A semi-analytical elastic stress–displacement solution for notched circular openings in rocks

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Abstract

A semi-analytical plane elasticity solution of the circular hole with diametrically opposite notches in a homogeneous and isotropic geomaterial is presented. This solution is based on: (i) the evaluation of the conformal mapping function of a hole of prescribed shape by an appropriate numerical scheme and (ii) the closed-form solutions of the Kolosov–Muskhelishvili complex potentials. For the particular case of circular notches––which resemble to the circular cavity breakout in rocks––it is demonstrated that numerical results pertaining to boundary stresses and displacements predicted by the finite differences model FLAC2D, as well as previous analytical results referring to the stress-concentration-factor, are in agreement with analytical results. It is also illustrated that the solution may be easily applied to non-rounded diametrically opposite notch geometries, such as “dog-eared” breakouts by properly selecting the respective conformal mapping function via the methodology presented herein. By employing a stress-mean-value brittle failure criterion that takes into account the stress-gradient effect in the vicinity of the curved surfaces in rock as well as the present semi-analytical solution, it is found that a notched hole, e.g. borehole or tunnel breakout, may exhibit stable propagation. The practical significance of the proposed solution lies in the fact that it can be used as a quick-solver for back-analysis of borehole breakout images obtained in situ via a televiewer for the estimation of the orientation and magnitude of in situ stresses and of strain–stress measurements in laboratory tests.

Introduction

Notched shaped configurations of rock cavities such as that shown in Fig. 1 are often encountered in mining, petroleum and geophysical engineering practice. They are present in deep boreholes as breakouts and in underground openings exhibiting sidewall spalling. In Fig. 1, the formation of the breakout cross-section by successive spalling can be clearly seen. Field practice shows that the geometry of such notched configurations can greatly influence the load bearing capacity of the rock structure and consequently its stability.

The excavation process reduces the confining stress to zero on the boundary of the opening. Thus, tunnel and borehole breakouts occur under high in situ stresses when the tangential stress at the excavation wall overcomes the uniaxial compressive strength of the rock. Under unequal principal stresses rock failure that is manifested by axial splitting and spalling fractures that are often extensional in nature, is observed in two diametrically opposed zones parallel to the minor principal stress forming a notched shaped excavation. An idealized breakout geometry is depicted in Fig. 2 where the initial breakout is shown by the dashed curve. Such breakouts are valuable indicators of the direction of action of the minimum compressive stress, while their size and shape, recorded via dipmeters and more precisely now by televiewers, may provide information about the magnitudes of the maximum and minimum stresses relative to the strength of the rock. As the in situ strength of rock and its state of stress are difficult to determine at great depth, observations of the size and shape of the breakouts and conditions under which they form could lead to estimation of these parameters, provided that a thorough understanding of the mechanisms involved in breakout formation become clear. According to Guisiat and Haimson (1992), Hottman et al. (1979) were perhaps the first investigators to attempt to estimate magnitudes of horizontal principal stresses from borehole breakouts. The good agreement of their estimation with other stress indicators led both Gough and Bell (1982) and Zoback et al. (1985) to investigate the mechanisms of borehole breakouts from field and laboratory experiments. They concluded that size, shape and depth could be predicted by using Kirsch’s (1898) elasticity solution and a linear Mohr–Coulomb failure criterion. In another more elaborate type of model (Zheng et al., 1988) the stresses around the borehole were compared to those required to cause failure according to a spalling criterion for the immediate zone around the borehole wall and the Mohr–Coulomb criterion for the remaining rock.

Thus, underground excavations in rock in the form of deep boreholes and tunnels or caverns display usually some degree of deviation from the circular shape either due to optimization of their shape or due to sidewall failure of the rock mass. Almost all relevant modeling work has been based on Kirsch’s solution for elastic stresses around circular holes. Since a circular solution cannot describe an actual notched-hole geometry it is evident that a completely new solution is necessary. As it is noted by Cheatham (1993) an exact elasticity solution for a breakout configuration that is quite similar to the initial circular cavity breakout shape (Fig. 2) has been published by Mitchell (1966). By using the method of complex potential funtions φ(z), ψ(z) of Muskhelishvili (1963), Mitchell (1966) has solved the stress-concentration-problem for a doubly symmetrical hole whose boundary consists of three intersecting circles. However, it would appear that the evaluation of the full-field stress distribution around the hole, and consequently the stress-gradient effect, has not been pursued by the author since he did not present the solution for the second potential function ψ(z) of the complex variable z=x+iy. Therefore, at this point it should be made clear that Mitchell has not presented the complete solution for the elastic stresses and displacements1 around the doubly symmetric hole. Further, the same investigator did not consider the effect of internal pressure. However, one may argue that the exercise of finding the complete representation of the stresses and displacements around notched openings analytically is still warranted in view of availability of many accurate and easy to use finite element, finite difference, or boundary element computer codes. In order to reply to this argument, we simply refer here to the statement made by Carranza-Torres and Fairhurst (1999):

…Although the complex geometries of many geotechnical design problems dictate the use of numerical modeling to provide more realistic results than those from classical analytical solutions, the insight into the general nature of the solution (influence of the variables involved, etc.) that can be gained from the classical solution is an important attribute that should not be overlooked. Some degree of simplification is always needed in formulating a design analysis and it is essential that the design engineer be able to assess the general correctness of a numerical analysis wherever possible. The closed-form results provide a valuable means of making this assessment…

Based on the above considerations, a semi-analytical solution of the notched circular cylindrical opening in elastic, isotropic and homogeneous rock is presented in Section 3, that is based on the powerful conformal mapping method2 in conjunction with a numerical scheme (Section 2) and closed-form complex function solutions. Subsequently in Section 4, Mitchell’s stress-concentration-solution for the doubly symmetric hole is compared successfully with the proposed solution. Moreover, stress and displacement analysis results of the present solution are compared with those predicted by the FLAC2D finite differences code. In the same section, it is also illustrated that this solution may be applied to solve non-rounded diametrically opposed notches, such as the dog-eared (wedge-shaped) notches. In Section 5, the effect of internal pressure on borehole stability is demonstrated. Also, a theoretical background is developed, based on the stress-gradient effect due to stress concentrations caused by extremely curved surfaces, that may lead to a better understanding of the stability of cavities in rock. The capability of this theory to capture the stability of breakout formation is also demonstrated in this section. Finally, Section 6 presents the main conclusions of this work.

Section snippets

The conformal mapping representation of notched openings

The methodology starts with the conformal mapping3 of the boundary of the notched hole C and its exterior region S (Fig. 3a) into the interior of the circle with unit radius (region Σ in Fig. 3b). The position of every point in the physical z-plane with z=x+iy=reia is mapped into the unit circle in

Solution of the traction boundary value problem of the notched hole

In the frame of the theory of complex potential functions, the boundary condition for the case that forces are prescribed along the contour γ of the unit disk takes the form (Muskhelishvili, 1963)φ0(ζ)+12πiγω(σ)ω(σ)φ0(σ)dσσ−ζ=12πiγf0dσσ−ζin which σ=eiθ denotes an arbitrary point of the contour γ, andf0(σ)=f(σ)−Γα0σ+ω(σ)ω(σ)Γᾱ0σ2Γᾱ0σAlso,Γ=14(N1+N2),Γ=−12(N1−N2)with N1, N2 referring to the principal stresses at infinity acting along Ox- and Oy-axes, respectively (e.g. Fig. 3a). The

Comparison of the closed-form solution for the doubly symmetric hole with existing analytical stress-concentration solutions and the FLAC finite differences code

In order to compare the present solution with the Mitchell’s (1966) solution pertaining to the stress-concentration factor (SCF) at the notch, we consider the doubly symmetric hole that is subjected to far-field uniaxial compression N2. Graphical results pertaining to the variation of the SCF, Kt=σθ(θ=0°)/N2, with the radii ratio R2/R1, are presented in Fig. 7. As it was noted by Mitchell, for very small values of the notch radius, i.e. R2→0, the local stress distribution will be the same as

Effect of internal pressure on borehole collapse

One of the borehole stability controlling parameters is the borehole pressure, as it influences the stress-concentration effect directly. In this paragraph we investigate the stress-concentration effect in circular and notched-circular holes by employing the semi-analytical solution presented in Section 3. When the absolute pressure P that is exerted on the boundary of a borehole drops, the tangential stress at the wall increases, and it results in a borehole collapse if this stress exceeds the

Conclusions

A semi-analytical solution of the notched hole has been presented that is based on the conformal mapping representation combined with Muskhelishvili complex potentials. The solution can be used for the optimum design of openings in rocks (i.e. boreholes, tunnels, galleries etc.) under given in situ stress field, rock deformability and strength data, or for back-analysis of borehole breakout data for the estimation of in situ stresses. It is a valuable tool for conceptual understanding of how

Acknowledgements

The authors gratefully acknowledge the comments made by Dr. C. Carranza-Torres as well as the financial support of the European Union Environment Programme 3F––Corinth (Faults, Fractures & Fluids) under contract ENK6-2000-0056.

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