The concepts of limit state and critical state proposed by Roscoe, Schofield and Wroth between 1958 and 1968   have been shown to apply well to natural agile, as shown in this illustration by Tavenas  in Figure 1.
This identification is repeated for two lateritic soils from Senegal from simple mechanical tests and the results are compared with results from shear tests on samples from the same locality   .
Figure 1. Representation of the model of cam clay in the (e, p, q) and (e, lnp)  plans.
Since the purpose of using Cam Clay’s model is to correctly describe a loading path, to make an elastoplastic study and to arrive at finite element method calculations, we present in this article the continuation of the study started on the two above-mentioned samples.
For this purpose, a verification study of the tests on Optum G2  was followed by structural calculations using Cast3M . It should be noted that, for the experimental test procedure, the reader is invited to refer to the publication .
2. Verification of the Casagrande’s Shear Box Tests
On OptumG2 , Finites Elements calculation software, we verified our mechanical results by simulating the shear and oedometric tests.
Under the hypothesis of plane stresses, the 2D specimen is modelled by a 60 × 20 mm rectangle, made of a material to which Cam Clay’s modified law is applied. The upper half-box is modelled by a set of rigid plates whose rotation is blocked in order to best approximate the real behavior of the ground during the test (Figure 2).
By a limit analysis for each normal stress, one can plot the evolution of the shear stress τ as a function of the normal stress σ and thus determine the cohesion c and the internal angle of friction φ.
The initial conditions of the constituent material have no influence on the limit state of the structure and since the structure is small in size, gravity can be neglected.
The limit analysis runs in two steps in order to approach the actual solution:
1) The calculation of the lower bound: the mesh is made of triangles with three nodes (only one Gauss point for stress integration) allowing the discontinuity of stresses between elements (but not of displacements). This step underestimates the stiffness matrix and allows the calculation of the lower bound of the solution.
Figure 2. Presentation of the casagrande shear box on Optum G2 with locking system boundary conditions.
2) Calculation of the upper bound: the mesh is made up of triangles with 6 nodes. The 3 Gauss points of these elements have the particularity here of being confused with the vertices of the triangle. Thus the interpolation of the displacements is quadratic, and that of the constraints is linear. They ensure the continuity of constraints and displacements. This step allows to calculate the upper bound of the solution by overestimating the stiffness matrix.
3. Influence of the Size of the Model
The size of the Casagrande box does not usually have inﬂuence on the test results . However, as OptumG2 is designed for calculations on large structures, calculations are faster and more accurate by increasing the size of the model. We present a small comparison in Table 1.
The values of the shear stresses at specimen vary little for larger scale models, which is why the calculations in this study are performed on a 10 scale model (60 * 20 cm specimen), in order to best respect the real test conditions (6 * 2 cm) and to facilitate the resolution of numerical problems.
3.1. Numerical Results of Shear Box Tests and Comparisons with Experimental Results
Calculations are performed using the cohesions and friction angles experimentally obtained during tests on the two soils . The names of these samples are: Sindia and Keur Samba (Table 2 and Table 3).
It is clear that the shear stress increases as it converges towards the limit value (τ_rup), obtained with the limit analysis. Furthermore, the behavior of the sample was first contracting and then expanding. This is a behavior observed in the experimental tests . The mechanical characteristics (c and φ) are similar in both cases, even if there are some discrepancies in the cohesion values that are not very noticeable.
3.2. Numerical Results of Oedometer Tests and Comparisons with Experimental Results
Taking into account the cell (cylinder of 8 cm diameter and 2 cm height), the
Table 1. Illustrative values of the influence of the size of the model.
Table 2. Values of the cohesions and angle of internal friction for the specimen of SINDIA.
Table 3. Values of the cohesions and internal rubbing angle for the sample of Keur Samba.
material and the load, the sample is modelled axisymmetrically by a 4 × 2 cm^2 rectangle.
• On the left edge of the cell, located on the axis of symmetry, normal displacements are blocked.
• On the right edge, glued to the cell wall, normal displacements and flow are blocked.
• On the lower edge, the material is drained and the normal displacements are also blocked.
• At the upper edge, the material is drained and the load is applied by a constant distributed force.
Several elasto-plastic analyses are carried out, corresponding to the different loading levels. From certain vertical stress values eﬀectives a stepwise discharge is performed, in order to obtain loading-unloading curves. Look at the representation in Figure 3.
Cc (coefficient of compressibility) and Cg (coefficient of swelling) (or λ (slope of the isotropic virgin curve)) and k (slope of the unloading-reloading curve)) are sought by these several elasto-plastic analyses. But first, the influence of friction on the wall can be checked by using the “Reduction Factor” on the Sindia sample, shown at Figure 4.
The friction at the interface with the material is translated by the “Reduction Factor” (RF = 1 corresponding to perfect adhesion, and RF = 0 to frictionless
Figure 3. Presentation of the oedometer cell in Optum G2.
Figure 4. Influence of the friction of the wall, tested on the Sindia specimen.
sliding). In the frictionless case, the vertical displacement is homogeneous over the entire upper edge. Otherwise, an average height variation on the upper edge is calculated by exporting the results of the analysis in Matlab for example. It is noted that the friction is only influent very little on the value of the average displacement on the upper edge. To facilitate the calculations, the elastoplastic analyses were subsequently performed from the model without plates with boundary conditions (equivalent to the model with frictionless plates).
The numerical results obtained are consistent with the experimental results. If significant gaps can be observed for the lateritic Keur Samba sample (Table 5), the orders of magnitude are correct. The results are obtained for much higher stresses than in the actual tests, which is probably due to a much lower initial pre-consolidation stress in the actual sample. The difference between the experiment and the numerical simulation gives a good appreciation of the consistency of the results .
4. Oedometer Test Modelling Using Cast3M
In these calculations, the elastoplastic model chosen is the modified Cam Clay
Table 4. Values of the slopes of the virgin and unloading-reloading curves of SINDIA.
Table 5. Values of the slopes of the virgin and unloading-reloading curves of Keur Samba.
model and the modeling parameters are those derived from identification with the model from simple mechanical tests (oedometer and shearing  ). The methodology and principle of testing is well supported  and  for compressible soils.
With Cast3M we choose an imposed displacement loading which consists in creating a point field.
In OPTI DIM3, the type of elements chosen here is Cube 8 nodes (CUB8) allowing the use of the “VOLU” operator. The global mesh includes 3437 nodes and of 5080 elements cubic cub8. The different steps A, B, C and D of the our oedometer cell are presented in Figure 5.
At the end of the calculation, objects of types MCHAML (element fields) and CHPOINT (point fields) (constraints and displacements) are grouped together in a table. From this table, we can calculate the maximum and minimum principal stresses using the PRIN operator (Figure 6).
From these same stress data stored in the table, we calculate the stress invariants I1 and J2 using the INVA operator which in turn will allow us to calculate the mean (p = I1/3) and deviation (q = ) stresses. The plasticity criterion (Equation (1)):
Can thus be traced and stress states with elastic or plastic behaviour can be traced (Figure 7).
The implementation of the Cam Clay model in Cast3M with a single load surface did not allow to take into account strain-hardening in a single calculation. As a result, two separate calculations are made, each with a chosen pre-consolidation pressure.
The strain-hardening of the material is noticeable by a decrease in the plastic zone (Figure 7(b)) due to an increase in the elastic behavior zone (negative criterion), because the pre-consolidation pressure has a higher value in the second calculation.
Figure 5. Mesh of the oedometer cell (a), material (b), cell + material (c), cell + material + piston (d).
Figure 6. Minor (a) and maximum (b) principal stress diagrams for the Sindia sample. (a) Sigma_33; (b) Sigma_11.
Figure 7. Plasticity criteria for Pcon1 (a) and Pcon2 (b) for the Sindia sample.
For Sindia laterite, the maximum principal isostresses range from −1.62 × 105 Pa to 1.65 × 105 Pa, while the minimum principal isostresses are between −2.7 × 105 Pa and 1.03 × 105 Pa (Figure 6).
Figure 7 shows the elasto-plastic stress state of the Sindia lateritic material. The first limit state point which is Pcon1 = 22.11 kPa determined from the oedometric results  leads to the plasticity criterion (Figure 7(a)) with 2.02 × 108 Pa as the plastic value closest to zero and 2.7 × 1010 Pa as the largest. The values for which f(p, q, pcr) < 0 are between −6.69 × 109 Pa and −1.52 × 109 Pa. For this stress range, the behavior of the material is purely elastic. For the criterion (Figure 7(b)) corresponding to the second preconsolidation pressure Pcon2 = 53.69 kPa , a decrease in the range of plastic behavior is observed due to the strain-hardening of the material. The value of the smallest plastic stress decreases from 2.02 × 108 Pa to 2.33 × 109 Pa. Below this value, the behavior remains elastic.
For the Keur Samba laterite sample, the calculations give maximum principal stresses between −2.31 × 105 Pa and 2.53 × 105 Pa (Figure 8(b)) and minimum principal stresses of values between −3.46 × 105 Pa and 1.62 × 105 Pa (Figure 8(a)). This is an important range of values when compared with the Sindia sample.
The first pre-consolidation pressure Pcon1 = 24.02 kPa and the second Pcon2 = 53.83 kPa  give the plasticity criteria f(p, q, pcr) in Figure 9. The range of elastic behavior (f < 0) varies between −1.63 × 1010 Pa and −1.91 × 109 Pa for Pcon1 and between −1.7 × 1010 Pa and −4.61 × 109 Pa for Pcon2. Naturally, the elasticity increases with the pre-consolidation pressure. The material collapses and increases its reversibility zone.
Figure 8. Minor (a) and maximum (b) principal stress diagrams for the Keur Samba sample. (a) Sigma_33; (b) Sigma_11.
Figure 9. Plasticity criteria for Pcon1 (a) and Pcon2 (b) for the Keur Samba sample.
In this publication, we have been able to establish the continuation of the work begun, which consisted in identifying tropical soils with complex behavioral models based on simple mechanical tests . However, in this article, we have verified the methodology of our previous study by simulating the laboratory tests using the finite elements software OptumG2. This verification has allowed us to believe that the tests were carried out correctly and that the methodology was appropriate.
Finally, a finite element calculation using the Cast3M software, allowed the application of the modified cam clay model on lateritic samples which were, in fact, identified with this model in the previous study  . This allowed us to interpret an elastoplastic behaviour based on the writing of the plasticity criterion. The knowledge of the stress invariants, stored in the Cast3M table after the calculations, allows us to determine the spherical pressure and deviatoric stresses. This leads to plotting the load surfaces.
This is a result that we consider interesting because it allows us to know all the behaviours of the studied material by applying the mathematical modelling that the modified cam clay model includes; namely the elastoplastic deformations (volumetric and deviatoric), the contracting and expanding behaviour as well as the level of stresses on the critical state line as well as their assumption in the dimensioning. This will probably be the subject of the next publication.
I deeply thank the Laboratory of Mechanics and Modelling of the UFR Sciences de l’Ingénieur of the University of Thiés (Sénégal) and the Laboratory of D’Alembert of the Sorbonne University (ex Pierre et Marie Curie) of Paris (France).
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