Modified Mohr–Coulomb criterion for non-linear triaxial and polyaxial strength of intact rocks

https://doi.org/10.1016/j.ijrmms.2011.02.004Get rights and content

Abstract

Triaxial or polyaxial strength of rocks is required while analysing many civil and mining engineering structures in rocks. Mohr–Coulomb criterion is the most widely used strength criterion in rock engineering problems. In its present form the criterion suffers from two major limitations. Firstly, it represents the strength of rock as a linear function of confining pressure. Secondly, the effect of intermediate principal stress is not considered by this criterion. In the present study, this criterion is modified to take into account the non-linearity and effect of intermediate principal stress on strength behaviour. Barton's [1] critical state concept for rocks has been employed for this purpose. The applicability of the proposed simple non-linear triaxial and polyaxial strength criteria has been verified by applying them to experimental results for the intact isotropic rock material available from literature and comparing the prediction with the other popular criteria in vogue. The agreement has been found to be excellent. The applicability of the concept to jointed rocks will be discussed in separate publication.

Introduction

The rocks generally encountered in civil and mining engineering projects are subjected to in-situ stress fields. Excavation of an underground opening in such a rock redistributes the stresses and, failure of rock takes place under the influence of a complex multiaxial stress state. Strength criteria are used to define the strength of rocks subjected to given stress field and also to determine the extent of plastic zone if failure occurs. The strength criteria are, therefore, very important in designing structures in rocks.

A number of strength criteria have been proposed for intact rocks by various researchers in past. Many of them are based on sound principles of mechanics. But for practical applications, it is more important, how easily the parameters of a strength criterion can be obtained in the field. This is probably the reason that, despite having numerous theoretical strength criteria, the conventional Mohr–Coulomb strength criterion is the most popular and widely used strength criterion. The criterion however suffers from two major limitations: (a) it is a linear criterion and expresses the strength of the rock as a linear function of confining pressure or normal stress, and (b) in its present form, the criterion ignores the effect of intermediate principal stress σ2. There is ample evidence available that, in general, the intermediate principal stress does have substantial influence on the strength of rocks [2], [3], [4], [5], [6], [7], [8], [9], [10], barring a few cases of non-dilatant rocks [9].

An attempt has been made in this paper to overcome the above-mentioned limitations. The non-linear response has been incorporated in the Mohr–Coulomb criterion. The shear strength parameters c and ϕ, obtained from triaxial strength tests performed at low confining pressure, are directly used in the modified criterion to determine the triaxial strength of the rock at higher confining pressure. Furthermore, the proposed criterion has been extended to determine strength under polyaxial stress conditions. The applicability of the criterion has been validated by applying it to extensive databases of triaxial and polyaxial stress strength tests from available literature. The applicability of the proposed strength criterion to jointed rocks and rock masses will be discussed in future publication.

Section snippets

Critical state concept for rocks

If a rock is tested under confined condition, its strength increases with increase in confining pressure. The rate of increase in strength is high at low confining pressure. As confining pressure is increased, the rate of increase in strength decreases. The instantaneous friction angle is, therefore, high at low confining pressure conditions. The reason behind this is the dilatant and brittle behaviour exhibited by rocks at low confining pressure. The micro-cracks, which exist in a rock, open

General

Let us start with the triaxial stress conditions to suggest a non-linear strength criterion. Fig. 2 shows a plot of Mohr–Coulomb linear criterion in (σ1σ3) vs. (σ3) space. The Mohr–Coulomb linear criterion may be expressed in terms of σ3 and σ1 as follows:(σ1σ3)=2cicosϕi1sinϕi+2sinϕi1sinϕiσ3where ci and ϕi are Mohr–Coulomb shear strength parameters of the intact rock; the term (σ1σ3) is the deviatoric stress at failure; and σ3 and σ1 are the minor and major effective principal stresses at

Polyaxial stress condition

Engineering structures in rocks are generally subjected to polyaxial stress conditions rather than the triaxial stress conditions. It has been observed that the intermediate principal stress σ2 does have substantial effect on the strength of the rock σ1 except for some non-dilatant rocks [9]. A need for an appropriate polyaxial strength criterion has generally been felt by various researchers. Some of the studies which have addressed this issue are reported in literature [2], [3], [4], [5], [6]

Concluding remarks

The strength of rocks under triaxial and polyaxial stress conditions is required in many rock engineering applications. Mohr–Coulomb criterion, which is the most widely used strength criterion in geotechnical engineering, not only neglects the effect of intermediate principal stress but also considers the strength behaviour to be linear. In present study, the Mohr–Coulomb criterion has been modified to take into account the non-linear behaviour of rocks under triaxial conditions. Barton's

Acknowledgement

The authors are grateful to the unknown reviewers for critically reviewing and suggesting improvements in the manuscript. Special thanks are due to the Reviewer #2 who suggested many points related to critical state concept, especially with reference to Fig. 1a from [1]. The authors are also thankful to Bezalel Haimson from the University of Wisconsin, for readily sharing polyaxial tests data for SAFOD granodirotie and TCDP siltstone.

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