Grainsize dependence of clastic yielding in unsaturated granular soils

Abstract

We use a recent reformulation of the Breakage Mechanics theory explaining comminution in wet granular assemblies. By using a dataset for sands, we quantify the relation between a geometric descriptor of the assembly (i.e., the mean grainsize) and the model constants that control the suction air-entry value and the stress threshold at the onset of crushing. Such relations are used to define two contrasting scenarios for the coupling between degree of saturation and yielding. In the first scenario, the suction air-entry value scales inversely with the mean grainsize, while the energy input for comminution is assumed to be independent of the size of the particles. The outcome of this assumption is that changes in degree of saturation are predicted to play a more intense role in finer gradings. Conversely, if we assume that also the energy input for grain breakage scales inversely with the size of the particles, the effect of the degree of saturation is predicted to be stronger in coarser assemblies. In other words, the deterioration of the yielding stress due to grainsize scaling effects is predicted to exacerbate the water sensitivity of unsaturated crushable soils. This result provides an interpretation for the evidence that solid–fluid interactions have a noticeable role in the compression response of assemblies made of coarse brittle particles (e.g., gravels or rockfill), while they tend to play little or no role in granular materials characterized by a finer grading (e.g., sands).

Appendices

Appendix A: Grading indices for uniform grainsize distributions

If the probability for a given particle of size $x$ to exist within the interval $(D_m, D_M)$ is assumed to be constant:

$$\begin{aligned} p\left( x \right) = \frac{1}{{{D_M} - {D_m}}} \end{aligned}$$

(20)

it is possible to show that the corresponding cumulative grain size distribution by number is given by:

$$\begin{aligned} F\left( x \right) = \int \limits _{{D_m}}^x {p\left( y \right) } dy = \frac{{x - {D_m}}}{{{D_M} - {D_m}}} \end{aligned}$$

(21)

By further assuming that the mass of each particle $x$ is proportional to $x^3$, the cumulative GSD by mass is can be derived as

$$\begin{aligned} {F^*}\left( x \right) = \frac{{\int _{{D_m}}^x {p\left( y \right) } {y^3}dy}}{{\int _{{D_m}}^{{D_M}} {p\left( y \right) } {y^3}dy}} = \frac{{{x^4} - {D_m}^4}}{{{D_M}^4 - {D_m}^4}} \end{aligned}$$

(22)

Then the corresponding probability density function by mass can be derived as

$$\begin{aligned} g\left( x \right) = \frac{{d{F^*}\left( x \right) }}{{dx}} = \frac{{4{x^3}}}{{{D_M}^4 - {D_m}^4}} \end{aligned}$$

(23)

Equation (22) shows good agreement with the initial grading of the commercial sands such as Hostun sand, as shown in Fig.  2. As far as the probability density function of the GSD at complete breakage, it is possible to use the expression suggested by Einav (2007a):

$$\begin{aligned} {g_u}\left( x \right) = \frac{{(3 - \alpha ){x^{2 - \alpha }}}}{{{D_M}^{3 - \alpha } - {D_m}^{3 - \alpha }}} \end{aligned}$$

(24)

where $\alpha$ is the fractal dimension (usually set to be 2.7). It is interesting to note that Eq. (23) is a particular case of equation Eq. (24), obtained by setting $\alpha$ equal to $-1$.

The grading indices associated with the previously defined GSD curves can be expressed as:

$$\begin{aligned} {\vartheta _M}&= 1 - \frac{{\int _{{D_m}}^{{D_M}} {{g_u}\left( x \right) {x^2}} dx}}{{\int _{{D_m}}^{{D_M}} {{g_0}\left( x \right) {x^2}} dx}} \nonumber \\&= 1 \!-\! \frac{3}{2}\left( {\frac{{3 \!-\! \alpha }}{{5 \!- \!\alpha }}} \right) \left( {\frac{{{D_M}^4 \!-\! {D_m}^4}}{{{D_M}^6 \!-\! {D_m}^6}}} \right) \left( {\frac{{{D_M}^{5 \!-\! \alpha } \!- \!{D_m}^{5 - \alpha }}}{{{D_M}^{3 - \alpha } \!-\! {D_m}^{3 - \alpha }}}} \right) \nonumber \\ \end{aligned}$$

(25a)

$$\begin{aligned} {\vartheta _H}&= \frac{{\int _{{D_m}}^{{D_M}} {{g_u}\left( x \right) {x^{ - 1}}} dx}}{{\int _{{D_m}}^{{D_M}} {{g_0}\left( x \right) {x^{ - 1}}} dx}} - 1 \nonumber \\&= \frac{3}{4}\left( {\frac{{3 {-} \alpha }}{{2 {-} \alpha }}} \right) \left( {\frac{{{D_M}^4 - {D_m}^4}}{{{D_M}^3 - {D_m}^3}}} \right) \left( {\frac{{{D_M}^{2 - \alpha } {-} {D_m}^{2 - \alpha }}}{{{D_M}^{3 - \alpha }{-} {D_m}^{3 - \alpha }}}} \right) {-}1\nonumber \\ \end{aligned}$$

(25b)

Both indices can be determined by specifing the maximum and minimum particle size (the latter being usually assumed to be 1 mm). In particular, by using Eq. (22) it is possible to express the maximum grainsize $D_M$ as a function of the mean grainsize $D_{50}$, as follows:

$$\begin{aligned} {D_M} = {\left( {2{D_{50}}^4 - {D_m}^4} \right) ^{1/4}} \end{aligned}$$

(26)

As a result, the two grading indices given in Eq. (25) can be plotted as a function of $D_{50}$ (1).

Appendix B: Yield stress upon one-dimensional compression

The vertical stress associated with the onset of comminution upon one-dimensional compression, ${\left( {{\sigma ' _V}} \right) _{CR_{0}}}$ can be computed by enforcing the yielding condition for a $K_0$ compression path. In this case, the stress path implies that $q = 3\left( {1 - {K_0}} \right) p'/\left( {1 + 2{K_0}} \right)$ or ${\sigma _v}' = 3p'/\left( {1 + 2{K_0}} \right)$ . Substituting these relations in Eq. (6), the yield stress for a saturated linear elastic medium is given by:

$$\begin{aligned} {\left( {{\sigma _V}'} \right) _{CR _{0}}} = \frac{3}{{1 + 2{K_0}}}\sqrt{\frac{{2{E_c}\left( {1 - \frac{{{\eta _{K0}}^2}}{{{M^2}}}} \right) }}{{{\vartheta _M}\left( {\frac{1}{K} + \frac{{{\eta _{K0}}^2}}{{3G}}} \right) }}} \end{aligned}$$

(27)

where ${\eta _{K0}} = {{3\left( {1 - {K_0}} \right) } \mathord {\Big /} {\left( {1 + 2{K_0}} \right) }}$. In the case of pressure-dependent elasticity, the yield function can be obtained by combining Eqs.  (5),(6) and (11):

$$\begin{aligned}&\frac{{{\vartheta _M}{{\left( {1 \!-\! B} \right) }^2}}}{{{E_c}}}\left[ {\frac{{{p_r}}}{{\bar{K}\left( {2 - m} \right) }}\frac{{\zeta {{\left( {p',q} \right) }^{2 - m}}}}{{{{\left( {1 - {\vartheta _M}B} \right) }^{2 - m}}}} \!+ \!\frac{{{q^2}}}{{6{p_r}\bar{G}}}\frac{{\zeta {{\left( {p',q} \right) }^{ - m}}}}{{{{\left( {1 - {\vartheta _M}B} \right) }^2}}}} \right] \nonumber \\&+ {\left( {\frac{q}{{Mp'}}} \right) ^2} \le 1 + \frac{{{\vartheta _H}}}{{{E_c}}}\psi _r^H\left( {{S_r}} \right) {\left( {1 - B} \right) ^2} \end{aligned}$$

(28)

where

$$\begin{aligned} \zeta \left( {p',q} \right) = \frac{{p'}}{{2{p_r}}} + \sqrt{{{\left( {\frac{{p'}}{{2{p_r}}}} \right) }^2} - \frac{{\bar{K}m}}{{6\bar{G}}}{{\left( {\frac{q}{{{p_r}}}} \right) }^2}} \end{aligned}$$

(29)

The $\left( {{\sigma ' _V}} \right) _{CR_{0}}$ for pressure-dependent elasticity can again be obtained by assuming a $K_0$ compression path, as follows:

$$\begin{aligned} {\left( {{\sigma _V}'} \right) _{C{R_{0}}}} = \frac{3}{{1 + 2{K_0}}}{\left[ {\frac{{{E_c}\left( {1 - \frac{{{\eta _{K0}}^2}}{{{M^2}}}} \right) }}{{{\vartheta _M}\left( {\frac{{{p_r}{\zeta _{K0}}^{2 - m}}}{{\bar{K}\left( {2 - m} \right) }} + \frac{{{\eta _{K0}}^2{\zeta _{K0}}^{ - m}}}{{6{p_r}\bar{G}}}} \right) }}} \right] ^{\frac{1}{2-m}}}\nonumber \\ \end{aligned}$$

(30)

where

$$\begin{aligned} {\zeta _{K0}} = \frac{{\zeta \left( {p',q} \right) }}{{p'}} = \frac{1}{{2{p_r}}} + \sqrt{{{\left( {\frac{1}{{2{p_r}}}} \right) }^2} - \frac{{\bar{K}m{\eta _{K0}}^2}}{{6\bar{G}{p_r}^2}}} \end{aligned}$$

(31)

The above relations have been used to identify the optimal values of $E_c$ allowing the mathematical capture of numerous compression experiments available in the literature (Fig. 13).