INTRODUCTION

The constitutive model of Extended Drucker-Prager has been widely used in research related to computer simulation of the mechanical response of agricultural soils, the tire-soil interaction and soil-turning plow (Xiang & Jin, 2004; Jafari et al., 2006; Davoudi et al., 2008; Herrera et al., 2008, 2013; Yu et al., 2008; Xia, 2011; González et al., 2012, 2013; Armin et al., 2014; Ibrahmi et al., 2014; Moslem & Hossein, 2014). Its use is primarily because it takes into account the flow rules, considering that the soil can flow associated or non-associated to the yield surface, In addition, this model allows predicting changes of tensions resulting from deformation by soil softening or hardening, it appears implemented in most commercial software used in the simulation by the finite element meth- od, requiring only six parameters as input data, which may be determined by conventional assays in mechanic of soils laboratories.

However, Egil & Risnes (2001); Xiang & Jin (2004); Helwany (2007), Yin et al. (2009), SIMULIA (2008a) report that this model presents a limitation or restriction on values of friction angle under 22 degree, if the convexity is assured. This limitation makes it inconsistent for friction angle val- ues greater than 22 degree, when setting the parameters of Drucker-Prager to Mohr-Coulomb ,which will be reflected in the accuracy of predictions. On the other hand, Colmenares & Zoback (2002), Al-Ajmi & Zimmerman (2006), Herrera et al. (2008), Alejano & Bobet (2012), concluded, by comparing the experimental data with predictions, that the model of Drucker-Prager tends to overestimate the diverters efforts. Other researchers like Otarawanna et al. (2004 ), Grujicic et al. (2009) report that the assumption of the associated flow rule leads to excessive dilation ¹.

In Cuba it has been used by Herrera et al. (2008) in the simulation of the mechanical response of a clay soil (Oxisol) ², and in modeling the compaction caused by the traffic of agricultural vehicle tires over these same soils. Both authors found the model Drucker-Prager Extended suitable to simulate these phenomena, although errors in predictions ranged from 9,66 to 22,97% (Herrera et al., 2008) and 9,6 to 19,15% 2, depending on the level of soil moisture and density. These authors report that the model has some inaccuracies unable to represent the soil brittle failure and stress changes. In an analysis using the Drucker-Prager model in a sandy soil the angle of dilatancy has great influence in predicting diverters efforts once the soil begins to plastically flow. This effect is closely related to the density and angle of internal friction, as these properties relate to dilatancy. Given the foregoing the present work was aimed adjusting constitutive model parameters Drucker-Prager Extended defining the phase of plastic deformation of soil in simulating the mechanical response of clayey soil.

METHODS

Drucker-PragerExtended Model

In this investigation the soil was considered as an elastoplastic material. So the yield criterion used was the Linear Drucker-Prager one, and it is written as:

(1)

Where:

F = the yield function, t = diverting stresses given by Equation (3), p = the diverting stress given in Equation (2), B = the angle of internal friction of Drucker-Prager, and d = the cohesion of the material.

(2)

Where:

(3)

(4)

(5)

Where:

K, is the coefficient relating diverting efforts obtained in triaxial extension with those obtained in triaxial compression (0.788 ≤ . ≤ .); when k= 1, t= q, which means that the yield surface is the circle of Von Mises represented in the plane of deviatory stress (Figure 1); σ1, σ2 and σ3 are compressivestresses in the triaxial test, and r is the third invariant of thedeviatory efforts.

The potential of plastic flow in the Linear Extended Drucker-Prager Model is expressed as shown in equation 6.

(6)

FIGURE 1.
Yield surfaces of the linear model in the plane of deviatory stresses (SIMULIA, 2008b).

Where:

Gf- is the potential of plastic flow, y dilatancy angle in the plane p - t .

In the case of granular materials such as soil, not associated flow rule ψ<b is applied; the soil does not flow in the normal direction to the yield surface (Figure 2).

Associated flow rule is applied to establish thaty=b, where the soil flows to normal direction of the yield surface. When the angle of dilatancy takes value ψ=0, inelastic deformation is incompressible. However; if it takes a value greater than zero ψ=0 the soil expands.

FIGURE 2.
Yield surface and flow direction of the Linear Extended Drucker-Prager Model in the linear plane p-t (SIMULIA, 2008b).

Properties and parameters required by DruckerPrager’s Extended constitutive model

The mechanical properties and constitutive parameters were taken from the realized investigations by de la Rosa et al. (2011), de la Rosa & Herrera (2013).

Virtual model implementation

The process of the implementation of the model in computational tool ABAQUS CAE 6.8-1 appears described in research by de la Rosa et al. (2013). The simulation comprised nine humidity and soil density defined during experimentation, guaranteeing checking the validity of the model of Drucker Prager Extended to predict the mechanical response of soil at different levels of moisture and density. In total 54 simulations were performed to evaluate the different configurations that can take the model, considering the soil as a dilatant materials (ψ=b) or not dilatant (ψ=0), taking the coefficient that relates the deflecting efforts the following values K= 0,80; k= calculated; k= 1.

Method used for the adjustment of parameters

The method of approximation of nonlinear functions or curve fitting Levenberg-Marquardt was used to adapt the constitutive Extended Drucker-Prager Model. This one is a widely used method for estimating variables in different study fields such as in estimating soil moisture rainforest (Truong-Loï et al., 2015). Levenberg (1944), Marquardt (1963), Moré (1978), report that this adjustment method is actually a combination of two methods of minimization: the method of gradient descent and the Gauss-Newton method.

The Drucker-Prager (B) failure plane angle dominates the trajectory of the diverting efforts once the soil begins to deform plastically (Figure 3), from the yield stress up to it reaches the breakage stress. The dilatancy angle (Ψ) defines the path of the deviatory efforts when the soil reaches its breaking stress.

FIGURE 3
Effort-deformation curve

The effort-deformation curve was divided into two parts, and equations describing its trajectory were sought. For the particular case of this study the method was programmed in Matchad software, version 14.0.

Initial data from of these two parameters correspond to the experimentally determined data. From this on, the approximate values of both the internal friction angle of the failure plane for the Extended Drucker-Prager Model, and the angle of dilatancy, will be calculated.

The value of the coefficient affecting the angle of friction of Drucker-Prager (B) depends on the type of failure that describes this curve, as it is shown in the equations 7 and 8.

For the plastic failure, where a well-defined break point was seen, and the soil was plastically deformed by hardening, the equation would be:

(7)

For the plastic failure, where a well-defined break point was not seen, and the soil was plastically deformed by softening, the equation would be:

(8)

Likewise, the value of dilatancy angle (ψ) was determined satisfying the accuracy in predictions (equations 9 and 10).

For the plastic failure, where a well-defined break point was observed, and soil was plastically deformed by softening, the equation would be:

(9)

For plastic failure, where no well-defined break point was observed, and the soil was plastically deformed by tightening, the equation would be:

(10)

An example of the approximate distribution of the exponential functions, depending on the internal friction angle of Drucker-Prager and on dilatancy (equation 7, 8, 9, and 10), is shown in Figure 4; it is shown that a suitable approximation of the predicted values in respect to the experimental ones, is reached for both the plastic failure (where a well-defined break point deformed by softening was observed), and the plastic failure deformed by hardening (where a well-defined break point was not observed).

FIGURE 4.
Approximation of functions to determine the angle of internal friction of Drucker-Prager. a) Plastic failure with deformation by hardening w = 27%, g = 1,05 g cm-3; b) Plastic failure with deformation by softening w = 51%, g = 1,09 g cm-3.

RESULTS AND DISCUSSION

Adjust of the plastic parameters of DruckerPrager’sExtended model

In the case of internal friction angle of Drucker-Prager a nonlinear relationship was found that allows correlating the values of the angle of internal friction of the soil (ψ)with the estimated angle (B ) from the function approximation, so the relationship implemented in ABAQUS to relate the frictional parameters of Drucker-Prager criterion with the ones of Mohr-Coulomb. The adjusted model graph is shown in Figure 5a. In the case of dilatancy angle a polynomial relationship was obtained, allowing correlating the levels of dilatancy angle (ψ ) with the friction angle of Mohr-Coulomb (φ). The graph of the model can be seen in Figure 5b. Both models are considered moderately good, because the values of the determination coefficient ranged from 84,60 to 90,00% ².

FIGURE 5
Graph of the adjusted model for best and yest.

After determining the optimal values of the parameters related to the stage of plastic deformation of the Extended Drucker-Prager Model, the next step was the checking of the adjustments made to the constitutive parameters of the Extended Drucker-Pager Model.

In the case where the moisture content was near the plastic limit, and the soil exhibited a plastic failure with hardening (Figure 6a) it could be seen that the Extended Drucker-Prager Model is able to predict accuracy of deviatory efforts in both the elastic and plastic phases, once the adequacy of its parameters is done. A decreased mean absolute error (MAE) from 12,68 to 1,86% was achieved (Table 1).

FIGURE 6
Stress-strain curves of the soil. a) w = 27% y g = 1,05 g cm-3; b) w = 51% y g = 1,09 g cm-3

TABLE 1.

Values of the angles of internal friction and dilatancy; statistics of Kolmogorov-Smirnov and MAE

TABLE 1. continuation

Values of the angles of internal friction and dilatancy; statistics of Kolmogorov-Smirnov and MAE

When the moisture content exceeded the limit of adhesion, and the soil showed a plastic softening failure (Figure 6b), it was observed that the model is able to predict the mechanical response of the soil for the entire range of deformation. Similarly, when the soil exhibited a plastic hardening failure, a decreased of the mean absolute error (MAE) from 15,42 to 2,98% was achieved (Table 1).

The lower value of the error was reached when the content of moisture was 27% and the density 1.05 g cm-3 (dry and loose soil). Here the soil exhibited a hardening plastic failure. On the other hand; the statistical analysis showed no statistically significant differences between experimental distributions and those predicted through simulation (Table 1); since the Pvalue for the cases under study was Pvalue ≥ 0,05.

CONCLUSIONS

• The constitutive parameters of the Extended Drucker-Prager Model that define friction and dilatancy of soil in the phase of plastic deformation were adapted, and there was achieved a reduction of errors below 4,35% was achieved.

• The adaptations made to the constitutive parameters of the Extended of Drucker-Prager Model by means of the function approximation method of Levenberg-Marquardt, made possible to accurately predict the mechanical response of Vertisol, regardless of their state of moisture, densification, and the type of failure the soil shows.

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Notes