2.1.1. Anisotropic Bounding Surface

Based on the two-dimensional bounding surface model developed by Hu and Liu [26] for the cyclic dynamic analysis of saturated clay, a three-dimensional form is established by the generalized Mises criterion method, *q* = q 3 *<sup>s</sup>ij* <sup>−</sup> *<sup>p</sup>*α*ijsij* <sup>−</sup> *<sup>p</sup>*α*ij* /2. The mathematical equation of the bounding surface is described as: *2.1. Constitutive Model*  2.1.1. Anisotropic Bounding Surface Based on the two-dimensional bounding surface model developed by Hu and Liu [26] for the cyclic dynamic analysis of saturated clay, a three-dimensional form is established by the

$$F\_m = (\overline{p} - \boldsymbol{\xi}\_p^{(m)})^2 - (\overline{p} - \boldsymbol{\xi}\_p^{(m)})p\_c^{(m)} + \frac{3}{2(M - a\_0^2)} \left[ \overline{s}\_{ij} - \boldsymbol{\xi}\_{ij}^{(m)} - (\overline{p} - \boldsymbol{\xi}\_p^{(m)})a\_{ij}^0 \right] \left[ \overline{s}\_{ij} - \boldsymbol{\xi}\_{ij}^{(m)} - (\overline{p} - \boldsymbol{\xi}\_p^{(m)})a\_{ij}^0 \right] \tag{1}$$

in which *<sup>p</sup>*,*sij* denotes the image stress point on the bounding surface, as shown in Figure 2; *p<sup>c</sup>* is the reference size of the bounding surface; ξ (*m*) *p* , ξ (*m*) *ij* is the coordinate of the endpoint; *M* represents the slope of the critical state line (CSL) in triaxial space; and α<sup>0</sup> denotes the inclination of the bounding surface, which represents the degree of soil anisotropy: 2 0 <sup>⋅</sup> *(m) 2 (m) (m) (m) (m) 0 (m) (m) 0 m p p c ij ij p ij ij ij p ij <sup>3</sup> F = (p - <sup>ξ</sup> ) - (p - <sup>ξ</sup> )p + s - <sup>ξ</sup> - (p - <sup>ξ</sup> )<sup>α</sup> s - <sup>ξ</sup> - (p - <sup>ξ</sup> )<sup>α</sup> 2(M - <sup>α</sup> )* (1) in which ( ) *ij p, s* denotes the image stress point on the bounding surface, as shown in Figure 2; *<sup>c</sup> p* is the reference size of the bounding surface; () () ( ) *m m ξp ij ,ξ* is the coordinate of the endpoint; *M*

$$a\_0 = \sqrt{\frac{3}{2} a\_{ij}^0 a\_{ij}^0} \,\tag{2}$$

**Figure 2.** Illustration of the bounding surface. **Figure 2.** Illustration of the bounding surface.

3 0 0 For consolidated samples, the initial stress ratio is *K*<sup>0</sup> = σ 0 3 /σ 0 1 . The anisotropic tensors can be defined as follows:

$$
\alpha\_{11}^{0} = \frac{2(1 - \mathcal{K}\_0)}{1 + 2\mathcal{K}\_0}, \; \alpha\_{22}^{0} = \alpha\_{33}^{0} = \frac{(\mathcal{K}\_0 - 1)}{1 + 2\mathcal{K}\_0} \tag{3}
$$

*0*

*(K - 1)*

*0 0*

*2(1- K )*

*0*

*11*

For consolidated samples, the initial stress ratio is *0 0 K = 0 31 σ σ* . The anisotropic tensors can be
