Novel Soil Strength Criterion Compared with Conventional Criteria

A novel soil strength criterion is proposed based on the shear stress ratio on a new spatially mobilized plane, where the cube root of principal stresses is constant. The strength failure surface depicted in the principal stress space by this criterion was smoothly conical, with a curved triangle shape on the octahedral plane. A comparative analysis of the strength failure surfaces of the Mohr-Coulomb (M-C), the Drucker-Prager (D-P), the Matsuoka-Nakai (M-N), the Lade-Duncan (L-D), the new criteria, and the shear strength laws of different criteria with parameter b on the π plane showed that the L-D criterion and the new spatially mobilized plane strength criterion were comparable, which revealed the physical essence of the L-D criterion. Comparing the new strength criterion with the measured results of true triaxial tests of 4 kinds of intact loess under conditions of consolidation and drain, the strength law of loess could be described by the new strength criterion under complex stress conditions, and the rationality and reliability of the strength criterion were verified by the correspondence between the criterion and experimental values.


Introduction
Four criterion models have been used extensively in rock and soil mechanics, including the Mohr-Coulomb (M-C) [1], the Drucker-Prager yield (D-P) [2], the Matsuoka-Nakai (M-N) [3], and the Lade-Duncan (L-D) [4].Among those, the M-C, D-P, and M-N models follow the principle that shear stress follows within the main stress element at material failure, with the assumption that the shear plane is in different types.The M-C model assumes that the major and minor principal stress planes introduced at failure will be perpendicular to the shear plane, which also forms an angle of 45 major principal stress, and the shear stress on it is in a linear relationship with the normal stress.Neither shear nor normal stresses are related to the intermediate principal stress, as shown in Figure 1.Furthermore, true triaxial compression test and plane strain test results usually show a shear failure envelope that is nearly identical to this theoretical shear plane.
The D-P strength criterion model is based on the assumption that the octahedral shear stress at failure depends linearly on an octahedral normal stress through material constants.The octahedral shear plane forms the same angle with each plane on which every 2 of the principal stress axes fall (Figure 2).
Based on the M-C criterion's geometrical description of the shear failure plane, the M-N failure criterion takes into account the intermediate principal stress, leading to the concept of the spatially mobilized plane [3] (Figure 3).In this criterion, the soil body is discretized into a series of cubes of hexahedral elements.When the 3 edges overlap the axes of the 3-dimensional rectangular coordinates, the corresponding intersections of the spatially mobilized plane and the coordinate axes are   The twin-shear strength criterion [6] proposed by Yu Maohong and L. N.He could calculate the shear stress on both shear planes at material failure, and could also be called a twin spatially mobilized plane criterion.
In addition to all the models above demonstrating that shear stresses on spatially mobilized planes are in a linear relationship with normal stress, the L-D criterion established the nonlinear relationship between shear stress and mean normal stress described by a power function.However, the L-D criterion does not establish a corresponding spatially mobilized plane.A nonlinear strength criterion was established by Yangping Yao et al. [7] by introducing a power function relationship between shear stress and spherical stress at material failure, and by combining generalized von Mises criterion (corresponding to the circumscribed circle of the M-C strength failure surface) with the M-N strength criterion.
To better describe the strength of loess, this paper introduces a new spatially mobilized plane model that generates the axis intersections as A comparison between the test results and this theoretical prediction is given below.

The 3 σ Spatially Mobilized Plane
Similar to the M-N proposal of a spatially mobilized plane, the new model is called the 3 σ spatially mobilized plane (Figure 4).
The spatially mobilized plane could be determined by According to the geometric relationship shown in Figure 4, the directional cosine of this spatially mobilized plane is calculated as, assuming OCB θ ∠ = ( ) ( ) The cosine of the normal direction of the spatially mobilized plane relating to the axis I is The cosine of the normal direction of the spatially mobilized plane relating to axes II and III are respectively: Or, the equations could be combined as From which the component of stress on the 3 σ spatially mobilized plane along the coordinate axes are derived as , , The resultant force on the 3 σ spatially mobilized plane is Also, the normal stress N σ and shear stress N τ would be:

The 3 σ Spatially Mobilized Plane Strength Criterion
Since shear stress to normal stress on the spatially mobilized plane is a constant value Under triaxial compression ( ) f k in (11) can be replaced with (12): With Formula (13), the strength failure envelope in the principal stress space and the strength failure curve on the octahedral plane could be plotted as in Figure 5.
The failure criteria are related only to the internal friction angle for noncohesive soil.
After testing soil samples with internal friction angles φ of 5˚, 15˚, 25˚, 35˚, and 45˚, the authors plotted the corresponding strength failure circle on the π plane as shown in Figure 6.
Generalizing the 3 σ spatially mobilized plane to cohesive soil, the shear stress and normal stress on the spatially mobilized plane would be and and the strength criterion would be

The L-D Criterion and the 3 σ Spatially Mobilized Plane Criterion versus Other Conventional Criteria
The D-P criterion can be generated by changing the circumcircle in the M-C criterion.It is also known as a generalized von Mises criterion: The M-N criterion's criterion is The L-D criterion generated the fitting curve of the failure points, which formed a curved triangle on the octahedral plane, based on the true triaxial test result.The strength curve is in a linear relationship with the average spherical stress on the meridional plane that can be described as very similar to that of the 3 σ spatially mobilized plane criterion, and they are all be- tween the D-P and M-N criteria's strength planes.
Further, a comparison between the L-D and 3 σ spatially mobilized plane criteria versus the M-C, D-P, and M-N criteria is plotted on the octahedral plane with the same meaning principal stresses in Figure 8.It shows the same result: the strength circles of the L-D and 3 σ spatially mobilized plane criteria fall between the D-P criterion strength circle and the M-C and M-N strength circles.In addition, the L-D criterion's strength curve is identical with that of the 3 σ spatially mobilized plane criterion.

The 3 σ Spatially Mobilized Plane Criterion and the L-D Criterion
The strength failure surface on the principal stress plane and the strength failure circle on the π plane can be derived from the L-D criterion.While the soil internal friction angle increases at 5˚, 10˚, 15˚, 20˚, 25˚, 30˚, 35˚, and 40˚, the strength failure surface and strength failure circle increase their radius at the same time, as shown in Figure 9.
Similarly, for the 3 σ spatially mobilized plane criterion, the test result is shown in Figure 10.
The following study reveals how soil shearing resistance changes along with the change of principal stress ratio b under different combinations of soil internal friction angles φ and spherical stresses on the π plane.As shown in Figure 11, the 3 σ spatially mobi- lized plane and the L-D criteria yield identical soil shearing resistances.

The 3 σ Spatially Mobilized Plane Criterion and the True Triaxial Test
Loess soil structure strength is highly correlated with the soil's outstanding physical and structural properties.However, its strength stability drops significantly when it loses its   In this study, the authors conducted true triaxial tests of 4 kinds of intact loesses under conditions of consolidation and drain.Tests were completed with the newly developed true triaxial apparatus, with 70 mm × 70 mm × 140 mm specimens, from the Xi'an University of Technology [9].The basic physical indices of different loesses are shown in Table 1, in which the liquid limit (LL) and plastic limit (PL) were tested by a conical weight with a 76 g weight penetrating a prepared soil sample to depths of 2 mm and 10 mm.The procedure adhered to the national standard for soil test methods of the People's Republic of China.The plastic index is the difference between the LL and the PL.The flow index is the ratio of the difference between the moisture content and the PL to the plastic index.
Three confining pressures were applied, 100 kPa, 200 kPa, and 300 kPa, and the intermediate principal stress ratios were controlled at 0.00, 0.25, 0.50, 0.75, and 1.00 (Table 2).The first step of the test procedure was to compress the loess sample with a certain confining pressure until its consolidation.The second step was to separate the drained true triaxial compression with a certain intermediate principal stress ratio until its deconstruction.
Four sets of test results were plotted for each type of loess sample.Figure 12 shows that normal stress is highly linearly correlated with shear stress on the 3 σ spatially mobilized plane for every sample set.
Figure 13 exhibits the theoretical 3 σ spatially mobilized plane curve along with the true triaxial test result curve on the π plane for each set.They all have same average spherical stress of 600 kPa, demonstrating consistency between the criterion and the experimental values, and the rationality and reliability of the 3 σ spatially mobilized plane criterion.

Conclusions
The normal stress plane intersected with each of the major, intermediate, and minor normal stress axes at points 3 1 k σ , 3 2 k σ , and 3 3 k σ .The surface that all 3 inter- section points fell on was defined as the 3 σ spatially mobilized plane.The 3 σ spatially mobilized plane strength criterion projected a strength failure envelope in a conical shape in the stress space and formed a curved triangle shape of strength failure surface on the octahedral plane.It falls on the exterior of the M-C envelope and the M-N cone, but the interior of the D-P cone is identical to the L-D envelope in the stress space.
On the octahedral plane, the shearing resistances of the L-D criterion and the 3 σ spatially mobilized plane strength criterion changed, along with the principal stress ratio b in the same form.This indicated that the L-D criterion and the 3 σ spatially mobilized plane strength criterion were essentially identical.The L-D strength criterion was an approximate 3 σ spatially mobilized plane strength criterion with a constant shearing stress ratio.
True triaxial test results verified the linear relationship between shear stress and normal stress on the 3 σ spatially mobilized plane.The test results were also comparable with the 3 σ spatially mobilized plane strength criterion on the π plane, which dem- onstrated a rational and reliable relationship of that strength criterion to the strength of loess.

3 1k σ , 3 2 k σ , and 3 3
k σ , in the same way the spatially mobilized plane does in the M-N criterion.The new model is comparable to the L-D criterion, showing that the L-D criterion is approximative with the strength criterion based on the 3 σ spatially mobilized plane, which reveals the physical basis of the L-D strength criterion.The rationality and reliability of the new model were demonstrated by a series of true triaxial tests on intact structural loess.

Figure 5 .
Figure 5.The cube root spatially mobilized plane criterion strength failure envelope.

Figure 7 .
Figure 7.Comparison of the L-D being similar to 3 σ spacial mobilization plane criteria with the D-P and M-N criteria in the principal stress space.

Figure 8 .
Figure 8.Comparison of the Lade-Duncan being similar to 3 σ special mobilization plane criteria with conventional criteria on the octahedral plane.

Figure 9 .
Figure 9.The strength failure surfaces of the Lade-Duncan criterion.
Figure 12.The shear and normal stresses of intact loess on the 3 σ spatially mobilized plane.
Figure 13.Comparison between the 3 σ spatially mobilized plane strength criterion and results meas- ured by a true triaxial test of loess on the octahedral plane.

Table 1 .
Basic physical indices of loess.

Table 2 .
The moisture content and dry density of intact loess.