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Cavity Expansion in Rock Masses Obeying the “Hoek–Brown” Failure Criterion

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Abstract

An unified approach is presented for the analysis of the expansion of both cylindrical and spherical cavities in an infinite elastic–perfectly plastic “Hoek–Brown” (H–B) material. The H–B failure criterion expressed in scaled form is adopted with a plastic flow rule characterized by a constant dilatancy angle \(\psi\). Closed form expressions are given for the extent of the plastic region and the related stress. Solutions of the displacement field in the plastic region are provided based on both small-strain and large-strain theories. An original relationship between the cavity pressure and its expansion is derived. The developed closed-form solutions are validated employing the finite element method. For comparison purposes, an approximate solution is presented by neglecting the elastic strains in the plastic region which reveals that the assumption of no elastic strains does not influence the results for strong rocks in contrast with weak rocks. For practical purposes, design charts are provided allowing easy and accurate estimates of the limit pressure for cavity expansion in rock masses. The cavity expansion solution is finally validated against results obtained using the Finite Element modelling.

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Abbreviations

\(r_{i}\) :

Instant cavity radius

\(r_{i0}\) :

Original cavity radius

\(r\) :

Radial coordinate

\(r_{p}\) :

Plastic radius

\(u\) :

Radial displacement

\(u_{\text{EPB}}\) :

Radial displacement at the elastic plastic boundary

\(P_{i}\) :

Internal cavity pressure

\(P_{y}\) :

Yield pressure

\(P_{0}\) :

Far field pressure

\(P_{ \lim }\) :

Limit pressure

\({\text{GSI}}\) :

Geological strength index of the rock

\(m_{i}\) :

Strength parameter of the intact rock

\(D\) :

Disturbance factor of the rock

\(s, a, m_{b}\) :

Hoek–Brown-derived parameters

\(E_{i}\) :

Deformation modulus of the intact rock

\(E_{rm}\) :

Deformation modulus of the rock

\(G\) :

Shear modulus

\(I_{r}\) :

Rigidity index

\(\varepsilon_{r}^{e}\) :

Elastic radial strain

\(\varepsilon_{\theta }^{e}\) :

Elastic circumferential strain

\(\varepsilon_{r}^{p}\) :

Plastic radial strain

\(\varepsilon_{\theta }^{p}\) :

Plastic circumferential strain

\(\sigma^{\prime}_{1}\) :

Major principal stress

\(\sigma^{\prime}_{3}\) :

Minor principal stress

\(\sigma_{r}\) :

Radial stress

\(\sigma_{\theta }\) :

Circumferential stress

\(\sigma_{ci}\) :

Uniaxial compressive stress of the intact rock

\(\sigma_{c}\) :

Uniaxial compressive strength

\(\sigma_{t}\) :

Uniaxial tensile stress of the intact rock

\(\nu\) :

Poison ratio

\(\psi\) :

Dilatancy angle of the rock mass

\(\omega\) :

Dilatancy coefficient

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Authors and Affiliations

  1. Institute of Mechanics, Materials and Civil Engineering (iMMC), Civil and Environmental Engineering Department, Université Catholique de Louvain (UCLouvain), Place du Levant 1, 1348, Louvain La Neuve, Belgium

    Haythem Gharsallaoui, Mustafa Jafari & Alain Holeyman

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  1. Haythem Gharsallaoui
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  2. Mustafa Jafari
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  3. Alain Holeyman
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Correspondence to Haythem Gharsallaoui.

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Gharsallaoui, H., Jafari, M. & Holeyman, A. Cavity Expansion in Rock Masses Obeying the “Hoek–Brown” Failure Criterion. Rock Mech Rock Eng 53, 927–941 (2020). https://doi.org/10.1007/s00603-019-01920-7

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