DEM Simulation of Direct Shear: 1. Rupture Under Constant Normal Stress Boundary Conditions

Abstract

A particle-based distinct element method and its grain-based method are used to generate and simulate a synthetic specimen calibrated to the rupture characteristics of an intact (non-jointed) low-porosity brittle rock deformed in direct shear. The simulations are compared to the laboratory-generated ruptures and used to investigate rupture at various normal stress magnitudes. The fracturing processes leading to shear rupture zone creation and the rupture mechanism are found to be normal stress dependent (progressing from tensile splitting to shear rupture) and show partial confirmation of rupture zone creation in nature and in experiments from other materials. The normal stress dependent change is found to be due to the orientation of the major principal stress and local stress concentrations internal to the synthetic specimens being deformed. The normal stress dependent rupture creation process results in a change to the rupture zone’s geometry, shear stress versus horizontal displacement response, and thus ultimate strength.

1 Introduction

The fracturing processes and mechanisms leading to shear rupture are investigated in intact (non-jointed) brittle rock deformed in direct shear under constant normal stress boundary conditions for the purpose of: (1) evaluating a particle-based distinct element method (DEM) and its embedded grain-based method (GBM), and (2) to provide a better understanding of the rupture of intact brittle rock in direct shear. The investigation is completed using the commercially available particle-based DEM, Particle Flow Code in Two Dimensions (PFC2D, v4.00-190) (Itasca 2011) and its embedded GBM (Potyondy 2010; Itasca 2011). A synthetic specimen is generated numerically and calibrated to the shear rupture characteristics of Lodève sandstone (a brittle low-porosity, <2 %, rock) deformed in direct shear under constant normal stress boundary conditions as reported by Petit (1988) and Wibberley et al. (2000). The GBM in PFC2D has not been previously reported in the literature for investigating the rupture of intact brittle rock specimens deformed in direct shear.

First the DEM, PFC2D and its GBM are introduced followed by the methodology for the generation and calibration of the synthetic specimen. Next, the fracturing processes and mechanisms leading to shear rupture zone creation and the synthetic specimen’s overall rupture mechanism are summarized with a detailed analysis of the internal fracture orientations occurring in the shear rupture zone for two synthetic specimen aspect ratios (1:1 and 1.5:1). The results from the investigations provide insight into: (1) the ability of the DEM–GBM to capture the various rupture characteristics of Lodève sandstone; (2) the fracturing processes leading to rupture zone creation; (3) the shear rupture mechanism and rupture zone geometry dependence on normal stress; and (4) the influence of specimen aspect ratio on shear rupture zone creation. A companion paper, Bewick et al. (2013), presents the influence of grain and grain boundary strength change on various rupture characteristics.

Limited research has focused on shear rupture zone creation in intact brittle rock (Obert et al. 1976; Petit 1988), intact brittle analogue materials (Lajtai 1969; Archambault et al. 1992; Cho et al. 2008), and concrete (Sonnenberg et al. 2003; Wong et al. 2005) subjected to direct shear under constant normal stress boundary conditions. The results of the previous investigations suggest that the shear rupture mechanism and fracturing processes leading to rupture zone creation are normal stress dependent (Lajtai 1969; Petit 1988; Sonnenberg et al. 2003) resulting in different rupture mechanisms and rupture zone geometries. The reasons for these changes are not well understood and are potentially related to stress rotation and magnitudes in the specimens.

The numerical simulations reported next allow one to track the internal state-of-stress during synthetic specimen deformation which is: (1) not possible in specimens deformed in the laboratory; and (2) largely assumed for the interpretation of the rupture process. In addition, fracturing (location, orientation, mechanism, and type, e.g. grain boundary and intra-grain), and particle displacement or velocity vectors (which indicate the failure mechanism of fracture systems) during shear rupture zone creation are tracked during synthetic specimen deformation. All of the above provide an opportunity to improve the understanding of shear rupture zone creation in brittle rock deformed in direct shear under constant normal stress boundary conditions.

2 Rock Type used for Investigation

Petit (1988) investigated peak shear strength and normal stress dependent rupture zone creation in specimens of intact Lodève sandstone, a brittle low-porosity (<2 %) rock, deformed in direct shear under constant normal stress boundary conditions. Wibberley et al. (2000) investigated the resulting fracture systems in the rupture zones generated by Petit (1988) using scanning electron microscope (SEM) images. This rock type was chosen for this investigation because of its brittleness, and it is the only well described direct shear testing series that has been completed on an intact brittle low-porosity rock in the literature allowing one to compare various shear rupture zone characteristics to numerical simulations.

The Lodève sandstone tested by Petit (1988) is not a geological unit in itself. The sandstone specimens were taken from inter-bedded sandstone—shale—cinerite beds (each approximately 10 cm–1 m thick) in an approximate 300-m thick sequence of Lower Permian lacustrine shoreline and deltaic deposits located in Southern France. The sandstone was the strongest (most cohesive) rocks not affected by Permian and later faulting in the area (Wibberley 2011, personal communication). The sandstone is fine to medium grained (Petit 1988; Wibberley et al. 2000) and consists of calcite (cementing component originating primarily from early precipitation of pore waters), orthoclase feldspar, albite feldspar, and quartz. The calcite cementation resulted in the low porosity of <2 % (Petit 1988). The mineral composition of the Permian Lodève sandstone based on Comte et al. (1985) is as follows: 45–55 % feldspar; 20 % carbonates; 15–20 % quartz; and 10 % argillaceous material. According to Petit (1988) and Wibberley et al. (2000), their sandstone does not contain argillaceous material. The composition described by Comte et al. (1985), suggesting argillaceous content, maybe related to the overall geologic sandstone in the area and may not be for the specific sandstone bed sampled by Petit (1988).

3 Introduction to Adopted DEM

PFC2D (Itasca 2011) models the movement and interaction of particles which are represented as rigid circular disks that can overlap at contacts. The particles abide by Newton’s laws of motion, and a force–displacement law is applied to each particle–particle contact. Boundary conditions (such as constant stress or velocity) are applied along walls which are rigid borders. Particles are not bonded to walls and shear can occur at particle–wall interfaces depending on the assigned particle micro-parameters: (1) coefficient of friction, μ, (2) normal to shear stiffness ratio, k _n/k _s; and (3) modulus, E _c. PFC2D uses an explicit finite difference method. The calculation cycle in PFC2D is summarized as follows (Schöpfer et al. 2006): (1) contacts from known particle and wall locations are updated; (2) a force–displacement law is applied to each contact to update particle–particle contact forces; and (3) a law of motion to each particle is applied to update particle velocity. The calculation cycle is carried out using a time-stepping routine where at each time step a cycle is carried out.

The GBM in PFC2D v4.00-190 (Potyondy 2010) is used to generate a synthetic material forming a synthetic specimen that mimics deformable, breakable, polygonal grains cemented along their adjoining sides. A two-dimensional disk-packing methodology is used to generate the polygonal grain structure as follows (Fig. 1): (1) a bonded particle model with no walls is generated (Fig. 1a) with particles of the desired grain size and variability based on the following parameters for each grain type: (a) percent volume composition; (b) minimum grain radius, $\overline{R}_{ \hbox{min} }$; (c) maximum to minimum grain radius ratio, $\overline{R}_{ \hbox{max} } /\overline{R}_{ \hbox{min} }$; (2) void centroids are identified between particles (Fig. 1b black ‘dots’); (3) the centroids of the voids (black ‘dots’ in Fig. 1b) are then joined with lines forming a polygonal network (Fig. 1c, d); and (4) the GBM is generated by overlaying the polygonal network on a bonded particle model of smaller particles filling the polygonal network (Fig. 1e).

Fig. 1

Grain structure generation. a Initial particle packing and contacts depicted as lines joining particles from their centres. b Void centroids (black ‘dots’). c Creation of polygonal network with nodes at void centroids. d Polygonal network. e Polygonal network overlaid on particle assembly (a–d modified from Potyondy 2010)

A grain, as shown in Fig. 2a, is composed of grain boundaries which are represented using smooth joint contacts (Mas Ivars 2010). The internal structure of a grain (Fig. 2a, b), bound by the smooth joints, consists of a cemented granular material bonded together by parallel bonds (Potyondy and Cundall 2004). Smooth joint bond breakage is representative of grain boundary fracture, and parallel bond breakage of intra-grain fracture in a synthetic specimen. Internal to the grain shown in Fig. 2a are 68 parallel bonded particles. Smooth joints remove the previous limitation in PFC2D where discontinuities or planar contacts were simulated as unrealistically rough and bumpy (due to the particle-based nature of the DEM); particles that are on adjacent sides of a smooth joint can pass through each other during sliding, forcing the sliding path along the smooth joint contact (Fig. 2c) opposed to riding over the particles along the sliding path, as in Fig. 2d. Parallel bonds, schematically illustrated in Fig. 2b, allow both a force and moment to be resisted between individual particles. Once a smooth joint or parallel bond breaks (in either shear or tension), the contact transitions to frictional behaviour depending on the assigned smooth joint residual coefficient of friction or particle coefficient of friction. Examples of broken parallel bonds and smooth joints are shown in Fig. 2e. A smooth joint is defined by the following micro-parameters: (1) normal and shear stiffness factors, $\overline{k}_{\text{n}}$ and $\overline{k}_{\text{s}}$; (2) tensile strength, σ _c , and shear strength (defined by a linear strength criterion), τ _c  = c + σ _ntanϕ (where c is cohesion, ϕ the friction angle of the smooth joint, and σ _n is normal stress); and (3) a residual coefficient of friction, μ _r. A parallel bond is defined by the following micro-parameters: (1) normal to shear stiffness ratio, $\overline{k}^{\text{n}} /\overline{k}^{\text{s}}$; (2) bond modulus, $\overline{E}_{c}$; (3) tensile strength, $\overline{\sigma }_{c}$, and shear strength (defined by a linear strength criterion), $\overline{\tau }_{c} = \overline{c} + \sigma_{\text{n}} { \tan }\overline{\phi }$ (where $\overline{c}$ is cohesion and $\overline{\phi }$ the angle of internal friction of the parallel bond); and (4) bond radius multiplier, λ. A bond radius multiplier of 1 completely fills the gap between two particles and if the multiplier approaches zero, the material behaves as a granular material. Bond strength standard deviation can be assigned if desired. A particle is defined by the following micro-parameters: (1) minimum radius, R _min; (2) maximum to minimum radius ratio, R _max /R _min; (3) normal to shear stiffness ratio, k _n/k _s; (4) density, ρ; (5) modulus, E _c; and (6) coefficient of friction, μ. Different modulus values can be assigned to the particles (E _c) and parallel bonds ($\overline{E}_{c} )$.

Fig. 2

Elements forming a grain in the GBM. a Example single grain showing smooth joint contacts forming the grain boundaries and the internal parallel bonded particles (not showing the parallel bonds for clarity). b Schematic representation of a parallel bond. c Schematic representation of the behaviour of a smooth joint. d Schematic representation of the behaviour of a contact without a smooth joint. e Examples of broken parallel bonds (black) and smooth joints (orange) (color figure online)

4 Grain Structure

All current numerical models have limitations when simulating actual grain structures of rock and simplifications are required (Lan et al. 2010). The grain structure simulated is not intended to exactly match that of the Lodève sandstone but to account for some of its grain size geometric heterogeneity and mineral composition. The synthetic grain structure generated for the Lodève sandstone is represented by three mineral types; feldspar (50 %), quartz (20 %), and calcite (30 %) (composition based on Comte et al. 1985 grouping the 2 feldspar mineral types together). SEM images from Wibberley (2011) were used to estimate grain sizes for each mineral grain type by visually outlining grains with a polygon around its border and recording the largest and smallest dimensions for two separate populations of grain diameters as follows: (measurement set 1) from a single SEM image; and (measurement set 2) from multiple SEM images. Images could not be used for input into a computer software tool for the automatic picking of grain types based on pixel colour. The resulting D₁₀, D₅₀, and D₉₀ values are summarized in Table 1 and the histograms for measurement set 2 (which also contains measurement set 1) are shown in Fig. 3. The grain sizes summarized in Table 1 are small (average 50 % passing grain diameter of 0.27 mm, D₅₀, Table 1, measurement set 1) compared to other previous PFC2D simulations (e.g. Potyondy and Cundall 2004 and Hazzard et al. 2000). During initial attempts to simulate the values for measurement set 1 (Table 1) using the GBM, model generation difficulties occurred because each grain is required to be represented by a number of bonded particles resulting in an unmanageable number of particles in a model (ref. Sect. 3 for a description of the GBM composition). In an effort to maintain the grain scale heterogeneity and keep the grain size as small as reasonably possible, the minimum grain size was eventually set at five times the estimated D₁₀ value (Table 1, ‘5 × D₁₀’ column, for measurement set 1) with the maximum (D _max) to minimum (D _min) grain size ratio set at the values listed in Table 1 (‘D _max/D _min’ column, measurement set 1). The simplified grain sizes and maximum to minimum grain ratios result in a synthetic specimen with an average grain size of 1.4 mm which is of medium grain size (Press and Siever 1998) and within the limits of that described by Petit (1988). The grain structure also accounts for some of the grain size heterogeneity (i.e., D _max/D _min ratio) and three different mineral types. Considering that the actual grain scale and structure cannot (yet) be simulated, this is considered to be a reasonable simplification. Measurement set 1 was used over set 2 because of the slightly larger estimated grain sizes. An example polygonal grain distribution resulting from the parameters discussed is shown in Fig. 4a, b in comparison with a representative SEM image from Wibberley (2011) (Fig. 4c). While the Lodève sandstone grains are not dominantly polygonal, visual inspection suggests that the grain composition generated is a reasonable representation of reality and a better grain representation than conventional bonded particle modelling with only circular shaped grains. Of course, a more representative synthetic specimen could be generated by developing a routine that ‘maps’ a grain network into a particle-based DEM such as that used by Katsaga (2010). However, the SEM images provided by Wibberley (2011) were found to not be adequate for the automatic picking of grain boundaries and grains as previously noted above. The synthetic specimen (Fig. 4a) shows the overall grains (~1,405) and does not show the 41,388 particles which make up the grains, the smooth joints (which make up the grain boundaries), or the parallel bonds between the particles in each grain.

Table 1 Estimated grain sizes by mineral type

Fig. 3

Combined histograms and cumulative “percent passing” curves for grains measured in ‘measurement set 1 and 2’ using SEM images provided by Wibberley (2011)

Fig. 4

Comparison between: a the simplified GBM generated for Lodève sandstone; b close-up view of boxed in area in a; and c a representative SEM image provided by Wibberley (2011). b, c Show a similar number of grains along both the horizontal and vertical dimensions. Some mineral grains are labelled in b, c for clarity. Darkest grey quartz (Qz), light grey feldspar (F); and white calcite (C). Scales as shown

Mineral properties to aid in the selection of micro-parameters for each mineral type (i.e., feldspar, quartz, and calcite) in the synthetic specimen were compiled from the literature and are listed in Table 2. Not all aspects of a mineral grain’s properties can be accommodated by the GBM such that the material properties in Table 2 should be thought of as overall bulk mineral mechanical properties.

Table 2 Compiled mineral properties

5 Simulation Procedure

The numerical direct shear setup (Fig. 5) is a simplification of the laboratory setup used by Petit (1988). Lodève sandstone specimens with aspect ratios of 0.66:1 and 0.42:1 were used with a 5-mm wide gap left around the specimens between the upper and lower shear box walls for partial observation of fracturing. In the numerical simulations, a tall specimen is not needed to reduce the tendency of a laboratory specimen to rotate or apply boundary forces and a 1:1 specimen aspect ratio is primarily used. The 1:1 shear box with constant normal stress boundary conditions created in PFC2D (shown in Fig. 5) is composed of a 50 × 50-mm synthetic specimen bound by an upper portion of the shear box which has two separate fixed lateral walls (Walls 3 and 4, Fig. 5) and a lower portion simulated as a single wall, of ‘U’ shape (Wall 1) which moves in the horizontal direction (Fig. 5). The contacts between all of the walls and the synthetic specimen are frictionless. A 5-mm gap is introduced to match the shear box gap reported by Petit (1988). Constant normal (vertical) stress is first applied to the synthetic specimen through applied constant velocity to the top wall of the shear box (Wall 2, Fig. 5). When the desired constant normal stress magnitude is achieved throughout the synthetic specimen, the velocity of the top wall is stopped. Shear displacement is then applied to the synthetic specimen through a constant velocity movement of the lower wall (Wall 1, Fig. 5) with rotation of the shear box restricted and a constant normal stress maintained at the top wall (Wall 2). Diederichs (1999) found that constant velocity loading influenced both the peak strength and post-peak load–displacement response of synthetic specimens in PFC2D. Therefore, the constant shear velocity of the lower wall was chosen to ensure that both the peak shear strength and post-peak shear stress versus horizontal displacement response were not influenced by slower constant shear velocities. A constant shear velocity of 0.04 m/s was selected which relates to a displacement of the lower wall of approximately 2.75e⁻⁷ mm per time step. This velocity resulted in quasi-static loading conditions.

Fig. 5

Constant normal stress direct shear simulation schematic showing the measurement circle in the centre of the specimen

The average shear stress is determined through the reaction forces acting along Wall 4 divided by the synthetic specimen length. Horizontal displacement is measured based on the movement of the lower wall (Wall 1). Based on the logic adopted by Cho et al. (2008), the principal stress magnitudes and the orientation of the major principal stress in the synthetic specimen are determined using the measurement circle (10 mm diameter) shown in Fig. 5 in the centre of the synthetic specimen. All stress components (σ _xx, σ _yy, σ _xy) are monitored for every particle within the measurement circle.

6 Calibration to Lodève Sandstone Characteristics

DEM models must be calibrated such that the assigned micro-parameters in the models produce the desired macro-properties (such as strength) and various characteristics (such as load–displacement and fracture response) of the material being simulated. Here, calibration is first done against reported characteristics of the Lodève sandstone deformed in direct shear (i.e., tensile strength, linear Coulomb strength envelope, post-peak shear stress versus horizontal displacement response, rupture zone geometries, and fracture angles at various normal stresses). The shear deformation modulus is not available for calibration (Petit and Wibberley 2011, personal communication). Given the missing shear deformation modulus, the resulting calibration parameters were then used to simulate biaxial compression tests (confining stress from 0 to 64 MPa) to determine if the Lodève sandstone calibration micro-parameters resulted in Young’s modulus and strength values (uniaxial and triaxial) of typical strong (uniaxial compressive strength ≥100 MPa), brittle, low-porosity (≤10 %), sandstones whose properties were compiled from the literature (summarized in Table 4). The micro-mechanical fracturing process aspects at the grain scale were not considered in the calibration but are discussed later and investigated in a companion paper (Bewick et al. 2013).

6.1 Calibration Results

Six normal stress magnitudes were simulated 5, 15, 25, 40, 60, and 90 MPa to cover the range of normal stresses reported by Petit (1988). The direct shear PFC2D GBM simulation results when compared to the laboratory test results of Petit (1988) are essentially identical (Coulomb strength envelope, Fig. 6a, and post-peak shear stress versus horizontal displacement response, Fig. 6b). The simulated tensile strength is also almost identical to that reported by Petit (1988). These calibration aspects are summarized as follows:

• (a) Linear Coulomb strength envelope (Fig. 6a):

  $${\text{Laboratory}}:\tau = 2 8 + \sigma_{\text{n}} 0. 8 1$$

  (1)
  $${\text{Simulation}}:\tau = 2 7 + \sigma_{\text{n}} 0. 7 6R^{ 2} = 0. 9 9$$

  (2)

• (b) Tensile strength:

  • Laboratory: −17 MPa

  • Simulation: −16 MPa

• (c) Post-peak shear stress versus horizontal displacement response:

  In both laboratory and simulation: brittle instantaneous shear stress drop at low normal stress magnitudes with maintained peak shear stress at the highest normal stress magnitudes (Fig. 6b).

Fig. 6

Strength and deformation calibration results. a Linear Coulomb strength envelope for laboratory data (grey circles) as reported by Petit (1988) compared to DEM simulation results (white squares). b DEM simulation shear stress versus horizontal displacement response in comparison to descriptions provided by Petit (1988) and Wibberley et al. (2000)

Qualitatively, the PFC2D final rupture zone geometries can be compared to the SEM images of the ruptures formed in the Lodève sandstone laboratory specimens away from the shear box walls for normal stresses 15, 60, and 90 MPa (87 MPa for the laboratory test) (Figs. 7, 8, 9). The simulation results appear to be similar to the SEM images (Figs. 7, 8, 9) showing changing rupture mechanisms from more tensile to shear, the developed cataclastic material, similar micro-fault orientations, and branching micro-fault systems. Also, clear resemblance is evident by comparing simulation results with the descriptions from Petit (1988) and Wibberley et al. (2000) where:

Fig. 7

15 MPa normal stress rupture. a PFC2D simulation showing shear rupture zone. b Backscattered SEM image of part of the sample ruptured at a normal stress of 15 MPa (courtesy of Wibberley 2011, personal communication). c, d Close-ups of boxed in areas in a and b, respectively. c, d Show a similar number of grains along both the horizontal and vertical dimensions. In PFC2D rupture images, orange grain boundary, black intra-grain tensile fracture. Grains not shown in c for clarity (color figure online)

Fig. 8

60 MPa normal stress rupture. a PFC2D simulation showing shear rupture zone. b Backscattered SEM image montage of part of sample ruptured at a normal stress of 60 MPa. c Close-up view of boxed in area in b. d Close of view of boxed-in area in a. c, d Show a similar number of grains along both the horizontal and vertical dimensions. In PFC2D rupture images, orange grain boundary, black intra-grain tensile fracture. b, c Courtesy of Wibberley (2011). Grains not shown in d for clarity (color figure online)

Fig. 9

90 MPa normal stress rupture. a PFC2D simulation showing shear rupture zone. b Close-up view of boxed-in area in a. c Backscattered SEM image of part of the specimen ruptured at a normal stress of 87 MPa (courtesy of Wibberley 2011). b, c Show a similar number of grains along both the horizontal and vertical dimensions. In PFC2D rupture images, orange grain boundary, black intra-grain tensile fracture. Grains not shown in b for clarity (color figure online)

• (a) the rupture zone geometry is dependent on normal stress (as evident in Fig. 10a, b where the 25 and 90 MPa rupture zone geometries are different);

  Fig. 10

  

  Comparison of fracture system angles. a, b DEM simulation rupture zone images for 25 and 90 MPa normal stress magnitudes, respectively (orange grain boundary, black intra-grain tensile fracture). c, d Orientation of fracture systems in a and b, respectively (counter clockwise from horizontal left half, clockwise from horizontal right half of rose diagram). e, f Orientation of micro-faults and tensile fractures in the Lodève sandstone specimen ruptured in direct shear as measured in SEM images by Wibberley et al. (2000) for 23 and 87 MPa normal stress magnitudes (color figure online)

• (b) at low normal stress magnitudes the rupture zone is thin with shallower fracture angles; and

• (c) at high normal stress magnitudes the rupture zone is relatively wide with steeper fracture angles (compare Fig. 10a, b where the 25 MPa normal stress rupture zone is visually thinner and has shallower angles relative to the 90 MPa normal stress rupture zone geometry).

The general process of micro-fault formation is also similar to that described by Wibberley et al. (2000) where the calibration resulted in the generation of only tensile fractures along grain boundaries and in grains in the synthetic specimen and these tensile fractures formed fracture systems to create larger fractures that were either of a tensile or shear mechanism (as determined by particle displacement or velocity vectors which explicitly show the mechanism of fracture systems at the time of creation and as it changes over time).

The fracture system angles generated in the synthetic specimens for the 25 and 90 MPa normal stresses were also compared to those measured by Wibberley et al. (2000) in specimens of Lodève sandstone ruptured at 23 and 87 MPa normal stress. The angles of the fracture systems in the two rupture zone ‘snap shots’ (Fig. 10a, b) were determined by sketching linear trends along them using scaled images and measuring the angle of the line relative to the horizontal (counter clockwise positive 0°–90°) (PFC2D can directly output the individual fracture angles but cannot output the angles of a fracture system, i.e., angles of larger fractures composed of multiple smaller fractures). An example showing two fracture systems and the resulting sketched fractures is illustrated in Fig. 11. The fracture systems were divided into two classes to facilitate comparison to the angles measured by Wibberley et al. (2000): (1) tensile fracture systems which predominately formed pre-peak strength (with particle velocity vectors indicating opening at time of creation); and (2) shear fracture systems which are composed of linked tensile fracture arrays which have shear displacement as evident from particle displacement or velocity vectors at time of creation. The measured angles in the synthetic specimens are presented in Fig. 10c, d as rose diagrams. The fracture system angles in the synthetic specimens (Fig. 10c, d) are compared to the angles of micro-faults and tensile fractures measured by Wibberley et al. (2000) (Fig. 10e, f). ‘Micro-fault’ is the terminology used by Wibberley et al. (2000) for fracture systems which showed measurable shear displacement at the scale of the SEM image. The fracture system angles from the simulations (Fig. 10c, d) show the trend of increasing angles with increasing normal stress magnitudes and generally similar overall angles compared to the micro-faults and tensile fractures measured by Wibberley et al. (2000) (compare rose diagrams in Fig. 10e to Fig. 10c and Fig. 10f to Fig. 10d). One difference of note is in Fig. 10d where there are two tensile fracture orientation peaks (35° and 55°) for the synthetic specimen, while there is only one dominate fracture orientation trend (44.7°) in the laboratory ruptured Lodève sandstone (Fig. 10f). The tensile fracturing in the synthetic specimens is predominately along grain boundaries (i.e., smooth joints). Reviewing the rupture images it was found that the polygonal grain geometry has some control on fracture orientation such that there are few grain boundaries with an orientation between 35° and 55° creating the bi-modal tensile fracture orientations in Fig. 10d. The average between the bi-modal orientations is 45° which is essentially the desired 44.7° tensile fracture angle.

Fig. 11

Example of fracture systems in a generated through a grain (black fractures) and along grain boundaries (orange fractures) from a number of individual tensile fractures. The fracture systems in a are shown in b with the overlaid fracture system sketch (dashed lines) (color figure online)

One can therefore conclude that the selected aspects of the Lodève sandstone’s mechanical response to direct shear deformation under constant normal stress can be captured by the calibrated PFC2D GBM synthetic specimen. The resulting micro-parameters for the model components and each grain type (which are assigned separate micro-parameters) are summarized in Table 3 along with the relevant mineral data (from Table 2) in parentheses. The simulated modulus values for the particles forming grains are comparable to the range of Young’s modulus values for quartz and feldspar except for calcite which was simulated with a lower modulus value and thus is more compliant compared to the quartz and feldspar minerals (in line with observations of calcite mineral grain behaviour, Wibberley et al. 2000). The tensile strength of the parallel bonded particle assembly forming each mineral is similar to the uniaxial compressive strengths of the minerals. This is consistent with other PFC2D calibrations where tensile bond strengths are comparable to the rock’s compressive strength (e.g. Potyondy and Cundall 2004). The densities of the particles are similar and the coefficients of friction for the particles are slightly higher than the compiled values for mineral gouge.

Table 3 Calibration micro-parameters (values in parentheses from Table 2 for comparison)

The micro-parameters in Table 3 were then used to simulate synthetic specimens in biaxial compression and the results were compared to the complied testing data for other strong brittle low-porosity sandstones. Table 4 summarizes the compiled uniaxial compressive strength, Young’s modulus, and modulus ratio (Young’s modulus to uniaxial compressive strength) and Fig. 12 summarizes the compiled triaxial strength data. The micro-parameters determined from calibrating to the Lodève sandstone specific characteristics resulted in typical properties of the compiled strong brittle low-porosity sandstone data (Table 4). Since the micro-parameters from the calibration resulted in the typical characteristics summarized in Table 4 and reproduced the selected Lodève sandstone characteristics in direct shear, it is concluded that the model is calibrated. The sensitivity of the calibration to the intra-grain and grain boundary tensile bond strengths is explored in a companion paper (Bewick et al. 2013).

Table 4 Compiled strong brittle low-porosity sandstone data

Fig. 12

Compiled strong brittle low-porosity sandstone triaxial data (grey circles) plotted in normalized principal stress space (σ ₁/UCS versus σ ₃/UCS) in comparison to DEM biaxial simulation results (white squares). Compiled data from Franklin and Hoek (1970), Santarelli and Brown (1989), and Kovari et al. (1983)

7 Forces, Mechanisms, and Fracture System Development

First, the distribution of forces in the synthetic specimens is investigated followed by the fracturing processes leading to rupture zone creation and the rupture mechanism for various applied normal stress magnitudes and for aspect ratios of 1:1 and 1.5:1. This is followed by an investigation into the internal stress path as determined from the measurement circle (10 mm diameter) in the centre of the synthetic specimen (see Fig. 5).

7.1 Distribution of Forces in Synthetic Specimen with two Aspect Ratio

No studies have been completed on how shear box aspect ratios influence the rupture zone characteristics in intact brittle rocks. Wang and Gutierrez (2010) recommend shear box aspect ratios of 1.5–2.0 for granular materials and concluded that when the aspect ratio of the shear box is small (i.e., H approaching L; towards at 1:1 geometry), global failure of the entire specimen occurs because the contact force chains that develop from the shear box lateral boundaries propagate further into the centre of the specimen avoiding the influence of the top and bottom boundaries of the shear box. Force chains show how a particle assembly is transmitting compressive forces along arrays of contacting particles with thicker black lines in Fig. 13 indicating higher particle contact forces. Large aspect ratios (long thin shear box geometries) were found to have contact force chains that are influenced by the top and bottom boundaries, with failure of the specimen occurring via propagating shear bands from the shear box lateral boundaries towards the centre of the specimen (Wang and Gutierrez 2010).

Fig. 13

Force chain networks in the 1:1 and 1.5:1 aspect ratio synthetic specimen at the indicated horizontal displacement of the lower shear box wall (δ _h). a 5 MPa normal stress. b 25 MPa normal stress. c 90 MPa normal stress. d From Wang and Gutierrez (2010). e From Zhang and Thornton (2007). f From Cho et al. (2008). g From Dyer and Milligan (1984). h From Allersma (2005). Light to black low- to high-compressive forces except g which is opposite. For additional description see text

The force chains in 1:1 and 1.5:1 synthetic specimens for normal stress magnitudes of 5, 25, and 90 MPa are shown in Fig. 13a–c for the indicated horizontal displacement and are shown along with the 1.57:1 aspect ratio specimen of Wang and Gutierrez (2010) (Fig. 13d). The force chain networks in the synthetic specimens in Fig. 13a–c show that they develop towards the centre, are minimally influenced by the top and bottom shear box boundaries, concentrate in the lower left and upper right corners, and are inclined at increasing angles about the horizontal due to the applied normal stress. The force chain distributions described are similar to that of Wang and Gutierrez (2010) (Fig. 13d). The force chains are also comparable to the DEM results of Zhang and Thornton (2007) (Fig. 13e) and Cho et al. (2008) (Fig. 13f) and laboratory experiments for crushed glass (Dyer and Milligan 1984) (Fig. 13g) and glass particles (Allersma 2005) (Fig. 13h). In summary, there are minimal differences between the force chain networks which develop in the 1:1 and 1.5:1 synthetic specimens, they are both comparable to other DEM and granular material assemblies, and they show symmetry of loading in the synthetic specimen. Based on the force chains, the 1:1 shear box aspect ratio is suitable for simulation.

7.2 Fracturing Process Leading to Rupture and Rupture Mechanism

Under constant normal stress, the rupture zone creation process and overall rupture mechanism are dependent on normal stress (σ _n) with rupture zone creation occurring in the post-peak region of the shear stress versus horizontal displacement curve. The results are grouped into normal stress ranges where similar fracturing processes and rupture mechanisms occur in the synthetic specimen: low (5–15 MPa or σ _n/UCS 0.03–0.10) where tensile splitting dominates; moderate (25–40 MPa or σ _n/UCS 0.17–0.28) where shear rupture via en échelon tensile fracture arrays dominates; and high (60–90 MPa or σ _n/UCS 0.41–0.62) where shear rupture via en échelon shear fracture arrays dominates (simulation UCS from Table 4). The 5, 25, and 90 MPa normal stress simulations are representative of the three groups and are used here to illustrate the fracturing process leading to rupture zone creation and the resulting rupture mechanism in each group.

In each representative simulation, fracture development is assessed in a number of stages relative to the shear stress versus (applied) horizontal displacement curve. In the DEM simulations, at each stage intra-grain (parallel bond) and grain boundary (smooth joint) fracture development are recorded (number, location, mechanism, and orientation) providing a description of the process leading to rupture zone creation. Fracture orientation is for the fractured bond between two particles or along a smooth joint contact. Since fracture systems form from a number of individual fractures in the synthetic specimen, systems of fractures were sketched (as previously discussed in Sect. 6.1 and shown in Fig. 11) to determine fracture trends at each stage. Orientations of the sketched fractures are plotted on rose diagrams and summarized statistically (average and standard deviation) where positive angles are counter clockwise and negative angles clockwise from the horizontal. The shear box geometry influences the stress conditions near the lateral walls. Therefore, the fractures that develop near the shear box lateral walls are not considered.

The representative simulations are graphically displayed in the same format in Figs. 14, 15, 16, which show from top to bottom: (1) shear box with rupture zone geometry and area indicated where detailed fracture system angles are assessed (AA) (top left); (2) chart with the shear stress versus horizontal displacement curve, orientation of major principal stress (σ ₁, determined as outlined in Sect. 5), and grain boundary (GB) and intra-grain (IG) cumulative fracture counts (top right); (3) rupture zone detail at indicated horizontal displacement and internal σ ₁ orientation (BB and CC indicate the locations of where particle displacement or velocity vectors are viewed); (4) sketches of the rupture zone detail; (5) rose diagrams of the fracture system angles based on sketched detail; and (6) mechanism of fracture and or rupture plotted using particle displacement or velocity vectors which show the explicit movement of particles in the simulations and thus mechanism (bond rupture mechanisms cannot be used because all initial fracturing is tensile in the model). They are used to classify the fractures forming in the synthetic specimens as follows: (1) tensile fracture—a fracture or system of fractures formed when particle velocity and or displacement vectors indicate opening; and (2) micro-fault or shear—a fracture or system of fractures formed when particle velocity or displacement vectors indicate a shear condition (a detailed example on how the vectors show the various mechanisms is shown in Fig. 21).

Fig. 14

5 MPa normal stress rupture zone creation and mechanism (1:1 aspect ratio) (orange GB, black IG tensile fracture). In the fracture system sketches, black are new and grey are precursory fractures from the previous rupture sketch. See text for detailed discussion (color figure online)

Fig. 15

25 MPa normal stress rupture zone creation and mechanism (1:1 aspect ratio) (orange GB, black IG tensile fracture). In the fracture system sketches, black are new and grey are precursory fractures from the previous rupture sketch. See text for detailed discussion (color figure online)

Fig. 16

90 MPa normal stress rupture zone creation (1:1 aspect ratio) (orange GB, black IG tensile fracture). In the fracture system sketches, black are new and grey are precursory fractures from the previous rupture sketch. See text for detailed discussion. See Fig. 17 for displacement vectors in locations BB and CC (color figure online)

7.2.1 1:1 Aspect Ratio

At low normal stresses (5–15 MPa or σ _n/UCS 0.03–0.10) where tensile splitting of the specimen dominates (see Fig. 14):

• prior to peak shear strength, grain boundary (GB) fracturing occurs oriented on average 15° predominately in line with the internal orientation of the major principal stress (13°–25°);

• at or just after peak, tensile rupture occurs across the synthetic specimen (indicted by particle velocity vectors at time of rupture creation, Fig. 14 BB). The major principal stress is oriented parallel to the rupture at creation, 13°. The rupture zone is thin, relatively continuous, and composed of both grain boundary and intra-grain tensile fractures (orange and black fractures, respectively);

• with continued horizontal displacement post-peak strength, fracturing leads to increased rupture zone connectivity, a transition from a tensile rupture to one that shows shear displacement (as indicated by particle velocity vectors on the lower half of the rupture which have rotated to indicate shear along the rupture surface opposed to predominately opening, Fig. 14 CC), with increasing occurrence of intra-grain (IG) fractures as evident from the cumulative fracture counts; and

• a fracture related to the shear box boundary is evident but does not influence the results. The boundary fracture occurs at low normal stress magnitudes because tensile stresses and low compressive force chains develop in the area where the fracture initiates. This is not a unique stress condition to the simulations and is also evident in, for example, the force chains of Wang and Gutierrez (2010) (Fig. 13d) where the light areas are zones of low compressive force chains and zones with tensile stress potential.

At moderate normal stresses (25–40 MPa or σ _n/UCS 0.17–0.28) where shear rupture via en échelon tensile fracture arrays dominates in the specimen (see Fig. 15):

• rupture zone creation occurs in a progressive manner;

• up to peak shear strength, the fracture angles average 22° and are in line with the internal major principal stress (25°–35°) when being created;

• initially, an en échelon tensile fracture array develops (as indicated by particle displacement vectors showing opening at time of array fracture system creation, Fig. 15BB) composed of both GB and IG fractures;

• with continued horizontal displacement, the en échelon tensile fractures turn into shear structures (as evident from the displacement vectors which have rotated and show shear along the initially tensile fracture system array, Fig. 15CC). Fractures from the shear structure tips in the array then propagate and other shallow angle fracture systems develop creating linkage across the array; and

• relative to rupture at low normal stresses (Fig. 14), the rupture zone is created in a more progressive manner, is thicker, and contains steeper fracture angles. Fractures related to the shear box boundaries are not as evient. Fractures propagating from the shear box gaps are evident but fracturing in the centre of the synthetic specimen occurs first.

At high normal stresses (60–90 MPa or σ _n/UCS 0.41–0.62) where shear rupture via en échelon shear fracture arrays dominates in the specimen (see Figs. 16, 17):

Fig. 17

90 MPa normal stress rupture mechanism (1:1 aspect ratio) as determined from particle displacement vectors (as shown). a Shallow angle fracture system showing synthetic sense of shear (boxed-in area) (BB in Fig. 16). b Steep angle fracture system showing antithetic sense of shear (boxed-in area) (CC in Fig. 16) (orange GB, black IG tensile fracture) (color figure online)

• fractures propagate from the shear box gaps at the first peak as indicated by (I);

• fracturing in the centre of the synthetic specimen eventually occurs well after the first peak. Fractures are initially oriented 36° on average and are in line with the internal major principal stress oriented 34°–37°;

• with continued horizontal displacement, first steep and then shallow fracture systems develop. They are themselves composed initially of en échelon tensile fracture arrays that then coalesce to form individual fracture systems with particle displacement vectors indicating shear at time of creation (Figs. 17, 21). The shallow angle fracture systems have a synthetic shear sense and are related to the applied principal displacement of the lower shear box wall while the steep angle fracture systems have antithetic shear sense. Antithetic refers to a sense of shear opposite while synthetic refers to a sense of shear in the same direction as that applied;

• the fracture systems eventually form a linked rupture zone across the synthetic specimen; and

• relative to the rupture zone that is created under moderate normal stresses (Fig. 15), the high normal stress rupture zone develops in an even more progressive manner, has initial fractures which propagate from the shear box gaps, is thicker, and contains even steeper fracture angles. Fractures related to the shear box boundaries are not evient.

7.2.2 1.5:1 Aspect Ratio

The synthetic specimens are 75 × 50 mm (L × H) and are composed of 62,082 particles. Normal stress magnitudes of 5, 25, 40, and 90 MPa were simulated. The shear stress versus horizontal displacement curves and linear Coulomb strength envelope are shown in Fig. 18. Comparing the shear stress versus horizontal displacement response and linear Coulomb strength envelopes of the 1.5:1 to the 1:1 aspect ratio synthetic specimens, additional horizontal displacement is needed to reach peak shear strength in the 1.5:1 aspect ratio specimens (i.e., lower shear modulus) (Fig. 18a) but the linear Coulomb strength envelopes are essentially identical (Fig. 18b). The fracturing process and mechanisms leading to rupture zone creation in the 1.5:1 synthetic specimens are similar to those described for the 1:1. Table 5 summarizes the pre-peak shear strength fracture angles and orientation of the major principal stress for the 1:1 and 1.5:1 aspect ratios to illustrate that they are nearly identical.

Fig. 18

Comparison of synthetic specimens with 1:1 and 1.5:1 aspect ratios. a Shear stress versus horizontal displacement responses (grey lines 1:1 and black lines 1.5:1 aspect ratios). b Linear Coulomb strength envelopes

Table 5 1:1 and 1.5:1 aspect ratio pre-peak fracture angles and major principal stress orientation summary

7.2.3 Internal Principal Stress Path

The internal principal stress path determined from the measurement circle in the centre of the synthetic specimen is shown in Fig. 19. At increasing normal stresses, the path becomes more compressive. The change in internal stress path supports the observed change in rupture mechanism from tensile to dominantly shear at higher normal stresses and is one of the dominate reasons for the mechanism change.

Fig. 19

Internal principal stress path determined from the measurement circle in the centre of the 1:1 aspect ratio synthetic specimen

8 Discussion

8.1 Rupture Mechanism and Geometry Dependence on Normal Stress

As normal stress is increased in direct shear under constant normal stress boundary conditions, the fracturing process leading to rupture zone creation and the related rupture mechanism change. The tensile splitting and brittle mode of rupture at low normal stresses which transitions to a more progressive, ductile, shear rupture zone creation process at higher normal stresses is supported by shear box laboratory experiments conducted by Lajtai (1969) using plaster-of-Paris, Sonnenberg et al. (2003) and Wong et al. (2005) using concrete, and by the rupture change reported by Petit (1988) in specimens of Lodève sandstone.

The simulations show that the rupture mechanism change is a result of the internal orientation of the major principal stress which tends to shallower angles at low and steeper angles at higher normal stresses and the internal principal stress path which transitions from dominantly tensile to compressive at the scale measured with the internal measurement circle (i.e., 10 mm diameter). The change in fracturing process leading to rupture zone creation and the resulting mechanism generates different rupture zone geometries with respective shear stress versus horizontal displacement (load–displacement) behaviour changing from brittle (loss of shear stress post-peak strength) to overall ductile (maintained shear stress post-peak strength). This demonstrates, as expected, that the load–displacement behaviour of a shear rupture zone (one of its characteristics) is not only a function of the properties of the rock in which the rupture zone is created but also the mechanism leading to a rupture zone’s geometry. Importantly, from a field observation perspective, rupture zones created in otherwise identical rocks will look different depending on the state-of-stress at the time of creation.

8.2 Shear Rupture Process

The shear rupture creation process in the synthetic specimens has similarities with the rupture processes proposed for the brittle crust by Crider and Peacock (2004), and observed in intact rock subjected to torsion (Cox and Scholz 1988a, b). In the numerical simulations at normal stresses of 25 and 40 MPa, initial en échelon fracture system arrays are created and are predominately of a tensile mechanism at their time of creation (see Fig. 15). These arrays then transition into shear structures which connect as stresses in the specimens rotate (Figs. 14, 15, 16). At normal stresses of 60 and 90 MPa, initial en échelon fracture system arrays are created and are predominately of a shear mechanism at their time of creation (but are created from individual tensile fractures as shown in Fig. 21 and summarized in Sect. 7.2.1). In both normal stress divisions for the simulation results, once the initial array of fractures are acting as shear structures, the rupture zone creation process generally follows the trend of the torsion experiments (Fig. 22) of Cox and Scholz (1988a, b) where with continued deformation, fractures propagate from the tips of the fractures in the en échelon arrays, after a period of applied displacement these tip fractures stop growing (Fig. 22a, b) and new shallower angle fracture systems interconnect the initial arrays of en échelon fractures (Fig. 22b, c). Finally, a continuous horizontal displacement fracture is created isolating lenses of material accommodating all of the displacements associated with the applied shear.

Fig. 20

90 MPa shear rupture zone in the specimen with 1.5:1 aspect ratio. Boxed-in area is location of rupture detail in Fig. 21

Fig. 21

Shear rupture process for one fracture system in the initial en échelon array of fractures as observed in the synthetic specimen at 90 MPa normal stress in Fig. 20. a–d Show the creation of an antithetic micro-fault in the synthetic specimen. First an array of en échelon tensile fractures develops (a). This array is composed of tensile intra-grain (black) and grain boundary (orange) fractures. Stresses then rotate due to shear box lower wall displacement and local interactions of fractures in b. Some precursory tensile fractures begin to shear (b, c). Once the array links the now larger fracture system begins to show shear displacement (c, d). e, f Show the explicit fracture mechanisms (shown schematically in a–d) using particle velocity vectors at 0.41 and 0.45 mm of horizontal displacement, respectively. In e ‘I’ shows a fracture being created in tension and ‘II’ shows an initially tensile fracture starting to show shear displacement. In f ‘III’ shows shearing along the upper portion of the fracture and ‘IV’ shows new fractures being created in tension (color figure online)

Fig. 22

Sequential rupture zone creation (a–c) in the torque experiments of Cox and Scholz (1988a) at increasing twists (a proxy for displacement). Redrawn from Cox and Scholz (1988a). For additional description see text

8.3 Grain Scale Micro-Mechanics

The dependence of the geometry of the fracture systems to applied normal stress magnitudes compared to those reported by Wibberley et al. (2000) are clearly predicted by the numerical simulation framework presented in this article. Although, in the simulations, first grain boundary fractures initiate during shear displacement which are then subsequently linked by intra-granular fractures. This is opposite to the grain scale failure mechanism reported by Wibberley et al. (2000) where trans-granular fractures dominated the rupture process in the Lodève sandstone specimens and these trans-granular fractures were typically linked via grain boundary fractures. The fact that the geometry of the fracture systems are reproduced in the simulations despite the different grain scale micro-mechanical rupture process suggests that the results are not highly dependent on the grain system (i.e., size, shape, and mechanical properties) but on the boundary condition.

The GBM was developed for particle-based DEMs to capture the appropriate tensile strength to uniaxial compressive strength ratio of hard brittle rocks (Potyondy 2010) which was previously limited to a ratio of around 0.25 (Cho et al. 2007). Thus, grain boundaries were introduced to allow a mechanism of tensile splitting to occur in a synthetic specimen at a lower strength (i.e., grain boundaries control the strength in tension while the grains control the strength in compression). The results here show that, as a consequence of the lower grain boundary strength, the grain scale failure mechanism becomes dominated by grain boundary opposed to intra-grain failure (i.e., fracture development in the grains is inhibited by the grain boundaries). The influence of grain boundary and intra-grain bond strength and their influence on the synthetic specimen’s grain scale failure process, compressive strength, direct tensile strength, rupture zone geometry, shear stress versus horizontal displacement response, and peak and ultimate direct shear strength envelopes are investigated in a companion paper (Bewick et al. 2013).

9 Conclusions

This investigation focused on understanding shear rupture zone creation in intact brittle rock deformed in direct shear under constant normal stress boundary conditions using a particle-based DEM and its embedded GBM. Numerical simulations were completed using a synthetic specimen that was calibrated to the strength (shear and tensile), post-peak shear stress versus horizontal displacement response, and fracture angle characteristics of Lodève sandstone, a brittle low-porosity rock, when deformed in direct shear.

It was found that:

• At low normal stresses (5–15 MPa or σ _n/UCS 0.03–0.10), the synthetic specimen ruptures in a predominantly tensile splitting mode; a process that occurs at or just after peak shear strength.

• At higher normal stresses (25–90 MPa or σ _n/UCS 0.17–0.62), the synthetic specimen ruptures progressively in a shear mode. First, an array of en échelon fractures develops followed by linkage of the array across the synthetic specimen leading to a shear rupture with related damage zone.

• The GBM was able to capture the tensile strength, linear Coulomb strength envelope, post-peak shear stress versus horizontal displacement response, and fracture angles of Lodève sandstone ruptured in direct shear. The grain scale micro-mechanical failure process was not captured and the reasons for this discussed.

At higher normal stresses, rotation of the internal major principal stress in the synthetic specimen tends to higher angles about the horizontal and the internal stress field becomes increasingly compressive. This causes fracturing at higher angles about the horizontal and limits fracture length with increasing applied normal stress. The shallower orientations of the major principal stress and a predominately tensile stress field at low normal stresses facilitate splitting while the higher major principal stress orientations and predominately compressive stress field at higher normal stresses do not allow for splitting of the synthetic specimen. As a result, a progressive shear rupture fracturing process is required to generate a rupture across the synthetic specimen.