Numerical modelling of a Brazilian Disc test of layered ...geogroup.utoronto.ca/wp-content/uploads/rockeng09/Session8/4148... · 1 INTRODUCTION Brazilian test (first developed by

1 INTRODUCTION

Brazilian test (first developed by Berrenbaum & Brodie 1959) is a common means of indirectly measuring the tensile strength of brittle materials including rocks and concrete. The test proce-dure and sample preparation, as suggested by the International Society for Rock Mechanics (ISRM 1978), is relatively simple for this test.

The indirect tensile strength of a cylindrical sample with radius R and thickness t (as shown in Figure 1), is given as

RtP

t πσ max= (1)

However, this equation is obtained analytically assuming the rock to be isotropic and homo-geneous. Unfortunately, it is almost always to opposite. Sedimentary rocks and schistose rocks are inhomogeneous and anisotropic materials. This means that typical analytical solutions based on the assumption of homogeneity cannot be applied since the strength of the bedding or schis-tose planes are the weakest point. Therefore, it is necessary to undertake experimental and nu-merical studies to correctly assess the strength of these materials.

The combined finite-discrete element method (FEM/DEM) is a numerical technique capable of dealing with mechanics of discontinuum. In this study, a modified FEM/DEM research code (Munjiza 2004) has been used to study the behaviour of layered rock samples under standard Brazilian tests. The effect of layering and the direction of loading on the samples behaviour are studied. The results show a promising potential for the use of FEM/DEM for such studies.

Numerical modelling of a Brazilian Disc test of layered rocks using the combined finite-discrete element method

O. K. Mahabadi & G. Grasselli University of Toronto, Toronto, Canada

A. Munjiza Queen Mary, University of London, London, UK

ABSTRACT: The scope of this study is to simulate the behaviour of a layered rock sample un-der standard laboratory Brazilian Disc test using an innovative combined finite-discrete element method (FEM/DEM) research code. The influence of layering and the direction of loading on the samples behaviour are studied through various layering and loading configurations. A para-metric study including the effect of loading rate on the tensile strength of samples has been car-ried out. This paper demonstrates the suitability of the numerical approach to explicitly model rock deformation and failure.

ROCKENG09: Proceedings of the 3rd CANUS Rock Mechanics Symposium, Toronto, May 2009 (Ed: M.Diederichs and G.Grasselli)

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Figure 1. Schematic of a Brazilian Disc for indirect measurement of tensile strength.

2 NUMERICAL SIMULATIONS 2.1 The combined finite-discrete element method

The existence of discontinuities in rocks together with fracture and fragmentation processes limits the applicability of continuum-based models to address rock engineering problems. Therefore, various discontinuum approaches have been developed in the past to overcome these limitations. They include the use of discrete elements methods (DEM) (Goodman et al. 1968; Cundall 1971; Cundall & Strack 1979; Williams & Mustoe 1993; Mustoe & Williams 1989; Lemos et al. 1985), discontinua deformation analysis (DDA) (Shi & Goodman 1988), and combined finite-discrete element method (FEM/DEM) (Munjiza et al. 1995). In contrast to continuum models which are based on constitutive laws, discontinuum models are based on interaction laws (Munjiza 1999). Thus, contact detection and interaction between individual bodies, deformability and fracture of the bodies are the key processes in discontinuum methods. In FEM/DEM each discrete element is discritised into finite elements which means that there is a finite element mesh associated with each discrete element. These meshes define the shape of discrete elements, contact between them and their deformability. Thus, continuum behaviour is modelled through finite elements while discon-tinuous behaviour is analysed by discrete elements (Munjiza & John 2002).

In the context of the combined finite-discrete element method, transition from continua to discontinua is done through fracture and fragmentation processes (Munjiza 2004). A combined single and smeared crack model is implemented in the FEM/DEM code used for this study. In this model, a typical stress-strain curve is divided into two sections. The first part corresponding to strain hardening prior to reaching the peak stress (i.e. tensile strength) is implemented through the constitutive law as in any standard finite element method. The second part, related to the post-peak behaviour, refers to strain-softening and is formulated in terms of stress and displacements (Munjiza et al. 1999). The softening stress-displacement relationship is modelled through a single crack model. A bonding stress is generated due to the separation of the edges. This stress is assumed to be a function of the size of separation (or crack opening). Further de-tails can be found in Munjiza (2004).

2.2 Simulation cases Using the Y-GUI (Mahabadi et al. 2009), four distinct models are considered for this study: Model A as shown in Figure 2a corresponds to a homogeneous disc, while models B to D shown in Figure 2c-d represent a layered rock sample. The difference between models B to D is the angle between the direction of loading and the orientation of the bedding planes, which is at 0, 60°, and 90° (from the horizontal axis), respectively. The radius of the discs is 50 mm and their thickness is assumed to be 1 mm. The same material properties are assigned to all layers in


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models B-D. The layers are separated by weak interfaces (bedding planes). The assigned mate-rial properties are summarized in Table 1. Note that the high shear strength value of the rock sample ensures that the rock material does not fail in shear, but rather in tension. A constant ver-tical velocity is applied to the loading platens.

Figure 2. Simulation models for this study: (a) Model A – homogeneous, (b) Model B (layered) – inclina-tion angle, β = 0, (c) Model C (layered) – inclination angle, β = 60°, (d) Model D (layered) – inclination angle, β = 90°. Note: The colours only represent layering of the sample.

Table 1. Material properties of the rock sample, the joint between the layers and the loading platens

Parameter Rock sample Joint Loading platens Young’s Modulus (GPa) 50 - 193Poisson’s ratio (-) 0.25 - 0.29Friction coefficient (-) 0.2 - 0Density (kg/m3) 2700 - 8030Shear strength (MPa) 3.15 × 1010 30 -Tensile strength (MPa) 5 3.5 -Fracture Energy (N/m) 50 50 -

2.3 Results The results of the various models are presented here. Unless otherwise mentioned, the loading rate applied is 0.1 m/s.

2.3.1 Homogeneous rock Fracture propagation of the homogeneous rock sample (Model A) is shown in Figure 3. Accord-ing to this figure, cracks initiate at the centre of the disc when tensile stresses exceed the tensile strength of the material. These cracks then propagate toward the loading platens (Figure 3b) completing primary fracturing. At later stages (Figure 3c, d), further cracks appear on the sides of the primary crack (secondary fracturing). Figure 4 shows the experimental fracture propaga-tion pattern for a Brazilian test (Malan et al. 1994 after Colback 1966). Comparison of the two figures reveals that the numerical simulations are capable of capturing primary and secondary fracturing properly. However, probably due to the very large shear strength assigned to the rock, tertiary fracturing does not occur in the model.

The reaction load on the platens together with the platens vertical displacement are recorded and illustrated in Figure 5. The sample shows a perfect linear behaviour until the peak is reached where the cracks initiate at the centre of the disc. This is in agreement with experimen-tal results. This stage is quickly followed by the crack propagation towards the two platens meaning that the load-bearing capacity of the disc suddenly decreases. For this model, the simu-lated tensile strength is 6.26 MPa which is greater than the input value (5 MPa). This is mainly due to the fact that the applied loading rate (0.1 m/s) is much higher than those used in labora-tory experiments (~0.01 mm/s). It is known than the dynamic strength of materials is always higher than their quasi-static one (for instance refer to Zhao & Li 2000; Brara et al. 2001; Cho et


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al. 2003; or Zhu & Tang 2006 for a list of other references). To numerically support this, a pa-rametric study of the influence of loading rate on the results is given in a later section of this manuscript. Also, a finer mesh would reduce the strength of the disc. However, due to con-straints on computation time, higher strain rates and coarser mesh (3000 elements) were applied.

Figure 3. Fracture propagation at different time steps for the homogeneous model (Model A) after (a) 2.1 ms, (b) 2.16 ms, (c) 2.25 ms, and (d) 3.6 ms. Colours represent vertical stresses (σyy).

(a) (b) (c)

Figure 4. Typical fracture propagation in a Brazilian test of a homogeneous rock sample; (a) primary ten-sile cracking along the loading diameter; (b) secondary cracking from the sides parallel to the primary cracks; (c) tertiary cracking due to shear failure near the platens (Malan et al. 1994 after Colback 1966).

Figure 5. Load – displacement curve for the homogeneous rock (Model A). Note that the platens were not in contact when the test started and moved 0.05 mm to reach the disc.


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2.3.2 Layered rock This study aims at simulating the behaviour of anisotropic schistose rocks. In order to explicitly model it, we used weak joint interfaces between layers of same material. These weak interfaces can fail both in tension and shear. Such mixed behaviour allows the sample to fail in tension at the centre of the disc and in shear along the interfaces close to the platens. Note that these joints are neither straight nor smooth.

Figure 6 presents the comparison between the load-displacement curves of the three layered

models (B-D) and that of Model A. As this figure shows, the pre-peak behaviour of the models is very similar. However, the peak loads and the platen displacements needed to reach the peak decrease as the angle β increases. The simulated tensile strength of Models B, C, and D are 6.21 MPa, 5.33 MPa, and 5.00 MPa, respectively.

Similar to the homogeneous rock (Model A), the failure mechanism for β = 0 and 90° is ten-

sile. Cracks initiate at the centre of the disc and propagate towards the loading plates. In con-trast, for β = 60° firstly shear failure occurs at the interface between layers near the platen where high shear stresses are developed, Figure 7a. This is because, compared to the intact rock mate-rial, the layer interfaces have a much lower shear strength. Continuing the test, tensile cracks appear in the centre of the specimen, Figure 7b. This continues until total failure of the disc is reached, Figure 7c. These findings are in reasonable agreement with experimental results of Chen et al. (1998) and the numerical simulations of Cai & Kaiser (2004).

Note that, as shown in Figure 8, the pre-peak behaviour of the layered Brazilian disc at β = 0

is approximately the same as the homogeneous model. Once the peak is reached, one dominant vertical crack appears in the model while secondary and tertiary cracking does not occur and the disc shows less strength. This is shown in Figure 8d.

For β = 90° however, the weaker joint interface between the layers plays a major role in the

failure initiation, but this time in tension. Most cracks initiate along the joint interface and then propagate both on the joints and inside the layers, as shown in Figure 9. Model D is the weakest because the joint has a lower tensile strength (3.5 MPa compared to 5 MPa of the material).

Figure 6. Load – displacement curve for the four models. Note that the platens were not in contact when the test started and moved approximately 0.05 mm to reach the disc.


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Figure 7. Fracture propagation at different time steps for Model C – layered rock inclined at 60°; after (a) 1.8 ms, (b) 2.0 ms, and (c) 2.4 ms. Colours represent vertical stresses (σyy).

Figure 8. Fracture propagation at different time steps for Model B – layered rock inclined at 0° after (a) 2.1 ms, (b) 2.16 ms, (c) 2.25 ms, and (d) 3.6 ms. Colours represent vertical stresses (σyy)..

Figure 9. Fracture pattern at 2 ms for Model D – layered rock inclined at 90°. Note the relation between cracks and layering.


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2.3.3 Effect of loading rate The strength of the rock discs varies in accordance to the applied loading rate. It was claimed that one of the major reasons for obtaining higher tensile strength in this study was the high loading velocity. This was supported by references from experimental and numerical studies. To verify the ability of the FEM/DEM code to capture this behaviour, two other loading velocities (0.5 m/s and 1.0 m/s) were used for testing homogeneous samples. The results of these two simulations are compared with that of 0.1 m/s. A comparison graph is presented in Figure 10. This figure shows that, as expected, the peak load and hence the tensile strength increases with the increase of strain rate. Tensile strength values are 8.12 MPa, 7.00 MPa, and 6.26 MPa for loading rates of 1.0 m/s, 0.5 m/s, and 0.1 m/s, respectively.

Figure 10. The influence of loading rate on the load-displacement curves of the homogeneous rock.

2.3.4 Validity of the results Every numerical simulation is dependent on the choice of input parameters including material

properties, boundary conditions, numerical parameters (such as penalty terms), and the mesh geometry. Therefore, a careful choice of these variables is needed for the simulations results to be acceptable and physically sound. In this study, material properties are taken as typical values for a brittle hard rock. Most of the numerical parameters including the penalty term and viscous damping were estimated through an iterative calibration process. Critical viscous damping was applied using the recommended value of ρEh2 where h is the average size of elements, E the Young’s modulus of elasticity, and ρ the density of the rock material. Penalty was assigned to be almost equal to E (Munjiza 2004).

In order to simulate standard Brazilian tests, a slow loading velocity, typically in the order of 0.01 mm/s, should be applied to the loading platens. Considering the very small time step size needed for the numerical technique to be stable, i.e. 2.5 × 10-5 ms, there need to be approxi-mately 400 million time steps to break the disc, which would result in several weeks of comput-ing time. Therefore, faster loading rates of 0.1 m/s were applied which resulted in an overesti-mation of the tensile strengths with respect to the quasi-static conditions.

Another major factor in the quality of results is the mesh geometry. In general, the finer the mesh is, the more accurate the simulation results are. However, a finer mesh demands smaller time step sizes (note that the time integration scheme of the FEM/DEM code is explicit). In practical applications, the mesh should be sufficiently refined to an extent that the results are not significantly altered by the mesh size. For the presented application, we used a mesh size equal to one half of the average grain size of a coarse granitic rock.


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3 SUMMARY AND CONSLUSIONS

The applicability of the combined finite-discrete element method to model laboratory scale ex-periments was shown through the simulations of Brazilian disc tests. Both a homogeneous rock and a layered rock representing schistosity were used. The simulation results suggest that the presence of layers or bedding planes plays a major role in mechanical behaviour of the models. Also, direction of loading with respect to those planes of weakness is of great importance. The main failure mechanism for the homogeneous disc and those layered and inclined at 0° and 90° was tensile splitting. In contrast, the layered disc aligned at 60° showed mixed tensile splitting and shear failure. This is due to the weak joint interface which is approximately aligned along the direction of large shear stresses near the loading platens.

As explained before, the simulated tensile strength values were generally higher than the in-

put. Several reasons including loading rate and mesh sensitivity were given for this issue. How-ever, further research is needed to overcome this problem.

REFERENCES

Berenbaum, R., & Brodie, I. 1959. Measurement of the tensile strength of brittle materials. British Jour-nal of Applied Physics, 10(6), 281-287.

Brara, A., Camborde, F., Klepaczko, J. R., & Mariotti, C. 2001. Experimental and numerical study of concrete at high strain rates in tension. Mechanics of Materials, 33(1), 33-45.

Cai, M., & Kaiser, P. K. 2004. Numerical simulation of the Brazilian test and the tensile strength of ani-sotropic rocks and rocks with pre-existing cracks. International Journal of Rock Mechanics and Min-ing Sciences, 41(Supplement 1), 478-483.

Chen, C., Pan, E., & Amadei, B. 1998. Determination of deformability and tensile strength of anisotropic rock using brazilian tests. International Journal of Rock Mechanics and Mining Sciences and Geome-chanics Abstracts, 35(1), 43-61.

Cho, S. H., Ogata, Y., & Kaneko, K. 2003. Strain-rate dependency of the dynamic tensile strength of rock. International Journal of Rock Mechanics and Mining Sciences, 40(5), 763-777.

Colback, P. S. B. 1966. An analysis of brittle fracture initiation and propagation in Brazilian test. Pro-ceedings of the 1st Congress of the International Society of Rock Mechanics, Lisbon. , 1 385-391.

Cundall, P. A. 1971. A computer model for simulating progressive, large-scale movements in blocky rock systems. Paper ii-8, Rock Fractures.

Cundall, P., & Strack, O. D. L. 1979. A discreet numerical model for granular assemblies. Geotechnique, 29(1), 47-65.

Goodman, R. E., Taylor, R. L., & Brekke, T. L. 1968. A model for the mechanics of jointed rock. Jour-nal of the Soil Mechanics and Foundations Division, 94, 637-660.

ISRM. 1978. Suggested methods for determining tensile strength of rock materials. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 15(3), 99-103.

Lemos, J. V., Hart, R. D., & Cundall, P. A. 1985. A generalized distinct element program for modelling jointed rock mass; a keynote lecture; proceedings of the International Symposium on Fundamentals of Rock Joints, Bjorkliden, Sweden.

Mahabadi, O. K., Grasselli, G., Munjiza, A. 2009. Y-GUI: A Graphical user interface and pre-processor for the combined finite-discrete element code, Y2D, incorporating material inhomogeneity. Submitted to Computers & Geosciences Journal on August 11, 2008 – ref. no. CAGEO-D-08-00220.

Malan, D. F., Napier, J. A. L., & Watson, B. P. 1994. Propagation of fractures from an interface in a Bra-zilian test specimen. International Journal of Rock Mechanics and Mining Sciences, 31(6), 581-596.

Munjiza, A., Andrews, K. R. F., & White, J. K. 1999. Combined single and smeared crack model in combined finite-discrete element analysis. International Journal for Numerical Methods in Engineer-ing, 44(1), 41-57.

Munjiza, A., & John, N. W. M. 2002. Mesh size sensitivity of the combined FEM/DEM fracture and fragmentation algorithms. Engineering Fracture Mechanics, 69(2), 281-295.

Munjiza, A., Owen, D. R. J., & Bicanic, N. 1995. A combined finite-discrete element method in transient dynamics of fracturing solids. Engineering Computations, 12(2), 145-174.

Munjiza, A. 1999. Fracture, fragmentation and rock blasting models in the combined finite-discrete ele-ment method, In: Aliabadi, MH. (Ed.), Fracture of Rock, Computational Mechanics Publications, pp. 125-166.

Munjiza, A. 2004. The Combined Finite-Discrete Element Method. Hoboken, NJ: Wiley. 333 pp.


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Mustoe, G. & Williams J. R. 1989. Proc. 1st U.S. Conf. on Discrete Element Methods, Golden, Colorado. Shi, G., & Goodman, R. E. 1988. Discontinuous deformation analysis; a new method for computing

stress, strain and sliding of block systems; key questions in rock mechanics; proceedings of the 29th U. S. symposium on Key Questions in Rock Mechanics, Minneapolis, MN, United States.

Williams, J. R., & Mustoe, G. G. 1993. 2nd International conference on discrete element methods (DEM), Cambridge, Massachusetts, March 18-19, 1993.

Zhao, J., & Li, H. B. 2000. Experimental determination of dynamic tensile properties of a granite. Inter-national Journal of Rock Mechanics and Mining Sciences, 37(5), 861-866.

Zhu, W. C., & Tang, C. A. 2006. Numerical simulation of brazilian disk rock failure under static and dy-namic loading. International Journal of Rock Mechanics and Mining Sciences, 43(2), 236-252.


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Numerical modelling of a Brazilian Disc test of layered ...geogroup.utoronto.ca/wp-content/uploads/rockeng09/Session8/4148... · 1 INTRODUCTION Brazilian test (first developed by