Figure 1.
(a) Particle distribution image and (b) contact network image in the idealized 2D experiments (lamellar particles distributed in monolayer representing the 2D particle assembly).
Figure 1.
(a) Particle distribution image and (b) contact network image in the idealized 2D experiments (lamellar particles distributed in monolayer representing the 2D particle assembly).
Figure 2.
The corner recognition results of (a) FAST algorithm, (b) HARRIS algorithm, and (c) SUSAN algorithm.
Figure 2.
The corner recognition results of (a) FAST algorithm, (b) HARRIS algorithm, and (c) SUSAN algorithm.
Figure 3.
The specific process of the recognition and determination technique.
Figure 3.
The specific process of the recognition and determination technique.
Figure 4.
Segmentation result of the OTSU algorithm.
Figure 4.
Segmentation result of the OTSU algorithm.
Figure 5.
Greyscale assigned image without boundary lines.
Figure 5.
Greyscale assigned image without boundary lines.
Figure 6.
Grayscale distribution (a) around the boundary with two greyscales and (b) around the corner with three greyscales.
Figure 6.
Grayscale distribution (a) around the boundary with two greyscales and (b) around the corner with three greyscales.
Figure 7.
Spatial positions of Ni and Nik.
Figure 7.
Spatial positions of Ni and Nik.
Figure 8.
Recognition effect of the Q-Y algorithm.
Figure 8.
Recognition effect of the Q-Y algorithm.
Figure 9.
The corner recognition results of (a) FAST algorithm, (b) HARRIS algorithm, and (c) SUSAN algorithm for the greyscale assigned image without boundary lines (the bule, green, and red points are the recognized corners by the three algorithms respectively).
Figure 9.
The corner recognition results of (a) FAST algorithm, (b) HARRIS algorithm, and (c) SUSAN algorithm for the greyscale assigned image without boundary lines (the bule, green, and red points are the recognized corners by the three algorithms respectively).
Figure 10.
(a) Recognition results of corner pixels (the green points are the recognized corner pixels); (b) the adjacent corner pixels are considered as a corner; (c) the geometric type of contact loops is determined by the number of corners; and (d) the statistical results of the technique (number and area of each geometric type).
Figure 10.
(a) Recognition results of corner pixels (the green points are the recognized corner pixels); (b) the adjacent corner pixels are considered as a corner; (c) the geometric type of contact loops is determined by the number of corners; and (d) the statistical results of the technique (number and area of each geometric type).
Figure 11.
(a) Particles within the initial rectangular area; (b) the stress state for generating the dense assembly; and (c) the stress state for the biaxial compression test.
Figure 11.
(a) Particles within the initial rectangular area; (b) the stress state for generating the dense assembly; and (c) the stress state for the biaxial compression test.
Figure 12.
The variation curves of (a) loop number, deviatoric stress, and (b) volumetric strain with axial strain.
Figure 12.
The variation curves of (a) loop number, deviatoric stress, and (b) volumetric strain with axial strain.
Figure 13.
(a) The variation curves of loop number and average coordination number with axial strain; (b) the diagrams of the relationship between Z and Nl for two examples.
Figure 13.
(a) The variation curves of loop number and average coordination number with axial strain; (b) the diagrams of the relationship between Z and Nl for two examples.
Figure 14.
The variation curves of (a) deviatoric stress, (b) volumetric strain, and number percentage indexes (ω3, ω4, ω5, and ω6+) with axial strain.
Figure 14.
The variation curves of (a) deviatoric stress, (b) volumetric strain, and number percentage indexes (ω3, ω4, ω5, and ω6+) with axial strain.
Figure 15.
The variation curves of (a) deviatoric stress, (b) volumetric strain, and area percentage indexes (A3, A4, A5, and A6+) with axial strain.
Figure 15.
The variation curves of (a) deviatoric stress, (b) volumetric strain, and area percentage indexes (A3, A4, A5, and A6+) with axial strain.
Figure 16.
Multicollinearity analysis results among the eight independent variables.
Figure 16.
Multicollinearity analysis results among the eight independent variables.
Figure 17.
The relationship between meso-structural indexes, principal components, and macro-mechanical indexes.
Figure 17.
The relationship between meso-structural indexes, principal components, and macro-mechanical indexes.
Figure 18.
The fitting results between the calculated values and the actual values for (a) strain and (b) deviatoric stress.
Figure 18.
The fitting results between the calculated values and the actual values for (a) strain and (b) deviatoric stress.
Table 1.
Contact parameters used in the DEM simulations.
Table 1.
Contact parameters used in the DEM simulations.
Parameters | Values |
---|
Interparticle friction coefficient, μp | 0.5 |
Wall-particle friction coefficient, μw | 0.0 |
Contact effective modulus, Ec (Pa) | 1.0 × 108 |
Normal-to-tangential stiffness ratio, kn/ks | 4/3 |
Damping constant, β | 0.7 |
Table 2.
Mesostructural indexes in the hardening stage.
Table 2.
Mesostructural indexes in the hardening stage.
Axial Strain (%) | Meso-Structural Indexes |
---|
ω3 | ω4 | ω5 | ω6+ | A3 | A4 | A5 | A6+ |
---|
0 | 34.22% | 40.80% | 19.86% | 5.13% | 12.28% | 38.49% | 32.04% | 17.19% |
0.1 | 30.86% | 44.14% | 19.19% | 5.80% | 10.45% | 40.43% | 30.43% | 18.69% |
0.2 | 26.59% | 45.89% | 18.67% | 8.85% | 8.10% | 37.76% | 27.02% | 27.13% |
0.3 | 24.26% | 45.21% | 19.01% | 11.52% | 6.90% | 33.49% | 26.11% | 33.50% |
0.4 | 22.40% | 44.57% | 19.45% | 13.58% | 6.07% | 30.85% | 25.16% | 37.92% |
0.5 | 21.38% | 44.62% | 19.40% | 14.61% | 5.62% | 29.12% | 24.29% | 40.97% |
0.6 | 20.57% | 44.81% | 18.82% | 15.79% | 5.27% | 28.03% | 23.08% | 43.63% |
0.7 | 19.62% | 44.49% | 19.34% | 16.55% | 4.84% | 26.90% | 23.10% | 45.17% |
0.8 | 19.85% | 43.39% | 19.56% | 17.20% | 4.78% | 25.35% | 22.61% | 47.26% |
0.9 | 19.52% | 43.23% | 19.06% | 18.19% | 4.66% | 24.80% | 21.48% | 49.06% |
1.0 | 18.74% | 43.00% | 19.65% | 18.60% | 4.27% | 23.77% | 21.45% | 50.51% |
1.1 | 18.56% | 42.76% | 19.45% | 19.22% | 4.31% | 23.34% | 21.12% | 51.24% |
1.2 | 18.05% | 42.60% | 20.07% | 19.28% | 4.04% | 22.77% | 21.65% | 51.53% |
1.3 | 18.09% | 42.32% | 19.00% | 20.60% | 4.02% | 22.26% | 19.39% | 54.32% |
1.4 | 17.63% | 42.62% | 18.98% | 20.77% | 3.84% | 22.22% | 19.82% | 54.12% |
1.5 | 17.44% | 42.15% | 19.18% | 21.23% | 3.71% | 21.52% | 20.01% | 54.75% |
1.6 | 17.40% | 42.07% | 18.71% | 21.82% | 3.78% | 21.38% | 19.14% | 55.70% |
1.7 | 16.84% | 41.81% | 18.84% | 22.51% | 3.52% | 20.85% | 18.59% | 57.04% |
1.8 | 16.73% | 42.29% | 18.63% | 22.36% | 3.63% | 21.18% | 18.65% | 56.54% |
1.9 | 15.96% | 41.37% | 19.31% | 23.37% | 3.31% | 20.40% | 18.82% | 57.47% |
2.0 | 15.81% | 41.71% | 19.39% | 23.10% | 3.34% | 20.08% | 19.07% | 57.51% |
Table 3.
Total variance explained of the first three principal components.
Table 3.
Total variance explained of the first three principal components.
Component | Initial Eigenvalues | Rotation Sums of Squared Loadings |
---|
Total Variance | Percentage of Variance (%) | Cumulative Contribution Rate (%) | Total Variance | Percentage of Variance (%) | Cumulative Contribution Rate (%) |
---|
F1 |
6.123
|
76.534
|
76.534
|
5.692
|
71.156
|
71.156
|
F2 |
1.243
|
15.536
|
92.070
|
1.247
|
15.581
|
86.737
|
F3 |
0.624
|
7.797
|
99.868
|
1.050
|
13.130
|
99.868
|
Table 4.
Rotated component matrix between the principal components and variables.
Table 4.
Rotated component matrix between the principal components and variables.
Variables | Principal Components |
---|
F1 | F2 | F3 |
---|
ω3 |
0.994
|
-
|
-
|
ω4 |
-
|
0.960
|
-
|
ω5 |
-
|
-
|
0.985
|
ω6+ |
−0.949
|
-
|
-
|
A3 |
0.997
|
-
|
-
|
A4 |
0.942
|
-
|
-
|
A5 |
0.955
|
-
|
-
|
A6+ |
−0.967
|
-
|
-
|
Table 5.
Component score coefficient matrix between meso-structural indexes and principal components.
Table 5.
Component score coefficient matrix between meso-structural indexes and principal components.
Variables | Principal Components |
---|
F1 | F2 | F3 |
---|
ω3 | 0.233 | −0.207 | −0.090 |
ω4 | −0.212 | 1.017 | 0.133 |
ω5 | −0.120 | 0.163 | 1.027 |
ω6+ | −0.144 | −0.086 | −0.027 |
A3 | 0.249 | −0.265 | −0.108 |
A4 |
0.150
|
0.094
|
−0.065
|
A5 |
0.151
|
0.028
|
0.102
|
A6+ |
−0.171
|
−0.006
|
0.022
|
Table 6.
Values of principal components under different axial strain.
Table 6.
Values of principal components under different axial strain.
Axial Strain/% | F1 | F2 | F3 |
---|
0 | 2.92098 | −2.36115 | 1.03283 |
0.1 | 2.2062 | 0.16253 | −0.32189 |
0.2 | 1.25432 | 1.55719 | −1.41124 |
0.3 | 0.72174 | 1.31898 | −0.46048 |
0.4 | 0.29222 | 1.12591 | 0.71207 |
0.5 | 0.06105 | 1.18091 | 0.61629 |
0.6 | −0.00741 | 1.11319 | −0.88492 |
0.7 | −0.27536 | 1.15016 | 0.50242 |
0.8 | −0.27167 | 0.39496 | 0.98101 |
0.9 | −0.22868 | 0.06347 | −0.37464 |
1.0 | −0.51139 | 0.19866 | 1.20855 |
1.1 | −0.46837 | −0.07702 | 0.66345 |
1.2 | −0.68439 | 0.10974 | 2.32144 |
1.3 | −0.48282 | −0.5935 | −0.60714 |
1.4 | −0.55272 | −0.33883 | −0.58048 |
1.5 | −0.59307 | −0.5942 | −0.09144 |
1.6 | −0.49762 | −0.87187 | −1.35984 |
1.7 | −0.62178 | −0.9802 | −1.03366 |
1.8 | −0.60195 | −0.71639 | −1.56138 |
1.9 | −0.79447 | −1.06359 | 0.20003 |
2.0 | −0.86479 | −0.77897 | 0.44901 |
Table 7.
Partial regression coefficient test results for Equation (13).
Table 7.
Partial regression coefficient test results for Equation (13).
Principal Components | β | 95%CI | p-Value |
---|
F1 | −0.993 | (−71.288, −65.210) | <0.001 |
F2 | −0.059 | (−7.121, −1.043) | 0.011 |
F3 | −0.058 | (−7.032, −0.953) | 0.013 |
Table 8.
Partial regression coefficient test results for Equation (14).
Table 8.
Partial regression coefficient test results for Equation (14).
Principal Components | β | 95%CI | p-Value |
---|
F1 | −0.596 | (−0.373, −0.185) | <0.001 |
F2 | −0.663 | (−0.405, −0.216) | <0.001 |
F3 | −0.223 | (−0.199, −0.010) | 0.032 |
Table 9.
Partial regression coefficient test results for Equation (15).
Table 9.
Partial regression coefficient test results for Equation (15).
Principal Components | β | 95%CI | p-Value |
---|
F1 | −0.814 | (−0.582, −0.428) | <0.001 |
F2 | −0.5 | (−0.388, −0.233) | <0.001 |
F3 | −0.167 | (−0.181, −0.026) | 0.012 |