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Characteristic parameters of soil failure criteria for plane strain conditions – experimental and semi-theoretical study

Justyna Sławińska-Budzich
Justyna Sławińska-Budzich
   | 30 set 2021
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Article Category: Original Study
Pubblicato online: 30 set 2021
Pagine: 237 - 254
Ricevuto: 12 gen 2021
Accettato: 29 apr 2021
DOI: https://doi.org/10.2478/sgem-2021-0015
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© 2021 Justyna Sławińska-Budzich, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1

Two-dimensional and three-dimensional soil stress states: a) cylindrical sample in axisymmetric stress conditions, σ2 = σ3 and b) rectangular sample in true triaxial conditions, σ1 ≠ σ2 ≠ σ3.
Two-dimensional and three-dimensional soil stress states: a) cylindrical sample in axisymmetric stress conditions, σ2 = σ3 and b) rectangular sample in true triaxial conditions, σ1 ≠ σ2 ≠ σ3.

Figure 2

The principal stress and axial strain curves for the selected test in plane strain conditions: a) q(ɛ1) and b) σ1(ɛ1), σ2(ɛ1) and σ3(ɛ1).
The principal stress and axial strain curves for the selected test in plane strain conditions: a) q(ɛ1) and b) σ1(ɛ1), σ2(ɛ1) and σ3(ɛ1).

Figure 3

Comparison of the results from drained triaxial and plane strain tests on sand [18] and true-triaxial tests on Skarpa sand.
Comparison of the results from drained triaxial and plane strain tests on sand [18] and true-triaxial tests on Skarpa sand.

Figure 4

Failure surfaces in the deviatoric plane, see Georgiadis et al. (2004). In plane strain conditions, Lode angle varies roughly from θ = 10° to θ = 20°.
Failure surfaces in the deviatoric plane, see Georgiadis et al. (2004). In plane strain conditions, Lode angle varies roughly from θ = 10° to θ = 20°.

Figure 5

Layout of the soil sample under plane strain conditions in EMTTA.
Layout of the soil sample under plane strain conditions in EMTTA.

Figure 6

Components of EMTTA, used in the study.
Components of EMTTA, used in the study.

Figure 7

a) The GDS EMTTA chamber with a sample prepared for the test. The role of the side plates is to prevent soil deformations in the x2direction, b) proximity transducer on the doors of the measurement cell (test chamber).
a) The GDS EMTTA chamber with a sample prepared for the test. The role of the side plates is to prevent soil deformations in the x2direction, b) proximity transducer on the doors of the measurement cell (test chamber).

Figure 8

Results of the experimental tests listed in Table 1: deviator stress as a function of the axial strain q(ɛ1).
Results of the experimental tests listed in Table 1: deviator stress as a function of the axial strain q(ɛ1).

Figure 9

Results of the experimental tests listed in Table 1: maximum principal stress as a function of the axial strain σ1(ɛ1).
Results of the experimental tests listed in Table 1: maximum principal stress as a function of the axial strain σ1(ɛ1).

Figure 10

Results of the experimental tests listed in Table 1: principal stress in the direction of fixed strain (ɛ2 = 0) as a function of the axial strain σ2(ɛ1).
Results of the experimental tests listed in Table 1: principal stress in the direction of fixed strain (ɛ2 = 0) as a function of the axial strain σ2(ɛ1).

Figure 11

Results of the experimental tests listed in Table 1: volumetric strain as a function of the axial strain ɛv(ɛ1).
Results of the experimental tests listed in Table 1: volumetric strain as a function of the axial strain ɛv(ɛ1).

Figure 12

Relations between principal stress components, corresponding to peak soil strength: σ1max (σ3) and σ2(σ3).
Relations between principal stress components, corresponding to peak soil strength: σ1max (σ3) and σ2(σ3).

Figure 13

Relation between Lode angle θ and intermediate stress σ2.
Relation between Lode angle θ and intermediate stress σ2.

Figure 14

The intermediate stress σ2, obtained for Drucker–Prager (D-P), Matsuoka–Nakai (M-N) and Lade–Duncan (L-D) failure criteria, assuming plane strain condition and the associated flow rule, as function of the measured σ2 (Table 3): (a) σ2calc (σ2exp) and (b) R(σ2exp), where R = σ2calc/σ2exp.
The intermediate stress σ2, obtained for Drucker–Prager (D-P), Matsuoka–Nakai (M-N) and Lade–Duncan (L-D) failure criteria, assuming plane strain condition and the associated flow rule, as function of the measured σ2 (Table 3): (a) σ2calc (σ2exp) and (b) R(σ2exp), where R = σ2calc/σ2exp.

Figure 15

Dependence of the intermediate stress σ2 (Tables 3 and 4) on the initial relative density of Skarpa sand.
Dependence of the intermediate stress σ2 (Tables 3 and 4) on the initial relative density of Skarpa sand.

Figure 16

Dependence of the ratio of intermediate stress σ2 to confining pressure σ3 (Table 4) on the initial relative density of Skarpa sand.
Dependence of the ratio of intermediate stress σ2 to confining pressure σ3 (Table 4) on the initial relative density of Skarpa sand.

Figure 17

Parameters of Mohr–Coulomb (M-C), Drucker–Prager (D-C), Matsuoka–Nakai (M-N) and Lade–Duncan (L-D) failure criteria depending on soil relative density: (a) friction angle ϕ, (b)–(d) comparison of κD-P, κL-D and κM-N, obtained by Eqs. (16)–(18) (full stress state measurement) and Eqs. (25)–(27) (plane strain condition – Vikash and Prashant approach).
Parameters of Mohr–Coulomb (M-C), Drucker–Prager (D-C), Matsuoka–Nakai (M-N) and Lade–Duncan (L-D) failure criteria depending on soil relative density: (a) friction angle ϕ, (b)–(d) comparison of κD-P, κL-D and κM-N, obtained by Eqs. (16)–(18) (full stress state measurement) and Eqs. (25)–(27) (plane strain condition – Vikash and Prashant approach).

Average relative difference of parameters κ and intermediate principal stress σ2, determined by the two approaches: full set of principal stresses and Vikash and Prashant proposal, for Drucker–Prager, Lade–Duncan and Matsuoka–Nakai failure criteria.

vκ vσ2
Drucker–Prager νκD−P=21.21 % \nu _\kappa ^{{\rm{D}} - {\rm{P}}} = 21.21\,\% νσ2D−P=119.0 % \nu _{{\sigma _2}}^{{\rm{D}} - {\rm{P}}} = 119.0\,\%
Lade–Duncan νκL−D=5.99 % \nu _\kappa ^{{\rm{L}} - {\rm{D}}} = 5.99\,\% νσ2L−D=55.5 % \nu _{{\sigma _2}}^{{\rm{L}} - {\rm{D}}} = 55.5\,\%
Matsuoka–Nakai νκM−N=0.66 % \nu _\kappa ^{{\rm{M}} - {\rm{N}}} = 0.66\,\% νσ2M−N=19.4 % \nu _{{\sigma _2}}^{{\rm{M}} - {\rm{N}}} = 19.4\,\%

The linear fits κexp(ϕps)and the corresponding statistics Pearson's correlation coefficients.

Linear fit Pearson's coefficient r
Drucker–Prager κD−Pexp=0.006353 φps+0.001497 \kappa _{{\rm{D}} - {\rm{P}}}^{\exp } = 0.006353\,{\varphi _{{\rm{ps}}}} + 0.001497 rD-P = 0.970
Lade–Duncan κL−Dexp=1.5 φps−10.613 \kappa _{{\rm{L}} - {\rm{D}}}^{\exp } = 1.5\,{\varphi _{{\rm{ps}}}} - 10.613 rL-D = 0.993
Matsuoka–Nakai κM−Nexp=0.2605 φps+2.767 \kappa _{{\rm{M}} - {\rm{N}}}^{\exp } = 0.2605\,{\varphi _{{\rm{ps}}}} + 2.767 rM-N = 0.997

Characteristics of peak strength state for the tested samples.

Test σ1max [kPa] σ2 [kPa] σ3 [kPa] p [kPa] q [kPa] b [ − ] θ [ ° ]
009_17_MC_5 1402 653 391 815 909 0.26 14.46
033_17_MC_14 1072 479 293 615 705 0.24 13.21
012_18_MC_21 1184 459 292 645 821 0.19 10.14
013_18_MC_22 678 262 146 362 485 0.22 11.97
010_18_MC_19 902 355 195 484 642 0.23 12.46
001_18_MC_15 870 332 191 464 621 0.21 11.35
010_15_MC_1 1291 529 278 699 914 0.25 13.76
009_18_MC_18 1483 528 292 768 1092 0.20 10.78
008_18_MC_17 1396 506 295 732 1012 0.19 10.40
028_17_MC_12 287 109 52 149 212 0.24 13.44
031_17_MC_13 508 191 99 266 372 0.22 12.38

Initial test conditions.

Test e ID σc3[kPa] ec IDc {\boldsymbol {I}}_{\boldsymbol {D}}^{\boldsymbol {c}} nc
009_17_MC_5 0.585 0.376 391 0.563 0.465 0.36
033_17_MC_14 0.559 0.482 293 0.548 0.527 0.354
012_18_MC_21 0.541 0.555 292 0.532 0.592 0.347
013_18_MC_22 0.519 0.645 146 0.514 0.665 0.339
010_18_MC_19 0.517 0.653 195 0.508 0.690 0.337
001_18_MC_15 0.521 0.637 191 0.499 0.727 0.333
010_15_MC_1 0.496 0.739 278 0.490 0.763 0.329
009_18_MC_18 0.488 0.771 292 0.480 0.804 0.324
008_18_MC_17 0.489 0.767 295 0.476 0.820 0.322
028_17_MC_12 0.467 0.857 52 0.462 0.878 0.316
031_17_MC_13 0.469 0.849 99 0.462 0.878 0.316

Characteristic parameters of Drucker–Prager, Matsuoka–Nakai and Lade–Duncan soil failure criteria, obtained from direct stress measurements (A) and the associated flow rule assuming plane strain conditions (B).

Test A. Direct stress measurements Eqs. (15)–(18) B. Flow rule and plane strain condition Eqs. (25)–(27)
ϕps κD−Pexp \kappa _{{\rm{D}} - {\rm{P}}}^{\exp } κM−Nexp \kappa _{{\rm{M}} - {\rm{N}}}^{\exp } κL−Dexp \kappa _{{\rm{L}} - {\rm{D}}}^{\exp } σ2D-P \sigma _2^{{\rm{D - P}}} σ2M-N \sigma _2^{{\rm{M - N}}} σ2L-D \sigma _2^{{\rm{L - D}}} κD-Pflow rule \kappa _{D{\rm{ - }}P}^{f{\rm{low}}\,{\rm{rule}}} κM-Nflow rule \kappa _{M{\rm{ - }}N}^{f{\rm{low}}\,{\rm{rule}}} κL-Dflow rule \kappa _{L{\rm{ - }}D}^{f{\rm{low}}\,{\rm{rule}}}
009_17_MC_5 34.3° 0.21 11.7 40.9 1181.5 740.4 896.5 0.179 11.7 39.6
033_17_MC_14 34.8° 0.22 11.9 41.7 904.8 560.4 682.5 0.181 11.8 40.0
012_18_MC_21 37.2° 0.25 12.5 45.7 1007.5 588.0 738 0.190 12.3 42.5
013_18_MC_22 40.2° 0.26 13.2 49.4 583.8 314.6 412 0.202 13.1 46.3
010_18_MC_19 40.1° 0.26 13.1 49.0 776.3 419.3 548.5 0.201 13.1 46.2
001_18_MC_15 39.8° 0.26 13.1 49.0 747.8 407.6 530.5 0.200 13.0 45.7
010_15_MC_1 40.2° 0.25 13.1 48.6 1111.5 599.1 784.5 0.202 13.1 46.3
009_18_MC_18 42.1° 0.27 13.8 53.4 1287.1 658.1 887.5 0.209 13.7 49.1
008_18_MC_17 40.6° 0.27 13.4 50.9 1203.9 641.7 745.5 0.203 13.2 46.9
028_17_MC_12 43.9° 0.27 14.3 55.3 251.0 122.1 169.5 0.215 14.3 52.0
031_17_MC_13 42.4° 0.27 13.8 52.9 441.3 224.3 303.5 0.209 13.7 46.5

The linear fits κexp(IDc) {\kappa ^{\exp }}\left( {I.D^c} \right) and the corresponding Pearson's correlation coefficients.

Linear fit Pearson's coefficient r
Drucker–Prager κD−Pexp=0.16IDc+0.13 \kappa _{{\rm{D}} - {\rm{P}}}^{\exp } = 0.16I_D^c + 0.13 rD-P = 0.94
Lade–Duncan κL−Dexp=26.33IDc+31.64 \kappa _{{\rm{L}} - {\rm{D}}}^{\exp } = 26.33I_D^c + 31.64 rL-D = 0.96
Matsuoka–Nakai κM−Nexp=9.20IDc+5.48 \kappa _{{\rm{M}} - {\rm{N}}}^{\exp } = 9.20I_D^c + 5.48 rM-N = 0.96

Parameters of Skarpa sand.

Specific density [kg/m3] 2650
Mean particle size [mm] D50 - 0.42
Uniformity coefficient [ − ] U = 2.5
Minimum void ratio [ − ] emin = 0.432
Maximum void ratio [ − ] emax = 0.677
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