TECTO-126391; No of Pages 15 Tectonophysics xxx (2014) xxx–xxx
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Shear rupture under constant normal stiffness boundary conditions R.P. Bewick a,b,c,⁎, P.K. Kaiser b, W.F. Bawden a a b c
University of Toronto, Department of Civil Engineering, 35 St. George, Rm 105, Toronto, ON M5S 1A4, Canada Centre for Excellence in Mining Innovation (CEMI), 935 Ramsey Lake Road, Willet Green Miller Centre, Sudbury, ON P3E 2C6, Canada Mirarco Mining Innovation, Laurentian University, 935 Ramsey Lake Road, Sudbury, ON P3E 2C6, Canada
a r t i c l e
i n f o
Article history: Received 14 January 2014 Received in revised form 12 June 2014 Accepted 16 July 2014 Available online xxxx Keywords: Intact brittle rock Direct shear Constant normal stiffness Rupture zone creation Distinct element method (DEM)
a b s t r a c t A grain based Distinct Element Method and its embedded Grain Based Method are used to simulate the fracturing processes leading to shear rupture zone creation in a calibrated massive (non-jointed) brittle rock specimen deformed in direct shear under constant normal stiffness boundary conditions. Under these boundary conditions, shear rupture zone creation relative to the shear stress versus applied horizontal displacement (load–displacement) curve occurs pre-peak, before the maximum peak shear strength is reached. This is found to be the result of a normal stress feedback process caused by the imposed shear displacement which couples increases in normal stress, due to rupture zone dilation, with shear stress, producing a complex normal-shear stress-path that reaches and then follows the rock's yield (strength) envelope. While the yield envelope is followed, the shear strength increases further and shear stress oscillations (repeated stress drops followed by re-strengthening periods) in the load–displacement curves occur due to fracture creation as the rupture zone geometry smoothens. Once the maximum peak strength is reached (after a series of shear stress oscillations) the largest stress drops occur as the ultimate or residual shear strength is approached. The simulation results provide insight into the fracturing process during rupture zone creation and improve the understanding of the shear stress versus applied horizontal displacement response, as well as the stick-slip behaviour of shear rupture zones that are being created under constant normal stiffness boundary conditions. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Shear rupture of massive (intact, non-jointed) brittle rock may occur under a variety of boundary conditions ranging from constant stress (without normal stress changes during shear displacement) to constant stiffness (with boundary resistance causing normal stress increases during shear displacement) (e.g., Goodman, 1976; McKinnon and Garrido de al Barra, 1998; Obert et al., 1976). While different boundary conditions (e.g., constant stress versus constant stiffness) in reality exist and may actually change during the shear rupture process, the fracture and shear rupture of massive rock in the brittle field are typically studied under constant stress boundary conditions (e.g., Lockner et al., 1991). Under these conditions, the fracturing process typically leads to the creation of a shear rupture zone at or near peak strength with a continuous shear rupture surface created in the post-peak region of the stress–strain curve. There is, however, evidence suggesting that shear rupture zone creation can occur during yield before the maximum peak strength is reached under constant stiffness boundary conditions (e.g., Hallbauer et al., 1973). This suggests that boundary conditions
⁎ Corresponding author at: Golder Associates Ltd., 1010 Lorne Street, Sudbury, ON P3C 4R9, Canada. Tel.: +1 705 524 6861; fax: +1 705 524 1984. E-mail address:
[email protected] (R.P. Bewick).
influence the shear rupture zone creation process and characteristics in massive brittle rocks. In this article, shear rupture zone creation in massive brittle rock is investigated in direct shear under constant normal stiffness boundary conditions. The results are discussed in comparison to shear rupture under constant normal stress boundary conditions (Bewick et al., 2013a,b). Through a series of simulations with a calibrated synthetic brittle rock specimen (synthetic specimen) generated and deformed using a two dimensional particle based distinct element method (DEM) and its embedded grain based method (GBM), it is shown that the boundary condition under which a shear rupture zone is created influences its characteristics (i.e., shear rupture zone geometry, load– displacement response, and shear rupture zone creation relative to the load–displacement curve). In other words, it is demonstrated that the characteristics of a shear rupture zone are not only a function of the rock or rock mass properties but the boundary conditions under which the rupture zone is created. The DEM-GBM synthetic specimen was calibrated (Bewick et al., 2013a) to various rupture characteristics of Lodève sandstone which is a low porosity brittle rock that has been ruptured in the laboratory in direct shear under constant normal stress boundary conditions (Petit, 1988; Wibberley et al., 2000). This calibrated synthetic specimen is used to investigate the impact of constant normal stiffness boundary conditions on the shear rupture process and the previously noted shear rupture zone characteristics.
http://dx.doi.org/10.1016/j.tecto.2014.07.016 0040-1951/© 2014 Elsevier B.V. All rights reserved.
Please cite this article as: Bewick, R.P., et al., Shear rupture under constant normal stiffness boundary conditions, Tectonophysics (2014), http:// dx.doi.org/10.1016/j.tecto.2014.07.016
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1.1. Background Shear rupture occurs in massive rocks under confined conditions when uncontrolled crack propagation is inhibited. The fracturing processes in brittle rocks leading to shear rupture are dilatant (Brace et al., 1966; Hallbauer et al., 1973; Peng and Johnson, 1972; Scholz, 1968) because of newly created fractures opening and pre-existing and newly created discontinuities shearing on or overriding asperities. When a shear rupture zone is being created and is surrounded by rock that resists dilatant deformation, confining (i.e., normal) stresses will increase during shear rupture zone creation and, to a lesser extent, during subsequent shearing along the newly formed shear rupture surface. The related increases in confining stress and associated stress-path during fracturing and shear differs from that under constant stress boundary conditions (e.g., Indraratna et al., 2005) and, in the extreme, can be represented by constant stiffness boundary conditions (Archambault et al., 1992; Goodman, 1976; Indraratna and Haque, 1997; Johnston and Lam, 1989; McKinnon and Garrido de al Barra, 1998; Obert et al., 1976). An example of the stress-path difference under constant stress and stiffness boundary conditions normal to a shear rupture zone during its creation is illustrated schematically in Fig. 1a. Also shown are the related schematic load–displacement curves (Fig. 1b), showing that the peak strength is typically reached when the shear stress reaches the yield (rupture or strength) envelope under constant stress conditions, with yield occurring long before the peak strength is reached under constant stiffness conditions. The shear rupture process is typically studied under constant stress boundary conditions (e.g., Lajtai, 1969; Lockner et al., 1991; Petit, 1988; Sonnenberg et al., 2003; Wong et al., 2005). In direct shear under constant
a) Ib
Constant normal stiffness, stress-path
Shear stress, τ
II Ia
Normal stress, σn
Constant normal stress, stress-path (arrows actually overlap)
b) Shear stress, τ
Ib
Constant normal stress
Constant normal stiffness
II Ia
Horizontal displacement, δh
Fig. 1. (a) Schematic stress-paths under constant normal stress and normal stiffness boundary conditions: Ia = peak and yield points, Ib = yield point, II = maximum peak strength. (b) Schematic load–displacement curves for the stress-paths under constant normal stress and stiffness boundary conditions also showing Ia, Ib, and II.
normal stress boundary conditions, the rupture mechanism, rupture zone geometry, and shear stress versus horizontal displacement response of an intact brittle specimen are dependent on the normal stress to uniaxial compressive strength ratio (σn/UCS) (Bewick et al., 2013a,b) as illustrated by Fig. 2: • At low ratios (σn/UCS b0.17) (Fig. 2a), specimens rupture in a predominantly tensile splitting mode; a process that occurs at or just after the yield point and peak shear strength is reached. The load–displacement response is brittle with a large post-peak strength drop and the rupture zone is relatively thin and planar; and • At higher ratios (σn/UCS 0.17 b 1.0), specimens rupture progressively in a shear mode. First, an array of en échelon fractures develop (consisting of either tensile or shear mechanism at time of creation depending on the σn/UCS ratio) followed by linkages of the fracture array across the specimen leading to a shear rupture surface with a relatively wide damage zone. The load–displacement response is strain-weakening at the lower limit of the range of σn /UCS (Fig. 2b) to one with no strength drop post-peak strength at high σn/UCS (Fig. 2c). The rupture zone is relatively wide, discontinuous, and irregular. Previous investigations on the effect of constant normal stiffness boundary conditions in direct shear on intact specimens (Archambault et al., 1992; Obert et al., 1976) focused on the strength characteristics (strength envelope shape) of the materials tested. They showed that under constant normal stiffness boundary conditions the associated normal-shear stress-path generally follows the strength envelope generated from tests under constant normal stress (as in the schematic stress-path for normal stiffness boundary conditions in Fig. 1). Only a few investigations have been completed on intact brittle rocks deformed under non-constant stress boundary conditions. Hallbauer et al. (1973), using copper jacketed cylindrical specimens of quartzite deformed in triaxial compression, stopped tests at predetermined locations along the loading path and removed the specimens for sectioning and microscope observations. In these experiments, the lateral stress magnitudes were not kept constant during loading (a result of the copper jacket) and increased during deformation. They found that shear rupture in the specimens initiated and began to propagate pre-peak strength (Jaeger and Cook, 1976). Their test results provide some insight into brittle rock specimen rupture under non-constant stress boundary conditions; shear ruptures were generated before peak as opposed to post-peak strength as determined from constant stress boundary conditions. The majority of the experiments on shear rupture of massive rock in the brittle field have been conducted using constant stress boundary conditions. This boundary condition may not prevail in nature (during earthquakes) or in mining (during rockbursts) when fracturing processes leading to shear rupture zone creation are constrained (i.e., away from free surfaces). A limited number of experiments using intact brittle rock have been completed on specimens under constant stiffness boundary conditions. Thus, the understanding of shear rupture zone creation under different boundary conditions is yet incomplete. In the following, the adopted DEM and simulation setup for investigating normal stiffness boundary conditions are outlined followed by a description of results in terms of: (1) normal-shear stress-paths and shear strength envelopes; (2) rupture zone creation processes; (3) rupture zone creation relative to the shear stress versus applied horizontal displacement response and stress-path; and (4) shear stress oscillatory behaviour before and after maximum peak shear strength. The results are then discussed relative to rupture observed in direct shear under constant normal stress boundary conditions. Insight is provided into the mechanics of fracturing under constant normal stiffness boundary conditions as well as stick-slip behaviour initiation. Stick-slip is an instability process which is repetitive where periods of shear locking are followed by a sudden shear displacement with an associated shear stress drop (Scholz, 2002).
Please cite this article as: Bewick, R.P., et al., Shear rupture under constant normal stiffness boundary conditions, Tectonophysics (2014), http:// dx.doi.org/10.1016/j.tecto.2014.07.016
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75mm (i)
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Idealized elastic-brittle loaddisplacement response
Idealized strain weakening loaddisplacement response
σn/UCS
