Formulation of Anisotropic Strength Criterion for Geotechnical Materials

School of Civil and Transportation Engineering, Ningbo University of Technology, Ningbo 315211, China Engineering Research Center of Industrial Construction in Civil Engineering of Zhejiang, Ningbo University of Technology, Ningbo 315016, China Ningbo Rail Transit Group Co., Ltd., Ningbo, Zhejiang 315211, China Beijing Municipal Engineering Research Institute, Beijing 100037, China NingboTech University, Ningbo 315100, China Guangxi Key Laboratory of Disaster Prevention and Engineering Safety, Guangxi University, Nanning, China


Introduction
e research of strength criterion theory [1][2][3][4][5][6][7][8][9][10][11] is an important topic in geotechnical engineering. ere are usually three types of methods for establishing strength criteria, namely, theoretical methods, empirical methods, and combined theoretical-empirical methods. eoretical methods are usually based on a hypothesis, and there is a model to interpret the failure mechanism of geomaterials. All the model parameters in the strength criteria established by this method have clear physical meanings. is type of strength criterion uses the law of friction to explain the failure of geomaterials, including the Mohr-Coulomb strength theory, the D-P criterion [11], the spatially mobilized plane (SMP) criterion [2], and the twin shear unified strength theory [9,12,13]. Strength criteria established by empirical methods are usually based on the fitting of test data and include the famous Willam-Warnke criterion [14] and the Hoek-Brown criterion [6,15]. Compared to the criteria that are based on theoretical methods, the physical meanings of some of the parameters of the strength criteria established by empirical methods are often not very clear. Finally, combined theoretical-empirical methods are usually used to establish special strength criteria for specific material properties. is type of criterion applies the Mohr-Coulomb strength theory to jointed rock materials [16] by introducing additional parameters [17][18][19].
However, the above strength criteria are associated with a single shape in the π plane and are thus unable to reflect the factors of material strength that may vary with the change of internal factors. To address this problem, researchers [10,11,[20][21][22][23][24][25][26] have proposed to unify the strength criteria. e common method is to introduce some parameters to change the size of the shape function. e above criteria are all isotropic strength criteria and cannot depict the strength anisotropy of geomaterials. However, numerous geomaterials exhibit transverse isotropy due to deposition; that is, the strength of the material is the same in the deposition surface but different in the depositional surfaces of different directions. e directional shear test of most soils shows that the change in the direction of the principal stress also leads to strength anisotropy, reflecting the initial anisotropy of the material. erefore, it is necessary to establish a uniform strength criterion that considers the depositional characteristics or initial anisotropy of geomaterials. e 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. e new criterion is approximated as a series of smooth curves between the SMP curved triangle and the von Mises circle in the π plane to unify the strength criteria. Based on the fabric tensor, a nonlinear anisotropic strength criterion is proposed. e correctness of the criterion is verified using sand, clay, and rock materials. [2]) has been widely used to check the strength of geomaterials because of its simple form and clear physical meaning. In the present study, the SMP criterion is modified to make it more widely applicable. e SMP criterion is shown in the following equation:

Linear Unified Strength (LUS) Criterion. SMP criterion (Matsuoka and Nakai
where I 1 , I 2 , and I 3 are stress invariants: C is a model parameter. Under conventional triaxial compression (σ 2 � σ 3 ), the sine of the friction angle φ 0 of a granular material is Substituting equations (2) and (3) into equation (1) gives Using the trigonometric identity, the expression of q is obtained as where where p is the mean stress and q is generalized shear stress. us, the shape function of the SMP criterion is where q 0 is the data for q when θ � 0. erefore, the forms of p and q of the SMP criterion are where g 0 (θ) is obtained by equation (10) and M is the coefficient of friction: e 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: where M is obtained from equation (12), σ 0 which is the triaxial tensile strength of the material is given by and p and q are the average stress and the generalized shear stress, as defined by equations (9) and (10), respectively. erefore, where and α is the model parameter determined by the strength ratio under the triaxial compression and extension conditions. e value of α ranges from 0 to 1. One has 2 Advances in Civil Engineering where L 2 and L 3 are obtained from equations (6) and (7), respectively. Figure 1 illustrates the influences of parameters φ 0 and α on linear criterion in the deviatoric plane. As Figure 1(b) shows, the shape of the criterion is shifted from the SMP curved triangle to the von Mises circle when α gradually increases from 0 to 1.

Nonlinear Isotropic Strength (NIS) Criterion.
In the triaxial test of the rockfill material tests [27], the large confining pressure causes the granules to break [28][29][30], and the material exhibits remarkable strength nonlinearities. erefore, to obtain the nonlinear of the rockfill material, the above new criterion equation (13) need be further modified as expressed in the following equation: where p r is reference stress usually taken as p r � 101 kPa; M f is usually obtained from the intercept of equation (18b) and the shape function; and n is a curvature parameter.
In the space of ln(q/p r ) − ln(p + σ 0 /p r ), ln[M f g(θ)] is the intercept and n is the slope.
When n � 1, where n is the curvature parameter, and the shape function in the deviatoric plane is

Nonlinear Unified Strength (NUS) Criterion.
Numerous geomaterials exhibit transverse isotropy due to deposition, as shown in Figure 2; that is, the strength of the materials is the same within the depositional surface but different in the depositional surfaces of different directions, reflecting the initial anisotropy of the material. erefore, it is necessary to establish a strength criterion that considers the depositional characteristics or initial anisotropy of the geomaterial. e premise of establishing the anisotropic strength criterion is to quantify the degree and direction of the initial anisotropy of the geomaterial. e fabric tensor proposed by Brewer is a good choice. e anisotropic function in the present study refers to the form in [31][32][33]: where A is the anisotropic parameter and A 0 is the value that A takes from b � 0 and α � 0. η 1 and η 2 in the anisotropic function are selected to depict the anisotropy of the material. e anisotropic function and the nonlinear isotropic strength criterion are combined to obtain an anisotropic strength criterion, as shown in the following equation:

Effects of η 1 and η 2 on ANUS.
e effect of isotropic parameters on the shape of the strength criterion in the deviatoric plane has been discussed above.
is 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 η 1 and η 2 in the anisotropic function are selected to depict the anisotropy of the material. Different formulas can be selected for the anisotropic parameter A in equation (22) according to the true triaxial test or the hollow torsional shear test. In the present study, the anisotropic strength in the true triaxial test is mainly discussed.
In the true triaxial test, when the stress tensor and the fabric tensor are coaxial (β � 0), as shown in Figure 2, the shape of the strength criterion in the π plane varies with the parameters η 1 and η 2 , as shown in Figures 3(a), 3(c), and 3(e). In this section, with η 2 fixed at −0.25, −0.333, and 0, the effect of varying η 1 on the strength curve in the π plane is discussed. As shown in Figures 3(a) and 3(b), with η 2 � −0.25, the strength values of the anisotropic criterion and the isotropic criterion at θ � 180°are equal with different η 1 values, and the criterion is symmetric about the σ z axis. e criterion expands in the π plane when η 1 > 0 and shrinks in the π plane when η 1 < 0, and the degrees of expansion or shrinkage are related to the value of η 1 . e anisotropic function variation in the π plane is consistent with that of the strength criterion, as shown in Figure 3(b). With η 2 � −0.333, as shown in Figure 3(c), the strength values of the anisotropic criterion and the isotropic criterion at θ � 120°and 240°, respectively, are equal with different η 1 values, and the criterion is symmetric about the σ z axis. Relative to the isotropic criterion, the anisotropic criterion expands in regions I and II and shrinks in region III of the π plane when η 1 > 0, and it shrinks in regions I and II and expands in region III of the π plane when η 1 < 0, with the degrees of expansion or shrinkage being related to the value of η 1 . e anisotropic function variation in the π plane is consistent with that of the strength criterion, as shown in Figure 3(d). When η 2 � 0, as shown in Figure 3(e), the criterion at different η 1 values is symmetric about the σ z axis; when η 1 > 0, the anisotropic criterion expands in the π plane relative to the isotropic criterion; and when η 1 < 0, the anisotropic criterion shrinks in the π plane relative to the isotropic criterion. e degrees of expansion or shrinkage are related to the value of η 1 . e anisotropic function variation in the π plane is consistent with that of the strength criterion, as shown in Figure 3(f ).
When the stress tensor and the fabric tensor are not coaxial (β ≠ 0), the changes in the shape of the strength criterion in the π plane at different included angles are shown in Figures 4(a) and 4(c). With fixed η 1 and η 2 values (η 1 � −0.08, η 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 π plane is discussed. When β � 0, as shown in Figure 4, the strength curve is symmetric about the σ z axis in the π plane. When the deposition surface is rotated in the x − z plane, as shown in Figure 4(a), the strength curve is symmetric with respect to the σ y axis in the π plane when β � 90°. e changes in the strength curve when β � 30°and 60°are shown in Figure 4

Parameter Determination.
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. e 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. [34] performed a large number of conventional triaxial tests and true triaxial tests on Pietrafitta clay to determine all the parameters in the anisotropic criterion. In the present study, the parameters are determined by conventional triaxial test data, and other test data are used for model verification. e specific determination steps are as follows: (1) Determination of M f , n, and σ 0 . e p − q curve is drawn using the failure data of the conventional  (2) Determination of η 1 and η 2 , η 1 , and η 2 can be determined by the q value of other stress paths with b ≠ 0. Usually, the strength of triaxial extension is taken as the reference value for calibration. In most triaxial tests, only one of the terms in equation (21) is needed to satisfy the prediction accuracy requirement. For this reason, only the η 1 term is taken in this test. In the triaxial extension test, when b � 1 and θ � 180°A-A 0 � 4. Using q e � f(B)M f g(θ)(p + σ 0 ), q e � 254.41 kPa (θ � 180°p � 250 kPa) can be obtained using the same method. erefore, g(θ) � 0.8462, f(A) � exp(4η 1 ), and η 1 � 0.0417.
Using the above steps, all parameters of the anisotropic criterion for Pietrafitta clay are determined. e test parameter determination method is shown in Figure 5(a). e comparison of the prediction results by the isotropic criterion and the anisotropic criterion in Figure 5(b) shows that the anisotropic criterion proposed in the present study better predicts the test data.

Strength Nonlinearity.
A series of large triaxial tests for rockfill materials were conducted to research strength nonlinearity of this material [27]. In the meridional plane (p − q), the test results show the strength nonlinearities. erefore, it is necessary to use nonlinear strength criteria to depict the strength development pattern. e two strength criteria are compared by using them to predict the test data, as shown in Figure 6. As the figure shows, the nonlinear strength criterion proposed this paper can depict the strength nonlinearity of this material. Advances in Civil Engineering 5

True Triaxial Test of Soil.
In addition to the above true triaxial test of Pietrafitta clay, fine glass-bead sand is then used to verify the anisotropic criterion. Haruyama [35] conducted a large number of conventional triaxial tests and true triaxial tests to study the initial anisotropy of fine glassbead sand. e effective stress of the test is p � 294 kPa. e conventional triaxial compression test and the triaxial tensile test are used to determine the parameters. e parameters determined by the test data are as follows: M � 1.142, σ 0 � 0, n � 1, η 1 � −0.038, and η 2 � 0. e prediction results and test data in the deviatoric plane and b-φ plane are shown in Figure 7. It can be seen from the figure that the anisotropic criterion of the present study can reasonably predict the strength of the fine glass sand in the π plane. However, the isotropic criterion fails to predict the strength of regions II and III in the deviatoric plane and overestimates the material strength. In addition, in the b-φ plane, the isotropic criterion can only predict one variation curve of the friction angle, whereas the anisotropic criterion results in a continuous prediction curve in each region, reflecting the anisotropy of the material. erefore, the anisotropic criterion can reasonably predict the peak strength of the fine glass-bead sand compared to the isotropic criterion.

Torsional Shear Tests for Soils.
e 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. [37] was analyzed using the proposed ANUS criterion. e effective stress of the test is p � 200 kPa. e parameters determined by the test data are as follows: for glass ballotini, M � 1.43, σ 0 � 0, n � 1, η 1 � −0.368, and η 2 � 0.1187; for medium dense LB sand, M � 1.805, σ 0 � 0, n � 1, η 1 � −0.2463, and η 2 � 0.0593; for dense LB sand, M � 1.91, σ 0 � 0, n � 1, η 1 � −0.2668, and η 2 � 0.0655. e prediction results (ANUS criterion and ALD criterion [38]) and test data in the α-φ plane are shown in Figure 8. As shown in Figure 8, under torsional shear tests, the sand shows obvious anisotropy, the new anisotropic criterion can predict the failure friction angle well, and the prediction curve is smooth.

Rock Triaxial Test considering Deposition Surface.
is 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. e isotropic strength criterion and the anisotropic strength criterion were used for   Figure 7: Verifications of the two criteria for glass beads in (a) the octahedral plane; (b) the b-φ diagram (test data from [35]; this figure is reproduced from [36] As shown in Figure 9, the anisotropic criterion can reasonably predict the strength failure line with different deposition surface angles in the p − q plane. In contrast, the isotropic strength criterion can only predict the failure line when β � 0°, cannot reflect the strength anisotropy, and overestimates the strength values with different deposition surface angles, especially for β � 45°. Meanwhile, the anisotropic criterion can reasonably predict the trend of strength variation under different confining pressures at different deposition surfaces, as shown in Figure 9(a). It can be seen from Figure 9(b) that the strengths of rocks with different deposition surface angles differ considerably. e peak strength of the rock is the largest when β � 0°or 90°and the smallest when the deposition surface angle is approximately 45°. In contrast, the isotropic criterion is a straight line in the β − q plane   Figure 9: Comparison of the two criteria for Tournemire shale in (a) the p − q diagram and (b) the β − q diagram (test data from [39]; this figure is reproduced from [40]). 8 Advances in Civil Engineering and, hence, cannot reflect the strength anisotropy caused by the direction of the deposition surface.

Conclusion
(1) In this paper, a new strength criterion is proposed on the basis of the SMP criterion. In the π plane, the new criterion is a series of smooth curves and can unify the strength criteria. Data Availability e underlying data used in the presented study were obtained from the literature.

Conflicts of Interest
e authors declare that they have no conflicts of interest.