DEM assessment of scaling laws capturing the grain size dependence of yielding in granular soils

Abstract

Experimental evidences show that the pressure at which granular soils exhibit a sharp increase of their compressibility depends on the size of the particles that constitute their skeleton, thus reflecting the role of micro-scale fracture events on the macroscopic compression of granular systems. In this paper, the distinct element method (DEM) is used to test the validity of scaling laws relating the macroscopic energy at which the grains of a soil matrix crush collectively to the energy at which individual grains subjected to diametrical compression undergo tensile fracture. Oedometric compression tests on uniformly graded specimens with different values of particle size have been simulated by considering two deterministic fracture models and a probabilistic criterion based on the Weibull weakest link theory. It has been shown that the constants of proportionality between grain-scale and assembly-scale crushing thresholds depend considerably on the statistical variability of the particle strength, and that a larger variability exacerbates the departure between the scaling constants pertaining to deterministic and probabilistic models. Nevertheless, for the chosen set of initial conditions and loading paths, the simulations have suggested the applicability of a proportional scaling between the energy stored in the assembly at the moment of yielding and that required to fracture a single grain. In particular, the simulations revealed that the scaling constants relating the microscopic and macroscopic energy thresholds fall within a rather narrow range and do not depend significantly on the grain size. The Breakage Mechanics theory has been used to further explore such connection between length scales, finding a good agreement between the DEM simulations and the yielding stress computed by the theory whenever its parameters were defined on the basis of the scaling constants computed from the DEM model. These results confirm the interplay between the statistical variability of the particle strength and the grain size dependence of the yielding pressure, stressing at the same time the usefulness of energy scaling arguments in incorporating the effect of micro-scale fracture events into continuum models.

Appendix

For the linear contact model, the size-dependency of $E_{pc}$ can be expressed as follows:

$$\begin{aligned} \hbox {Weibull theory }\quad \quad E_{pc}= & {} \overline{E_{pc,0} } \left( {\frac{d}{d_0 }} \right) ^{-6/\mathrm w} \end{aligned}$$

(17a)

$$\begin{aligned} \hbox {Centre crack }\quad \quad E_{pc}= & {} \overline{E_{pc,0} } \left( {\frac{d}{d_0 }} \right) ^{-1} \end{aligned}$$

(17b)

$$\begin{aligned} \hbox {Contact crack }\quad \quad E_{pc}= & {} \overline{E_{pc,0} } \left( {\frac{d}{d_0 }} \right) ^{-3} \end{aligned}$$

(17c)

where $\overline{E_{pc,0} }= 2.4 \times 10^5\,\hbox {J}/\hbox {m}^{3}$ is the value of specific energy threshold at particle crushing used for the reference grain size $d_0 = 2.0\,\hbox {mm}$. Such value is a direct outcome of the particle strength and contact stiffness parameters chosen for the DEM simulations discussed in the previous sections.

Similarly, the following relations hold for the case of Hertzian contact model:

$$\begin{aligned} \hbox {Weibull theory }\quad \quad E_{pc}= & {} \overline{E_{pc,0} } \left( {\frac{d}{d_0 }} \right) ^{-5/\mathrm w} \end{aligned}$$

(18a)

$$\begin{aligned} \hbox {Centre crack }\quad \quad E_{pc}= & {} \overline{E_{pc,0} } \left( {\frac{d}{d_0 }} \right) ^{-0.8} \end{aligned}$$

(18b)

$$\begin{aligned} \hbox {Contact crack }\quad \quad E_{pc}= & {} \overline{E_{pc,0} } \left( {\frac{d}{d_0 }} \right) ^{-2.5} \end{aligned}$$

(18c)

where $\overline{E_{pc,0} }=2.6 \times 10^5\,\hbox {J}/\hbox {m}^{3}$ is the value used for particles with a reference grain size $d_0 = 2.0\,\hbox {mm}$.