Multiscale numerical simulation of rock slope instabilities

Multiscale numerical simulation of rock slope instabilities

131 Multiscale numerical simulation of rock slope instabilities Mauro Borri-Brunetto, Bernardino Chiaia * Department of Structural and Geotechnical E...

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Multiscale numerical simulation of rock slope instabilities Mauro Borri-Brunetto, Bernardino Chiaia * Department of Structural and Geotechnical Engineering, Politecnico di Torino, 10129 Torino, Italy

Abstract The apparent shear strength of rock discontinuities is lower than that of small scale samples. At the same time, the sliding behavior is characterized, in situ, by marked instabilities. Numerical algorithms permit to calculate contact forces at any point, and to describe the stick-slip transition. On the other hand, the critical aspects are not captured by classical theories. Multiscale simulations show that the contact domain between rough surfaces is a lacunar set. This explains the size-dependence of the apparent friction coefficient. By applying an increasing tangential force, the regime of partial-slip comes into play. However, the continuous and smooth transition to full-sliding predicted by the Cattaneo-Mindlin theory is not occurring in real situations. We implement a numerical renormalization group technique, taking into account the redistribution of stress consequent to partial-slip. This permits the critical value of the tangential force to be found. The critical force is less than the one predicted by Coulomb's theory, and depends on the specimen size and on the topology of the interface. Keywords: Multiscale contact algorithm; Stick-slip transition; Renormalization group; Fractal geometry

1. Introduction October 9, 1963, 10.39 pm - The Vajont Dam in northem Italian Alps. 275 million m^ of rock slid into the 115 million m^ of water in the reservoir, which was about two thirds full. The sliding mass travelled as a sheet of undisturbed rock formations at maximum speed that has been variously estimated between 90 and 250 km/h. A water wave overtopped the dam by 70 m, obliterating several villages and killing 1925 people. Between the end of construction (1960) and the tragic event (1963), rock creep (i.e. partial slips) had been observed, in the direction of the slope, in the dolomitic limestones on the southern flank of the valley. Initially, the rate of creep was 1 cm/week, but gradually it accelerated to 1 cm/day and finally to about 80 cm/day. In the following years similar high-velocity rock slides occurred in the Italian Alps (e.g. events in Valtellina, 1987), characterized by apparent unpredictability and huge loss of human lives. Two aspects can be evidenced in these phenomena. First, the dynamics of sliding activation follow a typical critical behavior, e.g. partial slips occurred, with progressively larger extent, before the final catastrophic sliding. Secondarily, when the shear strength of rock samples was measured in the laboratory, it appeared * Corresponding author. Tel.: +39 (11) 564 4866; Fax: +39 (11) 564 4899; E-mail: [email protected] © 2003 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics 2003 K.J. Bathe (Editor)

significantly higher than the strength activated in the rock slides, e.g. sliding should not have occurred according to the measured values. The classical laws of Coulomb and Amontons state that the frictional force is proportional to the normal load and does not depend on the apparent area of contact A„. Indeed, it has long been known that the shear strength of rock discontinuities depends on roughness. An empirical quantity named joint roughness coefficient (JRC) is the most used in practice [1]. However, remarkable size effects on the JRC have been early reported and not yet explained.

2. Lacunar contact domains When two rough bodies are pushed into contact, a set of contact points, with a complex pressure distribution, is generated. If the interfaces are self-affine (like fracture surfaces in natural rocks [2]), the contact domain is not a compact area but a self-similar lacunar set (Fig. Ic, d, e). We have developed a multiscale algorithm to determine the set of points where contact occurs as well as the pressure distribution [3]. Solving the problem at different scales r, and comparing the contact domains Sr (Fig. 1), the concept of true contact area appears to be not univocal, since it tends to zero as r ^- 0. Correspondingly, the real mean pressure Pr tends to infinity as r -> 0. It is therefore

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advantageous [4] to introduce the fractal measure A J of the domain S, defined by the noninteger dimension [L]^^. Correspondingly, a fm\iQ fractal mean pressure p* = P/A*^ can be defined as the true scale-independent quantity. If b is a characteristic linear size of the interface, the following

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3. Multiscale algorithm for sliding instabilities The continuous transition to full-sliding predicted by the Cattaneo-Mindlin theory does not occur in real situations. A simple Renormalization Group (RG) calculation [5] permits the critical value of the shear force which provokes the sliding instability to be found. Assuming Coulomb's friction at the asperity level, the limit shear stress fp is known at any point. Given an incremental tangential force Q, the algorithm calculates the tangential stress q at any point. The point with the highest q/fp ratio is determined. The external load is scaled to make this point exactly reach the critical condition. Once this point slips, a certain amount of tangential stress needs to be redistributed. The transfer of stress to the neighbor asperities is the essential feature of the model, and can be ascribed to the elastic energy release consequent to micro-slip or to the drop of the friction coefficient (see Fig. 3e). The multiscale model is built by overlapping box-coverings upon the contact domain (Fig. 3). The first-order cell (microscale) corresponds to the finest resolution, subsequent orders are obtained by dyadic sequence. At the rth scale of observation, a generic contact point corresponds to several contact points at the lower

scale (r — l)th. Once a point has slipped, the friction coefficient drops to a value f = pf, where ^ < 1 is a redistribution coefficient. An amount of stress equal io {I— P)fp has to be redistributed in the cell. If another point, in the same cell, reaches its critical condition, it slips too, and some extra stress must be transferred. According to renormalization, an rth order cell slips only if all the corresponding points at the (r — l)th level have slipped. The calculations are repeated at larger scales, the collective behavior at a certain level representing the input for the larger one. As Q increases, micro-slip events can be confined below a certain scale {partial-slip), or spread virtually to any scale {full-sliding). In the former case, the correlation length § is finite. The separating value of the shear force is the critical shear force Qcru^ which can be calculated numerically. When Q = Qcrit^ ? diverges and full-slip occurs. The critical value is reached before all the points have attained their critical condition. Therefore, the limit shear value is smaller than the sum of the local shear strengths, i.e., the (apparent) global friction coefficient {fgiob = Qcru/P) is smaller than the microscopic one / . A graphical representation of the RG procedure is shown in Fig. 3f. Nonlinear regression of the numerical data yields a family of ^-shaped curves. The points 0 and 1 are stable fixed points, corresponding respectively to full-stick and full-sliding. The straight line corresponds to: Nsiip{r)/Ntot{r) = Nsiip{r + l)INUr + 1). The iterative relation crosses the straight line at point C, which is another fixed point, i.e. the critical point separating the basin of (stable) partial-slip from the basin of (unstable) full-sliding. The critical value depends on the density of the contact domain and on the microscopic friction mechanisms. It is interesting to compare the critical force Qcru to Qnom = fPFor the contact domain depicted in Fig. 3a, in the case of ^ = 0.8 we have obtained Qcrit = 0.S9Qnom- In the case of P = 0.7, Qcrit is equal to 0.76gnom, and for ^ = 0.6, Qcru is only 0.49Qnom-

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Numerical calculations show that, the larger A5, the larger the value of Qcrit and the smaller the ratio Qcrit/Qnom (see Fig. 3g). Thus, in the case of rarefied domains, stress transfer to larger scales is hindered by the presence of zones (lacunae) where contact does not occur, and transition to full-sliding is more ductile. To validate the theory, the experiments made by Bandis et al. [6] are shown in Fig. 4. Notice that, not only do larger specimens show lower shear strength under the same normal load, but also do the postpeak curves become steeper (i.e. more unstable) as the nominal contact area increases. Power-law fitting of the experimental data eventually permits the fractal dimension A 5 of the contact domain to be found (Fig. 4b,c).

References [1] Barton N. Review of a new shear strength criterion for rock joints. Eng Geol 1973;7:287-332. [2] Mandelbrot BB. Self-affine fractals and fractal dimension. Phys Scripta 1985;32:257-260. [3] Borri-Brunetto M, Chiaia B, Ciavarella M. Incipient sliding of rough surfaces in contact: a multiscale numerical analysis. Comput Methods Appl Mech Eng 2001;190:6053-6073. [4] Carpinteri A. Fractal nature of material microstructure and size effects on apparent mechanical properties. Mech Mater 1994;18:89-101. [5] Chiaia B. On the sliding instabilities at rough surfaces. J Mech Phys Solids 2002;50:895-924. [6] Bandis S, Lumsden AC, Barton NR. Experimental studies of scale effects on the shear behavior of rock joints. Int J Rock Mech Min Sci Geomech Abstr 1981;18:1-21.