Modelling brittle failure of rock

Abstract

Observations of brittle failure at the laboratory scale indicate that the brittle failure process involves the initiation, growth, and accumulation of micro-cracks. Around underground openings, observations have revealed that brittle failure is mainly a process of progressive slabbing resulting in a revised stable geometry that in many cases take the form of V-shaped notches. Continuum models with traditional failure criteria (e.g. Hoek–Brown or Mohr–Coulomb) based on the simultaneous mobilization of cohesive and frictional strength components have not been successful in predicting the extent and depth of brittle failure. This paper presents a continuum modelling approach that captures an essential component of brittle rock mass failure, that is, cohesion weakening and frictional strengthening (CWFS) as function of rock damage or plastic strain.

Introduction

Brittle failure is the product of the creation, growth and accumulation of micro- and macro-cracks. Many researchers have reported slabbing and spalling as a dominant failure mode around underground excavations in massive to moderately jointed rock masses subjected to high in situ stresses [1], [2], [3]. Unlike openings at shallow depth, or at low in situ stress, in which failure is controlled by discontinuities, at greater depth, the extent and depth of failure is predominantly a function of the in situ stress magnitudes relative to the rock mass strength [4], i.e., the stress level.

Traditional approaches of modelling rock mass failure are often based on a linear Mohr–Coulomb failure criterion or on a non-linear criteria such as the Hoek–Brown failure criterion.

The generalized Hoek–Brown failure criterion for jointed rock masses is defined by

$$\text{σ′}{}_1\text{=σ′}{}_3\text{+σ}{}_{\mathrm{<mtext>ci</mtext>}}\text{m}{}_{\mathrm{<mtext>b</mtext>}}\text{σ′}{}_3\text{σ}{}_{\mathrm{<mtext>ci</mtext>}}\text{+s}{}^{\mathrm{a}}\text{,}$$

where σ₁′ and σ₃′ are the maximum and minimum effective stresses at failure, m_b is the value of the Hoek–Brown constant m for the rock mass, s and a are constants which depend upon the rock mass characteristics, and σ_ci is the uniaxial compressive strength of the intact rock.

The Coulomb criterion, relating normal and shear stresses,

$$\text{τ=c+σ′}\text{tan}\text{φ}$$

can be determined by the method proposed by Hoek and Brown [5]. It is defined by

$$\text{τ=Aσ}{}_{\mathrm{<mtext>ci</mtext>}}\text{σ′}{}_{\mathrm{<mtext>n</mtext>}}\text{−σ}{}_{\mathrm{<mtext>tm</mtext>}}\text{σ}{}_{\mathrm{<mtext>ci</mtext>}}{}^{\mathrm{B}}\text{,}$$

where A and B are material constants, σ′_n is the normal effective stress, and σ_tm is the ‘tensile’ strength of the rock mass.

In both criteria, it is implicitly assumed that the cohesive and the normal stress-dependent frictional strength components are mobilized simultaneously, i.e., they are assumed to be additive as illustrated by the Terzaghi model in Fig. 1a.

Even when strain-softening models with residual strength parameters are chosen, the two strength components are assumed to be simultaneously mobilized and then lost in the post-peak range. These approaches with typical strength parameters have not been successful in predicting the depth and extent of failed rock in deep underground openings in hard rocks [6], [7], [8], [9], [10] or of borehole breakouts in deep boreholes.

As Schofield [11] pointed out “there is no true cohesion on the dry side of critical state”. In dense soil pastes, the peak strength is due to interlock and friction among particles and not due to the chemistry of bonds. While friction is immediately mobilized and the frictional strength component is proportional to the normal or confining stress, at low normal stress, the interlock resistance can be mobilized and then lost, leading to the typically observed strain-softening behavior of dense soils (the Taylor model; Fig. 1b). In other words, Eq. (2) should be written in a form whereby both strength terms are a function of plastic strain ε:

$$\text{τ=c(ε)+σ′(ε)}\text{tan}\text{φ.}$$

The pioneering work of Schmertmann and Osterberg [12] showed that in some soils, these two strength components (cohesional and frictional) are not necessarily mobilized simultaneously. They showed that the maximum of the cohesional component of strength c(ε) was mobilized early in the test, while the frictional component $\text{σ′(ε)}\text{tan}\text{φ}$ required 10–20 times more straining to reach full mobilization as shown in Fig. 2.

Similarly, there is no “true” and permanent cohesion in rocks, at least not in brittle rocks at relatively low confinement, where the cohesional strength component is gradually lost when the rock is strained beyond its peak strength. This is illustrated by the gap model of Fig. 1c, representing an analog for brittle rock. This analog illustrates that the cohesion at the bottom of the sliding wedge must be overcome before the frictional strength can be mobilized when the gap between the two wedges is closed. Only after this gap-closure deformation has taken place, will the normal stress, symbolized by the spring, be activated and a strain-dependent effective stress build up to create a frictional resistance. Consequently, the shear strength equation must be written in the form of Eq. (4), with a strain-dependent cohesion and a strain-dependent effective stress terms. In the low confinement range, the stress path will retract after reaching the cohesive strength surface as illustrated in Fig. 1b.

Martin and Chandler [13] demonstrated that the frictional strength component of granite is only mobilized after a significant amount of the rock's cohesional strength is lost. Originally, it was thought that the friction rather than effective contact stresses, producing the frictional strength component, needed to be mobilized. In otherwords, variable friction at a constant stress was implied rather than variable stress with constant friction.

This difference in interpretation is important as it is not the friction that depends on strain but the effective normal stress σ′(ε) causing a gradual development of the frictional strength component. Damage is induced in brittle rock when it is stressed beyond a damage initiation threshold. As a result, the effective normal stress σ′ inside the rock is highly variable and at some locations, less than the stress applied at the boundary. This is illustrated by the stress cone in Fig. 3, produced with a particle flow code (PFC) model by Diederichs [14], showing internal stress variations at a constant external confinement of 20 MPa. Hence, the frictional strength is not everywhere fully mobilized to the level determined by the average stress represented by the center of the cone [15], [29].

In this paper results from continuum models using conventional failure criteria and the strain-dependent cohesion weakening-frictional strengthening (CWFS) concept are compared.