation zone around the circular excavation exceed the values obtained by Equation (14). The convergence exists only at the crushing zone thickness of 2 m and inclination angle of up to 15 degrees.

5. Discussion

The results of numerical modeling (Figure 4, Figure 5, Figure 6 and Figure 7) allow us to draw conclusions about the degree and nature of the influence of parameters of the fractured rock zone on the stress–strain state of the massif in the vicinity of the vertical excavation.

In order to predict the size of potential collapse zones around the unfixed excavation, the results of calculations were analyzed using the second-order polynomial regression. The following dependence was obtained:

l_{p c z} = (13.73 + 0.278 m - 0.739 G S I - 0.109 α + 0.0299 m^{2} + 0.0106 {G S I}^{2} + 0.0017 α^{2} - 0.0081 m \cdot G S I + 0.0078 m \cdot α + 0.0022 G S I \cdot α) + R,

(22)

where

l_{p c z}

—maximum relative size of the plastic deformation zone;

m

—fractured rock zone thickness;

α

—fractured rock zone inclination angle;

G S I — G S I

index value of the fractured rock zone, and R—radius of the circular cavity.

The coefficient of determination R², which measures the proportion of explained variance of the target variable, amounted to 0.955. This means that the model explains 95.5% of the variation in the data.

The obtained expression (22) was then converted to a relative value to allow comparison of the results with earlier studies [32] (Figure 4a, Figure 5a, Figure 6a and Figure 7a, red dashed line). According to the results of the comparison of the two approaches to estimate the plastic deformation zone size, it can be concluded that the correlation of the obtained results exists only at small values of the crushed zone thickness and inclination angle (up to 2 m and up to 15 deg). Calculation results may differ up to two times (Figure 7a). As the inclination angle and thickness of the crushing zone rise, the discrepancy in the values of the plastic deformation zone increases; therefore, it is necessary to take into account these parameters of the disturbed zone in the calculations.

It is important to define restrictions when using expression (22): it is applicable in tectonically complicated fractured rock mass with brittle fracture and does not apply to other types of massive rock. It is practically applicable to the calculation of hazardous zones of plastic deformations when shafts cross fractured rock layers more than 2 m thick and an angle of inclination of more than 15 degrees.

6. Conclusions

The presented study proposes an approach to assess the influence of variable characteristics of disturbed crushed zones intersected by vertical excavation in the massif on the size of plastic deformation zones, which cause collapses into the excavated space. A theoretical Equation (22) is suggested for predicting the size of the failure zone around a circular excavation crossing the crushed zone in tectonically stressed massive rock, based on the strength and geometry of this zone.

A comparison with the existing methodology [32] for calculating the plastic deformation zone revealed that not accounting for crushed zone parameters may lead to an underestimation of the size of the potentially hazardous area up to 50% (Figure 7a), which may provide the appearance of an unexpected hazardous area during excavation, or insufficient support strength in the future. Therefore, when the inclination angle of the crushed zone is more than 15 degrees and its thickness is more than 2 m, it is recommended to use expression (22) to take into account the characteristics of crushed zones in the vicinity of the excavation.

The reliability of the results is supported by verification of the stress field specified in the numerical model with field data [7,43] and convergence of the proposed calculations under simple conditions (small crushed zone thickness and small inclination angle) with existing studies [32]. In the future, while constructing deep shafts in the study area (Talnakh deposit), experimental data will be accumulated on the basis of which it will be possible to verify the proposed approach by taking into account crushed zone parameters when constructing vertical shafts. During the construction, it will be necessary to refine the dependencies with monitoring data obtained by back analysis.

The dependencies obtained in this study are important for assessing the stability of mine excavations in complex mining and geological conditions, in particular, in tectonically disturbed areas with a disturbed structure and horizontal stress prevalence in the massive rock. The proposed Equation (22) is a theoretical approach to account for the size of the plastic deformation zones around circular excavations to develop further a methodology for designing vertical shafts’ supports crossing tectonically disturbed areas in rock mass.

Author Contributions

Conceptualization, P.A.D. and E.L.R.; methodology, E.L.R.; validation, P.A.D.; formal analysis, P.A.D.; investigation, E.L.R.; resources, E.L.R.; data curation, E.L.R.; writing—original draft preparation, E.L.R.; writing—review and editing, P.A.D.; visualization, E.L.R.; supervision, P.A.D.; project administration, P.A.D. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Definition of the influence zone of an opening intersecting the fractured rock zone: (a) design scheme; (b) stress fields in the plastic region around a circular opening, or area of potential collapse [compiled by authors]. Figure explanations:

l_{p c z} —

relative horizontal size of a potential collapse zone,

h_{p c z}

—absolute vertical size of a potential collapse zone;

m

—fractured rock zone thickness;

α

—fractured rock zone inclination angle.

l_{p c z} —

relative horizontal size of a potential collapse zone,

h_{p c z}

—absolute vertical size of a potential collapse zone;

m

—fractured rock zone thickness;

α

—fractured rock zone inclination angle.

Figure 2. Variation in stress in rocks [50] (data selected from [47,48]): (a) vertical stress variation with depth; (b) lateral stress coefficient variation (K) with depth. Red circles display values of the rock mass subject stress field components.

Figure 3. Model Verification [compiled by authors].

Figure 4. Plastic zone proportion size

ξ

for a fractured rock zone 2 m thick: (a) depending on the inclination angle of the fractured rock zone; (b) depending on the GSI parameter [compiled by authors].

Figure 4. Plastic zone proportion size

ξ

for a fractured rock zone 2 m thick: (a) depending on the inclination angle of the fractured rock zone; (b) depending on the GSI parameter [compiled by authors].

Figure 5. Plastic zone proportion size

ξ

for a fractured rock zone 4 m thick: (a) depending on the inclination angle of the fractured rock zone; (b) depending on the GSI parameter [compiled by authors].

Figure 5. Plastic zone proportion size

ξ

for a fractured rock zone 4 m thick: (a) depending on the inclination angle of the fractured rock zone; (b) depending on the GSI parameter [compiled by authors].

Figure 6. Plastic zone proportion size

ξ

for a fractured rock zone 6 m thick: (a) depending on the inclination angle of the fractured rock zone; (b) depending on the GSI parameter [compiled by authors].

Figure 6. Plastic zone proportion size

ξ

for a fractured rock zone 6 m thick: (a) depending on the inclination angle of the fractured rock zone; (b) depending on the GSI parameter [compiled by authors].

Figure 7. Plastic zone proportion size

ξ

for a fractured rock zone 8 m thick: (a) depending on the inclination angle of the fractured rock zone; (b) depending on the GSI parameter [compiled by authors].

Figure 7. Plastic zone proportion size

ξ

for a fractured rock zone 8 m thick: (a) depending on the inclination angle of the fractured rock zone; (b) depending on the GSI parameter [compiled by authors].

Table 1. The ranges of investigated parameters variation [compiled by authors].

Crushed Rock Zone Parameter	Unit	Ranges	Increment Step
GSI	-	15–30	2.5
thickness	m	2–8	2
angle of inclination to the intersected excavation	$Deg, °$	0–60	15

Table 2. Physical and mechanical parameters of the massif and fractured rock zone [compiled by authors].

Rock Parameter	Unit	Rock Massif	GSI 15	GSI 17.5	GSI 20	GSI 22.5	GSI 25	GSI 27.7	GSI 30
Modulus of deformation of rock, E	MPa	74,000	1782	1866	1963	2084	2237	2426	2661
$Compressive strength, σ_{c}$	MPa	115	4.1	4.7	5.3	6.0	6.7	7.4	8.1
Cohesion, C	MPa	15.2	0.8	0.9	1.1	1.2	1.3	1.4	1.6
Internal friction angle, φ	$Deg, °$	50	17	19	20	21	23	24	25
Poisson’s ratio, μ	-	0.25	0.38	0.37	0.37	0.36	0.36	0.35	0.35

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Demenkov, P.A.; Romanova, E.L. Regularities of Plastic Deformation Zone Formation Around Unsupported Shafts in Tectonically Disturbed Massive Rock. Geosciences 2025, 15, 23. https://doi.org/10.3390/geosciences15010023

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Demenkov PA, Romanova EL. Regularities of Plastic Deformation Zone Formation Around Unsupported Shafts in Tectonically Disturbed Massive Rock. Geosciences. 2025; 15(1):23. https://doi.org/10.3390/geosciences15010023

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Demenkov, Petr A., and Ekaterina L. Romanova. 2025. "Regularities of Plastic Deformation Zone Formation Around Unsupported Shafts in Tectonically Disturbed Massive Rock" Geosciences 15, no. 1: 23. https://doi.org/10.3390/geosciences15010023

APA Style

Demenkov, P. A., & Romanova, E. L. (2025). Regularities of Plastic Deformation Zone Formation Around Unsupported Shafts in Tectonically Disturbed Massive Rock. Geosciences, 15(1), 23. https://doi.org/10.3390/geosciences15010023

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Regularities of Plastic Deformation Zone Formation Around Unsupported Shafts in Tectonically Disturbed Massive Rock

Abstract

1. Introduction

2. Materials and Methods

3. Material Characteristics

4. Results

5. Discussion

6. Conclusions

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