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Strength of industrial sandstones modelled with the Discrete Element Method

Piotr Klejment
Piotr Klejment
, 
Robert Dziedziczak
Robert Dziedziczak
 and 
Paweł Łukaszewski
Paweł Łukaszewski
   | Oct 09, 2021
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Article Category: Research Article
Published Online: Oct 09, 2021
Page range: 346 - 365
Received: Dec 31, 2020
Accepted: Jul 08, 2021
DOI: https://doi.org/10.2478/sgem-2021-0020
Keywords
© 2021 Piotr Klejment et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1

The scheme of DEM simulation (after O’Sullivan 2011).
The scheme of DEM simulation (after O’Sullivan 2011).

Figure 2

The interaction of the two particles by the force–displacement law. The figure is redrawn from Ferdowsi (2014). Particles a and b of radii Ra and Rb are at a distance D with overlap C. Pa and Pb are the points of intersection of the line connecting the disc centres with the boundaries of the discs; d and t are unit vectors. Derivatives of r and θ stand linear damping and rotational damping, respectively.
The interaction of the two particles by the force–displacement law. The figure is redrawn from Ferdowsi (2014). Particles a and b of radii Ra and Rb are at a distance D with overlap C. Pa and Pb are the points of intersection of the line connecting the disc centres with the boundaries of the discs; d and t are unit vectors. Derivatives of r and θ stand linear damping and rotational damping, respectively.

Figure 3

Forces and moments between the particles bonded through rotational elastic-brittle bonds (Abe et al. 2014).
Forces and moments between the particles bonded through rotational elastic-brittle bonds (Abe et al. 2014).

Figure 4

Location of the active quarries on the map of Poland from which the tested sandstones originated.
Location of the active quarries on the map of Poland from which the tested sandstones originated.

Figure 5

Schematic representation of the uniaxial compression test. Sample is compressed from the top and bottom until failure. On the left – simplified scheme of the test, on the right – laboratory experiment.
Schematic representation of the uniaxial compression test. Sample is compressed from the top and bottom until failure. On the left – simplified scheme of the test, on the right – laboratory experiment.

Figure 6

Cracked samples after uniaxial compression test.
Cracked samples after uniaxial compression test.

Figure 7

Schematic representation of different possible failure patterns of the sample under uniaxial compression extracted and proposed by Basu et al. (2013).
Schematic representation of different possible failure patterns of the sample under uniaxial compression extracted and proposed by Basu et al. (2013).

Figure 8

Comparison between numerical and laboratory stress-strain curves for selected samples: (A) Brenna BR2, (B) Mucharz MU1, (C) Radków RAD1 and (D) Tumlin TU2.
Comparison between numerical and laboratory stress-strain curves for selected samples: (A) Brenna BR2, (B) Mucharz MU1, (C) Radków RAD1 and (D) Tumlin TU2.

Figure 9

Scaled number of bonds Nbonds/Nbonds max (number of bonds divided by maximum number of bonds) inside the sample as a function of axial displacement for all four types of sandstones.
Scaled number of bonds Nbonds/Nbonds max (number of bonds divided by maximum number of bonds) inside the sample as a function of axial displacement for all four types of sandstones.

Figure 10

Scaled number of fractures (Nfrac/Nfrac max-number of fractures divided by maximum number of fractures) as a function of axial strain with the stress–strain curve in the background: (A) Brenna, (B) Mucharz, (C) Radków, (D) Tumlin.
Scaled number of fractures (Nfrac/Nfrac max-number of fractures divided by maximum number of fractures) as a function of axial strain with the stress–strain curve in the background: (A) Brenna, (B) Mucharz, (C) Radków, (D) Tumlin.

Figure 11

(A) Scaled potential energy of bonds Epot/Epot max and (B) scaled kinetic energy Ekin/Ekin max for all four types of sandstones.
(A) Scaled potential energy of bonds Epot/Epot max and (B) scaled kinetic energy Ekin/Ekin max for all four types of sandstones.

Figure 12

Comparison between two scaled different components of kinetic energy (Eckin/Eckin max – kinetic energy divided by maximum kinetic energy) plotted as a function of strain: (A) Brenna, (B) Mucharz, (C) Radków, (D) Tumlin.
Comparison between two scaled different components of kinetic energy (Eckin/Eckin max – kinetic energy divided by maximum kinetic energy) plotted as a function of strain: (A) Brenna, (B) Mucharz, (C) Radków, (D) Tumlin.

Figure 13

Four different components of scaled potential energy of bonds (Ecpot/Ecpot max – potential energy of bonds divided by maximum potential energy of bonds) plotted together: (A) Brenna, (B) Mucharz, (C) Radków, (D) Tumlin.
Four different components of scaled potential energy of bonds (Ecpot/Ecpot max – potential energy of bonds divided by maximum potential energy of bonds) plotted together: (A) Brenna, (B) Mucharz, (C) Radków, (D) Tumlin.

Figure 14

Exemplary presentation of the numerical slab after two different tests: (A) failure resistance test, (B) impact resistance test. View of bonds.
Exemplary presentation of the numerical slab after two different tests: (A) failure resistance test, (B) impact resistance test. View of bonds.

Figure 15

Schematic representation of all four tests: (A) failure resistance test, (B) impact resistance test, (C) vibration resistance test and (D) abrasion resistance test. View of the cross-section of the numerical slab.
Schematic representation of all four tests: (A) failure resistance test, (B) impact resistance test, (C) vibration resistance test and (D) abrasion resistance test. View of the cross-section of the numerical slab.

Figure 16

Results of all four tests: (A) failure resistance test, dependence between scaled maximum force F/Fmax (force divided by maximum force) and sample thickness; (B) impact resistance test, dependence between scaled number of fractures Nfrac/Nfrac max (number of fractures divided by maximum number of fractures) which appeared in the sample after the test and sample thickness; (C) vibration resistance test, number of fragments into which the sample has fallen apart as a function of sample thickness; (D) abrasion resistance test, bulk friction coefficient of the sandstone.
Results of all four tests: (A) failure resistance test, dependence between scaled maximum force F/Fmax (force divided by maximum force) and sample thickness; (B) impact resistance test, dependence between scaled number of fractures Nfrac/Nfrac max (number of fractures divided by maximum number of fractures) which appeared in the sample after the test and sample thickness; (C) vibration resistance test, number of fragments into which the sample has fallen apart as a function of sample thickness; (D) abrasion resistance test, bulk friction coefficient of the sandstone.

Figure 17

Network of fractures which appeared in the sample after impact resistance test. View of bonds.
Network of fractures which appeared in the sample after impact resistance test. View of bonds.

Number of grains after failure.

Total number of grains Number of the smallest grains Number of middle grains Number of the biggest grains
Brenna 19,888 19,879 8 1
Mucharz 23,384 23,377 4 3
Radków 33790 33,764 14 12
Tumlin 152,953 152,944 8 1

Comparison of failure patterns in DEM modeling (initial model, and model just after the failure - view of bonds between particles) and in laboratory (laboratory failure).

Type Initial numerical model Numerical failure Laboratory failure
Brenna sample BR2
Mucharz sample MU1
Radków sample RAD1
Tumlin sample TU2

Range of results of geomechanical laboratory tests (mean values in brackets).

Lithological type ρ (kg/m3) UCS (MPa) E (GPa) ν (−)
Gogula sandstones from Brenna 2391 – 2444 (2429) 99 – 107 (102) 20.2 – 21.8 (20.8) 0.21 – 0.25 (0.22)
Krosno sandstones from Mucharz 2627 – 2668 (2655) 138 – 153 (143) 26.6 – 31.8 (29.2) 0.18 – 0.23 (0.19)
Joint sandstones from Radków 2119 – 2172 (2142) 55 – 60 (58) 22.5 – 24.9 (23.4) 0.20 – 0.29 (0.24)
Sandstones from Tumlin 2355 – 2480 (2448) 115 – 128 (123) 26.2 – 29.4 (27.4) 0.22 – 0.28 (0.25)

Detailed parameters of numerical models of sandstones.

Sandstone 1 Brenna Sandstone 2 Mucharz Sandstone 3 Radków Sandstone 4 Tumlin
Sample geometry Cylinder:height – 75 mm;radius – 18.75 mm Cylinder:height – 75 mm;radius – 18.75 mm Cylinder:height – 75 mm;radius – 18.75 mm Cylinder:height – 75 mm;radius – 18.75 mm
Particles radii 0.25 – 2.0 mm 0.25 – 0.5 mm 0.25 – 1.0 mm 0.125 – 0.5 mm
Particles density 2429 kg/m3 2655 kg/m3 2142 kg/m3 2448 kg/m3
Initial number of particles 82,266 339,324 150,653 1,155,851
Initial number of bonds 297,847 821,488 500,024 3,831,657
Time step 1.23×10−5s 8.32×10−6s 1.06×10−5s 5.14×10−6s
Parameters of bonds :- Young's modulus Eb (MPa)- Poisson's ratio vb (dimensionless)- cohesion cb (MPa)- tangent of internal friction angle φb (dimensionless) Eb = 5 284 MPavb = 0.25cb = 13.59 MPatan φb = 1.0 Eb= 50 258 MPavb= 0.25cb = 205.33 MPatan φb = 1.0 Eb = 12 492 MPavb = 0.25cb = 21.19 MPatan φb = 1.0 Eb = 15 144 MPavb = 0.25cb = 44.7 MPatan φb = 1.0
Stroke rate 0.2 mm/s 0.2 mm/s 0.2 mm/s 0.2 mm/s
Computational time 287 min 667 min 335 min 1440 min
Number of processors 92 228 93 320

Comparison between crucial macroscopic parameters used for model calibration. ‘Sim’ stands for simulation result and ‘lab’ stands for result from the laboratory experiment.

Selected test E (GPa) UCS (MPa) ez cr (%) ν (−)
Brenna sample BR2 Sim: 20.8Lab: 20.5 Sim: 98Lab: 99 Sim: 0.51Lab: 0.52 Sim: 0.21Lab: 0.25
Mucharz sample MU1 Sim: 29.2Lab: 31.8 Sim: 136Lab: 138 Sim: 0.47Lab: 0.49 Sim: 0.19Lab: 0.23
Radków sample RAD1 Sim: 23.4Lab: 22.8 Sim: 61Lab: 60 Sim: 0.28Lab: 0.29 Sim: 0.21Lab: 0.20
Tumlin sample TU2 Sim: 27.4Lab: 26.2 Sim: 118Lab: 115 Sim: 0.46Lab: 0.51 Sim: 0.26Lab: 0.22
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