Numerical simulation of lumpy soils using a hypoplastic model

Abstract

The lumpy soil is a by product of the open-pit mining. A composite-lumpy material (in which, the lumps are randomly distributed in the reconstituted soil) is being created due to the degradation of the initial granular structure. In the present study, the compression and failure behaviour of an artificial lumpy material with randomly distributed inclusions are investigated using the finite element method. The computation results show that the stress ratio, defined as the ratio of the volume average stress between the lumps and the reconstituted soil within the inter-lump voids, is significantly affected by both the volume fraction and the preconsolidation pressure of the lumps under an isotropic compression path, while the volume fraction of the lumps plays a minor role under a triaxial compression path. Based on the simulation results, a homogenization law was proposed utilizing the secant stiffnesses.

Appendix: Homogenization law using the tangent stiffnesses under isotropic compression load

Analogous to the secant stiffnesses, the tangent ones (hydrostatic $\bar{K}_{h}$ and deviatoric $\bar{G}_{d}$ parts, together with their counterparts $\bar{K}_{d}$ and $\bar{G}_{h}$) are defined as

$$\bar{K}_{h} = \frac{{{\text{d}}p}}{{{\text{d}}\varepsilon _{v} }};\quad \bar{G}_{d} = \frac{{{\text{d}}q}}{{{\text{d}}\varepsilon _{{\text{s}}} }};\quad \bar{K}_{d} = \frac{{{\text{d}}p}}{{{\text{d}}\varepsilon _{{\text{s}}} }};\quad \bar{G}_{h} = \frac{{{\text{d}}q}}{{{\text{d}}\varepsilon _{v} }}$$

(22)

The corresponding homogenization law is expressed as

$$\begin{aligned} \left[\begin{array}{cc} \log \bar{K}_{h} &{} \log \bar{K}_{d} \\ \log \bar{G}_{h} &{} \log \bar{G}_{d} \\ \end{array}\right]= n_e* \left[\begin{array}{cc} \log \bar{K}_{h} &{} \log \bar{K}_{d} \\ \log \bar{G}_{h} &{} \log \bar{G}_{d} \\ \end{array}\right]_{nc}+ (1-n_e)* \left[\begin{array}{cc} \log \bar{K}_{h} &{} \log \bar{K}_{d} \\ \log \bar{G}_{h} &{} \log \bar{G}_{d} \\ \end{array}\right]_{oc} \end{aligned}$$

(23)

The relationships between the tangent stiffnesses are shown in Fig. 22. Obviously, the simulation results from Sect. 3.1 fit excellently the homogenization law during isotropic loading in terms of the tangent stiffness.

Fig. 22

Relationship between the overall tangent stiffness and those of the constituents during triaxial shear loading. a $\bar{G}_d$. b $\bar{K}_h$. c $\bar{K}_d$. d $\bar{G}_h$