Predicting pillar burst by an explicit modelling of kinetic energy
Introduction
Violent pillar failure is a problem commonly encountered in underground mines, such as those exploited with the room-and-pillar method. It refers to the quick collapse of an isolated pillar, or of part of it, sometimes leading to the fragmentation and expulsion of rock pieces from the pillar. The phenomenon is known as “strain burst” or “pillar crush” depending on the severity of damage and of the magnitude of the associated seismicity (see Ortlepp's classification1). The more general term “pillar burst” is also used.2 Because it is difficult to predict, pillar violent failure can be the source of serious disorders for mining operations, particularly when it is at the origin of a “cascading pillar failure”3, 4 finally causing the collapse of a large mine panel and its overburden. Since the studies published by Cook,5 Starfield & Fairhurst6 and Salamon7, which can be considered as reference works on this topic, violent pillar failure has been considered as a mechanical instability affecting the host rock – pillar system during the post-peak phase of the pillar's behaviour, and not only the pillar itself.
Instability somehow is a catchall concept whose interpretation depends on the specific context in which it is used.8 At the microscopic scale, the theoretical definition of instability commonly used in rock mechanics derives from the original expression of the tensile strength of a pre-cracked material, such as proposed by Griffith.9 He showed that a pre-existing crack in an elastic plate will spontaneously propagate when its length is such that the rate of decrease of the elastic energy stored in the plate is higher than the rate of increase of the surface energy due to the crack growth. Trefftz10 formalized another definition of instability applied to elastic structures submitted to conservative forces – the equilibrium of such a structure is unstable if the total potential energy is at a local maximum. Hill's definition11 for elastoplastic solids states that the equilibrium of such a solid is unstable if the work done by constant (dead) external forces applied at its surface is greater than the energy stored or dissipated within it due to small virtual displacements of its free boundaries (compatible with the system's geometrical constraints).
In the end, regardless of the definition we consider, an unstable system is one that spontaneously moves away from an equilibrium position when a small displacement is applied on it. In other words, it is a system whose kinetic energy spontaneously increases when submitted to a constant external loading12. This definition falls within the general mathematical framework of the Lyapunov13 stability approach.
A literature review has allowed us to identify three major aspects of the pillar instability and burst phenomenon that require further attention. First, in the knowledge of the authors, the analytical 1D criterion for pillar instability developed by Cook,5 Starfield & Fairhurst6 and Salamon7 has never been quantitatively compared to more realistic 2D or 3D calculations. Second, the relationship between rockburst damage and kinetic energy release has been studied numerically14 for explaining how micro-seismicity could indicate an imminent pillar-burst. However, the generation, propagation and dissipation of the excess kinetic energy at the scale of one failing pillar and its close environment, including the host rock and the neighbour pillars, have been the subject of only few studies up to now.15, 16 Third, the kinetic energy is an indicator of the seismicity traditionally measured experimentally but, as it was highlighted by Spottiswoode17 and cited by Ortlepp,18 there are limited interactions between numerical modelling and seismicity in mines. More generally, there is a lack of comparison between the analytically, numerically and empirically estimated magnitudes of excess kinetic energy during pillar instability and burst. As a consequence, the use of numerical modelling for assessing sectors submitted to rockburst hazard in mines is not as developed as it could be, even if it is crucial for mining risk management.
Based on these observations, the present paper tackles three main objectives. I) understanding where kinetic energy is released and how it propagates during pillar failure, II) comparing 2D local numerical modelling solutions with a graphical (analytical) solution and an empirical classification, and III) proposing an easily reproducible modelling procedure, based on continuum mechanics only, for a rough prediction of the zones prone to burst in mines. For this purpose we used the explicit time-marching modelling scheme of the FLAC software (Itasca C.G. Inc.), the benefits of which will be highlighted.
The following Section 2 defines the concept of instability by referring to energy calculations, and then exposes its application to the problem of pillar stability. In Section 3, an explicit numerical method for calculating the damped kinetic energy at the local scale, based on the calculation scheme of the FLAC software, is succinctly presented and it is applied to calculate the kinetic energy generated and dissipated during the failure of a strain-softening pillar, as well as its distribution in time and space. The total amount and local density of damped energy as well as its relationship with the global pillar behaviour are then analysed and compared to Ortlepp's classification of rockbursts1 in Section 4. Finally, conclusions are drawn about the practical significance of the results and their applicability for rockburst prediction.
Section snippets
Mechanical energy balance
According to the first law of thermodynamics, the mechanical energy balance of a closed system can be written as follows:where Δ denotes a variation from one mechanical state to another. Ek is the kinetic energy of the system, Egp is its gravitational potential energy, Uc is the elastic strain energy stored in the system, Wdiss is the dissipated mechanical energy between the two considered states (always positive) and Ws is the work done by the boundary (surface) forces.
Numerical calculation of Wk during pillar failure
The damped kinetic energy Wk can be determined by at least three methods. I) Graphical calculations –they derive from an analytical method such as that suggested in Section 2.3. (Eq. (6)) and are only applicable to idealized systems. II) Implicit numerical method – it consists of applying the energy balance of Eq. (1) within finite element or boundary element models. Then, the terms Ws is calculated by using the boundary forces and displacements and ΔEgp, ΔUc and ΔWdiss, are calculated by using
Contribution of the horizontal forces to Wk
The damped energy calculated with the local method has been decomposed into the sum of vertical (v) and horizontal (h) forces contributions on every node i of the models such as:
Table 4 shows that the vertical forces contribute to about 90% of the total damped energy Wk in every model. This is consistent with the fact that the pillars are mainly loaded vertically under the effect of gravity. The 10% of horizontal contribution are related to the variation of
Conclusions
We have first presented the concept of mechanical instability and reviewed a classical instability criterion commonly considered at pillar scale and initially demonstrated by Salamon7 on the basis of a simple 1D model. We then have highlighted that the instability of the pillar – host rock system is characterized by a certain amount of excess kinetic energy to be damped.
Then, we have used the explicit numerical modelling scheme of the FLAC software in order to calculate the damped kinetic
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2021, Underground Space (China)Citation Excerpt :Indeed, the failure of rock is driven by energy activities, including absorption, evolution, dissipation, and the release of strain energy (Huang & Li, 2014; Li, Sun, Xie, Li, & Ranjith, 2017; Peng et al., 2015; Weng, Huang, Taheri, & Li, 2017; Zhou, Xu, Karakus, & Shen, 2019). Over the last few decades, extensive relevant researches have been conducted to gain a better understanding of the mechanism of strain burst and the energy evolution using laboratory tests (Bruning, Karakus, Akdag, Nguyen, & Goodchild, 2018a; Hauquin, Gunzburger, & Deck, 2018; He, Jia, Coli, Livi, & Sousa, 2012; Hua & You, 2001; Li, Du, & Li, 2015; Meng, Zhang, Han, Pu, & Nie, 2016; Xu & Karakus, 2018). Li, Huang, and Li (2014) studied the strain rate dependency of rock strength, deformation and strain energy conversion.