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A new nonlinear unified strength (NUS) criterion is obtained based on the spatially mobilized plane (SMP) criterion and Mises criterion. New criterion is a series of smooth curves between SMP curved triangle and Mises circle in the

The research of strength criterion theory [

However, the above strength criteria are associated with a single shape in the

The above previous studies show that the strength criterion is researched through a simple single form to a unified strength theory system with a relatively wide scope of application. Consequently, it is important to explore a widely applicable strength theory system. In the present paper, a new anisotropic nonlinear unified strength criterion is established based on Mises criterion and SMP criterion. The new criterion is approximated as a series of smooth curves between the SMP curved triangle and the von Mises circle in the

SMP criterion (Matsuoka and Nakai [_{1}, _{2}, and _{3} are stress invariants:

_{2} = _{3}), the sine of the friction angle _{0} of a granular material is

Substituting equations (

Using the trigonometric identity, the expression of _{0} is the data for

The shape function of the SMP criterion cannot reflect the influence of internal factors on material; therefore, the application of the criterion is limited. In this report, the shape function is rewritten as the following expression:_{0} which is the triaxial tensile strength of the material is given by

Therefore,_{2} and _{3} are obtained from equations (

Figure _{0} and

Variations of criterion with _{0} and

In the triaxial test of the rockfill material tests [

That is,_{r} is reference stress usually taken as _{f} is usually obtained from the intercept of equation (

In the space of

When

Numerous geomaterials exhibit transverse isotropy due to deposition, as shown in Figure _{0} is the value that _{1} and _{2} in the anisotropic function are selected to depict the anisotropy of the material. The anisotropic function and the nonlinear isotropic strength criterion are combined to obtain an anisotropic strength criterion, as shown in the following equation:

Application of true triaxial tests on cross-anisotropic soils in the octahedral plane.

The effect of isotropic parameters on the shape of the strength criterion in the deviatoric plane has been discussed above. This section focuses on the effect of anisotropic parameters on the shape of the strength criterion. According to the characteristics of the test data, the parameters _{1} and _{2} in the anisotropic function are selected to depict the anisotropy of the material. Different formulas can be selected for the anisotropic parameter

In the true triaxial test, when the stress tensor and the fabric tensor are coaxial (_{1} and _{2}, as shown in Figures _{2} fixed at −0.25, −0.333, and 0, the effect of varying _{1} on the strength curve in the _{2} = −0.25, the strength values of the anisotropic criterion and the isotropic criterion at _{1} values, and the criterion is symmetric about the _{z} axis. The criterion expands in the _{1} > 0 and shrinks in the _{1} < 0, and the degrees of expansion or shrinkage are related to the value of _{1}. The anisotropic function variation in the _{2} = −0.333, as shown in Figure _{1} values, and the criterion is symmetric about the _{z} axis. Relative to the isotropic criterion, the anisotropic criterion expands in regions I and II and shrinks in region III of the _{1} > 0, and it shrinks in regions I and II and expands in region III of the _{1} < 0, with the degrees of expansion or shrinkage being related to the value of _{1}. The anisotropic function variation in the _{2} = 0, as shown in Figure _{1} values is symmetric about the _{z} axis; when _{1} > 0, the anisotropic criterion expands in the _{1} < 0, the anisotropic criterion shrinks in the _{1}. The anisotropic function variation in the

Effect of _{1} on the failure loci and _{2} in the octahedral plane: (a, c, e) the failure curves and (b, d, f) the

When the stress tensor and the fabric tensor are not coaxial (_{1} and _{2} values (_{1} = −0.08, _{2} = 0), the effect of the change in the included angle between the deposition surface and the vertical stress on the strength curve in the _{z} axis in the _{y} axis in the _{x} axis in the

The variations of the failure surface and

To facilitate the application of the criteria, the model parameters should be determined as much as possible using conventional triaxial compression or extension tests. In this section, the proposed ANUS criterion is applied to various geomaterials. The geomaterial model parameters need to be determined only by conventional triaxial compression or extension tests. In this section, the strength parameters and anisotropy parameters in the anisotropic criterion are mainly determined. Callisto et al. [

Determination of _{f}, _{0}. The _{0} is the friction angle. From the fitted linear relation in the figure, it can be seen that, for the Pietrafitta clay, _{0} = 14.478°, and _{0} = _{0} = 193.58 kPa can be obtained from the values of _{0}, taking

Determination of _{1} and _{2}, _{1}, and _{2} can be determined by the _{1} term is taken in this test. In the triaxial extension test, when _{0} = 4. Using _{e} = 254.41 kPa (_{1}), and _{1} = 0.0417.

Using the above steps, all parameters of the anisotropic criterion for Pietrafitta clay are determined. The test parameter determination method is shown in Figure

Illustration on calibrating the parameters _{f}, c, _{0} (b) Comparison of the two criteria for natural Pietrafitta clay (test data from [

A series of large triaxial tests for rockfill materials were conducted to research strength nonlinearity of this material [

Verifications of the two criteria. (a) Parameter determination. (b) Test data in the

In addition to the above true triaxial test of Pietrafitta clay, fine glass-bead sand is then used to verify the anisotropic criterion. Haruyama [_{0} = 0, _{1} = −0.038, and _{2} = 0. The prediction results and test data in the deviatoric plane and

Verifications of the two criteria for glass beads in (a) the octahedral plane; (b) the

The failure friction angle measured from torsional shear tests on Leighton Buzzard sand of different densities and on spherical glass beads conducted by Yang et al. [_{0} = 0, _{1} = −0.368, and _{2} = 0.1187; for medium dense LB sand, _{0} = 0, _{1} = −0.2463, and _{2} = 0.0593; for dense LB sand, _{0} = 0, _{1} = −0.2668, and _{2} = 0.0655. The prediction results (ANUS criterion and ALD criterion [

Verifications of ANUS criterion for different granular materials (data from [

This test is mainly used to verify the peak strength of the rock considering the different angles between the rock deposition surface and the principal stress. The isotropic strength criterion and the anisotropic strength criterion were used for prediction, and a comparison of the prediction results with the test data shows the superiority of the anisotropic strength criterion. Here, based on the schist triaxial tests with different deposition surface angles in the literature, the parameters of the anisotropic strength criterion are _{0} = 8 Mpa, _{1} = −2.47, and _{2} = −0.33. It can be seen from the test results that the two types of rock exhibit strong strength anisotropy in different deposition directions. In comparison, the use of the isotropic strength criterion to predict strength variations can lead to large errors, making it difficult to promote the isotropic criterion in engineering applications.

As shown in Figure

Comparison of the two criteria for Tournemire shale in (a) the

In this paper, a new strength criterion is proposed on the basis of the SMP criterion. In the

The new criterion can depict the 3D strength variation of the material and reflect the effect of the intermediate principal stress and the nonlinear characteristics of the material strength.

The unified anisotropic strength criterion based on the pattern of the fabric evolution of granular materials can reflect the strength anisotropy caused by the deposition characteristics of the material. This criterion can be applied to the true triaxial test to consider the direction angle of the deposition surface. The correctness of the criterion was verified using sand, clay, and rock materials.

The underlying data used in the presented study were obtained from the literature.

The authors declare that they have no conflicts of interest.

This work was sponsored by Ningbo Natural Science Foundation Project (2019A610394) and the Initial Scientific Research Fund of Young Teachers in Ningbo University of Technology (2140011540012) and supported by the Systematic Project of Guangxi Key Laboratory of Disaster Prevention and Structural Safety (no. 2019ZDK005) and the Ningbo Public Welfare Science and Technology Planning Project (no. 2019C50012). These financial supports are gratefully acknowledged.