International Journal of Rock Mechanics & Mining Sciences 59 (2013) 57–69 Contents lists available at SciVerse ScienceDirect International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms Universal criteria for rock brittleness estimation under triaxial compression Boris Tarasov a,n, Yves Potvin b a Centre for Offshore Foundation Systems, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia b Australian Centre for Geomechanics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia article info abstract Article history: Brittleness is one of the most important mechanical properties of rock; however, the concept of Received 3 November 2011 brittleness in rock mechanics is yet to be precisely defined. Many brittleness criteria have been Received in revised form proposed to characterise material behaviour under compression, but there is no consensus as to which 19 September 2012 criteria is the most suitable and reliable. Accepted 18 December 2012 This paper considers brittleness at compression as the rock capability to self-sustaining macro- Available online 18 January 2013 scopic failure in the post-peak region due to elastic energy accumulated within the loaded material. Keywords: The applicability of various criteria for assessing rock brittleness from this point of view is analysed. It is Rock brittleness criteria shown that only two of many existing criteria can describe properly the intrinsic material brittleness within Confined compression the whole range of brittleness variation from the absolute brittleness to ductility. These criteria rely upon Post-peak energy balance post-peak energy balance and are based on sound physics principles. Unlike other existing criteria they Scale of brittleness Intrinsic and relative brittleness allow for the representation of two classes of rock behaviour (Class I to Class II) in the form of continuous, monotonic and unambiguous scale of brittleness. The effect of confining pressure on rock brittleness is analysed where rock behaviour can be changed from Class I to Class II and then to Class I again. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction The degree of post-peak instability caused by solely elastic energy stored within the loaded material is classified as intrinsic Brittleness is a very important mechanical property of intact brittleness, while the degree of post-peak instability in relation to rock because it has a strong influence on the failure process and on elastic energy accumulated in the loading system is classified as the rock mass response to mining or tunnelling activities. Large relative brittleness. The paper focuses mainly on criteria charac- seismic events are often produced when rock masses are submitted terising intrinsic rock brittleness. The concept takes into account to triaxial compression generating violent shear failures. The correct the fact that elastic energy distributed quite evenly within the determination of brittleness at such loading conditions is important material body at peak stress represents the source of energy for to better predict and mitigate these dynamic events. However, the localised post-peak failure in the form of shear rupture typical for concept of brittleness in rock mechanics is yet to be precisely confined compression. Features of the post-peak energy balance defined. Several brittleness criteria have been proposed to char- at the localised failure are discussed. acterise material behaviour under compression [1–19], but there is The paper analyses how different existing brittleness indices no consensus as to which criteria is the most suitable and reliable. reflect the degree of post-peak instability. It is shown that only The following approach for brittleness estimation at confined two of many existing criteria can describe properly the intrinsic compression (s14s2¼s3) is used in the paper. material brittleness within the whole range of brittleness varia- Experiments show that when loading a rock specimen, the tion from the absolute brittleness to ductility. These criteria allow specimen deformation is always macroscopically stable and for the representation of two classes of rock behaviour (Class I controllable, before the peak stress is reached. Macroscopic and Class II) in the form of continuous, monotonic and unambig- instability associated with strength degradation in the form of uous scale of brittleness. This universal scale is used to illustrate spontaneous failure can only take place in the post-peak region. some features of brittleness variation as a function of rising The post-peak instability can be treated as a manifestation of rock confining pressure for different rocks where rock behaviour can brittleness. In this paper, the degree of post-peak instability be changed from Class I to Class II and then to Class I again. estimated on the basis of post-peak energy balance is used for rock brittleness characterisation at compression. 2. Principles of brittleness estimation on the basis of post- peak energy balance n Corresponding author. Tel.: þ61 8 9380 7368; fax: þ61 8 9380 1044. Informative characteristics of intrinsic material properties, E-mail address: [email protected] (B. Tarasov). before and after the peak stress is reached, can be obtained from 1365-1609/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2012.12.011 58 B. Tarasov, Y. Potvin / International Journal of Rock Mechanics & Mining Sciences 59 (2013) 57–69 Class I B B B 3 3 3 – σ – σ – σ 1 1 1 σ = σ = σ = dW r C E ε ε AD ε Class II dWe B B B 3 3 3 – σ – σ – σ 1 1 1 dWa σ = σ = σ = C dWr ε ε D A ε Fig. 1. Illustration of the post-peak energy balance for rocks of Class I and Class II behaviour. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.) the complete stress–strain diagrams. It is known that rock beha- brittleness indices actually characterise the degree of intrinsic viour in the post-peak region can be characterised by negative or instability of the material at failure. The graphs in Fig. 1 can positive post-peak modulus M¼ds/de,wheres is the differential therefore be used to determine brittleness from energy para- stress (s1–s3)ande is the axial strain. In accordance with meters. For simplified estimation of the elastic energy dWe with- classification proposed in [20] Mo0 corresponds to Class I beha- drawn from the material specimen during the post-peak failure viour while M40 corresponds to Class II behaviour. process between points B and C (red area on the right), it is The complete stress–strain curves in Fig. 1 illustrate Class I and assumed that the elastic modulus E¼ds/de is the same at both Class II behaviour in the post-peak region. The graphs show the points. It should be noted that the modulus E represents the energy balance at three stages of deformation: at the peak stress unloading elastic modulus (point B), at an intermediate post-peak stage, and at the complete 2 2 sBÀsC failure (point C). Areas of the red triangles here correspond to dWe ¼ ð1Þ elastic energy stored within the specimen at the three mentioned 2E stages of deformation. The grey areas represent the post-peak The graphs show that the post-peak rupture energy dWr is rupture energy. determined by the amount of withdrawn elastic energy dWe plus The graphs illustrate the dynamics of transforming the elastic the additional energy corresponding to the grey area ABCD in the energy accumulated within the specimen material at peak stress, case of Class I behaviour, or minus the released energy corre- into post-peak rupture energy. The red areas (elastic energy) are sponding to the yellow area ABCD in the case of Class II behaviour. partly replaced in the graphs by the grey areas (rupture energy). The additional (or released) energy is described by The elastic energy represents the source of the post-peak failure 2 2 sBÀsC process and provides the physical basis for the post-peak failure dWa ¼ ð2Þ regime. For Class II, the fracture development occurs entirely due 2M to the elastic energy available from the material. The failure here the post-peak modulus M is negative for Class I and positive process has a self-sustaining character, with the release of excess for Class II behaviour. energy, corresponding to the yellow area (ABCD). The released The post-peak rupture energy dWr is described by energy can be transformed into the failure process dynamics, ÀÁ 2 2 particularly associated with fragmentation, flawing fragments, sBÀsC ðÞMÀE dWr ¼ dWeÀdWa ¼ ð3Þ seismicity, heat, etc. For Class I, the amount of elastic energy 2EM available from the material is not sufficient to produce failure, and some additional amounts of energy (the grey area ABCD) are This equation takes into account the sign of post-peak modulus required to support this process. M for Class I and Class II behaviour. The brittleness index K1 below Brittleness indices based on the ratio between the elastic is determined by the ratio between the post-peak rupture energy energy withdrawn from the material during the failure process and the withdrawn elastic energy [1–3] and the post-peak rupture energy (or released energy) can be used to characterise the capability of the rock for self-sustaining dWr MÀE K1 ¼ ¼ ð4Þ failure due to the elastic energy available from the material. Such dWe M B. Tarasov, Y. Potvin / International Journal of Rock Mechanics & Mining Sciences 59 (2013) 57–69 59 The brittleness index K2 represents the ratio between the the pre-peak parts of the curves are the same. Areas defined by the released and the withdrawn elastic energy [1–3] large red triangles correspond to elastic energy We stored within the rock material at peak stress, while the smaller red dotted dW E K ¼ a ¼ ð5Þ triangles on the right side of the curves represent the unconsumed 2 dW M e portion of the stored elastic energy, within the material, after It is known that the unloading elastic modulus E and the failure.