A critical-state constitutive model for considering the anisotropy in sandy slopes

Abstract

In this paper, a novel approach that associates the behavior of granular material with the void space and plastic deformations is applied to evaluate the strength changes with the loading directions. For this target, a critical state constitutive model based on Imam et al. (2005) is adopted to quantify the mobilized strengths. In this strategy, the tensor of fabric becomes co-direction with the tensor of loading to reach a constant magnitude of the unit at the compact state. In this state, the interlocks among the sand particles are strictly high, so that slip cannot happen. It is supposed that at this stage, the sample’s reaction to the stress changes is isotropic. The validation is performed against some datasets of monotonic element tests and cyclic triaxial tests. Then, the different aspects of the proposed model and their influences on the dynamic-induced displacements in the slopes are investigated. The results reveal the prominent role of the fabric in capturing the anisotropic behavior of sand, especially in slope stability analysis. Also, the results show that fabric evolution significantly affects the strength characteristics and the deformation of slopes. In the bedding plane orientations from 45 to 90°, the extended model ensures a higher magnitude for displacements in comparison to the basic model. The lack of considering the evolution of the fabric to the void ratio leads to a non-conservative analysis of the slopes, and it is a critical item in determining the displacements in dynamic loadings.

Appendix A

To determine the values of du_z, the incremental movement of sand particles in the finite difference mesh, the equation between the strain and stress increments can be presented in a form of:

$$\left[\begin{array}{c}\textrm{d}{\sigma}_{xx}\\ {}\textrm{d}{\sigma}_{zz}\\ {}\textrm{d}{\sigma}_{xz}\end{array}\right]=\left[\begin{array}{ccc}{c}_{11}& {c}_{12}& {c}_{13}\\ {}{c}_{21}& {c}_{22}& {c}_{12}\\ {}{c}_{31}& {c}_{32}& {c}_{33}\end{array}\right]\left[\begin{array}{c}\textrm{d}{\varepsilon}_{xx}\\ {}\textrm{d}{\varepsilon}_{zz}\\ {}\textrm{d}{\varepsilon}_{xz}\end{array}\right]$$

(A1)

Therefore, the Eq. (28) can be calculated in a finite difference form as:

$$\frac{\dot{u}\left(i+1,j\right)-\dot{u}\left(i,j\right)}{\varDelta z}+\frac{\dot{v}\left(i+1,j\right)-\dot{v}\left(i,j\right)}{\varDelta x}=\alpha (z)\frac{\sum \dot{u}\left(i,j-1\right)-2\dot{u}\left(i,j\right)+\dot{u}\left(i,j+1\right)}{{\left(\varDelta t\right)}^2}\varDelta \varsigma -\frac{1}{G^{\textrm{ep}}(z)}{\int}_0^z\frac{\partial {\dot{\sigma}}_{xx}}{\partial x}\textrm{d}x$$

(A2)

Substituting from Eq. (A2) for ${\dot{\sigma}}_{xx}$ yields:

$$\frac{\dot{u}\left(i+1,j\right)-\dot{u}\left(i,j\right)}{\varDelta x}+\frac{\dot{v}\left(i+1,j\right)-\dot{v}\left(i,j\right)}{\varDelta z}=\alpha (z)\frac{\sum \dot{u}\left(i,j-1\right)-2\dot{u}\left(i,j\right)+\dot{u}\left(i,j+1\right)}{{\left(\varDelta t\right)}^2}\varDelta \varsigma -\frac{1}{G^{\textrm{ep}}(z)}{\int}_0^z\frac{\partial \left({c}_{11}{\dot{\varepsilon}}_{xx}+{c}_{12}{\dot{\varepsilon}}_{zz}+{c}_{13}{\dot{\varepsilon}}_{xz}\right)}{\partial x}\textrm{d}x$$

(A3)

Substituting from Eq. (A3) for ${\dot{\varepsilon}}_{xx},{\dot{\varepsilon}}_{zz}$ and ${\dot{\varepsilon}}_{xz}$ yields:

$$\frac{\dot{u}\left(i+1,j\right)-\dot{u}\left(i,j\right)}{\varDelta x}+\frac{\dot{v}\left(i+1,j\right)-\dot{v}\left(i,j\right)}{\varDelta z}=\alpha (z)\frac{\sum \dot{u}\left(i,j-1\right)-2\dot{u}\left(i,j\right)+\dot{u}\left(i,j+1\right)}{{\left(\varDelta t\right)}^2}\varDelta \varsigma -\frac{1}{G^{\textrm{ep}}(z)}\int_0^z\frac{\partial \left[{c}_{11}\frac{\partial \dot{u}}{\partial x}+{c}_{12}\frac{\partial \dot{v}}{\partial z}+{c}_{13}\frac{1}{2}\left(\frac{\partial \dot{u}}{\partial z}+\frac{\partial \dot{v}}{\partial x}\right)\right]}{\partial x}\textrm{d}x$$

(A4)

Therefore:

$$\frac{\dot{u}\left(i+1,j\right)-\dot{u}\left(i,j\right)}{\varDelta x}+\frac{\dot{v}\left(i+1,j\right)-\dot{v}\left(i,j\right)}{\varDelta z}=\alpha (z)\frac{\sum \dot{u}\left(i,j-1\right)-2\dot{u}\left(i,j\right)+\dot{u}\left(i,j+1\right)}{{\left(\varDelta t\right)}^2}\varDelta \varsigma -\frac{1}{G^{\textrm{ep}}(z)}\int_0^z\left[{c}_{11}\frac{\partial^2\dot{u}}{\partial {x}^2}+{c}_{12}\frac{\partial^2\dot{v}}{\partial z\partial x}+{c}_{13}\frac{1}{2}\left(\frac{\partial^2\dot{u}}{\partial x\partial z}+\frac{\partial^2\dot{v}}{\partial {x}^2}\right)\right]\textrm{d}x$$

(A5)

Thus, for a typical node, the corresponding ID equation is:

$${\displaystyle \begin{array}{l}\frac{\dot{u}\left(i+1,j\right)-\dot{u}\left(i,j\right)}{\varDelta x}+\frac{\dot{v}\left(i+1,j\right)-\dot{v}\left(i,j\right)}{\varDelta z}=\alpha (z)\frac{\sum \dot{u}\left(i,j-1\right)-2\dot{u}\left(i,j\right)+\dot{u}\left(i,j+1\right)}{{\left(\varDelta t\right)}^2}\varDelta \varsigma -\frac{c_{11}}{G^{\textrm{ep}}(z)}\frac{\sum \dot{u}\left(i,j-1\right)-2\dot{u}\left(i,j\right)+\dot{u}\left(i,j+1\right)}{{\left(\varDelta x\right)}^2}\varDelta x-\\ {}\frac{c_{12}}{G^{\textrm{ep}}(z)}\frac{\sum \dot{v}\left(i+1,j+1\right)-\dot{v}\left(i+1,j-1\right)-\dot{v}\left(i-1,j+1\right)+\dot{v}\left(i-1,j-1\right)}{4\varDelta x\varDelta z}\varDelta x-\frac{c_{13}}{G^{\textrm{ep}}(z)}\frac{\sum \dot{u}\left(i+1,j+1\right)-\dot{u}\left(i+1,j-1\right)-\dot{u}\left(i-1,j+1\right)+\dot{u}\left(i-1,j-1\right)}{4\varDelta x\varDelta z}\varDelta x-\end{array}}$$

$$\frac{c_{13}}{2{G}^{\textrm{ep}}(z)}\frac{\sum \dot{v}\left(i,j-1\right)-2\dot{v}\left(i,j\right)+\dot{v}\left(i,j+1\right)}{{\left(\varDelta x\right)}^2}\varDelta x$$

(A6)

Similarly, this manner is adopted for Eq. (A6) to provide two equations with two unknowns as u̇ and v̇. Through solving this system, the deformations of slope under dynamic loads can determine.

${\displaystyle \begin{array}{l}\frac{\dot{u}\left(i+1,j\right)-\dot{u}\left(i,j\right)}{\varDelta x}+\frac{\dot{v}\left(i+1,j\right)-\dot{v}\left(i,j\right)}{\varDelta z}=-\frac{c_{22}}{G^{\textrm{ep}}(z)}\frac{\sum \dot{v}\left(i,j-1\right)-2\dot{v}\left(i,j\right)+\dot{v}\left(i,j+1\right)}{{\left(\varDelta z\right)}^2}\varDelta z-\\ {}\frac{c_{23}}{2{G}^{\textrm{ep}}(z)}\frac{\sum \dot{v}\left(i+1,j+1\right)-\dot{v}\left(i+1,j-1\right)-\dot{v}\left(i-1,j+1\right)+\dot{v}\left(i-1,j-1\right)}{4\varDelta x\varDelta z}\varDelta z-\frac{c_{21}}{2{G}^{\textrm{ep}}(z)}\frac{\sum \dot{u}\left(i+1,j+1\right)-\dot{u}\left(i+1,j-1\right)-\dot{u}\left(i-1,j+1\right)+\dot{u}\left(i-1,j-1\right)}{4\varDelta x\varDelta z}\varDelta z-\end{array}}\frac{c_{23}}{2{G}^{\textrm{ep}}(z)}\frac{\sum \dot{u}\left(i+1,j+1\right)-\dot{u}\left(i+1,j-1\right)-\dot{u}\left(i-1,j+1\right)+\dot{u}\left(i-1,j-1\right)}{4\varDelta x\varDelta z}\varDelta x-\frac{c_{13}}{2{G}^{\textrm{ep}}(z)}\frac{\sum \dot{u}\left(i,j-1\right)-2\dot{u}\left(i,j\right)+\dot{u}\left(i,j+1\right)}{{\left(\varDelta z\right)}^2}\varDelta z$ (A7)

The numerical scheme employed to evaluate the governing equation is simple and will be demonstrated briefly here. In this scheme, the solution domain is covered by a finite difference mesh, which is shown in Fig. 17. It is assumed that the meshes have j_max, columns and i_max rows which spaced at uniform intervals of Δt, and Δz, respectively. Δz is kept constant thus Δϛ= Δz. The evaluation of the u̇ at every point of the mesh is the main analysis object. This is compiled through deriving the coefficients of the linear formulation for $\dot{u}(1),\dot{u}(2),\dots, \dot{u}\left({n}_{\textrm{max}}\right),$ where n_max = (i_max − 1)j_max + j_max = i_maxj_max is the total of unknowns that should be computed. Consider the situation in (i, j) by a node identification, n_max = (i − 1)j_max + j.. To find the values of constants a(n, 1), a(n, 2) ... a(n, n), ..., a(n, n_max + 1), the equation corresponded to the node n could be found as:

$$a\left(n,1\right)\dot{u}(1)+a\left(n,2\right)\dot{u}(2)+\cdots +a\left(n,n\right)\dot{u}(n)+a\left(n,n+1\right)\dot{u}\left(n+1\right)+\cdots =a\left(n,{n}_{\textrm{max}}+1\right)$$

(A8)

with the aid of the last results and recalling Eq. (A2) one can write:

$$\dot{u}\left(i+1,j\right)-\dot{u}\left(i,j\right)=\sum \limits_{m=1}^ja\left(m,j\right)\left[\dot{u}\left(m,j-1\right)-2\dot{u}\left(m,j\right)+\dot{u}\left(m,j+1\right)\right]$$

(A9)

where:

$$a\left(m,j\right)=\alpha (z)\frac{\left(m-1\right){\left(\varDelta z\right)}^2}{{\left(\varDelta t\right)}^2}=\frac{\gamma \left(m-1\right){\left(\varDelta z\right)}^2}{gG^{\textrm{ep}}(z){\left(\varDelta t\right)}^2}$$

(A10)

and the nodal number assigned to the point (m, j) is s = (m − 1)j_max + j.. The node directly below the (m, j) will have a nodal number of r = mj_max + j.. It remains to express Eq. (A10) in a form suitable for coding. That is, to reduce it to several simultaneous algebraic equations. Thus Eq. (A10) can be rewritten as:

$$\dot{u}(r)-\dot{u}(n)=\sum \limits_{s=1}^n\alpha (s)\left[\dot{u}\left(s-1\right)-2\dot{u}(s)+\dot{u}\left(s+1\right)\right]$$

(A11)

The scheme is fully explicit. Equation (A11) must be computed subject to the appropriate boundary conditions as:

$$z=z\to \dot{u}={a}_0\omega \cos \left(\omega t\right)@\left|\begin{array}{c}t=0\to \dot{u}=0,\sigma =\gamma z\\ {}z=0\to \dot{u}=0\end{array}\right.$$

(A12)

for periodic loading and u_base for dynamic loading, where a₀ is the amplitude of the wave of the periodic sine function, and ω is the angular wave frequency (radians/s).

Fig. 17

The mesh adopted in the present study to simulate the slopes