Numerical simulation of root reinforcement for herbs in Loess Plateau based on asymptotic homogenization theory

Abstract: Loess Plateau is the most severe soil and water loss area in the world, as well subjected to shallow-landslide disaster in China. Currently, grass system has been widely distributed in the Loess Plateau, while the root system shows obviously periodic characteristics in spatial distribution. In order to accurately evaluate the influence of root groups on root reinforcement, and the coupling relationship between roots and soils, a constitutive relation of the "root-soil" composite was constructed via the "unit cell" of the root and soil based on the asymptotic homogenization theory. The "root-soil" composite in the nature similar to the reinforced concrete, can be regarded as a "new composite material", with a strong coupling relationship between roots and soil. In the deduction, some numerical methods including the perturbation method, periodic boundary conditions, subsection integral method and divergence theorem, were used to derive the expression of the equivalent stiffness matrix, and the equivalent density of the root-soil composite. The detailed solution to these functions was also given based on the finite element method. The two-dimensional elastic parameters of the root-soil composites were calculated by MATLAB program. The three-dimensional equivalent elastic parameters of the root-soil composite were eventually obtained, where the two-dimensional plane strain problem can be extended to three-dimensional one by additional equations. The calculation accuracy and stability of the present method are better than those of the simplified method, particularly on calculating the three-dimensional constitutive relationship. There was a certain deviation (up to 29.4%) in the calculation of equivalent Poisson's ratio υxy, whereas, the calculation errors of other equivalent parameters are less than 7.1%. To illustrate the accuracy and efficiency of the homogenization method in root reinforced slopes, the influence of the Elymus dahuricus's roots on the stress and strain field of the slope was analyzed based on the finite element software ANSYS. Three types of numerical models were constructed, including the slope model without grass, root reinforced slope with separated root elements, and root reinforced slope based on the present homogenization theory. The slope safety factor was calculated using the strength reduction method considering the penetrated equivalent plastic strain zone or not. The results show that: (1) The asymptotic homogenization theory can accurately construct the constitutive relation of "root-soil" composites, while reduce the calculation work (the element number reduced up to 95.58%). The error of υxy has little effect on the stress distribution of the simulated slopes with homogenized materials. (2) Elymus dahuricus's root system can modify the stress field of the shallow slope, indicating more uniform of the shear stress in the root distribution zone. Therefore, the slope stability can be improved. (3) If the slope angle is small (30°), the safety factor of the slope without grass is large (F = 4.28). The root system has a small effect on the slope stability (an average increase is only 2.92%). If the slope angle increases up to 45°, the safety factor of the slope without grass reduces to F = 2.90, while the averaged safety factor of slopes with grass increases by 13.45%, indicating the dominated reinforcement effect of root system on slopes. These findings can open up a new way to set "root-soil" periodic composites for root reinforced slopes.