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The main objective of this research is to study the mechanical behaviour of tropical soils using elasto-plastic constitutive equations in the so-called limit and critical states. Indeed, researchers of the Cambridge University had noticed that during their various experiments, the rate of volumetric deformation ( ) of the sample tending to zero every time the rupture of the specimen is reached during a test performed on a clay specimen Roscoe et al., 1958. To better understand and clarify this mechanical behaviour, a description has been proposed in the (e, p, q) representation that means void ratio, volumetric stress (spherical pressure) and deviatoric stress. This frame of theoretical study and apprehension is called: the theory of the Critical State. One of the major problems met at the time of our present research is the non-availability of triaxial apparatus allowing us to achieve some tests on tropical soils (samples from Senegal in West Africa) and to describe the behaviour of these materials easily like the researchers of the university of Cambridge in the theory of the critical state. To by-pass this difficulty, we decided to consider two very classical and simple mechanical tests: shear-box and the oedometer test as well as the interrelationship of the results given by the tests and some theoretical calculations. This is a way to identify an elasto-plastic model (the modified Cam Clay model) without any triaxial experiment. Indeed it supposes the model to be suitable to describe the mechanical behaviour of the considered clays.

The concepts of limit and critical states have been established initially by the researchers of the University of Cambridge from both isotropic consolidation and triaxial tests on clays reconstituted in the laboratory [

All the samples that we studied here come from Senegal in West Africa “

The preliminary analyses showed that these clays are low plastic lateritic gravels with the following characteristics “

We used a square shear-box (60 mm) to make the whole set of tests on our samples. Our samples have all been compacted to the middle energy of Proctor (theoretically) in Proctor moulds as specimen of more or less 70 mm in height. These specimens were realized in several copies at the same time and then packed in tight films. Before the compaction of our samples in the Proctor moulds, we made them pass to the sifter of the module 44 (20 mm) in accordance with the norm French Industrial Standards Authority (AFNOR) to knead to the contents in water of the Proctor modified then found for every sample.

After this modus operandi, we realised the tests in three phases of respective loads, corresponding to initial normal constraints of 196 kPa, 392 kPa and 784 kPa with the speed fixed to 0.3 mm/min for all the tests. The characteristics of shearing that we have found are summarized in the “

The lateritic soil of Sindia presents a strong cohesion due to the fact that the effect of compaction cements the sol. The evolution of the tangential stress is monotonic and becomes rather constant when the resistance to the maximal shearing is reached. It is due to the fact that the compactness does not reach 95%

Characteristics | Sindia | Keur Samba Kane |
---|---|---|

LL (Liquid Limit) | 30.7 | 31.1 |

LP (Plastic Limit) | 15.7 | 17.6 |

IP (Plasticity Index) | 15 | 13.4 |

CBR (California Ratio) | 56.2 | 79.4 |

γ_{S} (kN/m^{3}) (Specific Density) | 27 | 26 |

γ_{d}_{max} (kN/m^{3}) (Dry Bulk Density) | 21.1 | 21.4 |

W_{OPM} (%) (Opt. Water Content) | 8.2 | 8.2 |

Characteristics | Sindia |
---|---|

C (kPa) | 117 |

ϕ (˚) | 18 |

ɣ_{h} (kN/m^{3}) | 21.5 |

ɣ_{d} (kN/m^{3}) | 17.8 |

ω_{i} (%) | 13.3 |

ω_{f} (%) | 20.9 |

Characteristics | Keur Samba Kane |
---|---|

C (kPa) | 211 |

ϕ (˚) | 14.9 |

ɣ_{h} (kN/m^{3}) | 23.5 |

ɣ_{d} (kN/m^{3}) | 19.4 |

ω_{i} (%) | 7.87 |

ω_{f} (%) | 20.96 |

of the OPM [

For a normal initial stress of 196 kPa, the resistance to the maximal shearing is of 228 kPa, whereas for a normal initial constraint of 784 kPa, the resistance to the maximal shearing developed by the lateritic material of Keur Samba Kane in a compacted state is of 535 kPa. The curves of the vertical displacement versus the horizontal displacement all show a contracting behaviour of the material all along the test. This reduction in the volume is mainly caused by the presence of water at the time of the test that favours the settling [

Usually we observe for lateritic soil a dilating (resp. contracting) behaviour for low (resp. large) constraints. This transition between dilation and contraction is also observed when water content of compaction varies [

The effect of the compaction is noticed well for the two samples compacted lateritic gravels at the time of the loadings to the oedometer. We do not observe an elastic slope at the beginning of the loading; the void ratio decreases after a certain level of stress “

Characteristics | Sindia | Keur Samba Kane |
---|---|---|

e_{0 } | 0.9 | 0.8 |

Cc | 0.11 | 0.14 |

Cs | 0.01 | 0.01 |

σ_{P}_{1}_{ } | 28 | 29 |

σ_{P}_{2} | 68 | 65 |

oedometer loading-unloading, we got the following set of parameters (

To study the mechanical behaviour of a soil by the model of Cam Clay (_{λ}, λ, k, M, e_{0}, ν and P_{C}_{0}) that should be given from triaxial test.

λ: slope of the isotropic virgin curve

k: slope of the elastic curve (unloading-reloading curve)

M: slope of the Critical Line State

e_{0}: initial void ratio

ν: Poisson coefficient

P_{c}_{0}: pressure of pre-consolidation

e_{λ} and e_{k}: void ratio for a pressure of reference of 1 kPa.

Here, we recover these seven parameters from the shear-box and oedometer tests according to the following relations:

λ = C c / ln 10 . (1)

k = C s / ln 10 . (2)

P C 0 = ( σ v + 2 ∗ σ H ) / 3 . (3)

σ v = σ p and σ H = k 0 ∗ σ v . (4)

λ | k | M | e_{0 } | ν | P_{c}_{01} (kPa)_{ } | P_{c}_{02} (kPa) | e_{λ}_{ } | e_{k}_{1}_{ } | e_{k}_{2} | |
---|---|---|---|---|---|---|---|---|---|---|

Sample of Sindia | 0.05 | 0.004 | 0.7 | 0.9 | 0.3 | 22.11 | 53.69 | 0.86 | 0.8 | 0.76 |

Sample of Keur Samba Kane | 0.061 | 0.004 | 0.56 | 0.89 | 0.3 | 24.02 | 53.83 | 0.84 | 0.76 | 0.7 |

k 0 = 1 − sin φ . (5)

M = 6 sin φ / ( 3 − sin φ ) . (6)

The Poisson coefficient ν, is supposed equal to 0.3 for the two materials. The constraints matrix is given by:

σ = ( σ p − − − σ H − − − σ H ) (7)

We get the following results for the two compacted lateritic materials

The yield surface in the plane (p, q) of modified Cam Clay model is given by this expression:

f = q 2 + M 2 ( p 2 − p p c r ) = 0. (8)

The Critical Line State is given by this equation:

q = M p . (9)

While applying these expressions to the results of our tests on the two compacted lateritic materials, we obtain the “

The projection of these curves of the plane (q, p) onto the plane (e, p) permits us to pass to the plane (e, lnp) to see the isotropic virgin curve and the critical state line defined by the equations:

e = e λ − λ ln p , (10)

e = Γ − λ ln p , (11)

with

Γ = e λ − λ + k . (12)

This projection is made in the same system of axis and permits to really describe the loading path for the oedometer test.

The isotropic virgin curve (named as virgin consolidation curve) is the curve of loading that would be obtained during the isotropic triaxial test. Soil is normally consolidated all along this curve and is in a plastic state “

To validate our approach and to make a comparison between the results given by the shear-box and the oedometer tests and the results given by triaxial tests, we used the results of three samples made of compacted lateritic gravel from Senegal. This set of experimental results has been provided in [

The results of the consolidation tests “

The results of the following triaxial tests for various pressures of confinement gave the values of p and q “

We proceeded thus to a possible comparison with the surfaces given from consolidation tests and shear box. This triaxial tests also give the slope of the critical state line η = q/p for the set of confinement constraints “

We have results of shearing, oedometer and triaxial tests for three samples made

M | σp_{1} (kPa) | σp_{2} (kPa) | P_{c}_{01} (kPa) | P_{c}_{02} (kPa) | Φ (˚) | |
---|---|---|---|---|---|---|

Ndienne | 2.05 | 210 | 2100 | 102.8 | 1028 | 50 |

Sebikhotane | 1.9 | 210 | 2100 | 102.4 | 1076 | 47 |

Yenne mer | 2.2 | 250 | 2200 | 116.8 | 1028 | 53 |

Ndienne | Sebikhotane | Yenne | σ ′ 3 C (kPa)_{ } | |||
---|---|---|---|---|---|---|

p (kPa) | q (kPa) | p (kPa) | q (kPa) | p (kPa) | q (kPa) | |

252.3 | 513.4 | 182.4 | 355.2 | 270.1 | 508.3 | 50 |

419.4 | 819.8 | 264.1 | 465 | 328.3 | 743.5 | 100 |

513 | 973.7 | 346.5 | 630.4 | 380.1 | 743.7 | 150 |

532.3 | 973.4 | 519.9 | 1052 | 461.5 | 848.6 | 200 |

953.3 | 1743 | 710.4 | 1325 | 891 | 1697 | 400 |

1410 | 2648 | 1041 | 1788 | 1317 | 2408 | 600 |

η (50 - 150 kPa) | η (200 - 600 kPa) | Φ_{cu}_{ } | |
---|---|---|---|

Sample of Ndienne | 1.77 | 1.9 | 4.7 |

Sample of Sebikhotane | 1.66 | 1.41 | 41.9 |

Sample of Yenne mer | 2.29 | 1.82 | 27.8 |

of compacted lateritic gravels of Ndienne, Sebikhotane and Yenne. These results permit us verification while drawing the yield surfaces and respective CSL “Figures 13-15”. That is to say, for a couple of results shear-box, oedometer test, we draw the loading surface and the CSL; for results given by the triaxial tests on the same material we draw the yield surface also and the CSL. For every material, we put the two loading surfaces on the same figure and at the same scale to compare the results.

The remark is that with the means of simple tests (shear-box and oedometer tests) we succeeded to describe the behaviour in the critical state but with less precision compared to the identification of the results given by triaxial tests.

For the samples of Ndienne and Yenne, we see that the spherical pressure is very lower (difference of 400 kPa) than the one of the triaxial test. It can be a consequence of the method of calculation of the pressure in the case of the oedometer test. Indeed in this case the lateral stresses is unknown and very often one uses the empirical law k_{0} = 1 − sinφ (k_{0} is the coefficient of earth pressure at- rest) while considering that the materials are normally consolidated. If we observe the sample of Sebikhotane, we even see that the spherical mean pressure is larger in the case of the method of the simple tests. This sample of Sebikhotane presents the weakest angle of friction (biggest value of the empiric constant k_{0}).

I deeply thank all researchers from the Laboratory of Mechanics and Modelling of the UFR Sciences de l’Ingénieur of the University of Thiès and the Laboratory of D’Alembert of the University of Pierre et Marie Curie-Paris.

Ndiaye, C. and Berthaud, Y. (2018) Identification of a Cam Clay Model through Shear-Box and Oedometer Tests. Application to Lateritical Soils from Senegal (West Africa). Geomaterials, 8, 1-13. https://doi.org/10.4236/gm.2018.81001