Prediction of the limit state and dilatancy of rocks around mine workings

Funding

The work was financially supported by a grant from the Russian Science Foundation (Project N 23-17-00144).

Introduction

With increasing depth of mineral deposit exploitation, a decrease in the stability of rock exposures is observed [1-3], accompanied by the development of plastic strains in the limit state zone [4]. The solution to the problem of predicting the limit state zone and displacements around rock exposures located in an elastoplastic rock mass is of considerable interest in the development of underground space and mineral deposits [3, 4]. This involves: studying the mechanical properties of rocks [5-8]; investigating the rock mass  characteristics [9, 10], formulating new rock strength criteria [11-13]; developing models to predict the geomechanical state of the rock mass around mine workings – both based on its natural stress state and taking into account internal pressure that may arise in the mined-out area. Based on the solutions obtained, measures are developed to bring rock exposures to a stable state [14].

The modelling of limit state zones around mine workings in a plane‑strain formulation reduces to solving two-dimensional elastoplastic problems [15, 16] with a priori undetermined boundary that separates the elastic and plastic regions in the rock mass. In these regions, the strain of the media is governed by different sets of equations. Analytical solution of such problems poses certain difficulties, since the shape and size of the plastic zone are not known beforehand and must be determined as part of the solution [17-19].

The first exact solution to the elastoplastic problem of stress distribution around a circular hole in an infinite plane was provided by L.A.Galin. This solution can be used to estimate limit state zones around a working in a rock mass subjected to constant stresses at infinity and with normal forces applied at its contour. Displacements in the plastic zone for Galin’s problem were determined using D.D.Ivlev’s method of a small parameter. N.I.Ostrosablin obtained an exact solution of the system of equations for displacements in this zone. Further studies of the plane elastoplastic problem involved changing the hole shape from circular to elliptical.

Most existing analytical and semi‑analytical solutions for predicting the size of limit state zones around a mine working are based on considering an infinite plane in the vicinity of a rock exposure as an elastoplastic medium, using the Coulomb plasticity condition [20, 21]. A number of analytical solutions were obtained for predicting the stress-strain state using nonlinear strength criteria for rocks [22-25]. The main focus was on the formation of the stress-strain state around a circular mine working. Solution to this problem was derived for an ideal elastoplastic medium [26-28], an elastoplastic medium accounting for isotropic hardening or softening [29-31]. Analytical dependencies were presented for predicting the size of the limit state zone and strains of the rock exposure contour – both for small and large strains [32, 33]. Some studies addressed the prediction of the stress-strain state around a working located in an anisotropic rock mass [34] or in a viscoplastic medium [35]. Recent works found wide application in predicting the stress-strain state of rock masses prone to large strains. Specific problems include predicting the stress-strain state of a rock mass around non-circular tunnels [36] or predicting the state around twin tunnels [37].

Despite these results, analytical solutions were obtained for simple strength conditions, predominantly linear ones. It is known that many rocks exhibit nonlinear envelopes of limit Mohr’s circles. This fact should be considered when performing geomechanical predictions. The present work focuses on deriving an analytical solution and developing a numerical model to predict the stress-strain state of a rock mass around a circular mine working, based on A.N.Stavrogin’s strength criterion.

Research methods

A.N.Stavrogin’s strength criterion

The shape of the envelope of the limit Mohr’s circles for rock is curvilinear, which is related to the complex nature of its failure under different types of stress states. In special cases, the envelope can be simplified and represented as a linear relationship, referred to as the Coulomb strength criterion. To describe the nonlinear envelope, [25] proposes to use various curves, including a parabola, an exponential function, a cycloid, and others – either in combination with straight-line segments or without them. A.N.Stavrogin proposed strength criteria for the limit elastic state and the strength limit of the following form:

where equation (1) is the condition for elastic state limits; equation (2) is the condition for strength limits; $\begin{array}{c}{\text{τ}}_{\text{п}}=\left({\sigma }_1^{\text{п}}-{\sigma }_3\right)/2\end{array}$ are the elastic and strength limits; τ_e⁰, τ_p⁰ are constants representing the elastic limit and strength limit under uniaxial compression; B, A are constants reflecting the hardening of rocks with increasing hydrostatic pressure; C=σ₃/σ₁ is a parameter characterizing the type of stress state.

From the equality condition τ_e=τ_p, we find the coordinate:

The value C_p, substituted into (1) or (2), allows obtaining the second coordinate of the intersection point of the limit curves lnτ_p. After that, we can calculate the tensile strength using the formula

Statement of the problem of predicting the limit state zone around a mine working

The prediction of the limit state zone around a mine working is considered in the plane strain formulation (Fig.1). The size of the mine working is taken as r₀, and the size of the computational domain is R >> r. The external loading on the boundaries of the computational domain is specified as pressure equal to γH (vertical component) and  λγH (horizontal component). As a result of the external loading, a stress state forms in the computational domain, defined by the components of the stress tensor σ_r, σ_θ, τ_rθ. When determining the size of the limit state zone, it is assumed that the zone size is R_L. In this case, the analytical solution is meaningful only if the limit state zone fully encompasses the contour of the mine working. The numerical solution does not have such limitations.

A.N.Stavrogin’s strength criterion is not implemented in software products for strength computations using numerical analysis methods. Plasticity conditions for geomaterials in such software are limited to linear forms. To incorporate A.N.Stavrogin’s plasticity condition, a special procedure must be developed. This procedure should update the stress state during elastoplastic strain of the material. By dividing A.N.Stavrogin’s nonlinear plasticity condition into a set of linear segments, its implementation can be achieved without developing a full model that includes all procedures characteristic of the numerical theory of plastic flow.

As the linear plasticity condition, the widely validated Coulomb – Mohr plasticity criterion is adopted:

where c is the adhesion; φ is the internal friction angle.

To convert A.N.Stavrogin’s strength parameters to equivalent Coulomb strength parameters (i.e., equivalent cohesion and internal friction angle), we use an approximation of the exponential envelope of the limit stress circles (2) by a piecewise-linear function. The accuracy of the approximation is determined solely by the number of points selected on the original exponential envelope.

For two consecutive points on the exponential envelope, which represent the initial and final points of a single linear segment on the approximating piecewise-linear envelope (with constant values of cohesion and internal friction angle), the values are determined according to the expressions:

where τ_M, τ_N, σ_M, σ_N are the coordinates of adjacent points M and N on the exponential envelope.

Research results

Analytical solution for predicting the stress-strain state of a rock mass using A.N.Stavrogin’s criterion around a circular mine working

Research [26] considers the issues of modelling the zone of plastic strains around a circular mine working for the case of a hydrostatic natural stress state. The boundaries of the limit state zone are determined by A.N.Stavrogin’s plasticity condition.

The size of the limit state zone is obtained using the method of a small parameter and can be written as:

where ε=0.5(1-λ); θ is the angular coordinate;

λ₁=0.5(1+λ); H – depth of the working; γ – specific weight of the rock mass; R₀ – radius of the working; $N_1=\ln \frac{{\lambda }_1\gamma Hk}{{\tau }_{\text{п}}^{\text{0}}}.$

The value k=a/r₀² is determined from the equation

In formulas (9) the constant С₁ is equal to

τ₁ is determined from the equation

where p₀ is the support back-pressure.

The stress components in the elastic zone around the working are expressed as:

where a, b, d₂ are coefficients accounting for the boundary conditions.

Let us now proceed to determine the displacements around a mine working for the case of a geostatic natural stress state, when the lateral pressure coefficient λ differs from unity. In this case, displacements in the rock mass around the working depend on the radial coordinate r and the angular coordinate φ. When solving the problem, we restrict ourselves to a linear approximation of dilatancy.

The dilatancy condition in polar coordinates can be written as:

where u and v are the radial and circumferential displacements.

Experimental data show that for some rock types, the coefficient d in linear approximation varies from 0 to –0.86.

Due to isotropy, the volume change during plastic strain of rocks does not depend on direction and does not affect shear strains. Therefore, in the limit zone near the working contour, the shear strains

Then, to determine the displacements u and v, we must solve equations (16) and (17). For this, we use the semi-inverse solution method. Restricting ourselves to a first approximation for the small parameter ε, we will seek the displacements in the form:

where u₀ is a constant function; f (r), φ(r) are the desired functions.

By introducing functions (18) and (19) into equations (16) and (17) and equating expressions for the same powers of ε, we obtain the system

Let us reduce this system of equations to the Euler differential equation:

where $N_2=\frac{1-d}{1+d}$ and the characteristic equation is written as

has roots

Given that the coefficient d, according to experimental data, is in the range -√3/2<d≤3, we will consider only the case of complex roots. Then the function φ(r) of equation (21) will be written as:

Determining from equation (25) φ(r) and substituting (25) into (18) and (19), we write the displacements u and v as follows:

where

In the elastic zone, stresses and displacements are related by Hooke’s law:

where E, ν is the modulus of elasticity and Poisson’s ratio of the rock mass, respectively.

We find the displacements in the specified zone by introducing stresses (13) and (14) into (28) and (29), integrating the resulting expressions, and limiting ourselves to a first approximation for the small parameter ε, we obtain:

where

The unknown coefficients u₀, c₁, c₂ are determined from the continuity conditions for the displacements u and ν at the boundary of the region of limit strains (equations (26) and (27). By pairwise equating the expressions (26) and (30), (27) and (31), we obtain the following:

where

From system (34) we obtain:

The displacement of the working contour is obtained from expression (30) by substituting the derived coefficients c₁, c₂, and r = 1:

where

Discussion

To assess the applicability of A.N.Stavrogin’s criterion for describing rock strength, we consider test results obtained under triaxial stress conditions according to the Karman scheme (σ₁ > σ₂ = σ₃). Figure 2, a shows the complete stress-strain diagrams of the rock obtained at different levels of lateral pressure. The horizontal axis shows longitudinal (axial) strain ε₁ and lateral strain ε₃. The vertical axis shows the magnitude of the major axial stress σ₁. Next to each curve, the values of parameter C are indicated – this parameter characterizes the stress state under which the laboratory experiment was conducted. Figure 2, b, c presents the experimental plots of limit elastic states, limit strength states in the coordinates (ln τ – С), using marble sample data.

The experimental dependencies show that the studied rocks exhibited deviations from Hooke’s law during strain. The largest deviations were observed in samples of two varieties of white marble with identical mineral composition (marble N 1 and marble N 2). The results of processing expe-rimental studies on marble samples are presented in Fig.3. Analysis of the experimental results allowed determining principal relative elastic strains ε₁^e, ε₂^e=ε₃^e, inelastic strains ε₁^p=ε₁-ε₁^e, ε₂^p=ε₂-ε₂^e, as well as total θ=ε₁+2ε₂, elastic θ^e=ε₁^e+2ε₂^e, and inelastic θ^p=ε₁^p+2ε₂^p=θ-θ^e volume change.

The coefficient of inelastic lateral strain is determined by the formula

In the studied rocks, the dependence of  θ on σ shows deviations of the curves (at different C values) from the hydrostatic compression line. This indicates the formation of microcracks in the tested samples. The strains were divided into elastic and residual using the method described above. Subsequently, dependencies between residual strains and the stress state parameter C were constructed.

The values of the ultimate residual volume change θ^p at the failure stress level, as a function of parameter C, are presented in Fig.2. Similar dependencies were obtained for other tested rocks. For all tested rocks, a characteristic feature is the presence of a maximum in volume change, the location of which along the C axis depends on the specific rock properties. For marble N 1, the value of  θ^p at the maximum was the highest among all tested rocks. The coefficient of lateral residual strain μ, at C values close to zero, reaches its highest value in all cases – and for some rocks it exceeds 2. As C increases, the value of μ gradually decreases and tends to 0.5. When μ = 0.5, the residual volume change equals zero. A similar dependence is observed for the other investigated rocks.

At C values where μ → 0.5 and θ^p → 0, the shear angle tends to 45°. Under uniaxial compression, α is significantly less than 45°. Although individual data points are scattered, the overall dependence of  α on C is clearly observable in all cases. The absolute values of limit strain ε₁^p increase, indicating a rise in the material’s plasticity.

The experimental results obtained cannot be explained by assuming that the nature of residual strains in rocks consists solely of shear processes. Evidence against this approach includes: an increase in  θ^p and values of the coefficient μ far exceeding 0.5. The deviation of the shear angle α from 45° is only formally explained in Mohr’s theory, without revealing its physical essence.

The listed experimental results can be explained based on a simplified hypothetical scheme of the strain and failure process in an isotropic material with heterogeneous structure. When a sample deformed by principal stresses σ₁ and σ₂ fails, two features form: a macroscopic shear plane inclined to the sample axis at an angle α and tensile microcracks that develop along planes normal to the principal stresses. The formation of negative dilatancy (loosening) is associated with the development of tensile microcracks. The decrease in dilatancy with increasing C as μ → 0.5 is explained by the closure of tensile microcracks.

Along with strain-induced dilatancy, which is associated with strain and failure processes during stress relief, the initial effective porosity of rocks also influences dilatancy. Analysis of experimental data on dilatancy changes in marble N 1 shows that dilatancy reaches a maximum at C = 0.313 and is negative – i.e., rock loosening occurs. When C varies from 0 to 0.313, dilatancy increases; in the range from 0.313 to 0.5, dilatancy decreases to zero.

Experimental data for various rock types show that plastic dilatancy depends on C and is a nonlinear function:

Considering the relationship between C and the stresses σ and τ, the dilatancy condition can be written as:

Experimental data show that rock dilatancy is proportional to the maximum shear strain γmax=ε₁^p-ε₃^p and can be described by the following relationship:

where λ(C) is a known function determined from experimental data.

For some rock types, the experimental dependence is close to linear (Fig.4). For such rocks, the generalized expression for dilatancy can be written as

where d, d₁ are the constants of linear interpolation for the dependence of plastic dilatancy on the maximum shear.

Let us consider the issues of using Stavrogin’s criterion to solve applied problems. Based on the presented theoretical provisions for predicting limit state zones in the rock mass around a mine working and displacements of the rock contour, we propose an algorithm for modelling them using the derived analytical formulas that consider Stavrogin’s criterion (see Table) in relation to a test problem. It is necessary to estimate the dimensions of the limit state zone, radial displacements of the contour of an unsupported circular mine working with the following parameters: excavation radius r₀ = 5 m (see Fig.1); depth H = 1000 m from the ground surface; rock mass with the following physical and mechanical properties γ = 25 kN/m³; E₀ = 1000 MPa; ν = 0.3; τ_p⁰= 8 MPa; A = 5. The problem is solved for lateral pressure coefficients in the rock mass of 0.6, 0.8, and 1.0. Additionally, the following variants are considered: a rock mass that is not subject to dilatancy during plastic strain; a dilating rock mass with parameters d = –0.268 and d₁ = 0.615. The stress state is specified through the horizontal σ_x and vertical σ_y components.

Algorithm for predicting the stress-strain state of a rock mass around a mine working

The results of modelling the limit state zone around the mine working are presented in Fig.5. Based on the equations used in the modelling, it follows that dilatancy parameters – within the hypotheses embedded in the analytical methodology – do not affect the size of the limit state zone. Therefore, only the influence of the lateral pressure coefficient on this size was examined. For convenience of analysis, the linear dimensions of the limit state zone are presented in the graphical dependencies as fractions of the working radius.

The largest dimension of the limit state zone under a gravity field of natural stresses is oriented horizontally: the lower the value of the lateral pressure coefficient λ, the more elongated the zone becomes in this direction. Analysis of the performed computations shows that full envelopment of the mine working by the limit state zone is achieved only at lateral pressure coefficients of at least 0.6. This fact limits the applicability range of the analytical methodology.

The analytical methodology presented in this work is characterized by a linear inverse dependence of the maximum linear dimension of the limit state zone on the lateral pressure coefficient and on the mine working depth. This dependence is confirmed by the results presented in Fig.5, b.

The results of modelling displacements of the mine working contour, both without and with accounting for rock dilatancy, are presented in Fig.6, a, b. For convenience of result analysis, the displacement values in the graphs are normalized to the radius of the mine working. Under conditions of a gravity stress field, the greatest radial displacement values occur at the crown of the mine working when the lateral pressure coefficients differ from unity. Accounting for dilatancy in the forecast of radial displacements leads to a significant increase in their magnitude; however, the relationships with the dilatancy parameters d and d₁ are not identical. Analysis shows that the modelled value of radial displacements of the contour is nonlinearly inversely proportional to the parameter d and linearly directly proportional to the parameter d₁.

Implementing A.N.Stavrogin’s strength criterion within numerical modelling allows significantly expanding the range of solvable problems in geomechanics and mechanics of underground structures. Figure 6, c illustrates the process of forming a zone of limit state around the mine working. Due to the iterative nature of computations when obtaining a numerical solution, each image corresponds to a specific portion of the total stresses implemented in the model.

At the initial stages of solving the problem, with small changes in the stress‑strain state of the rock mass, the zone of limit state has an open shape (Fig.6, d), i.e., it does not fully encompass the entire working contour. Such a pattern cannot be obtained using the analytical solution discussed earlier. Consequently, the numerical solution has a broader range of applicability and can be used for forecasting tasks in low-stress or high-strength rock masses, as well as in conditions with low lateral pressure coefficients λ < 0.6.

Let us compare the results of modelling radial displacements obtained using the analytical method and the finite element method. For clarity, the diagrams of displacement of the mine working contour, obtained when solving the problem without accounting for rock dilatancy, are superimposed in Fig.6, b, c. If we compare the forecasted radial displacements by their extreme values, the discrepancy between the forecast data ranges from 3 to 11 %. On average, along the perimeter of the mine working contour, the discrepancy falls within the 6-7 % range. Given the assumptions adopted when deriving the equations for the analytical solution and the simplifications embedded in transforming A.N.Stavrogin’s nonlinear strength condition into a piecewise-linear function within the user subroutine for numerical solution, this result can be considered acceptable, and the convergence of the two solutions can be deemed satisfactory.

It is important to note that within the framework of the numerical solution, it was not possible to account for plastic dilatancy of rocks, since the adopted rock strain model does not allow reproducing phenomena characteristic of a rock mass. Consequently, the model should be improved by modifying the plastic potential function depending on the achieved values of shear and volumetric strains.

Conclusion

The improvement of methods for solving problems in geomechanics and mechanics of underground structures aims to enhance forecast reliability through the gradual complication of applied models and approaches. This is achieved by systematically abandoning certain assumptions and simplifications. The fact of a nonlinear relationship between average stresses and rock strength was found long ago, but its implementation in practical problems has become feasible only recently due to significant advancements in the computational capabilities of numerical methods. Implementing A.N.Stavrogin’s strength criterion in numerical modelling enables obtaining solutions for predicting displacements of mine working’s contours and sizes of limit state zones in their vicinity, which are comparable in accuracy to analytical solutions. Such solutions do not have significant limitations in terms of applicability that are typical for analytical solutions and stem from initial premises embedded in the derivation of equations. Furthermore, the finite element method allows solving problems within a spatial formulation, which is unattainable using analytical methods.

During the present work, we identified the necessity to refine the rock strain model into a form that fully accounts for the features of volumetric strain under plastic shear. This research direction constitutes a natural continuation of the work.

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