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3D DEM simulations of basic geotechnical tests with early detection of shear localization

Aleksander Grabowski
Aleksander Grabowski
 and 
Michał Nitka
Michał Nitka
   | Dec 04, 2020
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Article Category: Original Study
Published Online: Dec 04, 2020
Page range: 48 - 64
Received: Jun 23, 2020
Accepted: Oct 16, 2020
DOI: https://doi.org/10.2478/sgem-2020-0010
Keywords
© 2021 Aleksander Grabowski et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1

Two spheres in contact with forces and momentum acting on them ( F→n{\vec F_n} - normal contact force, F→s{\vec F_s} – tangential contact force, M→n{\vec M_n} - contact moment force and n→\vec n - contact normal vector) [48].
Two spheres in contact with forces and momentum acting on them ( F→n{\vec F_n} - normal contact force, F→s{\vec F_s} – tangential contact force, M→n{\vec M_n} - contact moment force and n→\vec n - contact normal vector) [48].

Figure 2

Mechanical response of (a) tangential (b) normal and (c) rolling contact model laws [45].
Mechanical response of (a) tangential (b) normal and (c) rolling contact model laws [45].

Figure 3

Model set-up for numerical simulations of triaxial test.
Model set-up for numerical simulations of triaxial test.

Figure 4

Discrete simulations of homogeneous triaxial compression test compared to experiments by Wu [1]: A) vertical normal stress σ1 and B) volumetric strain ɛv versus vertical normal strain ɛ1 from a) numerical results and b) experimental results (e0 = 0.53, d50 = 5.0 mm, σ0 = 50, 200 and 500 kPa).
Discrete simulations of homogeneous triaxial compression test compared to experiments by Wu [1]: A) vertical normal stress σ1 and B) volumetric strain ɛv versus vertical normal strain ɛ1 from a) numerical results and b) experimental results (e0 = 0.53, d50 = 5.0 mm, σ0 = 50, 200 and 500 kPa).

Figure 5

Vertical normal stress σ1 versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial void ratios e0: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 5.0 mm).
Vertical normal stress σ1 versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial void ratios e0: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 5.0 mm).

Figure 6

Volumetric strain ɛv versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial void ratios e0: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 5.0 mm).
Volumetric strain ɛv versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial void ratios e0: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 5.0 mm).

Figure 7

Void ratio e versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial void ratios e0: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 5.0 mm).
Void ratio e versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial void ratios e0: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 5.0 mm).

Figure 8

Internal friction angle ϕw versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial confining stress: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 5.0 mm).
Internal friction angle ϕw versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial confining stress: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 5.0 mm).

Figure 9

Volumetric strain ɛv versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial confining stress: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 5.0 mm).
Volumetric strain ɛv versus vertical normal strain ɛ1 from discrete simulations of homogeneous triaxial compression test for different initial confining stress: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 5.0 mm).

Figure 10

Model set-up for numerical simulations of direct shear test.
Model set-up for numerical simulations of direct shear test.

Figure 11

Internal friction angle ϕw versus horizontal displacement ux from discrete simulations of direct shear test for different initial void ratios: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 0.5 mm).
Internal friction angle ϕw versus horizontal displacement ux from discrete simulations of direct shear test for different initial void ratios: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 0.5 mm).

Figure 12

Volumetric strain ɛv versus horizontal displacement ux from discrete simulations of direct shear test for different initial void ratios: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 0.5 mm).
Volumetric strain ɛv versus horizontal displacement ux from discrete simulations of direct shear test for different initial void ratios: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 0.5 mm).

Figure 13

Front view of the specimen at the final state for different initial void ratios: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 0.5 mm) (the dark/light grey stripes were perfectly vertical at the beginning of the tests).
Front view of the specimen at the final state for different initial void ratios: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 0.5 mm) (the dark/light grey stripes were perfectly vertical at the beginning of the tests).

Figure 14

Distribution of sphere rotations at the final state of the test for different initial void ratios: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 0.5 mm) (red colour – clockwise rotations, blue colour – anti-clockwise rotations) (colour online)
Distribution of sphere rotations at the final state of the test for different initial void ratios: a) e0 = 0.53, b) e0 = 0.60 and c) e0 = 0.75 (σ0 = 200 kPa, d50 = 0.5 mm) (red colour – clockwise rotations, blue colour – anti-clockwise rotations) (colour online)

Figure 15

Internal friction angle ϕw versus horizontal displacement ux from discrete simulations of direct shear test for different vertical load: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 0.5 mm).
Internal friction angle ϕw versus horizontal displacement ux from discrete simulations of direct shear test for different vertical load: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 0.5 mm).

Figure 16

Volumetric strain ɛv versus horizontal displacement ux from discrete simulations of direct shear test for different vertical load: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 0.5 mm).
Volumetric strain ɛv versus horizontal displacement ux from discrete simulations of direct shear test for different vertical load: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 0.5 mm).

Figure 17

Void ratio e0 versus horizontal displacement ux from discrete simulations of direct shear test for different vertical load: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 0.5 mm).
Void ratio e0 versus horizontal displacement ux from discrete simulations of direct shear test for different vertical load: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 0.5 mm).

Figure 18

Distribution of sphere rotations at the final state of the test for different vertical load: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 0.5 mm) (red colour – clockwise rotations, blue colour – anti-clockwise rotations) (colour online).
Distribution of sphere rotations at the final state of the test for different vertical load: a) σ0 = 50 kPa, b) σ0 = 200 kPa and c) σ0 = 500 kPa (e0 = 0.60, d50 = 0.5 mm) (red colour – clockwise rotations, blue colour – anti-clockwise rotations) (colour online).

Figure 19

Distribution of horizontal sphere displacement u’x across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).
Distribution of horizontal sphere displacement u’x across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).

Figure 20

Distribution of vertical sphere displacement u’y across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).
Distribution of vertical sphere displacement u’y across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).

Figure 21

Distribution of sphere rotations ω across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (negative value corresponds to the clockwise rotation; dark vertical line demonstrates 5% limit).
Distribution of sphere rotations ω across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (negative value corresponds to the clockwise rotation; dark vertical line demonstrates 5% limit).

Figure 22

Distribution of void ratio e across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).
Distribution of void ratio e across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).

Figure 23

Distribution of coordination number n across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).
Distribution of coordination number n across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).

Figure 24

Evolution of displacements fluctuations ( (u→i−u→i,avg)\left( {{{\vec{u}}}_{i}}-{{{\vec{u}}}_{i,avg}} \right) ) in the entire specimen for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (the arrows are multiple by 10 due to readability; red lines show the estimated localization shape).
Evolution of displacements fluctuations ( (u→i−u→i,avg)\left( {{{\vec{u}}}_{i}}-{{{\vec{u}}}_{i,avg}} \right) ) in the entire specimen for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (the arrows are multiple by 10 due to readability; red lines show the estimated localization shape).

Figure 25

Force chain distribution in the entire specimen for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (red colour corresponds to the force chain above the mean value) (colour online)
Force chain distribution in the entire specimen for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (red colour corresponds to the force chain above the mean value) (colour online)

Figure 26

Distribution of the normal forces fx acting on spheres across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).
Distribution of the normal forces fx acting on spheres across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).

Figure 27

Distribution of the normal forces fy acting on spheres across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm).
Distribution of the normal forces fy acting on spheres across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm).

Figure 28

Distribution of the maximal normal forces fx,max acting on spheres, in the averaging cell, across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).
Distribution of the maximal normal forces fx,max acting on spheres, in the averaging cell, across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).

Figure 29

Distribution of the maximal normal forces fy,max acting on spheres, in the averaging cell, across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).
Distribution of the maximal normal forces fy,max acting on spheres, in the averaging cell, across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).

Figure 30

Distribution of the resultant moment m acting on spheres across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).
Distribution of the resultant moment m acting on spheres across the normalized specimen height h/d50 at the specimen centre for: a) ux = 0.25 mm, b) ux = 0.50 mm, c) ux = 0.75 mm, d) ux = 1.00 mm, e) ux = 1.25 mm, f) ux = 1.50 mm, g) ux = 1.75 mm, h) ux = 2.00 mm and i) ux = 10.00 mm (e0 = 0.60, σ0 = 200 kPa, d50 = 0.5 mm) (dark vertical line demonstrates 5% limit).

Early predictor summation.

Predictorux (mm)ts (mm)
Horizontal displacement ux0.7520 × d50
Vertical displacement uy0.7520 × d50
Rotations ω1.5019 × d50
Void ratio e2.0025 × d50
Coordination number n0.75-
Displacement fluctuations1.756 × d50
Normal force fx0.75-
Normal force fy--
Maximal force fx,max1.00-
Maximal force fy,max0.7525 × d50
Moments m0.5030 × d50

Material micro-parameters for discrete simulations.

Material micro-parametersValue
Modulus of elasticity of grain contact Ec (MPa)300
Normal/tangential stiffness ratio of grain contact vc (−)0.3
Inter-particle friction angle μ (°)18
Rolling stiffness coefficient β (−)0.7
Moment limit coefficient η (−)0.4
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